As I said, I'm reading from Carroll's textbook, and he discusses the idea of "coupling fields" without ever defining what fields are or what it means for fields to couple. Can you please briefly describe those concepts?
From Carroll (2004)
It is possible to derive the Einstein Equations (with ##c=1##) via functional variation of an action
$$S=\dfrac{S_H}{16\pi G}+S_M$$
where
$$S_H= \int \sqrt{-g}R_{\mu\nu}g^{\mu\nu}d^4 x$$
and ##S_M## is a corresponding action representing matter. We can decompose ##\delta...
The Schwarzschild Metric (with ##c=1##),
$$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$
can be adjusted to a form involving three rectangular coordinates ##x##, ##y##, and ##z##:
$$ds^2 =...
I'm trying to prepare to read The Large Scale Structure Of Space-time by Hawking and Ellis. I've been reading a General Topology textbook since the authors say "While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavored to write this book so...
I'm aware. I was interested in determining which of the Riemann tensor components were not identically vanishing, rather than which components were not identically zero for a diagonal matrix. As a result, many of the non identically zero components I calculated were zero by the diagonal property...
I'm not doing these as homework problems. I was just inspired by a question (that I answered correctly, thank you very much) asking how many independent components there were. Here's how I solved the problem while users on this thread were answering me with unhelpful questions. I was working...
What 20 index combinations yield Riemann tensor components (that are not identically zero) from which the rest of the tensor components can be determined?
Sorry, a more clear statement is:
Pressure affects a gravitational field, but only when it is extremely large. An example is a neutron star, which is a star so dense that the atoms get mushed together. High pressure would have effects on space time within the boundaries of the star.
Much of General Relativity is devoted to the discovery of post-Newtonian effects, or effects that don't exist in the Newtonian approximation. A prime example is that of active gravitational mass. In general relativity, ##(\rho+3p)## replaces ##\rho## in the Newtonian equations. The ##\rho## part...
The Ricci Tensor (##R_{\mu\nu}##) is a tensor that represents curvature. It was invented by mathematicians, and satisfies a number of useful properties. The tensor ##R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}## is called the trace reverse of the Ricci Tensor, and satisfies the following very important...
Sure, it's possible to explain GR in words, but the words will essentially be replacements for or simplifications of the math.
1. You mentioned the acronym FLRW, which is a solution of the principle equations of GR that holds for an isotropic expanding fluid. That essentially means that the...