Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Kryptid on 06/04/2024 21:40:07
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This was inspired a bit by something I had been corrected about earlier. There was a discussion in another thread about black holes where I stated that black hole singularities had infinite density. I was corrected about that assertion in that relativity doesn't predict a density for a singularity (the math doesn't give a number for an answer). I then began to think about my other conceptions regarding relativity...
I had been under the assumption that time is frozen for light because it travels at the speed of light. Likewise, I had assumed that the entire universe was also length contracted to zero along the light's direction of movement, meaning that it was simultaneously at its starting and ending points (thus explaining how it can arrive at its destination despite being frozen in time). Then I took a look at the equations again.
It seems that length contraction doesn't give an answer when the speed is put in as the speed of light. Doing so yields a zero in the denominator, and dividing by zero doesn't give a sensible answer. The same problem pops up with the time dilation equation.
So would it be more correct to say that special relativity doesn't predict the length contraction or time dilation for light? Or am I missing something?
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This was inspired a bit by something I had been corrected about earlier. There was a discussion in another thread about black holes where I stated that black hole singularities had infinite density.
That is not correct of course. The exact nature of a black hole singularity depends on the type of black hole it is (charged, rotating, etc), and of course depends on a unified theory which is lacking. In the simplest case, a Schwarzschild black hole is a vacuum solution, so there is no material at all in it at any density. Time simply ends at the singularity, and anything that gets there has no future light cone at all, so it cannot persist as compressed anything.
Things falling into black holes are pulled apart, not squished together. There's not a point where that changes.
I had been under the assumption that time is frozen for light because it travels at the speed of light.
Time is not frozen for it. Time and distance is just not meaningful for light since there is no meaningful reference frame for it.
Likewise, I had assumed that the entire universe was also length contracted to zero along the light's direction of movement, meaning that it was simultaneously at its starting and ending points
Which I suppose is one way to show the absurdities that result from attempting to define a frame where light is both stationary and also by definition moving at c.
So would it be more correct to say that special relativity doesn't predict the length contraction or time dilation for light?
It is not meaningful for a coordinate system in which light (or anything) has an undefined local speed. Length contraction is something meaningful between a pair of inertial frames where things like distance and simultaneity are meaningful.
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Krytptid's thoughts here mirror mine very closely and it seems I have a major gap in my understanding of SR(not surprising really). If one could travel at the speed of light surely one's rapidity would be infinite and hence time and space would have no existence- why does this not apply to light?. I have heard several physicists state that "light does not experience time or space" or something similar expressed more accurately: I cannot remember any names other than the mathematician Roger Penrose(who can be controversial at times). This question of mine is a bit muddled but I could not find a better wording.
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Krytptid's thoughts here mirror mine very closely and it seems I have a major gap in my understanding of SR(not surprising really). If one could travel at the speed of light surely one's rapidity would be infinite and hence time and space would have no existence- why does this not apply to light?. I have heard several physicists state that "light does not experience time or space" or something similar expressed more accurately: I cannot remember any names other than the mathematician Roger Penrose(who can be controversial at times). This question of mine is a bit muddled but I could not find a better wording.
"If one could travel at the speed of light ..." That's the issue. Anything that travels at c would be in the same boat as light.
Consider the equation for relativist KE:
KE = mc^c (1/sqrt(1-v^2/c^2)-1) if v=c, then it reduces to :
KE mc^2(1/0-1)
Note the 1/0. 1/0 is undefined, thus the KE of a mass traveling at c would be undefined. Just like the singularity, the math doesn't give an answer for this scenario.
So when you post "If one could travel at the speed of light ...", you are essentially saying, "If Relativity doesn't hold..."
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So when someone asks, "What does the universe look like from light's perspective?", the answer is that there is no answer? Or that space and time don't apply?
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Indeed Janus, for an object with mass as it approaches the speed of light both it's mass and KE will increase without bound and will be undefined at v=c. I am just thinking out loud, toying with the idea, to get a better understanding of light in space and time. It still holds that if one could travel at a speed arbitrarily close to the speed of light one could traverse a finite universe in a short time.
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Hi.
I basically agreed with @Halc and therefore didn't need to say anything. However, it's always pleasant to hear from a moderator (Kryptid) and since you give up a lot of time to answer others, we ought to give up some of our time to reply to your post.
I had been under the assumption that time is frozen for light because it travels at the speed of light.
This is usually a result people obtain or propose just by extrapolating the Lorentz Transformation with v→c. So it can be useful to have as many ways as possible to guard against it or see where it might fail. It's especially helpful to go along with their line of reasoning and identify some potential flaws in it, which is what I'll try to do first (below). It's also possible to go straight against the line of reasoning but then people will wander around trying to figure out why their line of reasoing wasn't right and whether you might just be wrong or trying to brush a potential flaw in physics under the carpet.
Situation 1: People often personify light and assume that this result (extrapolating LT with v→c) tells you how things would look or seem to the light.
Problem 1: We shouldn't personify light but let's be tolerant.
Problem 2: Whenever people make the claim 'this is how it seems to "Fred" ', they are always telling you how Fred reports things with co-ordinates of an inertial frame where Fred is at rest. For example, no one would dream of telling you how Fred would report things or identify events in a co-ordinate system where Fred himself is still moving at a speed of 0.133 c, or c/2, or -0.3c why would any of those frames seem natural to Fred? It is clear that we're interested in how Fred reports things with co-ordinates derived from a frame of reference where he is at rest and we would probably be prepared to accept that a frame where Fred is at rest is the one that seems most natural to him. Provided Fred does use the co-ordinates of his own rest frame, then his wrist watch is measuring the co-ordinate time that he will use, that seems natural. If he used any other frame then his wrist watch is not counting the co-ordinate time he would be using (and that isn't a natural choice for Fred, if he's anything like me and you).
So the problem with light is that we have no guarantee that light is at rest in any inertial frame. People have usually just tried to extrapolate the LT with v→c but if you start trying to do this, there is no evidence that light is ever slowing down. Go along with a simple approach of just following a pattern....
With boost parameter v =0.9c , light would move at speed c in the primed co-oridnates.
with v = 0.99c, light would move at c in the primed co-ordinates.
with v =0.999c, light moves at c in the primed co-ords.
with v =0.9999c, light moves at c in the primed...
.....
with v =0.999999999999c, light still moves at c in the primed co-ordinates.
The important point is: There's nothing here suggesting that light is slowing down in the primed co-ordinates as you increase the parameter v towards c.
Now you could even use some algebra and pull out the relativistic velocity addition formula. You can easily show that light is always travelling at c in the primed co-ordinates, regardless of how close that LT boost parameter v gets to c.
This is very diferent to what happens with Fred. If you tried the same game with Fred, then as you increase the v parameter in the Lorentz transformation (LT), his speed in the primed co-ordinates is slowing down. Moreover you can even over-shoot the mark and have the parameter v set so high that Fred is travelling backwards. Specifically, we know that we can always find an inertial frame where Fred would be at rest. However, we have no such guarantee with light, it never seems to slow down in the primed co-ordinates regardless of how high we set the v parameter in the LT.
Now when people (who have just extrapolated the LT with v→c) suggest that "this set of things" (like time being frozen and length contacting to 0) is how it would seem to the light, you at least have something you can come back with as a reply or counter-argument. You'd say "but this isn't how it seems to Fred, even though he could have chosen to use the same frame. I'll assume that's because Fred would naturally choose to use a frame where he is at rest. However, we have no guarantee that light would be at rest in this frame you are proposing for light and so you cannot claim that this frame would be the natural one for light to use or that this is in some way how things would seem to the personified light."
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Situation 2: You just try to remind them of the limits of SR.
SR can be derived in various ways, however the most common method is just to assume light has a speed c in all inertial reference frames.
So it is clear that any frame where light is at rest cannot be an inertial frame and therefore isn't a situation that SR ever aimed to or claimed to say anything about.
They probably will have arguments about extrapolating the LT with v→c and suggest or insist that it ought to mean something or be valid. You could try to remind them that it's valid while v<c but abruptly stops being valid when v=c. They won't like it but you can remind them that this happens often in physics.
There's a whole set of favourite examples you might have, here's just a few you can use:
(i) A position on planet earth can be uniquely defined by spherical polar co-ordinates everywhere and perfectly continuously right up until you hit the North or South poles. At these poles, the situation abruptly changes, the azimuthal angle can be anything you like and yet you're still describing the same position and not a different position. You can't do this anywhere else, a different azimuthal angle would certainly be describing a different position but at the poles, it's fine. This isn't a continuous change, it's a discontinuous or abrupt one.
(ii) A graph of y=1/x behaves perfectly sensibly with x gettting as small as you like. However at x=0 it's undefined, worse than this it's very badly behaved. It might seem that it's becoming positive infinity but you only have to look at small negative values of x and you realise it's a strange and slippery thing. It somehow doesn't come back from positive infinity, it seems to come back from negative infinity. Any pattern you may try to follow as x → 0 has got to break at x=0.
(iii) The helicity of a Dirac particle will be a fixed unvarying quantity (+/- 1 for an electron) in all frames of reference right up until you overtake the particle, in those frames the helicity changes sign (would be - or resp. + 1). It's an abrupt change, it doesn't slowly slide down to +1 from -1. Extrapolating the behaviour for frames of reference with boost parameter v < particle speed (in the lab frame) would not have predicted this behaviour. That'll do as an example and you could leave it there. If you know what helicity is then it's apparent that it must change sign when you overtake the particle. However, this is why I picked this example because we can push this one a bit further: Light is an interesting particle in that it really does always have a constant helicity in any inertial frame of reference. This is a result you can obtain from Quantum Mechanics. We can only understand that result classically by recognising that light cannot be overtaken (or even brought to rest) in any inertial reference frame.
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Situation 3: Again you go straight against any attempt to extrapolate the LT with v→c and their claim that this is an indication of what would happen in a "frame" where v=c and light might be at rest. However, this time you target the tacit or implicit assumption that you would have anything you can call a valid frame.
A co-ordinate system can be almost anything but it MUST serve to uniquely identify different events, this is part of the formal definition of what a co-ordinate system must be and is non-negotiable. Just take all of their claims at face value. So they claim that light starting it's journey over here and the event where the light ended it's journey over there (which could clearly be identified as different events in the lab frame) are given the same x,y,z,t co-ordinates in the rest "frame" of the light. That automatically and necessarily dictates that this "co-ordinate system" obtained from the "rest frame" of the light is actually NOT a valid co-ordinate system.
We have no experience of using a degenerate pseudo-co-ordinate system and it is not a natural choice for Fred to try and use a degenerate pseudo-co-ordinate system. There are plenty of degenerate systems Fred could have chosen to use but he didn't, he chose a co-ordinate system where time was the thing on his watch and distance was something which would also help to keep different events separated, only that seemed like a natural choice of co-ordinate systems to Fred. So it's not sensible to suggest that anything reported in the degenerate pseudo-co-ordinate system you propose for the "rest frame" of light indicates how things would seem for the light.
We then have two possibilities:
(i) There may be some co-ordinate system for light where its notion of time is the thing that passes on its hypothetical wrist watch and its notion of space stuff also helps to seperate different events in some way. However, the Lorentz Transformation does not seem to be the thing connecting these co-ordinates for the rest frame of light to the co-ordinates of space and time that we use. (We can see that precisely because it would suggest that light is left with a set of numbers or pseudo-co-ordinates that fail to separate different events).
(ii) There just may not be any such co-ordinate system or a suitable rest frame for light.
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That's too long, even I'm fed up with listening to myself. I hope some of it is of some use.
Best Wishes.
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Very good, ES, that is the best explanation I have ever received on this subject.