The Physics of Infinity
H. de Bruijn
Re: Infinity as a proper mathematical concept (sci.philosophy.tech)
Re: What doe it mean that "actual infinities" exist? (sci.logic)
Re: Actual infinities DO exist! (sci.logic)
Leif Sterner writes (sci.logic):
> I would say that it's pretty clear that infinities actually do exist.
And later on:
> ........ . An interesting question is if actual infinities could arise in the
> physical world. Recently I made some postings to the sci.physics newsgroup
> on this subject. ( ... stuff deleted ... ), ... if anyone is interested I can
> send the articles by @mail.
Very strange kind of reasoning ! There ARE actual infinities, but you quest if
they could arise in everyday (physical) REALity. You seem to assume that there
exists some other "reality" out there. Let's call it the Heaven of Mathematics,
or even better: Plato's good old World of Ideas.
Anyway, I'm certainly interested in these sci.physics articles I missed.
^^^^^^^^^^^
Gene Ward Smith writes:
> Of course, outside of pure mathematics the trouble with dismissing infinity
> is even greater. As was pointed out, the universe might very well be infinite.
> Quantum field theory has to deal with infinite numbers ... and "renormalize".
And again to Eli Chiprout:
> I notice you did not deal with the point that actual, physical infinities
> seem to be apart of the physical universe. What about it ?
The answer of Eli could have been mine:
> They do ? That's a new one to me ! What part of the physical universe have
> you discovered that I don't know about ?
Serious now.
Of course, there is NO physical evidence that infinities actually do exist.
Actually, there IS physical evidence that actual infinities do NOT exist.
I have done my best to collect some substantial, physical, arguments.
Most of these are certainly NOT my own. (Remember !)
1. If the universe was infinite in space and time, then it would'nt become dark
at night. The whole sky would be as bright as the sun itself. There would be
an infinite number of stars, and all their light would have reached us, since
there would be an infinite time for it to travel.
2. What do you think of the ever increasing entropy in an infinite universe?
3. Infinite vector spaces ARE used in Quantum Mechanics. However, they all have
a complete base, AS IF they were only vectorspaces with a very BIG dimension.
In fact, these Hilbert spaces cannot be distinguished from a big, but finite
vectorspace, as every physicist knows from practice.
Therefore, from a physical point of view, the Universe seems to be FINITE.
Big Bang, Creation or Whatsoever ...
If it is finite in space, it must also be finite in time, and vice versa:
the theory of relativity leaves no other choice to us.
So far about the infinitely BIG things. What about the infinitely SMALL?
4. Fluid flow is described by a set of coupled partial differential equations.
However, it is very clear that this must be considered as a mere illusion.
The "differential" volumes are NOT REALLY infinitely small. Far from that !
They should contain A LOT OF molecules, for the approximation to be valid.
There is a beautiful text about this (due to Perrin) in Benoit Mandelbrot's
"The Fractal Geometry of Nature", first few pages (6,7).
5. Space and time seem to be the only things that remain infinitely divisible.
But there are also strong arguments against that ! Quoting without permission
from Landau and Lifshitz "Quantum Electrodynamics", Introduction, page 2+3:
> In the rest frame of the electron, the least possible error in the measure
> ment of its coordinates is:
> dq = h/mc.
>
Here q=position, h=Planck's constant, m=electron mass, c=velocity of light.
If you try to measure the position of an electron, you must send for example
a photon to it. The electron is disturbed by this photon, which is known as
the Compton effect. As a consequence of this, there is a definite bound on
the accuracy with which you can locate the electron. This is expressed by
the above formula. Any attempt to locate the electron *within* this interval
is "severely punished" by:
> ..... the (in general) inevitable production of electron-positron pairs in
> the process of measuring the coordinates of an electron. This formation of
> new particles in a way which cannot be detected by the process itself
> renders meaningless the measurement of the electron coordinates.
6. There is another (exellent) book, written by Leon Brillouin. It is called
"Relativity Reexamined", Academic Press (1970). Quoting without permission:
> Einstein's clocks were supposed to emit extremely short signals and to
> measure accurately time intervals between signals emitted and received.
> In a word, *an Einstein clock was a radar system*, and its requirements
> were thus very different from those of a frequency standard. ...
The modern atomic clocks are referred to by the above "standard".
It can be shown by a simple thought-experiment that, if one tries to measure
an electron by such an Einstein radar pulse, this pulse allways must be much
WIDER than the following:
dt = h/mc^2 (t=time)
It thus is definitely impossible to measure (electron) time more accurately
than this amount, due to the same Compton effect as in (5).
7. There are some pathological theories in physics where infinities DO arise,
like Quantum Electro Dynamics (QED). In these theories, notions like "point"
charges, with infinitely small spatial dimensions, are used explicitly. But
what happens here is NOT a proof that "Actual infinities DO exist". Instead
it should be looked upon as a
SEVERE WARNING:
|| Absolute mathematical concepts like "point", "continuity" and "Identity" :-)
|| (yes!) are not void of danger if they become an integral part of a difficult
|| physical theory. Then the foundations of mathematics suddenly change into
|| a piece of real world physics. Once our mathematical axioms are absorbed by
|| a physical theory, they become, in fact, assumptions about Nature.
I don't think most mathematicians did design their basics for such a purpose.
Therefore, in my opinion, physics should UNDERLY mathematics, in some sense.
See signature below.
QED suffers from its infamous divergencies ever since it came into existence.
Folks like Richard P. Feynman have devised some dirty "renormalization" tricks,
in order to cure the very worst symptoms of this "infinity" disease. At least,
QED nowadays gives SOME of the desired answers. But, as Feynman himself points
out in the famous "Feynman's Lectures on Physics" (part II), the difficulties
already showed up in classical electromagnetic theory. The self-energy of point
charges is infinite. Electrons cannot move, if "actual infinities DO exist" !
Physical theories CAN suffer from an inadequate foundation of their mathematics,
I think. Somebody should have the guts to postulate the following
Basic axiom: EVERYTHING IS FINITE.
==================================
Challenge to everyone:
|| Give me ONE valid counter-example: a piece of evidence that the Infinite is
|| actually useful in physics, in a theoretical or practical sense (independent
|| of wishful mathematical thinking, of course).
--
* Han de Bruijn; computer graphics | "A little bit of Physics *
* TU Computing Centre; P.O. Box 354 | would be NO idleness in *
* 2600 AJ Delft; The Netherlands | Mathematics" (HdB). *
* Fax: +31 15 78 37 87 ============================================
Jeffrey Weiss
which is HOTTER than good old positive temperatures. As it cools T
decreases to T= - oo = + oo and then T decreases further until
reaching equilibrium. Thus, the spin system will pass through a stage
where its temperature is infinite. This, however, is sort of a cheat,
since the physically relevent quantity is 1/T which is zero
when T is infinite, and infinite when T is absolute zero, an
unnattainable state (?).
Jeffrey Weiss
jwe...@cgdra.ucar.edu
Kenneth Arromdee
>Actually, there IS physical evidence that actual infinities do NOT exist.
> at night. The whole sky would be as bright as the sun itself. There would be
> an infinite number of stars, and all their light would have reached us, since
> there would be an infinite time for it to travel.
Bogus. Assumes uniform distribution. Suppose the stars were gathered in
clumps, then clumps of clumps, then clumps of clumps of clumps, etc....
>2. What do you think of the ever increasing entropy in an infinite universe?
So? What's wrong with this?
--
"The workers ceased to be afraid of the bosses. It's as if they suddenly
threw off their chains." -- a Soviet journalist, about the Donruss coal strike
Kenneth Arromdee (UUCP: ....!jhunix!arromdee; BITNET: arromdee@jhuvm;
INTERNET: arro...@crabcake.cs.jhu.edu)
Nichael Cramer
>In article <9...@dutrun.UUCP> rct...@dutrun.UUCP (H. de Bruijn) writes:
>>Actually, there IS physical evidence that actual infinities do NOT exist.
>> [OLBER'S PARADOX DELETED]
>Bogus. Assumes uniform distribution. Suppose the stars were gathered in
>clumps, then clumps of clumps, then clumps of clumps of clumps, etc....
As stated, I don't think this works this simply. I.e. simply arguing for a
simple non-uniform distribution doesn't win. All you need for the original
argument to hold is *one* star, *somewhere* along every line of sight.
On the one hand, an infinite universe could be far from uniform and this
would still be the case. On the other hand, one could contrive an infinite
universe in which this *didn't* happen, but it would need to be much more
than simply "non-uniform"; it would have to be pretty highly ordered.
Aephraim M. Steinberg
>>Actually, there IS physical evidence that actual infinities do NOT exist.
>>1. If the universe was infinite in space and time, then it would'nt become dark
>> at night. The whole sky would be as bright as the sun itself. There would be
>> an infinite number of stars, and all their light would have reached us, since
>> there would be an infinite time for it to travel.
>
>Bogus. Assumes uniform distribution. Suppose the stars were gathered in
>clumps, then clumps of clumps, then clumps of clumps of clumps, etc....
>
I would have you consider what it really means to call something infinite.
I have trouble conceiving of any truly infinite set of clumps where the
summation out to r=oo doesn't make all directions look equivalent.
Not to say that I consider this argument alone a disproof of infinities.
The resolution I had heard, and which I believe to be orthodox, is that due
to the expansion of the universe, light coming from far away has been red-
shifted (i.e., has lost energy, relative to us, at all events) and therefore
contributes less than 1/r^2.
>>2. What do you think of the ever increasing entropy in an infinite universe?
Aephraim Steinberg.
Ice
>H. de Bruijn writes:
>>Actually, there IS physical evidence that actual infinities do NOT exist.
>>1. If the universe was infinite in space and time, then it would'nt become
>>dark at night. The whole sky would be as bright as the sun itself. There
>>would be an infinite number of stars, and all their light would have
>>reached us, since there would be an infinite time for it to travel.
>Bogus. Assumes uniform distribution. Suppose the stars were gathered in
>clumps, then clumps of clumps, then clumps of clumps of clumps, etc....
I'm doubtful this is the explaination. I prefer to use the following
as possible reasons why it ain't so bright:
1) Light losing energy in transit,
2) Dark matter.
Just out of curiousity, why can't there be another explanation why the light
from many faraway objects is shifted? I don't mean to imply that red and
blue shifting due to velocity doesn't occur, but that there could be some
other phenomena that also could contribute to the shifting. Especially in
the case of far away objects. Could it be that the light simply loses a tiny
fraction of it's energy, and hence the frequency is downshifted, as it
travels distances on the order of many light-years? Is Doppler shifting the
only possibility? How could we verify it, other than actually going out
there and measuring the velocities of far away objects (specifically the
velocity along the line between us and them, since any other component won't
contribute to Doppler shifting from our standpoint)?
Further, what if there was a lower limit of a sort, that kept the energy of
the light from going appreciably lower than some frequency (maybe it's an
assymptotic approach), perhaps because light at lower frequencies are
affected less by this phenomenon, and say that frequency corresponded with
the background radiation at around 3K? Could that explain what we see just
as well as the Big Bang does?
Just some thoughts...
- Carl Johnson
Michael B. Brooks
Subject: The Physics of Infinity
Date: 8 Nov 89 09:53:04 GMT
Organization: Delft University of Technology, The Netherlands
Actually, there IS physical evidence that actual infinities do NOT exist.
H. de Bruijn writes:
>3. Infinite vector spaces ARE used in Quantum Mechanics. However, they all have
> a complete base, AS IF they were only vectorspaces with a very BIG dimension.
> In fact, these Hilbert spaces cannot be distinguished from a big, but finite
> vectorspace, as every physicist knows from practice.
> Basic axiom: EVERYTHING IS FINITE.
==================================
>Challenge to everyone:
>|| Give me ONE valid counter-example: a piece of evidence that the Infinite is
>|| actually useful in physics, in a theoretical or practical sense (independent
>|| of wishful mathematical thinking, of course).
--
>* Han de Bruijn; computer graphics | "A little bit of Physics *
>* TU Computing Centre; P.O. Box 354 | would be NO idleness in *
>* 2600 AJ Delft; The Netherlands | Mathematics" (HdB). *
>* Fax: +31 15 78 37 87 ============================================
I do have a problem with point #3 above. Thanks Han, for an interesting post.
Since QM based on Hilbert spaces is founded on linear superposition, one
can make the argument that the infinite nature of the vector space is
required, and that a finite vector space cannot work. Completeness may be
described for a set of eigenfunctions of such a space as the property
that "an arbitrary continuous function can be expanded in terms of them."
[Schiff, p50] If one limits the basis to finite size, it is not
necessarily the case that this latter condition can be met.
This is mathmatical gibberish in a sense, since for practical analytical
solutions, or numerical ones, we mostly limit the number of terms in our
solutions. One might say that this is due to the Uncertainty Principle,
which as you pointed out, limits effectiveness of experimental
verification of theory. One might stand the whole argument on its head
and conclude that a set of infinite states does exist, that lying between
the bounds of h(bar)/2 and zero experimental error. To be fair, we cannot
know that this does or does not fit "reality", since our model, bounded
by the Uncertainty Principle, doesn`t permit it. To make this more
apparent, consider the continuum distribution of states for a simple
scattering experiment, and observe that we can`t assess occupancy of
momentum states that differ by less that delta p = h(bar)/2 /(delta x).
Point here is that while I haven't given experimental verification of
the existence of an infinity of states, computationally and
theoretically it is much easier to assume that these exist, and then
simplify the model to suit experimental data, afterwards.
Mike Brooks/Stanford Electronics Labs (solid state)/SU
Gene W. Smith
>Of course, there is NO physical evidence that infinities
>actually do exist.
Not enough matter has been found to "close up" the universe and
make it finite. This is not conclusive, but it is physical
evidence.
>Actually, there IS physical evidence that actual infinities do NOT exist.
>I have done my best to collect some substantial, physical, arguments.
>Most of these are certainly NOT my own. (Remember !)
>If the universe was infinite in space and time, then it would'nt
>become dark at night. The whole sky would be as bright as the sun
>itself. There would be an infinite number of stars, and all their
>light would have reached us, since there would be an infinite
>time for it to travel.
This is Olber's famous paradox. It doesn't work, because in the
standard Big Bang cosmologies, there is not an infinite time for
the light to travel, so we see only a finite number of stars.
Moreover, what we do see is red-shifted.
>What do you think of the ever increasing entropy in an infinite
>universe?
You want to be careful of treating the whole of an infinite
universe as a single system. Otherwise, what I think of it is
about the same as what I think of the ever-increasing entropy of
a finite universe.
>Space and time seem to be the only things that remain infinitely
>divisible. But there are also strong arguments against that !
[What follows does not seem to me an argument against the
infinite divisibility of space and time. It is an argument that
there is a limit to how accurately the position of an electron
can be measured.]
>There are some pathological theories in physics where infinities
>DO arise, like Quantum Electro Dynamics (QED). In these theories,
>notions like "point" charges, with infinitely small spatial
>dimensions, are used explicitly. But what happens here is NOT a
>proof that "Actual infinities DO exist".
I wasn't trying to use them as proof of the existence of actual
infinities. I was trying to use them as (rather weak) evidence
that such actual infinites might in some sense exist, in the
sense of infinite numbers of different Feynman diagrams, etc.
>I don't think most mathematicians did design their basics for
>such a purpose. Therefore, in my opinion, physics should UNDERLY
>mathematics, in some sense. See signature below.
I do number theory and Galois theory. Why should physics
"UNDERLY" these topics, and what would it mean if it did?
> Basic axiom: EVERYTHING IS FINITE.
> ==================================
When you run out of arguments, assume your conclusion?
>Challenge to everyone: Give me ONE valid counter-example: a piece
>of evidence that the Infinite is actually useful in physics, in a
>theoretical or practical sense (independent of wishful
>mathematical thinking, of course).
You already gave the example of Hilbert space. What's wrong
with that?
--
ucbvax!garnet!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720
"A good punch in the nose IS often effective communication"-- Ken Arndt
Gene W. Smith
>I'm doubtful this is the explaination. I prefer to use the following
>as possible reasons why it ain't so bright:
>1) Light losing energy in transit,
>2) Dark matter.
Dark matter won't work, because the stars would heat up the
dark matter. What is wrong with the finite age of the universe as
an explanation?
>Just out of curiousity, why can't there be another explanation why the light
>from many faraway objects is shifted?
This "tired light" stuff has occurred to others. You might try
asking in sci.astro instead for what the current status is.
--
ucbvax!garnet!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720
"We should consider it as one of the most astonishing errors of the present
age that so many people listen to the words of pseudoprophets who, in place
of the dogmas of religion offer scientific dogmas with medieval impatience
but without historical justification." --Baron Lorand von Eotvos
Ice
>Dark matter won't work, because the stars would heat up the dark matter.
Sure, but would the heated dark matter necessarily re-radiate the light in
the visible spectrum (for fun, might the re-radiated light be about 3K?)?
Should be akin to black body radiation, I imagine.
>What is wrong with the finite age of the universe as an explanation?
The first problem is that the original statement was assuming infinite time,
but if you want to waive that, you still have another problem. Since you're
assuming a Big Bang type universe (from your other post), the matter would
have to travel infinitely fast to create the infinite universe from a Big
Bang. Is that what you had in mind?
Your idea would work if the universe is repeating (ie. on the surface of a
four-dimensional sphere, for example), but whether that's considered an
infinite universe is probably a personal choice. I'd also wonder what it
would be like where the expansion meets itself.
I'll stick with red-shifting and/or tired-light for now, until I understand
better what you're saying, and since they work with infinite time, too.
- Carl Johnson
Steve Stevenson
>In article <9...@dutrun.UUCP>, rctthdb@dutrun (H. de Bruijn) writes:
>>Of course, there is NO physical evidence that infinities
>>actually do exist.
> Not enough matter has been found to "close up" the universe and
>make it finite. This is not conclusive, but it is physical
>evidence.
I'm not sure this latter statement helps. If we buy the ``Big Bang'' then
only a finite time has passed since the beginning of the universe. We
keep expanding --- but how is this an actual completed infinity? :-)
--
Steve (really "D. E.") Stevenson st...@hubcap.clemson.edu
Department of Computer Science, (803)656-5880.mabell
Clemson University, Clemson, SC 29634-1906
Chris Phoenix
As I understand the argument, one can't interpose dust clouds between
the stars and you to darken things, because the dust clouds will
absorb so much energy they'll eventually glow as brightly as the stars
behind them.
But: What is the possibility that if a dust cloud gets too excited, it
starts to break down helium (lithium etc) into hydrogen, more than it
fuses hydrogen into helium? My intuition says nothing and I don't have
the math--in an infinite universe, where every point has a constant influx
of energy, is it possible for cool spots to develop where the energy is
stored up in nuclei configurations?
I can see a system where novae produce helium clouds, which break down
to hydrogen and then coalesce, producing more helium and light... and
a temporary dark spot.
Is this totally off the wall or otherwise impossible?
--
Chris Phoenix | A harp is a nude piano.
cpho...@csli.Stanford.EDU | "More input! More input!"
For every idiot-proof system, a new improved idiot will arise to overcome it.
Disclaimer: I want a kinder, gentler net with a thousand pints of lite.
Ethan Tecumseh Vishniac
> gsm...@garnet.berkeley.edu (Gene W. Smith) writes:
> >In article <9...@dutrun.UUCP>, rctthdb@dutrun (H. de Bruijn) writes:
> >>Of course, there is NO physical evidence that infinities
> >>actually do exist.
> > Not enough matter has been found to "close up" the universe and
> >make it finite. This is not conclusive, but it is physical
> >evidence.
>
> I'm not sure this latter statement helps. If we buy the ``Big Bang'' then
> only a finite time has passed since the beginning of the universe. We
> keep expanding --- but how is this an actual completed infinity? :-)
I believe the point here was that the homogenous, isotropic cosmological
model which best matches the available evidence is of infinite spatial
extent, although of finite age.
Of course, it requires a rather large leap of faith to believe
that the model should be extrapolated infinitely beyond our current
particle horizon. Some theories which purport to explain the origin
of the universe (like Eternal Chaotic Inflation) suggest a topologically
more complicated, but still infinite universe (in fact, infinite in
time as well as in space) but their contact with observations is
weak at best.
--
I'm not afraid of dying Ethan Vishniac, Dept of Astronomy, Univ. of Texas
I just don't want to be {charm,ut-sally,emx,noao}!utastro!ethan
there when it happens. (arpanet) et...@astro.AS.UTEXAS.EDU
- Woody Allen (bitnet) ethan%astro.as....@CUNYVM.CUNY.EDU
These must be my opinions. Who else would bother?
Doug Merritt
>apparent, consider the continuum distribution of states for a simple
>scattering experiment, and observe that we can`t assess occupancy of
>momentum states that differ by less that delta p = h(bar)/2 /(delta x).
>
>Point here is that while I haven't given experimental verification of
>the existence of an infinity of states, computationally and
>theoretically it is much easier to assume that these exist, and then
>simplify the model to suit experimental data, afterwards.
If you're looking for an infinity of states, look at atomic spectra.
Although photon emission is limited to discrete quantum levels, (A) there
are an infinite number of such discrete levels in each atomic line spectrum
in each of the series (e.g. Balmer, Lyman, Paschen etc), each asymptotic
to a particular limit; (B) velocity is not by itself quantized, despite
your mention of Heisenberg limitations on determination of (momentum,position).
There are thus an infinite number of possible velocities for particles,
and this is directly observed in the arbitrary doppler shift of emitted
quanta from atoms & molecules due to thermal motion, resulting in a continuum
of observed levels. This may be a weaker example than (A), since one could
argue that any observed "black body" in fact consists of a finite number
of atoms, so at any given instance there will be a finite number of
observed frequencies. The counter argument is that there are still an
infinite number of possible states for these frequencies, but admittedly
you'd have to extend your observation time to infinity to see them all.
Still it is not a reasonable approximation to impose an arbitrary finitude
on the number of possible levels, since this is equivalent to saying that
momentum alone is quantized, but it is not...in this context there is
no attempt to observe position, leaving momentum unhampered by Uncertainty.
See for example "Atomic Spectra and Atomic Structure" Gerhard Herzberg
(c) 1937.
Doug
--
Doug Merritt {pyramid,apple}!xdos!doug
Member, Crusaders for a Better Tomorrow Professional Wildeyed Visionary
Aephraim M. Steinberg
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From: aephraim
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To: aephraim
Status: R
In article <19...@pasteur.Berkeley.EDU> c18...@katerina.UUCP (Ice) writes:
>Kenneth Arromdee writes:
>>H. de Bruijn writes:
>
>>>Actually, there IS physical evidence that actual infinities do NOT exist.
>>>1. If the universe was infinite in space and time, then it would'nt become
>>>dark at night. The whole sky would be as bright as the sun itself. There
>>>would be an infinite number of stars, and all their light would have
>>>reached us, since there would be an infinite time for it to travel.
>
>>Bogus. Assumes uniform distribution. Suppose the stars were gathered in
>>clumps, then clumps of clumps, then clumps of clumps of clumps, etc....
>
>I'm doubtful this is the explaination. I prefer to use the following
>as possible reasons why it ain't so bright:
>
>1) Light losing energy in transit,
>2) Dark matter.
I believe that the actual definition of dark matter is stuff which has mass
yet NEITHER emits NOR absorbs electromagnetic radiation, despite the name.
>
>Just out of curiousity, why can't there be another explanation why the light
>from many faraway objects is shifted? I don't mean to imply that red and
>blue shifting due to velocity doesn't occur, but that there could be some
>other phenomena that also could contribute to the shifting. Especially in
I actually read an article on this by a scientist whose name I've forgotten
but who was described as the "father of radioastronomy" in a magazine run
either by Lyndon LaRouche or by some of his friends. I'm sorry I forgot what
it was called as well, but it had articles on topics like "Nuclear Power Now,"
the claim that if we retune our instruments to the "natural" scale with middle-C
at 256 Hz, DNA will resonate with it (23-odd octaves higher), the AIDS virus
as an antenna, and a theory of a static universe where all the "redshifts" we
observe are actually due to Compton scattering with interstellar matter.
I can't believe this, but I must admit that I have not worked out just how
much scattering would be necessary to produce this result. Naturally, I assume
that it would both require an unreasonable amount of "dust" and that it would
scatter light so much that stars could not possibly look like points. But if
you're really interested, you might confirm these guesses on your own.
Aephraim Steinberg.
Gene W. Smith
>Gene W. Smith writes:
>>Dark matter won't work, because the stars would heat up the dark matter.
>Sure, but would the heated dark matter necessarily re-radiate the light in
>the visible spectrum (for fun, might the re-radiated light be about 3K?)?
>Should be akin to black body radiation, I imagine.
Think about the Olbers paradox again. Stars would be in all
directions, so the matter would be at least as hot as the average
photosphere (of course, actually the stars would heat each other
up and it would all go completely to hell).
>>What is wrong with the finite age of the universe as an explanation?
>The first problem is that the original statement was assuming
>infinite time, but if you want to waive that, you still have
>another problem.
I was the one who made the "original statement", which amounted
to a claim that the universe might be spacially infinite and that
there was (weak) evidence that is, so this is not *my* problem.
>Since you're assuming a Big Bang type universe (from your other
>post), the matter would have to travel infinitely fast to create
>the infinite universe from a Big Bang. Is that what you had in
>mind?
Of course it isn't. In the "standard" isotropic Big Bang
universe, the space-slices of the expanding cosmos are a space of
constant curvature--which can be non-positive. In that case, the
expanding universe is infinite from the beginning.
>Your idea would work if the universe is repeating (ie. on the surface of a
>four-dimensional sphere, for example), but whether that's considered an
>infinite universe is probably a personal choice. I'd also wonder what it
>would be like where the expansion meets itself.
The surface of a four-dimensional sphere is a space of constant
curvature of *positive* curvature, and is *not* what I mean by an
infinite universe. There is no reason to believe, or at least no
evidence, that such a universe "repeats", in any case.
I also wonder what would happen if the expansion meets itself,
since I don't even know what the hell you mean by that.
>I'll stick with red-shifting and/or tired-light for now, until I understand
>better what you're saying, and since they work with infinite time, too.
This reminds me of how you plan to stick to what you "know"
about Hitler, despite not having any evidence. It's a bad habit--
a better one is to *find out*. There are books on cosmology just
as there are books on the Nazis.
--
ucbvax!garnet!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720
"To name the unnamable, to point at frauds, to take sides, start arguments,
shape the world and stop it from going asleep". -- 'The Satanic Verses'
2011...@uwovax.uwo.ca
[quoted article deleted...]
With regard to (A):
The infinite number of states in a discrete spectral series is not a good
example.
The number of levels in a discrete spectral series is infinite only for a
theoretical atom (i.e. an atom completely isolated and alone in the
universe). In a real atom, the energy levels are affected by neighboring
atoms; the interactions with neighbors broaden the energy levels so that
the high members of the series are absorbed into the continuum. Here is
one way of looking at it: the mean radius of the orbital increases
with principle quantum number n; for hydrogen, a_n = a_0 n**2 where a_0
is the Bohr radius. For some value of n, the mean diameter of the orbital
becomes comparable to the mean separation between atoms. A state with
larger mean radius will be more strongly affected by neighboring atoms than
by the original atom and cannot be considered to be a bound state of the
original atom.
Allowing an infinite number of discrete levels gets you into trouble in
another way: the partition function diverges, which can be interpreted as
meaning that in thermodynamic equilibrium, atoms are *always*ionized*,
contrary to observation! The resolution: only states with mean radius
less than the mean atomic separation should be included in the partition
function sum; this makes the partition function finite and gets those
electrons back down their ground states where they belong. (Rudolf Peierls
gives a nice discussion of the paradox and its resolution in his book
"Surprises in Theoretical Physics" (Princeton U. Press, 1979).)
The finite number of bound states in atoms in a gas has a practical
application in astrophysics: in situations in which all the bound states
are populated, the principle quantum number of the highest series member
actually present in the spectrum can be used to obtain a rough estimate
for the gas density.
--
Terry Gaetz -- ga...@uwovax.bitnet -- ga...@uwovax.uwo.ca
Astronomy Dept. --
U. Western Ontario -- "The simplest explanation is this:
Canada -- it doesn't make any sense." Prof. Buechner
Kevin Bagley
>In article <19...@pasteur.Berkeley.EDU>, c188-br@katerina (Ice) writes:
>>I'm doubtful this is the explaination. I prefer to use the following
>>as possible reasons why it ain't so bright:
>>1) Light losing energy in transit,
>>2) Dark matter.
> Dark matter won't work, because the stars would heat up the
>dark matter. What is wrong with the finite age of the universe as
>an explanation?
>>Just out of curiousity, why can't there be another explanation why the light
>>from many faraway objects is shifted?
> This "tired light" stuff has occurred to others. You might try
>asking in sci.astro instead for what the current status is.
Several months ago, Sky&Telescope ran an article specifically related to
this. I don't have the article handly, or remember which issue it was, but
the title was something like "Why Is The Sky Dark?".
Basically, what the article was about, was that the sky is NOT dark.
The sky radiates energy from all directions. This energy is thought to be
the 'background' radiation from the 'Big Bang'. In theory, as you look
further into space, you see further back into time. At some point, you
should be able to look at the Big Bang itself. The direction you look
doesn't matter.
The 'Tired Light' idea was mentioned in the article, but it seems that
there is currently no evidence for this.
One thing that came to mind as I read some of this thread, is that angular
diameter has some bearing on this. I realize the concept of 'infinite' time
and distance, but you also have to keep in mind that a reciever on earth
will only receive a photon from a star at infinite distance in an infinite
amount of time.
>ucbvax!garnet!gsmith Gene Ward Smith/Brahms Gang/Berkeley CA 94720
--
_____ Kevin Bagley Global Technology Mukilteo WA 98275 (206)742-9111
)__) _ _ _ UUCP:uw-beaver!uw-nsr!gtisqr!kevin
_/__) (_(_(_)_/_)_ ARPA:uw-nsr!gtisqr!ke...@beaver.cs.washington.edu
____________/ Disclaimer: "I did not say this. I am not here."
H. de Bruijn
> Point here is that while I haven't given experimental verification of
> theoretically it is much easier to assume that these exist, and then
> simplify the model to suit experimental data, afterwards.
with some of these common opinions. Think about it: are you really sure
that infinities are more "easy" to handle than the FINITE things you can
simply lay your hands on? Sounds a little bit strange, is'nt it?
Or is the real reason that we just accept everything that mathematicians
do for us, without thinking of the physical consequences?
You probably have heard of Constructivism and Intuitionism.
I wonder if there also exists someting like *Physicism* in mathematics!
H. de Bruijn
> This is Olber's famous paradox. ...
Thank you. I forgot the name.
Gene W. Smith:
> [What follows does not seem to me an argument against the
> infinite divisibility of space and time. It is an argument that
> there is a limit to how accurately the position of an electron
> can be measured.]
In fact, it IS an argument against infinite divisibility of space and time.
Because electrons and positrons ARE created as soon as you try to measure ANY
object within dimensions of h/mc (space) or h/mc^2 (time).
> I do number theory and Galois theory. Why should physics
> "UNDERLY" these topics, and what would it mean if it did?
with number theory and Galois fields.
>> Basic axiom: EVERYTHING IS FINITE.
>> ==================================
> When you run out of arguments, assume your conclusion?
sightest attempt to argue anything? Finitism of nature CANNOT be proven as a
mathematical theorem, of course. Axioms merely serve as a STARTing point,
remember?
> You already gave the example of Hilbert space. What's wrong with that ?
the "infinite" number of states in a "real" atom. Read it!
john baez
>You express the common opinion, Michael. But I just happen NOT to agree
>with some of these common opinions. Think about it: are you really sure
>that infinities are more "easy" to handle than the FINITE things you can
>simply lay your hands on? Sounds a little bit strange, is'nt it?
Perhaps strange, perhaps not so strange. Try doing the sum
from 1 to 100,000,000 of 1/n^2 . This is hard. Try doing
the sum from 1 to infinity. This is easier - (pi^2)/6.
Try doing the integral from 0 to 100,000,000 of exp(-x^2).
It's hard. Try the integral from 0 to infinity. It's easy.
So, one can say `if infinity did not exist one would have to
invent it' - and perhaps we have!
>Or is the real reason that we just accept everything that mathematicians
>do for us, without thinking of the physical consequences?
>You probably have heard of Constructivism and Intuitionism.
>I wonder if there also exists someting like *Physicism* in mathematics!
As a mathematical physicist, I don't what I'd do without the
concept of infinity. After hours it's fun to worry about it,
but while busy solving problems I just use it. Take statistical
mechanics - it's easier to understand the limiting behaviour of
n objects as n -> infinity than it is to understand what 7 objects
do. So, I'll have to say - there IS such a thing as physicism
in mathematics... it's the mathematics we have today!!!! Think about
it: Newton, Leibniz, Laplace, Lagrange, D'Alembert, Bernoulli,
Euler, Gauss, Fourier, von Neumann, etc. (if I list living ones
someone will be upset at being left out!) - they were all great
physicists as well as mathematicians.
H. de Bruijn
> So, one can say `if infinity did not exist one would have to
> invent it' - and perhaps we have!
I fully agree with you if by "infinite" you mean: taking the limit of
something finite. Considering the examples you give, I think I have to
subscribe your opinion a great deal. See below: "Potential Infinity"
is perfectly acceptable from a physical point of view.
> ... So, I'll have to say - there IS such a thing as physicism
> in mathematics... it's the mathematics we have today!!!! .....
There is no such thing as THE mathematics we have today. Mathematics
is splitted up in different schools. I think this is not well known
to many physicists. They call themselves Formalists, Constructivists
or Intuitionists. This discussion actually started in 'sci.logic' and
'sci.philosophy.tech'. It is not so much about the useless of the whole
concept of Infinity, as well as WHICH of the concepts one should adopt:
the Potential Infinite of the Constructivists school (which I think is
the more acceptable one from a physical point of view) OR the so called
Actual Infinite, which is accepted by the Formalist school (Cantor and
Hilbert: set theory, transfinite cardinals). It is ONLY the latter kind
of Infinity I have my objections against. At this point, I thought that
Physics might have something substantial to contribute.
I DON'T think that making the wrong choice can do no harm to a physical
theory, especially if this theory is very far from everyday experience.
Michael B. Brooks
infinite levels in atomic spectra, why this is a bad example, and the like.
My original post mentioned a collision problem, not a bound state, and I
indicated that the continuous nature of the eigenvalues as a possible
"infinity". The issue was raised, "well, what good is the concept of
infinity", not with-standing what I pointed out. OK, I`ll let a real
expert try to describe the utility of "infinity" in these problems.
From PAM Dirac, The Principles of Quantum Mechanics, pg 185-6, on
Collision Problems. [apply to my simple scattering example]
"Thus only the relative probabilities of the particle being in different
finite volumes will be physically significant, their absolute values being all
zero. The total energy of the system has a continuous range of eignevalues,
since the initial energy of the particle can be anything. Thus a ket Is> say,
corresponding to a stationary state, being an eigenket of the total energy,
must be of infinite length. We can see a physical reason for this, since if
Is> were normalized and if Q denotes that observable--a certain function of the
position of the particle--that is equal to unity if the particle is in a given
finite volume and zero otherwise, then <sIQIs> would be zero, meaning that the
average value of Q, ie. the probability of the particle being in the given
volume, is zero. Such a ket [s> would not be a convenient one to work with.
However, with [s> of infinite length, <sIQIs> can be finite and would then give
the relative probability of the particle being in the given volume."
Again one can argue that this is a "theoretical issue", and I am not pretending
to present experimental evidence demonstrating that an "infinity of
states" exists. I only point out that infinity can enormously simplify
calculations that are later modified to account for experimental observations.
Ice
>Think about the Olbers paradox again. Stars would be in all directions, so
>the matter would be at least as hot as the average photosphere (of course,
>actually the stars would heat each other up and it would all go completely
>to hell).
Three possible answers:
1) Sure, stars would be in all directions, but, again, dark matter (and/or
tired light) could prevent the dark matter, the stars, and the earth from
being heated up, by smothering Olber's paradox. It's a self-supporting
position basically/unfortunately.
2) Or, as someone else said, though I'm not certain of it's validity, if dark
matter neither absorbed nor emitted light, then it wouldn't heat up either.
3) Finally, my "for fun" possibility would just have all light and heat be
converted by the dark matter to 3K radiation, which isn't all that hot.
About Gene's answer "finite age" in conflict with the original statement:
>I was the one who made the "original statement", which amounted to a claim
>that the universe might be spacially infinite and that there was (weak)
>evidence that is, so this is not *my* problem.
What??? Gene and I must be thinking of different statements. Both Ken and
I were responding to H. de Bruijn's statements: "Actually, there IS physical
evidence that actual infinities do NOT exist. 1. If the universe was
infinite in space and time, then it would'nt become dark at night..." So I
think Gene's answer is for a different question.
>In the "standard" isotropic Big Bang universe, the space-slices of the
>expanding cosmos are a space of constant curvature--which can be
>non-positive. In that case, the expanding universe is infinite from the
>beginning.
I've never heard of this '"standard"' model, but then I'm not a very active
cosmology nut. Could Gene possibly describe this model a bit more?
Everything I've heard so far has the Big Bang universe as a finite size (e.g.
Hawking). Apparently the books I've read on cosmology aren't the ones
Gene's read. Some pointers? (And let's be civil, please?)
- Carl Johnson
William H Katzman
{stuff deleted!}
>
>>In the "standard" isotropic Big Bang universe, the space-slices of the
>>expanding cosmos are a space of constant curvature--which can be
>>non-positive. In that case, the expanding universe is infinite from the
>>beginning.
>
>I've never heard of this '"standard"' model, but then I'm not a very active
>cosmology nut. Could Gene possibly describe this model a bit more?
>Everything I've heard so far has the Big Bang universe as a finite size (e.g.
>Hawking). Apparently the books I've read on cosmology aren't the ones
>Gene's read. Some pointers? (And let's be civil, please?)
>
>- Carl Johnson
he says, it doesn't matter if anything existed before the Big Bang, because
there is no way that we can tell what would have existed if anything did, so
we will define time as having started at the Big Bang. It is sort of like
saying that if all of space is totally empty, and either totally concentrated,
or totally flat, then it is inconsequential, because there is nothing there
to measure time/space.
-thekat
(Geez, I believe that's it, but I could be wrong - then again I might be....
.....but....... Oh Geez.)