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For a small amount of matter falling into an initially and finally stationary rotating black hole, the Physical-process first law for a rotating black hole isLet $q^a$ be an affinely parametrized horizon generator, with affine parameter $\lambda =0$ at the background bifurcation surface. The horizon Killing vector field iswhere constancy of $\kappa$ is the Zeroth law of black-hole mechanics. The flux of the conserved Killing current through the horizon is

On the stationary background, expansion and shear vanish. To first order in the perturbation, the quadratic optical terms in the Null Raychaudhuri equation may be dropped, and the Einstein equation givesImpose the teleological final condition $\theta (\infty )=0$. Integration followed by reversal of integration order givesComparison with the Killing-energy flux proves the stated law.

With $u=t-z$, the quadratic geodesic Lagrangian isIts transverse Euler-Lagrange equations areThe longitudinal equations areafter using the difference of those same equations. In particular,This conserved quantity also follows from the Killing vector field ${\partial }_t+{\partial }_z$ and the geodesic conserved quantity from a Killing vector.

Write $f(r)=1-r_{+}^2/r^2$ and let a dot denote differentiation with respect to an affine parameter $\tau$. The time-translation and $z$-translation Killing vector fields, together with rotational symmetry of the unit $S^3$, give the conserved quantitiesDefine $\epsilon =-g_{ab}{\dot{x}}^a{\dot{x}}^b$, so $\epsilon =1,0,-1$ for timelike, null and spacelike geodesics respectively. The normalization equation becomesIt therefore has the effective potential form

The normal to a surface of constant $r$ has squared normwhich vanishes at $r=r_{+}$. Thus this surface is a null hypersurface. The stationary Killing vector fieldhas $K^2=g_{tt}=-f$, so it becomes null there and generates a Killing horizon. Since $f$ has a simple zero, its surface gravity isA horizon cross-section has topology $S^3\times S^1$, where the circle is the periodic $z$ direction. Including a complete generator, the null hypersurface has topology

A Killing vector field $K$ obeys ${\mathcal{L}}_{K}g=0$, or equivalently ${\nabla }_{(a}{K}_{b)}=0$. Along an affinely parametrized geodesic with tangent $U$,The second term vanishes by the geodesic equation, while the first contracts the symmetric tensor $U^a{U}^b$ with the antisymmetric part selected by the Killing equation. Thus $g(K,U)$ is a geodesic conserved quantity from a Killing vector. In Minkowski spacetime, translational Killing fields give conserved energy-momentum and rotational or boost Killing fields give the corresponding angular-momentum and boost charges; the same construction works for any spacetime isometry.

The flow of a Killing vector field consists of local isometries. Isometries preserve both the metric and its unique torsion-free metric-compatible Levi-Civita connection, hence ${\mathcal{L}}_K\nabla =0$. Set $X=K$ and $Y=Z=U$ in part (b). Since ${\nabla }_{U}U=0$,and antisymmetry of the first two curvature arguments givesThis is the geodesic deviation equation with connecting field $K$: applying the isometry to the original geodesic produces a neighboring geodesic, and curvature determines their relative acceleration.

The four independent Killing vector fields are the stationary field ${\partial }_t$ and the three generators of spatial rotations on the two-spheres. In the Schwarzschild interior, $f(r)<0$, soEvery rotational Killing field is tangent to the positive-definite round-sphere metric. All four fields are tangent to a surface of constant $r$, whose induced metricis positive definite. Consequently every nonzero linear combination of the Killing fields is spacelike wherever it does not vanish. A pure rotational Killing field can vanish on its rotation axis, but it is nowhere timelike.

The metric is the left-invariant metricIts right-invariant vector fields areTheir flows act by left translations, which preserve a left-invariant metric. Directly,so both are Killing vector fields and generate one-parameter isometry groups.

There is an additional Killing field. Put $y=e^a>0$ and $x=b$; thenthe hyperbolic plane of constant curvature $-1$. Its isometry algebra is three-dimensional, whereas the space of right-invariant fields here is two-dimensional. For example, the third independent Killing field can be writtenwhich is not right invariant. Hence the answer is yes.

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 $q=A^{\prime \prime }/A=a^{\prime }+a^2$.

A connected nonvacuum perfect fluid with equation of state $p=\rho /3$ obeys the stationary radiation density law $\rho =C/N^4$ with $C>0$. On a compact region of a regular spacetime, a smooth Killing vector field has bounded norm. Consequently $\rho$ cannot tend to zero at a finite regular free surface. A vanishing timelike norm instead makes the density diverge. A radiation-filled cavity can have a material wall with nonzero boundary pressure; this is a different boundary condition.

A symmetric two-tensor satisfying this equation gives a quadratic geodesic first integral. Examples include the Riemannian metric, symmetric products of metric-dual Killing vector fields, and the square of a Killing-Yano two-form.

For $M>0$ and $M\ge |Q|$, put $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The stationary Killing vector field normalized at infinity generates the outer Killing horizon. In an ingoing null chart its covector is $dr$ on that horizon; the Killing equation then gives ${\nabla }_{k}k=\frac{1}{2}{f}^{\prime }(r_{+})k$, where $f=(r-r_{+})(r-r_{-})/r^2$. Thus the positive nonextremal surface gravity is $(r_{+}-r_{-})/(2r_{+}^2)$. It tends to zero when $M=|Q|$, where the horizon becomes a degenerate Killing horizon. The past horizon branch has the opposite signed nonaffinity for the same time-translation field.

A Riemannian metric on a Lie group is right-invariant when every right translation is an isometry. It is determined by a positive-definite inner product at the identity, extended with a coframe of right-invariant differential forms. Its translation-generated Killing vector fields are left-invariant, rather than the right-invariant vector fields dual to that coframe.

For a Schwarzschild black hole of mass $M$ in geometrized units, $f=1-2M/r$ gives $\kappa =f\prime (2M)/2=1/(4M)$. The Killing vector field is normalized at infinity; changing that normalization changes $\kappa$.

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.

For the future-directed central-time Killing vector field in a static patch of de Sitter spacetime, the outgoing null chart gives the future branch coefficient $+1/R$ in ${\nabla }_{k}k=\kappa k$. The ingoing chart gives $-1/R$ on the past branch. Both follow from $d(k^2)=-2\kappa k^{♭}$; the cross-term sign changes the horizon covector. The physical Hawking temperature uses $|\kappa |$, not a signed negative temperature.

The equation of state of isotropic blackbody radiation is $p=\rho /3$. The barotropic hydrostatic first integral gives $\rho N^4=C$. The Stefan–Boltzmann law $\rho \propto T^4$ then yields the Tolman–Ehrenfest relation $TN=\mathrm{constant}$. Here $N=\sqrt{|K^a{K}_a|}$ for the stationary timelike Killing vector field; its normalization only changes the constants.

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.

The surface gravity $\kappa$ measures the failure of a horizon-generating Killing vector field to be affinely parametrized. For a static metric $d{s}^2=-f(r)d{t}^2+f(r{)}^{-1}d{r}^2+\cdots$ with a simple horizon at $r=r_{+}$, $\kappa =|f^{\prime }(r_{+})|/2$.

A timelike vector field assigns a timelike vector continuously or smoothly to every event in its domain. A globally defined such field gives a time orientation, but it need not be a Killing vector field or a timelike geodesic vector field.