How Are Gravitational Waves Derived from Einstein's Field Equations?

[Dale]

How do you get gravitational waves or gravitons out of the EFE? It certainly doesn't look like a wave equation. Are there some second derivatives hidden in the Einstein tensor?

 

[cristo]

Yes, there are second derivatives hidden in the Einstein tensor. If you think of what it's made up of: Riemann tensor, which is made up of derivatives of the Christoffel symbols, which are made up of derivatives of the metric tensor.

 

[joelperr]

You don't get gravitons out of Einstein's field equations; those only show up when you attempt to generate an effective quantum field theory that includes gravity.
For gravitational waves, the easiest method would be to use the weak-field equations in the transverse gauge, and set the energy-momentum tensor to zero (which corresponds to solutions of the equation infinitely far away from the originating source term). After a few lines of basic tensor analysis, you're left with the curved-spacetime version of the homogeneous wave equation, in terms of the d'Alembertian operator.