 #1
Wannabe Physicist
 17
 3
 Homework Statement:

Imagine that space (not spacetime) is actually a finite box, or in more sophisticated
terms, a threetorus, of size ##L##. By this we mean that there is a coordinate system ##x^\mu = (t,x,y,z)## such that every point with coordinates ##(t,x,y,z)## is identified with every point
with coordinates ##(t,x+L,y,z)##, ##(t,x,y+L,z)##, and ##(t,x,y,z+L)##. Note that the time coordinate is the same. Now consider two observers; observer A is at rest in this coordinate system (constant spatial coordinates), while observer B moves in the xdirection with constant
velocity ##v##. A and B begin at the same event, and while A remains still, B moves once
around the universe and comes back to intersect the world line of A without having to
accelerate (since the universe is periodic). What are the relative proper times
experienced in this interval by A and B? Is this consistent with your understanding of
Lorentz invariance?
 Relevant Equations:
 ##(\Delta \tau^2) = (\Delta t)^2(\Delta x)^2(\Delta y)^2(\Delta z)^2##
Let us denote the events in spacetime before the trip has started by subscript 1 and those after the trip is over by subscript 2. So before the trip has begun, the coordinates in spacetime for A and B are
##A = (t_{A_1},x,y,z)## and ##B = (t_{B_1},x,y,z) = (t_{A_1},x,y,z)##.
After the trip is over,
##A = (t_{A_2},x,y,z)## and ##B = (t_{B_2},x+L,y,z)##.
Note that ##\Delta t_A = t_{A_2}t_{A_1} = \frac{L}{v}## and ##\Delta t_B = t_{B_2}t_{B_1}##. Now, by the time dilation formula, ##\Delta t_B = (\Delta t_A)/\gamma##
Thus if ##\Delta \tau_A## and ##\Delta \tau_B## are proper time intervals of A and V respectively
$$(\Delta \tau_A)^2 = (\Delta t_A)^2  0  0 0 \implies \Delta \tau_A = \Delta t_A$$
$$(\Delta \tau_B)^2 = (\Delta t_B)^2  L^2  0 0 = \frac{(\Delta t_A)^2}{\gamma^2}  L^2 = \frac{(\Delta \tau_A)^2}{\gamma^2}  L^2 $$
I feel that this must be right. I, with my naive eyes, am not able to see any flaw in my argument. But this blog post: https://petraaxolotl.wordpress.com/chapter1specialrelativityandflatspacetime/ seems to be telling a different story.
It says $$(\Delta \tau_B)^2 = (\Delta t_B)^2(\Delta x)^2 = (\Delta \tau_B)^2(1v^2) = (\Delta \tau_A)^2(1v^2) = \frac{(\Delta \tau_A)^2}{\gamma^2}$$
Am I making any mistake?
##A = (t_{A_1},x,y,z)## and ##B = (t_{B_1},x,y,z) = (t_{A_1},x,y,z)##.
After the trip is over,
##A = (t_{A_2},x,y,z)## and ##B = (t_{B_2},x+L,y,z)##.
Note that ##\Delta t_A = t_{A_2}t_{A_1} = \frac{L}{v}## and ##\Delta t_B = t_{B_2}t_{B_1}##. Now, by the time dilation formula, ##\Delta t_B = (\Delta t_A)/\gamma##
Thus if ##\Delta \tau_A## and ##\Delta \tau_B## are proper time intervals of A and V respectively
$$(\Delta \tau_A)^2 = (\Delta t_A)^2  0  0 0 \implies \Delta \tau_A = \Delta t_A$$
$$(\Delta \tau_B)^2 = (\Delta t_B)^2  L^2  0 0 = \frac{(\Delta t_A)^2}{\gamma^2}  L^2 = \frac{(\Delta \tau_A)^2}{\gamma^2}  L^2 $$
I feel that this must be right. I, with my naive eyes, am not able to see any flaw in my argument. But this blog post: https://petraaxolotl.wordpress.com/chapter1specialrelativityandflatspacetime/ seems to be telling a different story.
It says $$(\Delta \tau_B)^2 = (\Delta t_B)^2(\Delta x)^2 = (\Delta \tau_B)^2(1v^2) = (\Delta \tau_A)^2(1v^2) = \frac{(\Delta \tau_A)^2}{\gamma^2}$$
Am I making any mistake?