Formula sheet · PU-403

General Relativity — formula sheet

Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.

D-384

Equivalence Principle Forces a Curved Metric

\[ \boxed{\; \eta_{\mu\nu}\ \longrightarrow\ g_{\mu\nu}(x),\qquad \frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma^{\lambda}{}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=0,\qquad \Gamma^{\lambda}{}_{\alpha\beta}=\tfrac{1}{2}g^{\lambda\rho}\big(\partial_\alpha g_{\rho\beta}+\partial_\beta g_{\rho\alpha}-\partial_\rho g_{\alpha\beta}\big)\;} \]

Universality of free fall means the "force of gravity" can be transformed away locally by falling with the field. What cannot be transformed away globally is the way neighbouring free-fall frames fail to align — encoded in a position-dependent metric \(g_{\mu\nu}(x)\). Free particles then move on geodesics, and the connection \(\Gamma^\lambda{}_{\alpha\beta}\), built from first derivatives of \(g\), plays the role of the gravitational field. Gravity is not a force on flat spacetime; it is the curvature of the metric. (Genuine curvature — irremovable at second order — is the tidal part; see Discussion.)

D-385

Geodesic Equation from Extremal Proper Time

\[ \frac{d^2 x^\alpha}{d\tau^2} + \Gamma^\alpha_{\mu\nu}\,\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0,\qquad \Gamma^\alpha_{\mu\nu} = \frac{1}{2}g^{\alpha\beta}\!\left(\partial_\mu g_{\beta\nu} + \partial_\nu g_{\beta\mu} - \partial_\beta g_{\mu\nu}\right) \]

The four-acceleration measured in the coordinate basis is not zero in curved spacetime; instead it is exactly cancelled by the connection term \( \Gamma^\alpha_{\mu\nu}\dot x^\mu\dot x^\nu \). The invariant statement is that the covariant acceleration vanishes, \( \dot x^\nu \nabla_\nu \dot x^\alpha = 0 \): the four-velocity is parallel-transported along its own worldline. The particle "feels" nothing — it is in free fall — yet its coordinate path curves because the coordinate grid itself is curved. The Christoffel symbols encode how basis vectors twist from event to event; they are built from first derivatives of the metric alone, so the connection is fully determined by, and compatible with, the geometry.

D-386

Covariant Derivative and the Metric Connection

\[ \boxed{\;\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12\,g^{\lambda\rho}\!\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right)\;}\qquad \nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda} \]

Among all connections, exactly one is simultaneously torsion-free and metric-compatible, and its coefficients are built purely from first derivatives of the metric. The covariant derivative it defines turns \(\partial_{\mu}\) into a genuine tensor operation: the \(\Gamma\)-terms are the "fictitious force" bookkeeping that accounts for how the coordinate basis vectors themselves twist and stretch from point to point.

D-387

Riemann Tensor from the Covariant Commutator

\[ [\nabla_\mu,\nabla_\nu]V^\rho = R^\rho{}_{\sigma\mu\nu}V^\sigma, \qquad R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma} \]

Parallel-transporting a vector around an infinitesimal coordinate parallelogram spanned by the \(\mu\) and \(\nu\) directions rotates it by an amount \(R^\rho{}_{\sigma\mu\nu}\) per unit area; the tensor packages that rotation. The four algebraic symmetries \(R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}=-R_{\rho\sigma\nu\mu}=R_{\mu\nu\rho\sigma}\) plus the first Bianchi identity cut the naive \(n^4\) components down to \(\tfrac{1}{12}n^2(n^2-1)\) — 20 in four dimensions. The second Bianchi identity is the differential constraint that, once contracted, guarantees \(\nabla_\mu G^{\mu\nu}=0\) and hence local energy-momentum conservation.

D-388

Parallel Transport and Geodesic Deviation

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

The relative acceleration of two infinitesimally separated free-falling particles equals minus the Riemann tensor contracted twice with their common four-velocity and once with their separation. Flat spacetime (\(R^\mu{}_{\nu\alpha\beta}=0\)) gives zero relative acceleration — parallel free-fallers stay parallel, exactly the Newtonian statement that a uniform field produces no tides. A non-zero right-hand side is the invariant, frame-independent presence of a genuine gravitational field. The map \(\xi^\alpha\mapsto -R^{\mu}{}_{\nu\alpha\beta}u^\nu u^\beta\) is the relativistic tidal tensor; its symmetry (from the pair-symmetry of Riemann) guarantees principal tidal axes.

D-389

Contracted Bianchi Identity and the Einstein Tensor

\[ \nabla^{\mu}R_{\mu\nu}=\tfrac12\,\nabla_\nu R \qquad\Longleftrightarrow\qquad \nabla^{\mu}G_{\mu\nu}=0,\quad G_{\mu\nu}=R_{\mu\nu}-\tfrac12\,g_{\mu\nu}R \]

The divergence of the Ricci tensor is not zero but is locked to half the gradient of the scalar curvature. The particular combination that subtracts \(\tfrac12 g_{\mu\nu}R\) exactly cancels this "leftover," producing the unique (up to scale and a cosmological term) symmetric tensor whose covariant divergence vanishes identically on every metric. This is a geometric identity, true independently of any field equation.

D-390

Stress-Energy Tensor as the Conserved Source

\[ T^{\mu\nu}=\frac{2}{\sqrt{-g}}\,\frac{\delta S_{\text{m}}}{\delta g_{\mu\nu}}=T^{\nu\mu},\qquad \nabla_\mu T^{\mu\nu}=0\ \Longleftarrow\ \nabla_\mu G^{\mu\nu}\equiv 0 \]

Translation invariance produces a conserved canonical current, the stress-energy tensor. Lorentz invariance and the Belinfante-Rosenfeld improvement symmetrize it without changing its charges; the metric-variation (Hilbert) definition delivers the same symmetric object directly. General covariance of the matter action then forces \( \nabla_\mu T^{\mu\nu}=0 \) — precisely the identity the contracted Bianchi identity demands of the Einstein source. The geometry (left side of Einstein's equations) and the matter (right side) are mutually consistent by construction, not by coincidence.

D-391

Field Equations from the Einstein-Hilbert Action

\[ \boxed{\,G_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2}R\,g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}\,} \]

Spacetime curvature (the Einstein tensor on the left, a specific combination of the Ricci tensor and scalar) is sourced by the local energy, momentum, and stress of matter (the stress-energy tensor on the right). The proportionality constant \(8\pi G/c^4\) is tiny in SI units, so it takes enormous energy densities to curve spacetime measurably. Because \(\nabla^\mu G_{\mu\nu}=0\) identically, the equation enforces \(\nabla^\mu T_{\mu\nu}=0\): geometry itself conserves energy–momentum.

D-392

Newtonian Limit Fixes the Coupling Constant

\[ G_{\mu\nu}=\frac{8\pi G}{c^4}\,T_{\mu\nu},\qquad g_{00}=-\left(1+\frac{2\Phi}{c^2}\right),\qquad \nabla^2\Phi=4\pi G\rho \]

The undetermined constant in Einstein's equations is not free: forcing gravity to reproduce Newton in the weak, slow, static corner of parameter space fixes it to \(8\pi G/c^4\). Geometrically, the Newtonian potential lives in the time–time part of the metric, and its Laplacian is (up to \(1/c^2\)) the \(00\) Ricci curvature; the \(8\pi\) is the fingerprint of the trace-reversal, while the \(1/c^4\) sets the extreme rigidity of spacetime.

D-393

Schwarzschild Solution from Spherical Vacuum

\[ ds^2 = -\left(1-\frac{2GM}{c^2 r}\right)c^2\,dt^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \]

Outside any spherical mass \( M \) the geometry depends on a single parameter, the Schwarzschild radius \( r_s=2GM/c^2 \). Time runs slower and radial rulers stretch as \( r\to r_s \); at \( r=r_s \) the coordinates \( t,r \) break down (a coordinate, not curvature, singularity), while \( r=0 \) is a true curvature singularity where \( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_s^2/r^6\to\infty \). Birkhoff guarantees this holds even while the source pulsates spherically.

D-394

Perihelion Precession from Schwarzschild Orbits

\[ \boxed{\;\Delta\phi = \frac{6\pi GM}{c^2\,a(1-e^2)} = \frac{3\pi\,r_s}{a(1-e^2)}\;}\qquad\text{radians per orbit} \]

Each orbit the ellipse rotates prograde (in the direction of motion) by an angle set entirely by the ratio of the Schwarzschild radius \(r_s=2GM/c^2\) to the semi-latus rectum \(p=a(1-e^2)\). Tighter orbits (small \(p\)), more massive centres, and higher eccentricity all amplify the advance; the effect vanishes as \(c\to\infty\), recovering the closed Keplerian ellipse.

D-395

Light Deflection and Gravitational Redshift

\[ \boxed{\;\delta = \frac{4GM}{c^2 b}\;}\qquad\qquad \boxed{\;\frac{\nu_r}{\nu_e} = \sqrt{\frac{1-r_s/r_e}{1-r_s/r_r}} \;\approx\; 1-\frac{GM}{c^2}\!\left(\frac{1}{r_e}-\frac{1}{r_r}\right)\;} \]

A light ray grazing a mass \(M\) at closest approach \(b\) is bent toward the mass by an angle \(4GM/c^2b\) — exactly twice the Newtonian "falling corpuscle" value \(2GM/c^2b\), the extra factor coming from the curvature of space (the \(g_{rr}\) term), not just of time. Independently, a spectral line emitted deep in the potential well and received higher up is shifted to lower frequency (redshifted) by a fractional amount equal to the difference in Newtonian potential per \(c^2\). Both are first-order in \(GM/c^2\).

D-396

Event Horizon and Maximal Extension

\[ ds^2 = -\frac{32\,G^3M^3}{r}\,e^{-r/2GM}\,dU\,dV + r^2(U,V)\,d\Omega^2,\qquad UV = \left(1-\frac{r}{2GM}\right)e^{r/2GM} \]

In Kruskal–Szekeres coordinates the metric coefficient is finite and non-zero at \(r=2GM\), so the horizon is a smooth null surface, not an edge of spacetime. Radial light cones stay at \(45^\circ\) everywhere (lines \(U=\text{const}\), \(V=\text{const}\)), making the causal structure manifest: inside region II every future-directed causal curve has increasing \(r\)-decreasing character and must hit \(UV=1\) (\(r=0\)). The horizon \(UV=0\) is crossed by an infaller in finite proper time, while a static exterior observer sees the crossing red-shifted to \(t\to\infty\).

D-397

Linearized Gravity and the Wave Equation

\[ \Box\,\bar h_{\mu\nu}=-\frac{16\pi G}{c^{4}}\,T_{\mu\nu}\qquad\xrightarrow{\ \text{vacuum}\ }\qquad \Box\,\bar h_{\mu\nu}=0,\quad \omega=c|\mathbf{k}| \]

In the weak-field, Lorenz-gauge limit each component of the trace-reversed metric perturbation obeys an ordinary flat-space wave equation sourced by the local energy–momentum. Where matter is absent the perturbation propagates as a free wave whose null wavevector forces the phase and group speed to equal \( c \). Fixing the residual gauge (TT) strips the ten components down to the two transverse, traceless polarizations \( h_{+} \) and \( h_{\times} \) that a distant observer can measure as a quadrupolar stretch-and-squeeze of proper distances.

D-398

Quadrupole Formula for Radiated Power

\[ \boxed{\,P=\frac{G}{5c^5}\left\langle \overset{\smash{\cdots}}{Q}_{ij}\,\overset{\smash{\cdots}}{Q}_{ij}\right\rangle\,},\qquad Q_{ij}=\int\rho\left(x_i x_j-\tfrac13\delta_{ij}r^2\right)d^3x. \]

Gravitational luminosity is set by the third time derivative of the traceless mass quadrupole, squared and summed over the two free indices, then averaged over several wave periods (the brackets). There is no monopole term (total mass is conserved) and no dipole term (its second derivative is the conserved total momentum's rate of change, which vanishes) — the quadrupole is the leading radiating multipole, and the ferocious \(c^{-5}\) is why gravity radiates so feebly.

D-399

Friedmann Equations from the FLRW Metric

\[ \left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3},\qquad \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda c^2}{3} \]

The first equation says the squared expansion rate \(H^2\equiv(\dot a/a)^2\) is sourced additively by mass–energy density, spatial curvature (acting like a negative energy for \(k>0\)), and the cosmological constant. The second says gravity decelerates expansion whenever \(\rho+3p/c^2>0\): pressure gravitates, so radiation (\(p=\rho c^2/3\)) decelerates more strongly than dust at equal density, while any component with \(p<-\rho c^2/3\) (e.g. \(\Lambda\), \(w=-1\)) drives acceleration.

D-400

Cosmological Redshift and Hubble's Law

\[ \boxed{\,1+z = \dfrac{a(t_o)}{a(t_e)}\,}\qquad\qquad \boxed{\,v = H_0\,d \;\;(z\ll 1)\,} \]

The first relation is exact: light emitted when the universe was a factor \(a(t_e)/a(t_o)\) smaller arrives stretched by exactly the inverse of that factor. A galaxy at \(z=1\) emitted its light when the universe was half its present size. The second relation is the nearby limit: recession speed grows linearly with distance, with slope the present expansion rate \(H_0\). Redshift is not a Doppler shift through space but the accumulated stretching of the wave by the expansion of space itself.

D-401

Raychaudhuri Equation and Geodesic Focusing

\[ \frac{d\theta}{d\tau}=-\tfrac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b,\qquad \frac{1}{\theta(\tau)}\ge\frac{1}{\theta_0}+\frac{\tau}{3}. \]

The fractional rate of change of a comoving 3-volume, \(\theta\), is driven down by three effects that all focus — its own square (self-focusing of the flow), shear (anisotropic tidal stretching), and the Ricci curvature sourced by matter — and driven up only by rotation, which centrifugally resists collapse. Remove the rotation and satisfy the energy condition, and convergence, once begun, runs away to a caustic in bounded proper time.

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