Inside the star we may use the Schwarzschild interior metric.
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star 1 star 2
What if we have two stars?
Can we find an approximate solution?
Let us write the Schwarzschild metric for a single star
g_M + g_S,
where g_M is the standard Minkowski metric (-1, 1, 1, 1) and g_S is a "small" deformation of the Minkowski metric. Let T_S be the associated stress-energy tensor.
One may conjecture that
g_sum = g_M + g_S1 + g_S2,
where g_S1 and g_S2 are the deformations caused by the stars 1 and 2, is an approximate solution of the Einstein equations.
Since the Einstein equations are nonlinear, the simple sum g_sum above probably is not an exact solution. If it were, that would be in the literature. Also, the stars attract each other. The solution cannot be a sum of two static solutions.
If we calculate the stress-energy tensor T_sum for g_sum, using the Einstein equations, then T_sum differs slightly from
T_S1 + T_S2.
The difference ΔT may contain matter of positive or negative density throughout the universe, maybe some pressure, momentum, and even shear stresses. If it is possible for such matter to exist, then by adding that matter to the spacetime, we would have an exact solution of the Einstein equations.
However, we are looking for an exact solution where the space is a vacuum outside the stars.
As far as we know, no one has found an exact solution for two stars, or proved that a solution exists. Since the solution is not static, there will be gravitational radiation from the stars. That complicates the situation further.
We are not aware of anyone finding an exact solution of two stars embedded in an FLRW universe, either, if there is no matter between the stars. The solution of C. Gilbert requires that each star is alone in its "hole" in the uniform FLRW universe.
Since we do not know if any solution exists, we cannot say that g_sum is an "approximate" solution.
The reasoning above shows that Stephen R. Green and Robert M. Wald have not proved that they have found an approximate solution for a realistic FLRW universe, because they did not prove that any solution exists at all.
What if we assume that a solution exists? Can we somehow show that g_sum is close to that solution? That looks like a hard task. If we would be able to show that some iterative process of refining the candidate solution converges, then we might be able to prove something about the closeness of g_sum.
Green and Wald in their papers define a family of metrics indexed by λ, and they probably mean that when λ goes to zero, then the limit metric is an exact solution of the Einstein equations. We need to check if the resulting limit metric is trivial. Since no one has been able to prove the existence of exact solutions in the lumpy case, the limit metric probably is trivial.
Constructing exact solutions by using shells of negative or positive mass
If we have a star of a Schwarzschild mass m, then we can "reset" its gravitational field at some distance r, by enclosing it into a spherical shell which carries a negative mass -m.
( ● )
star m shell -m
The gravitational field is zero outside the shell.
We can form various exact solutions by embedding the star-shell structure to an ordinary Minkowski space.
We avoid the non-linearity of the Einstein equations by isolating the field of each star, so that the fields do not overlap anywhere.
If we have two stars of a mass m, we can embed each into a shell, and then put a spherical shell of the mass 2m around the whole system. In that way we get an exact solution which mimics Newton's gravity of two stars in much of the space.
But the field is zero in a certain zone, and the use of negative mass makes the solution very unphysical. Our model is not a good approximation of Newton's gravity. Can we improve the model somehow?
Our example suggests that it may be impossible to mimic Newton's gravity with exact solutions of the Einstein equations.
A rubber model of gravity can be made linear?
It may be that the differential equations of the laws of nature have to be linear, so that the existence of solutions is guaranteed. Can we make a rubber model of gravity such that the model is linear at the fundamental level?
The nonlinear behavior of gravity close to a black hole would be a high level phenomenon and would not reflect nonlinearity in the deeper level of the theory.
Cosmological models and dark energy
Observations suggest that the expansion of the universe started accelerating about 5 billion years ago.
The currently observed spatial flatness of the universe cannot be explained with the known matter and dark matter.
These facts suggest that there is dark energy which is accelerating the expansion.
The above facts assume that the FLRW model of general relativity is approximately right for inhomogeneous matter content. But we do not know if general relativity has any solution for such a matter content.
We need to check what exactly is the evidence for an accelerating expansion. If we assume that the spatial metric has been roughly flat for the past 5 billion years, how does the acceleration exhibit itself?
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