Speed of Light -- Implications

[vexx32]

We all know the speed of light is a universal limit. Many of you probably have some idea about how approaching the speed of light affects perception from all frames of reference, often with varying results. I.e., Relativity.

There is a strange sort of implication in this bit, though. If distance contracts as you approach the speed of light, and time dilates... hypothetically, what do the equations say if you move at the speed of light? Well, nothing that makes sense, to be honest. The length contraction formula implies that from the frame of reference of anything travelling at the speed of light, length is meaningless. You end up dividing by zero if you try to find the length of anything from that reference frame. The time dilation formula basically ends up being rather similar.

However, there's an interesting difference. Distance contracts and gets smaller and smaller as you approach light speed, whilst time dilates and gets more and more stretched out as you approach the speed of light.

So, to something moving at light speed (hypothetically, of course), distance will have shrunk into a meaningless nonentity smaller than a Planck Length, whereas time will have stretched out so far as to be effectively if not literally stopped.

This, to me, implies something very odd. Time and space are thought to be inseparable -- gravity distorts space-time, as though it's a membranous thing which consists of both distance and duration, as if the two are inextricably tied to one another. But these equations in relativity appear to indicate that if you go fast enough, you distort space and time separately -- space shrinks whilst time expands. This doesn't really make sense, unless time and space are simply entirely different entities... but there is so much in physics that points to them being inextricable from each other, and meaningless without each other.

The other thing it suggests to me is that to an observer that is travelling at light speed (shall we say a hypothetical observer that is a conscious construct of photons?), space and time do not even exist. I mean, when you think about it, all the funny quantum effects where photons behave as though they are in two or more places at once work with this sort of thinking -- after all, if they do not have to worry about time or distance, they could be all across the universe at any given time or place. Every photon in the universe could the the very same photon, just popping up in our perception at different times and places.

The question that arises when one considers the fact that a thing travelling at light speed has no limitations based on space or time is... well... how does that speed even exist? If space and time are shrunken and stretched in and out to infinity, would we not even be able to measure the speed? Should it not appear to travel instantaneously and not have a speed at all? It doesn't make sense that just by the very fact of us claiming light speed as a speed limit implies that the speed of the thing we are measuring (light) should be infinite, if its own speed is indeed the "limit" that we think it is.

Your thoughts, guys?

[Stochasty]

This is going to get a little math heavy, because there's no other way to satisfactorily answer your questions.

However, there's an interesting difference. Distance contracts and gets smaller and smaller as you approach light speed, whilst time dilates and gets more and more stretched out as you approach the speed of light.

So, to something moving at light speed (hypothetically, of course), distance will have shrunk into a meaningless nonentity smaller than a Planck Length, whereas time will have stretched out so far as to be effectively if not literally stopped.

This, to me, implies something very odd. Time and space are thought to be inseparable -- gravity distorts space-time, as though it's a membranous thing which consists of both distance and duration, as if the two are inextricably tied to one another. But these equations in relativity appear to indicate that if you go fast enough, you distort space and time separately -- space shrinks whilst time expands. This doesn't really make sense, unless time and space are simply entirely different entities... but there is so much in physics that points to them being inextricable from each other, and meaningless without each other.

Just because we physicists treat space and time on the same footing doesn't mean we think they are the same thing. Causality is a big deal in physics, and maintaining causality requires us to distinguish between space and time.

Mathematically, what you are describing is a consequence of the change in metric signature between space coordinates and time coordinates. To illustrate, the Minkowski space (special relativity) distance element is given by:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.

This means that the special relativity manifold is not Euclidean, but rather hyperbolic (also called Lorentzian in the literature).

Here's a way to think about the implications of this fact. Imagine a 2-dimensional Euclidean spacetime defined by one direction which we'll call "time" and another direction which we'll call "space." Velocity vectors are, by convention, defined to be unit vectors on this spacetime, so the the velocity vector of a "stationary" object (one not moving in the spatial dimension) is a vector of length one which points entirely in the time direction. Now, imagine that we apply acceleration in the space direction - this will add a spatial component to our velocity vector, but the vector must stay normalized so what we're actually doing is rotating that vector. It's relatively easy to see that if we continue to apply acceleration the allowed velocity vectors inscribe a circle on spacetime.

However, here's where causality comes into play: a "trajectory" is an integration over some arbitrary function of velocity. Think about it as starting at a point, and moving at a constant rate in the direction of your velocity vector while at the same time allowing that vector to rotate according to this arbitrary function. It's pretty easy to see that, if velocity vectors are allowed to inscribe a complete circle, it is possible to define a function which will eventually return one to one's starting point. This is a problem, though, because we have defined one of the directions to be "time" - if you are allowed to visit the same point in space and time more than once you have violated causality (there are some caveats to this, but they are beyond the scope of the present illustration). Basically, by treating space and time identically the way a Euclidean spacetime would suggest, we lose causality, and therefore we lose any notion of "time" itself.

However, the SR metric is hyperbolic. What that means is that, as we rotate the velocity vector in the method I described above, the shape we trace out is not a circle but a hyperbola with asymptotes along the lines x=t and x=-t. This confines our allowed velocities to a 90 degree cone around the time axis, and thus means that it is impossible to generate a trajectory which visits the same spacetime point more than once - hence, causality is preserved.

The other thing it suggests to me is that to an observer that is travelling at light speed (shall we say a hypothetical observer that is a conscious construct of photons?), space and time do not even exist.

This is pretty much the case. Null vectors are weird. There are no non-singular transforms between time-like and null vectors, and the tangent space of a null vector is ill-defined.

Edited May 7, 2013 by Stochasty
too many typos

[Nikolai]

Some quick thoughts:

* If you have rest mass, you're not getting to the speed of light. It would require an infinite amount of energy to get there. From every inertial reference frame it's possible for you to get to, the speed of light would appear to be invariant.

* Light in vaccum never goes slower than c. What I mean by that is that it's not as if light accelerates to c from some slower speed. It either exists -- at speed -- or it doesn't. Indeed, even when something is reflected, modern physics holds that one photon is absorbed and another emitted; it's not as if the photon "bounces" off the surface (though we can pretend it does to model things like momentum transfer).

* Thus, in many ways, the vantage point of a photon is unique. We cannot make claims about how it will appear to our inertial reference frames based on how things appear to it.

Does that help?

[Cesrate]

It's surely odd when thinking relativistic effect on only space or only time.

[vexx32]

This is going to get a little math heavy, because there's no other way to satisfactorily answer your questions.

Just because we physicists treat space and time on the same footing doesn't mean we think they are the same thing. Causality is a big deal in physics, and maintaining causality requires us to distinguish between space and time.

Mathematically, what you are describing is a consequence of the change in metric signature between space coordinates and time coordinates. To illustrate, the Minkkowski space (special relativity) distance element is given by:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.

This means that the special relativity manifold is not Euclidean, but rather hyperbolic (also called Lorentzian in the literature). Here's a way to think about the implications of this fact. Imagine a 2-dimensional Euclidean spacetime defined by one direction which we'll call "time" and another direction which we'll call "space." Velocity vectors are, by convention, defined to be unit vectors on this spacetime, so the the velocity vector of a "stationary" object (one not moving in the spatial dimension) is a vector of length one which points entirely in the time direction. Now, imagine that we apply acceleration in the space direction - this will add a spatial component to our velocity vector, but the vector must stay normalized so what we're actually doing is rotating that vector. It's relatively easy to see that if we continue to apply acceleration the allowed velocity vectors inscribe a circle on spacetime.

However, here's where causality comes into play: a "trajectory" is an integration over some arbitrary function of velocity. Think about it as starting at a point, and moving at a constant rate in the direction of your velocity vector while at the same time allowing that vector to rotate according to this arbitrary function. It's pretty easy to see that, if velocity vectors are allowed to inscribe a complete circle, it is possible to define a function which will eventually return one to one's starting point. This is a problem, though, because we have defined one of the directions to be "time" - if you are allowed to visit the same point in space and time more than once you have violated causality (there are some caveats to this, but they are beyond the scope of the present illustration). Basically, by treating space and time identically the way a Euclidean spacetime would suggest, we lose causality, and therefore we lose any notion of "time" itself.

However, the SR metric is hyperbolic. What that means is that, as we rotate the velocity vector in the method I described above, the shape we trace out is not a circle but a hyperbola with asymptotes along the lines x=t and x=-t. This confines our allowed velocities to a 90 degree cone around the time axis, and thus means that it is impossible to generate a trajectory which visits the same spacetime point more than once - hence, causality is preserved.

This is pretty much the case. Null vectors are weird. There are no non-singular transforms between time-like and null vectors, and the tangent space of a null vector is ill-defined.

Thanks for all that

I think I got all of it. I'm not familiar with the special relativity distance element formula, but I sort of get what it indicates. So... if both space and time end up being null vectors at light speed, how does light even appear to have a limit? I mean, to my way of thinking, if space and time are reduced to nothing or are meaningless, it doesn't follow that light has a measurable speed.

Some quick thoughts:

* If you have rest mass, you're not getting to the speed of light. It would require an infinite amount of energy to get there. From every inertial reference frame it's possible for you to get to, the speed of light would appear to be invariant.

* Light in vaccum never goes slower than c. What I mean by that is that it's not as if light accelerates to c from some slower speed. It either exists -- at speed -- or it doesn't. Indeed, even when something is reflected, modern physics holds that one photon is absorbed and another emitted; it's not as if the photon "bounces" off the surface (though we can pretend it does to model things like momentum transfer).

* Thus, in many ways, the vantage point of a photon is unique. We cannot make claims about how it will appear to our inertial reference frames based on how things appear to it.

Does that help?

Not at all. I understand all of that, I was merely saying that because of the reference frame of a photon being at the speed limit of all things in the universe, it does not quite follow that it has a speed limit at all.

It's surely odd when thinking relativistic effect on only space or only time.

Relativity is weird. It does make a weird sort of sense, though, but it tends to take a good deal of puzzling out.

[Person012345]

Is "time expanding" even a sensible concept? I mean, what definition of expanding are we using? It certainly may not imply the opposite of space shrinking as far as I'm aware (although I can't into math), because "expanding" usually refers to spacial measurements, as it does in the case of "distance shrinking". I'm not aware that time is measured in 3 dimensional spacial measurements.

But yes, things are weird there. To get to c requires infinite energy and when you get to infinities in nature things always start to get weird.

[vexx32]

"Time expanding" just basically means that an object progresses through time slower, or effectively that for the person that time seems to be "expanding", one second lasts longer than it would appear for others who're travelling slower.

Yeah, infinite energy is all well and good, but photons have a very specific energy and apparently a zero mass (or very close to zero; I've heard some few people say that they have infinitesimal mass). Even if they can travel at light speed, being able to do so throws up some interesting ideas for me, that's all

[Person012345]

"Time expanding" just basically means that an object progresses through time slower, or effectively that for the person that time seems to be "expanding", one second lasts longer than it would appear for others who're travelling slower.

What I'm saying is that, as far as I am aware, that is not the opposite of space "shrinking" necessarily (it may be, I just may not be aware of it). It would only seem opposite because you're using "expanding" to describe it.

Yeah, infinite energy is all well and good, but photons have a very specific energy and apparently a zero mass (or very close to zero; I've heard some few people say that they have infinitesimal mass). Even if they can travel at light speed, being able to do so throws up some interesting ideas for me, that's all

0 mass = unintuitive in the first place so idk. I don't understand them. Would an "at rest" photon (if that is indeed a sensible notion) be bound to stay where it is? I mean, it would have no inertia would it? Idek how that would work. Photons are weird.

If they have a very small mass then they won't be traveling at c, which makes the point moot.

[Stochasty]

I think I got all of it. I'm not familiar with the special relativity distance element formula, but I sort of get what it indicates. So... if both space and time end up being null vectors at light speed, how does light even appear to have a limit? I mean, to my way of thinking, if space and time are reduced to nothing or are meaningless, it doesn't follow that light has a measurable speed.

Space and time only become "meaningless" from the perspective of an observer travelling along a null trajectory. From the perspective of a time-like observer, they still have meaning, and it is easy to compute the coordinates of an object travelling along a null trajectory. "Velocity" within the frame of the time-like observer is just distance over time, which are both well defined quantities for him.

The "speed limit" concept arises from the fact that the Lorentz transforms mean that all null trajectories appear to have the same velocity when viewed by any inertial observer.

Edit: I'll try to give a better description of what is meant by the Minkowski distance element and how it applies to velocity vectors.

The distance element formula describes how infinitesimal distances are measured in a spacetime. Thus, the distance between two points is the integral over ds following any curve between those two points. For Minkowski space, this simplifies to an easy coordinate representation:

d(p1,p2)^2 = -(t1-t2)^2 + (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2,

where d is the distance function, p1 and p2 are the two points, and t1, t2, etc. are the coordinate representations of each of those points. Compare this to the Euclidean distance formula,

d(p1,p2)^2 = (t1-t2)^2 + (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2.

Velocity vectors are unit vectors, which means they have magnitude one. In the two-dimensional Euclidean example I gave above, the allowed velocity vectors are all vectors which satisfy

v.v = 1,

implying that

v_x^2 + v_t^2 = 1,

(Where v_x and v_t are the spatial and time coordinates, respectively). This is obviously the equation for a unit circle.

For the Lorentzian signature, allowed velocity vectors satisfy

v.v = -1,

giving

v_t^2 - v_x^2 = 1,

which is the equation for a hyperbola. The two branches correspond to travel forwards and backwards through time.

Edited May 7, 2013 by Stochasty
Woah! Caught a big mistake. Fixed now.

[vexx32]

What I'm saying is that, as far as I am aware, that is not the opposite of space "shrinking" necessarily (it may be, I just may not be aware of it). It would only seem opposite because you're using "expanding" to describe it.

0 mass = unintuitive in the first place so idk. I don't understand them. Would an "at rest" photon (if that is indeed a sensible notion) be bound to stay where it is? I mean, it would have no inertia would it? Idek how that would work. Photons are weird.

If they have a very small mass then they won't be traveling at c, which makes the point moot.

I don't really know. To my understanding of time, it appears that the effect on space and time are the inverse of each other. Whether that's actually the case I do not know.

A photon cannot be at rest because that idea means that you know both that it has a zero velocity and you know its exact position. The uncertainty principle states that you can't know both at the same instant, so you can never actually have a photon "at rest"

Space and time only become "meaningless" from the perspective of an observer travelling along a null trajectory. From the perspective of a time-like observer, they still have meaning, and it is easy to compute the coordinates of an object travelling along a null trajectory. "Velocity" within the frame of the time-like observer is just distance over time, which are both well defined quantities for him.

The "speed limit" concept arises from the fact that the Lorentz transforms mean that all null trajectories appear to have the same velocity when viewed by any inertial observer.

So, essentially, what you're saying is... because the trajectory is null, the speed always appears identical, which is why the velocity never actually changes (light slows down in water, for example, but its actual velocity is still c; it's just got to get through the water, which impedes the movement by essentially creating a semipermeable barrier for it in a vague sense)... and the reason that no object with mass can reach the speed of light is because it would require that it also has a null velocity vector, which isn't possible for objects with mass...

Okay, I kind of understand how it works. Relativity is a bit of a weird one.

I guess the only other thing I'm still unsure of is how an object with a null velocity can appear to have a very definite position or speed to all observers not having a null velocity vector... I guess I'll have a look at the Lorentz transforms at some point. Thankyou very much

EDIT: Okay, that math was interesting... So the equation ends up being a hyperbola for the velocity... I need to think on this more when it's not 2am xD Thankyou very much for all this, though

Edited May 7, 2013 by vexx32

[Person012345]

I don't really know. To my understanding of time, it appears that the effect on space and time are the inverse of each other. Whether that's actually the case I do not know.

A photon cannot be at rest because that idea means that you know both that it has a zero velocity and you know its exact position. The uncertainty principle states that you can't know both at the same instant, so you can never actually have a photon "at rest"

:eh:

Uncertainty principle = Quantum Mechanics = OMG WHAT THE HELL IS GOING ON YOU MAKE NO SENSE BUT ARE ALSO CORRECT.

[vexx32]

Oh, quantum mechanics makes sense all right, it just takes a while to get your head around it.

[Stochasty]

how an object with a null velocity can appear to have a very definite position or speed to all observers not having a null velocity vector

This is the crux of relativity. Everything is observer dependent - including the coordinate frames in which you do your measurement. Just because the coordinate frame of a null observer is ill-defined does not mean that that observer does not have a well-defined trajectory in the frame of a second, time-like observer. So, while the first (null) observer would get a divide-by-zero error when trying to compute his own velocity, the second observer has no such difficulty in computing the velocity of the first.

[vexx32]

True. Once again I've been making the assumption that each particle has a position and velocity that is its own... It's more that there is no intrinsic velocity or anything, it's just something we measure.

It does seem kind of weird to think that something could be trying to compute its own velocity and actually fail to do so, when someone standing by and watching them pass can just punch some numbers and find it easily enough. xD

[Person012345]

Oh, quantum mechanics makes sense all right, it just takes a while to get your head around it.

I doubt you can actually wrap your head around it without very advanced maths capability. I tried once, almost thought I was getting it and then realised that I actually didn't get it at all. It's not that I can't understand it in a basic sense, it's that I cannot fit it into my understanding of reality in any way. Which is an important step in saying I can actually understand it and it's effects and so on. Since I can't do the math.

It doesn't help that quantum mechanics itself is not fully understood yet even by those in the field (there's lots of discoveries still to be made in quantum mechanics). Maybe if I was actually schooled in it I might get it, but I am not.

[Person012345]

I do accept that it is right though, even if I don't understand how it is right.

[Stochasty]

I doubt you can actually wrap your head around it without very advanced maths capability.

The math behind QM is actually not all that difficult, at least compared with the math behind GR. There's a reason that Introductory QM is typically taught in the second year, whereas the first introduction most students have with GR is not until grad. school (and many physicists actually obtain PhDs without ever having had a formal course in GR).

The trick to understanding QM is being able to conceptualize what is meant by state vectors and Dirac notation. That's the stumbling block for most students, in my (somewhat limited) experience.

[vexx32]

The important thing to remember is that trying to apply something that affects primarily the things that occur on scales so small a human mind has trouble envisioning the actual size to a world that we can see and interact with is a great deal too complex to try to do. It's useful for understanding how phenomena happen, but for the everyday world most people live in, it's not very useful. I can generally understand it if I put my mind to it, but trying to relate it back to macroscopic scales is a bit of a futile endeavour, I've found.

Quantum effects are often specialised cases that only apply under certain conditions, too.

[Stochasty]

One of my current research projects in on exactly that - generalized Quantum Brownian Motion (basically, the study of how quantum interactions between a system and its environment produce noise).

[Nikolai]

I do accept that it is right though, even if I don't understand how it is right.

I think I can safely say that nobody understands quantum mechanics. -- Richard Feynman, The Character of Physical Law, chapter 6, "Probability and Uncertainty"

This was a man who worked deeply with the subject matter and excelled at explaining deep concepts to people. I think he would have agreed with your statement perfectly.

[WestAir]

Another interesting implication of the speed of light is the jump in causality and the break of entropy caused by the mere existence of force carrier particles. I'm not as knowledgeable as Sochasty or K^2, so maybe they'll have a more sensible retort, but picture this:

All the forces in the Universe (Weak, Strong, Gravity, Electromagnetic) propogate along with specific force carrier particles. Not all of those particles are mass-less, and not all of them travel at C, but all of them are strictly limited to C as a maximum velocity. So, when gravity bends space and time to a point where a photon sphere, and an event horizon are formed, it becomes impossible for any particle to escape the event horizon on any outward trajectory, because (from the perspective of all particles within the event horizon), there is no speed or trajectory that points away from the singularity. Visually, it would be as if the singulairty was actually a spherical wall surrounding particles, constantly closing in and getting closer.

That in mind, a question I had a while back was that for the force carrier particle for gravity (gravitons) to convey the gravity of a black hole to any outside particles or observers, and for the gravity waves to propagate away from the singularity to suck anything else in, it apparently must escape the event horizon to convey its force to particles outside the event horizon. That means a graviton has to violate causality and entropy because it has to travel faster than C to convey its force to particles outside the event horizon.

While we don't know how gravitons or gravity works, the fact is that space and time are looped into a (from an interior observers perspective) closed universe within that event horizon. No vector line within an event horizon leads out. Since black holes do have gravity, the question I have to ask is... how? A second question I have is: If gravitons can escape, then can't we also invent FTL communication by wiggling some mass around and using a detector to detect it at what must be faster than light speeds, allowing us to send messages faster than C?

[K^2]

I'm not actually sure if it's me or Stochasty that should be explaining that. But there are two factors here. First of all, there is no such thing as a graviton in the standard model. It's a particle that shows up when you try to quantize gravity, but you should also keep in mind that no successful quantum gravity theory exists. In other words, if such a thing as graviton really exists, we have no theory to properly describe it. Though, there are some indication of its properties if it does, in fact, exist.

The slightly more relevant comment is that what you describe is relevant to on-the-shell particles. These are particles that satisfy E²/c² - p² = m²c². (This is called a mass shell, because possible values of (E, p) form a hyper-shell in 4-momentum space. This is true even if m is zero.) Any particle that freely propagates through space must satisfy this relationship, but a force carrier particle does not. These are virtual particles, and they generally exist off the shell. A particle off the shell can, in principle, propagate across the event horizon, traverse untraversable wormholes, and do many other interesting things.

This is where I kind of run into limit of my own understanding, because I don't deal with paritcle interactions in curved space-time. It's more of Stochasty's domain. But any description of quantum interaction in curved space-time would have to take this into account, and I wouldn't be surprised if this is the explanation for how black hole interacts with stuff.

[Stochasty]

This is where I kind of run into limit of my own understanding, because I don't deal with paritcle interactions in curved space-time. It's more of Stochasty's domain. But any description of quantum interaction in curved space-time would have to take this into account, and I wouldn't be surprised if this is the explanation for how black hole interacts with stuff.

This is pretty close to the limit of my understanding, too; a satisfactory answer here really requires a quantum gravity theory.

Part of the problem is that we always talk about gravitons as perturbations around a background metric - but in the black hole case Schwarzschild (or Kerr, etc.) is the background metric, which means that the information regarding the propagation of gravity away from the black hole is already contained in the background. Attempting to describe the singularity itself perturbatively runs into renormalization issues and generally fails. You can consider propagation of gravity waves due to a change in the position of the black hole - this is the subject of the numerical relativists working on binary black hole mergers in conjunction with LIGO on the experimental side - but even those calculations are considering perturbations well away from the two black holes and the event horizons; the collision itself is handled non-perturbatively.

However, K^2 is correct that virtual particles are not confined to behavior on the mass shell. This is one of the ways to think about Hawking radiation - you have correlations between virtual particles on either side of the horizon which result in an energy flux across the horizon. Gravitons can be approximated by a massless, minimally coupled scalar field which certainly obeys the properties K^2 describes.

Now, if we want to get speculative and start talking quantum gravity, things change. There's some indication, based on generic properties of candidate quantum gravity theories (and supported by some of my own, soon-to-be-published research in stochastic gravity) that spacetime is not four dimensional on small scales, but rather two-dimensional. The geometry may be something like a 2-complex (to understand what a 2-complex is, think of an arbitrary graph in 3 dimensions - containing vertices connected by edges - and allow that graph to evolve in time such that each vertex traces out a world line and each edge a world sheet, but also vertices are allowed to merge and split so that the graph changes shape from time slice to time slice). If this is the case, then (on a given time slice) a black hole would look like a graph where every vertex has many connections leading towards the singularity and only a few leading away, such that any particle random walking the graph has an overwhelming probability of falling towards the singularity. "Gravitons" in this picture are the connections in the graph itself, so the singularity is still gravitationally connected to the outside universe. Of course, this is only on such picture (one suggested by the Loop Quantum Gravity approach, but also not incompatible with several others); it is by no means the only picture, nor is there any reason yet to believe that this is the way things really are.

Edited May 7, 2013 by Stochasty

[Person012345]

The geometry may be something like a 2-complex (to understand what a 2-complex is, think of an arbitrary graph in 3 dimensions - containing vertexes connected by edges - and allow that graph to evolve in time such that each vertex traces out a world line and each edge a world sheet, but also vertices are allowed to merge and split to that the graph changes shape from time slice to time slice).

That sounds too complex.

Ba dum dum tish.