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I don't have a clear understanding of what a dynamical solution of the Einstein equations would look like. So maybe my first question is “what is a dynamical solution of the Einstein equations?”

And why can the linearized Einstein equations have a dynamical solution, while the complete ones can't?

My question arises because of the paper Some Mathematical and Physical Errors of Wald on General Relativity by C. Y. Lo.

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    $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Mar 29 at 7:11
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I think a reasonable answer to your first question would be that a dynamic solution to the Einstein field equations is one which is not either stationary or, more strongly, static. I need to define those terms!

A stationary solution is one which admits a Killing vector field which is asymptotically timelike.

A static solution is a stationary solution in which the Killing vector field is in addition everywhere orthogonal to a family of spacelike hypersurfaces.

Both of these definitions really should have caveats saying 'everywhere where the solution is nonsingular'.

Both of these rely on the notion of a Killing vector field. A Killing vector field is a way of expressing a symmetry of a manifold. In particular the notion it expresses is that, if you drag the metric tensor along a Killing vector field, then it is unchanged. Formally you can express this by saying that $\mathcal{L}_\vec{v}\mathbf{g} = 0$, where $\mathbf{g}$ is the metric tensor, and $\mathcal{L}_\vec{v}$ is the Lie derivative along $\vec{v}$, which is the Killing vector field. The nice thing about Lie derivatives is that you don't need a metric to define them.

So if a solutions admits a Killing vector field, this means that it has an isometry: as you move along the field nothing changes. If it has a Killing vector field which is timelike then it has an isometry in time: things are the same at different times. That's a pretty good way of capturing what it means to be 'not dynamic', I think.

As examples: the Schwarzschild solution is static, while the Kerr solution is stationary.

So the question then is: are there solutions to the EFEs which are neither static nor stationary? Yes, there are: for instance the FLRW metric which is a pretty important solution.


As a note: I have not read the paper you refer to in any detail, and I don't know anything about the author. But it looks confused to me, at best. Perhaps I am wrong about this.

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Time plays a fundamentally different role in GR compared to any other field theory like eg electrodynamics. In field theories, time is part of the Minkowski space on which the theory is defined. In GR, time itself is a dynamical variable, encoded in the metric. In the absence of an externally defined time, the solutions of the theory can’t evolve with time. Instead of a single notion of time valid for any solution, in GR each solution of the Einstein equations has its own flow of time.

This feature of GR is known in the Quantum Gravity literature as “problem of time” (though it is also there in the classical theory, and it is not really a problem — more like a feature).

The situation is different for linearized gravity. Here we expand around a flat space solution, which means that we have a notion of external time coming from that flat space solution. Linearized gravity is only an approximation. When the gravity field is strong, the flat space time loses its physical meaning.

UPD: to address issues raised by Bruce Lee.

There exist static solutions to Einstein's equations, which don't change with time and are irrotational. Any other solution which is non-static is dynamical (more loosely, you can also define a broader class of stationary spacetimes, which basically means that there exist asymptotic timelike Killing vectors; and define dynamical as whatever isn't stationary).

This is fair. I assumed a different definition of "dynamical" – in my definition a dynamical solution is one which evolves with an external time parameter, and the evolution can be calculated solving the Hamilton's equation using a model-specific Hamiltonian. In retrospect, Bruce Lee's definition of "dynamical" may be more inclined with what OP had in mind.

In quantum gravity the Wheeler-deWitt equation says that the wavefunction of the universe doesn't change, however as claimed in the other answer, this really doesn't talk about the classical solution of the Einstein equations, but talks about the wavefunction of the universe. See de Witt's original paper on quantum theory of gravity for more details.

The classical (ADM) Hamiltonian formulation of GR, the Hamiltonian is a linear combination of constraints, hence vanishes on-shell. One can still use it to evolve quantities with respect to coordinate time, if one arbitrarily chooses the values of Lagrange multipliers corresponding to the constraints. The nonphysical choice of the Lagrange multipliers corresponds to the absense of a notion of physical time that is universal for all solutions – time in GR exists for each classical solution, but not for all of them.

Have a look at this sciencedirect.com/science/article/pii/0003491674904047. The Hamiltonian as a surface term's derivation is given here. I don't see the difficulty in computing the time evolution.

  1. This claim is only valid for a subclass of solutions that satisfy the condition of asymptotic flatness. There are important examples of solutions that don't satisfy asymptotic flatness, a notable subclass coming from cosmology.
  2. If we discuss Quantum Gravity, there are more reasons to reconsider the condition of asymptotical flatness. Loosely speaking, quantum operators are noncommutative deformations of functions over the phase space, which is the space of solutions. It is not clear whether it is correct to restrict these functions to asymptotically flat solutions prior quantization. The act of restricting them would for example eliminate the topology changing amplitudes. For a toy model of Quantum Gravity theory with topology changing amplitudes, see Witten's Chern-Simons formulation of 3d GR.
  3. Even if we restrict GR to asymptotically flat solutions, the causal structure within space-time will differ from that of a flat space. By computing the time evolution one will get the evolution with respect to the unique time-like parameter coming from the flat Minkowski space at infinity, but there is still no universal notion of physical time in the bulk that is independent of the solution (I infer from Bruce's answer that there is no disagreement here).

Hi, your answer is wrong

This is unfair, and it is an overstatement at the least. In his answer, Bruce agrees with my main point:

It is true that there is no "canonical notion" of time in GR as one has in special relativity. There is a canonical notion of time which makes sense under the isometry group of Minkowski space (the Poincare group). However Einstein Field equations transform covariantly under diffeomorphisms, so different coordinate systems don't agree on a unique direction of time.

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  • $\begingroup$ I've deleted a number of obsolete comments and/or responses to them. $\endgroup$ – David Z Mar 30 at 8:28
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    $\begingroup$ "This is unfair, and it is an overstatement at the least." Sorry for the late reply, I was out of station. I see your point, and should have phrased it in a better way. I didn't mean your answer's content was wrong (all your points are correct independently), but meant that yours didn't answer OP's question. Also I believe we were unnecessarily getting confused in the comments. Apologies. $\endgroup$ – Bruce Lee Mar 31 at 12:54
  • $\begingroup$ @BruceLee no worries, thanks for clearing the confusion $\endgroup$ – Prof. Legolasov Mar 31 at 19:53
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It is a false assertion that there don't exist dynamical solutions to the Einstein equations. Trivial counter-examples are solutions to Friedmann equations.

There exist static solutions to Einstein's equations, which don't change with time and are irrotational. Any other solution which is non-static is dynamical (more loosely, you can also define a broader class of stationary spacetimes, which basically means that there exist asymptotic timelike Killing vectors; and define dynamical as whatever isn't stationary).

It is true that there is no "canonical notion" of time in GR as one has in special relativity. There is a canonical notion of time which makes sense under the isometry group of Minkowski space (the Poincare group). However Einstein Field equations transform covariantly under diffeomorphisms, so different coordinate systems don't agree on a unique direction of time.

In quantum gravity the Wheeler-deWitt equation says that the wavefunction of the universe doesn't change, however as claimed in Prof. Legolasov's answer, this really doesn't talk about the classical solution of the Einstein equations, but talks about the wavefunction of the universe. See de Witt's original paper on quantum theory of gravity for more details.

The paper you linked to in your question deals with non mainstream physics as it denies established notions like Singularity theorems and Positive energy theorem as a consequence of its false assertions. Strictly speaking its outside the scope of this Stack Exchange.

Update: Prof. Legolasov's answer is now more clarified to include our discussions. IMO, his earlier version (had correct points but) didn't answer OP's question.

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  • $\begingroup$ There are currently three answers, and may be more in the future. "The other answer" is thus an ambiguous reference. Better to refer to the answer author if you refer to another answer. $\endgroup$ – Lawnmower Man Mar 30 at 4:25
  • $\begingroup$ @LawnmowerMan True, though I'd add that the best way to refer to another answer is to link to it in addition to identifying it by author. $\endgroup$ – David Z Mar 30 at 8:29
  • $\begingroup$ @DavidZ Thanks, for some stupid reason I didn't do so. $\endgroup$ – Bruce Lee Mar 31 at 13:08
  • $\begingroup$ @LawnmowerMan Thanks and done! $\endgroup$ – Bruce Lee Mar 31 at 13:08

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