A working reference sheet · General Relativity

Use this appendix to retrieve equations after their derivations. The formulas follow the book’s metric signature and curvature convention; their stated domains and approximation assumptions remain part of each result.

B.1 The geometry ladder#

Interval and clock time

ds^2=g_{\mu\nu}dx^\mu dx^\nu, \qquad d\tau^2=-ds^2/c^2\quad\text{on a timelike worldline}.

Inverse and index conversion

g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu{}_\nu, \qquad V_\mu=g_{\mu\nu}V^\nu, \qquad V^\mu=g^{\mu\nu}V_\nu.

Levi-Civita connection in a coordinate basis

\Gamma^\rho{}_{\mu\nu} =\frac12g^{\rho\sigma} \left(\partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu}\right).

Covariant derivatives

\nabla_\mu f=\partial_\mu f, \qquad \nabla_\mu V^\nu=\partial_\mu V^\nu +\Gamma^\nu{}_{\mu\rho}V^\rho,

\nabla_\mu\omega_\nu=\partial_\mu\omega_\nu -\Gamma^\rho{}_{\mu\nu}\omega_\rho.

An upper tensor index gets a plus connection term; a lower index gets a minus term. This rule applies to ordinary tensors; tensor densities have an additional weight term.

Geodesic with affine parameter $\lambda$

\frac{d^2x^\mu}{d\lambda^2} +\Gamma^\mu{}_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda}=0.

Proper time is affine on a timelike geodesic. Null geodesics have no proper-time parameter.

Riemann curvature

R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}.

Contractions

R_{\mu\nu}=R^\rho{}_{\mu\rho\nu}, \qquad R=g^{\mu\nu}R_{\mu\nu}, \qquad G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}.

Tidal acceleration

\frac{D^2\xi^\mu}{d\tau^2} =-R^\mu{}_{\alpha\nu\beta}u^\alpha\xi^\nu u^\beta.

This form assumes a neighboring family of affinely parameterized geodesics with the usual commuting tangent and separation fields.

Volume, divergence, scalar wave operator

dV_4=\sqrt{-g}\,d^4x, \qquad \nabla_\mu V^\mu= \frac1{\sqrt{-g}}\partial_\mu(\sqrt{-g}V^\mu),

\Box f=\nabla_\mu\nabla^\mu f =\frac1{\sqrt{-g}}\partial_\mu \left(\sqrt{-g}g^{\mu\nu}\partial_\nu f\right).

In flat inertial coordinates, $\Box=-c^{-2}\partial_t^2+\nabla^2$ .

B.2 Matter, action, and dynamics#

Observer energy

E_{(U)}=-p_\mu U^\mu, \qquad U^\mu U_\mu=-c^2.

Perfect fluid

T^{\mu\nu}= \frac{\epsilon+p}{c^2}u^\mu u^\nu+pg^{\mu\nu}, \qquad T=-\epsilon+3p.

Einstein equation and trace reversal in four dimensions

G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}, \qquad \kappa=\frac{8\pi G_N}{c^4},

R=4\Lambda-\kappa T, \qquad R_{\mu\nu} =\kappa\left(T_{\mu\nu}-\frac12Tg_{\mu\nu}\right) +\Lambda g_{\mu\nu}.

Action and metric variation, with $x^0=ct$

S_{\mathrm{EH}}=\frac{c^3}{16\pi G_N} \int d^4x\sqrt{-g}(R-2\Lambda),

\delta\sqrt{-g}=-\frac12\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}, \qquad \delta S_m=-\frac1{2c}\int d^4x\sqrt{-g} T_{\mu\nu}\delta g^{\mu\nu}.

The total variational problem also includes its appropriate boundary terms and boundary conditions. The bulk expression alone is not a universal boundary prescription.

Identities and balance laws

\nabla_\mu G^{\mu\nu}=0, \qquad \nabla_\mu T^{\mu\nu}=0

for matter compatible with the Einstein equation and constant $\Lambda$ . The first is a geometric identity; the second is an on-shell matter balance law in the standard coupled theory. Neither is the assertion that all components are constant.

Weak Newtonian limit

g_{00}\simeq-\left(1+\frac{2\Phi}{c^2}\right), \qquad \frac{d^2\mathbf{x}}{dt^2}\simeq-\boldsymbol\nabla\Phi, \qquad \nabla^2\Phi=4\pi G_N\rho

for slow test motion, weak approximately static fields, negligible pressure, and negligible $\Lambda$ on the scales considered.

B.3 Vacuum is a family of possibilities#

Condition	Consequence in four-dimensional GR	What is still possible
$T_{\mu\nu}=0$ , $\Lambda=0$	$R_{\mu\nu}=0$ , $R=0$	Weyl curvature, black-hole exterior tides, gravitational waves
$T_{\mu\nu}=0$ , $\Lambda\ne0$	$R_{\mu\nu}=\Lambda g_{\mu\nu}$ , $R=4\Lambda$	Additional Weyl curvature; maximal symmetry is an extra condition
$T=0$ , $\Lambda=0$	$R=0$	Nonzero Ricci from trace-free matter, such as a classical Maxwell field
$R^\rho{}_{\sigma\mu\nu}=0$ on an open region	Locally flat geometry	Curvilinear/accelerated coordinates; possible global topology beyond a local chart

B.4 Six distinctions to keep beside your notebook#

Do not merge	The distinction
Coordinates and observers	A chart labels events; an observer is a physical timelike worldline, with additional frame choices for measurements
Connection and curvature	The connection compares nearby tangent spaces; curvature measures a particular failure of path-independent transport
Spatial curvature and spacetime curvature	Spatial slices depend on a slicing; the four-dimensional curvature describes spacetime
Local conservation and global conserved energy	A covariant balance law does not automatically provide a global time-translation symmetry or a total energy
Coordinate singularity and geometric singularity	A failing chart can be replaced; spacetime incompleteness requires a different analysis
Mathematical identity and equation of motion	An identity holds for every field of the relevant class; an equation of motion restricts the physically allowed fields

B.1 The geometry ladder#B.1 The geometry ladder

B.2 Matter, action, and dynamics#B.2 Matter, action, and dynamics

B.3 Vacuum is a family of possibilities#B.3 Vacuum is a family of possibilities

B.4 Six distinctions to keep beside your notebook#B.4 Six distinctions to keep beside your notebook

B.1 The geometry ladder#

B.2 Matter, action, and dynamics#

B.3 Vacuum is a family of possibilities#

B.4 Six distinctions to keep beside your notebook#