# Frame of reference of a freely falling observer in General Relativity

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?

• so you're looking for something like the boosted schwarzschild metric? Commented Dec 17, 2022 at 17:38
• 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. Commented Dec 17, 2022 at 17:48
• @Yukterez can you explain what a "boosted" (..) is? Commented Dec 17, 2022 at 18:35
• 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 Commented Dec 17, 2022 at 18:42
• Have a look at: en.wikipedia.org/wiki/Fermi_coordinates Commented Dec 17, 2022 at 21:29