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

Use units $c=1$ and signature $(+---)$. For a massive test particle, write its four-momentum as $p^i=m{u}^i$, where $u^i=d{x}^i/d\tau$, $\tau$ is Minkowski proper time, and $u^i{u}_i=1$. The stated flat-background scalar-force theory of gravity then givesContract with $p_i=m{u}_i$. The two terms cancel:Thus rest mass is conserved and the acceleration law isThere is no mass or composition parameter in this equation. Particles released at the same event with the same velocity consequently have identical trajectories. In a static weak field at small speed, its spatial part becomes $d^2\mathbf{x}/d{t}^2=-\nabla \Phi$, since raising a spatial index introduces a minus sign. The field equation likewise reduces to ${\nabla }^2\Phi =4\pi G\rho$ for nonrelativistic matter, because $T^i{}_i\simeq \rho$ and $\square =-{\nabla }^2$ in the static limit.

Therefore yes: the model agrees with the Eötvös test of universal free fall. In the point-particle/test-body approximation, inertial and passive gravitational masses have the same universal ratio. This is the weak equivalence principle tested by the Eötvös experiment; it is not a claim about strongly self-gravitating bodies or a proof of all aspects of the Einstein equivalence principle.

For a photon, $p^i{p}_i=0$ and the null displacement along the ray is parallel to $p^i$. Consequently $p_{j}d{x}^j=0$, and the first term of the momentum law vanishes. Write $d{x}^i=p^{i}d\lambda$ with an appropriate path parameter. ThenFor a static potential, a photon emitted and received by laboratories at rest in the background frame has energy proportional to $p^0$. Using $E=h_{\mathrm{P}}\nu$, the scalar-force gravitational redshift isRestoring $c$, the leading fractional shift is $\Delta \nu /\nu =-\Delta {\Phi }_N/c^2$, where ${\Phi }_N$ is the dimensional Newtonian gravitational potential. Near Earth's surface, an upward displacement $H$ has $\Delta {\Phi }_N=gH>0$, hence $\Delta \nu /\nu =-gH/c^2$: the receiver sees a redshift.

Thus yes: the predicted weak-field frequency shift agrees with the Pound-Rebka experiment. The calculation concerns stationary emitter and receiver, with identical local transition-energy standards; it does not add a Doppler shift due to their relative motion. Agreement with this gravitational redshift measurement alone does not establish the tensor theory of general relativity.

The null momentum equation gives $p^i=C^i{e}^{-\Phi }$, with a constant null vector $C^i$. Since the tangent is parallel to $p^i$, every spatial direction ratio is constant:The changing potential changes the scale of the four-momentum, not its direction. Photon paths are therefore straight lines in the background Minkowski spacetime, up to reparametrization. This is also consistent with conformal preservation of null geodesic paths: the particle motion law can be expressed through a conformally flat metric, which cannot change the unparametrized null trajectories of the background.

Hence no: the theory predicts zero solar light deflection, in disagreement with the observed nonzero bending of light. For comparison, the leading Schwarzschild light deflection is $4GM/(b{c}^2)$ at impact parameter $b$. The scalar theory's nonzero gravitational redshift does not rescue this failed directional prediction.