In Re Our 4D Universe
4/8/2002:
The
generalrelativistic 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 generalrelativistic 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.
4/7/2002:
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 Renantiomer of alphalipoic acid. Apparently, most racemic mixtures
are actually primarily comprised of the Renantiomer, and apparently, the
racemic mixtures work almost as well as the pure Renantiomers, 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 alternativemedicine aficionados, including aloe vera, jojoba oil, eucalyptus,
lavender oil, orange oil, vitamin E, collagen, retinyl palmitate, and retinol,
Vitamin C, alphalipoic acid, and acetyllcarnitine 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
generalrelativistic 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 alphalipoic acid, and 250 mg. a day of acetyllcarnitine. 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 alphalipoic acid has
a warning to diabetics and hypoglycemics that alphalipoic 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
generalrelativistic integral
is essentially
complete.
4/5/2002:
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((x1)/x), or of its reciprocal, sqrt(x/(x1)),
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. Hmmm... 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, lawabiding 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.... )
4/3/2002:
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,
I haven’t had time today to return
to this topic.
I haven’t followed up on the
zeropoint 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 mindboosting and (we would like to believe) lifeextending dietary
supplements. I had mentioned that we both seemed to have experienced insomnia
on doses of 100 mg. per day of alphalipoic acid. (Dr. Sahelian said that
he suffered from insomnia at doses above 40 mg. per day of alphalipoic 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 Renantiomer
of a
lipoic acid is also an open question.
4/2/2002:
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 2.5kilowatt 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 thirdworld 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 underfunded 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 generalrelativistic 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 bestkept secrets in recent history.
Stay tuned.
3/28/2002:
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 hardworking 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: rnseitz@ultrahiq.net
. I also have email addresses at
rnseitz@megafoundation.org
and at rnseitz@megasociety.net
. For some undoubtedlyunhappy 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 nonprofit organization
to seek ways to better utilize the talents of the ultragifted for society's
benefit, as well as helping satisfy the desires of the ultragifted 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. .
3/27/2002:
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 simplelooking 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.
3/26/2002:
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
onepersecond 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 vitaminE,
gammatocopherol
, may play an important role in cancer prevention and ageretardation.
There are also
discussions
of the fact that the
alpha lipoic acid + acetyllcarnitine
that I've been touting requires the
Renantiomer
of
alphalipoic acid
rather than the usual overthecounter 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 20202030 time frame.)
3/24/2002:
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
.
3/23/2002:
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
alipoic acid and/or the acetyllcarnitine were causing us insomnia. I’m
playing guinea pig by resuming 25 milligrams of alipoic 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 "multigeneration
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 antisenescence 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 Microsoftmade
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: PulseCounting
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.
3/19/2002:
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 righthandside 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 10^{11}
gees acceleration on the surface of a solarsized, 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,
Brain Boosters Again.
Tommie and I have slept like
stones the past two nights, lending further currency to the idea that the
alipoic acid and/or the acetyllcarnitine were causing us insomnia. Im
playing guinea pig by resuming 25 milligrams of alipoic 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
alipoic 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 lightyears. Only oneeighth 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.
3/16/2002:
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 speedlet'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 clockshe'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 lightyears. 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
lightspeed.
During her outward flight, since
she's traveling at 7/8ths of the speed of light, she'll receive only a fraction
of the terrestrialclock 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 Earthbound 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 lightyears 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 4.87year 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 headon
into all the pulses that had been chasing her. Now, all the pulses that are
in transit plus all the pulses issued during her 4.87year 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 4D 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 (r^{2} = x^{
2}  y^{2} instead of r^{2} = x^{2} + 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.)
3/15/2002:
Where Is Our New SpaceTime 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 blackhole density when it was
about 2 lightyears 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 spacetime or to expand existing spacetime. From
a fourdimensional viewpoint, the universe just is. It's a static sculpture
in four dimensions. What are the rules for the creation of spacetime by
matter? Does it require energy? We don't see matter creating new spacetime
or expanding old spacetime around us. And what does this portend for the
constants and the laws of physics? Is energy stored in curved spacetime?
And woul that mean that spacetime has mass? (I've never asked these questions
before. This was never discussed in graduate school.)
(To be continued)
3/14/2002:
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 twodimensional 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 3^{rd}, 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, acetyllcarnitine and alphalipoic 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.)
3/13/2002:
To summarize, I've been wrestling with two basic models of the universe
this past week. One is the traditional BigBang 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 threedimensional surface of a fourdimensional
hypersphere closed by the gravitational curvature of spacetime.
Our absolute motion relative to the cosmic background radiation is about
350 kilometers per second... about 0.1% of lightspeed, and comparable to
the vector sum of our solar, galactic, galactic cluster, and galactic supercluster
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 lightyears, 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 lightspeed 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 wellentrenched
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 lightyears. A lightyear 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 lightyears 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 threedimensional spherical volume that envelopes a fourdimensional
hypersphere embedded in a fivedimensional 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 twodimensional
planar world). Well see lines coming in from all directions in threedimensional
space because we can imagine three dimensions.
If we imagine that Ms. Flatland
can see the twodimensional crosssections as she moves infinitesimally south
of the North Pole, the first crosssection shell see will by an infinitesimal
circle where the sphere on which shes standing is sliced by a horizontal
plane to give her its twodimensional crosssection. The circle is a curved
onedimensional 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 twodimensional 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 twodimensional, diskshaped 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 threedimensional, spherical universe.)
The answer would be a larger spherical surface.
As Ms. Flatland approached the equator, the expansion of the circle
that bounded her diskshaped crosssection 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 curvatureinthethird 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 crosssection
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 curvatureinthefourth 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 threedimensional 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 ThreeDimensional CrossSections of a Hypersphere
expanding from the left
the center of the enclosing sphere.
Incidentally, the toruslike 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 threespace analogous to the surface of a sphere, we can
go in any direction and still come back to our starting point. Duhhh!
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 FourDimensional Rather Than FiveDimensional
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 twodimensional crosssections showing
where the lines of longitude cross the planar crosssections. 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) coplanar 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 4psteradian 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 rederived the expression
for the
hypervolume
of a hypersphere, getting the same formula I did previously, 1/2
p^{2}r^{4}, and
the same expression for its hypersurface, 2
p^{2}r^{3}: 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.
392002:
Tonight, my picture of the volume that encloses the hypersphere
is that it consists of continuouslyoverlapping 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 halfplane 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 sidebyside semicircles on top with a single semicircle with
twice the diameter (spanning both of the upper semicircles) on the bottom.
More in the morning.
382002:
Tonight's specialty is a set of articles on telomeres, under
Prolongevity
.
Visualizing the threedimensional spherical volume that envelopes a fourdimensional
hypersphere embedded in a fivedimensional 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 seeminglystraight 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 seeminglystraight 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 threespace but not fourspace 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 straightandnarrow 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.
372002:
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,
'BraneStorm' Challenges Part of Big Bang Theory
 Space.com
, suggests a fivedimensional manifold
in which fourdimensional hypermembranes drift, so discussions of fourdimensional
hyperspheres embedded in fivedimensional manifolds may be an educational
exercise. At the same time, it's weighty news that our universe seems to
be fourdimensional 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 1213 billion years and an
absolute radius of 1213 billion lightyears 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 observerthat 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 threedimensional spherical volume that envelopes a fourdimensional hypersphere
embedded in a fivedimensional 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
onedimensional 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 seeminglyinfinite 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 fourdimensional spherical sphere and, with a back
of the envelope calculation, have arrived at 2
p^{2} r^{3}. This
number needs to be rechecked, but the second factor of
p appears because it involves an
integral of sin^{4} q
, and this introduces a second
p factor. Writing 2
p^{2} r^{3} as 6/3
p^{2}
r^{3}, and dividing by 4/3
pr^{3} yields a ratio of
1.5 p
. for the volume of the "hyperspherical surface" enclosing a spherical sphere.
(To be continued)
362002:
Visualizing the threedimensional spherical volume that envelopes a fourdimensional
hypersphere embedded in a fivedimensional manifold
Suppose that there were someone
by the name of Linus who lived on a very, very, very long linea line 24
billion lightyears longin a onedimensional 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 lightyear circumference. In that case, each time Linus crawled 24
billion lightyears, he would find himself retracing the same old "terrain"
he had covered before.
Note that we use a onedimensional
line to generate a twodimensional circle by rotating it about an axis in
a third direction, and that Linus is crawling along the circle's onedimensional
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 twodimensional
circle to generate a threedimensional 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 lightyears. They
would pass through a fourth spatial dimension of which we would be unaware.
They would fill all of our universe' threespace. 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
fourdimensional 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 reexamine it tomorrow.
The problem I posed last night
is similar, except that, in keeping with the generalrelativistic 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 FlatWorlder. We've said
that a spherical surface requires three dimensions to contain it, and by extension,
a hyperspherical surface would require four dimensions to contain it. Like
the FlatWorlder 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 fourdimensional
universe depicted above is the correct picture rather than a universe closed
on itself in a fivedimensional continuum. This also justifies our somewhatapproximate
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 threedimensional spherical volume that envelops a fourdimensional hypersphere
embedded in a fivedimensional manifold.
352002:
Visualizing the threedimensional spherical volume that envelopes a fourdimensional
hypersphere embedded in a fivedimensional 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 infinitesimallythick
disks or circular laminae, with steadily increasing radii. As such, it's
a conically shaped volume.
But...
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 twodimensional 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
threedimensional spherical volumes of infinitesimal duration with steadilyincreasing
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 conicalsphericallyshaped
hypervolume.
But...
It can also be regarded as just
a threedimensional static volume that is the container of all the instantbyinstant
threedimensional shells from the moment of the Big Bang to the present time,
but that doesn't incorporate all the instantbyinstant threedimensional
volumes themselves, although it requires a fourdimensional manifold to contain
it. In other words, it's the form without the content. It's fourdimensional
because it includes the spherical outlines or boundaries of all the instantbyinstant
spherical volumes of the universe' existence.


The problem I posed last night
is similar, except that, in keeping with the generalrelativistic 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 FlatWorlder. We've said
that a spherical surface requires three dimensions to contain it, and by extension,
a hyperspherical surface would require four dimensions to contain it. Like
the FlatWorlder 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 fourdimensional
universe depicted above is the correct picture rather than a universe closed
on itself in a fivedimensional continuum. This also justifies our somewhatapproximate
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 threedimensional spherical
volume that envelopes a fourdimensional hypersphere embedded in a fivedimensional
manifold.
342002:
Abundance of Ice on Mars!
I should certainly mention the
stunningly wonderful news that
waterice 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 northfacing 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 selfsustaining industrial base on Mars. This will entail taking maximum
advantage of Mars' endemic resources from the getgo, shipping only hightech
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 remotelyoperated equipment to be transshipped
to Mars. One might start with a microindustrial base, and then, using small
devices to build larger devices, might bootstrap to a largelyautonomous
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 298plus on the StanfordBinet
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 StanfordBinet
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 Floridabased 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 oneinfivebillion ratioIQ
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 exprodigies, and as adults,
they're phenomenal, with virtually perfect scores on adult IQ tests, and
with obviouslyextraordinary talents.
Visualizing
the threedimensional spherical volume that envelopes a fourdimensional hypersphere
embedded in a fivedimensional manifold
No, I haven't yet. Have you?
(I mean, like, "Gosh, it looks a little tough!")
332002:
Mind
Boosters
Last Sunday, I posted an
article
that makes unusual antiaging claims for rats fed the supplementary
nutrients acetyllcarnitine and alphalipoic 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 coauthors was
the wellknown 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 acetyllcarnitine
and alphalipoic 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 alphalipoic 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 wellbeing 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
alipoic 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?
322002:
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 twodimensionalplustime universe that considers the twodimensional
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. Hmmm. 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 spacetime 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. Hmmm...
Any ideas? (For my part, I'm going to do a little homework investigating
the cosmological concepts that cosmologists are employing.)
312002:
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 farreaching) 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 andEinsteinian?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 lightyears). 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 blueshifted in front of you and
redshifted 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 cosmicbackground microwave radiation would be Dopplershifted
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 stayathomes who hadn't chased the horizon.
The Universe
as Spherical Surface of a Hypersphere
Yesterday's twodimensionsplustime
model treats the universe at any point in time as a disk (or, by extension
to four dimensions, as a static threedimensional sphere)
Another model that might better
fit the demands of general relativity would be to replace the disk
in the twodimensionsplustime model below by a threedimensional 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 threedimensional sphere, we would need the spherical
volume of a fourdimensional spherical sphere.
Sacre bleu!
There was a fivedimensional model
of spacetime proposed by Kaluza and Klein. Maybe this is the reason for
it.
(To be continued)
2282002:
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 threedimensional spacetime,
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 lightcone 12 billion lightyears in radius (24 billion lightyears
in diameter) that stretches 12 billion lightyears along its time axis.
Thinking of this in terms of special relativity only, the "edge" of our
twodimensional
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)
2252002:
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 billionvery 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
6milesperhourpercentury 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 lightyears.
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, gravitationallyproduced 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/c^{2} by E/c^{2}
, where E is the
total energy in the universe. That would give us
GE/c^{4} 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!)
2242002:
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 pulsee. g., a single square or sawtooth 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 sinewave and cosinewave 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
lightyears, it would have to possess a mass of about
382 X 10^{20} solar masses, compared to a very tentative
current estimate of about 60 X 10^{20}
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 10^{10}
, 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 healthfoodstore
supplements, acetyllcarnitine and alphalipoic acid. Carnitine
is an aminoacid found in meat (as in chili con carne), and alphalipoic
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 saleslady 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. Acetyllcarnitine 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 RetinA, these prolongevity
agents will probably rapidly escape the confines of the diseaseoriented
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.
2232002:
Been
rebuilding the science news backlog today. Will next attempt a derivation
of the Heisenberg Uncertainty Principle.
2222002:
.
It will probably come as no surprise that I didn't accomplish
a lot, physicswise, 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 electronvolts, 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 10^{8} meters/second. Then
n = v/
l. Given a wavelength of 0.6 X 10^{
6} meters, the frequency,
n = = v/
l = 3 X 10^{8}/.6 X 10^{
6} = 0.5 X 10^{15} 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 10^{15} Hertz, or
const. = 3.1 X 10^{19} joules/0.5 X 10^{15} Hertz = 6.2 X 10^{34} jouleseconds.
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}
jouleseconds would be very close to the constant that Planck had used in
1901 to fit blackbody 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 = mc^{2} 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/c^{2}. To say it another way, the total energy,
E, carried away by the
photons = mc^{2}
. 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
Mc^{2}, since that's how much energy we could (in principle)
take out of it.
2162002:
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 reread 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 reread 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
The gravitational potential will reach 2 when r =
½ r_{s}., 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:
_{
}
_{
, or multiplying both sides by 2,}
_{
. }_{
Thus, for the nonrelativistic 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 nonrelativistic approximation.}
2142002:
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:
Einstein's formula for the Newtonian
gravitational potential is
His formula for the relativistic
potential energy is given by,
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
.
2132002:
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 fourdimensional
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 highvoltage xray 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 fourdimensional 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=m_{0}c^{2} 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, fourdimensional 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 fourdimensional structures
ourselves, and that we arent moving down the time axis, since we already
exist as fourdimensional 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.)
2122002:
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 mv^{2}, since the potential
energy is all converted to kinetic energy. Canceling common terms (½ and
m), we get
Dividing both sides by c^{2} to get the velocity as a fraction of
the speed of light yields
But
Therefore, the expression
for the nonrelativistic 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.
2112002:
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} jouleseconds, 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 19^{th}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 1/3^{rd} 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 1/3^{rd} 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
= hn
, that means that the potential = ½ c^{2} and the potential energy
= ½ mc^{2} (where m is a little test mass) at r = 4/3rds the radius
of the black hole, r_{S}. 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,
On February 5^{th}, when I embarked on this ramble, I
used for the mass of the sun the value given in the linked vignette,
10^{10} solar masses per average galaxy, and 60 billion galaxies
,.2 X 10^{30} 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 10^{30} kilograms to calculate the Schwartzchild radius of other
objects.
I have figured out where the
above fudge factor of 1.35 X 10^{30} kilograms (which I got empirically
by dividing the suns mass by its Schwartzchild radius) comes from. The fudge
factor is given by
In order for this relativistic
formula to work, it has to reduce to the classical expression for gravitational
potential, f
=
The general relativistic expression
for gravitational potential energy is
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 kilowatthours, or 40,000 joules per day. There are 86,400 seconds
in a day, so the average household uses about 3.456 X 10^{9} joules
a day. With 365 days a year, that becomes .1.26144 X 10^{12} joules
a year. Assuming 100,000,000 (10^{8}) 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 10^{13} kilowatts falls on the Earth
24 hours a day. Thats about 6,000 kilowatts per person.)
(To be continued)
2102002:
General Relativistic Effects
Rulers
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, r_{S} 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
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 1/3^{rd} 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 gravitywell clocks for every 2 seconds
that elapse on our clocks.
As seen by us from flatspace, 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 violetshifted 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.
292002:
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 zaxis, 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
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 fellowtraveler 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
282002:
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 newlyinvented 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
.
272002:
One way to address the problem of gravitation is to consider what
happens in Isaac Newton's prerelativistic 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 Earthmeteorite
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 10^{24} kilograms) to infinitesimally increase the total mass
of the Earthmeteorite 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 layerbylayer
out of its (diminishing) gravitational field.
So in prerelativistic 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 = mc^{2} was
unknown to prerelativistic 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 pitchblack "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 preatomic 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 spacetime in the neighborhood of massive bodies, but such bodies must
interact with the spacetime around them. If the bodies move, the spacetime
around them distorts accordingly. If a gravitational pull develops on a black
hole, altering the spacetime around it, somehow, the spacetime 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 lightspeed, as presumably, do changes in gravitational
"force". So how can there be communication across the event horizon between
the black hole and the surrounding spacetime? (Of course, the expression,
E = hn
, 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)
262002:
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(1r_{
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 redshift (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(1r_{s}/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 LorentzFitzgerald contraction.)
For a body falling into a black hole, r_{s}/r is equivalent
to the classical potential energy. The speed of a body falling into the hole
will be given by v/c = sqrt(r_{s}/r). Consequently, for a freely
falling body, r_{s}/r = v^{
2}/c^{2}, and 1/sqrt(1r_{s}/r) becomes our familiar
Lorentz contraction factor, 1/sqrt(1v^{
2}/c^{2}). 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/(1r_{s}/r) rather than by 1/sqrt(1r_{
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?
252002:
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 realworld 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 10^{30} 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 10^{30} kilograms, giving it a Schwartzchild radius of about
3 kilometers. If we imagined a 300,000,000,000star, MilkyWaygalaxysized
black hole, it would have a Schwartzchild radius of about 300,000,000,000
X 3 kilometers, or about 900,000,000,000 kilometersbig, but less than 1/10th
of a lightyear 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 10^{45} cubic meters. Its mass would be about 2 X 10^{
30} kilograms X 3 X 10^{11} stars = 6 X 10^{41} kilograms.
Dividing 3 X 10^{45} cubic meters by 3 X 10^{41} 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 10^{10}) lightyears and we use
10,000,000,000,000 (10^{13}) kilometers per lightyear as a conversion
factor, the astronomicallyestimated 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 lightyears
in radius: 400
X 10^{20} solar masses
. One crude current estimate of the mass of our galaxy assumes
10^{10} solar masses per average galaxy, and 60 billion galaxies
, (=
60 X 10^{20} solar masses
).
Hmmm....
Some arguments militating against
this possibility would seem to be:
(1) The formula I'm usingradius (in kilometers) = mass (in kilograms)/0.6745
X 10^{30} kilogramsapplies 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 universeasablackhole model would be incompatible with our expandinguniverse
model unless the value for the universal gravitational constant, G, is increasing
by one part in 10^{10} per year, or the speed of light is decreasing
by the same rate. (Presumably, our universe' total massenergy 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 onepartin100,000,000 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 blackhole 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 r^{2}, and there are billions
of them. A backoftheenvelope 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 spacetime 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.)