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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.
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.
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.
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.
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.
