How to prove that zero Weyl tensor predicts no deflection of light?

Question

There is Nordstrom theory, which can be given as $$ C_{\mu \nu \alpha \beta} = 0. $$ The solution of Einstein equations for this case is conformally flat metric: $$ g^{\mu \nu} = e^{\epsilon \varphi (x)}\eta^{\mu \nu}. $$ How to show that this theory predicts no deflection of light?

Answer 1

In the solar system, there is only a weak gravitational field outside the sun. So for practical purpose, you can expand the metric to the first order in $\epsilon$(and I guess that why have this parameter in the definition),

$$g_{\mu\nu} = e^{-\Phi }\eta_{\mu\nu} = e^{-\epsilon\phi} \eta_{\mu\nu} \approx -(1-\epsilon\phi) dt^2 + (1-\epsilon\phi ) (dx^2 +dy^2 + dz^2 ) $$ Notice that spatial part of the perturbation metric has the opposite side to the linearized Einstein theory. That's exactly why you don't have light bending. It is a good exercise to do, since the derivation is completely parallel to the usual linearized theory, see for example, MTW, exercise 18.6.

However, the result is true even beyond linear order. You know the equation of motion of photon is geodesic equation, $$ \nabla_p p = 0 $$

where momentum $p$ is the tangent vector of the geodesic curve and for photon it is a null vector. In components, $$ \frac{d}{d\tau} p^{\mu} - p^{\mu} p^\nu \partial_{\nu} \Phi = 0 $$ where I have used the explicit form of connection and $p^{\mu}p_{\mu} =0$(please verify it).

This equation can be integrated out, $$ \frac{d}{d\tau} p^{\mu} - p^\mu \frac{d}{d\tau} \Phi = e^{\Phi} \frac{d}{d\tau}(e^{-\Phi} p^\mu) = 0 $$ so $e^{-\Phi}p^\mu$ is a constant alone the geodesic.

Following the standard procedure, we compare 4-momentum at emission and receiving; in both cases the photon are very far away from the star in the center, and thus $\Phi \sim 0 $. Therefore we can conclude that the 4 momentum doesn't change in the asymptotic flat space, i.e. no light bending.

You can read MTW exercise 7.1. That's a problem starting from an action of a scalar(Nordstorm theory) gravitational field, and there are also some useful hints and comments in the text.

Added: The conserved quantities are due to the four conformal Killing vectors: $\partial_{\mu} $. $$\mathcal{L}_{\partial_\mu} g = -\partial_{\mu}\Phi g $$ Let $\xi= \partial_\mu$, then $g(p,\xi)= p^\nu \xi_\nu$ is a conserved quantity. That's because $$\nabla_p g(p,\xi) =(\nabla_p g)( p, \xi ) +g( \nabla_p p, \xi) + g(p, \nabla_p \xi ) = g(p, \nabla_p \xi )\\= g(p, [p,\xi] ) + g(p, \nabla_{\xi} p ) = - g( p, \mathcal{L}_{\xi} p ) + \frac{1}{2} \nabla_{\xi}g(p,p ) \\= -\frac{1}{2} \mathcal{L}_{\xi} g(p,p) + \frac{1}{2} (\mathcal{L}_{\xi} g)( p, p ) = -\partial_\mu \Phi g( p, p ) = 0 $$

A quick check shows that $p^\mu \xi_\mu = p^\mu e^{-\Phi } $ are just what we have derived.