# Transform TT gauge -> orthonormal comoving frame (MTW 35.5)

1. Apr 18, 2010

### IAmAZucchini

1. The problem statement, all variables and given/known data

Introduce a TT coordinate system in which, at time t=0, the two particles are both at rest. Use the geodesic equation to show that subsequently they both always remain at rest in the TT coordinates, despite the action of the wave. This means that the vontravariant components of the separation vector are always constant in the TT coordinate frame:
$$n^{j}=x_{B}^{j} - x_{A}^{j} = const$$
Call this constant
$$x_{B(0)}^{j}$$.
Transform these components to the comoving orthonormal frame, the answer should be
$$x_{B}^{\hat{j}}(\tau)=x_{B(0)}^{\hat{k}}(\delta_{jk}+\frac{1}{2}h_{jk}^{TT})_{(at position of A)}$$

2. Relevant equations
I've got the first part down, I just need to show the transformation, which I for some reason cannot do. I think the relevant equation is
$$h_{jk}^{TT}=P_{jl}P_{mk}h_{lm}-\frac{1}{2}P_{jk}(P_{ml}h_{lm})$$
and what I have right now is, in the TT gauge,
$$x_{B}^{j}(\tau)=x_{B(0)}^{j}$$
but I don't know where to go from there...

3. The attempt at a solution
Perhaps multiply the equation by $$x_{B(0)}^{j}$$, but not sure this will help.

I think this is an incredibly simple problem and I am just missing something fundamental.

Thanks for any help!