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Every time a major news story about dark matter and/or dark energy comes out, like the recent release of data from the Dark Energy Survey, I also come across articles and links about alternative theories to the 'dark' stuff, like MOND (Modified Newtonian Dynamics).....

When I typed Modified General Relativity into Google, this paper came up twice, near the top:

Subjects: General Relativity and Quantum Cosmology (gr-qc) Journal reference: Gen Relativ Gravit (2019) 51: 53 DOI: 10.1007/s10714-019-2537-y, https://arxiv.org/abs/1904.10803 Author: Gary Nash

Here's the gist:

An orthogonal decomposition of symmetric tensors can be constructed in terms of the Lie derivative along $X$ of the metric and a product of the unit vectors; and a linear sum of divergenceless symmetric tensors. A modified Einstein equation of general relativity is then obtained by using the principle of least action, the decomposition and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor $\Phi_{\alpha\beta}$ which describes the energy-momentum of the gravitational field. It completes Einstein's equation and addresses the energy localization problem.

Not that understand everything in this paper, but I've never heard of the 'energy localization problem'.

What is the 'energy localization problem' with general relativity?

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    $\begingroup$ In GR, gravitational energy is a non-local quantity, you can't express gravitational energy as a local density (i.e. gravitational energy is zero everywhere but shows up in total energy contribution). So you can't have a local $T_{\mu\nu}$ for gravity. See Penrose's Quasi Local Momentum and Angular Momentum Twistor construction for more details $\endgroup$
    – paul230_x
    Commented Jun 11, 2021 at 19:06

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Here is one way to think about this problem.

Consider an apple far from the Earth (like Moon is) moving in Earth's gravity field towards the Earth. In General Relativity all frames are acceptable for expressing laws of physics. This means the Earth-fixed inertial frame where the apples accelerates is fine, and the free-falling frame where the apple is at rest is fine to.

In the first frame, gravitational field is present (non-zero) everywhere, and thus field-proportional gravitational energy is present everywhere around the Earth, including vicinity of the apple. Increasing kinetic energy of the apple can be related to this gravity energy decreasing.

But in the second frame there is no gravity field near the apple (in its vicinity, the gravity field has been transformed out), and the apple is not gaining kinetic energy. Instead, the Earth is gaining kinetic energy. So in this frame, we can't say there is field-proportional energy near the apple, but there is one near the Earth.

Some people are bothered by this immense difference in where the energy is present, depending on our choice of frame of reference. Mathematically, it turns out that one cannot define gravitational energy-momentum tensor, only "pseudotensor" which takes into account this dependence of energy location on the reference frame.

This really stems from the desideratum that all frames be equally acceptable, something that was never accepted in pre-relativistic mechanics. But the tenet of GR is that all frames are equally valid, and then the "relativity of energy location" is the necessary implication.

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  • $\begingroup$ Do you know if there's any work trying to understand this energy location issue in GR in the context of the uncertainty principle in quantum theory? $\endgroup$ Commented Jun 7, 2021 at 14:53
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    $\begingroup$ @DaddyKropotkin I don't. I do not see how these things could be related. $\endgroup$ Commented Jun 7, 2021 at 15:07
  • $\begingroup$ Thanks. Just a naive question :) $\endgroup$ Commented Jun 7, 2021 at 15:08
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One manifestation of the 'energy localization problem' is the difficulty to define an stress-energy-momentum tensor for the gravitational field, cf. e.g. the Landau-Lifshitz pseudo-tensor. This is discussed in more details in e.g. this related Phys.SE post.

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It is well known in General Relativity (GR) that a tensor describing the energy-momentum of the gravitational field does not exist; GR is not complete. Einstein realized (with Grossmann) in 1913 that gravity gravitates and there must be a tensor that describes that phenomenon. However, he could not produce that entity and introduced a pseudo-tensor instead in 1915. The futility and infinity of pseudo-tensors and other approaches to describe local gravitational energy led to the general statement first mentioned above. However, gravity still gravitates and we must be able to produce a tensor to describe that important fact. That was accomplished by extending a classic theorem of differential geometry in a Riemannian spacetime, the Berger-Ebin theorem, to a Lorentzian spacetime called the Orthogonal Decomposition Theorem (ODT) in the paper Modified General Relativity, and updated in the recent paper Modified General Relativity and quantum theory in curved spacetime published in Int J Mod Phys A. The ODT states that an arbitrary symmetric tensor in a Lorentzian spacetime $w_{ab}$ can be decomposed into a linear sum of divergenceless tensors $v_{ab}$ plus another tensor $\Phi_{ab}$, which belongs to a subspace orthogonal to that of $v_{ab}$: $w_{ab}=v_{ab}+\Phi_{ab}$. By returning to Einstein's original postulate of a total e-m tensor $T_{ab}$, which must be divergenceless and locally conserved, the matter e-m tensor $\tilde{T}_{ab}$ is no longer divergenceless. A constant multiple of it can be set equal to an arbitrary symmetric tensor and decomposed by the ODT to give: $k\tilde{T}_{ab}=v_{ab}+\Phi_{ab}$. Lovelock's theorem demands that in 4-D spacetime, the only divergenceless symmetric tensors consisting of a concomitant of the metric and its first two derivatives are the metric and the Einstein tensor. Thus, we arrive at Einstein's equation $k\tilde{T}_{ab}=\Lambda g_{ab}+G_{ab}+\Phi_{ab}$ with a new tensor that describes the energy-momentum of the gravitational field. Why can I say that? $\Phi_{ab}$ is constructed from the Lie derivative of both the metric and a product of unit line element covectors. Lie derivatives have the unique property that a tensor constructed from them has the same value when the Lie derivative is expressed with covariant or partial derivatives. Thus, when the connection coefficients (Gamma's) vanish under free-fall, $\Phi_{ab}$ is invariant. Gravitational energy can be localized. It is easy to prove that $\Phi_{ab}$ vanishes if and only if X, the line element vector along which the Lie derivative is calculated, is a Killing vector. Of course, in general there are no Killing vectors unless a symmetry is involved. Line element vectors are virtually never used in the literature except for a few theorems on the evolution of time. It is imperative to understand that a Lorentzian spacetime does not exist without a line element field (X,-X): a non-compact paracompact manifold admits a Lorentzian metric $g_{ab}$ if and only if it admits a line element field (Hawking and Ellis 1973).

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For dark matter, it's difficult to find out where its mass-energy-momentum is exactly present. The effects are there but to find out from where this comes exactly is difficult as it can't be seen directly. Observations on the Bullet Cluster seem to give some indication, but the exact nature of the dark matter (and hence its location) is difficult to find out.

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