# Mass term inside a metric

I'm not very fund of the subject, but what i know is that the Schwarzschild metric and other known solutions for Einstein's action have some constant representing mass. However, I encountered some solutions where no such constant existed. Can someone explain to me what does this mean exactly? In fact, does it mean that there is no mass in the geometry? and if so, how do we get a curved space-time when general relativity states that the curvature is determined by the distribution of mass?

Dale
Mentor
2020 Award
how do we get a curved space-time when general relativity states that the curvature is determined by the distribution of mass?
The source of gravity in GR is not just mass (a scalar) but the stress-energy tensor (a rank-two tensor). Additionally, the Riemann curvature tensor is a rank-four tensor so the boundary conditions are important and certain boundary conditions can result in curvature even when the stress-energy tensor is 0 everywhere. In fact, the Schwarzschild solution is an example of such a solution. It is a vacuum solution, the mass itself is a boundary condition and the solution only applies to the region of vacuum outside the mass.

K^2
Mass in the metric just means that the problem was solved as a function of that mass. If there is no mass parameter, it doesn't mean that no mass was involved. It just means the solution is only valid for a specific problem with specific masses.

There are also a bunch of metrics that are simply used as examples, and nobody even bothers to try and figure out what combination of masses results in such a metric.

If there is no mass parameter, it doesn't mean that no mass was involved. It just means the solution is only valid for a specific problem with specific masses.

Is there a way of knowing what this specific mass is? I mean of course of the solution is derived from some general metric without an explicit value of the mass being given.

atyy
In GR, localized mass-energy is not the only "source" of curvature.

The gravitational field (which is the only classical field that does not have localized mass-energy) obeys non-linear equations and so is itself a "source" of curvature.

Thus the mass parameter in the maximally extended vacuum Schwarzschild solution does not represent localized mass-energy. It is an example of a vacuum solution in which the only field present is the gravitational field. All vacuum solutions are Ricci flat. In a slightly different context than classical general relativity, the Calabi-Yau manifolds are examples of Ricci flat solutions that are curved.

In the Schwarzschild solution, if the mass parameter is set to zero, we recover flat Minkowski spacetime.

In most practical cases, we do not use the maximally extended vacuum Schwazrschild solution. Only part of the vacuum Schwarzschild solution is used (outside the star), while a different non-vacuum solution is used inside the star. The boundary conditions then specify the Schwarzschild mass parameter as an integral over the localized mass-energy of the star (however, the integral is not over elements of proper volume). Details are given in 10.41 of http://books.google.com/books?id=qhDFuWbLlgQC&dq=schutz+general+relativity&source=gbs_navlinks_s.

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