The Cauchy problem is to solve a partial differential equation for the given initial values (Cauchy data). The values are given on a hypersurface of spacetime.
The Dirichlet problem is a similar problem where the values of a single unknown function are specified on the boundary of a volume in R^n.
Cyclic time
If the cyclic time Gödel universe can be defined for varying initial values, then a Cauchy problem is to find a solution where the universe magically returns to its original configuration after one cycle of time.
Intuitively, it is hard, or impossible, to find initial values which would have a solution. A problem is that if entropy grows at the start, how can we return the entropy back to the low value? If a black hole forms in the universe, how can we return it back to ordinary matter after a cycle?
Cyclic space
What is the difference of cyclic time to the "cyclic space" of a de Sitter universe? Is it easy to find initial values for some time point t_0 after the Big Bang, such that there exists a solution of the universe after t_0. Or a solution before t_0? Is it easy to find initial values which have a solution?
Building a solution for a chosen metric g
Let us start building a solution for general relativity this way:
1. first choose some metric g at some "moment of time", then
2. calculate the stress-energy tensor, and then
3. place in the spacetime the required mass density, momenta, pressure, and shear stresses.
After that, we can use some foliation and a time step to develop the solution back in time and forward in time.
Do we get a physically realistic solution for general relativity that way? What could go wrong?
The first problem is finding realistic matter which can fulfill the conditions of the stress-energy tensor.
The second problem is if there exists a physically realistic history from the Big Bang to the configuration we found.
If all chosen metrics g require "exotic" matter or an unrealistic history, then general relativity does not have any realistic solution at all.
The standard Schwarzschild interior and exterior metric require exotic matter, that is, incompressible fluid. Furthermore, the metric is static, asymptotically Minkowski, and cannot arise from the Big Bang.
We already noted that Birkhoff's theorem bans lagrangians L_M which do not conserve energy.
The formation of singularities can be interpreted as having no solution?
It is not known if a singularity can form in a realistic collapse of matter in general relativity.
For an artificial problem of a spherically symmetric collapse of dust, a singularity is inevitable if we choose to extend the spacetime maximally, so that we calculate the development also behind the horizon. The familiar Penrose diagram shows the singularity.
We could interpret the result in the way that general relativity does not have a solution for the collapse of the dust. If we ban singularities in a solution, then there is no solution.
What is the ultimate reason why a singularity forms? The reason probably is the equivalence principle, which implies that a freely falling observer will see the dust fall with him at all times.
In newtonian gravity, the dust would collapse into a point. There would be a singularity in newtonian gravity, too.
Reza Mansouri (1979) calculated a more realistic model where a fluid sphere has the pressure a function of mass-energy density p = p(ϱ) only. Mansouri's conclusion is that general relativity has no solution for the collapse of such a fluid sphere.
Suppose that we introduce a new theory of static electricity. The theory predicts that a singularity forms when an electron meets a positron. We would suspect that the theory is wrong. In general relativity, 104 years have taught people to accept a singularity.
Albert Einstein himself did not approve of a singularity but tried to argue in his 1939 Einstein cluster paper that a singularity cannot form.
The formation of a singularity (in an artificial, highly symmetric setup) may be one of the symptoms that general relativity does not have any solution for a dynamical system. For a static system we have the Schwarzschild solution.
A related question is if a singularity can form in a rubber sheet model of gravity. If it obeys the equivalence principle, then dust can collapse into a point, and put an infinite strain on the rubber.
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