Mechadragon
Level: Delver
Rank Points: 61
Registered: 10-30-2004
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Re: I Want to Learn General Relativity (+2)
Special relativity in a nutshell:
Postulate 1: All inertial (non-accelerating) frames of reference are subject to the same laws of physics. This means that a game of billards of billards in a bar would proceed the same as a game of billards on a train with a perfectly straight track. This is known as the principle of relativity. No inertial frame of reference is 'special' since they are all equivalent.
Postulate 2: All inertial frames of reference observe light travelling at the same speed, c.
This leads to some interesting consequences, but only at -very- high speeds. Imagine that, say, a train sped through a station at relativistic speeds (meaning very close to c, say 0.5 c or so) with an operator on board. Exactly as he reaches the middle of the station he activates a lamp which proceeds to trigger two light-sensitive doors on either side of the train. As he is in the same frame of reference as the train he'll see light from the lamp proceed at equal speeds in both directions, and from his frame of reference both doors open at once. However, a person on the station observes light travelling at c with respect to his frame of reference, and as the train speeds off the light has further to go in one direction than in another; he will observe the rear door open first, then the front door. As no inertial frame of reference is special we are forced to conclude that events are only simultaneous with respect to some inertial frame of reference and not others.
Similarly, let's run another relativistic train by with length L at velocity v. This time it has a lamp on one end of the train next to a light sensor with a mirror on the far end. When the operator runs it, he sees that it takes 2L/c seconds to activate the sensor. However, a person on the station will see that the light has to make a longer trip one way (since it is chasing the front of the train as it moves away) then a shorter trip the other way (since the back of the train is moving towards it).
The forward trip has L + vt = ct, ie t = L/(c-v)
The return trip has L - vt = ct, ie t = L/(c+v)
This means that the total T is L(c+v)/(c-v)(c+v) + L(c-v)/(c-v)(c+v) or 2Lc/(c^2+v^2) which can be arranged to equal 2L/c(1+v^2/c^2).
Both times must be the same or, well, the universe would desync (that's bad!) so the person on the station must observe their length (Ln) to be lower than the original length (Lo)
2Ln/c(1+v^2/c^2) = 2Lo/c
Ln = Lo(1+v^2/c^2)
I actually screwed this up though since you're supposed to find time dilation before length dilation so that you can substitute the first into the second and get the right formula. See if you can imagine these ones through on your own:
-A train runs through a station with a lamp set up on its floor and a mirror on the ceiling, a light sensor set up just beside it to receive the pulse on the return trip. The operator activates it just as it passes through the station. How long does it take for the operator and how long does it take for a person watching from the station?
-Now solve the length dilation equation with the time dilation one with it.
-Let's say that two spacecraft with equal and opposite momentums along the x axis collide, sending them back with again equal and opposite momentums (law of conservation of momentum) but with a slight y component. Given length and time dilation however, what does an observer on one spacecraft see about the other spacecraft's momentum? Given that the law of conservation of momentum applies in all frames of reference, what does this mean about what one spacecraft sees about the other's mass?
-Given that energy is also conserved and you can't accelerate past c, what happens when an object is accelerated to relativistic speeds? (This is where Einstein's famous mass-energy equivalence equation comes from)
Sorry, I'm just too lazy to write these all out in full. Here's a nutshell: If you're in frame of reference A and frame of reference B is travelling with relativistic (close to c) speed relative to you, you will see its length contract along the direction of travel, its time will slow down relative to yours and its mass will increase. I can work through this properly if there's enough of a demand.
General relativity in a nutshell:
Postulate 1: Everything in special relativity is assumed to hold true.
Postulate 2: A frame of reference that is accelerating is equivalent to one that is in a gravitational field, pulling everything down with an equal amount of acceleration as the first frame of reference. That is, the laws of physics are the same in both.
This equivalence between acceleration and experiencing the effects of gravity is used to 'port' phenomena over from accelerating frames to frames in a gravitational field.
Consequences:
1. If you are accelerating and send a light pulse in the direction of the acceleration it will red shift and be spread out. In the opposite direction it will be blue shifted and bunched in. This slower/faster communication means that observers in an 'acceleration well' experience time slower then those higher up in them, and since acceleration and gravity produce equal effects observers in a gravity well have their time dilated.
2. If you considered the position of an object in space and time as a 4-dimensional line you would notice that it curved towards gravitational fields, i.e. by matter. This means that the geometry of spacetime can be considered to be 'curved' by matter in the form of gravity. If it's hard to picture, imagine a trampoline with balls on it; the balls push down on the fabric and other balls follow these depressions inwards when they get close. In this geometry, non-accelerating bodies are straight lines; light in particular always goes in a straight line, which means that if it passes through a gravitational field it will curve around the body, leading to a phenomena known as gravitational lensing where the light from celestial bodies shifts if it goes near another body, making it appear to move in the sky, or even multiply!
It predicts other things like the tidal effect and the precession of Mercury but I'm not sure how to explain those. Try wikipedia?
[Last edited by Mechadragon at 10-01-2008 04:59 AM]
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