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## The Question

(Submitted July 01, 1997)

Time dilation is at least several times stated to be a function of acceleration rather than velocity. The Lorentz transformations do not deal with acceleration, just velocity. I started to calculate the time dilation effect of mass and velocity as functions of energy in the system. I wonder if anyone has an analysis I can read. My math stops short of tensor calculus, but I can usually comprehend other's work.

Time dilation is a function of velocity. It is a function of the relative velocity of two inertial reference frames. However, to get from one reference frame to another, you have to change your velocity, you have to accelerate.

In the twin paradox, Romulus stays home on Earth, while Remus travels at a good fraction of lightspeed to Alpha Centauri, then turns around and comes back. When they get back together again, they find that Remus is younger than Romulus. However, during the out bound leg of the flight, Remus sees Romulus aging more slowly than him, and during the return leg, Remus also sees Romulus aging more slowly than him. Since Remus always sees Romulus aging more slowly, how did Romulus get to be older? (Romulus always sees Remus aging more slowly, so he has no reason to be surprised.)

The explanation is that when Remus turns around and comes back, he shifts from a frame where Romulus is younger than Remus, to a frame where Romulus is older. It is this changing of frames, rather than the physical effects of acceleration, that results in Remus ending up younger than his stay-at-home twin.

Understanding this type of explanation makes it easier to understand the confusing explanations often found in books.

Tom Bridgman and David Palmer
for "Ask an Astrophysicist"

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