Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 1

Choose local coordinates $y^1,\dots ,y^{D-1}$ on the hypersurface $\Sigma$, and let ${\Phi }_s$ be the local flow of the vector field $X$. The map $F(s,y)={\Phi }_s(p(y))$ follows the integral curves of a vector field starting at $p(y)\in \Sigma$. At $s=0$, its differential maps the $s$ direction to $X$ and the remaining directions to a basis of $T_p\Sigma$. Transversality makes these vectors linearly independent, so the inverse function theorem makes $F$ a local manifold chart.

Set $x^0=s$ and transport the $y^i$ unchanged along each integral curve of a vector field. Then the adapted coordinates satisfyThe flow-box theorem gives this construction locally. The parameter is the flow parameter, and need not be arc length; a global manifold chart would require additional hypotheses.

In a stationary spacetime chart, take $V={\partial }_0$. Its squared length is $g_{00}<0$, so $V$ is a timelike vector. Since its components are constant, the Lie derivative of a tensor field obeys $({\mathcal{L}}_{V}g{)}_{\mu \nu }={\partial }_0{g}_{\mu \nu }=0$. For the Levi-Civita connection,so $V$ satisfies the Killing equation.

Conversely, a timelike Killing vector field is nowhere zero. Apply the flow-box theorem to choose coordinates with $V={\partial }_0$. The Killing equation then gives ${\partial }_0{g}_{\mu \nu }=0$, and $g_{00}=g(V,V)<0$. Thus local stationarity is equivalent to the existence of a timelike Killing vector field.

Use the standard meaning of static spacetime: a stationary spacetime chart whose time-space components also vanish. The printed static definition needs to inherit the timelike requirement $g_{00}<0$; without it, the requested timelike conclusion does not follow.

The preceding argument shows that $V={\partial }_0$ is a timelike Killing vector field. Its metric-dual one-form is $v=V_{\mu }d{x}^{\mu }=g_{00}d{x}^0$. Its exterior derivative satisfiesFor a torsion-free connection, antisymmetrizing the covariant derivative of a one-form gives its exterior derivative. Consequently the Killing vector has hypersurface orthogonality:Geometrically, the hypersurfaces $x^0=\mathrm{constant}$ are orthogonal to $V$.

Write $F=V_{\rho }{V}^{\rho }<0$ and $A_{\mu \nu }={\nabla }_{\mu }{V}_{\nu }$. The Killing equation makes $A_{\mu \nu }$ antisymmetric. Expanding hypersurface orthogonality and contracting with $V^{\alpha }$ givesHere metric compatibility gives ${\partial }_{\mu }F=2V^{\rho }{A}_{\mu \rho }$. Thereforeand the normalized Killing one-form satisfiesThus the required exponent is $\boxed{n=-2}$. The notation $|V{|}^2$ in this paper denotes the negative scalar $F$, rather than a positive squared norm; the answer uses only its real reciprocal. More generally, for a multiplier $h(F)$ the antisymmetric derivative is $(h/F+h^{\prime })(V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F)$, so $h=C/F$ is the universally valid choice. If $F$ is constant, other exponents can work in that particular metric.

The normalized Killing one-form $\eta =F^{-1}{V}_{\mu }d{x}^{\mu }$ is a closed differential form. The Poincaré lemma gives a local scalar potential $\varphi$ with $\eta =d\varphi$, hence $V_{\mu }=F{\partial }_{\mu }\varphi$. Choose the adapted chart of part (i), so $V={\partial }_0$. ThenThe closed differential form condition supplies the integrability of these spatial derivatives. For a fixed $g$ and $V$, $f$ is determined locally up to an additive constant; its derivatives cannot be chosen freely despite the wording in the paper.

Use ${\bar{x}}^0=\varphi$, ${\bar{x}}^i=x^i$. The Jacobian determinant is one, so this is a local coordinate change. Holding ${\bar{x}}^i$ fixed gives $V={\partial }_{\bar{0}}$, while its metric-dual one-form is $v=Fd{\bar{x}}^0$. Thus ${\bar{g}}_{00}=F<0$ and ${\bar{g}}_{0i}=0$. The Killing equation still holds, so ${\partial }_{\bar{0}}{\bar{g}}_{\mu \nu }=0$. The metric is locally static:A closed differential form need not be globally exact; neither a global potential nor a global static chart follows without additional topological assumptions.