Past exam of the general relativity and cosmology course of the University of Oxford / 2013 / 2

Although ${\nabla }_{\beta }f={\partial }_{\beta }f$ for a scalar field, its first derivative is a covector field. ThereforeCommuting partial derivatives and using the torsion-free connection gives $\boxed{[{\nabla }_{\alpha },{\nabla }_{\beta }]f=0}$. For a vector field, differentiate the $(1,1)$ tensor ${\nabla }_{\beta }{V}^{\mu }$ with both its indices included:Subtract the expression with $\alpha ,\beta$ exchanged. Second derivatives of $V$ cancel; the terms multiplying first derivatives cancel pairwise, and the final lower-index connection terms cancel by symmetry. The remaining coefficient of $V^{\nu }$ isThis is the curvature commutator. On a covariant index the sign reverses, giving the curvature commutator on a covariant tensor used in the next part.

For a torsion-free connection, the connection terms in the antisymmetrized covariant derivative cancel: $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. Thus the electromagnetic field tensor is the exterior derivative of the four-potential. It is a tensor, is invariant under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\chi$, and obeys the homogeneous Maxwell equations ${\nabla }_{[\alpha }{F}_{\beta \gamma ]}=0$. The covariant divergence equation is the generally covariant form of the local source equation; use its minus sign as specified in the paper.

Expand that equation before imposing the Lorenz gauge condition:The curvature commutator on a covariant tensor givesHere the contraction $g^{\mu \rho }{R}^{\alpha }{}_{\mu \rho \nu }=-R^{\alpha }{}_{\nu }$ accounts for the second sign reversal. Hence in Lorenz gaugeThis curved-spacetime electromagnetic potential wave equation contains a Ricci tensor term even in a locally inertial coordinate system: a connection can vanish at one point while its derivatives and curvature remain nonzero.

Write $\mathcal{H}=a^{\prime }/a$, with the prime denoting ${\partial }_0=c^{-1}{\partial }_{\eta }$. The metric and inverse metric are $g_{\mu \nu }=a^2{\eta }_{\mu \nu }$ and $g^{\mu \nu }=a^{-2}{\eta }^{\mu \nu }$. Since $a$ depends only on $x^0$, the Christoffel symbols reduce toConsequentlyFor the stated de Sitter spacetime scale factor, $\mathcal{H}=-1/(c\eta )$. The factor $c^{-1}$ is required because the prime is not differentiation with respect to $\eta$ itself.

For a minimally coupled massless scalar field, the covariant wave operator isThe Fourier mode $\varphi =u(\eta )e^{i\mathbf{k}\cdot \mathbf{r}}$ therefore satisfiesSubstituting $u=(A+B\eta )e^{-i\omega \eta }$ leaves $2(i\omega A-B)e^{-i\omega \eta }/\eta$. ThusThis is an unrescaled de Sitter massless scalar mode. For $k\ne 0$, the opposite-frequency time solution supplies the second independent mode; a real scalar field combines appropriate conjugate spatial modes. For $k=0$ the particular ansatz gives only the constant solution, while the full homogeneous solution is $u=C+D{\eta }^3$. The requested ansatz was not a general zero-mode parametrization.