- #1

Gleeson

- 30

- 4

- TL;DR Summary
- Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.

I have been learning a bit about Fermi normal coordinates in Eric Poisson's "A Relativist's Toolkit". Problem 1.10 in this book is to express the Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.

I know that in the Fermi normal coordinates (denoted {T, X^a} where a = 1, 2, 3), the metric to quadratic order will be:

$$g_{TT} = -1 + R_{TaTb}(T)X^aX^b $$

$$g_{Ta} = -2/3R_{Tbac}(T)X^bX^c $$

$$ \delta_{ab} - 1/3 R_{abcd}(T)X^c X^d $$

T is the proper time of along the geodesic.

So to solve this I believe I need the above components of the Riemann tensor, in the Fermi normal coordinates, evaluated on the geodesic of interest.

I'm not sure where to start, and would appreciate some suggestions please.

I was thinking that I should first consider Schwarzschild metric in Schwarzschild coordinates. In these coordinates then I should find the radially infalling geodesic(s). Then, still in these coordinates, find the orthonormal basis attached to this geodesic (the time like vector corresponding to tangent vector to the geodesic). Then I should calculate:

$$ R_{Tabc}(T) = R_{\mu \nu \alpha \beta}e^{\mu}_T e^{\nu}_a e^{\alpha}_b e^{\beta}_c $$

where the greek indices denote the Scharzschild coordinates.

Does this seems correct? Is there a better way to do this?

I know that in the Fermi normal coordinates (denoted {T, X^a} where a = 1, 2, 3), the metric to quadratic order will be:

$$g_{TT} = -1 + R_{TaTb}(T)X^aX^b $$

$$g_{Ta} = -2/3R_{Tbac}(T)X^bX^c $$

$$ \delta_{ab} - 1/3 R_{abcd}(T)X^c X^d $$

T is the proper time of along the geodesic.

So to solve this I believe I need the above components of the Riemann tensor, in the Fermi normal coordinates, evaluated on the geodesic of interest.

I'm not sure where to start, and would appreciate some suggestions please.

I was thinking that I should first consider Schwarzschild metric in Schwarzschild coordinates. In these coordinates then I should find the radially infalling geodesic(s). Then, still in these coordinates, find the orthonormal basis attached to this geodesic (the time like vector corresponding to tangent vector to the geodesic). Then I should calculate:

$$ R_{Tabc}(T) = R_{\mu \nu \alpha \beta}e^{\mu}_T e^{\nu}_a e^{\alpha}_b e^{\beta}_c $$

where the greek indices denote the Scharzschild coordinates.

Does this seems correct? Is there a better way to do this?