What Form Must H Take for a Vacuum Plane Gravitational Wave Metric?

[joseamck]

Hello,

I'm having problems solving this problem I got in class.
I want to learn the concept and how to approach the solution.

Here it is:

Consider the metric

ds=dx^2+dy^2-dudv+2H(x,y,u)du^2

What form must the function H have for this metric to represent a plane gravitational wave propagating in vacuum?




This is how I'm approaching the problem:

first I let the metric be g_{σβ}=[1,0,0,0;0,1,0,0;0,0,2H,-1;0,0,-1,0]

and I know that in vacuum the Einstein equation in far outside the source's field leads to
T^{σβ} = 0
and I know that a metric representing a plane gravitational wave propagating in vacuum is
g_{σβ=[1,0,0,0;0,1,0,0;0,0,0,-1;0,0,-1,0]

coming from the metric ds^2=-dudv+dx^2+dy^2

Then I don't know how to put the pieces together. I was thinking relating the Einstein equation in terms of the ricci tensor and solving for the Christoffel symbols.

Not sure. I need to understand better the problem and how to approach this problem.
Thanks.

 

[latentcorpse]

Can't you LaTeX this?

If the metric has a -dudv term then I think the matrix should have -0.5 in dudv and dvdu positions

 

[joseamck]

the metric is

$ds=dx^2+dy^2-dudv+2H(x,y,u)du^2$

I know that in vacuum
$T^{σβ} = 0$

 

[latentcorpse]

the metric is

$ds=dx^2+dy^2-dudv+2H(x,y,u)du^2$

I know that in vacuum
$T^{σβ} = 0$

I think you should derive the vacuum field equations.

Consider the trace of the EFE. This should allow you to prove that $R_{\mu \nu}=0$

So it looks like you're going to have to work out a bunch of Christoffel symbols. Although it looks as though everything will vanish except for the situations when you have to take a derivative of $g_{uu}$

Hopefully...