UPDATE Nov 9, 2020: Reza Mansouri did NOT prove the non-existence of a solution. He only proved that no "uniform" metric, where spatial directions at each point are stretched uniformly, is a solution. More complicated solutions might exist.
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R. Mansouri: On the non-existence of time-dependent fluid spheres in general relativity obeying an equation of state.
Ann. Inst. Henri Poincare vol. XXVII no. 2 (1977) pp. 175 - 183.
"It is shown that there are no solutions of Einstein's field equations representing a collapsing (or bouncing) fluid sphere obeying an equation of state p = p(ϱ) except for the trivial case p = 0 [identically]."
This interesting paper deserves a blog post of its own. If the result is correct, then Einstein's field equations cannot handle a very simple dynamic process where compressible fluid contracts under its own weight, except in the case where the "fluid" is dust and has the pressure zero.
The classic Oppenheimer-Snyder paper from year 1939 assumes a collapse of dust. The authors of that paper state that the case with a non-zero pressure would be hard to calculate.
Newton's gravity certainly can handle the process of collapsing fluid. Since the low-energy limit of Einstein's field equations is Newton's gravity, the problems in finding a solution must happen with very dense objects like neutron stars.
We have been suspecting in this blog that Einstein's field equations are too restrictive, and no solutions might exist for physically realistic systems.
Fluid in the real world consists of particles or fields. Is the Mansouri result valid if we model the fluid with fields?
Mansouri at the end if the paper suggests that we should use an equation of state where
p = p(ϱ, t, r)
depends on the time and the radial coordinate. Does such an equation make sense? The pressure, of course, also depends on how deep we are in the fluid sphere, that is, r. But why the density ϱ is not enough to tell us also that factor? The fluid compresses and its density grows with pressure.
https://www.researchgate.net/publication/259392422_There_is_no_Slow_Uniform_Contraction_of_a_Fluid_Sphere_obeying_an_Equation_of_State
In this 1980 paper Reza Mansouri wonders how general relativity "knows" about thermodynamical aspects which may require the pressure to be a function of both the density ϱ and the radius r.
Another way to interpret the result is that Einstein's field equations are too strict and do not have a solution for a realistic problem.
The Christodoulou and Klainerman nonlinear stability theorem
D. Christodoulou and S. Klainerman proved that the Minkowski space metric is stable in general relativity with respect to a small perturbation of the initial data (initial values).
The authors state in the paper that:
"it is not even known whether there are, apart from the Minkowski spacetime, any smooth, geodesically complete solution which becomes flat at infinity on any given spacelike direction."
They refer to the theorems of Lichnerowicz and Birkhoff. According to the authors, the theorem of Lichnerowicz implies that a static solution which is geodesically complete and flat at the infinity on any spacelike hypersurface must be flat.
Geodesically complete means that all geodesics extend to all time.
Geodesically complete means that all geodesics extend to all time.
Gödel's cyclic universe
In the Gödel metric, there are closed timelike curves.
Suppose that we try to form a realistic model of the universe with the Gödel metric. Then the random distribution of matter makes the metric very complicated.
Are there closed timelike curves also in this random metric? If yes, how can we make the "ends meet", so that everything happens in the exact same way when we have traveled one full cycle? At the first glance, the probability is zero that the ends meet for random initial data.
In de Sitter models of the universe, there are no closed timelike curves. But there are spacelike curves which "run around the universe". Can we make the ends meet for such curves?
Discussion
The existence of any solution for any realistic physical system seems to be an open problem in general relativity.
In an earlier blog post we remarked that if the matter field lagrangian L_M does not conserve energy in a spherically symmetric closed system, then Birkhoff's theorem implies that general relativity does not have any solution for such a system.
In our blog we have suggested that the Einstein-Hilbert action should be replaced with a rubber sheet model which would be more flexible, and the existence of a solution for virtually any reasonable initial data would be intuitively clear.
What do we know about the validity of general relativity?
Empirical data shows that the limit of general relativity, the newtonian gravity, is a very good approximation.
Gravitational lensing and the waves observed by LIGO prove that the linear limit of general relativity works well.
Binary pulsars lose energy in waves at the rate predicted by general relativity. This shows that the theory is on the right track for strong gravitational fields outside neutron stars.
General relativity works very well outside heavy bodies, that is, in the vacuum. What about inside heavy bodies, like a neutron star or a forming black hole?
The interaction of the lagrangian L_M and the metric is important inside heavy bodies. If general relativity handles the interaction in a wrong way, that might have measurable implications for star collapses into a neutron star or a black hole.
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