Nordström's theory of gravitation

Neither of Nordström's theories are in agreement with observation and experiment. Nonetheless, the first remains of interest insofar as it led to the second. The second remains of interest both as an important milestone on the road to the current theory of gravitation, general relativity, and as a simple example of a self-consistent relativistic theory of gravitation. As an example, this theory is particularly useful in the context of pedagogical discussions of how to derive and test the predictions of a metric theory of gravitation.

He then derived an expression for the stress–energy tensor of the gravitational field in Nordström's second theory,

which he proposed should hold in general, and showed that the sum of the contributions to the stress–energy tensor from the gravitational field energy and from matter would be conserved, as should be the case. Furthermore, he showed, the field equation of Nordström's second theory follows from the Lagrangian

Since Nordström's equation of motion for test particles in an ambient gravitational field also follows from a Lagrangian, this shows that Nordström's second theory can be derived from an action principle and also shows that it obeys other properties we must demand from a self-consistent field theory.

We can immediately write down the general vacuum solution in Nordström's theory:

in Nordström's theory, electromagnetic field energy does not gravitate!

In any Lorentzian manifold (with appropriate tensor fields describing any matter and physical fields) which stands as a solution to Nordström's field equations, the conformal part of the Riemann tensor (i.e. the Weyl tensor) always vanishes. The Ricci scalar also vanishes identically in any vacuum region (or even, any region free of matter but containing an electromagnetic field). Are there any further restrictions on the Riemann tensor in Nordström's theory?

leaves the trace-free Ricci tensor entirely unconstrained by algebraic relations

which constrains the semi-traceless part of the Riemann tensor (the one built out of the trace-free Ricci tensor).

otherwise gravitation would not, according to this theory, be a long-range force capable of propagating through a vacuum

We can tabulate the most basic differences between Nordström's theory and general relativity, as follows:

At this point, we could show that in the limit of slowly moving test particles and slowly evolving weak gravitational fields, Nordström's theory of gravitation reduces to the Newtonian theory of gravitation. Rather than showing this in detail, we will proceed to a detailed study of the two most important solutions in this theory:

We will use the first to obtain the predictions of Nordström's theory for the four classic solar system tests of relativistic gravitation theories (in the ambient field of an isolated spherically symmetric object), and we will use the second to compare gravitational radiation in Nordström's theory and in Einstein's general theory of relativity.

The static vacuum solutions in Nordström's theory are the Lorentzian manifolds with metrics of the form

Adopting polar spherical coordinates, and using the known spherically symmetric asymptotically vanishing solutions of the Laplace equation, we can write the desired exact solution as

These are exactly the same vector fields which arise in the Schwarzschild coordinate chart for the Schwarzschild vacuum solution of general relativity, and they simply express the fact that this spacetime is static and spherically symmetric.

The geodesic equations are readily obtained from the geodesic Lagrangian. As always, these are second order nonlinear ordinary differential equations.

where to first order in m we have the same result as for the Schwarzschild vacuum. This also shows that Nordström's theory agrees with the result of the Pound–Rebka experiment. Second, we have

It makes sense to ask how much force is required to hold a test particle with a given mass over the massive object which we assume is the source of this static spherically symmetric gravitational field. To find out, we need only adopt the simple frame field

Then, the acceleration of the world line of our test particle is simply

In other words, nearly circular orbits will exhibit a radial oscillation. However, unlike what happens in Newtonian gravitation, the period of this oscillation will not quite match the orbital period. This will result in slow precession of the periastria (points of closest approach) of our nearly circular orbit, or more vividly, in a slow rotation of the long axis of a quasi-Keplerian nearly elliptical orbit. Specifically,

and to first order in m, the long axis of the nearly elliptical orbit rotates with the rate

This can be compared with the corresponding expression for the Schwarzschild vacuum solution in general relativity, which is (to first order in m)

Thus the coordinate time from the first event to the event of closest approach is

Here the elapsed coordinate time expected from Newtonian theory is of course

We can summarize the results we found above in the following table, in which the given expressions represent appropriate approximations:

which we can interpret in terms of the propagation of a gravitational plane wave.

This Lorentzian manifold admits a six-dimensional Lie group of isometries, or equivalently, a six-dimensional Lie algebra of Killing vector fields: