Orbis non sufficit
Wednesday, March 12, 2008
Spacetime
"that same unseen power of the world which impels everyone from age 20 to 40, and from 40 to 80, impels objects inside the horizon irresistably towards the singularity."
-Wheeler etc in "the big black book", describing an objects fall into the singularity of a black hole.
He says this because once you are inside the horizon of a black hole, there exist no reference frames that are at rest relative to the singularity, i.e. no possible act of man or god can halt your fall to the end of your world line. It's a pretty intense thing to say if you consider the subtleties of it.
It made me think about something else though: The magnitude of your 4-velocity (velocity through spacetime, a combination of your regular spatial velocity and your inexorable velocity through time) is constant, in all reference frames. In fact, it is equal to the speed of light.
Consider that briefly. You are hurtling unstoppably through spacetime at the speed of light. Intense.
I think this idea makes time dilation a bit more intuitive though. In your rest frame, the one you cannot help but observe the universe from, you have no spatial velocity, so your plunge through spacetime is entirely in the time direction. Aging at the speed of light.
If you observe someone else from a frame in which they appear to be moving, some of their 4-velocity is in the spatial directions, relative to you, so their plunge through time is slower than yours. Not all their 4-velocity is devoted to time. The symmetry of the situation is an interesting, and classic, consideration; namely that from THEIR point of view time is moving more slowly for YOU. Think about it. It will definately mess with your brain.
"that same unseen power of the world which impels everyone from age 20 to 40, and from 40 to 80, impels objects inside the horizon irresistably towards the singularity."
-Wheeler etc in "the big black book", describing an objects fall into the singularity of a black hole.
He says this because once you are inside the horizon of a black hole, there exist no reference frames that are at rest relative to the singularity, i.e. no possible act of man or god can halt your fall to the end of your world line. It's a pretty intense thing to say if you consider the subtleties of it.
It made me think about something else though: The magnitude of your 4-velocity (velocity through spacetime, a combination of your regular spatial velocity and your inexorable velocity through time) is constant, in all reference frames. In fact, it is equal to the speed of light.
Consider that briefly. You are hurtling unstoppably through spacetime at the speed of light. Intense.
I think this idea makes time dilation a bit more intuitive though. In your rest frame, the one you cannot help but observe the universe from, you have no spatial velocity, so your plunge through spacetime is entirely in the time direction. Aging at the speed of light.
If you observe someone else from a frame in which they appear to be moving, some of their 4-velocity is in the spatial directions, relative to you, so their plunge through time is slower than yours. Not all their 4-velocity is devoted to time. The symmetry of the situation is an interesting, and classic, consideration; namely that from THEIR point of view time is moving more slowly for YOU. Think about it. It will definately mess with your brain.
Comments:
That's awesome!
I've never heard of the concept of 4-velocity. You're right - it does make time-dilation more intuitive! I love it!
It also helps when considering hypothetical situations of 'travelling at the speed of light'. Essentially this would mean a zero velocity in the time dimension, or instantaneous travel. Hey... is that correct...?
Also saying that if you travelled faster than the speed of light that you could travel back in time. BUT you couldn't - if your 4-vel is fixed, you can't travel faster than that in ANY dimension. Having a greater component of your 4-vel in spatial dimensions doesn't make the travel in time negative - because (as you know) velocity is a vector - magnitude + direction - with fixed magnitude, direction can only be altered.
I've pictured a unit circle, x-axis = time vel, and y-axis = spatial vel. Most of the time people are near (1,0), or just above it. High speed particles would beb some where up closer to (0,1), but of course only light is actually AT (0,1). (Does that mean that light doesn't travel in the time dimension?) If you were to travel back in time you would be in the second quadrant - so in effect, you would be travelling backwards in time, but spatially travelling slower than the speed of light. Also, this would mean to travel back in time (at reverse 1x speed) you would have to be absolutely stationary. Oh, and there is ne way to travel faster than 1x either forward or backwards.
What about the 3rd and 4th quadrants? I don't think they really make any sense. Given that I defined my y-axis as the magnitude of spatial velocity, it can't be negative! Is the same to be said of the x-axis, time velocity? If so we work in just the first quadrant. But is time velocity just a single axis, which we may only travel in one direction along?
Talk to me!
Rick
=:-)
I've never heard of the concept of 4-velocity. You're right - it does make time-dilation more intuitive! I love it!
It also helps when considering hypothetical situations of 'travelling at the speed of light'. Essentially this would mean a zero velocity in the time dimension, or instantaneous travel. Hey... is that correct...?
Also saying that if you travelled faster than the speed of light that you could travel back in time. BUT you couldn't - if your 4-vel is fixed, you can't travel faster than that in ANY dimension. Having a greater component of your 4-vel in spatial dimensions doesn't make the travel in time negative - because (as you know) velocity is a vector - magnitude + direction - with fixed magnitude, direction can only be altered.
I've pictured a unit circle, x-axis = time vel, and y-axis = spatial vel. Most of the time people are near (1,0), or just above it. High speed particles would beb some where up closer to (0,1), but of course only light is actually AT (0,1). (Does that mean that light doesn't travel in the time dimension?) If you were to travel back in time you would be in the second quadrant - so in effect, you would be travelling backwards in time, but spatially travelling slower than the speed of light. Also, this would mean to travel back in time (at reverse 1x speed) you would have to be absolutely stationary. Oh, and there is ne way to travel faster than 1x either forward or backwards.
What about the 3rd and 4th quadrants? I don't think they really make any sense. Given that I defined my y-axis as the magnitude of spatial velocity, it can't be negative! Is the same to be said of the x-axis, time velocity? If so we work in just the first quadrant. But is time velocity just a single axis, which we may only travel in one direction along?
Talk to me!
Rick
=:-)
Ok, from the beginning:
1. If you could somehow travel at the speed of light, then in a sense you would be able to travel anywhere "instantaneously". For you, anyway. Time would be observed differently for other people, so for instance if you sped away from the Earth at the speed of light and then came back, it'd be instant for you but hundreds of years might have passed here. An alternative perspective is that you wouldn't think you went anywhere (because lengths were contracted to nothing for you) while others would see you fly off at the speed of light then come back.
It's a bit trippier tho coz you would also have infinite mass and momentum... You might be able to closely approximate the situation tho if you put in enough energy. It's a bit different for photons since they have no rest mass, and not even a rest frame (because they would have to go at the speed of light in that frame...equals paradoxial badness)
2. Indeed, travelling faster than the speed of light is theoretically impossible. Locally. Wormholes are another story because you can locally travel at normal speeds (but they create other problems)
3. What time means to light is pretty messed up, see 1. You're kind of right though.
As for your unit circle idea, it seems like a good idea but I think the problem lies in the fact that you need to consider multiple reference frames somehow. You could draw one of these diagrams for each reference frame though. Being in the quadrant you speak of would mean having a negative time component of your 4-velocity which I think would be uncool. I'll think about that. As for not going faster than 1x forward or back through time, you can certainly go forward through time (from some reference frames perspective) faster than 1x, see 1. however you can't go through spacetime at any other speed than 1x.
4. Yeah you're axes are a little screwed up, you can certainly have negative spatial velocities though, it's just about your direction relative to your coordinate system. Time is a trick thing, the laws of physics don't say much about why we go in the direction we do through time. The equations all work just as well with time going the other way. Some think it's just a trick of our brains, an illusion to make spacetime understandable (rather than experiencing it all at once in some kind of timeless fashion. This however brings up some deep philosophical questions which are certainly debatable)
1. If you could somehow travel at the speed of light, then in a sense you would be able to travel anywhere "instantaneously". For you, anyway. Time would be observed differently for other people, so for instance if you sped away from the Earth at the speed of light and then came back, it'd be instant for you but hundreds of years might have passed here. An alternative perspective is that you wouldn't think you went anywhere (because lengths were contracted to nothing for you) while others would see you fly off at the speed of light then come back.
It's a bit trippier tho coz you would also have infinite mass and momentum... You might be able to closely approximate the situation tho if you put in enough energy. It's a bit different for photons since they have no rest mass, and not even a rest frame (because they would have to go at the speed of light in that frame...equals paradoxial badness)
2. Indeed, travelling faster than the speed of light is theoretically impossible. Locally. Wormholes are another story because you can locally travel at normal speeds (but they create other problems)
3. What time means to light is pretty messed up, see 1. You're kind of right though.
As for your unit circle idea, it seems like a good idea but I think the problem lies in the fact that you need to consider multiple reference frames somehow. You could draw one of these diagrams for each reference frame though. Being in the quadrant you speak of would mean having a negative time component of your 4-velocity which I think would be uncool. I'll think about that. As for not going faster than 1x forward or back through time, you can certainly go forward through time (from some reference frames perspective) faster than 1x, see 1. however you can't go through spacetime at any other speed than 1x.
4. Yeah you're axes are a little screwed up, you can certainly have negative spatial velocities though, it's just about your direction relative to your coordinate system. Time is a trick thing, the laws of physics don't say much about why we go in the direction we do through time. The equations all work just as well with time going the other way. Some think it's just a trick of our brains, an illusion to make spacetime understandable (rather than experiencing it all at once in some kind of timeless fashion. This however brings up some deep philosophical questions which are certainly debatable)
Awesome conversation going on here. I actually never realised that 4-velocity was a constant. That is cool. I'll have to take a gander at the big black book sometime.
Antiparticles are in your second quadrant :P
That still trips me out sometimes.
It's pretty fucked to try to think about how the world would look if your travelling backwards in time even for a point particle.
Antiparticles are in your second quadrant :P
That still trips me out sometimes.
It's pretty fucked to try to think about how the world would look if your travelling backwards in time even for a point particle.
hmm yeah, antiparticles. Damn those crazy things. I'm not totally sure I believe the travelling backwards through time interpretation tho. I'll need more convincing. And I've only been reading about 4-velocity in special rel, I dunno if gravity messes with it's constantness. I guess it shouldn't really. But yeah, i think all tensors have the same "magnitude" in all reference frames, like their determinant or whatever. I could be wrong, I didn't check before writing this. Steve? If i'm right then its easy to show 4-velocity is constant:
U = γ(c,ū)
is the definition of 4-velocity, where c is the time component, ū is the 3 regular spatial components and γ is the relativistic speed factor thingo.
So in the rest frame γ=1 (no crazy time dilation crap) and ū=0 so we have
U = (c,0,0,0)
or
|U| = c
q.e.d.
U = γ(c,ū)
is the definition of 4-velocity, where c is the time component, ū is the 3 regular spatial components and γ is the relativistic speed factor thingo.
So in the rest frame γ=1 (no crazy time dilation crap) and ū=0 so we have
U = (c,0,0,0)
or
|U| = c
q.e.d.
Maybe that should be Q.E.F:
Quod erat faciendum - "which was to be done.", since it's not such a strong proof of what I was talking about...
Quod erat faciendum - "which was to be done.", since it's not such a strong proof of what I was talking about...
Hmm yes we spoke about this the yesterday, I want to get to the bottom of it now.
We should start a geekspeak forrum and get kent and co onto it so we can pose questions and hopefully someone can answer them
We should start a geekspeak forrum and get kent and co onto it so we can pose questions and hopefully someone can answer them
If you'd like I could chuck a Simple Machines forum on my hosting for you instead, although your free host seems reasonably performant and doesn't seem to be ad-ridden.
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