# Revised Field Equations for GR Explain Dark Energy and Dark Matter

#### AllenMcC.

Did anybody else hear the news?

A recent paper examines the derivation of Einstein's field equations and proposes that the original assumption of a divergence-free energy-momentum tensor may not be valid anymore due to the discovery of dark matter and dark energy. The authors derive new field equations in which $T_{\mu\nu}$ isn't divergence-free.

This step has several striking consequences. For example, the new Schwarzschild solution contains extra terms that describe the effect of dark matter, and even the repulsive effect of dark energy. The result fits observation, as it correctly describes Rubin's law for galactic rotation curves, and a rough estimation done in the paper predicts the real gravitational mass to be about 4 times the amount of visible mass. Also, the distances at which dark matter and dark energy effects become significant appear to fit observation as well. Furthermore, according to the revised theory, the nature of dark matter and energy has to do with a new quantity in the field equations, a scalar potential energy density, that is shown to reflect the non-uniform distribution of matter in the universe.

Here's the paper: http://arxiv.org/abs/1206.5078 (Conclusions on page 31)

I've read this paper several times by now and am quite excited about these new ideas. I have long suspected there might be something slightly but fundamentally wrong with general relativity. Now dark energy and dark matter naturally pop out of the new equations and are part of the normal gravitational field of matter in the universe.

Related Special and General Relativity News on Phys.org

#### bcrowell

Staff Emeritus
Gold Member
The paper starts by asserting that there are a bunch of terrible problems with GR. The authors don't give any evidence for these claims, and most of them don't seem at all plausible to me. The numbers in brackets below are mine.

[1] the Einstein field equations failed to explain the dark matter and dark energy
[2] the equations are inconsistent with the accelerating expansion of the galaxies
[3] we can prove that there is no solution for the Einstein field equations for the spherically symmetric case with cosmic microwave background (CMB)
[4] the discontinuities of T give rise to the same discontinuities of the curvature and the discontinuities of space-time. This is certainly an inconsistency which needs to be resolved.
[5] it has been observed that the universe is highly non-homogeneous as indicated by e.g. the ”Great Walls”, filaments and voids. However, the Einstein equations do not appear to offer a good explanation of this inhomogeneity.
[1] This is silly. GR doesn't try to explain the properties of the matter fields. It takes their properties as inputs.
[2] This is false. Just take $\Lambda\ne 0$.
[3] Where have they published this proof?
[4] A discontinuity in curvature is not the same as a "discontinuity of space-time," whatever that means.
[5] There are open problems in the application of GR to formation of structure. That doesn't mean that GR is defective or inconsistent with observation.

The paper only has three references to anything but the authors' own work, and these appear to be textbooks.

So we've got two hallmarks of a kook paper: (a) overly broad and undocumented claims that something is terribly wrong with the current state of physics, and (b) a lack of connection to the state of the art of research in the field.

There are lots of scalar-tensor theories out there. Apparently they've constructed a new one. It would be nice if they would just present it without the ridiculous hyperventilation.

One specific that that caught my eye and seemed bogus was this:

Also, this scalar energy density is conserved with mean zero:
$\int_MΦ dM = 0$
Scanning through the paper, they don't seem to prove this anywhere, just assert it, and they don't seem to say what they mean by the notation. This doesn't make a lot of sense as a physical law, regardless of whether M refers to all of spacetime or to a spacelike 3-surface. In a spatially infinite universe, you can't test observationally what is the result of an integral that extends over all of space, or all of spacetime.

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#### AllenMcC.

I'm not sure why you have a problem with their work. All they are saying is that the energy-momentum tensor need not be divergence free and they explore the consequences of that possibility. It gives rise to slightly different field equations and thus to different solutions, and these solutions are promising.
This is what theoretical physics is all about.

#### Ben Niehoff

Gold Member
All this paper has done is used a bunch of extraneous mathematical gobbledy-gook (some of which may be true, I don't know) to say that dark matter and dark energy can be modeled by adding certain kinds of matter fields. Just define

$$8 \pi \tilde{T}_{\mu\nu} \equiv 8 \pi T_{\mu\nu} + \nabla_\mu \nabla_\nu \varphi$$
and you've accomplished almost as much as this paper. They do go on to find a spherically-symmetric solution with this source (which I haven't read closely enough to tell if it is exact and correct).

They're not really saying that local energy-momentum conservation is violated; they're saying that there is an additional matter field that takes a certain form. I don't know if the exact thing they've put in has been used before, but it is well-known that adding a scalar field to the Lagrangian with a certain potential can lead to solutions that asymptotically behave as though they have a cosmological constant. This happens a lot in supergravity theories.

Edit: I do agree with Ben that this paper shows a certain lack of understanding of the current state of research in GR and "beyond GR".

#### Mentz114

Gold Member
I have agree with the criticisms above. I don't see how the oft-repeated claim that DαDβ∅ represents the inhomogeneity of the matter can be justified. It's a point measure isn't it ? To describe inhomogeneity one needs a distribution function of some kind.

This paper is in the gen-phys category in arXiv. I doubt if it would survive refereeing in a good journal.

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#### bcrowell

Staff Emeritus
Gold Member
Scalar-tensor theories of gravity have a long history, going back at least to Brans-Dicke gravity in the 60's. The standard modern reference seems to be Fujii and Maeda, The Scalar-Tensor Theory of Gravitation. This may also be useful: Brans, "The roots of scalar-tensor theory: an approximate history," http://arxiv.org/abs/gr-qc/0506063

#### Mentz114

Gold Member
...I don't see how the oft-repeated claim that DαDβ∅ represents the inhomogeneity of the matter can be justified. It's a point measure isn't it ?
On reflection I want to withdraw that remark. The distribution of matter and energy has a big effect on the magnitude of the potential, at least in Newtonian terms. Attributing the change in the potentials entirely to inhomogeneity is a bit speculative.

#### Messenger

[2] This is false. Just take $\Lambda\ne 0$.
Not disputing any other comments, but a non-zero cosmological constant has been discussed since well before the discovery of accelerating expansion (i.e. Zel'dovich). I was not aware that even one of these discussions ever included in the possibility of a positive acceleration. Would be very interested in reading any reference you have for this.

#### PAllen

Not disputing any other comments, but a non-zero cosmological constant has been discussed since well before the discovery of accelerating expansion (i.e. Zel'dovich). I was not aware that even one of these discussions ever included in the possibility of a positive acceleration. Would be very interested in reading any reference you have for this.
Not sure what you mean. The best fit lambda-cdm model for 'accelerated expansion' has a small positive cosmological constant. This produces accelerated expansion. Lambda=0 produces asymptotically flat expansion.

#### Messenger

Not sure what you mean. The best fit lambda-cdm model for 'accelerated expansion' has a small positive cosmological constant. This produces accelerated expansion. Lambda=0 produces asymptotically flat expansion.
I certainly understand that a linear term is the best fit for the data but to say it "produces" an accelerated expansion is a hypothesis after the fact of the discovery. The old cosmological constant problem had nothing to do with an accelerating expansion. The existence of a non-zero Lambda seems to raise many more questions than it answers so I do not follow bcrowell's logic concerning [2]. His statement would seem to verify the original point, not refute it.

#### PAllen

I certainly understand that a linear term is the best fit for the data but to say it "produces" an accelerated expansion is a hypothesis after the fact of the discovery. The old cosmological constant problem had nothing to do with an accelerating expansion. The existence of a non-zero Lambda seems to raise many more questions than it answers so I do not follow bcrowell's logic concerning [2]. His statement would seem to verify the original point, not refute it.
Only in the minds of some. Search for threads on this in the cosmology forum. The original history is:

1) The field equations naturally contain this constant as a constant of integration. Einstein noticed that setting just the right value for it produced a static universe, which he philosophically preferred at the time (and no one realized, at the time, that such a static universe was an unstable equilibrium).

2) When expansion was discovered, setting the constant to zero was an adequate fit for the data, and seemed simplest. There was no deep reason for believing zero was the answer, and the data did not constrain the value to be exactly zero.

3) When accelerated expansion was discovered, a small positive constant fits the data so far.

Note that if the acceleration is of the wrong form, a constant lambda would not fit, and GR would then be broken because there is no freedom for lambda to be non constant in the GR field equations. Thus, GR is making a falsifiable predition here - the expansion must be consistent with lambda.

The only 'problem' with lambda (IMO) is the general desire to 'explain' constants in theories especially constants with values that seem funny to us (almost an integer; almost zero). Claims that QFT suggests lambda should be huge rely on a false argument (or additional assumption) about what lambda is 'supposed' to represent. It is that unwarranted supposition which creates a problem.

#### bcrowell

Staff Emeritus
Gold Member
The only 'problem' with lambda (IMO) is the general desire to 'explain' constants in theories especially constants with values that seem funny to us (almost an integer; almost zero). Claims that QFT suggests lambda should be huge rely on a false argument (or additional assumption) about what lambda is 'supposed' to represent. It is that unwarranted supposition which creates a problem.
But what they say is this: "the equations are inconsistent with the accelerating expansion of the galaxies." This just seems obviously wrong. I don't see any way to interpret it that makes it true. FLRW solutions with nonzero cosmological constant (a) are solutions to the field equations, and (b) have accelerating expansion.

The paper falls into a pattern that is very common with kook papers. They start by asserting one or more things that no non-kook expert believes are true. They talk as though these assertions had already been established, but they don't give any references to show where they think they've been established. Their assertions are also not phrased in exact language, and the lack of references makes it impossible to infer whether the thing being asserted really is what it appears to be.

The lack of references in these cases is also generally related to the fact that the kooks are out of touch with the current state of the art in the field in which they claim to have made Amazing! New! Discoveries!

#### Messenger

Only in the minds of some. Search for threads on this in the cosmology forum. The original history is:

1) The field equations naturally contain this constant as a constant of integration. Einstein noticed that setting just the right value for it produced a static universe, which he philosophically preferred at the time (and no one realized, at the time, that such a static universe was an unstable equilibrium).

2) When expansion was discovered, setting the constant to zero was an adequate fit for the data, and seemed simplest. There was no deep reason for believing zero was the answer, and the data did not constrain the value to be exactly zero.

3) When accelerated expansion was discovered, a small positive constant fits the data so far.

Note that if the acceleration is of the wrong form, a constant lambda would not fit, and GR would then be broken because there is no freedom for lambda to be non constant in the GR field equations. Thus, GR is making a falsifiable predition here - the expansion must be consistent with lambda.

The only 'problem' with lambda (IMO) is the general desire to 'explain' constants in theories especially constants with values that seem funny to us (almost an integer; almost zero). Claims that QFT suggests lambda should be huge rely on a false argument (or additional assumption) about what lambda is 'supposed' to represent. It is that unwarranted supposition which creates a problem.
I do understand the history and your opinion (and perhaps bcrowell is of the same), but I think it should be pointed out that your conclusion (and possibly the consensus on PF) is not the mainstream one. Do you not think the best position for PF mentors to take is that dark energy is not understood, and so any positions that are unpublished should be off topic? It would seem that your personal opinion might steer someone away from the mainstream consensus of an open question.

#### PAllen

I do understand the history and your opinion (and perhaps bcrowell is of the same), but I think it should be pointed out that your conclusion (and possibly the consensus on PF) is not the mainstream one. Do you not think the best position for PF mentors to take is that dark energy is not understood, and so any positions that are unpublished should be off topic? It would seem that your personal opinion might steer someone away from the mainstream consensus of an open question.
I am not saying accelerated expansion (the observation, rather than the ludicrous name 'dark energy') is not an open problem. Any day the data could fail to be explicable by GR. Further, there is value in reducing the number of free parameters in theories.

What I interpret Bcrowell as criticizing was the statement from the paper:

"[1] the Einstein field equations failed to explain the dark matter and dark energy
[2] the equations are inconsistent with the accelerating expansion of the galaxies"

because [1] is silly and [2] is false. These critiques are true irrespective of whether you desire a deeper explanation of the accelerated expansion. Note, [1] is silly because there is already reasonably clear evidence that dark matter is clumpy, and its center may not coincide with the COM of visible matter in a galaxy. Thus it is simply matter of an unknown type, and is thus, by definition, outside the purview of GR (which deals only in how matter couples to gravity).

#### Messenger

I am not saying accelerated expansion (the observation, rather than the ludicrous name 'dark energy') is not an open problem. Any day the data could fail to be explicable by GR. Further, there is value in reducing the number of free parameters in theories.

What I interpret Bcrowell as criticizing was the statement from the paper:

"[1] the Einstein field equations failed to explain the dark matter and dark energy
[2] the equations are inconsistent with the accelerating expansion of the galaxies"

because [1] is silly and [2] is false. These critiques are true irrespective of whether you desire a deeper explanation of the accelerated expansion. Note, [1] is silly because there is already reasonably clear evidence that dark matter is clumpy, and its center may not coincide with the COM of visible matter in a galaxy. Thus it is simply matter of an unknown type, and is thus, by definition, outside the purview of GR (which deals only in how matter couples to gravity).
I think I see your main point, in that the equations need a linear term for the acceleration and there is one available irrespective of any physical theory.
I see that the Arxiv doesn't note it has been peer reviewed...how is it gaining press coverage?

#### PAllen

But what they say is this: "the equations are inconsistent with the accelerating expansion of the galaxies." This just seems obviously wrong. I don't see any way to interpret it that makes it true. FLRW solutions with nonzero cosmological constant (a) are solutions to the field equations, and (b) have accelerating expansion.

The paper falls into a pattern that is very common with kook papers. They start by asserting one or more things that no non-kook expert believes are true. They talk as though these assertions had already been established, but they don't give any references to show where they think they've been established. Their assertions are also not phrased in exact language, and the lack of references makes it impossible to infer whether the thing being asserted really is what it appears to be.

The lack of references in these cases is also generally related to the fact that the kooks are out of touch with the current state of the art in the field in which they claim to have made Amazing! New! Discoveries!
I completely agree, as implied in the earlier bullets you didn't quote. The part you quoted was aimed that segment of mainstream physicists that (IMO) exaggerate or mis-characterize the issues around 'dark energy'.

#### bcrowell

Staff Emeritus
Gold Member
Not disputing any other comments, but a non-zero cosmological constant has been discussed since well before the discovery of accelerating expansion (i.e. Zel'dovich). I was not aware that even one of these discussions ever included in the possibility of a positive acceleration. Would be very interested in reading any reference you have for this.
de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis", Proc. Kon. Ned. Acad. Wet. 19: 1217–1225
de Sitter, W. (1917), "On the curvature of space", Proc. Kon. Ned. Acad. Wet. 20: 229–243

These are the references given in the WP article on de Sitter space: http://en.wikipedia.org/wiki/De_sitter_space

I see that the Arxiv doesn't note it has been peer reviewed...how is it gaining press coverage?
Is it being written about in the popular press? Where?

I've only heard about it via PF.

#### Messenger

de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis", Proc. Kon. Ned. Acad. Wet. 19: 1217–1225
de Sitter, W. (1917), "On the curvature of space", Proc. Kon. Ned. Acad. Wet. 20: 229–243

These are the references given in the WP article on de Sitter space: http://en.wikipedia.org/wiki/De_sitter_space

Is it being written about in the popular press? Where?

I've only heard about it via PF.
Looking a bit closer, guess popular press would be bit of a stretch. Seems the source of the article, besides their university, is http://www.tgdaily.com/general-sciences-features/65997-new-theory-unifies-dark-matter-and-dark-energy

I see where deSitter talks about redshift etc, but I don't see where he mentions that Lambda would lead to a positive acceleration.
http://www.dwc.knaw.nl/DL/publications/PU00012216.pdf
http://www.dwc.knaw.nl/DL/publications/PU00012455.pdf

#### PAllen

Looking a bit closer, guess popular press would be bit of a stretch. Seems the source of the article, besides their university, is http://www.tgdaily.com/general-sciences-features/65997-new-theory-unifies-dark-matter-and-dark-energy

I see where deSitter talks about redshift etc, but I don't see where he mentions that Lambda would lead to a positive acceleration.
http://www.dwc.knaw.nl/DL/publications/PU00012216.pdf
http://www.dwc.knaw.nl/DL/publications/PU00012455.pdf
What, exactly, are you looking for? The equations of GR fist published included the cosmological constant. The lambda-CDM model uses these equations (along with particular assumptions about the stress energy tensor). Nothing about the original equation had to be changed, except choosing a small positive value for lambda (cosmological constant).

Does this help?

http://relativity.livingreviews.org/Articles/lrr-2001-1/fulltext.html [Broken]

While I don't agree with all philosophical slants in the above, the physics and math are the accepted standard, and the agreement with observation is covered, showing the blatant falsity of a main claim of the paper in the OP. (Note, if this paper had argued that their new field equations were better in some demonstrable way, that would be different; but the grandiose claim about incapability of GR to account for accelerated expansion is crazy).

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#### ApplePion

The paper only has three references to anything but the authors' own work, and these appear to be textbooks.So we've got two hallmarks of a kook paper: (a) overly broad and undocumented claims that something is terribly wrong with the current state of physics, and (b) a lack of connection to the state of the art of research in the field.
Einstein's most important Special Relativity paper had only one reference to a paper other than his own, and in general Einstein did not often reference papers other than his own.

Do you realize you are implying that Einstein was a kook?

Something that I have noticed is that when someone refers to someone else as a kook or crackpot, the accuser usually ends up not looking good.

P.S. I also note that you said one looks like a kook if one is not aware of the state of the art research in the field. Einstein claimed he was not aware of the Michaelson Morley experiment while developing Special Relativity.

#### PAllen

Einstein's most important Special Relativity paper had only one reference to a paper other than his own, and in general Einstein did not often reference papers other than his own.

Do you realize you are implying that Einstein was a kook?

Something that I have noticed is that when someone refers to someone else as a kook or crackpot, the accuser usually ends up not looking good.

P.S. I also note that you said one looks like a kook if one is not aware of the state of the art research in the field. Einstein claimed he was not aware of the Michaelson Morley experiment while developing Special Relativity.
Well, Einstein was an exception to most rules (and has been justly criticized for lack of references). Easily 99% of people who have tried to emulate the 'Einstein method' have been kooks, as proved by history. Further, one could even argue that 'late Einstein' was a kook - he ceased being an exception.

#### ApplePion

Well, Einstein was an exception to most rules (and has been justly criticized for lack of references).
.

What rules are Einstein an exception to? Did Newton cite a lot of sources? Did Maxwell? Did Schrodinger?

I suspect that the actual rule is that people who have important things to say concentrate on the physics rather than on making their work appear "scholarly".

I will also note that the works of those people did not pass peer review, peer review being another thing highly respected by "scholarly" people.

Easily 99% of people who have tried to emulate the 'Einstein method' have been kooks, as proved by history.
That's factually correct, but it is the converse situation that matters. While most people who behave in a non-standard way are likely just kooks, most people who do important things would seem to be kooks to the Physics Forums enthusiasts.

(and has been justly criticized for lack of references).
.

I think the actual physics was what was important.

#### Nugatory

Mentor
Einstein's most important Special Relativity paper had only one reference to a paper other than his own, and in general Einstein did not often reference papers other than his own....
Einstein claimed he was not aware of the Michaelson Morley experiment while developing Special Relativity.
Several reasons why I find this comparison unconvincing:
1) Citations are for bringing to the readers' attention supporting work that they might otherwise be unaware of, for supporting unusual claims of fact, and for demonstrating reasonable familiarity with the state of the art. If, as I expect, you are referring to the 1905 "On the electrodynamics of moving bodies" Einstein started from a base of generally accepted science that was already completely familiar to his intended audience. It's not at all obvious what if anything in that paper needed a formal citation, nor how such citations would have improved the exposition. This is most assuredly not the case for http://arxiv.org/abs/1206.5078.

2) Einstein did not claim to have uncovered and resolved a basic flaw in contemporary physics that had somehow been overlooked for decades. Instead he advanced a solution to a well-recognized unsolved problem, namely how to reconcile the principle of relativity and the laws of electricity and magnetism.

3) Einstein may not have known about the results of the M-M experiment, but he was nonetheless deeply familiar with the state of theoretical physics at the time. It is simply historically inaccurate to point to him as an example of a lone genius laboring away in splendid isolation to come up with new brilliancies that defied the establishment.

#### ApplePion

Einstein may not have known about the results of the M-M experiment, but he was nonetheless deeply familiar with the state of theoretical physics at the time. .
What caused you to conclude that?

It is simply historically inaccurate to point to him as an example of a lone genius laboring away in splendid isolation to come up with new brilliancies that defied the establishment.
He was a clerk in a patent office who could not get a job in physics.

#### PAllen

What caused you to conclude that?

He was a clerk in a patent office who could not get a job in physics.
He was also a reviewer for Annalen der Physik, and was known to have read recent work by Poincare and Lorentz. I have read a good number histories of the period and Einstein biographies; it is now thoroughly established that Einstein was familiar with current developments in physics.

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