Question on alternatives to GR

[skippy1729]

Over the years there have been many alternative theories of gravity: Brans-Dicke, quadratic Lagrangians, F(R) &ct. Do any of these share the nice property of GR that the equations of motion are a consequence of the field equations and do not have to be separately postulated?

Thanks, skippy

 

[bcrowell]

What do you mean by "equations of motion" in contradistinction to "field equations?" To me, the field equations *are* the equations of motion of the theory.

By equations of motion, do you mean the equations of motion for a test particle? If so, then any metric theory has the property you're talking about, since test particles move along geodesics determined by the metric. For example, Brans-Dicke gravity is a metric theory.

-Ben

 

[skippy1729]

What do you mean by "equations of motion" in contradistinction to "field equations?" To me, the field equations *are* the equations of motion of the theory.

By equations of motion, do you mean the equations of motion for a test particle? If so, then any metric theory has the property you're talking about, since test particles move along geodesics determined by the metric. For example, Brans-Dicke gravity is a metric theory.

-Ben

That test particles (which produce negligible back-reaction on the metric) follow geodesics was originally an added postulate to the field equations. Point particles of non-negligible mass moving on arbitrary world lines will not in general satisfy the field equations. The path that they must follow for the field equations to be satisfied is determined by the equations of motion. This is what I mean by equations of motion. Note that the original calculations of Mercury's orbit were done under the assumption that Mercury was a test particle. The problem for General Relativity was not solved until the 1938 paper by Einstein, Infeld and Hoffmann "The Gravitational Equations and the Problem of Motion" Annals of Mathematics Vol. 39 No. 1 1938.

Skippy