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- What is the formula for potential energy of an object, falling with relativitic velocity in a pseudo-gravitational field in an accelerated frame?

Reference frame is an accelerated frame in SR (uniformly accelerated with "g" in flat spacetime). An object is falling with relativitic velocity of up to 0.8 c in the pseudo-gravitational field in this frame.

From Newton's theory, I know the formula for potential energy in such a scenario:

##W = m * g * h##.

For small velocities, this formula should also be usable in SR in the mentioned scenario. But what is at relativistiv velocities?

Shall I use in the formula for potential energy in hight "h" the rest mass (m₀) or the formerly called "relativistic mass" (m₀ *

##W = m_0 * g * h## or

##W = m_0 * \gamma * g * h## ?

Reason for the question: In the following paper about the "twin paradox", I don't understand the reason for including a factor

https://arxiv.org/ftp/arxiv/papers/1002/1002.4154.pdf

From Newton's theory, I know the formula for potential energy in such a scenario:

##W = m * g * h##.

For small velocities, this formula should also be usable in SR in the mentioned scenario. But what is at relativistiv velocities?

Shall I use in the formula for potential energy in hight "h" the rest mass (m₀) or the formerly called "relativistic mass" (m₀ *

*γ*)?##W = m_0 * g * h## or

##W = m_0 * \gamma * g * h## ?

Reason for the question: In the following paper about the "twin paradox", I don't understand the reason for including a factor

*γ*in the formula for pseudo-gravitational potential, see equation (8), then compare with equation (3) in:https://arxiv.org/ftp/arxiv/papers/1002/1002.4154.pdf

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