# Wave equation in curved spacetime

by Pacopag
Tags: curved, equation, spacetime, wave
 P: 197 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.
 P: 197 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.
 P: 448 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.)
P: 303
Wave equation in curved spacetime

 Quote by shoehorn 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
P: 448
 Quote by Antenna Guy 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.

 Quote by Antenna Guy [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

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

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