# Instantaneus Age Changes in the Twin "Paradox"

#### MikeFontenot

The apparent paradox in the twin "paradox" scenario arises because it would seem that each twin should conclude that the other twin is ageing more slowly, due to the well-known "time-dilation" result, during the entire trip, except for the single instant at the instantaneous turnaround ... "surely" nothing could happen to peoples' ages during a single instant. But that assumption is wrong: the traveling twin must conclude that the home twin's age instantaneously increases during the instantaneous turnaround. There are various ways to obtain this result, but by far the easiest and quickest way is to use the equation described in this posting.

The change in the home-twin's (her) age, before and after an instantaneous velocity change at some instant t in the traveler's (his) life, is given by the very simple "delta_CADO_T equation":

delta_CADO_T(t) = - L(t) * delta_v(t),

where

delta_v(t) = v(t+) - v(t-),

and where t- and t+ are the instants of his life immediately before and immediately after his instantaneous velocity change at t. The quantities v(t+) and v(t-) are their relative speeds at the instants t+ and t-, according to her. v is positive when the twins are moving apart, and negative when they are moving toward each other. The quantity L(t) is their distance apart when he is age t, according to her.

So, getting the change in her age during an instantaneous velocity change by him is very simple: you just multiply the negative of their distance apart (according to her) by the change in his velocity. Couldn't be simpler.

For example, take a case where their relative velocity right before his turnaround is v = 0.9 ly/y (they are moving apart), and right after his instantaneous velocity change their relative velocity is v = -0.8 ly/y (they are moving toward one another). Then

delta_v = (-0.8) - (0.9) = -1.7 ly/y.

Suppose that their distance apart at the turnaround is 20 ly. Then

delta_CADO_T = - 20 * (-1.7) = 34.0 years,

so he says that she instantaneously got 34 years older during his instantaneous turnaround. Couldn't be simpler.

Now, suppose that at some later instant t in his life, he decides to instantaneously change his velocity again, this time from -0.8 ly/y to 0.7 ly/y. So this time, he is instantaneously changing from going toward her to going away from her. In this case, we have

delta_v = (0.7) - (-0.8) = 1.5 ly/y.

Suppose their distance apart now 18 ly. Then

delta_CADO_T = - 18 * (1.5) = -27.0 years,

so he says that she instantaneously got 27 years younger during his instantaneous turnaround. Couldn't be simpler.

The above information was intentionally designed to be as concise and "narrowly-focused" as possible. Much more complete and wide-ranging information about the traveler's perspective in the twin "paradox" is contained in my webpage:

#### HallsofIvy

The apparent paradox in the twin "paradox" scenario arises because it would seem that each twin should conclude that the other twin is ageing more slowly, due to the well-known "time-dilation" result, during the entire trip, except for the single instant at the instantaneous turnaround ... "surely" nothing could happen to peoples' ages during a single instant.
Why would you say '"surely" nothing could happen to peoples' ages during a single instant" when you have just proposed that that there is an "instantaneous turn around" (involving a deceleration from a very high speed to 0 then an acceleration back to a high speed in the other direction)? One is exactly as non-realistic as the other.

But that assumption is wrong: the traveling twin must conclude that the home twin's age instantaneously increases during the instantaneous turnaround. There are various ways to obtain this result, but by far the easiest and quickest way is to use the equation described in this posting.

The change in the home-twin's (her) age, before and after an instantaneous velocity change at some instant t in the traveler's (his) life, is given by the very simple "delta_CADO_T equation":

delta_CADO_T(t) = - L(t) * delta_v(t),

where

delta_v(t) = v(t+) - v(t-),

and where t- and t+ are the instants of his life immediately before and immediately after his instantaneous velocity change at t. The quantities v(t+) and v(t-) are their relative speeds at the instants t+ and t-, according to her. v is positive when the twins are moving apart, and negative when they are moving toward each other. The quantity L(t) is their distance apart when he is age t, according to her.

So, getting the change in her age during an instantaneous velocity change by him is very simple: you just multiply the negative of their distance apart (according to her) by the change in his velocity. Couldn't be simpler.

For example, take a case where their relative velocity right before his turnaround is v = 0.9 ly/y (they are moving apart), and right after his instantaneous velocity change their relative velocity is v = -0.8 ly/y (they are moving toward one another). Then

delta_v = (-0.8) - (0.9) = -1.7 ly/y.

Suppose that their distance apart at the turnaround is 20 ly. Then

delta_CADO_T = - 20 * (-1.7) = 34.0 years,

so he says that she instantaneously got 34 years older during his instantaneous turnaround. Couldn't be simpler.

Now, suppose that at some later instant t in his life, he decides to instantaneously change his velocity again, this time from -0.8 ly/y to 0.7 ly/y. So this time, he is instantaneously changing from going toward her to going away from her. In this case, we have

delta_v = (0.7) - (-0.8) = 1.5 ly/y.

Suppose their distance apart now 18 ly. Then

delta_CADO_T = - 18 * (1.5) = -27.0 years,

so he says that she instantaneously got 27 years younger during his instantaneous turnaround. Couldn't be simpler.

The above information was intentionally designed to be as concise and "narrowly-focused" as possible. Much more complete and wide-ranging information about the traveler's perspective in the twin "paradox" is contained in my webpage: