- #1
etotheipi
I thought I had a decent basic understanding of this stuff, but it turns out I've got quite a lot of gaps. I drew 3 frames, S, S' and S", where S' and S" move at ##v_1## and ##v_2## w.r.t. S in the ##x## direction, and two different paths to a particular event P:
Now the proper time between two events, O and P, is dependent on the path taken through spacetime, and goes something like this with one space dimension $$\Delta \tau = \int_{P} \sqrt{1-\left(\frac{dx}{dt}\right)^2}dt = \int_{P} \frac{dt}{\gamma(t)}$$ and for any given path all different observers will calculate the same ##\Delta \tau##. But suppose that between O and P that S" measures a time interval of ##\Delta t''##, and we want to figure out what time S measures between those same events. Since O is at ##t=t'=t''=0##, we can just look at the times at P. The Lorentz transformations tell me that $$t = \gamma_2 (t'' + v_2 x'')$$ and it follows that ##\Delta t = \gamma_2 \Delta t''##.
However, I usually see this quoted as ##\Delta t = \gamma_2 \Delta \tau##. The only problem is that ##\Delta \tau = \Delta t''## only if the path is chosen so that the worldline is along the ##t''## axis, and in general there are infinitely many different paths with different ##\Delta \tau## that you could choose between any two events.
So I wondered if someone could clarify, since it would appear to only be valid to say "the time between two events in one frame equals ##\gamma## times the time between the two events in another frame whose time axis is a straight line between those points". Is that right, or is there something snappier/more correct? Thank you, sorry if it's a silly question!
Now the proper time between two events, O and P, is dependent on the path taken through spacetime, and goes something like this with one space dimension $$\Delta \tau = \int_{P} \sqrt{1-\left(\frac{dx}{dt}\right)^2}dt = \int_{P} \frac{dt}{\gamma(t)}$$ and for any given path all different observers will calculate the same ##\Delta \tau##. But suppose that between O and P that S" measures a time interval of ##\Delta t''##, and we want to figure out what time S measures between those same events. Since O is at ##t=t'=t''=0##, we can just look at the times at P. The Lorentz transformations tell me that $$t = \gamma_2 (t'' + v_2 x'')$$ and it follows that ##\Delta t = \gamma_2 \Delta t''##.
However, I usually see this quoted as ##\Delta t = \gamma_2 \Delta \tau##. The only problem is that ##\Delta \tau = \Delta t''## only if the path is chosen so that the worldline is along the ##t''## axis, and in general there are infinitely many different paths with different ##\Delta \tau## that you could choose between any two events.
So I wondered if someone could clarify, since it would appear to only be valid to say "the time between two events in one frame equals ##\gamma## times the time between the two events in another frame whose time axis is a straight line between those points". Is that right, or is there something snappier/more correct? Thank you, sorry if it's a silly question!