Travelling Again
I fear that I'm away from computers much of this week. It will means some great pictures soon. In the meantime, please see the comments on Asymptotia. Aloha!
posted by L. Riofrio at 11:19 AM
Tuesday, November 28, 2006
About Me
- Name: L. Riofrio
Full-time scientist. Before graduating I learned that the speed of light is slowing down and originated the "GM=tc^3" theory, which explains the dark energy problem and most physicists still can't explain. More recent work seeks Black Holes in some unexpected places, even within Earth. I've been working at NASA in Houston on studies of the Moon, and have an insider's view of the Space program. Actress in film, television and stages from Honolulu to Houston. In spare time I fight off hostile aliens, explore a strange world and unusual forms of life.
Previous Posts
- Solution to Einstein Equation
- Turkey Day
- Getting to GM=tc^3
- Exploding Supernova Evidence
- Fun In the Dark
- Firefly
- Your Tax Dollars At Work
- The Crab SupernovaCourtesy of ESA and the Hubble S...
- The Universe in 60 Seconds, Without "Dark Energy"B...
- Wings Over the World
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The Paper
by Clifford, at 3:02 pm, November 20th, 2006 in science, astronomy, cosmology, dark energy
I learned from Often In Error that the paper of Riess et al, reporting on the research that was in the recent NASA press release, is out. It is here.
(Aside:- I must use the term “cosmic jerk” in an everyday sentence one day…. probably not as a term of endearment….)
-cvj
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31 Responses to “The Paper”
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1 Louise
Nov 20th, 2006 at 3:25 pm
Redshifts are the only evidence of cosmic acceleration. The data is still consistent with a “c change” predicted by GM=tc^3. Concerning “cosmic jerk,” dependence on sqrt(1 + Z) ensures that redshifts increase linearly for low Z, where c change is negligible. High redshifts curve upward as GM=tc^3 predicts. This should be considered along with other priors.
2 Clifford
Nov 20th, 2006 at 9:05 pm
Thanks Louise…. I hope you’ve given some thought to my question (considerably refined by Jacques’ further questions) about what the physics origin of your changing-speed-of-light equation might be. It would help a lot to understand what underlies it, as opposed to being an empirical formula that seems to very much violate SR and GR.
Best,
-cvj
3 Louise
Nov 20th, 2006 at 9:52 pm
HI Clifford, and I will answer your questions as best I can.
1) Origins: Good question. What is c anyway? It is not just the speed of light, it is the speed of any massless particle and underlies Special Relativity even in a dark room with the lights out. It can be understood as the “conversion” factor in a Space/Time diagram. To summarise SR, Space and Time are one phenomenon related by c.
2) First Principle: Scale R of the Universe is its age multiplied by c, R = ct. Our distance from the “Big Bang” origin is so related to age of the Universe. That is why, as t increases, the Universe expands.
3) It can’t expand at the same rate c continuously, for gravity slows it down. c and t are further related by GM = tc^3. Derivation. Together, these two expressions form a solution to field equation of gravitation, expanding as t^{2/3}. Locally this reduces to SR, for change in c is only apparent at very high redshifts.
I hope this helps answer your curiosity. You no doubt have further questions, which I will also answer as best I can. This has all developed in response to such questions. I enjoy reading your posts too.
4 K
Nov 20th, 2006 at 10:21 pm
Hi Clifford,
(this may be little out of context!)
I am sure you are aware of the following papers. The authors of these papers argue, in there own words,
“it is not appropriate to evaluate cosmological parameters based on measurements from either WMAP or COBE. WMAP images do not meet accepted standards for imaging”.
http://ptep-online.com/index_files/2007/PP-08-01.PDF
http://ptep-online.com/index_files/2007/PP-08-02.PDF
http://ptep-online.com/index_files/2007/PP-08-03.PDF
I just want to know your opinion on this, if you happen to read this papers already. I see that they have a point here !!
K
5 Clifford
Nov 20th, 2006 at 10:29 pm
Hello,
I have not read those.
-cvj
6 Clifford
Nov 21st, 2006 at 10:07 am
Hi Louise….. The explanation seems riddled with misconceptions and non-physics to me. See the comments of Jacques on the other thread on this topic, and further see some of the objections rasied by Chirstine in the thread of the post on your blog that you pointed me to.
To my mind, there’s a lot of work to be done to get a foundation for your formula. Until you have something, it might be (as I said on the other thread) a bit premature to be going around lots of blogs saying that the standard cosmology is wrong, and that your “theory” explains all, etc etc.
Best,
-cvj
7 nc
Nov 21st, 2006 at 12:05 pm
Hi Clifford,
To prove Louise’s MG = tc^3:
(1) Einstein’s equivalence principle of general relativity:
gravitational mass = inertial mass.
(2) Einstein’s inertial mass is equivalent to inertial mass potential energy:
E = mc^2
This equivalent energy is “potential energy” in that it can be released when you annihilate the mass using anti-matter.
(3) Gravitational mass has a potential energy which could be released if somehow the universe could collapse (implode):
Gravitational potential energy of mass m, in the universe (the universe consists of mass M at an effective average radius of R):
E = mMG/R
(4) We now use principle (1) above to set equations in arguments (2) and (3) above, equal:
E = mc^2 = mMG/R
(5) We use R = ct on this:
c^3 = MG/t
or
MG = tc^3
Which is Louise’s equation. QED.
I’ve done and published (where I can) work on the detailed dynamics for such an energy-based mechanism behind gravity. The actual dynamics are slightly more subtle than above because of the definition of mass of universe and its effective radius, which are oversimplified above. In reality, the further you see, the earlier the phase of the universe you are looking at (and receiving gravity effects from), so the effective density of the universe is greater. There is however, a limiting factor which eventually offsets this effect when considering great distances, because big masses in the very dense early-time universe, receding from us rapidly, would exchange very severely redshifted gravitons (or whatever the gauge bosons of gravity are).
Cheers,
nc
8 Jacques Distler
Nov 21st, 2006 at 12:25 pm
Nigel,
I’m afraid you will need to learn General Relativity, before you can start monkeying around with modifications of it, in any intelligent fashion. None of the “equations” you wrote down above (nor any of the connecting prose) make any sense, in the context of GR.
One can’t “derive” Louise’s equation from anything, for the simple reason that it is incompatible with both GR and SR. She needs to write down a new theory, which replaces GR, a theory in which the quantity c(t)=a(t)/t actually turns out to be the speed of light. And then she has to show that this new theory is compatible with all of the experimental tests of GR.
It’s a tall order, but monkeying around with pseudo-Newtonian mumbo jumbo, as you have done above, is not going to get her there.
9 nc
Nov 21st, 2006 at 1:30 pm
Dear Professor Jacques Distler,
Louise’s varying c solution to her equation can’t be derived from anything I know, so I agree with you there. But see http://nige.wordpress.com/2006/09/30/keplers-law-from-kinetic-energy/ for Dr Love’s derivation of Kepler’s law from an analogous physical argument about energy! (He emailed it to me rather than send it to arxiv, possibly because he has had experience being treated as a monkey.)
As somebody working in extra-dimensional string theory, your judgement is very influential on me regards the fact that I do need to learn more general relativity (and more QFT too, and I’ll do that as I’m setting up a site about that), but I do know the basics of general relativity and its solutions from a course on cosmology and also I’ve studied quite a bit more about it independently; I address the dynamics behind general relativity at http://feynman137.tripod.com (gravity introduction) and also http://feynman137.tripod.com/#h for proof. I also did quantum mechanics and I can understand Dyson’s and more recent lecture notes in basic QFT.
Best wishes,
nc
10 astromcnaught
Nov 21st, 2006 at 1:40 pm
Jacques Distler says …It’s a tall order…
What a wonderful explanation of the task, and without resort to the Index too! Let’s just hope that the task is tractable, and that the Universe has not been designed by some Cosmic Jerk:)
11 Louise
Nov 21st, 2006 at 6:21 pm
Nigel’s quick derivation is entertaining. I did it the other way around, starting from R=ct and GM=tc^3 to show that total energy E + U = 0.
Since these expressions solve the field equations, in what way are they incompatible with GR? If someone thinks you can start with the assumption that c is constant and arrive at a theory where c is not, then that order is infinitely tall.
It is most pleasing that Jacques considers me capable of replacing GR. I will try not to disappoint you.
12 Elliot
Nov 21st, 2006 at 7:05 pm
Louise,
Without bothering to enumerate what is incorrect about your so called theory, let me point out an obvious error. You suggest that c is slowing down because the expansion of the universe is slowing. This is directly contradicted by the dicovery of dark energy and the acceleration of the expansion.
I would humbly suggest that you need to rethink your approach. I find it unlikely that the majority of professional physicists/cosmologist are wrong and you are the only one who has a handle on this.
Elliot (of course I could be wrong)
13 SteveM
Nov 21st, 2006 at 11:55 pm
Hi Louise,
My background is mathematical physics/biology and I admit I am no expert on the hard physical aspects of modern cosmology, but I think I am fairly well versed enough in GR and differential geometry to see some problems here:
[1] In your derivation you seem to confuse the FRW scale factor R with r, and you seem to equate R=ct with r,theSchwarzchild radius. This makes no sense to me. Any reasonable cosmological derivation or argument has to be based on a FRW metric.(Also R goes as t^(1/2), for k=0 and radiation? Can’t remember exactly.) You say “the gravitational influence is the same as if the source of the mass where at a central point”. In general relativity the source of the gravitational field is not just the mass M but ANY kind of energy, including the gravitational field itself that it generates. Thus even if we have a distribution of mass M confined to a finite region of space, or an idealised point, the very gravitational field that it generates fills that space and beyond so the source is not confined to that region and is no longer a point source. Source in GR=mass plus gravitational field; source in Newtonian gravity =mass only. This is why the Einstein field equations are nonlinear. The source of the field equations is the energy-momentum tensor of mass and energy. It is for this reason you can’t have a point mass as the source of the Schwarzchild solution; the singularity would also then be non-spacelike when it is in fact spacelike. (For example, see Class. Quant. Grav. 10 (1993), p2271.) In your derivation of your formula, you seem to have the Schwarzchild solution as a limiting case if M is confined to a point and M is nonzero? You can’t do this for the reasons just mentioned. (I don’t really follow everything you do in your derivation.)
[2] As Jacques points out, GR is based on Riemannian geometry with locally inertial Lorentz frames where the speed of light is fixed. When c varies you are no longer dealing with GR but need a new theory of gravitation, and one must pass the tests that GR has already passed.
[3] As Clifford points out, proposing a varying speed of light requires an underlying dynamical mechanism that causes it. Why would it change? What’s the physical reason for it being larger in the past?
[4] Even if a varying c can do away with dark energy there are surely many other problems with having c larger in the past and infinite at the Big Bang. Even if you can fix one problem like the dark energy you can then screw up everything else for which there is excellent, indeed overwhelming, evidence: for example, how does it affect electroweak symmetry breaking and phase transitions and the general thermal history of the early universe? How does it affect baryon asymmetry? How does it affect primordial nucleosynthesis? It seems to me that a lot of calculations involving Standard Model physics, statistical mechanics and nuclear physics (cross sections and reaction rates and stuff) really do need a fixed c. A higher c in the early universe would affect the ratio of the reaction rates going on to the expansion rate H–this ratio can depend on c. (See Weinberg’s classic book). The fine structure constant and weak coupling constant for example also depends on c etc. etc. etc. If c varies then these vary too. All these issues have to be addressed, not just the dark energy.
The dark energy is a hard hard problem–in fact everything in gravitation and cosmology is hard:) Keep thinking about it but a reasonable theory proposing a varying c idea has to start addressing a lot of these points in a technically detailed and rigorous way, and that’s a very tall order for anyone in the face of overwhelming evidence that c=1.
Best.
14 nc
Nov 22nd, 2006 at 3:16 am
Elliot you wrote:
“You suggest that c is slowing down because the expansion of the universe is slowing. This is directly contradicted by the dicovery of dark energy and the acceleration of the expansion.”
The acceleration of the expansion is the primary evidence for dark energy, as I’m sure you are aware. The acceleration observed is supposed to caused by dark energy.
Friedmann’s solution to GR says the effective dimension of the universe R should increase (for critical density) as t^{2/3}. The two-thirds power of time is due to the effect of gravity.
Now, via the October 1996 issue of Electronics World, I had an 8-pages paper analysing this, and predicting the results Perlmutter found two years later.
GR is widely accepted to not be the final theory of quantum gravity, and the discrepancy is in that a Standard Model type (ie Yang-Mills) quantum field theory necessitates gauge boson radiation (”gravitons”) being exchanged between the gravitational “charges” (ie masses). In the expanding universe, over vast distances the exchange radiation will suffer energy loss like redshift when being exchanged between receding masses.
This predicts that gravity falls off over large (cosmological sized) distances. As a result, Friedmann’s solution is false. The universe isn’t slowing down! Instead of R ~ t^{2/3} the corrected theoretical prediction turns out R ~ t, which is confirmed by Perlmutter’s data from two years after the 1996 prediction was published. Hence no need for dark energy; instead there’s no simply gravity to pull back very distant objects. Nobel Laureate Phil Anderson grasps this epicycle/phlogiston type problem:
‘the flat universe is just not decelerating, it isn’t really accelerating’ - Phil Anderson, http://cosmicvariance.com/2006/01/03/danger-phil-anderson/#comment-10901
“I find it unlikely that the majority of professional physicists/cosmologist are wrong and you are the only one who has a handle on this.” - Elliot.
Well, you’re right about the specific solution to the equation which has c varying, but not about Louise’s basic equation, which - correction factors for the definition of mass and time aside - is physically sound enough. Just as well Planck didn’t dismiss Einstein as being an egotist in 1905, when Einstein doubted aether.
15 Elliot
Nov 22nd, 2006 at 7:41 am
nc,
Alright lets just make all the constants of nature variable. That should make it really easy to fit the experimental data.
Your description of gravity is reminiscent of the “tired light” theories to explain increasing redshifts with distance.
Elliot
16 nc
Nov 23rd, 2006 at 8:57 am
No, the facts predicted the correct cosmological expansion in Oct 96 EW, two years before Perlmutter’s results from automated CCD observations on supernovae redshifts confirmed it.
Instead of this Yang-Mills quantum gravity winning, an ad moc dark energy was added to the CDM model. Tired light (1) mas no proved mechanism which makes correct predictions and (2) is plain wrong. See http://www.astro.ucla.edu/~wright/tiredlit.htm
17 Louise
Nov 23rd, 2006 at 1:33 pm
Hello all: Eliot, theory says that the universe is not accelerating, it decelerates as most cosmologists thought before 1998. Redshifts appear to accelerate because c has been slowing. I appreciate when informed questions are asked like yours. This is not “tired light” which says that redshifts themselves are caused by varying c. Expansion is predicted by R=ct, change in c makes that expqnsion apper to accelerate.
Steve, your question are very thoughtful and i will try to answer as many as possible. (1) R = ct is less than the Schwarzhild radius, which is referred to as r. Theory predicts R ~ t^{2/3} as in an Einstein-de sitter expansion of k = 0. Arguments have started over whether the gravitational influence of a spherical mass distribution is the same as a point mass, but the proof works in 3 or 4 dimensions. A gravitational field itself has no mass.
(2) If it pleases Jacques, we can call this a new theory that reduces to GR. GR with fixed c only works in an Einstein static universe, which experiment shows we are not living in.
(3) Gravity here provides a mechanism to change c, just as it affects light paths.
(4) It is most likely that the product hc and the fine-sturcture value are truly constant. Then planck value h ~ t^{1/3}. That avoids the problems you mention.
No doubt you have other questions and objections. I will take a swim in the reef and answer as many as possible. Please don’t get angry if I do not get to your question right away.
18 Clifford
Nov 23rd, 2006 at 10:00 pm
Louise. It is not a theory. You have not given us any basis for your empirical formula.
-cvj
19 Peter
Nov 24th, 2006 at 12:58 am
Clifford wrote
“Louise. It is not a theory. You have not given us any basis for your empirical formula.”
Oh I quite agree. A theory formulated on an empirical formula that has no basis is less preferable than a theory formulated on one that has a basis.
To me this is the main failing of Ptolemaic Astronomy. No one, then and now, can give a physical reason or a further interpretation as to why “all objects in the sky would rotate around the earth in a 24-hour period.”
Newton was certainly concerned with this problem with his theory of gravity. He said to Bentley, “You sometimes speak of gravity as essential and inherent to matter. Pray do not ascribe that notion to me, for the cause of gravity is what I do not pretend to know…(H.S. Thayer, Newton’s Philosophy of Nature 1953). Can anyone then and now give a physical reason as “why mass can attract mass?”
Since we are so beholden the General Relativity, we must ask has Einstein improved things with his modification of Newton’s theory? For instance, can anyone give a physical reason “why mass can warp space?”
If they cannot, then this puts the theory of General Relativity in the same class as Ptolemy’s theory. While this seems unreasonable, there is another somewhat of an analogy between the two theories.
Ptolemaic Astronomy finally met its demise when the telescope enabled people like Galileo to see the heavens much better and to realize that the new observations were better explained by Copernicus’s theory developed 50 to 75 years earlier. Now at the present time with satellites like the Hubble telescope enabling us to get above earth’s atmosphere the new observations from the satellites are only enabling us to posit that 95 % of the universe is made of dark matter and dark energy.
This analogy breaks down a little bit because Galileo had a reasonable theory to fall back on to explain such findings as the phases of Venus that the new telescope provide. With such findings as the acceleration of the universe and higher-than-expected rotation curves we do not have a reasonable theory to fall back on, accept if you want to consider my theory (http://infraforce.googlepages/infraredlevertheory), which like Copernicus’s, has taken 30 years to develop.
20 Louise
Nov 24th, 2006 at 10:29 am
Scale R of the Universe is its age t multiplied by c. Is this not a good basis to start from?
21 rgb
Nov 24th, 2006 at 11:56 am
No. The good basis to start from is the theory of gravity that you are using. ie. What are the variables, how are they connected to the observables, the equations of motion etc. I am sorry if I am adding noise here, as this has already been said.
Can one write the scale factor R as a number times t? Sure, the number in question which you call c is simply the time average of the derivative of R with respect to t, BTW, it might be useful to state that this number is changing with time in the standard theory. However, in standard theory it has nothing to do with the speed of light. In order to make such an identification (actually any identification of physical significance), you will have to tell us about a theory of gravity, answering at least the questions raised in comments above.
22 Kea
Nov 24th, 2006 at 1:26 pm
Will a derivation of the lepton masses do for a start?
23 Louise
Nov 25th, 2006 at 1:31 am
Actually rgb, Rdot = (2/3)c = (2/3t)ct, therefore:
(Rdot/R) = (2/3t), as we would expect for k = 0.
A solution to the standard field equation of gravitation has been reposted here. You can state objections to the first page first. Your comments are still very entertaining.
24 Clifford
Nov 25th, 2006 at 6:53 am
“Your comments are still very entertaining.”
Nice.
-cvj
25 rgb
Nov 25th, 2006 at 1:32 pm
Dear Louise, That is good since you have finally written down the GR field equations in your repost. In GR the speed of light in vacuum is a constant. This is not inconsistent with most of what you say, except that what you call c(t) has nothing to do with the speed of light. Indeed you did not give any reason why it should be so.
Also, the Friedmann equations you write down follow from the GR field equations if you assume the universe to be homogenuous and isotropic, otherwise you will have more/different equations (g varies with time). Finally your solution (r ~ ^{2/3}) is a different way of writing the solution for matter-dominated universes that have been studied for a while, and do not satisfy the data for low redshifts. I hope you will stop calling this a theory where the speed of light changes.
26 stevem
Nov 26th, 2006 at 7:47 pm
You ignore the fact that the Einstein field equations, your first equation, must satisfy the Bianchi identities, but they rule out any functional dependance like c(t). When you say ” we have the constant k=8 pi G” you forget the gravitational coupling is actually given by k=8pi Gc^{-4}, and the source term for the Einstein equations on the rhs is then actually 8 pi G c^{-4} T_{uv}(x). Usually we use units with c=1 but for your purposes it has to be restored. If c has any functional dependance such as c(x) or c(t) the Bianchi identies are NOT satisfied.The Bianchi identities are fundamental to GenRel and are:
\grad^{u} (R_{uv}(x)-1/2 g_{uv}(x)R(x)) = 0
where \grad^{u} is the covariant derivative, and this is always true for the Einstein tensor. The rhs of the field equations must then satisfy:
8 pi G c^{-4}\grad^{u} T_{uv}(x)=0 ,
which is tantamount to covariant conservation of energy. This is only possible if c is constant and G is constant. If you assume c has a general functional dependance c(x), with c(t) arising as a special case, then the Bianchi identities can never be satisfied, but in GenRel the Einstein tensor on the lhs must always satisfy them . When c=c(x)
\grad^{u}(R_{uv}(x)-1/2 g_{uv}(x)R(x))= 8 pi G (-4(c(x))^{-5}\grad^{u}c(x))T_{uv}(x)
+ 8 \pi G(c(x))^{-4}\grad^{u}T_{uv}(x) = 0
This can be only be satisfied for \grad^{u}c(x)=0 or c=(x)=c=constant along with the usual \grad^{u}T_{uv}(x)=0. This is a powerful constraint in GenRel and the only term Einstein could later add to his equations was a cosmological constant.
A constraint equation or energy conservation arising from the Bianchi identities also accompanies the usual Friedman equations of cosmology. A varying speed of light c(x) or c(t) is therefore NOT possible within pure GenRel and is incompatible with the Einstein equations–the Bianchi identities ensure no such solution exists. A VSL theory requires an alternative to GR, something like a scalar-tensor type theory where c(x) or c(t) is promoted to the status of a “field” with its own source term, right from the beginning.
27 Louise
Nov 28th, 2006 at 1:14 pm
Stevem, thank you for a fascinating and well-informed question. Enquiries like this are good practice. Obviously you know much about Relativity. One reason GR works so well is that it reduces to the earlier Newton theory.
The speed of light is not a factor in your first Bianchi identity:
\grad^{u} (R_{uv}(x) - 1/2 g_{uv}(x) R(x)) = 0
The c^2 and c^4 factors result from the convenience h = c = 1. When GR is normalised to the reality of Friedmann equations, as is done in the post, you get \kappa = 8 \pi G. Your second equation then becomes:
8 \pi G \grad{u} T_{uv}(x) = 0
Since c is not a factor, varying c does not violate Bianchi. Some books use a c^4 or a c^2 factor, but to do so will not balance the units. (Try it.) The Bianchi identity does place a constraint on varying-G theories. Thanks once more for your curiosity. This is a most interesting thread!
28 Louise
Nov 28th, 2006 at 1:18 pm
Whoops, that post is here. Wouldn’t it be nice if we could put proper equations on blogger?
29 anon
Nov 28th, 2006 at 6:40 pm
Um, the Einstein equation — relation of curvature to source — is
(R_{ab} - (1/2) g_{ab} R) = 8 pi (G/c^4) T_{ab}
You CANNOT get rid of the c^4 factor here; it only disappears in units in which c = 1. This is trivial to proove: In, for example, an orthonormal basis, components of the stress-energy tensor have the units of energy density. The curvature quantities on the right-hand side have the units length^(-2). The factor G/c^4 is exactly what is needed to convert energy density to inverse length squared. This is also trivial to see: Energy is mass times velocity squared; G/c^2 times mass is a length; G/c^4 times energy is a length; G/c^4 times energy density is inverse length squared. Your statement that you don’t need the c^4 factor is wrong.
It is true that the Bianchi identity merely tells us
Grad^a (R_{ab} - (1/2) g_{ab} R) = 0.
However, we must also have conservation of source:
Grad^a T_{ab} = 0
The requirement that the Bianchi identity be equivalent to this source conservation tells us that the quantity 8 pi G/c^4 must be constant. If it is NOT constant, then the Einstein equation is not consistent, and you are not doing general relativity.
That’s not necessarily a bad thing, but you need to figure out what the theory is.
30 stevem
Nov 28th, 2006 at 9:23 pm
Anon, yes I agree. Louise, clearly we have not convinced you that c is constant–yet:). A varying speed of light (VSL) theory will probably violate energy conservation anyway, but VSL theories in the literature are NOT–and cannot be–based on classical GenRel for this reason. For a gravitational theory in which G or c varies you cannot start from the usual Einstein field equations themselves but must begin with the fundamental action. The Einstein field equations are of course derived from the Einstein-Hilbert action (restoring c and ignoring the \sqrt{det[g(x)]} term):
S=(c^{4}/16 pi G)\INT d^{4}x[R(x)+16pi G c^{-4}L_{m)]
where L _{m}(x) is a matter source Lagrangian and R(x) is the curvature scalar. Varying the action such that \delta S=0 gives the usual field equations, which is a standard textbook calculation. But if you are considering c or G as no longer being constants but fields c(x) or G(x) then they have to go under the action integral and are part of the Lagrangian BEFORE you derive the field equations. When c=const and G=G(x) you basically get a Brans-Dicke scalar-tensor theory. For c(x) and G constant, you will not get the usual pure Einstein equations but a different theory where \Grad^{u}c(x) is not zero. The required action is then some sort of scalar-tensor type theory analogous to the Brans-Dicke theory:
S=(1/16 pi G) \INT d^{4}x [c^{4}(x)R(x)
+ (16pi G)L_{m}(x)+ L_{s}]
with a “field” c^{4}(x), and you could call this field \psi(x)=c^{4}(x) say, with a source L_{s}. Again, the crucial point is that c^{4}(x) now has to be INTEGRATED over and you get a different set of field equations as a starting point for a VSL theory, but not GenRel. It is of course conjectural at this point whether such an action even makes any sense but the papers that have proposed VSL cosmologies have all started with a general action of this form, and not general relativity which strictly requires energy conservation via Bianchi and fixed c. Also, one actually could/should have violation of energy conservation in a VSL theory.
Varying speed of light theories can be found in astro-ph/0305457, gr-qc0007036, astro-ph/0305457, astro-ph/0010591, Repts. Prog Physics 66 (2003). 2025 and are mostly due to a guy called J. Magueijo. They also consider such a theory with a field\psi(x)=log(c(x)/c) or c(x)=cexp(\psi(x)). I find the VSL stuff proposed in these papers very messy and artificial though and ultimately unconvincing. But I think you really need to review the literature and study what others have done here. I really don’t believe in these theories due to the outright violation of Lorentz invariance and the fact that Lorentz invariance has been such a very powerful guiding physical principle in modern physics, and there is so much experimental evidence supporting it. I would also say the success of primordial nucleosynthesis (explaining observed ratio of hydrogen to helium) puts pretty strict constraints on how much c could vary in the early universe before it relaxed to its current value.
31 Louise
Nov 30th, 2006 at 1:24 pm
anon first: Normalising to the Friedmann equations requires a factor 8 /pi G, without any c^2 or c^4. In the Einstein formulation, T_{ab} has units of mass density, “T_{44} = \rho the only element different from 0.” G times mass density is inverse time squared.
Therefore, radial scale has units of time, which is separation from a “big bang” origin. G/c^2 times mass density is inverse (c x time) squared. Restoring your c^2 gives us radial scale R = ct, which with GM = tc^3 forms a solution to the field equations.
stevem next: The Einstein-Hilbert action then becomes:
S = \int ((k R + L_m) d^{4}x)
Where k is again a multiple of G. You needn’t introduce any c^4 factors, making your problem much easier to integrate.
Theory fits BBN data, is locally Lorentz-invariant, explains the 4.507034% proportion of baryonic matter, CMB uniformity, lack of large-scale fluctuations, and supernova redshifts. For messy and artificial, see “dark energy.”
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