General Relativity — Foundations

Manual: General · Subject: General Relativity

Spacetime curvature, geodesics, and Einstein's field equations

Gravity as Curved Spacetime

The Equivalence Principle and Curved Spacetime

Einstein's equivalence principle: a uniform gravitational field is locally indistinguishable from acceleration. This leads to the conclusion that gravity is spacetime curvature.

The metric tensor $g_{\mu\nu}$ encodes distances: $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ . Flat spacetime (Minkowski): $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$ .

Einstein's Field Equations

$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$ where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor, $R_{\mu\nu}$ the Ricci tensor, $\Lambda$ the cosmological constant, and $T_{\mu\nu}$ the stress-energy tensor.

Free particles in curved spacetime follow geodesics (locally straight paths): $\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$ .

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Schwarzschild Metric

For a spherically symmetric mass $M$ in vacuum:<br> $ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2 + r^2d\Omega^2$ <br>where $r_s = 2GM/c^2$ is the Schwarzschild radius (event horizon).

Einstein's field equations relate:

Spacetime curvature (geometry) to mass-energy (matter) Force to acceleration Electric field to charge density Entropy to temperature

The Schwarzschild radius of Earth ( $M_\oplus = 6\times10^{24}$ kg) is approximately:

~9 mm ~9 km ~9 m ~900 km

What observational evidence supports general relativity? Name at least three.

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General Relativity — Foundations

Gravity as Curved Spacetime

The Equivalence Principle and Curved Spacetime

Einstein's Field Equations