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Time dilation arises from the relativity of simultaneity, which is responsible for length contraction also. The effect is simplest to explain in the case of a 2-d spacetime. In a given reference frame, lines parallel to the x-axis are lines of constant time- anywhere along such a line it is the same time t.
NOTE 1: We would not have been able to use the simple time dilation formula if we'd considered the transit of light only from one end of the rod to the other. NOTE 2: there are much simpler ways of deriving the length contraction equation using the time dilation formula than methods involving pulses of light!
That expression for gravitational time dilation comes from the Schwarzschild solution to Einstein's field equations from General Relativity. The Schwarzschild metric tell us about the space-time in a vacuum near a stationary , spherically symmetric mass.
The Time Dilation formula for the Kerr-Newman metric, in terms of a number of length parameters for a position ##(r,\theta)## away from center of mass and polar axis respectively of a gravitational body of mass ##M##, polar angular momentum ##J## and electrical charge ##Q##, is independent of azimuthal angle ##\phi## and expressed as: $$
$\begingroup$ I'm struggling to understand the difference between the two formulae in the question. I start by writing down the lorentz transformation equations for space and time coordinates between S, the rest frame, and S', the relatively moving frame.
A generalized version of the gravitational time dilation formula that takes motion into account comes from taking the integral of the magnitude of a worldline's tangent vectors parametrized by λ.
$\begingroup$ The time dilation is dependant on the gravitational potential not the gravitational force. $\endgroup$ – John Rennie Commented Nov 24, 2014 at 20:25
1. What is the time dilation formula for constant acceleration on a circular path? The time dilation formula for constant acceleration on a circular path is Δt' = Δt√(1 - (v^2/c^2)), where Δt is the time measured in the rest frame, Δt' is the time measured in the moving frame, v is the velocity of the moving frame, and c is the speed of ...
Because the time dilation formula can in fact be derived directly from the Lorentz transformation. See, e.g. this link. Special relativity is often first presented as a sequence of "effects" with their own formulae, sometimes justified with an accompanying thought experiment.
So “no time dilation due to acceleration” is correct and accurate. Saying “it would keep changing” is misleading IMO. It implies that there is some time dilation unaccounted for and that the general formula to account for it is unknown, neither of which is true. $\endgroup$ –