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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 dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.

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.

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.

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 physically reasonable matter fields. 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 dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.

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.

AVS says in a comment:

Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system.

This would seem to violate Lemma 4.3.1 in Hawking and Ellis, which they say on p. 94 can be interpreted to give this:

...if the energy-momentum tensor [assumed to satisfy the DEC] vanishes on [a Cauchy surface], then it also vanishes on the future Cauchy development [of that surface]

(Cf. Wald, p. 219.) It would be interesting to understand how the counterexample that you sketch evades this. Maybe you could develop this example a little more clearly as a separate question. Is the spacetime you propose nonsingular?

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 dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.

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.

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 physically reasonable matter fields. 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 dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.

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.

AVS says in a comment:

Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system.

This would seem to violate Lemma 4.3.1 in Hawking and Ellis, which they say on p. 94 can be interpreted to give this:

...if the energy-momentum tensor [assumed to satisfy the DEC] vanishes on [a Cauchy surface], then it also vanishes on the future Cauchy development [of that surface]

(Cf. Wald, p. 219.) It would be interesting to understand how the counterexample that you sketch evades this. Maybe you could develop this example a little more clearly as a separate question.

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 physically reasonable matter fields. 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 dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.

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.

AVS says in a comment:

Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system.

This would seem to violate Lemma 4.3.1 in Hawking and Ellis, which they say on p. 94 can be interpreted to give this:

...if the energy-momentum tensor [assumed to satisfy the DEC] vanishes on [a Cauchy surface], then it also vanishes on the future Cauchy development [of that surface]

(Cf. Wald, p. 219.) It would be interesting to understand how the counterexample that you sketch evades this. Maybe you could develop this example a little more clearly as a separate question. Is the spacetime you propose nonsingular?