- #91
Ibix
Science Advisor
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The reason for the differential aging of the twins is due to the difference in length of the space-time paths taken by the two twins. That is all the reason there is. It is your destination if you are attempting to understand the twin paradox scenario. Dismissing it as "far away from your destination" is just daft - why ask the question if you aren't going to listen to the answer?
There are basically two sets of transforms between inertial frames that are consistent with the notion that physics is the same in all of them: these are the Galilean transforms and the Lorentz transforms. The Lorentz transforms are consistent with observation while the Galilean ones are not. So the Lorentz transforms are the right ones (but we were fooled for centuries because we didn't have sensitive enough experiments to spot the errors in the Galilean transforms).
In a Euclidean space, the distance between points is given by Pythagoras' Theorem, and the answer does not dependent on the choice of coordinates. I might say that two points are ##\Delta x=L## meters apart in the x direction and ##\Delta y=W## meters apart in the y direction. You might say that they ae ##\Delta x'=W## meters apart in the x' direction and ##\Delta y'=-L## meters apart in the x direction. We are using coordinates rotated by 90° with respect to each other. However, we will both agree that the distance between the points is ##\sqrt{L^2+W^2}## meters.
The Lorentz transforms imply that space and time are one four-dimensional whole called space-time - but it does not follow Euclidean geometry. In space-time, the equivalent of Pythagoras' theorem is ##c\Delta\tau = \sqrt{c\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2}##. This quantity does not depend on the choice of coordinates - you and I might disagree on ##\Delta x## (that would be length contraction) or ##\Delta t## (that would be time dilation), but we will always come up with the same ##\Delta\tau## for any given path.
It's easy to see that, in your rest frame, ##\Delta\tau=\Delta t##, because in your rest frame you are not moving so for you ##\Delta x=\Delta y=\Delta z=0##. Someone at rest in one frame is moving in another, though, so in general ##\Delta\tau## is the time experienced by someone moving at constant speed from point A to point B, separated in space by (##\Delta x,\Delta y,\Delta z##), in time ##\Delta t##.
It's then easy to see that if I move from an event at (t,x,y,z)=(0,0,0,0) to (t,x,y,z)=(10,0,0,0) (i.e., stay put for ten years) while you travel from (0,0,0,0) to (5,3,0,0) then to (10,0,0,0) (i.e., take five years to get three light years away in the x direction, then turn round and come back) that the ##\Delta\tau##s are different - ##\sqrt{10^2-0^2}=10## years for me, ##\sqrt{5^2-3^2}+\sqrt{5^2-(-3)^2}=8## years for you.
I think that's a complete explanation of the twin paradox from top to bottom. Certainly you should learn the maths and be able to derive the Lorentz transforms if you intend to study SR. It will show you that what I have written above is self-consistent. But it will not give you any further insight into the twin paradox - this is the whole of the "why" there is differential aging.
There are basically two sets of transforms between inertial frames that are consistent with the notion that physics is the same in all of them: these are the Galilean transforms and the Lorentz transforms. The Lorentz transforms are consistent with observation while the Galilean ones are not. So the Lorentz transforms are the right ones (but we were fooled for centuries because we didn't have sensitive enough experiments to spot the errors in the Galilean transforms).
In a Euclidean space, the distance between points is given by Pythagoras' Theorem, and the answer does not dependent on the choice of coordinates. I might say that two points are ##\Delta x=L## meters apart in the x direction and ##\Delta y=W## meters apart in the y direction. You might say that they ae ##\Delta x'=W## meters apart in the x' direction and ##\Delta y'=-L## meters apart in the x direction. We are using coordinates rotated by 90° with respect to each other. However, we will both agree that the distance between the points is ##\sqrt{L^2+W^2}## meters.
The Lorentz transforms imply that space and time are one four-dimensional whole called space-time - but it does not follow Euclidean geometry. In space-time, the equivalent of Pythagoras' theorem is ##c\Delta\tau = \sqrt{c\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2}##. This quantity does not depend on the choice of coordinates - you and I might disagree on ##\Delta x## (that would be length contraction) or ##\Delta t## (that would be time dilation), but we will always come up with the same ##\Delta\tau## for any given path.
It's easy to see that, in your rest frame, ##\Delta\tau=\Delta t##, because in your rest frame you are not moving so for you ##\Delta x=\Delta y=\Delta z=0##. Someone at rest in one frame is moving in another, though, so in general ##\Delta\tau## is the time experienced by someone moving at constant speed from point A to point B, separated in space by (##\Delta x,\Delta y,\Delta z##), in time ##\Delta t##.
It's then easy to see that if I move from an event at (t,x,y,z)=(0,0,0,0) to (t,x,y,z)=(10,0,0,0) (i.e., stay put for ten years) while you travel from (0,0,0,0) to (5,3,0,0) then to (10,0,0,0) (i.e., take five years to get three light years away in the x direction, then turn round and come back) that the ##\Delta\tau##s are different - ##\sqrt{10^2-0^2}=10## years for me, ##\sqrt{5^2-3^2}+\sqrt{5^2-(-3)^2}=8## years for you.
I think that's a complete explanation of the twin paradox from top to bottom. Certainly you should learn the maths and be able to derive the Lorentz transforms if you intend to study SR. It will show you that what I have written above is self-consistent. But it will not give you any further insight into the twin paradox - this is the whole of the "why" there is differential aging.
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At any rate, I recommend that you do this first - I'm sure 99% of your problems will dissolve along the way! Of course, other users are welcome to suggest changes to this little chronology of concepts. I by no means proclaim that this list is comprehensive or exhaustive - the invaluable inputs of other users can help tailor this list, and you can finally say "I nailed SR!"