Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 61 / 2 / a / Solution

Scalar-force gravitational redshift

{title2=$dp^i=\partial^i\Phi\,p_jdx^j-(p^j\partial_j\Phi)dx^i$}

= Scalar-force gravity
{synonym}

This covariant scalar-force model uses the <Minkowski metric> and the displayed momentum law, with $\Box\Phi=-4\pi G T^i{}_i$. For a massive <test particle>, write $p^i=mu^i$, $u^iu_i=1$. Contracting the force law with $p_i$ gives constant <rest mass>; the acceleration is $du^i/d\tau=\partial^i\Phi-u^i(u^j\partial_j\Phi)$, independent of the mass. A static weak field obeys the <Poisson equation> and yields Newtonian acceleration $-\boldsymbol\nabla\Phi$. For a <photon>, $p\cdot dx=0$ and $dx$ is parallel to $p$, so $dp^i=-p^i d\Phi$: photon energies change but their unparametrized paths remain straight. The motion law has the same unparametrized curves as the conformal metric $e^{2\Phi}\eta$, but the field equation must not be silently identified with a different scalar-gravity theory.

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 Incoming links: Flat-background scalar-force theory of gravity