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Wave equation in curved spacetime

by Pacopag
Tags: curved, equation, spacetime, wave
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Pacopag
#1
Apr7-08, 01:00 PM
P: 198
Does anyone know how to derive the wave equation in curved spacetime?
[tex](-g)^{-1\over 2}\partial_\mu((-g)^{1\over 2}g^{\mu \nu}\partial_\nu \phi) = 0 [/tex]

A reference, or an outline of the derivation would be very helpful. Thanks.
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Pacopag
#2
Apr7-08, 02:59 PM
P: 198
It seems that just writing the d'Alembertian in covariant form
[tex]\Delta \phi = g^{\mu \nu}\phi_{;\mu \nu}=0[/tex] 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
#3
Apr7-08, 06:13 PM
P: 448
Given a vector [itex]V^\mu[/itex], can you think of any cute expressions for [itex]\nabla_\mu V^\mu[/itex]? (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
#4
Apr7-08, 07:37 PM
P: 303
Wave equation in curved spacetime

Quote Quote by shoehorn View Post
Given a vector [itex]V^\mu[/itex], can you think of any cute expressions for [itex]\nabla_\mu V^\mu[/itex]?
Just a guess, but : [itex]\nabla_\mu V^\mu=V^\mu[/itex]?

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

Regards,

Bill
shoehorn
#5
Apr7-08, 11:01 PM
P: 448
Quote Quote by Antenna Guy View Post
Just a guess, but : [itex]\nabla_\mu V^\mu=V^\mu[/itex]?
Nope. That can't work because the left-hand side is a scalar quantity, while the right-hand side is a vector.

Quote Quote by Antenna Guy
[addendeum: or is it [itex]\nabla_\mu V^\mu=\frac{\delta V^\mu}{\delta t}[/itex]?]

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 [itex](M,g)[/itex], 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 [itex]M[/itex]. The relationship between the covariant divergence of a vector and the partial derivative of the vector is given by

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


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