Past exam of the general relativity II course of the University of Oxford / 2015 / 1 / b

For a Conformal Killing vector field, the same metric compatibility calculation givesThus its Conformal Killing equation is ${\nabla }_{(\mu }{X}_{\nu )}=\lambda g_{\mu \nu }$. Taking the tensor contraction with the inverse metric, and using $g^{\mu \nu }{g}_{\mu \nu }=4$, determines the conformal Killing factor:Parentheses include the factor $1/2$ in tensor index symmetrization.

Use an affine parameter $s$ on the geodesic, so its tangent vector $U$ obeys ${\nabla }_{U}U=0$. The product rule, metric compatibility, and the Conformal Killing equation giveHenceIt is conserved on every null geodesic with an affine parameter, but generally not on other geodesics. On a timelike geodesic parametrized by proper time, $g(U,U)=-1$, so the derivative is $-\lambda$. A Killing vector field, or a geodesic on which $\lambda$ vanishes, supplies the exceptional conservation. This is the conformal Killing geodesic identity.

For an arbitrary parameter with ${\nabla }_{U}U=fU$, the right-hand side acquires $f(X\cdot U)$. Thus even for a null geodesic, the simple conserved expression presupposes an affine parameter.

In Cartesian coordinates on Minkowski spacetime, the Conformal Killing equation becomesDifferentiate the equations with index pairs $(\nu ,\rho )$, $(\mu ,\rho )$ and $(\mu ,\nu )$ by ${\partial }_{\mu }$, ${\partial }_{\nu }$ and ${\partial }_{\rho }$, respectively. Adding the first two and subtracting the third, Clairaut's theorem yields the flat conformal Killing integrability formulaApply ${\partial }^{\rho }$ and use ${\partial }^{\rho }{X}_{\rho }=4\lambda$. With the d'Alembert operator $\square ={\eta }^{\alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }$, this givesA tensor contraction with ${\eta }^{\mu \nu }$ gives $2\square \lambda =-4\square \lambda$, so $\square \lambda =0$. Returning to the preceding equation proves the stronger statementThus the conformal Killing factor is an affine function, not an arbitrary solution of the wave equation.

Write the affine function obtained in part (iii) as $\lambda =c+2b_{\mu }{x}^{\mu }$, with constant $c$ and $b_{\mu }$. A Conformal Killing vector field with precisely this conformal Killing factor isDirect partial differentiation verifies its Conformal Killing equation. Subtracting it from any other solution leaves a Killing vector field $Y$. The formula in part (iii), now with $\lambda =0$, says ${\partial }_{\mu }{\partial }_{\nu }{Y}_{\rho }=0$. Therefore $Y$ is affine, and the Killing equation makes its linear coefficient antisymmetric after lowering an index. Every solution is consequentlyThe four constant $a^{\mu }$ generate translations, the six independent ${\omega }_{\mu \nu }$ generate Lorentz transformations, $c$ generates a dilation, and the four $b^{\mu }$ generate special conformal transformations. Their constant, linear and quadratic terms are independent; within the linear terms, the antisymmetric coefficient cannot equal a nonzero multiple of the identity. Hence the conformal Killing algebra of Minkowski spacetime hasFor $\lambda =1$, the canonical choice is the dilation generator $D=x^{\mu }{\partial }_{\mu }$. Its flow of a vector field solves $d{x}^{\mu }/ds=x^{\mu }$, givingOne may add any Killing vector field without changing $\lambda =1$, so this choice is not unique. The displayed dilation gives a global one-parameter group of diffeomorphisms. The count of fifteen concerns independent Conformal Killing vector fields; some special conformal transformations have only local flows on all of affine Minkowski spacetime.