I have the following question to solve:Use the metric:$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$
Test bodies are arranged in a circle on the metric at rest at ##t=0##.
The circle define as $$x^2 +y^2 \leq R^2$$
The bodies start to move on geodesic when we have $$a(0)=0$$
a. we have to...
I understood it is not the same, this is why I used the derivation of a product rule and got the expression i got. (I attached a photo of how i got it).
Yes, i have expression for the Ricci tensors (if that's what you mean):
$$R_{tt} = \frac{2a^2(2a'^2 -aa''(a^4-1))}{(a^4-1)^2}$$
$$R_{xx} =...
Thank you very much for responding. i am not sure i understood what you are saying about the derivative of the area. I carry the integral all the way.
the integration of the integral should be:
for y: $$ -\sqrt{R^2-x^2}<y<\sqrt{R^2-x^2}$$
and for x: $$-R<x<R$$
That's how i understad it so farm...
i attach here the calculation in the photos.
This is how i understood to take the second derivative respected to time. I understood that dx and dy are also being effected.
I also show there the Geodesic equations i have calculated using the Christoffel symbols.
can you please explain what...
thank you for your reply.
I have calculated the Christoffel symbols using Mathmatica. are you sure there is a sign error? isn't it that: $$\Gamma^{y}_{ty} = \Gamma^{y}_{yt}$$?
If it's not the case i will try to calculate agin but it according to Mathmatica it should be the signs (else i don't...
I have the following question to solve:Use the metric:
$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$
Test bodies are arranged in a circle on the metric at rest at $$t=0$$.
The circle define as $$x^2 +y^2 \leq R^2$$
The bodies start to move on geodesic when we have $$a(0)=0$$
a. we have to...