# A Conceptual Problem With the Field Equations of General Relativity

I have two questions:

1. Suppose that we have an amount of energy in the form of a perfect fluid in the right hand side of Einstein field equations (energy momentum tensor), this will lead to a gravitational field, the gravitational field itself has energy, and this self energy also produces gravitational field ... in other words the gravity beget gravity !! ... due to this scenario we will have an infinite gravitational field!! ... what's wrong here?! is my reasoning wrong or is it the field equations that are not correct?

2. Has this non-linear behavior of gravity(or maybe graviton!) anything to do with the fact that when we try to quantize gravity we encounter infinities?

## 3 Answers

(1) Well, that's the basic intuition one should have when expanding out the metric as fluctuating about the flat Minkowski metric, i.e., writing $$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$$ where $h_{\mu\nu}$ contains all the information about the curvature, $\eta_{\mu\nu}$ the usual Minkowski metric. What usually happens in most classes is we approximate the inverse metric as $$g^{\mu\nu}\approx\eta^{\mu\nu}-h^{\mu\nu}.$$ This is technically wrong: the full answer should be an infinite series.

As we keep adding terms, the intuition should be that we are iteratively moving between "spacetime tells matter how to move" and "matter tells spacetime how to curl up".

(Edit: the first proof of this that was really given can be found freely online. S Deser, "Self-Interaction and Gauge Invariance". Gen.Rel.Grav. 1 (1970) 9-18. Eprint arXiv:gr-qc/0411023.)

(2) The usual arguments for nonrenormalizability boils down to: $G$ the coupling constant for gravity has geometric dimensions of length-squared, so power-counting tells us this results in a nonrenormalizable theory. You might be interested in:

• Assaf Shomer, "A pedagogical explanation for the non-renormalizability of gravity". Eprint arXiv:0709.3555, 10 pages.

This is the famous back-reaction problem in perturbative gravity. To avoid it, we typically only work to a few orders in a perturbative series (though the PPN people will go farther than seems sane when doing numerical work, but you can't blame them considering that radiation only shows up at 2.5 PPN). It is unclear whether perturbative methods in general relativity converge.

What is clear, however, is that you can safely have exact solutions to general relativity where you solve this back-reaction problem nonperturbatively. In particular, there is an existant proof that the classical self-energy of a charged ball is finite, due to a cancellation of the infinite electromagnetic self-energy against the infinite gravitational self-energy.

• I tried to choose both answers but i couldn't ... i vote up your beautiful answer ... Thanks. – user55867 Sep 5 '14 at 4:59
• A question: isn't this perturbative method the same method we use in some Quantum gravity theories? i think i have seen "perturbative quantum gravity" and "non perturbative quantum gravity" a couple of times , so i guess this back reaction may be related to them?!! ... besides that the same problem of infinite self-energy of electron exists in classical electrodynamics which can be solved by quantum field theory methods ... that's why i guessed that this may be related to quantum gravity ... – user55867 Sep 5 '14 at 6:42

As far as I know, the non-linear behaviour of gravity has nothing to do with getting infinities upon its quantisation (its non-renormalizability). In-fact pure Einstein gravity is 1-loop finite and Einstein gravity coupled to scalar field is 2-loop finite. In contrast, Yang-Mills theory for example (a special case of which lies at the heart of the Standard Model) is also non linear but is renormalizable (UV finite!) as was proven by t'Hooft in the 70's.

Regarding the first question one could give the following argument. Consider the Einstein field equations (EFE) :$$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G_N}{c^4} T_{\mu\nu}.$$

Notice that on the right hand side the stress energy tensor is the one sourced by matter fields. One solves the field equations for the metric $g_{\mu\nu}$ and gets the solution which we call "gravitational field". Now we would like to consider the effect of the gravitational field itself to the right hand side of the Einstein field equations. Following Landau-Lifshitz (vol. 2 section 96), the next step would be to construct the gravitational stress energy (pseudo-)tensor: $$t^{\mu\nu} = - \frac{c^4}{8\pi G_N}G^{\mu \nu} + \frac{c^4}{16\pi G_N (-g)}((-g)(g^{\mu \nu}g^{\alpha \beta} - g^{\mu \alpha}g^{\nu \beta}))_{,\alpha \beta}.$$ Notice that this quantity is NOT a tensor (see L&L vol. 2 for further details). However it should be such that when added to $T_{\mu\nu}$ it should give zero divergence (so that the Einstein tensor satisfies Bianchi identities or in other words the total energy momentum tensor (of matter$+$field) be conserved). Since $t_{\mu\nu}$, which encodes the information about the energy density of the gravitational field, is not a tensor (in particular if it vanishes at one point it may not vanish at some other point) it becomes physically meaningless to talk of whether or how much gravitational energy is there at a definite point in space-time. It is in-fact delocalised! This is consistent with the fact that locally we can "turn off" the gravitational field by a suitable choice of coordinate systems in that local neighbourhood! So if you say that EFE implies infinite gravitational energy it would contradict the equivalence principle - rendering GR self-inconsistent! In my opinion this basic argument seems to settle the issue about the seemingly recursive nature of the Einstein field equations.

• Thanks Orbifold, your answer gave good insights about nature of gravitation field energy, but i needed more direct answers including the intuitive idea of back-reaction problem ... besides that we have equivalent formulations of GR in which we can define a tensor quantity for gravitational energy-momentum tensor and yet we have the back-reaction problem ... – user55867 Sep 5 '14 at 6:39