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Question wave equation in curved spacetime ( Physics Help and Math Help - Physics Forums Special General Relativity )
Updated: 2008-04-20 01:23:57 (4)
wave equation in curved spacetime

Does anyone know how to derive the wave equation in curved spacetime?
Click to see the LaTeX code for this image

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

Answers: wave equation in curved spacetime ( Physics Help and Math Help - Physics Forums Special General Relativity )
wave equation in curved spacetime

Originally Posted by Antenna Guy
Just a guess, but : Click to see the LaTeX code for this image?
Nope. That can't work because the left-hand side is a scalar quantity, while the right-hand side is a vector.

Originally Posted by Antenna Guy
[addendeum: or is it Click to see the LaTeX code for this image?]

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 Click to see the LaTeX code for this image, 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 Click to see the LaTeX code for this image. The relationship between the covariant divergence of a vector and the partial derivative of the vector is given by

Click to see the LaTeX code for this image

shoehorn

wave equation in curved spacetime

Originally Posted by shoehorn
Given a vector Click to see the LaTeX code for this image, can you think of any cute expressions for Click to see the LaTeX code for this image?
Just a guess, but : Click to see the LaTeX code for this image?

[addendeum: or is it Click to see the LaTeX code for this image?]

Regards,

Bill

Antenna Guy

wave equation in curved spacetime

It seems that just writing the d'Alembertian in covariant form
Click to see the LaTeX code for this image 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.

Pacopag

wave equation in curved spacetime

Given a vector Click to see the LaTeX code for this image, can you think of any cute expressions for Click to see the LaTeX code for this image? (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.)

shoehorn

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