Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B (General Relativity)

PutUsing metric compatibility and the symmetric Hessian of a scalar field, the divergence of the proposed tensor isFor this to vanish for arbitrary solutions and local field data, the two coefficients must vanish. HenceThe resulting tensor is the Conserved Klein-Gordon scalar stress-energy ansatz.

Write the Einstein field equations in $D$ dimensions asTaking the trace givesTherefore the Trace-reversed Einstein field equations in D dimensions are

For the values from part (a), let $Y={\nabla }_{\rho }\varphi {\nabla }^{\rho }\varphi$. ThenandSubstitution cancels the $Y{g}_{\mu \nu }$ terms and yieldsFor the convention $\kappa =8\pi G$, the prefactor is $4\pi G$. This is the Ricci curvature sourced by a Klein-Gordon scalar field.

Because the condition is local and covariant, choose normal coordinates at the point and a Lorentz frame in which the timelike observer is $X^{\mu }={\delta }_0^{\mu }$. Rescaling a timelike vector by a positive factor does not affect the sign, so it suffices to use this unit vector. With signature $(-,+,\dots ,+)$,The measured energy density is thereforeThe value of $\varphi$, its time derivative, and its spatial first derivatives can be varied independently at a point. Nonnegativity for all such data is therefore equivalent to nonnegativity of all three coefficients:Thus the most general constraints areThese conditions are also sufficient in every timelike frame, because any timelike vector can be brought to the chosen rest frame. This is the weak energy condition for a quadratic scalar stress-energy ansatz.

The conserved values from part (a),satisfy these inequalities for real $m$, as required.