Wave equation in curved spacetime

[Pacopag]

Does anyone know how to derive the wave equation in curved spacetime?
(-g)^{-1\over 2}\partial_\mu((-g)^{1\over 2}g^{\mu \nu}\partial_\nu \phi) = 0

A reference, or an outline of the derivation would be very helpful. Thanks.

 

[Pacopag]

It seems that just writing the d'Alembertian in covariant form
\Delta \phi = g^{\mu \nu}\phi_{;\mu \nu}=0 does the trick.
This form is giving me the results I want, but I still don't know how to put it in the form written in my original post.

 

[shoehorn]

Given a vector V^\mu, can you think of any cute expressions for \nabla_\mu V^\mu? (Hint: you can write the covariant four-divergence of a vector in terms of the coordinate four-divergence much like the expression in the first post.)

 

[Antenna Guy]

Given a vector V^\mu, can you think of any cute expressions for \nabla_\mu V^\mu?

Just a guess, but : \nabla_\mu V^\mu=V^\mu?

[addendeum: or is it \nabla_\mu V^\mu=\frac{\delta V^\mu}{\delta t}?]

Regards,

Bill

 

[shoehorn]

Just a guess, but : \nabla_\mu V^\mu=V^\mu?

Nope. That can't work because the left-hand side is a scalar quantity, while the right-hand side is a vector.

[addendeum: or is it \nabla_\mu V^\mu=\frac{\delta V^\mu}{\delta t}?]

Regards,

Bill

Not this either. Perhaps it will be simpler if I just state the result and leave the proof as an exercise. Given an orientable Riemannian manifold (M,g), one has a preferred idea of a connection in the form of the Levi-Civita connection. This allows one to define, for example, a straightforward notion of covariant differentiation on tensor fields over M. The relationship between the covariant divergence of a vector and the partial derivative of the vector is given by

\nabla_\mu V^\mu = \frac{1}{\sqrt{\textrm{det}g_{\rho\sigma}}}\partial_\mu(\sqrt{\textrm{det}g_{\rho\sigma}}V^\mu)