# Proper time in Nordstrom gravity

This wikipedia article claims that there are two interpretations of Nordstrom's scalar theory of gravity:

1) A scalar field theory on flat space. The reason why an apple falls is that its mass is coupled to $\phi$, which causes it to deviate from a geodesic.

2) A theory of curved space-time. $R = k T$. The metric of space-time is $g = \phi^2(x) \eta$. The falling apple travels on a geodesic.

My question is, why are these considered two interpretations of the same theory, and not two separate theories? Although both give the same predictions for the trajectories of a particles, shouldn't the time elapsed on a watch differ by a factor of $\sim \phi$ between the two theories?

In other words, suppose that a particle gets displaced a small amount $\delta x$ in space-time. The proper time elapsed on its watch is according to 1) is $\delta \tau^2 \sim \eta \delta x \delta x$, and in 2) isn't $\delta \tau^2 \sim \phi^2 \delta x \delta x$? On the other hand, wiki says:

which metric is the one which according to Nordström can be measured locally by physical experiments? The answer is: the curved spacetime is the physically observable one in this theory (as in all metric theories of gravitation); the flat background is a mere mathematical fiction

Wiki says : "Einstein and Fokker observed that the Lagrangian for Nordström's equation of motion for test particles, $L = \phi^2 \, \eta_{ab} \, {u}^a \, {u}^b$ , is the geodesic Lagrangian for a curved Lorentzian manifold with metric tensor $g_{ab} = \phi^2 \, \eta_{ab}$ ."
[Remark : here $u^a = \frac{dx^a}{ds}$, so no "dot" here on the $u^a$, I think there is an error in wiki]
This means that the Euler-Lagrange equations applying to the lagrangian $L$ (in a Minkowski space-time), are equivalent to a geodesic equation with the metric tensor $g_{ab}$, so, in a curved space-time.
So, the Lagrangian point of view (in a flat space-time) is an alternative mathematical presentation (possible due to the simplicity of the metrics), but the physical point of view is that there is a curved spacetime. So, the correct physical proper time is given, as usual, by $ds^2= g_{ab} dx^a dx^b$
• The true theory (the second theory) is a curved space time with $R=kT$. It is an equation for the gravitational field. In the first theory, there is no equation for the gravitational field. There is only a mathematical correspondence, for the movement of a particle in the first theory, from Lagrange-Euler equations, and a geodesic in the second theory. – Trimok Nov 25 '13 at 19:35