- #1
joseamck
- 13
- 0
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.
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.