In Re Our 4-D Universe


Some Relevant Constants

4/8/2002:     The general-relativistic integral    has been corrected and updated again. What's significant about it is that what it seems to me to be saying that something can fall into a black hole almost as quickly as though no general-relativistic effects were present. At the same time, it will slow down without limit during the last few kilometers. and yet, somehow, its overall time to fall into a black hole will still be rapid from the standpoint of an external observer. Or at least, that's how I'm interpreting these results at the moment.
    I haven’t had much time for work today. Tommie’s under the weather with a ticklish cough (nothing serious.), and I’m having to doctor her.   
    One significant point concerns the R-enantiomer of alpha-lipoic acid. Apparently, most racemic mixtures are actually primarily comprised of the R-enantiomer, and apparently, the racemic mixtures work almost as well as the pure R-enantiomers, and are a lot cheaper. (I found it today at Walmart for $7 for 60 capsules.) It’s also beginning to appear in some skin creams. Vaseline Intensive Care Lotion is a biochemical soup containing many of the ingredients dear to the hearts of alternative-medicine aficionados, including aloe vera, jojoba oil, eucalyptus, lavender oil, orange oil, vitamin E, collagen, retinyl palmitate, and retinol, Vitamin C, alpha-lipoic acid, and acetyl-l-carnitine haven’t made it into Vaseline Intensive Care Lotion yet, but I feel confidant that they will.
    I found a little time this morning to work a little more on the general-relativistic integra l. I didn’t expect this integration to take more than an hour. Instead, it has taken many hours, spread over weeks. It seems important to me in terms of understanding what happens when something falls into a black hole, or onto a neutron star.
    The results have been unexpected.
    I’m still taking 100 mg. a day of alpha-lipoic acid, and 250 mg. a day of acetyl-l-carnitine. Is it working? It’s hard to tell, but I certainly have endless energy, without insomnia. One note of caution: I noticed today that the Walmart alpha-lipoic acid has a warning to diabetics and hypoglycemics that alpha-lipoic acid may enhance insulin sensitivity (increase the effectiveness of insulin), and should be used by diabetics (hyperglycemics) and hypoglycemics only under a doctor’s supervision.

4/6/2002:    Last night's program of finishing the pesky general-relativistic integral is essentially complete.
    I wasn't able to finish today's (long) editorial. However, I'm going to post what's done , with the idea that there's more to come.
4/4/2002:     The integral of sqrt((x-1)/x), or of its reciprocal, sqrt(x/(x-1)), look very simple, but instead, they’re both tricky little nippers. Russell Rierson has written with a result that he found in a book on general relativity. This is the same formula I derived night before last. Hm-m-m... My problem with it was that logarithms are very finicky about their arguments, demanding dimensionless numbers. This formula, like the one I derived night before last, has an argument with the dimensions of sqrt(r). I'm not sure that a decent, law-abiding logarithm would accept that very well. That dimensionality indigestibility was what led me to rework one of those integrals in a way that made   its arguments dimensionless.
    I guess we’re not through with these integrals quite yet. (Sigh.... )

 I've massaged last night's relativistic integral so that the argument of the logarithm is dimensionless, as it must be. However, I'm still not entirely content with it. The function we're trying to integrate,

    , increases without limit as r approaches r S, and approaches 1 for r very large.  Our integral approaches 0 as r sneaks up on rS, and increases approximately as r (as it should) for r very large. It doesn’t seem to me that our integral should fizzle away to 0 where our integrand is courting infinity.
    I haven’t had time today to return to this topic.
    I haven’t followed up on the zero-point energy machine today
    I am reducing the science news backlog so that the news will be more current. It's comforting to me to have a few days' or a week's backlog, but I don't think it's in your best interests.
    One further tidbit. I've mentioned that I've been your (and Tommie Jean's) canary with respect to trying some of these mind-boosting and (we would like to believe) life-extending dietary supplements. I had mentioned that we both seemed to have experienced insomnia on doses of 100 mg. per day of alpha-lipoic acid. (Dr. Sahelian said that he suffered from insomnia at doses above 40 mg. per day of alpha-lipoic acid.) I backed off to 25 mg. a day, and Tommie quit lipoic acid   altogether. However, when I read that the standard dose should be 100 to 600 mg. a day, I went back to 100 mg. a day. I'm happy to report that after two weeks of this, I'm experiencing no insomnia from it. I would also say that I have a great deal of energy (although whether that has anything to do with what I'm eating is an unanswered question). The role of the R-enantiomer of
a -lipoic acid is also an open question.
 Tonight’s lead science news article concerns drawing limitless free energy from “nothing”. “Nothing” in this case refers to the vast sea of negative energy that reposes on the negative side of the energy ledger in Dirac’s model of quantum mechanics. Momentary random fluctuations of this energy onto the positive side of the ledger have generated hopes that some of it might be harvested before it can return to its ground state. Tom Beardon claims to have done it.
    The article avers that he will market a 2.5-kilowatt device next year. Such a device, if it exists, would replace just about every conventional energy source. It would power homes, factories, and vehicles. It would provide cheap power with no infrastructure in third-world nations. It would very probably revolutionize interplanetary travel, opening up the colonization of Mars and the moon, and the mineral riches of the asteroid belt. It would be a dramatic step toward interstellar flight.
    If I were he, and I had something I thought was an utterly revolutionary power source, I believe that I would keep a low profile until I got my device on the market. Energy companies would be put out of business by such an invention. (Actually, they probably wouldn’t, but they’d certainly have to scramble, and to buy or steal rights to the invention. In the past, big corporations have run under-funded inventors off the road.)
    Tom Bearden lives in Huntsville. I know him more by reputation than through actual contact. I believe we were introduced a couple of decades ago. I didn’t find Magnetic Energy Limited in the latest Huntsville phone book. Tom is a retired Lt. Col. who lives around the corner from us on the road I travel every day to climb Suicide Hill, and within easy walking distance.  
    The gauge condition,


 to which they refer in their paper is a continuity condition relating the time rate of change of the magnetic potential to the electrostatic field. And, although I never thought about it before, I guess I see what they mean about the similarities (if not the outright identity) of the Maxwell Equations with hydrodynamic flow equations.
    I’m also linking to one version of the integral of the general-relativistic time or distance (depending upon the perspective of the viewer). It's incomplete because it generates an integral that contains the logarithm of a length, and that doesn't look right.
    I want to thank Ronald Penner and Russell Rierson for their contributions to these results.

3/29/2002:    It has occurred to me that if there are feverish attempts underway to identify and produce the agents that rejuvenate a fertilized ovum, their protagonists would do well to say no more about it than the minimum necessary to secure funding, and to say that only behind closed doors. For one thing, there would be an incalculable amount of money riding on it. A worldwide market of $200,000,000,000 a year would seem to me to be a conservative sales estimate. Competition could be fierce.
   If that weren't enough, there would be a lot of opposition to such a development (some of which might come from competitors, trying to crowd out their rivals). Last Sunday, in Parade Magazine, someone asked Marilyn vos Savant if she thought that there were any issue on which everyone would agree. She replied no, that no matter how worthy the cause, there are some people whose identity is defined by renitence. These are people who thrive on disagreeing with the majority opinion. And there are many people who would feel, or be, threatened (e. g., funeral directors) if everyone were to become physiologically youthful.
    Something this big might warrant different treatment than another remedy for baldness.
    The Manhattan Project was one of the best-kept secrets in recent history.
    Stay tuned.

     The Mega Foundation website is now back in operation. I want to offer my humblest apologies for the interruption in service.. For the past week, our hard-working webmaster has been updating the files on our Mega Foundation web server. We think and hope that this will be the last time this kind of interruption will be necessary, at least in the foreseeable future.
    If you would, please note the new email address: . I also have email addresses at and at .  For some undoubtedly-unhappy reason, Comcast seems unable to get their email service working. It will be four weeks on Friday.
New! The Mega Foundation has established new bulletin boards . If you think you might have any interests in the discussions that are underway, we would certainly welcome you, and would invite your participation. The currently active message boards or Conferences are "TOE's GUT's, CTMU & Related Topics", "Science Topics", Spiritual "Questions", and "Genius".
     I'd like to underscore the connection between this website and our Mega Foundation.
    The Mega Foundation was founded in January, 2000, by members of the "forgotten gifted" as a non-profit organization to seek ways to better utilize the talents of the ultra-gifted for society's benefit, as well as helping satisfy the desires of the ultra-gifted to contribute to it. The prodigiously gifted are the pool from which our greatest geniuses have sprung, and yet, there seems to be no organized effort to mobilize their talents to solve society's most  pressing problems. The Mega Foundation was established to seek ways of better ways of effecting this. . 
    I’ve been trying to integrate


.   I’ve succeeded, but I've produced a rather complicated result .

    I was interrupted by a supper outing, and haven’t finished it. However, this latter approach may have yielded a simpler form of the result. (Of course, one form must be transformable into any other.) I know that there are a number of math whizzes reading this page. It’s a simple-looking integral, and it’s certainly integrable in closed form. I thought maybe someone else might like to take a cut at this.
    I'm still unable to upload to the Mega Foundation website, although it's been repaired, and should soon be accessible.
New! The Mega Foundation has established new bulletin boards . If you think you might have any interests in the discussions that are underway, we would certainly welcome you, and would invite your participation.
    The computer just crashed, taking with it the last ten minutes' derivations. I'll have to continue in the morning.
   One comment about longevity research. It seems to me to be an unavoidable conclusion that nature has some way of cleaning up not only the genome but the entire cell when biogenesis occurs. Otherwise, if a baby were born 20 years old, and its child were born 40 years old, and its grandchild were born 60 years old... you can see where this is going. If it proves possible to replicate the enzymes and other biochemical agents that must operate within an oocyte immediately after fertilization, and to apply these to adult cells, all other longevity research would seem to be overtaken by events.
   Although I haven't seen discussions of research into these phenomena in the first few search returns I've found, that might not necessarily mean that it isn't taking place. With 1 in 3 people in the developed world approaching retirement, and 1.2 billion people retired by 2050, hundreds of billions of dollars a year could accrue to whatever corporation cuts this Gordian knot first. You could see why involved organizations might possibly choose caginess over candor.
3/25/2002:     Tonight's editorial considers a detailed analysis of the Twins Paradox from the standpoint of one-per-second light pulses sent by the stationary twin to the traveling twin, and by the traveling twin to the stationary twin.
   Tonight's other topic concerns longevity research, or "prolongevity". One excellent web page concerning this subject is offered by John Furber . . Following the leads listed on his page, I found that Bruce Ames, et al, is finding that the gamma form of vitamin-E, gamma-tocopherol , may play an important role in cancer prevention and age-retardation. There are also discussions of the fact that the alpha lipoic acid + acetyl-l-carnitine that I've been touting requires the R-enantiomer of alpha-lipoic acid rather than the usual over-the-counter form of the substance.
   The recommended daily dosage levels for alpha lipoic acid are in the 120 milligram to 1 gram range. I'm back to trying higher dosage levels.
   I believe that there may really be agents and protocols that will already extend lifespan. I think they're already showing up in our populations. (One in three people in the U. S. are projected to be retired in the 2020-2030 time frame.)
   In searching Google for info on longevity research, guess what? Item 9 turns out to be a news release from this news site. (Ray Sahelian's personal website was first on the list.) It brought home to me the range of readers who find these articles through search engines, and the importance of accuracy on my part. So I've spent the day reviewing longevity research. I'll try to carry follow through, and report on any intervention strategies that I find. I hadn't planned to go off more than one night on this tangent, but having tackled it, I'll try to research this sufficiently to do a reasonable job of it.
       I've added some material to Total Rejuvenation .
   Tonight's discussion is about Total Rejuvenation . The Mega Foundation science news site still can't be updated.
   The Mega Foundation science news site is still unavailable.
   Last night, just as I was preparing to upload to this website, the cable went out, and stayed out all night. (I know, because I checked it again at 4:30 this morning.) (It's 4:30 in the morning. Do you know where your cable modem is?) But now, it's back in service.
    Tommie and I have slept like stones the past two nights, lending further currency to the idea that the a-lipoic acid and/or the acetyl-l-carnitine were causing us insomnia. I’m playing guinea pig by resuming 25 milligrams of a-lipoic acid every other day.

3/22/2002:     New! The Mega Foundation has established new bulletin boards . If you think you might have any interests in the discussions that are underway, we would certainly welcome you, and would invite your participation.
   Tonight's editorial ran into a speed bump just as it was going to press. I'll have to postpone it until I can recheck my results.
    Tonight's science news underscores the ground swell of growing enthusiasm for space flight, and for the manned (and womanned) investment of Mars. There are discussions about "multi-generation arks" departing Earth before the end of this century. However, one fact is missing from these scenarios: the retardation, if not the actual reversal, of aging. (It needs to be stressed that the reversal of aging would not result in immortality or anything like it. Children and young adults die from various causes, and these fatalities wouldn't disappear even if one didn't age at all. There are other ravages of aging that probably wouldn't be ameliorated by anti-senescence treatments, such as bony protruberances caused by tight shoes.)
    Another interesting product that's on its way to us is "Mira", due by year's end. 'Mira" is a Microsoft-made gadget that would amount to a portable terminal linked by RF to your computer. You could carry it around the house or outside in the yard and work from there. It will cost about $500.

Back to the Twins' Paradox: Pulse-Counting
   It might help to give a detailed pulse count for the Twins' Paradox. I didn't try it last Friday because it was late.
   Because the traveling twin's clock is only running half as fast as the stationary twin's clock,
3/21/2002:     For some reason, the Mega Foundation website won't accept uploads today, so I'll have to update this site only. (The Mega Foundation page can't be uploaded.)
I've spent what little time I've had today reviewing the Maxwell Equations.
3/20/2002:    Tonight's editorial contains numerous equations, and is on a separate page.
      One curious fact about last nights discussion of relativity and electromagnetism is that there is no provision on the current side of the equation for charges moving at relativistic speeds.   The strength of a magnetic field is directly proportional to the total current flowing through a wire. The right-hand-side of the Maxwell equations is a function of current density but not of current velocity! Of course, when current is flowing down a wire, the positive and negative charge in the wire is balanced, with no net electric fields (for practical purposes) emanating from the wire. Otherwise, the electrostatic forces would be orders of magnitude greater than the magnetic fields. It may be that relativistic effects, even though they are extremely small, are enough to produce the effects we see in the everyday world (just as terrestrial gravity is strong by our standards, but is infinitesimal compared to the 2 X 1011 gees acceleration on the surface of a solar-sized, cold neutron star).
    Its clear that electromagnetic effects have to operate in accordance with relativity. Otherwise, wed be able to use them to detect absolute motion. Its also clear that all other fields and forces must also possess wave equations of the form,

for the same reason.
Brain Boosters Again.
    Tommie and I have slept like stones the past two nights, lending further currency to the idea that the a-lipoic acid and/or the acetyl-l-carnitine were causing us insomnia. Im playing guinea pig by resuming 25 milligrams of a-lipoic acid every other day.
3/18/2002:    Brain Boosters: As I mentioned about a week ago, Tommie Jean and I cut our doses of a -lipoic acid to 25 milligrams a day.
    Tommie Jean has been our canary. She's been sleeping like Rip van Winkle. But all of a sudden, she developed serious insomnia, so we're cutting back further on the
a-lipoic acid to, probably, 25 milligrams every other day.
   Today's editorial is " Hidden Relationships ",
3/17/2002:        A few remarks regarding last nights discussion of the Twins Paradox:
    Since the traveling twinss traveling at 7/8ths of the speed of light, shell be 7/8ths of the way to the Centauri system when the first light pulse from Earth reaches Centauri. It will take her 8/7ths as long as it takes light to reach the Centauri system, or about 4.9 light-years. Only one-eighth of the total number of pulses sent by the stationary twin will be intercepted by the traveling twin before she reaches Centauri. The other 7/8ths will still be in the pipeline between Earth and
a -Centauri.  But when she turns around and starts back, then everything is going to catch up with her. On the way back, all the pulses that are still heading her way from Earth plus all the new pulses that are emitted while shes returning home will wash over her. Since time is only passing half as fast for her as it is for the stationary twin, the frequency with which the terrestrial pulses are arriving at her ship will appear to be twice as great as it would if she werent experiencing time dilation.
    I wrote an Introduction to Relativity that Im not sure has been presented here. The logistics of converting diagrams and equations to html may have intimidated me.

    Russell Rierson has written to discuss the Twins Paradox of special relativity.
    The Twins Paradox arises when one member of a pair of twins travels at relativistic speeds to another star and then returns. The traveling twin will age less than the stationary twin who remains at home.
    The paradox arises because, in accordance with the special theory of relativity, while the traveling twin is traveling, it appears to her that the stationary twin's clock is running half as fast as her own, while to the stationary twin, it appears that it's the traveling twin's clock that is running half as fast as the stationary twin's clock! So why should the traveling twin be the one who ages? Why single her out?

   I've approached the Twins Paradox in three ways.

(1)   The Traveling Twin's Relativistic Speed Appears to Her to Shorten the Distance to Her Destination
    When the traveling twin reaches her cruising speed---let's say that it's about 7/8ths of the speed of light to make it a round number, so that her clock is running only half as fast as the stationary twin's clock---she'll discover that the Centauri system where she's headed appears to be only half as far away as she thought it was, or about 2.13 light-years. In fact, she'll discover that everything in front of her and everything behind her is only half as far away as it was before she started. Because the distance to Centauri is now only half as great as she thought it was before she embarked, she'll think that it only takes her half as long to get there as she'd expected. (Of course, from our perspective, her clock is running only half as fast as it should, and that's why she thinks it's taking her only half as long to get where she's going.) While she's traveling, it will look to her as though our clocks are only running half as fast as hers.
    From our perspective, she'll look only half as "thick" as she did before she accelerated to 7/8ths of light-speed.
   During her outward flight, since she's traveling at 7/8ths of the speed of light, she'll receive only a fraction of the terrestrial-clock laser pulses that have been beamed to her over the 4.87 years since she left home. The rest will still be in transit between the Earth and the Centauri system. It's on the strength of these light pulses that the traveling twin concludes that the stationary twin's Earth-bound clock is only running half as fast as the traveling twin's clock.
   Then when she reaches her destination and decelerates, she'll find that everything gets back to normal. The Earth is once again 4.26 light-years away, and everybody's clocks will appear to be running at the same speed..
    Now she accelerates back toward the solar system, and once again. everything seems half as far. On the way back, she'll run into all the light pulses that have been emitted by the Earth that she didn't pick up en route. The other 7/8ths of the pulses that are in transit plus all the pulses issued during her 4.87-year flight home will reach her receiver. When she nears the solar system and decelerates, clocks and distances will return to normal for her, but she has experienced only half the trip time (plus whatever it took to accelerate and decelerate) that her stationary twin recorded.
   The asymmetry here arises when the traveling twin decides to turn around and come back to the Earth, and then to decelerate and remain on the Earth. Had she chosen to continue on her way past the Centauri system, then she would have continued to measure terrestrial clock speeds that were half her own, and her terrestrial twin would have inferred clock speeds for her that were half those of terrestrial clocks. But when she turned around and started back, she began to run head-on into all the pulses that had been chasing her. Now, all the pulses that are in transit plus all the pulses issued during her 4.87-year flight home will reach her receiver.

(2)   The Traveling Twin's Clock Only Beams Half As Many Laser Pulses Toward the Stationary Twin As the Stationary Twin's Clock Sends to the Traveling Twin
     A second way to confirm this is with clocks and light pulses. Here, although you can run a detailed count of all the pulses emitted and received by both the stationary twin and the traveling twin (which I almost did above), the easy way to see it simply is to observe that the traveling twin's clock is running only half as fast as the stationary twin's clock. The traveling twin will emit only half as many pulses during the trip as will the stationary twin. Both will agree upon the numbers of pulses that each one transmits and that each one receives, but they only agree because the traveling twin turned around and intercepted all the pulses that the stationary twin's clock had broadcast throughout the traveling twin's entire flight. If the traveling twin had just kept going, the terrestrial twin's clock would have seemed to continue to run half as fast as the traveling twin's clock. It's turning around and coming back, and then remaining in the solar system that makes the process asymmetric.

(3)   Looking At It As Alternate Paths in a Wood
    The third way to tackle it is to think of it in terms of the way that I started to explain it in " Our 4-D Universe ".
    You and I are walking down a road together. We come to place where there's a fork in the road, with one road leading off at a 30° angle to the left and the other road veering off at a 30° angle to the right. You're on the right side, so you agree to take the road that angles off to the right, and I take the road on the left. As we walk along, it looks to me as though you're behind me on the right, and it looks to you as though I'm behind you on the left. After a while, you come to a side road that crosses your road at 90° and I come to a side road that leads off to the right at 90°. Then If I wave you in my direction and you come over to my road, it will look to you as though I'm up ahead, and you'll have to travel farther than I have to catch up with me (because you will have taken a detour). On the other hand, if you encourage me to catch up with you, and I do, then it will look as though you're ahead of me, and I'll be the one who has to travel farther to get where you are (because I took a detour).
     This is what happens in the Twins' Paradox, except that the twins are moving in the time direction (as are we all), with one of them (the traveling twin) heading off at the spatial equivalent of 60° for a while (accelerating to 7/8ths c) and then turning back at the spatial equivalent of a 120 angle (decelerating to 0 and then accelerating 7/8ths c in the opposite direction), returning to the path the rest of us are on (reaching the Earth), and turning the same direction we're pointed (decelerating)..
    If these were spatial rotations, the traveling twin would have traveled farther than the stationary twin and would be 8½ years older than the stationary twin, but because of the zany way that distances are measured in a hyperbolic space (r2 = x 2 - y2 instead of r2 = x2 + y 2), the traveling twin actually travels a shorter distance through time by detouring than does the stationary twin by remaining on the straight and narrow path. (I realize that's counterintuitive.)

- Where Is Our New Space-Time Coming From, And How Much Does It Cost?
  Maybe it's time to introduce a couple of questions that are puzzling me.
Presumably, the mass of the universe hasn't changed since the Big Bang. That means that the density of that early universe, containing about 1052 kilograms, must have been enormous. Neutron stars weigh in with a density of about 1015 grams per cubic centimeter. Our sun's Schwartzchild radius is 2.95 kilometers, or about 3 kilometers, or about 3,000 meters, or about 3 X 105 centimeters. That means its volume, if it were a black hole, would be about 1.13 X 1017 cubic centimeters. Its mass is about 2 X 1030 kilograms, or about 2 X 1033 grams. Dividing the one by the other (Oh, all right. If I must, I must. I'm dividing 2 X 1033 grams by 1.13 X 1017 cubic centimeters) yields a little less than 2 X 1016 grams/cubic centimeter. Scaling that up to the universe, it would have had the sun's black-hole density when it was about 2 light-years in radius, or very shortly after the blessed event.
Anyway, getting to the questions that are puzzling me, I'm wondering if the idea of an explosive expansion is appropriate to a situation in which matter has to create new space-time or to expand existing space-time. From a four-dimensional viewpoint, the universe just is. It's a static sculpture in four dimensions. What are the rules for the creation of space-time by matter? Does it require energy? We don't see matter creating new space-time or expanding old space-time around us. And what does this portend for the constants and the laws of physics? Is energy stored in curved space-time? And woul that mean that space-time has mass? (I've never asked these questions before. This was never discussed in graduate school.)
            (To be continued)

-   There are Actually No Axes
    In the real world of our experience, there are no axes. Width, depth, and height depend upon our orientation. Viewing something, and then walking a quarter of a circle around it will convert width to depth and depth to width. What really happens is that when we go from, viz., two dimensions to three, all of a sudden, our world blossoms from a plane to the rich, full panoply of three dimensions. Vision in a two-dimensional world would see, looking through an infinitesimal slit, only range and direction.   So how does it look when you add a fourth dimension to our familiar three? I cant even imagine. It would presumably be a breathtaking experience.
    I have been maundering about the time direction and about our moving down the time axis. &. only there arent any axes! Now its true that time is different from space when it comes to rotating the time axis about one of the spatial axes We do have temporal dimensions that arent accessible or visible to us. Id better think further about this before saying more. (Books on relativity show light cones and world lines, so Im not the first to employ these constructs.)

A few Minutes Later   After thinking about it, I think that time has to be as previously I've pictured it, with our temporal extensions roughly parallel with each other in a "time direction". It's a filamentary universe, in which objects stretch terribly much farther in one dimension than they do in the other three. This can be checked using two dimensions and time, and then easily generalized to three dimensions and time. I think the time axis extensions have to be somewhat parallel to fit together. I'll examine the alternatives, but at the moment, I'm thinking that it may be the way I've previously described it.    

Brain Booster Update    
   On March 3rd, I described my experiences while taking the food supplements described in the article, New Pill May Lead to Full Body Rejuvenation "  I said that the two supplements, acetyl-l-carnitine and alpha-lipoic acid, seemed to have upped my "energy" levels to the point of causing serious insomnia, and that I thought memory and other cognitive functions might have slightly improved. Close caucus with Dr. Ray Sahelian's " Mind Boosters " revealed that Tommie and I were taking excessive dosages. We cut back our dosage levels, and have now been taking these supplements for about two weeks. So far, I haven't seen any ponderable rejuvenation taking place. On the other hand, though it's hard to tell subjectively whether these supplements are making their marks as aides memoire, my best guess would be that they are. (But I still can't find what I did with my keys.)

-   To summarize, I've been wrestling with two basic models of the universe this past week. One is the traditional Big-Bang picture of a sphere that's expanding linearly with time. Its expansion rate seems to be accelerating smartly, apparently because of the presence of "dark matter" or "negative energy" that generates repulsive gravitational fields. It requires three spatial dimension plus one temporal dimension. The other is also a Big Bang scenario that differs from the traditional Big Bang concept in that it postulates that the universe is the three-dimensional surface of a four-dimensional hypersphere closed by the gravitational curvature of space-time.
Our absolute motion relative to the cosmic background radiation is about 350 kilometers per second... about 0.1% of light-speed, and comparable to the vector sum of our solar, galactic, galactic cluster, and galactic super-cluster velocities in the dance of the galaxies.
The fact that we have a small but measurable velocity relative to the cosmic background microwave radiation suggests to me that the simple spherical expansion scenario is the one favored by the experimental evidence.
As I read the runes, in the current cosmological picture, the universe has an absolute size of 12 to 16 billion light-years, and an absolute age of 12 to 16 billion years. (Both numbers are the same, presumably because the boundary of the universe has been expanding at light-speed since the Big Bang.) These absolute values are a stark contradiction to the philosophy of physics taught to me as a graduate student in the 1950's. I was taught that because it was impossible even in principle to detect absolute motion, it didn't exist. (As a student, I bought the idea that you couldn't measure absolute motion (within the framework of special relativity), but not the idea that absolute motion didn't necessarily exist... just that we couldn't measure it within the framework of special relativity.) This was hammered home and perhaps oversold during the early 20th century by a younger generation of physicists trying to get through to an older generation of physicists who were locked into the mechanistic subluminiferous ether of the 19th century. (I'm told that Einstein used to attend seminars where some well-entrenched older physicist would spend two hours calculating the tremendous forces that would be required to contract a rulers of various materials to half its length, and arguing that no conceivable mechanical force could cause a clock to slow down. Sitting in the audience, Einstein would smile and applaud.)
By recognizing time as a fourth dimension, special relativity "merely" extended Galileo's theory of relativity of inertial frames by adding time and motion to it. But what really set the cat among the pigeons was Einstein's realization that the speed of light will measure the same in any inertial system. (This is really only saying that rotations of the time axis about a spatial axis are "hyperbolic" rather than circular, and this was revealed to us by the Maxwell Equations.)
Our orbital speed about the center of the Milky Way galaxy is about 225 kilometers a second. (Let's check that number. Our radial distance from the center of our galaxy is about 33,000 light-years. A light-year is about 10 trillion (1010) kilometers, so we're looking at about 0.33 times 1015 kilometers. Multiplying that times 2 pi gives us an orbital circumference of about 2 time 1015 kilometers. There are about 31,536,600, or 0.315 times 108 seconds in a year. At 225 kilometers per second, the solar system would travel about 0.71 times 10 kilometers or about 0.71 light-years per year. At that rate, it would take the solar system about 2 X 1015/0.71 x 1010, or about 282,000 years to orbit the Milky Way galaxy. (For us here in Solar Sector, a galactic year is about 282,000 terrestrial years.)

3/12/2002: -  Eureka! I've got it!
What I described last night is correct. It just needs a little amplification.

Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold  
    Ms. Flatland can think of the universe as being comprised of two dimensions, plus time as the third dimension. That way, she can use her sense of the passage of time to help her visualize whats happening in the third dimension.
    We can think of the universe being comprised of three dimensions plus time as the fourth dimension. That way we can use our sense of the passage of time to help us visualize whats happening in the fourth dimension.
    At the North Pole, Ms. Flatland will see lines of longitude coming in from all directions (in her two-dimensional planar world). Well see lines coming in from all directions in three-dimensional space because we can imagine three dimensions.
    If we imagine that Ms. Flatland can see the two-dimensional cross-sections as she moves infinitesimally south of the North Pole, the first cross-section shell see will by an infinitesimal circle where the sphere on which shes standing is sliced by a horizontal plane to give her its two-dimensional cross-section. The circle is a curved one-dimensional object that is a closed curve and that requires two dimensions to contain it.

    In the same way, as we moved infinitesimally away from the focal point of all the possible geodesic curves, we would see an infinitesimal spherical surface enclosing us (assuming that we could squeeze inside it). It would be a curved two-dimensional object that is a closed surface, and that requires three dimensions to contain it.
    As Ms. Flatland moved farther along (and down) her curve of longitude, the circle would expand very rapidly at first, and then, slower and slower. One of the questions Ms. Flatland might ask would be: What lies outside the circle? (Ms. Flatland would be thinking in terms of a two-dimensional, disk-shaped universe.) The answer would be a larger circle.
    As we move farther along (and around) our curve of longitude (or geodesic), the sphere that encloses us would expand very rapidly at first, and then slower and slower. One of the questions we might ask would be, What lies outside this spherical surface? (We would be thinking in terms of three-dimensional, spherical universe.) The answer would be a larger spherical surface.

    As Ms. Flatland approached the equator, the expansion of the circle that bounded her disk-shaped cross-section would slow, until, at the equator, it would cease altogether. Then it would begin to shrink. Ms. Flatland would probably wonder at this. She would wonder even more when we told her that all the lines of longitude, which were spreading to all the points of the compass at the North Pole, were now all running parallel to each other. She would wonder how they could go from being so divergent to being parallel. The answer is that these lines of longitude are time lines or world lines. The bending of these world lines would have taken place in the third dimension, and would have manifested themselves to her as developments over time. She would have experienced their curvature-in-the-third dimension as changes in the rate at which the circle expanded.
    As we approached the equator, the expansion of the spherical surface that bounded our spherical cross-section would slow, until, at the equator, it would cease altogether. Then it would begin to shrink. We would probably wonder at this. We would wonder even more when we learned that all the lines of longitude, which were spreading in all directions at the focal point, were now all running parallel to each other. We would wonder how they could go from being so divergent to being parallel. The answer is that these lines of longitude are time lines or world lines. The bending of these world lines would have taken place in the fourth dimension, and would have manifested themselves to us as developments over time. We would have experienced their curvature-in-the-fourth dimension as changes in the rate at which the spherical surface expanded.

    Beyond the equator, Ms. Flatlands circle would shrink slowly and than faster and faster until she reached the South Pole. (looking to her just like the North Pole). Finally, she would go through the same process of seeing the circle expand and than contract as she rode along her line of longitude (or timeline) back to the North Pole.
    Beyond the equator, our spherical surface would shrink slowly and than faster and faster until we reached the South Pole. (looking to us just like the North Pole). Finally, we would go through the same process of seeing the spherical surface expand and than contract as we rode along her line of longitude (or timeline) back to the North Pole.
    Ms. Flatland would be able to see how the profile of her trip through the third dimension would show up as a circle. If she looked at all the lines of longitude from a point above the North Pole, she would see that they all extended out one radius away from the North Pole and then appeared to end. Of course, in reality, they were curving down into the third dimension, and where they appeared to end, they had instead reached the equator and were curving back toward the South Pole. Their apparent discontinuity in two dimensions masked continuity in the third dimension.
    So there you have it. Ms. Flatlands circles can be visualized as the lines of latitude on a world globe, and our universe could be visualized in the same way, except that in place of Ms. Flatlands circles, we would use three-dimensional surfaces.

This Fourth Dimension Is Spatial. Time Becomes a Fifth Dimension
    Of course, Ive used time and changes over time as an aid to understanding how this works. Really, though, this hypothetical hypersphere is strictly a spatial matter, just as it is for Ms. Flatlander.

Same for Any Point on the Sphere
    Any point on a sphere could have been designated as the North Pole for the purposes of understanding whats going on. Whats true at one point will be true at all the others.

Do We Need Four Dimensions to Represent a Hyperspherical Surface?
    One question might be: Do you really need four dimensions to represent a hyperspherical surface? I think the answer to that is: yes. To see that, you need only consider the case of how many dimensions it takes to represent a spherical surface. There might be some abstract way to define a spherical surface using only two dimensions, but to properly deal with it, I think you need three dimensions.

How Can We Visualize a Hypersphere?
    So how can we visualize a spherical sphere? I know of two ways to proceed. One is to visualize it is the spherical polar coordinate way, starting with a spherical dot at the center that expands very rapidly at first and then slower and slower until it stops expanding and begins to contract. It contracts slowly at first, and then faster and faster until it again becomes a point at the center.
    Another way to visualize it thats analogous to thinking of a sphere as being constructed of disks of varying radii is to imagine an enclosing sphere, with internal spheres that start with a dot at the left end of the sphere, and grow larger and larger as we move to the right until we reach the middle of the sphere. In the middle of the sphere, our growing sphere has reached its maximum size, and is the ssize as the enclosing sphere. Then as we look farther to the right, we see smaller and smaller spheres, culminating in a dot at the right side of the enclosing sphere. (Please see Figures 1 below.) This was the conceptual model I used to calculate the volume of a hypersphere (analogous to using straight lines to compute the area of a circle, and disks to determine the volume a sphere).


   Figure 1 The Three-Dimensional Cross-Sections of a Hypersphere








   Figure 1a The spheres are              Figure 1b The spheres have reached
   expanding from the left                       the center of the enclosing sphere.








   Figure 1c The spheres have reached the far end of the enclosing sphere .

   Incidentally, the torus-like drawings I displayed a few nights ago (please see below) were wrong. Sorry about that!










    To sum it up, the hypersphere behaves the same way I described a spherical sphere in talking about time as a fourth dimension. It starts out as a point, expands very rapidly and then slower and slower until it reaches its maximum size. Then it begins to shrink, slowly at first, and then faster and faster until it disappears. But its taken me a while to connect that with the idea that in a closed three-space analogous to the surface of a sphere, we can go in any direction and still come back to our starting point. Duh-h-h!

The Role of Time in a Hyperspherically Closed Universe
    This still does not include time, which would appear in this model as a fifth dimension. However, its relatively simple. All it adds to the picture is that the radius of the hypersphere weve been discussing would be increasing linearly with time.

Apparently, Our Real Universe Is Four-Dimensional Rather Than Five-Dimensional
    The fact that a dipolar Doppler shift has been identified in the cosmic background microwave radiation would seem to indicate that our real universe is just what it seems to bean expanding sphere, rather than a universe that, hyperspherically closed. If our universe were the hypersurface of a hypersphere (as weve been exploring above), there would be no preferred directions, any more than theres a preferred direction on the surface of a sphere. Still, Im pleased to have solved this problem because it might surface again. Then, too, Ive seen drawings that seemed to me to be advancing the idea that if you went off in any direction, you would follow a geodesic that would eventually bring you back to your starting place. If for some reason thats still the prevailing view, we now understand it.

3/11/2002: -  Hyperspheres again
    Perhaps I've made a little headway toward understanding the hypersphere.
    It may help if we imagine ourselves thinking like Ms. Flatlander when she's at the North Pole. Ms. Flatlander looks at the lines of longitude spreading out in all directions. It looks to her as though they're heading as far apart from each other as they possibly could. She can understand how the line on which she's standing can form a great circle that returns to the spot where she's standing, but what about the other lines of longitude? They're going off in other directions. How can they possibly bend down so fast that they meet the antipodal point on "her" circle? And how about the line that comes in from straight above? Will it go to the left or to the right? How can you visualize it? (Of course, she's picturing it the only way she knows how: in two dimensions. It's hard for her to avoid imagining them bending down in a plane.)
    One way for her to visualize the third dimension is to imagine her riding down along her great circle as it leaves the North Pole, and seeing two-dimensional cross-sections showing where the lines of longitude cross the planar cross-sections. In fact, she could imagine using time as the third dimension, and watching this unfold over time.
    Right at the Pole itself, lines would be coming in from all (horizontal or vertical) co-planar directions, but the moment she got an infinitesimal distance from the North Pole, the lines would form a tiny but expanding circle. The circle would continue to expand until it reached a maximum, when all the lines were parallel. Then it would begin to shrink again until the lines all converged again at the South Pole.
For us, the analogy might idea that we could imagine a spherical surface enclosing all the possible 4p-steradian paths leading away from our reference point. At the starting point itself, we would be like Ms. Flatland at the North Pole, except that instead of being at the center of horizontal lines of longitude, we would be at the center of lines of longitude coming from all directions. The moment we moved away from the focal point of all the longitudinal lines& our starting point or "North Pole"& we could imagine our location on "our" line of longitude defining the current radius of the spherical shell through which all the lines of longitude would pass. Of course, it would be encompassing all the "lines of longitude" (Gauss Theorem in action). As we watched, the spherical surface would expand until it reached a maximum radius. This maximum radius would be the maximum radius of our particular great circle, and it would also be the maximum radius of all the other great circles of all the other lines of longitude. Then the spherical surface would begin to contract, and it would continue to contract until it shrank to a point at the "South Pole" of our circle and of all the other circles.
    (To be continued)

Falling Inexorably Into the Black Hole
    With respect to something falling into a black hole or onto a neutron star, because rulers shrink and time intervals expand as the something approaches the black hole, it appears to an external observer that the object is falling slower and slower. To see this, imagine a neutron star with a radius that's 4/3rds its Schwartzchild radius. In this situation, time intervals will stretch out by a factor of 2. Suppose that someone on the neutron star times something that's moving parallel to its surface, and announces that it's moving 10 meters per second. The external observer says, "Since your clock is only running half as fast as our clocks are out here in the real world, whatever you're clocking at 10 meters in one second is really taking two seconds to go 10 meters, so it's really only moving at 5 meters a second."
    Now suppose that the same something is measured at a vertical speed of 10 meters per second by the guy on the neutron star. In that case, the fact that his rulers have shrunk to half their length also enters in. The guy at a distance says, "Aw, Mac, you're putting me on! Not only is your 10 meters is only 5 meters, but on top of that, you're taking 2 seconds to cover 5 meters. Your real speed is only 2½ meters a second." So an object falling into a black hole will slow down as he gets within a few kilometers of a black hole. At an altitude of 1/3rd the Schwartzchild radius of the hole, when we would expect our object to be traveling at about 7/8ths of the speed of light, it will be doing only about 1/4th that speed, or 7/32nds the speed of light.
I've been working on this quantitatively today, but as of the moment, I don't trust the quantitative answer I'm getting.
                                    (Also to be continued. Be sure to catch the next thrilling episode.)

3/10/2002: - Tonight, I'm going to try something I just discovered: that websites will display pages written in Microsoft Word. Netscape handles it as a file which you can open on the spot. I've had a great deal of trouble transferring equations and drawings to web pages. It will be very helpful if Microsoft documents can be displayed directly on the Internet. But this will be an experiment.
   I've re-derived the expression for the hypervolume of a hypersphere, getting the same formula I did previously, 1/2
p2r4, and the same expression for its hypersurface, 2 p2r3: I've also attempted to generate a side view and a top view of what I was describing last night, at the bottom of the page.

3-9-2002:   Tonight, my picture of the volume that encloses the hypersphere is that it consists of continuously-overlapping spherical surfaces that surround our reference point such that their centers lie throughout the lower 2 p steradians, and and whose (individual) hemispheres extend into the upper half-plane as a kind of torus, such that their "inner" surfaces pass vertically through the point where we're located. However, there are no spheres whose centers lie above the horizontal plane. Picture two side-by-side semicircles on top with a single semicircle with twice the diameter (spanning both of the upper semicircles) on the bottom.
    More in the morning.

   Tonight's specialty is a set of articles on telomeres, under Prolongevity .

   Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold
     I've "gotten" it, but I don't understand it well enough yet that it will be easy to explain.
    The key to understanding this seems to me to be the case of Ms. Flatlander. Lacking a simple drawing tool, I won't be able to present this graphically, but perhaps your imagination can fill in the missing images.
    Any point on a sphere can be considered a "North Pole" as easily as any other point, so we're going to imagine designating Ms. Flatlander's current location as the "North Pole". She's familiar with circles, so she can imagine the seemingly-straight line she's following continuing all the way around the rim of a disk and bringing her back to her starting point. When she looks at another line of longitude that makes an angle with her line, she intuitively thinks, "If that line is an arc of a circle that eventually curves back to cross my line at the South Pole, it's going to have to be longer than my line because my line is a straight line."
    If she now imagines looking at the lines of longitude from a point above the equator, her line will appear straight. The neighboring lines of longitude will curve out from the North Pole, be parallel to her line at the equator, and then converge again at the South Pole. And just as she expected, they'll all look longer than her straight line, with the longest lines of latitude being those rotated 90 degrees from her line, and approaching the poles perpendicular to her line.
    At the same time, she'll realize that when she's at the North Pole, she can choose another line of longitude. After all, there's no difference between one line of longitude and another. They're all there together at the North Pole. All she has to do is turn in a different direction. If she does that, then the new line she's chosen will appear to be the straight line, and all the other longitudinal lines will appear to be longer than that line. Obviously, there must be some kind of strange perspective effects occurring here, presumably having to do with the fact that somehow, these lines are curving into the mysterious third dimension.
    We could just as well imagine the axis of the sphere being horizontal instead of vertical, so that the other lines of longitude slant up and down from her horizontal line instead of left and right. And this suggests how seemingly-straight paths or trajectories can go not only left and right but also up and down.   a plane
    The equivalent situation with us who can instinctively visualize three-space but not four-space is one in which we see lines that can go in all directions and are all great circles that converge and cross each other at an antipodal point at the "opposite end" of the universe from where we're located. Only now it's we who are having to invoke perspective effects to explain how all these lines actually great circles that bend through both the third and fourth dimensions to wrap around a hypersphere.
    Whatever path we take will be a great circle around some spherical surface oriented so that we're starting at its "North Pole". For example, if we start along a curve going in the direction we're facing but tilted upward 1 degree, it will seem to us as though we're setting off to travel further to reach the opposite pole than we would if we stayed on the straight-and-narrow path we were taking before we changed directions. And yet, reason tells us that our original path is tilted one degree below ours, and that it now appears as though it should be the longer path to the opposite pole.
    Somehow, all of those spheres fold to gether to form the hypersphere with radius r and circumference 2pr.
    One important consideration has to do with inside and outside. We've been thinking of walking on the outside of the earth, and that it curves downward away from us. We could also have considered the situation in which we're inside a sphere that's curving upward in both directions. This latter situation would seem to fit better with our situation in our universe.

    Mindboosters : You can find the material regarding "Mindboosters " using the Intelligence Site Map (above) and clicking on " Updated 3/2/2002: My Experiences with Mind Boosters .".

   One of tonight's articles, 'Brane-Storm' Challenges Part of Big Bang Theory - , suggests a five-dimensional manifold in which four-dimensional hypermembranes drift, so discussions of four-dimensional hyperspheres embedded in five-dimensional manifolds may be an educational exercise. At the same time, it's weighty news that our universe seems to be four-dimensional after all. Cosmic background radiation exhibits a small bipole Doppler shift indicating that we are moving at 350 kilometers per second toward the constellation of Leo and away from the constellation of Pisces. It's upon the strength of this (and presumably other) experimental evidence that we can assign an absolute age of 12-13 billion years and an absolute radius of 12-13 billion light-years to the universe. In other words, there is absolute motion, absolute space, and absolute time! Special relativity reveals that a moving observer's clocks and rulers change in just such a way that the moving observer measures the same value for the speed of light as a fixed observer---that is, an observer who is moving at 350 kilometers per second away from Leo and toward Pisces. Such an observer will be at rest with respect to the universe' background radiation.

Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold
    The reason I wasn't very happy with the solution I gave yesterday (please see below) was this:
    We generated a circle from a line segment (bounded by two points) by rotating the original line segment around an axis through its center, basically rotating it into a second dimension. We can think of the disk it creates as consisting of an infinite number of one-dimensional line segments that extend left and right into the second dimension, contacting the original line only at the pivot point at their centers.
    We generated a a spherical surface from a circle by rotating it around an axis that passes through the center of the circle, basically rotating it into the third dimension. We generate a sphere by rotating a circular disk around an axis that passes through the center of the disk. We can think of the sphere that the disk generates as an infinite number of disks that extend into the third dimension, contacting the second dimension only along the line of rotation (axis) that passes through its center.
    We generate a hyperspherical surface (a volume) by rotating a spherical surface around a plane that passes through its center, basically moving it into a fourth dimension. We generate a spherical sphere by rotating a sphere around a plane of rotation that passes through the center of the sphere. We can think of the hypersphere it generates as an infinite number of spheres that extend into a fourth dimension, contacting the third dimension only along the plane of rotation that passes through their centers.
    It seemed to me that last night's approach involved rotating spherical surfaces around points in the surface of the sphere rather than around the center of the sphere.
    So what does this mean? We could imagine a seemingly-infinite horizontal plane, and we could understand how it might really be the surface of a sphere... e. g., the earth But how can we use a set of such spherical surfaces to handle moving up or down above or below our horizontal plane?
    Consider how a spherical surface would appear to a "Flatlander". When the Flatlander looked at her own line of longitude, she would see the kind of straight line she would expect, and she could imagine it going around the world and bringing her back to her starting point. But the only way she could visualize other lines of longitude would be in terms of their projections on "her" plane of longitude... that is, upon the disk that was enclosed by her line of longitude. Looking at it that way, she would see a family of ellipses, ranging from the great circle she was following to a straight line for the line of longitude perpendicular to her great circle.
    By analogy, we can easily imagine defining a spherical surface such that whatever direction to left or to right (or straight ahead) we traveled, we would be brought back to our starting point.... e. g., the earth. Then we might visualize the projections upon our volume of space of spherical surfaces that tilted up or down from our spherical reference surface to be ellipsoids of revolution, ranging from the spherical surface which we had taken for our reference to a vertical plane for a surface that intersected our surface straight up and down. (I still don't have this nailed down.)
    I've tried to calculate the volume of the space enclosing a four-dimensional spherical sphere and, with a back of the envelope calculation, have arrived at 2
p2 r3. This number needs to be rechecked, but the second factor of p appears because it involves an integral of sin4 q , and this introduces a second p factor. Writing 2 p2 r3 as 6/3 p2 r3, and dividing by 4/3 pr3 yields a ratio of 1.5 p . for the volume of the "hyperspherical surface" enclosing a spherical sphere.
                                                               (To be continued)

    Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold
    Suppose that there were someone by the name of Linus who lived on a very, very, very long line---a line 24 billion light-years long---in a one-dimensional universe. Linus could crawl up and down his line for a very, very, very long time, feeling different textures, smelling different smells, and tasting different tastes. But supposing that we rotated Linus' line aroujnd an axis in the middle and generated a circle. And Linus, instead of living on a diameter of that circle, lived on its 24 billion light-year circumference. In that case, each time Linus crawled 24 billion light-years, he would find himself retracing the same old "terrain" he had covered before.
   Note that we use a one-dimensional line to generate a two-dimensional circle by rotating it about an axis in a third direction, and that Linus is crawling along the circle's one-dimensional rim.
   Next, suppose we know a fellow nicknamed "Flats" who lives on what he thinks is a flat disk. But suppose, in reality, that his disk has been rotated around an axis that runs through its diameter to generate a sphere, and that "Flats", instead of living on a flat disk, lives on its spherical surface.
   Note that we use a two-dimensional circle to generate a three-dimensional spherical surface by rotating it about an axis in the plane of the circle.
    Finally, suppose that we live in what seems to us to be a spherical volume, but in reality, it has been rotated through a fourth dimension (e. g., time), and is really the hyperspherical surface of a hypersphere.
   Note that for each of the two preceding universes, we rotated it through the next higher dimension around an axis through its center. Note also that a great circle represents one path that either Linus or "Flats" can take on a spherical surface, and that we generate an infinite number of great circles that define the spherical surface.
  One way to imagine the hyperspherical surface of a hypersphere is to think of it as comprised of an infinite number of spherical surfaces passing through any given point on a spherical surface and tilted up at angles ranging from 0 to 180 degrees. Each of these spherical surfaces would have circumferences measuring 24 billion light-years. They would pass through a fourth spatial dimension of which we would be unaware. They would fill all of our universe' three-space. If we changed direction, then unbeknowns to us, we would be moving through this additional dimension without realizint it, just as "Flats", living on the surface of a sphere, could move through the third dimension without being aware of it, or realizing that it existed.
   Here, we would be generating a four-dimensional hyperspherical by rotating a three dimensional spherical surface around axes at every point in the spherical surface.

   I'm not very happy with this explanation. I'll try to re-examine it tomorrow.

    The problem I posed last night is similar, except that, in keeping with the general-relativistic notion that at any given instant, the universe is closed just like a spherical surface (e. g., the surface of the earth) would be to a Flat-Worlder. We've said that a spherical surface requires three dimensions to contain it, and by extension, a hyper-spherical surface would require four dimensions to contain it. Like the Flat-Worlder who would experience what would seem to be an infinite but cyclic surface, we would experience an infinite but cyclic volume.
    I will probably write a little more about this tomorrow night or so, but I found in reading Paul Davies' book "About Time: Einstein's Unfinished Revolution", (Simon & Schuster, Inc.) today the statement that recent results from the COBE (Cosmic Background Explorer) satellite show that we are moving at an absolute velocity of about 350 kilometers per second in a specific direction relative to the universe as a whole. This would indicate that the model of the four-dimensional universe depicted above is the correct picture rather than a universe closed on itself in a five-dimensional continuum. This also justifies our somewhat-approximate and labile value for the age of the universe, since the time dilatation would be negligible at a speed that is only about 1/1000th of the speed of light.
     In other words, we don't have to visualize the three-dimensional spherical volume that envelops a four-dimensional hypersphere embedded in a five-dimensional manifold.

  Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold
    Turns out it wasn't as tough as I thought it would be. Here's how I think it works.
    Consider the cone shown below in Figure 1. It can be considered to consist of a stack of infinitesimally-thick disks or circular laminae, with steadily increasing radii. As such, it's a conically shaped volume.
    it can also be considered a hollow shell. Both a conical volume and a conical shell look the same from the outside, but one is a volume and the other is basically a two-dimensional shell, although it requires three dimensions to contain it.

    In a similar vein, the spherical hypercone (Figure 2) can be considered to consist of an infinite number of three-dimensional spherical volumes of infinitesimal duration with steadily-increasing radii. (Think of an infinite number of spherical volumes at different instants in time that are steadily increasing in volume with time. They are occupying the same space but at different instants in time.) As such, it's a conical-spherically-shaped hypervolume.
    It can also be regarded as just a three-dimensional static volume that is the container of all the instant-by-instant three-dimensional shells from the moment of the Big Bang to the present time, but that doesn't incorporate all the instant-by-instant three-dimensional volumes themselves, although it requires a four-dimensional manifold to contain it. In other words, it's the form without the content. It's four-dimensional because it includes the spherical outlines or boundaries of all the instant-by-instant spherical volumes of the universe' existence.

Figure 1 (Above) - An ordinary cone

Figure 2 (Right) - A spherical hypercone the shape of our universe, showing the spherical universe at different moments in time. This would be its overall shape in four dimensions, as depicted in two dimensions, augmented by your imagination. It's just an expanding sphere, with its time history shown by the conical envelope depicting the time history of its steady inflation
    Most objects in the universe look like spider -silk in four dimensions, but the universe itself looks as broad as it does long.

    The problem I posed last night is similar, except that, in keeping with the general-relativistic notion that at any given instant, the universe is closed just like a spherical surface (e. g., the surface of the earth) would be to a Flat-Worlder. We've said that a spherical surface requires three dimensions to contain it, and by extension, a hyper-spherical surface would require four dimensions to contain it. Like the Flat-Worlder who would experience what would seem to be an infinite but cyclic surface, we would experience an infinite but cyclic volume.
    I will probably write a little more about this tomorrow night or so, but I found in reading Paul Davies' book "About Time: Einstein's Unfinished Revolution", (Simon & Schuster, Inc.) today the statement that recent results from the COBE (Cosmic Background Explorer) satellite show that we are moving at an absolute velocity of about 350 kilometers per second in a specific direction relative to the universe as a whole. This would indicate that the model of the four-dimensional universe depicted above is the correct picture rather than a universe closed on itself in a five-dimensional continuum. This also justifies our somewhat-approximate and labile value for the age of the universe, since the time dilatation would be negligible at a speed that is only about 1/1000th of the speed of light.
     In other words, we don't have to visualize the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold.

  Abundance of Ice on Mars!
    I should certainly mention the stunningly wonderful news that water-ice has been discovered in lavish abundance on Mars . Of course, this opens the garden gate to human colonization of Mars. For a number of reasons, we'll have to be troglodytes on Mars. Mars' lack of van Allen belts, an ozone layer, and a dense atmosphere exposes it to solar particle radiation, ultraviolet light, and micrometeorite bombardment. Also, with a pressure differential of at least 3/4ths tons/sq. ft. or 20 metric tons per sq. meter, about 8 feet (2.5 meters) of dirt will have to sit on top of a building's roof to offset the upward thrust of an inhabited building. (It might be possible to provide buildings with northern exposures and north-facing windows, since most micrometeorites and solar radiation would be confined to roughly the plane of the ecliptic.)

"Colonization of Mars" Participatory Website
    A couple of months ago, I set up a "participatory website " where anyone could contribute thoughts about the colonization of Mars   The problem is an exercise in logistics in which one tries to minimize the weight of payload that will have to be shipped to Mars in order to establish a self-sustaining industrial base on Mars. This will entail taking maximum advantage of Mars' endemic resources from the get-go, shipping only high-tech parts from the Earth. For example, one of the challenges will be to find limestone, sand, and water for the preparation of cement. Another challenge will be the transportation of minerals from remote locations to some central industrial complex. I thought about a computer game in which players try to optimize the selection and sequencing of remotely-operated equipment to be trans-shipped to Mars. One might start with a micro-industrial base, and then, using small devices to build larger devices, might bootstrap to a largely-autonomous local economy.

Justin Chapman: An Untold Story
    A couple of weeks ago, a newspaper article appeared in the "Rocky Mountain News" following up the story of Justin Chapman . Tommie Jean and I had first heard about Justin from the Kearneys a little over two years ago. At the time, Justin was auditing, with the instructor's permission, a physics class at the University of Rochester.   ,
    Last summer, when we met in Nashville, Kevin and Cassidy expressed concern over rumors they had heard regarding Justin's situation. Today, in our local newspaper, an article appeared entitled,
    "Mother admits she rigged results for son regarded as a boy genius."
    The article cites "a long list of Justin's purported accomplishments, including a perfect 800 on the math section of the Scholastic Aptitude Test, a genius score at age 3 on the Wechsler Intelligence Scale test, and an IQ score of 298-plus on the Stanford-Binet Intelligence Scale at age 6.
    The article continues:
    "The latter test was administered by Linda Silverman of the private Gifted Development Center in Denver. She described Justin as "the greatest genius to ever grace the earth."
    Justin's mother, Elizabeth Chapman , "told the News she had checked out a copy of the Stanford-Binet IQ test booklet and researched it with her son before Silverman administered the test. She told the New York Times that Justin himself had found the manual in the University of Rochester library and memorized the answers."
    "Chapman said that she had apologized to Silverman, who had helped her move to Colorado and had been one of Justin's staunchest advocates. She said she had apologized to other friends as well.
    "Chapman also acknowledged that she made a copy of of a neighbor's SAT scores. She said she altered the score so it appeared the perfect scores of 800 in math and 650 in verbal were achieved by Justin.
    "She said Justin never finished the Wechsler test at age 3 and that the score was a fake.
    "Still, Chapman said her son was highly gifted, even without the deception. She said Justin took the University of Rochester courses himself, and did the course work necessary to receive a high school diploma from Cambridge Academy, a Florida-based online school, where he was credited with a 3.75 grade average.
    "Chapman said her parents and the boy's father, James Maurer, had filed for custody of the boy, who now lives with a foster family. Maurer, who lives in Raleigh, N. C., confirmed he had filed for custody but declined further comment, the Times said."

    Poor Justin! Poor everyone involved.
    My calculations regarding the frequency distributions of ratio IQ's point to a one-in-five-billion ratio-IQ of about 256 for the brightest person on the planet. Someone might wonder if all of these reports of amazing precocity are fraudulent. The answer to that is a resounding "no". I know a few of these ex-prodigies, and as adults, they're phenomenal, with virtually perfect scores on adult IQ tests, and with obviously-extraordinary talents.

Visualizing the three-dimensional spherical volume that envelopes a four-dimensional hypersphere embedded in a five-dimensional manifold
    No, I haven't yet. Have you? (I mean, like, "Gosh, it looks a little tough!")

3-3-2002: Mind Boosters
    Last Sunday, I posted an article that makes unusual anti-aging claims for rats fed the supplementary nutrients acetyl-l-carnitine and alpha-lipoic acid in a study funded by the National Institutes on Aging. I took it seriously because the article was based upon three journal articles appearing in the Proceedings of the National Academy of Sciences, and because one of the co-authors was the well-known biologist, Bruce Ames. I also wrote it up last Sunday night.
    My message for tonight is that I think it might be working. For the past few nights, I've had serious insomnia. I've been able to weather it during the day without getting sleepy, but it's become unavoidably noticeable. Today, I looked up acetyl-l-carnitine and alpha-lipoic acid in Dr. Sahelian's Mind Boosters . He says this about them.
    Dr. Sahelian cautions, though, that high dosages can induce nausea, restlessness, and insomnia. He recommends dosages of 100 to 250 milligrams a day.
    Since last Saturday, Tommie and I have been taking 500 milligrams a day. We'll cut back to 250 mgs. a day.

    Concerning alpha-lipoic acid, Dr. Sahelian writes,

"The Author's Experience
    "Unlike most antioxidants such as vitamins C, E, and selenium, there is usually a noticeable effect from taking Lipoic Acid. I've observed a sense of relaxed well-being and slightly enhanced visual acuity. Higher dosages of 40 mg. of more, even when taken in the morning, cause me to have insomnia."

    Me, too. Tommie Jean and I have been taking 100 mg. a day, and something is causing me to have insomnia. We'll cut back to 20 milligrams of
a-lipoic acid a day. Also, it's hard to tell about such things, but it seems as though my vision and my cognitive powers might have improved. I don't generally look for, or think about such effects, so they have to jump out and trip me before I'm aware of them.
    I've only been taking these supplements for a week.
    But the most relevant thing about this is that, like the women at the health food store, I'm noticeably feeling the effects of these nutrients. My memory seems to be more powerful. And if these two nutrients really work, what about some of the others?

Problems with the Speed of Light
    After uploading last night's web pages, I realized that there's a fundamental dilemma with a model of the two-dimensional-plus-time universe that considers the two-dimensional universe to cover the curved surface of a sphere rather than the flat surface of a disk. The problem is that if the radius of the   sphere is expanding at the speed of light, the circumference of the sphere will be expanding at 2
p times the speed of light. And that's 2p too fast. Hm-m-m. We might argue that the radius of the sphere is only expanding by 1/2p times the speed of light. We also have to remember that the universe is the same age as what we measure for its age... ~12,000,000,000 years. Is an observed age of the universe of 12,000,000,000 years commensurate with an expansion rate of 1/2p X c  for the radius of the sphere? In other words, if space-time is circularly closed on itself, and we measure the speed of light to be 3 X 10 kilometers per second around our spherical shell (which is what we perceive as our universe), could that speed be independent of the speed at which the shell itself expands? It would mean that the angle between the time axis and the radius of the cone would have to be precisely 7.161972439 degrees (45/2p degrees) instead of 45 degrees. Hm-m-m...   Any ideas? (For my part, I'm going to do a little homework investigating the cosmological concepts that cosmologists are employing.)
Galilean Relativity
   Both Galilean and Newtonian relativity set forth in their models of the universe the principle that there is no such thing as absolute location, absolute orientation, or absolute motion. The laws of physics appear to be the same in all reference systems, whether "fixed" or in a uniform state of motion. The only codacil that Einstein added to these observations was the (revolutionary and far-reaching) conclusion that measurements of the speed of light will yield the same result in all reference systems.
How Can the Universe Have an Absolute Radius and an Absolute Date of Origin?
   Note that both Newtonian and--Einsteinian?--relativity refer to the mathematical (or physical) models that we use to describe our universe, but they are not the universe itself. ("The map is not the territory.") To my knowledge, there is nothing that says that there cannot be absolutes that apply to the universe. You can say that a sphere has no preferred axis, but our spherical Earth has a preferred direction defined by its axis of spin. And while the Earth's axis of rotation is a very localized orientation today, the Earth would have defined the entire mortal universe prior to1500.
    I'm mentioning this only because today, our universe is being assigned a fixed time of creation (12,000,000,000 B. C.), and a known radius (12,000,000,000 light-years). It might seem that this would violate the premise that there are no absolutes in relativity, but I'm arguing to myself that the situation is similar to other mathematical constructs that are applicable within their domains.
The Universe as a Squashed Ellipse
    Getting back to yesterday's conclusion that for someone traveling at high relativisitic speeds, the universe would become a squashed ellipse, you don't have to think about this long to realize that this would allow you to determine your absolute velocity. Looking to left and right, stellar densities would be lower than they would be fore and aft. And of course, starlight would be blue-shifted in front of you and red-shifted behind you. These observations would only tell you about your local conditions, but when you looked deep into space, you would see the farthest galaxies now appearing much nearer than do those that were perpendicular to you. Also, the cosmic-background microwave radiation would be Doppler-shifted by your motion. But an interesting consequence of this interpretation is that light would no longer be traveling on the light cone at 45 degrees from the time axis unless the age of the universe, as determined in your direction of motion, were smaller by the right amount. Since light must travel at the speed of light, we have to conclude that this is what would happen. However, when we do this, it would seem to me as though we have to give up an absolute value of ~12,000,000,000 years for the age of the universe.
   Interestingly, you wouldn't be traveling quite as fast as the light (and other "force fields") at the rim, so as the universe continued to expand, you would slowly drift farther back from the rim. On the other hand, you wouldn't drift as far from the rim as would the stay-at-homes who hadn't chased the horizon.
The Universe as Spherical Surface of a Hypersphere
   Yesterday's two-dimensions-plus-time model treats the universe at any point in time as a disk (or, by extension to four dimensions, as a static three-dimensional sphere)
   Another model that might better fit the demands of general relativity would be to replace the disk in the two-dimensions-plus-time model below by a three-dimensional surface . If we did that, it would truly be a closed surface, like the surface of the Earth, There would be no rim, and no preferred direction. You could go in any direction and come back to your starting point, although it would take a long time to travel around the circumference of a universe whose diameter is expanding at the speed of light while you circumnavigate it. (I guess you couldn't do it.) The only problem is that such a universe requires four dimensions just to house two dimensions plus time. Our real universe would require five dimensions to house three dimensions plus time. Instead of the spherical surface of a three-dimensional sphere, we would need the spherical volume of a four-dimensional spherical sphere.
    Sacre bleu!
   There was a five-dimensional model of space-time proposed by Kaluza and Klein. Maybe this is the reason for it.
                                                                        (To be continued)

    Our Location in the Universe

Some of the questions about the universe that inevitably crop up are:

(1) Where are we in the universe? Are we close to its edge? How come it looks the same in all directions?
(2) What does it mean to say that space is closed?
(3) Wouldn't the age of the universe depend upon how fast the solar system has been traveling since the Big Bang?

I've certainly asked these questions.
In an effort to arrive at answers, I've imagined a three-dimensional space-time, with two spatial dimensions and time. (Extending this to three spatial dimensions and time should be straightforward.) In this model, the universe is assumed to have exploded from a primordial point 12 billion years ago. Since its radiation has been expanding at the speed of light ever since, it would look like a light-cone 12 billion light-years in radius (24 billion light-years in diameter) that stretches 12 billion light-years along its time axis.
Thinking of this in terms of special relativity only, the "edge" of our two-dimensional

universe would be expanding at the speed of light. If we set out to reach the edge, we could never get there because even if we could move at the speed of light, we could never catch up with it. It would have had a head start. Furthermore, if we headed toward the rim of the circle, no matter how fast we went, it would always seem as though we were standing still, and the speed of light would be just as great as ever. Our path through both space and time would appear to an "outside" observer as though it lay nearly parallel to the light cone, though point ing inward somewhat. However, if we approached the speed of light, other galaxies would seem to be whizzing toward us, and distances would be foreshortened in our direction of motion. The universe, instead of being circular would appear to be more and more elliptical as it seemed to shorten in our direction of motion. In the limit, as we approached more and more closely the speed of light, we would appear to be closer and closer to the "rim" fo the universe, although over time, the rim would outrun us.
                                                   (To be continued)

  I misquoted the news article, Mystery Force Pulls Old Space Probe , yesterday. I had characterized it as being 1 part in 1 billion of the normal gravitational force. Instead, the article says that the mystery force is about one part in 10 billion--very close to the 1 part in 12 billion that is required if the universe is to be considered to be a black hole. In fact, using 6-miles-per-hour-per-century as the deceleration rate, I arrive at an annual rate of change of one part in 11.5 billion, and a radius for the universe of 11.5 billion light-years.
    There are problems with assuming that the universal gravitational constant is increasing with time. This would mean that gravitational attraction is increasing, tending to pull stars closer together, and yet, the Schwartzchild radius would also be increasing, and the density would be decreasing even as gravity continued to grow stronger. Conversely, looking back in time, gravity would have been weaker and weaker until, at the moment of the Big Bang, it would have been zero. This doesn't agree with current ideas about the four "forces" of nature being equal in magnitude immediately after the Big Bang. However, it would be consistent with the explosive expansion following the Big Bang. On the other hand, it isn't compatible with an accelerating expansion of the universe at the present time.
    Another major problem is that not only would velocities be increasing with time... the "force" of gravity itself (and therefore, the force of gravity) would be increasing with time! Thus, gravitationally-produced velocities ought to be increasing as the square of the time. However, the article seems to say that the mystery force is a force that's constant with time that adds to the sun's normal gravitational pull. It will be interesting to learn more about how this new mystery force varies with time and distance.
   Another way that the universe' Schwartzchild radius could be expanding would be if the universal gravitational constant remained constant and the speed of light were decreasing. Since mass is frozen energy, we might want to replace the mass in the equation GM/c2 by E/c2 , where E is the total energy in the universe. That would give us GE/c4 for the universe' Schwartzchild radius. In that case, the annual decrease in the speed of light would only be about 1 part in 50 billion.
    Finally, it's possible that G and c are constant over time, and that what's increasing is the total energy in the universe, and therefore, its total mass, M. How? Hey! My job is to think up these questions, not to answer them. (LOL!) 

   The Heisenberg Uncertainty Principle
    I haven't tried to present the derivation of the Heisenberg Uncertainty Principle today because it's going to require imparting a certain amount of mathematical background. (I spent yesterday rebuilding a backlog of science news pages.) But the Uncertainty Principle derivation  is in the pipeline. Basically, it draws upon the fact that almost any function can be approximated by a Fourier series, if it's periodic, or by a Fourier integral if it's aperiodic. It's characteristic of Fourier integrals that the shorter and sharper you make a pulse--e. g., a single square or saw-tooth wave, the broader the frequency spectrum that is required for a given accuracy of approximation. Since frequency is proportional to energy level for both electromagnetic and matter waves, this is tantamount to saying that the shorter the pulse, the wider the range of sine-wave and cosine-wave frequencies, and therefore, of energies, required to represent it.  

    We're Back in the Hole... the Black Hole .
    I have more accurately recalculated the parameters that our universe would have to possess in order qualify as a black hole. If its radius is 12 billion light-years, it would have to possess a mass of about 382 X 1020 solar masses, compared to a very tentative current estimate of about 60 X 1020 solar masses.
    In addition, since the Schwartzchild radius of the universe is increasing as it expands, either the mass of the universe or the universal gravitational constant, G, would have to be increasing annually by about 1 part in 1010 , or the speed of light would have to be decreasing by the same fraction. My guess would be that if the speed of light were changing by 1 meter per second every decade (one part in a billion), we would have detected it by now. So how about the universal gravitational constant?
    A few days ago, we saw the article, Mystery Force Pulls Old Space Probe . Might this be consistent with an increment in gravitational force of about one part in a billion? We need an increment of about one part in 10 billion per year, or one part in a billion per decade.
    One might ask whether an increase in G would cause planetary orbits to contract by 1 part in 12,000,000,000 per year.

Total Body Rejuvenation, Anyone?
    One article tonight,
New Pill May Lead to Full Body Rejuvenation - Cosmiverse , under Prolongevity, makes strong claims for two innocuous health-food-store   supplements, acetyl-l-carnitine and alpha-lipoic acid. Carnitine is an amino-acid found in meat (as in chili con carne), and alpha-lipoic acid is one of the body's fatty acids. Experiments conducted with old rats are said to have produced extraordinary gains in energy levels and cognitive functions, in keeping, I guess, with the expectations of the researchers. One of the two  researchers, Bruce Ames, is a leading U. S. biologist and gerontologist. (The other is Tory Hagen with the University of Oregon's Linus Pauling Institute.) This study was funded by the National Institutes on Aging, as opposed to nutritional supplement companies. The authors have just submitted three papers to the Proceedings of the National Academy of Sciences.
    After reading this article, I hied myself thither to Walmart to see if they carried these supplements. Unfortunately, I didn't find them there, so I went on to "Foods for Life". The sales-lady at the store advised me that they have had these products on their shelves for only a month, but that women have already besieged them. Several women said that their skin tightened shortly after they begin taking these supplements. So, of course, I bought some. Acetyl-l-carnitine is very expensive at nearly $1.00 a pill.
    The article contains no information regarding recommended dosage levels.
    The coming years should see a plethora of such products, some of which may actually work. The really effective agents will be available first only by prescription and only for pathological conditions. However, like Viagra, Rogaine, and Retin-A, these prolongevity agents will probably rapidly escape the confines of the disease-oriented prescription system and will probably become available to the general public within a year or two of their introduction. The amount of money to be made is staggering.
    We'll see.

 Been rebuilding the science news backlog today. Will next attempt a derivation of the Heisenberg Uncertainty Principle.

2-22-2002: .     It will probably come as no surprise that I didn't accomplish a lot, physics-wise, while we were on vacation in Atlanta. However, a couple of interesting tidbits emerged.

(1)   The photoelectric effect can be used to arrive at the expression E = h n, and to determine Planck's constant. .
    The photoelectric effect refers to the fact that light impinging upon a clean metal surface in a vacuum can evict electrons from it, and that there is a threshold frequency at which this occurs. The energy required to liberate electrons is called the "work function" of the metal. For cesium, which has the lowest work function of any metal, that energy is about 1.93 electron-volts, or about 3.1 X 10 -19 joules. I don't have the threshold   frequency of the light at which photoemission first begins, but I'm going to assume that it's in the middle of the visible spectrum at about 6,000 Angstroms (
l = 0.6 microns = 0.6 X 10-6 meters). The velocity of any kind of wave is given by v = nl. For light, the velocity, v, is 3 X 108 meters/second. Then n = v/ l. Given a wavelength of 0.6 X 10 -6 meters, the frequency, n = = v/ l = 3 X 108/.6 X 10 -6 = 0.5 X 1015 cycles/second (Hertz).
    It's obvious through consideration of the photoelectric effect that the energy of a photon is directly proportional to its frequency. So the energy of a photon with frequency of 0.5 X 10 15 Hertz must be  3.1 X 10 -19 joules. In other words, its energy, 3.1 X 10-19 joules = const. X 0.5 X 1015 Hertz, or

const. = 3.1 X 10-19 joules/0.5 X 1015 Hertz = 6.2 X 10-34 joule-seconds.

    But lo and behold, Planck's constant = 6.624 joule seconds! Anyone looking at this result in 1905 would have realized that 6.2 X 10-34 joule-seconds would be very close to the constant that Planck had used in 1901 to fit black-body radiation.
    Evidently, the threshold wavelength must be about 6,400 Angstroms (orange?).

(2)  I've been looking for a simple way to demonstrate that E = mc2 for solid matter.
    Gamma rays would have an energy h
n, and a momentum h n/ c. They also have inertial mass (though no rest mass) that should be given by h n/c 2, since mass = momentum/velocity = mv/v, and for gamma rays, v = c. If I have a radioactive photon emitter with mass M that emits gamma ray photons, and if we call the total amount of energy carried away by the gamma rays = nh n = E, then the gamma rays will carry away a total amount of mass, m = E/c2. To say it another way, the total energy, E, carried away by the photons = mc2 . But the fraction of the mass of the emitter carried away by the gamma rays can be arbitrarily large, including, in principle, all of the mass M. Therefore, the total energy stored in the emitter must be Mc2, since that's how much energy we could (in principle) take out of it.

    My computer is going around the bend. I'm going to have to ship it back to the factory for repairs under warranty. In the meantime, Windows ME is flaking out. Also, I'm going to need to re-read Einstein's 1916 paper on general relativity before I continue this "exposition". I think I may have missed some subtleties. That's why I haven't "add[ed] to this in the morning", or added to this tonight.
    We're preparing to leave for Atlanta on Sunday, returning on Wednesday. I'll try to set up the website for the three+ days we'll be gone.
Later:    Eureka! What raised the caution flag that I've described above is that last night, when I re-read Einstein's landmark 1916 paper on general relativity, I found that he seemed to be referring to the expressions that I've been using for the expansion of seconds,

and for the contraction of space,

, as far-field approximations. But that's exactly what we don't want, since we're interested in what happens near, and at a black hole. However, I remembered later that I've seen these derivations elsewhere, with no mention made of weak-field approximations. In particular, the Liebers derive them directly through tensor manipulations. I think the approximations arose when Einstein expanded

. as

, for r>>rs. This last expression gives the gravitational potential  as 1 at r>>r s. Note that

= ½ rs   

The gravitational potential will reach 2 when r = ½ rs., although this approximation will break down long before that occurs. However, we want to set the potential equal to 0 at large r. We can accomplish this by subtracting 1 from the above formula to get:

for the non-relativistic potential energy. Actually, we want the potential energy to be negative, since it will take energy to separate masses in a gravitational potential well. However, the above equation can be equated to the kinetic energy required to escape from such a well:

, or multiplying both sides by 2,

. Thus, for the non-relativistic approximation,


, as previously shown. It now appears to me that the expression I got for relativistic potential energy in a gravitational may well be correct, and that Einstein's formula is for the non-relativistic approximation.


  Newsflash! I've just received this London Telegraph article, " Mystery Force Pulls Old Space Probe ", from Dan Thompson.

    I goofed. I arrived at an expression for the potential energy in a gravitational "field" by relating the emission frequency of a light emitter at a given elevation in the "field" to the time dilatation described by the general theory of relativity to be:
  . This morning, I looked up Einstein's expression for this potential and found it to be identical with the Newtonian gravitational potential:

. Einstein includes the c2 in the denominator in order to convert G from "Babylonian time" to relativistic time, ct. (G has dimensions of mass-length3/sec.2 . A factor of c2 in the denominator is required to convert seconds into meters,)
    Einstein's formula for the Newtonian gravitational potential is 

Here, K is our G. I don't know what  is supposed to be. He uses it elsewhere as an index to select coordinates.
    His formula for the relativistic potential energy is given by,

. Since the 8 p's cancel out, the formulae are actually the same.
    This raises the question: why didn't my approach work? I'll examine that question later.

    You might check the
other website later, since I'll be able to update it after this one goes to press.
    It's interesting to note that there has been continued questioning of the existence of black holes:
New Theories Dispute the Existence of Black Holes and Hawking's Breakthrough Is Still an Enigma .    
    Previous discussions of relativity may be found here .

  Reviewing What Weve Done  

    As we enter the second week of these soliloquies upon relativity, it might be timely to sum up whats been said so far.
    One parenthetical topic that might warrant mention are the roles that light signals, measuring rods, and clocks play in most tutorials that deal with relativity. I have always found these discussions confusing& something you could learn but not without midnight lucubrations. And how could your ruler be shorter than my ruler and your clock be slower than my clock while at the same time, my ruler was shorter than your ruler and my clock was slower than your clock? At the time that the special theory of relativity was gestating in the minds of Lorentz, Poincare, Minkowski, and Einstein, the principal problem of the day was their failure to detect an ether drift. I think its a tribute to Einsteins prodigious powers of deductive reasoning that he was able to deduce the special theory of relativity from only two postulates:

(1)      that measurements of the speed of light will yield the same value in reference frames   (coordinate systems) that are in a state of uniform motion with respect to each other as they will if the reference frames are at rest with respect to each other, and

(2)      that the laws of physics will have the same form whether one is at rest with respect to another frame, or whether one is in a state of uniform relative motion.

   However, I fantasize that he went at it the hard way, without grasping in 1905 what Minkowski grasped in 1908, that time is a fourth dimension, and that the paradoxes of special relativity are rotational perspective effects arising from hyperbolic rotations of a moving systems time axis toward or away from our own time axis. (Of course, Einstein incorporated this four-dimensional concept into his future work.)
    Unfortunately, this emphasis upon the rotational perspective effects of relativity, driven by the exigencies of the day, has, I surmise, obscured the fact that the heart of relativity lies in the discovery that time is a fourth dimension, (to be measured in meters) and that rotations of the time axis are hyperbolic rather than circular. Once you know that, you can calculate rotational perspective effects to your hearts content, using tanhs, sinhs, and coshs in place of sines, cosines, and tangents, rather than using Lorentz contraction factors.
    The problem I see is that this focus upon light beams and pulses has confused generations of amateur physicists, who have gotten hung up on whether Einstein made a mistake in counting his pulses. However, in my opinion, that approach was overtaken by events when Minkowski published his landmark paper in 1908. In fact, the requirement that time be considered a fourth dimension and that rotations of the time axis must be hyperbolic was already incorporated into the Lorentz form of the Maxwell Equations before 1905. In principle, I believe that someone could have adduced the special theory from the time the Lorentz form of the Maxwell Equations first appeared (whenever that was). By now, the rotational perspective effects of special relativity are built into high- energy devices like particle accelerators and high-voltage x-ray tubes. So even if they were right and Einstein made a mistake in some particular, it wouldnt matter. Einstein was only the first to arrive at special relativity. Poincaré was baying at his heels, and Minkowski arrived at the four-dimensional interpretation three years later, presumably independently of the others. And as Ive mentioned, it was all there in the Maxwell Equations*.

* - Although it was all there in the Maxwell Equations, it wasnt all there in the equations of Newtonian mechanics. Einstein had to correct the equations of mechanics, incorporating the increase in inertial (and gravitational) masses that occurs when moving bodies approach the speed of light. He also had to derive his famous E=m0c2 relationship, and to reformulate the relationship between energy and momentum. So I guess its only obvious with 20/20 hindsight.

Ive never seen it written&
    I have never seen the kind of derivation Ive given here anywhere else. Ive never found a treatment of relativity that discussed what our everyday world looks like when the fourth dimension of time is incorporated into the picturethat is, a static, four-dimensional sculpture. Nor have I seen mention of the fact that the relativistic distortions of space and time are rotational perspective effects that arise when we rotate the time axis around one of the spatial axes, just like the rotational perspective effects that we get when we spatially rotate something. Ive never seen it mentioned that a velocity, v, is the slope of a worldline with respect to my worldline, or that its equal to the hyperbolic tangent. Ive never read that the hyperbolic cosine of that angle of tilt is the Lorentz contraction factor, revealed now to be the projection of a time unit in a moving system upon our own time axis. Ive never seen it mentioned that we are frozen four-dimensional structures ourselves, and that we arent moving down the time axis, since we already exist as four-dimensional objects throughout our pasts and our futures. Whatever is moving down the time axis is the moment of the present, or is/are our individual awarenesses, like film, which has a past and a future, and is animated when its run through a projector. Ive never seen it written that were moving down the time axis at light speed, c (which might more properly be called the speed of time?). How do I know that were moving at light speed? Because in every relativistic equation, c appears wherever t appears, as the constant that converts seconds to length. (An interesting corollary is that insofar as relativistic physics is concerned, there is no free will and our futures are frozen into geometrical structure, and are completely foreordained. Quantum mechanics may be signaling amendments to this rigid relativistic predeterminism, but relativity doesnt allow deviations from the script.)

    I should probably warn you that my daily derivations are being derived extemporaneously. I may make more mistakes than usual, since Im reporting intermediate results as I go along. Im basically thinking out loud. Also, this digression on black holes isnt part of what Id been planning to present, but all roads lead to Rome.
    For a body in free fall in a gravitational field, the gravitational potential energy, GMm/r, is equal to the kinetic energy of the falling body, ½ mv2, since the potential energy is all converted to kinetic energy. Canceling common terms (½ and m), we get


Dividing both sides by c2 to get the velocity as a fraction of the speed of light yields


, so that .

 Therefore, the expression

appears to be equivalent to, and interchangeable with


for the non-relativistic case where v<<c.

    In venturing an opinion about the relativistic situation, I need to do a little more homework before I venture where angels fear to tread.

2-11-2002: Revisiting Black Holes

    One of my ambitions for this treatment is to see if there isnt some way to greatly, greatly simplify the explication and understanding of general relativity. I have a partial example that I found in a reference book on relativity, but my treatment needs to be completed, and rewritten to make it easier to follow. (Foolishly, I didnt record the name of the library book, and now I cant find it to reference it.) But for an isolated sphere, the final results are so simple! Surely there must be some easy, intuitive route to these results.
    I think that what Einsteins general theory does at least in part is to update Newtons gravitational model so that it becomes compatible with special relativity. I guess that should come as no surprise, since general relativity is a superset of special relativity.

Potential Energy in Deep Gravity Wells
    Near black holes, I'm thinking that it might be possible to derive an expression for potential energy in deep gravity wells (as well as shallow wells) by noting that E = h
n*, and that n is going to depend upon the clock speed at the level at which it was emitted. So what is the expression for energy in a deep gravitational well?

* - where h is Plancks constant, 6.626 X 10-34 joule-seconds, and n is the frequency in cycles per second (Hertz).

    Newtonian physicists would say that light red shifts as it comes out of a deep potential well because it loses energy as it climbs the gravitational hill. But in general relativity, we would explain it by saying that its emission frequency is lower in the gravity well because clocks run slower down there. Of course, to an observer down in the well at the point the light is emitted, it would seem to him to exhibit its proper frequency, since his clocks are also running slow. As the light climbs toward us, clocks at higher and higher levels would run faster and faster, and the frequency of the light would seem lower and lower by comparison with those faster clocks. So it would appear as though it were losing energy climbing the hill, just as 19th-century physicists would expect.

Equating the Length of the Second to Gravitational Potential Energy
        In the example given above, when were at an altitude above a black hole thats 1/3rd the radius of the hole, seconds will seem to be twice as long as they are for us far away from the hole. That means that when a sodium vapor streetlight is glowing down in the well 1/3rd of a radius above the black hole, its light will look yellow to someone who lives there. But since her seconds last twice as long as ours, it follows that, from our perspective, only half as many waves will be emitted in one of our seconds as are emitted in one of hers. So we will perceive the frequency of her sodium lamp to be only half that of ours. (For us, that will put its light into the infrared.) And since E = h
n , that means that the potential = ½ c2 and the potential energy = ½ mc2 (where m is a little test mass) at r = 4/3rds the radius of the black hole, rS. So the potential energy is determined by the time interval expansion formula: .
    Actually, the frequency, and therefore, the potential energy, is going to be given by the reciprocal of this expression. The time interval,
, gets larger and larger as we get closer to the black hole, while the frequency (which is the reciprocal of the time interval or period of oscillation), gets smaller and smaller, so we want to use the reciprocal of to calculate the frequency. This is just n = nemission .   is dimensionless. It gives the gravitational potential, f, as a dimensionless fraction (ranging from ~0, when r is extremely large, to 1 when r = r S).

The Schwartzchild Radius: Redux

    On February 5th, when I embarked on this ramble, I used for the mass of the sun the value given in the linked vignette, 1010 solar masses per average galaxy, and 60 billion galaxies ,.2 X 1030 kilograms, and arrived at a value of 3 kilometers for the Schwartzchild radius of the sun.
    Also, I used the formula 2M/1.35 X 1030 kilograms to calculate the Schwartzchild radius of other objects.
    I have figured out where the above fudge factor of 1.35 X 1030 kilograms (which I got empirically by dividing the suns mass by its Schwartzchild radius) comes from. The fudge factor is given by , where G is the universal gravitational constant = 6.673X 10 -11 nt-m2/kg2. (Note that nt = kg.-m/sec 2. Combining units, we get: m3/kg- sec2.) The expression multiplies our units for G leaving us with . Then when we multiply by M, were left with units of length. So why must GM divided by c 2 to calculate the Schwartzchild radius? I believe thats because
f= , which is the classical Newtonian formula for f, is given in joules/kilogram. But joules are an arbitrary human invention that isnt based upon natural constants. is the natural conversion factor that translates f, in joules/kilogram, to f as a dimensionless quantity as a fraction of the rest mass of an object placed in its field. Apparently, Nature wants to measure energy as frozen energy (mass), given by .

    In order for this relativistic formula to work, it has to reduce to the classical expression for gravitational potential, f = , at values of r >> rS. (Ive included c2 in this expression so that we can use the same units as the general relativistic equivalent.)
    The general relativistic expression for gravitational potential energy is      This gives the potential energy measured up from the bottom of the potential well. Normally, in classical physics, we measure it down from the top. In that case, it would, I think, become:

. .

How Much Energy, by Weight, Do U. S. Homeowners Consume Each Year?
    Its interesting to estimate how much energy, by weight, U. S. homes consume per year.
    The average household requires about 40 kilowatt-hours, or 40,000 joules per day. There are 86,400 seconds in a day, so the average household uses about 3.456 X 109 joules a day. With 365 days a year, that becomes .1.26144 X 1012 joules a year. Assuming 100,000,000 (108) households in the U. S., our total annual household energy consumption would amount to about 1.26 X 10 20 joules per year, or about 1.4 metric tons of pure energy each year. (Dont worry. There are orders of magnitude more renewable energy available than that. Approximately 4 X 1013 kilowatts falls on the Earth 24 hours a day. Thats about 6,000 kilowatts per person.)

(To be continued)  


General Relativistic Effects

    In the vicinities of large masses, such as stars, rulers and meters shrink in the radial direction (though not perpendicular to it) by a factor  

. In this expression, rS is the Schwartzchild radius of the star, and is the radius at which the escape speed from the star becomes equal to the speed of light. Its also the radius of the black hole that the star would become if it, when it cooled, it were to undergo gravitational collapse and become a black hole. (The threshold for this is a few times the mass of the sun, so our sun wont ever become a black hole.) Our sun has a Schwartzchild radius of about 3 (or more precisely, 2.94) kilometers.

    At great distances from the star, when r is very large,  

will be approximately 1, and we wont observe any shrinkage.
    For our sun, with a radius of 697,000 kilometers, the contraction will be about 1 part in 117,000 or about 6 kilometers.
    Only when we get to neutron stars, where all the matter in a star may be compressed into an incredibly dense ball a few kilometers in diameter will we see obvious effects.
    Because rulers shrink in the radial direction (as seen by us at a distance from the star), it would look to us as though everything were squashed down in the vicinity of the star. To them, everything would look, and would be normal. However, because theyre using a smaller ruler, I think things in the radial direction would appear to be farther away from them than they are, as measured in flat (as opposed to curved) space far from the star. So they would see an asymmetric universe around them.
    This is just the opposite of what happens in special relativity , where, in a system thats moving relative to our own, meters and rulers appear to stretch in the direction of motion by the factor  

, or saying it in terms of the normalized velocity:



    Clocks are also affected in the vicinities of large masses. Time intervals and seconds stretch by a factor

 , as seen by us who are outside gravitational fields. (Of course, were never really outside gravitational fields, but anywhere in the solar system, general relativistic effects are virtually undetectable.)
    For example, if

 , then

    This is saying that at a distance above a black hole of 1/3rd the radius of the hole, seconds will be twice as long as they are for us farther away from the hole. In other words, only 1 second will tick by on the gravity-well clocks for every 2 seconds that elapse on our clocks.

    As seen by us from flat-space, seconds are longer near a neutron star than are ours. Consequently, it will appear to us as though clocks run slower in a neutron stars gravitational potential well, since, for instance, when 10 seconds have passed according to our clocks, only 5 seconds may have ticked by on their clocks.
    This is just like special relativity.
    I believe that this means that the light coming in from distant sources to the inhabitants of a neutron star will be shifted toward the violet, not because of a Doppler shift per se, but because the clocks on the neutron star are running much slower than those at the lights source. This means that light from all directions will be violet-shifted because this effect depends upon clock rates rather than upon spatial direction.
            However, one question that still needs to be investigated is the effect of the shortening of radial rulers.

2-9-2002: We'll need a brief comparison of special relativistic effects and general relativistic effects to proceed with black holes. Tonight's discussion is upon special relativistic effects.

Special Relativistic Effects

Rotational Perspective Effects
    When we tilt something toward us or away from us, it looks shorter than it is. Its such a trivial, intuitive thing that we pay no attention to it. And of course, theres an angle of tilt,
q , and a slope, tan q ., and if you want to get technical about it, the projection of the objects actual length (the apparent length we see) is given by cos q. The rotated object also has a projection on what Ill call the z-axis, and that projection is given by sin q . The slope, tan q , is given by z/x.

    Its also trivially obvious that if someone orients himself in the tilted direction and then looks back at an object tilted the way we are, its length will look normal to us, but shortened to him, since that's the way rotational perspective effects work.

How This Relates to Relativity
     The key insight in the theory of relativity was the realization that time is a fourth dimension very like the other three spatial dimensions. That wasnt obvious until Minkowski pointed it out in 1908. The reason probably was:

(1)      we cant see or move up and down the time axis the way we can see and move up and down the three spatial axes, and

(2)      were moving down the time axis at constant speed, and we cant stop, slow, or back up.

The moving finger writes, and having writ,
  Moves on, nor all our piety nor all wit,
  Can stay its hand one jot,
  Nor all our tears wipe out a word of it.
                                      --The Rubaiyat
                                         Omar Khaiyam/Thomas Fitzgerald

       Not only can we not see up and down the time axis& we can only see an infinitesimally thick slice of whats exactly perpendicular to the time axis. This is the moment we call the present. We see a smooth progression of static three-D images, just like an all-encompassing three-D movie.
Rotational Perspective Effects When We Tilt the Time Axis
     One of the consequences of time being a dimension is that the time axis can be rotated about another axis just like a spatial axis. The slope of such a tilt would be z/t. But we interpret z/t as a velocity! The reason is because were zipping down the time axis. If something that is traveling right beside us as we whip down the time axis stays right there beside us, well see it as stationary (like two interplanetary space ships that are traveling together side by side). But if something (e. g., the other space ship) starts to veer off, we would see it simply moving away from us. (If we were traveling through interplanetary space, we wouldnt be able to tell visually that we were moving, so if our fellow-traveler began to veer off on a diagonal course, it would look to us as though it were simply moving away from us.) What a shift in perspective! Motion is simply a tilt in your trajectory (world line) down the time axis relative to my trajectory (world line) down the time axis!
    There is one difference between rotations of spatial axes around other spatial axes, and rotations of the time axis around a spatial axis. Spatial rotations are circular rotations in which the radius follows the arc of a circle, and its length remains constant. Rotations of the time axis around a spatial axis are hyperbolic rotations in which the radius follows the arc of an hyperbola, and has to stretch to stay on the hyperbola. Here, the sine, cosine, and tangent are replaced by the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh). Where the sine and the cosine range between 0 and 1, the hyperbolic sine and the hyperbolic cosine range between 1 and infinity. The tan ranges between -infinity and +infinity, and so does thhe hyperbolic tangent. (Its hard to find a larger range than -infinity to +infinity, unless you go to aleph numbers.)
      As a result of this difference in rotations, circular rotations cause things to look shorter than they are, while hyperbolic rotations make them look longer than they are. When we apply this to the tilting of the time axis away from our own, it means that a second along a tilted time axis appears to us to be larger than our own second. Consequently, a clock in a tilted time framethat is, for something thats moving away from usappears to us to be running slower than our own clock. (It has to be moving away from us at speeds approaching the speed of light before the clock retardation becomes glaringly obvious.) Similarly, hyperbolically rotated rulers appear longer to us than our own unrotated rulers, even though, if we rotate them back parallel to ours, theyll once again appear to be the same length as our own rulers. (This is just the opposite from what happens with ordinary rotations. No wonder it took a long time to discover this!)
    And remember that a hyperbolic rotation is seen by us as a change in velocity.

Summing It Up
    To sum it up, if something is moving relative to us,

(1)     its clocks will appear to run slower than ours (since its seconds are longer than ours), and

(2)     its rulers will appear to be longer than ours (since its spatial units will also be longer than ours)

Tomorrow night: General Relativistic Effects

  I spent half the day today writing up tonight's discussion of relativity, only to have my computer freeze and wipe it all out. No, I hadn't periodically saved it. I usually do this, but the computer hasn't frozen for a while and I let down my guard. From now on, I'll work in Word, which, unlike FrontPage 2000, has an autosave feature. In the meantime, I've run out of time and heart to try to reconstruct it. Ill work on it again tomorrow.
    One very interesting topic is tonight's presentation of the harnessing of the Casimir Effect. What's been done is no more than a laboratory curiosity, but it suggests the possibility of something that would actually draw power from the void (see
Inventor touts power source, but skeptics abound   - MSNBC ). When Faraday was asked what his newly-invented electric generator was good for, he replied that it was like a baby...  its utility would grow over time
    It's interesting to note that there has been continued questioning of the existence of black holes:
New Theories Dispute the Existence of Black Holes and Hawking's Breakthrough Is Still an Enigma .
.    Later: The computer just finished eating the update I had written for this page. Somehow, it failed to save what I had written. This hasn't been my day. I'll update this tomorrow.
The last three days of relativity discussions may be found
here .


2-7-2002:   One way to address the problem of gravitation is to consider what happens in Isaac Newton's pre-relativistic gravitational model when a meteorite collides with the Earth. The Earth's gravitational field acts as a banker, loaning the meteorite kinetic energy in exchange for its negative potential energy as it falls ever deeper into the Earth's gravitational potential well. The meteorite blazes into the Earth's atmosphere, radiating away part of its kinetic energy into space.  When it hits the ground, almost all of the rest of its kinetic energy is transformed into heat, and eventually, radiated away into space by the Earth-meteorite system as a negligible part of the Earth's energy budget. Meanwhile, the meteorite has added its incomparably smaller mass to that of the Earth (which is about 6 X 1024 kilograms) to infinitesimally increase the total mass of the Earth-meteorite combination. So where did the energy come from to power the meteorite's blazing fall from Grace?
    I believe that the conventional answer was that planets (and objects in general) created gravitational potential wells of "negative energy" in their neighborhoods. Work must be done on an object on a planet in order to lift it out of the planet's gravitational potential well. It should be easy to calculate the (great) magnitude of the planet's total negative potential energy by calculating the total amount of work that would have to be done to remove all the planetary mass layer-by-layer out of its (diminishing) gravitational field.
    So in pre-relativistic physics, where did this "negative energy" reside? Not in the solid, stolid Earth, but in its gravitational field. The gravitational field was envisioned to be an invisible, elastic construction (the subluminiferous ether?), with energy stored in its "springs"
    Suppose that two planets collided. In the Newtonian gravitational model, there would be an enormous release of kinetic energy, in the form of heat. The combined bodies would have the sum of the masses of the two individual bodies. Since E = mc2 was unknown to pre-relativistic physicists, there would be no idea that the mass of the combined body would be slightly less than the sum of the masses of the two parent bodies by the equivalent mass of the energy radiated into space as heat. Of course, this would increase the gravitational field, drawing in additional matter, until all the loose matter in the vicinity of the planets had been gathered up.
    What would be the mass of the largest possible body? Presumably, it would have been one in which all the mass in the universe were gathered into one gigantic sphere. A great deal of heat would have been generated, with the body radiating as a star until it had cooled sufficiently and contracted sufficiently that no more gravitational energy could have been squeezed out of it. Then it would have become a gigantic cinder or "clinker". (This is the picture of a dying sun that H. G. Wells sets forth in "The Time Machine".) At some point along this progression, this super star's  gravitational potential would have become so large that its escape speed would have exceeded the speed of light, and light could no longer have escaped from it. Thereafter, it would have become a pitch-black "light sink"... a black hole. However, since the speed of light had no particular significance, gravitation could have continued to be felt. How large this sphere would have been would probably have been beyond the ability of pre-atomic physics to predict. Before the 20th century, matter might have been presumed to be incompressible.

    One of the curious questions about black holes is, "If light can't escape from them, how can gravitation?" I realize that general relativity reveals gravitation to be a compression of space-time in the neighborhood of massive bodies, but such bodies must interact with the space-time around them. If the bodies move, the space-time around them distorts accordingly. If a gravitational pull develops on a black hole, altering the space-time around it, somehow, the space-time around it will communicate this change across the event horizon to the black hole, and it will move in response to the tug on it.. Presumably, gravitational waves propagate at light-speed, as presumably, do changes in gravitational "force". So how can there be communication across the event horizon between the black hole and the surrounding space-time? (Of course, the expression, E = h
n , allows us to reduce the energy of a photon. I don't know just how this would work with a graviton.)
    My expectation would have been that if something undergoes a gravitational collapse and becomes a black hole, it would simply become undetectable by us.

    Another approach to the problem is to consider what would happen if we replace our black hole with a neutron star that is a few metric tons shy of becoming a black hole.

(To be continued)

2-6-2002:   Here's my conundrum for today.
    Suppose that we were to drop a grain of sand into a black hole. In accordance with the special theory of relativity, its mass would increase as it approached the black hole. In fact, as it crossed the event horizon, it would reach the speed of light and its mass would become infinite. Slightly before that, its mass would exceed the mass of the rest of the universe. So where would the energy come from that would permit such an increase in mass? It would seem to require more energy than exists in the universe to carry that grain of sand across the event horizon. Obviously, that's not going to happen. Something is missing from this description or no one would be seriously entertaining the idea of black holes. Could it be some effect derivable from general relativity that offsets the mass increase attributable to special relativity?
    General relativity predicts that, as measured by us who are far away, seconds will stretch in the neighborhood of a massive gravitational potential in accordance with the formula 1/sqrt(1-r s/r), where r is the Schwartzchild radius of the spherical, gravitating body which is generating the massive gravitational potential. Consequently, clocks will slow down at locations where the gravitational escape speed is approaching the speed of light. This is consistent with the fact that light will red-shift  (lose energy) as it struggles up out of the deep gravitational potential well.
    In contrast, meter sticks will appear to us to shorten in the radial direction as sqrt(1-rs/r) deep in the potential well. This is just the reverse of what happens in special relativity, where both time intervals and meter sticks lengthen as we approach the speed of light. (Meter sticks lengthen so that distances appear shorter. Thus, the Lorentz-Fitzgerald contraction.)
    For a body falling into a black hole, rs/r is equivalent to the classical potential energy. The speed of a body falling into the hole will be given by v/c = sqrt(rs/r). Consequently, for a freely falling body,  rs/r = v 2/c2, and 1/sqrt(1-rs/r) becomes our familiar Lorentz contraction factor,  1/sqrt(1-v 2/c2). At this point, as an astute reader, you're going to ask,
    "What about the fact that the mass of the falling body increases as it approached the speed of light? Won't its acceleration slow down?"
    And the answer is that gravitation differs from other forces in that, as an object's mass increases relativistically, so will its gravitational mass. Consequently, the force acting upon the object will increase in direct proportion to its mass, so that its acceleration remains constant.
    Looking at how the gravitational and relativistic effects interact, it appears to me as though the foreshortening of meter sticks that is introduced by general relativity exactly cancels out the lengthening of meter sticks predicted by special relativity. However, the slowing of clocks finds both influences working in the same direction, with clocks slowing by a factor of 1/(1-rs/r) rather than by 1/sqrt(1-r s/r).
    But offhand, I still don't see how this would sidestep the problem of objects falling into the black hole requiring infinite energy to reach the speed of light and cross the event horizon.  Any ideas?

2-5-2002:   Several of you good folk have been asking for more math content in the Daily Science News. For the past two weeks, I've been planning and working toward something that I'm hoping might challenge your math talents in the pursuit of important real-world problems. I've been working on some ideas that at a minimum should be interesting, and that, at a maximum, might lead on to new discoveries. It's taking a considerable dollop of writing and of the pursuit of relationships to get this ready. To give you a slight taste, my personal research topic du jour has been black holes. Could our universe be a massive black hole in some parent universe? One of the interesting characteristics of black holes is that the bigger they get, the lower their internal density becomes (as calculated by an external observer). The radius of a black hole  (Schwartzchild radius) is given by

  radius (in kilometers) = mass (in kilograms)/0.6745 X 1030 kilograms.   

    As you can see, the radius of a black hole is directly proportional to its mass, which means that the volume of a black hole increases as the cube of its mass. Consequently, its density varies inversely as the square of its mass. Increasing the mass of a black hole by a factor of 10 reduces its density by a factor 100!
    The mass of the sun is about 2 X 1030 kilograms, giving it a Schwartzchild radius of about 3 kilometers. If we imagined a 300,000,000,000-star, Milky-Way-galaxy-sized black hole, it would have a Schwartzchild radius of about 300,000,000,000 X 3 kilometers, or about 900,000,000,000 kilometers--big, but less than 1/10th of a light-year in radius. However, its total volume would be about 4
p/3 X (9 X 10 14)3 meters, or (reaching for my trusty calculator) about 3.016 X 1045 cubic meters. Its mass would be about 2 X 10 30 kilograms X 3 X 1011 stars = 6 X 1041 kilograms. Dividing 3 X 1045 cubic meters by 3 X 1041 kilograms yields a density of about 2 X 10-4 kilograms per cubic meter or 2 X 10-7 kilograms per liter. Air weighs about a kilogram per cubic meter, so this a little more than the density of air, but 1/600th the density of water.
    If we take the radius of the universe to be 12,000,000,000 (1.2 X 1010) light-years and we use 10,000,000,000,000 (1013) kilometers per light-year as a conversion factor, the astronomically-estimated radius for the universe is 1,200 X 10 20 kilometers. Dividing this by the Schwartzchild radius of the sun (3 kilometers), we arrive at the total mass (in solar masses) the universe would have to have in order to qualify as a black hole 12,000,000,000 light-years in radius:
400 X 1020 solar masses . One crude current estimate of the mass of our galaxy assumes 1010 solar masses per average galaxy, and 60 billion galaxies , (= 60 X 1020 solar masses ).
    Some arguments militating against this possibility would seem to be:
(1) The formula I'm using--radius (in kilometers) = mass (in kilograms)/0.6745 X 1030 kilograms--applies to black holes embedded within our universe... that is, viewed from the outside. However, I'm trying to use from the inside the same formula to size the black hole that I'm postulating our universe to be. Offhand, I wouldn't expect the same metric to apply inside a black hole that would apply outside it, but if so, the universe in which we are embedded would have to be truly Brobdingnaggian. (Of course, given another trillion years of expansion, our universe should be truly gargantuan in its own right. Also, black holes may become hugely larger in another trillion years, as they gradually gobble up stars and each other.)
(2) The universe-as-a-black-hole model would be incompatible with our expanding-universe model unless the value for the universal gravitational constant, G, is increasing by one part in 1010 per year, or the speed of light is decreasing by the same rate. (Presumably, our universe' total mass-energy has been constant since it first came into being 12,000,000,000 years ago.) We've measured the speed of light to sufficient accuracy that I would suspect that we'd know if it were changing by 10 centimeters per second per year or a meter per second per decade. It seems as though   we'd have observed such a one-part-in-100,000,000 shift in G in over the 20th century if it were occurring.
    Still, it's an intriguing coincidence that black holes scale in such a way that a black-hole model is within a factor of 10 of fitting our universe.
    Another reason for suspecting such a condition is related to Olbers paradox.
    The escape speed from the center of our galaxy is something like 1,000 miles (1,600 kilometers) per second. (We should be able to infer the escape speed fairly accurately if we can measure the orbital speed of stars close to the center of our galaxy. The escape speed will be sqrt(2) times the orbital speed.) In addition, there will be small contributions to the gravitational potential at the center of our galaxy from all the other galaxies in the universe, since gravitational potentials are additive. Now the gravitational potential falls off as 1/r, but the number of galaxies increases as r2, and there are billions of them. A back-of-the-envelope calculation suggests that the escape speed from the gravitational potential well that is the universe would approach the speed of light. What is the escape speed from our universe? (Presumably, it's the speed of light, but for arguments based upon the space-time curvature of general relativity rather than gravitational potential per se. Still, the gravitational potential outside our Milky Way galaxy would be less than it is at the heart of our galaxy.)