What does gamma tensor $\gamma_{\mu\nu}$ mean in General Relativity

Question

I was recently going through Einstein's 'The Meaning of Relativity', where in chapter 4, he describes:

To the second approximation we must then put $g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}$ where $\gamma_{\mu\nu}$ are to be regarded as small of the first order.

At first I thought that $\gamma_{\mu\nu}$ is just any other addition to the metric, but as I read along, I found that $\gamma_{\mu\nu}$ is used plenty of times, even while discussing the cosmological problem. Einstein, at one point also says that $\frac{\gamma_{44}}{2}$ can be identified as the gravitational potential.

My question is this: Is there a physical and/or mathematical meaning for $\gamma_{\mu\nu}$. And what is it's connection with the metric, and how can it be derived. Basically I want to know what $\gamma_{\mu\nu}$ means in the real world and how it is obtained mathematically. And I think mathematical rigour would be preferable for me to grasp the concept more fundamentally.

Answer 1

Since the universe (at any length scale) is not very symmetric, in almost all physical situations, we do not know the spacetime geometry exactly. What this means is that one can never analytically compute $g_{\mu \nu}$ for a given region of spacetime. Thankfully, we don't need to, since many phenomena in the universe can be described using a geometry that is approximately $g_{\mu \nu}$. This means that we can consider a 'small' deviation $\gamma_{\mu \nu}$ from a known and highly symmetric geometry $g_{\mu \nu}$, to describe physics to a good approximation.

One example is gravitational waves traveling large distances to reach the earth from some binary black hole collision. These waves have essentially been traveling in empty space (or flat space, approximately) for a long time. We can therefore fix $g_{\mu \nu}$ as flat space (= $\eta_{\mu \nu}$), and $\gamma_{\mu \nu}$ as gravitational waves traveling on this flat space.

As another example, consider a non-rotating black hole. It can be described exactly by a Schwarzschild metric. But this is a perfect scenario. In real life, black holes interact with stars, matter, and other black holes around them. So there is to be some deviation from pure Schwarzschild geometry in such dynamical processes. We then fix $g_{\mu \nu}$ as the Schwarzschild metric, and $\gamma_{\mu \nu}$ as some deviation around it. One then substitutes the approximation of the full metric $g_{\mu \nu} + \gamma_{\mu \nu}$ in Einstein's equations and obtains equations for $\gamma_{\mu \nu}$. These describe how exactly the black hole changes its shape from pure Schwarzschild geometry. This is just an example - in real life, astrophysical black holes are always rotating (because of angular momentum conservation) so you would take the Kerr metric instead of Schwarschild.