SPACE, TIME AND FUNDAMENTAL INTERACTIONS

Real meaning of the Einstein’s hypothesis in comparison of the Schwarzschild metric with the synchronous one

Meierovich B. E.

When comparing the properties of the Schwarzschild metric and the synchronous one, the deep meaning of Einstein’s hypothesis on the choice of a coordinate system is revealed. In the absence of a singularity, the determinants of the metric tensor and the signature of the metric are not invariant (they can change signs) when changing the coordinate system. In the Schwarzschild coordinates the signs of the metric tensor components are pre-fixed. As a result the problems (singularity, incompleteness of the reference system, limitations on the mass of a gravitating object) appear. If, based on Einstein’s hypothesis, we choose a synchronous coordinate system, then these problems do not exist. However, there is an opinion that (with rare exceptions) in a synchronous reference frame, matter cannot be in a static state. In this article I show that in the synchronous frame of reference matter can be in a static state. But only if it is compressed to the ultrarelativistic limit \(p = -\varepsilon/3\) by its own gravitational field. Negative pressure \(p < 0\) means that matter tends to compress, and not to expand. The static state of extremely compressed matter can exist with no mass limitation, and regardless of the internal structure of the object. Moreover, in a synchronous frame of reference the minimum and maximum sizes of extremely compressed matter depend only on the total mass of the object.