But is it just a lucky coincidence that a simple and beautiful metric exists for a spherically symmetric mass?
Our own Minkowski & newtonian gravity model reproduces the Schwarzschild metric in the case of a spherically symmetric mass M, if we tune the "shipping" of energy to be the radial distance r of the test mass m to the center of the mass M. This is probably just a coincidence. There is no obvious reason why our model would make Ricci curvature zero outside more complex mass distributions.
In the fall of 2023 we tentatively proved that changes in pressure "break" the Einstein field equations. The equations in that case do not have a solution, even though our own Minkowski & newtonian model does have a simple solution also in that case.
Nonlinear differential equations in physics
Question. Do the Einstein field equations have a solution for any complex mass distribution at all? For example, for a finitely long uniform cylinder?
If there is no solution to certain nonlinear equations, the symptom often is that an attempt to solve the equations leads a singularity. We know that general relativity is rife with singularities.
In physics, differential equations which describe an interacting system, are almost always nonlinear. However, the problem of nonlinearity often is not as bad as in general relativity:
1. There may be efficient iterative methods which find an approximate solution easily.
2. The system consists of atoms or molecules, and the system is microscopically totally different from its high-level differential equations. An example is the Navier-Stokes equations: they ignore molecules.
3. The system may be quantum, and we can only make a single measurement from it. The existence of a continuous solution which fills Minkowski space, is not too relevant if we cannot really probe that solution efficiently.
In general relativity, the metric is assumed to be a continuous function which fills the entire universe. A singularity in that metric would often have drastic physical consequences.
General relativity claims things about mathematics
Let us assume that the Einstein equations are a correct physical theory. That assumption implies that the equations do have solutions for all realistic mass distributions.
On the other hand, it is difficult or impossible to prove mathematically that such solutions do exist. It may even be more probable that solutions in most cases do not exist.
General relativity tries to be a mathematical theory. This does not sound reasonable!
Another question is if nature itself is able to solve difficult nonlinear differential equations. Do we expect too much of the mathematical capabilities of nature?
Hypothesis of simplicity in nature: there cannot exist "pathologically" nonlinear differential equations in nature
The hypothesis in the title claims that nature does not possess "magical" capabilities in solving equations. For every nonlinear equation occurring in nature, there has to exist an "efficient" method to generate approximate solutions.
Atomism
Our simplicity hypothesis is related to atomism.
Atomism claims that there is no truly "continuous" matter. There were conceptual difficulties in imagining how continuous substances could have chemical reactions and form chemical compounds. Greek philosophers Leucippus and Democritus, in the 5th century BC, solved the problem by assuming the existence of atoms.
Their solution turned out to be correct. John Dalton in his book in 1808 noted that chemical elements react in ratios p / q, where p and q are often small natural numbers. In 1808 there already was strong evidence that matter is divided into atoms. Quantum physics confirmed the existence of atoms in the early 20th century.
In the case of general relativity, the metric of spacetime is assumed to be a continuous "substance" which has astounding mathematical capabilities. It can magically solve nonlinear differential equations – maybe even in cases where there does not exist any solution!
Atomism wants to remove the inherent complexity of continuous substances. Our simplicity hypothesis has a similar goal.
Conclusions
Our simplicity hypothesis has ramifications throughout physics. In quantum field theory, the series of Feynman diagrams seems to diverge, but intermediate results produce astoundingly accurate predictions – up to 14 significant decimals. We will investigate the convergence problem in the future.
In the fall of 2023 we tentatively proved that the Einstein equations break in a pressure change. It might be that the equations do not produce meaningful results for any complex mass distribution. In that case, the correct "metric" of spacetime might have the Ricci curvature tensor non-zero even in vacuum locations.
We will next try to determine an approximate metric for a long cylinder and check if the Einstein equations can produce a reasonable metric.
The Levi-Civita metric around an infinitely long cylinder is very strange. The force of gravity is not ~ 1 / r around the cylinder, where r is the distance from the center of the cylinder.
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