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- TL;DR Summary
- Twin Paradox: Alice with inertial motion, Bob with hyperbolic motion

Alice rests at ##X=L+1## in the inertial frame (T, X).

Bob is at rest in the Rindler frame (t, x) at ##x=1## and has the proper acceleration ##\alpha=1##.

In the rest frame of Alice, Bob moves from event ##E_1=(-T_2, L+1)## over the distance of ##L## in negative X-direction to event ##(0, 1)## and then returns to event ##E_2=(+T_2, L+1)##.

Because of time dilation, Bob's elapsed proper time, when moving form event ##(0, 1)## to event ##E_2##, is:

##t_2=\int_0^{T_2} \sqrt{1-\frac{v(T)^2}{c^2}} \ dT \ \ \ \ \ (1)##

Bob's velocity, when moving form event ##(0, 1)## to event ##E_2##, is:

##v(T) = \frac{\alpha T}{\sqrt{1+(\alpha T/c)^2}} \ \ \ \ \ (2)##

Source:

https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)#Worldline

Plugging equation (2) into equation (1) and setting ##c:=1##

##t_2=\int_0^{T_2} \sqrt{1-\frac{\alpha^2 T^2}{1+\alpha^2 T^2}} \, dT =\int_0^{T_2} \sqrt{1-\frac{1}{1+1/(\alpha T)^2}} \, dT \ \ \ \ \ (3)##

Calculating the integral:

##t_2= \frac{\sinh^{-1} {(\alpha T_2)}}{\alpha}= \sinh^{-1} {(T_2)} ##

Example ##T_2 = 5##:

##t_2=\sinh^{-1} {(5)} \approx 2.31##

For symmetry reasons:

Alice ages by 10 years from event ##E_1## to event ##E_2##.

Bob ages by 4.62 years from event ##E_1## to event ##E_2##.

Bob is at rest in the Rindler frame (t, x) at ##x=1## and has the proper acceleration ##\alpha=1##.

In the rest frame of Alice, Bob moves from event ##E_1=(-T_2, L+1)## over the distance of ##L## in negative X-direction to event ##(0, 1)## and then returns to event ##E_2=(+T_2, L+1)##.

Because of time dilation, Bob's elapsed proper time, when moving form event ##(0, 1)## to event ##E_2##, is:

##t_2=\int_0^{T_2} \sqrt{1-\frac{v(T)^2}{c^2}} \ dT \ \ \ \ \ (1)##

Bob's velocity, when moving form event ##(0, 1)## to event ##E_2##, is:

##v(T) = \frac{\alpha T}{\sqrt{1+(\alpha T/c)^2}} \ \ \ \ \ (2)##

Source:

https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)#Worldline

Plugging equation (2) into equation (1) and setting ##c:=1##

##t_2=\int_0^{T_2} \sqrt{1-\frac{\alpha^2 T^2}{1+\alpha^2 T^2}} \, dT =\int_0^{T_2} \sqrt{1-\frac{1}{1+1/(\alpha T)^2}} \, dT \ \ \ \ \ (3)##

Calculating the integral:

##t_2= \frac{\sinh^{-1} {(\alpha T_2)}}{\alpha}= \sinh^{-1} {(T_2)} ##

Example ##T_2 = 5##:

##t_2=\sinh^{-1} {(5)} \approx 2.31##

For symmetry reasons:

Alice ages by 10 years from event ##E_1## to event ##E_2##.

Bob ages by 4.62 years from event ##E_1## to event ##E_2##.

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