stress energy tensor

[Sanford]

Someone emailed me the following statement that does not make sense to me.

 

"The Einstein theory is an example of a non-linear theory in which the stress energy tensor vanishes in a mass zero zone."

 

Is there any evidence of nonlinearity? How can the tress energy tensor vanish near a mass when the field does not vanish?

[ajb]

The field equations are clearly non-linear in the metric. This leads to technical difficulties in dealing with gravity but also is the origin of gravitational self-interactions.

 

Think of point charges in electromagnetic theory. The charge is isolated at a point but the electromagnetic field is non-trivial around the charge. The same thing happens in general relativity. The energy-momentum acts as a source of gravity. This does not mean that where we have no energy-momentum the geometry is necessarily trivial (flat).

 

The so called vacuum field equations reduce to the statement that the space-time is Ricci flat. ([math]R_{\mu \nu}=0 [/math] ). This does not mean that the Riemann curvature is zero.

 

In fact I am sure you know of several vacuum solutions, that is no energy-momentum on the RHS of the field equations. For example the Schwarzschild solution is one well-known example.