# Einstein field equation specific solution

Do Einstein's field equations admit a solution such that spacetime was empty in the past of a hypersurface of constant time say $$t =0$$, but in the future there exists a non-vanishing energy momentum-tensor $$T_{\mu\nu}$$?

If so how can we justify that?

Gravitational field carries energy and momentum yet the energy-momentum tensor of purely vacuum spacetime is zero. So it is possible to envision processess where the energy carried by the gravitational field is converted into the energy of matter, including situations where matter is created purely from the gravitational field.

A simple example could be formulated in the language of quantum field theory: a pair of gravitons produce a electron–positron pair. This is a gravitational analogue of two-photon pair production and is certainly permitted, provided that obvious constraints (for example, from energy conservation) are satisfied. Such process could (in principle) be formalized as a solution of Einstein field equations with an appropriate matter content.

Note, that characteristic wavelength of gravitational waves of such hypothetical solution of EFE's would be smaller than the Compton wavelength of electron. At these scales no classical energy condition could be imposed on the matter fields, so the lemma 4.3.1 of Hawking & Ellis is inapplicable. Another way to circumvent the lemma is to include non-minimal coupling of matter to gravitational fields. Just like scattering of light by light is absent in “minimal” Maxwellian electromagetism but appears if we use Euler–Heisenberg Lagrangian, production of electromagnetic waves from colliding gravitational wave would appear if we include in the action for Maxwell field non-minimal coupling terms like $$\sim R_{\mu\nu\lambda\rho}F^{\mu \nu}F^{\lambda\rho}$$.

Yet another possibility (and potentially simpler solutions of EFE's for the most basic of matter fields) is superradiant instability around rotating black holes: a wave impinging on a rotating black hole is amplified provided that certain conditions. The energy for this amplified wave is extracted from the black hole rotational energy contained outside its horizon. If we confine the wave so that it is amplified continuously we would obtain so-called black hole bomb: infinitesimally small perturbation would grow exponentially until it enters nonlinear final phase producing large amount of matter and gravitational radiation. A simplest such confining mechanism is provided by massive scalar field (Here is a recent paper discussing such mechanism, with references to earlier works). For such black hole bomb solution lemma 4.3.1 is not applicable because $$T_{\mu\nu}$$ is never zero at finite times, but moving back in time the energy of matter could be made arbitrarily small.

In general relativity, it doesn't really make sense to talk about a hypersurface of constant time as if that had some intrinsic physical meaning. Coordinates such as a time coordinate are arbitrary in GR. So to state your question correctly, what you really want to talk about is a spacelike surface or Cauchy surface. Let's call this surface S.

The answer to your question is no, for matter fields that satisfy the dominant energy condition (DEC). The field equations imply that the stress-energy T has zero divergence, which is a local statement of conservation of energy-momentum. We don't have a global Gauss's law in a curved spacetime, but local conservation is enough to rule out your scenario. For example, suppose that a hydrogen atom pops into existence at some point in spacetime, as in the old steady-state cosmological models. For a sufficiently small neighborhood of this point, curvature is negligible, and we can adopt Minkowski coordinates in which the atom is at rest. In these coordinates, $$\partial_\kappa T^{\kappa t}=\partial T^{tt}/\partial t\ne 0$$.

The idea here is that although it is possible to trade gravitational energy (which is not counted in the stress-energy) for the energy of matter fields, we must always do so in such a way that to a local, free-falling observer, energy appears to be conserved. This is the equivalence principle.

What I gave above is only an argument that rules out one specific example, in which a hydrogen atom spontaneously pops into existence. The only fact about this matter field that was used in the argument was that it was possible to define a local Minkowski frame in which the matter field was at rest. Referring to an answer to a very similar question, I think this is equivalent to assuming the DEC (as a strict inequality). The DEC ensures that the flow of energy is subluminal, so that we can define such a frame. The condition can be relaxed to the normal, less strict definition of the DEC as a non-strict inequality (see Hawking and Ellis, p. 94, or Wald, p. 219).

There are independent physical reasons why such a scenario is problematic. We expect the matter fields to obey some wave equation, but if the wave and all its derivatives are zero on a Cauchy surface, then it would seem to violate causality if the wave were later to be nonzero. Also, it will be impossible to avoid a violation of Lorentz invariance, since there is no preferred frame of reference in which a newly created particle should be at rest.

• Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system. – A.V.S. Nov 21 '20 at 17:10
• As it stands this answer is wrong. That $T$ has zero covariant divergence doesn't prevent you from writing down, say, a FLRW solution with $\dot a=0$ before some time and $\dot a\ne 0$ after. You need an energy condition to rule that out. Also (as A.V.S. implicitly said) gravitational waves break Lorentz invariance and couple to other fields. – benrg Nov 21 '20 at 17:59
• @benrg: Thanks for the comment. Re the energy condition, yes, you're right; please see the updated version of my answer. Re Lorentz invariance, no that's a local symmetry and is not broken by gravitational waves in classical relativity. – user280394 Nov 21 '20 at 18:08
• @A.V.S.: Thanks for your comment. The counterexample you sketch would seem to violate Hawking and Ellis's lemma 4.3.1 (see interpretation on p. 94, and cf. Wald p. 219), since the electron and positron do satisfy the DEC. I could be misunderstanding something, but this makes me think that the counterexample you sketch doesn't exist or has some nonclassical element. If you would like to post it as a separate question, that would be interesting. – user280394 Nov 21 '20 at 18:46
• it isn't a very similar question. It is exactly the same question. – MBN Nov 23 '20 at 8:22