An explanation of Relativity
For as long as I can remember many questions posed into the Athenaeum have to do with astrological concepts tied to Einstein's theory of relativity (whether General or Special Relativity): questions about time travel, wormholes, time dilation, etcetera. Since there was no real explanation for the phenomenon, I thought I'd write a guide for the layman (and refresh my physics courses in the progress) so I can refer people to it instead of making ad-hoc-replies all the time.
So, let's hit it...
Einstein's Postulates
Spoiler Alert, click show to read:The entire theory of relativity is based on Einstein's two postulates: deceptively simple statements but with grand consequences. They are, according to Wiki:
1) Principle of relativity: "The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion."
2) Invariance of c: "As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body."
Both are simple enough: the latter simply says that the speed of light is always the same (roughly 3.10^8 m/s), and the former is a fancy way of saying that the laws of nature are consistent in every inertial frame, in other words no matter at what speed you're moving or in what direction, gravity's force will be F=m.g, Maxwell's equations will remain the same and so on.
N.B. An inertial frame is again a fancy way of saying 'viewpoint' or a 'frame of reference'. Inertial frames can move relative to each other, for example if you're standing still and you're sitting in a car, your inertial frame (=yourself) is moving away from mine (myself) with a certain speed.
These two principles have quite profound consequences, and I'll illustrate both of them with two thought experiments.
Thought experiment 1: Breakdown of Simultaneity
Spoiler Alert, click show to read:One of the things that's quite hard to grasp with regards to relativity is that since time is relative, events that would seem to be simultaneous in one frame of reference, will not be simultaneous in another.
Example: person A is sitting in the middle wagon of a train that's going at a speed of 0.5 c (in other words, half the speed of light) and person B is standing still, watching the train go by.
Now suppose that just as person B is at the middle of the passing train, he sees two red lights starting to flash on each end of the train. Since he knows the speed of light, and since he can see that he's in the middle of the train, he's thinking:
- I saw the lights at the same time (in other words, the wavefronts of light reached my eyes at exactly the same moment)
- I'm at the same distance from the back end of the train as I am from the front end
Ergo, since I saw the lights at the same time, the red lights must indeed have flashed at the same time. Conclusion: person B deems the red lights to have flashed simultaneously.
However, that's not the conclusion person A will make. Since the train is moving at half the speed of light, he would see the wavefront of the red light at the front of the train first (since the train is 'moving into' the wavefront, so to speak) and he'll only see the light at the back of the train a short while later (since that wavefront has to 'chase' the train to get to person A).
Now here's the thing: person A is sitting perfectly still with regards to the train. From his perspective, it's person B that's moving away from him at half of the speed of light in reverse!
Person A will think:
- The wavefront of the light from the front of the train reached me sooner than the other wavefront
- I'm at the same distance from the back end of the train as I am from the front end
Ergo, the light on the front end of the train must have started flashing sooner. The events are not simultaneous.
The weird thing to accept now is that both are correct in their own frame of reference! We might assume that the viewpoint of the person watching the train go by has more validity, but that's really just an opinion: the person inside the train could theoretically not know about the train moving and so he will think that his frame of reference is the unchanging one. So we can't say that one is better than the other.
If that's hard to accept, just replace the train with a spaceship orbiting around the Earth. Who's to say that the spaceship is moving and the Earth is standing still? Isn't the Earth moving as well (rotating around the sun and stuff)? It's just as correct to say that the Earth is moving constantly and the spaceship is the point we should measure it from. Both frames are equivalent.
Ditto for the train: both observe from their own frame of reference and both make perfectly valid reasonings, yet still come to other conclusions.
This is an example of the principle of relativity: no single inertial frame is better than the other; they are all relative.
The only important thing according to Einstein is that the different laws of nature must be correct in both frames of reference. This clearly is not a problem: person B thinks both lights flashed simultaneously and everything makes sense according to him and person A thinks the front light flashed first and everything makes sense to him too. It's useless trying to figure out "who's right": they both are because it both makes sense from their viewpoint.
The only casualty in this elegant solution is the concept of simultaneity: we just killed it. After all, the above thought experiment shows us that when you see two events that are simultaneous (which is to say, simultaneous from your perspective), other people moving away from you or moving towards you might not perceive said events to be simultaneous. In fact, every single reference frame different from yours (and there's obviously an infinity of those) will disagree the observed events are simultaneous! Similarly, events that seem simultaneous in another frame of reference will not appear to be to you. So you see: the concept of simultaneity becomes utterly meaningless and subjective.
All this makes for the horribly bad joke physics professors love to tell whenever they're not on time for class: "I'm sure there was an inertial frame in which I was entirely on time!"
Thought Experiment 2: Newton's Failure
Spoiler Alert, click show to read:This consists of two quick scenarios, and I want you to use your basic physics gut feeling.
Scenario 1: you're in a spaceship that's moving at 10000 km/h away from the Earth. At some point you fire a torpedo that starts moving at 10000 km/h away from the spaceship (in the same direction that the spaceship is going).
Now the question is: how fast will observers on Earth measure the torpedo to be going?
The answer should be obvious: they will see the torpedo going at 10000 km/h + 10000 km/h = 20000 km/h (the torpedo moves at a speed of 10k away from the spaceship, and the spaceship moves at 10k away from the Earth, so we have to add the two speeds).
This intuitive feeling you had is actually Newtonian physics: Newton says that if something is going at a speed A in a certain frame of reference, then if we try to measure that speed in another frame of reference from which the first frame is moving away with speed B, the speed of the object as measured in the second frame will be A+B.
All of this matches with experiments, by the way. If you were to run that exact scenario you'd indeed get approximately 20000 km/h.
Now let's change the scenario a little bit to get scenario 2: same spaceship, but now instead of a torpedo we're going to fire a beam of light (which will be travelling at the speed of the light c, of course) from the spaceship. What will be the speed of the light beam measured from Earth? Newtonian physics says that we should measure it going at 300000 km/s + 100000 km/hour (about 28 km/second) = 300028 km/s (speed of light + the difference between the speed of the inertial frames).
But that's not what we see from EarthThe beam will not be going at 300000 km/s plus anything; we will measure the light beam travelling at exactly 300000 m/s and nothing else. That's precisely what Einstein's second Postulate tells us.
I don't know if you realise it, but we've just engaged in a paradox. We know that the lightbeam is moving at speed c away from the spaceship, and the spaceship itself is also moving at a certain speed away from the Earth. Yet from the Earth we still see the beam only going at speed c!!! In other words, the simple Newtonian adding of velocities clearly doesn't work anymore... now let's see what that teaches us.
Time dilation and the Relativity of Time
Spoiler Alert, click show to read:How do we solve the paradox? It's not our machinery failing us (these experiments have been done plenty of times), and it's not leprechauns.
Here's where it gets neat
The only way to resolve this paradox is to recognise that time in both inertial frames is running at different speeds. That way we realise that for us on Earth, the spaceship moves slower than we expect because from our perspective, time in the spaceship's frame moves slower.
This concept is called time dilation.
The main idea here is that the faster an object is moving away from your frame of reference, the slower time seems to run for that object (a clock on the object would appear to run slower for us, although for the object itself time would seem to be going on as normal). This, again, isn't just some theoretical conjecture: we can actually observe this phenomenon in a variety of ways, one of the most obvious ones being that clocks that went on high-speed space flights run a couple of seconds behind by the time they return to Earth.
If you think about it, you realise that it is this time dilation that stops us from breaking Einstein's Second Postulate: the closer an object gets to the speed of light, the slower time appears to be going when someone looks at that object and so it will never beat the speed of light (due to time running slower, it partly compensates the increase in speed an object might experience: this catch-22 leads to the object never breaking the speed of light)!
This time dilation is expressed by the following formula:
t is the amount of time something takes in a given frame of reference (frame A)
t' is the amount of time something takes in another frame of reference (frame B)
v is the speed at which frame B is moving away from frame A
c is the speed of light
As you can see in the graph and in the equation, the closer to the speed of light c an object gets (the 'object' here is inertial frame B, and so v is the speed of the object), the more time dilation it experiences.
Applied to our second thought experiment, the light beam should have been travelling at 300000 km/s + 10000 km/h, but the time dilation will compensate just enough for the observed velocity to remain 300000 km/s and nothing more.
Even if the spaceship itself also starts travelling at something close to the speed of light and then fires a light beam, the spaceship will experience lots of time dilation and the light beam even more so. For an observer the time dilation will cancel out the adding of velocities (A+B, remember) you'd except in Newtonian terms, and so you'll always observe the beam moving exactly c, no matter how you look at it.
Furthermore and finally, we can again apply Einstein's First Postulate (relativity). Just as we would see a clock on the spaceship moving slower when we observe it from Earth, the reverse is also true: people on the spaceship would see clocks on Earth running slower as well (because for them, their clocks are standing still and it's the Earth that's moving away).
Time dilation goes both ways, and there's again no way of determining which one is correct or which one is better; both are correct in their frame of reference and all the laws of nature (including Einstein's Postulates) apply in both situations.
Time is relative.
That's all I'll write for now. I'll continue later to expand on some more specific aspects of relativity
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Originally Posted by Exarch
