General relativity

For positive smooth $A$, this Lorentzian manifold has flat $(t,x,y)$ slices scaled by $A(z{)}^2$. Translation in each slice coordinate gives a Killing vector field. An orthonormal coframe is $e^0=Adt$, $e^1=Adx$, $e^2=Ady$, $e^3=dz$. Its Levi-Civita connection and Ricci tensor are controlled by $a=A^{\prime }/A$ and $q=A^{\prime \prime }/A=a^{\prime }+a^2$.

For the orthonormal coframe of a planar warped spacetime, Cartan's first structure equation gives ${\omega }^i{}_3=a{e}^i$ and ${\omega }^3{}_i=-s_{i}a{e}^i$, where $i=0,1,2$ and $(s_0,s_1,s_2)=(-1,1,1)$. All other connection 1-forms vanish. Cartan's second structure equation gives ${\Theta }^i{}_3=-q{e}^i\wedge e^3$ and ${\Theta }^i{}_j=-s_j{a}^2{e}^i\wedge e^j$ for $i\ne j$. Lowering the first index makes both the connection 1-forms and the curvature 2-forms antisymmetric. Contraction yields $R_{00}=q+2a^2$, $R_{11}=R_{22}=-q-2a^2$, and $R_{33}=-3q$.

Transforming a metric tensor and all matter fields together by the pullback of a diffeomorphism preserves the Einstein field equations and covariant matter equations. Passive coordinate changes only change the description of the fields; active pullbacks produce another representative of the geometry. The treatment of transformations acting on a boundary depends on the boundary conditions and possible charges. A diffeomorphism is not necessarily an isometry of a fixed metric.

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives $g(\dot{\gamma },\dot{\gamma })=-1$.

A coordinate singularity is a failure of a coordinate description which disappears in a regular chart. The horizon singularity of the Schwarzschild metric in $(t,r)$ coordinates is removable; its $r=0$ curvature singularity is not.

In a spherically symmetric spacetime, the areal radius $r$ is defined by the area $A=4\pi r^2$ of each symmetry sphere. It is independent of which radial coordinate chart is used.

Geometrized units set $G=c=1$, allowing mass, length, and time to be expressed in common units. Restore $GM/c^2$ for a mass parameter in a radial metric tensor.

A geometrized luminosity is dimensionless because in geometrized units an energy and a time have the same dimension. Starting from SI luminosity, multiply by $G/c^5$. The corresponding SI power unit is $c^5/G$. Radiating a fraction $f$ of a mass $M$ at constant SI luminosity $L$ takes $fM{c}^2/L$, or $f(GM/c^3)/L_{\mathrm{geom}}$ in seconds.

The ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time $v$ and areal radius $r$. In geometrized units it is $d{s}^2=-(1-2M(v)/r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$. A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.

The two radial null geodesic families are $v=\text{constant}$ and $dr/dv=(1-2M(v)/r)/2$. The first has affine radius, while the second has tangent $k={\partial }_v+(f/2){\partial }_r$ obeying ${\nabla }_{k}k=(M/r^2)k$, so advanced time is generally a nonaffine parameter. An affine parameter $\lambda$ for the outgoing family satisfies $d\log (dv/d\lambda )/dv=-M/r^2$.

An outgoing radial null geodesic of the Vaidya metric prescribed as $r=r_0(v)$ determines the mass along that curve by $M(v)=r_0(v)(1-2r_0^{\prime }(v))/2$. This inverse relation comes from the radial null condition. It determines a function of advanced time wherever the prescribed curve is differentiable and has positive radius.

Writing $f=1-2M(v)/r$, the radial Christoffel symbols are ${\Gamma }^v{}_{vv}=M/r^2$, ${\Gamma }^r{}_{vv}=fM/r^2+M^{\prime }/r$ and ${\Gamma }^r{}_{vr}=-M/r^2$. The angular couplings are ${\Gamma }^v{}_{\theta \theta }=-r$, ${\Gamma }^r{}_{\theta \theta }=-fr$, with ${\sin }^2\theta$ factors for the corresponding $\varphi \varphi$ terms, ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\varphi }{}_{r\varphi }=1/r$, ${\Gamma }^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\Gamma }^{\varphi }{}_{\theta \varphi }=\cot \theta$. The lower indices are symmetric. The plus sign of $M^{\prime }/r$ follows from the ingoing $+2dvdr$ cross term.

The equivalence principle states that gravity couples universally to energy-momentum and that a freely falling local frame removes uniform gravitational acceleration at a point. Tidal curvature remains measurable over a finite region.

The Equivalence principle allows Riemann normal coordinates in which the first derivatives of the metric tensor vanish at a chosen point. A purported local gravitational stress-energy tensor built from those first derivatives would vanish there in that chart and therefore, if tensorial, in all charts. Such an expression cannot encode the gravitational energy assigned by a coordinate-dependent field description. This rules out an ordinary universal local covariant gravitational energy density, not the construction of local curvature tensors or boundary and averaged notions of gravitational energy. The Riemann curvature tensor still describes invariant tidal gravity.

The spacetime dimension is the dimension of the smooth manifold used to model spacetime. Ordinary general relativity uses one temporal and three spatial dimensions.

A Killing vector field $K$ generates a local one-parameter family of isometries. Equivalently,

Where a Killing vector field is ${\partial }_x$ and its squared norm $V=h_{xx}$ is nonzero, an invariant metric tensor decomposes as $h=V(dx+A{)}^2+\gamma$. The quotient one-form has $A_i=h_{xi}/V$ and the quotient metric has ${\gamma }_{ij}=h_{ij}-h_{xi}{h}_{xj}/V$. All coefficients are independent of $x$. Changing the fibre coordinate to $x^{\prime }=x+\lambda$ sends $A$ to $A-d\lambda$. A null Killing direction requires a different decomposition.

For $h=\alpha d{x}^2+\beta d{y}^2+\chi (dz-ydx{)}^2$, the Kaluza-Klein decomposition along a Killing field ${\partial }_x$ has $V=\alpha +\chi y^2$, $A=-\chi ydz/V$ and $\gamma =\beta d{y}^2+\alpha \chi d{z}^2/V$. For a Riemannian metric, positive constants make the fibre norm and quotient metric positive everywhere.

Every Killing vector field on Euclidean space is the sum of a constant translation and a constant infinitesimal rotation. The second covariant derivative of a Killing vector vanishes in flat space, so its components are affine; the Killing equation makes their linear coefficient an antisymmetric matrix.

An infinitesimal rotation in one coordinate plane plus a constant translation has integral curves of a vector field that rotate about a shifted centre in that plane and move uniformly in the remaining directions. Nonzero transverse drift gives helices; zero drift gives circles, with zero radius allowing straight or stationary curves.

Killing's equation is ${\nabla }_{(a}{K}_{b)}=0$. It says that the infinitesimal flow of $K$ preserves the metric.

For a Killing vector field, set $F_{ab}={\nabla }_a{K}_b$ and $a^s=K^t{\nabla }_t{K}^s$. The Killing equation makes $F$ antisymmetric, and expansion of normalized three-index antisymmetrization gives the displayed identity. On a Killing horizon, hypersurface orthogonality removes its first right-hand term and relates $F^{ab}{F}_{ab}$ to surface gravity.

For a Killing vector field and the curvature convention $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\alpha }=R^{\alpha }{}_{\beta \mu \nu }{V}^{\beta }$,The identity follows by combining the Ricci identity, Killing equation, and first Bianchi identity.

Every Killing vector field $K$ preserves the scalar curvature: ${\mathcal{L}}_{K}R=K^{\alpha }{\nabla }_{\alpha }R=0$. One index proof rewrites this derivative as a doubly contracted curvature commutator acting on the antisymmetric tensor ${\nabla }_{[\mu }{K}_{\lambda ]}$; both terms vanish by symmetry of the Ricci tensor.

If $U$ is the tangent to an affinely parametrized geodesic and $K$ is a Killing vector, then $g(K,U)$ is constant along the geodesic.

For a geodesic tangent $u$ and a stationary Killing vector field $K$, the Killing energy is $E=-g(K,u)$. It is conserved along the geodesic. When $K$ is future timelike and $u$ is future causal, $E>0$ for a nonzero tangent; it need not be positive where $K$ becomes spacelike.

Suppose $V$ is a Killing vector field, $F$ is an electromagnetic field tensor satisfying $dF=0$ and ${\mathcal{L}}_{V}F=0$, and a particle of nonzero mass $m$ obeys the Lorentz force equation ${\nabla }_{u}u=(q/m)F^a{}_b{u}^b$. Cartan's magic formula gives $d({\iota }_{V}F)=0$, so the local potential of a closed differential one-form proof gives $d\Phi ={\iota }_{V}F$ locally. The Killing equation eliminates $u^a{u}^b{\nabla }_a{V}_b$, and the force term gives $d(V\cdot u)/d\tau =(q/m)d\Phi /d\tau$. Therefore $mV\cdot u-q\Phi$ is locally conserved. The sign assumes contraction in the first argument of $F$. A global potential additionally requires the closed one-form to have zero periods; invariance alone is insufficient without the homogeneous Maxwell equations.

If $T^{ab}$ is symmetric and covariantly conserved and $K$ is a Killing vector field, then $J^a=T^a{}_b{K}^b$ is conserved. Its flux through a spacelike hypersurface is the energy or momentum charge generated by the isometry.

A Komar integral constructs a conserved gravitational charge from the exterior derivative of the one-form dual to a Killing vector. In four dimensions, a stationary mass is conventionally normalized as

A Killing vector field gives an antisymmetric tensor ${\nabla }^{\mu }{\xi }^{\nu }$ whose divergence is this current, up to overall surface-orientation convention. The four-dimensional Einstein field equations express it as $8\pi G(T^{\mu }{}_{\nu }-\frac{1}{2}T{\delta }^{\mu }{}_{\nu }){\xi }^{\nu }$ at zero cosmological constant. The contracted Bianchi identity and Killing invariance of scalar curvature prove its divergence is zero. Stress conservation alone does not cancel the trace derivative without Killing invariance of the trace.

Komar angular momentum is the Komar charge associated with an axial Killing vector field in four-dimensional general relativity. In geometrized units its normalization is $(16\pi {)}^{-1}{\int }_S\star dm$, where $m$ is the metric-dual differential one-form. With zero cosmological constant, vacuum Einstein field equations make $d\star dm=0$, so homologous surrounding surfaces in the vacuum region give the same charge.

With zero cosmological constant and the component Hodge star operator convention $(\star j{)}_{abc}={ϵ}_{abcd}{j}^d$, the Einstein field equations and Killing equation give $d\star dm=-16\pi \star (T_{ab}{m}^b-\frac{1}{2}T{m}_a)$. If $m$ is tangent to a spacelike hypersurface, the trace term pulls back to zero. For $\partial \Sigma =S_{\infty }-{\bigcup }_i{S}_i$, Stokes theorem yieldsThe inner Komar charges cannot be dropped for a vacuum Kerr black hole, whose angular momentum is nonzero although its stress-energy tensor vanishes. A matter-only volume formula requires a regular filling with no inner boundaries.

Komar mass is the Komar charge associated with a stationary timelike Killing vector, using the normalization appropriate to gravitational mass.

For a spacetime with a non-flat reference background, replacing $k$ by $k-\bar{k}$ removes the background contribution and can make the Komar surface integral finite.

A conformal Killing vector field generates a local flow that preserves the metric up to a position-dependent scale factor.

In $d$ dimensions, the conformal Killing equation is

A gravitational wave is a propagating disturbance of spacetime curvature produced by changing mass-energy multipole moments.

An exact axial Killing vector field makes Komar angular momentum independent of the surrounding vacuum surface. Therefore gravitational waves in a configuration with exact axisymmetry carry no net angular momentum about that symmetry axis, although they can carry energy. This is specific to axial angular momentum; it is not an absence of radiation.

The gravitational-wave frequency is the temporal oscillation frequency of a radiative metric perturbation. A rigid source rotating with angular frequency $\omega$ and invariant under a half-turn radiates dominantly at angular frequency $2\omega$.

The trace-free mass quadrupole moment of a mass distribution isIt is the leading time-dependent mass multipole that can radiate in general relativity: mass conservation removes monopole radiation, while momentum conservation removes mass-dipole radiation.

The second mass moment tensor records the quadratic spatial distribution of mass. Its trace-free part is the mass quadrupole moment $Q_{ij}=I_{ij}-{\delta }_{ij}{I}_{kk}/3$. The mechanical moment-of-inertia tensor instead equals $J_{ij}={\delta }_{ij}{I}_{kk}-I_{ij}$. A rigid rotation $L$ transforms the second mass moment tensor as $I(t)=L(t)I(0)L(t{)}^T$.

The scalar second mass moment is the trace of the second mass moment tensor. The moment of inertia about a specified axis instead integrates squared distance from that axis. For an isotropic distribution each axial moment of inertia is $2I/3$; the sum over three orthogonal axes is $2I$.

For a slowly moving isolated source, the leading gravitational-wave luminosity iswhere $Q_{ij}$ is the trace-free mass quadrupole moment.

The dimensionless ratio of gravitational radius to orbital scale measures how relativistic a bound orbit is. In an equal-mass circular binary with individual orbital radius $R$, the leading quadrupole formula gives $P=(2/5)(c^5/G)[GM/(R{c}^2){]}^5$. Compact objects permit smaller noncontact orbits than extended stars and hence larger orbital compactness; near merger the Newtonian approximation must be replaced.

For two masses $M$ at opposite points of a circular orbit, each at radius $R$ from the centre of mass, ${\Omega }^2=GM/(4R^3)$. The trace-free mass quadrupole moment oscillates at twice the orbital frequency, and its third-derivative squared contraction is $128M^2{R}^4{\Omega }^6$. The quadrupole formula gives the displayed phase-independent luminosity. Using separation $2R$ rather than individual radius $R$ avoids a factor-of-32 error.

For two equal point masses at $\pm z$ falling from rest at $\pm z_0$ under Newtonian gravity, in geometrized units one has $\ddot{z}=-m/(4z^2)$ and $\dot{z}=-\sqrt{m(1/z-1/z_0)/2}$. The only nonzero component of the second mass moment tensor is $I_{zz}=2m{z}^2$, and its third derivative is $-m^2\dot{z}/z^2$. The trace-free mass quadrupole moment and quadrupole formula give the displayed power. From infinity, integrating down to $z=z_c$ gives $E=2\sqrt{2}{m}^{9/2}/(105z_c^{7/2})$. Relative to the total initial mass $2m$ the fraction is $\sqrt{2}(m/z_c{)}^{7/2}/105$. The weak-field slow-motion approximation must be distinguished from its extrapolation to a compact endpoint.