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## Main Question or Discussion Point

Imagine a non-rotating particle, one with zero angular momentum. Attach a tetrad of orthonormal vectors to the particle.

Imagine that we accelerate the particle via a force through it's center of mass, so that it follows some curve through space-time parametreized as [itex]x^i(\tau)[/itex] while continuing to have zero angular momentu.

In flat space the vectors "attached" to the particle all obey the Fermi-walker transport law.

[tex]\frac{dv^a}{d\tau} = (u^a \wedge a^b ) v_b[/tex]

here v is the vector to be transported (one of the orthonormal vectors attached to the particle), u is the 4-velocity of the particle[itex]{dx^i}/{d\tau}[/itex], and a is the 4-acceleration of the particle [itex]{d^2 x^i}/{d\tau^2}[/itex]. The symbol "^" represents the wedge product.

My question is whether or not this expression works correctly in curved space-times and arbitrary coordinate systems, and if it doesn't, what the correct expression is. I.e. suppose we have a particle with zero intrinsic angular momentum that's orbiting a black hole, given that we know the path of the orbit and the associated 4-acceleration, can we use this formula to transport the particle's coordinate tetrad?

Imagine that we accelerate the particle via a force through it's center of mass, so that it follows some curve through space-time parametreized as [itex]x^i(\tau)[/itex] while continuing to have zero angular momentu.

In flat space the vectors "attached" to the particle all obey the Fermi-walker transport law.

[tex]\frac{dv^a}{d\tau} = (u^a \wedge a^b ) v_b[/tex]

here v is the vector to be transported (one of the orthonormal vectors attached to the particle), u is the 4-velocity of the particle[itex]{dx^i}/{d\tau}[/itex], and a is the 4-acceleration of the particle [itex]{d^2 x^i}/{d\tau^2}[/itex]. The symbol "^" represents the wedge product.

My question is whether or not this expression works correctly in curved space-times and arbitrary coordinate systems, and if it doesn't, what the correct expression is. I.e. suppose we have a particle with zero intrinsic angular momentum that's orbiting a black hole, given that we know the path of the orbit and the associated 4-acceleration, can we use this formula to transport the particle's coordinate tetrad?