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Imagine a particle moving on a timelike curve in a general (any metric) 4-dimensional spacetime.

How does one mathematically construct a moving frame with coordinates on this curve (i.e. a rest frame for the particle, which doesn't have to be inertial)? I know that it requires one to identify the tangent vector to this curve and take it as the timelike vector in the moving frame. The other 3 spacelike vectors could be chosen arbitrarily. But how does one construct coordinates (of the rest frame) from these basis vectors? Do these vectors take the form $ \, \partial_i \, $ in the new coordinates?

These are not homework questions. The reason I am interested in them is that I am trying to prove that the proper time, defined by the formula $$ \quad \tau = \int \sqrt{-g(\dot{\gamma},\dot{\gamma})}d\lambda \quad $$ is really the time experienced by the particle (i.e. the time elapsed in its rest frame) moving on the curve $ \, \gamma(\lambda)$. My approach is to try to find the transformation law from the general coordinates to the particle rest frame coordinates and then see from there whether the formula for the proper time matches the time elapsed in these coordinates. But I can't find the transformation law, hence my questions. Also, even if this approach is not correct, or pointlessly complicated, I am still interested in the above questions.

Qmechanic
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2 Answers2

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Ok, I found the answers to my questions. The keyword is Fermi coordinates. I didn't know about them before, but they exist and are well-known. The mathematical construction of a moving frame and its associated coordinates (the Fermi coordinates) on a general timelike curve requires so-called Fermi-Walker transport. Explanation and derivation can be found in Eric Poisson's The Motion of Point Particles in Curved Spacetime. There have already been several questions asking about the construction:

Extent of coordinate freedom to set metric components along a spacetime path

Local inertial coordinates/Fermi normal coordinates (discusses Fermi coordinates for null geodesics)

but for me the most useful answer was from Valter Moretti to the question

Difference between Fermi and Riemann normal coordinates.

By the way I also found an article discussing the construction of Fermi coordinates for a spacelike curve. I hope somebody will find my response useful, somewhere on their future worldline.

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I would just calculate the motion in the rest frame assuming just the x,y, z cartesian coordinates. Then determine the transformation equations and substitute these values in. For example, for a pure translation, x -> x-x0 and so on depending on direction of movement of the object frame.

Natsfan
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