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Suppose I have an Einstein manifold (a manifold with a metric that solves the Einstein field equations. We can take $\Lambda = 0$ for simplicity in this example). The worldline of a freely falling observer would be parametrized, given some coordinates $(t, x^1, x^2, x^3)$, by:

$\gamma (\lambda) = \left( \gamma^0(\lambda), \gamma^1(\lambda), \gamma^2(\lambda), \gamma^3(\lambda) \right)$,

where $\lambda$ is some parameter (not necessarily the proper time).

My question is: how do I parametrize the manifold with the physical coordinates of this observer? By physical coordinates I mean coordinates such that the time coordinate is the time measured by the observer and the observer has fixed space coordinates.

My attempt:

My idea was to reparametrize $\gamma$:

$ \tilde{\gamma} (t) = \left( t, \tilde{\gamma}^1(t), \tilde\gamma^2(t), \tilde\gamma^3(t) \right) $,

and define new temporary coordinates $\tilde{x}^{\mu}$:

$ \tilde{x}^{\mu} = (t, \tilde{x}^1, \tilde{x}^2, \tilde{x}^3) $, with $\tilde{x}^i = x^i - \tilde{\gamma}^i(t)$.

The metric in the new coordinates is $\tilde{g}_{\mu \nu}$ an the time experienced by the observer is:

$T = \int_0^{t} \sqrt{-\tilde{g}_{00}(t', \pmb{0})} \; dt' $.

I have found a $T = T(t, \tilde{x}^i)$ and I can take:

$X^i = \tilde{x}^i$. These should be the coordinates of the freely falling observer according to my reasoning.

Is this correct? Also, I expect this system of coordinates to be locally inerzial (ie, the final metric should be Minkowskian around the point $X^{\mu} = (T, 0, 0, 0)$ ). Is there a way to see this explicitly?

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  • $\begingroup$ so you're looking for something like the boosted schwarzschild metric? $\endgroup$
    – Yukterez
    Commented Dec 17, 2022 at 17:38
  • $\begingroup$ I don't exactly know what a boosted Schwarzschild metric is to be honest, but I'll try to look into it. I think my question is simpler though. I just want to know if my approach to find the coordinates that describe the reference frame of a freely falling observer works. In case it does I also ask if there is a good way to see that the metric would be locally flat around (T, 0, 0, 0) . I don' really care about the explicit expression of the metric in these coordinates though. $\endgroup$ Commented Dec 17, 2022 at 17:48
  • $\begingroup$ @Yukterez can you explain what a "boosted" (..) is? $\endgroup$ Commented Dec 17, 2022 at 18:35
  • $\begingroup$ That would be a lorentz boost to change from the coordinates where the BH is at rest to the coordinates where the BH is moving $\endgroup$
    – Yukterez
    Commented Dec 17, 2022 at 18:42
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    $\begingroup$ Have a look at: en.wikipedia.org/wiki/Fermi_coordinates $\endgroup$
    – TimRias
    Commented Dec 17, 2022 at 21:29

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