The Einstein field equations (EFE) characterize how mass curves spacetime. $$ R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa T_{\mu\nu} $$ I try to understand the curvature of spacetime.

In the EFE, there is the Ricci tensor, which in many references is characterized as the volume gain.

I am stuck with the following question: When the Ricci tensor stands for the volume gain - is it possible that mass is the source of spacetime?

I'm familiar with electrostatic and electromagnetic fields (more than with general relativity) - there are sources and fields. Maybe I overestimate the similarities between both realms. Could you please help me to understand the differences, and understand where there are similarities - emphasizing that the Ricci-tensor stands for volume gain (which probably led me astray?)

A related question is: What's the difference between space itself and the gravitational field? If mass is not the source of space itself there seems to be a difference between space itself and the gravitational field. What is this difference? I would be interested in calculations and explanations.

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – SuperCiocia
    Dec 5, 2021 at 21:05

1 Answer 1


If the Ricci tensor is zero everywhere, gravitational waves are valid solutions for the EFE. That means: there is a spacetime, where for every event $X^{\mu}$ there is a metric tensor, even in the absence of any mass as a source.

It is similar to electromagnetism, where the Maxwell equations without sources allows EM waves as solutions. The equations don't lead us to any dependence of the fields on the sources. There are no sources but there are $E$ and $B$ fields.

The same for the EFE, where spacetime with defined metric are not related to any source.

  • $\begingroup$ Thank you for this insightful answer! However, what I still don't get is: if we emphasize the parallels between the EM field and spacetime and take into account that there is more volume with more mass, isn't a flat spacetime equivalent to a constant (not zero!) electromagnetic field? $\endgroup$ Dec 8, 2021 at 7:45

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