I’m having a small Relativity-related problem: I am having a hard time figuring out how the observations of independent observers moving relative to each other don’t lead to contradictions. I understand that the resolution to the Twins Paradox is the fact that one twin has to accelerate to return to the other twin, thereby defining him as the younger. What I don’t understand are the opinions of two observers approaching each other at high speed in empty space.
If Joe and Moe are approaching each other at high speed, Joe will observe that Moe is foreshortened and time-dilated (slow-Moe, if you will), while Moe attributes the same qualities to Joe. How can both these viewpoints be consistent? If Moe watches Joe drop an egg and it appears to take five minutes, how can Joe watching Moe while the egg drops not see Moe doing five minutes worth of stuff? If, as they go flying past one another, they hold up rulers in direction of motion, they each will think that the other’s ruler should be shorter than their own. How can two rulers each be shorter than the other? Please tell me how I’m looking at this wrong, because it’s giving me a headache.
Consider your ruler. Moe is flying at Joe holding his ruler up. Imagine a photon hits the ‘1 inch’ mark and the ‘2 inch’ mark at the same time and bounces off towards Joe (the 1 inch mark is closer to Joe). Since Joe is moving towards Moe he will see the 1 inch photon arrive followed by the 2 inch photon but he will shorten the distance between the 1 inch and 2 inch photon due to his own motion…the number 2 photon will arrive sooner than if they were both standing ‘still’. Thus the distance between 1 and 2 will seem less than it is.
Of course, this works in reverse too and Moe will see the same thing. Of course, everything seems perfectly normal to each observer and they assume it is the other guy who is moving. Assuming no other reference points neither can ever figure which one is moving and which is standing still (or if both are moving in any combination of speeds).
The whole idea behind relativity is that you can only look at things from one viewpoint at a time. There is no privileged observer outside the event who can look at the whole and say, huh, each ruler is smaller than the other. You can be in Joe’s point of view or Moe’s. In each, everything is consistent and there is no paradox.
It can’t be any other way because special relativity must be self-consistent. If either observer observed the system differently than the other, then physical law would be different for different inertial frames, and Einstein’s first postulate would be violated.
And as Whack-a-Mole says there is no preferred frame so both observers are perfectly free to consider themselves at rest, so they better see the same thing.