No, I believe that the light cone always has a slope of c. (Einstein’s Constant here - I’m not familiar with the notation of c vs. C.) Because that is the theoretical limit of information transfer, not the speed of light.
My understanding of Hawking’s time cone is that it is defined by C, no matter what the speed of light is in the medium surounding the point where you’re defining the cone from.
Sorry for the simulpost.
But the time cone is the circle (the radius of which is the distance traversed by light) as time progresses. The event is described by the light that started from the event, not the speed of light in vacuum as i understand from hawking.
No, it is defined by the fastest speed at which matter and information can travel, which is C.
It has been said by several others, but I will reiterate it with more words…
The way time and space comingle involves a constant. We’ll call it c. It happens to look like a velocity because we have distances and times in the math. It also happens that a massless particle moving through empty space – for example, light in a vacuum – travels a speed equal to this constant. In bulk media, photon propagation gets more complex so light doesn’t travel as fast, but that’s okay.
Again, light is not special. It happens to travel at c in a vacuum, so it is often convenient when discussing relativity to use examples involving light. Perhaps your source (Hawking?) gave such an example, and perhaps he used words that gave the impression that photons (light) were somehow special. They are not, and that is not what he intended.
The phrase “light cone” is unfortunate when talking about stuff in a bulk medium since in this situation light does not travel along the light cone. “Light cone” is just easier to say than “the surface in Minkowski space that separates those events etc etc…”
Yes, andy, you are wrong and remain wrong, because you stubbornly insist on assuming that you can substitute c in place of C in the equations. You cannot. Most theories assume that gravity propagates at C. So the beta particles are not exceeding the speed of gravitons.
Because your “understanding” is simply wrong, as several people have pointed out. The time cone is defined by C instead of c. (Achernar, I need some way to try to distinguish between the local speed of light and the speed of light in a vacuum. If there is another convention preferred by physicists, then let’s switch to that.)
Perhaps this is too great a generalization, but the only effect of a charged particle exceeding the local speed of light is that it gives off Cerenkov radiation. You can’t make it mean anything else.
It’s like when you’re driving on the freeway and there’s a motorcycle cop in an adjacent lane. Even though you’re not exceeding the “absolute” speed limit of 65mph, you still feel a little uneasy passing him at 60mph. If 65mph were Einstein’s constant, and the motorcyle cop were light, you would emit Cerenkov radiation when you exceed his speed in the medium (the freeway). Or something.
That’s as good as any I’ve seen; I suggest you keep using it. The only one I’ve actually seen, though, is using c and v, which is of course rather confusing.
Perhaps this will help: Photons always travel at c. The fact that light appears to travel at less than c is an artifact of our macroscopic perspective – the individual photons, all of them, are still travelling at c. How does this work? Let’s enlarge the water a billion times to take a closer look. At this scale we see that water (and, in fact, all earthly matter) is mostly empty space. There is empty space between the molecules, and, even more, there is empty space inside the atoms. If the electron shells are the diameter of a football stadium, the nucleus is an acorn at the center. Photons, having no mass, always travel at c. From this scale, travelling through water is similar to travelling through a vacuum: relatively larve distances to travel through, undisturbed. However, there is a difference: there is more stuff in the water. So as the electrons fly along, they occasionally encounter an electron, or a proton. When they do, they are absorbed. Some short time later, they are re-emitted. As your average photon travels through the medium, it has an average number of encounters with particles, which slows its journey by an average amount. So, while each photon is always travelling at c, the time for light to enter one side of the medium and exit the other is longer than it would have been in empty space. Our coarse perceptions only see this average, and it looks as if the velocity of light in a medium is slower than c.
It looks like andy_fl might just be confused, still. So, let’s see if I can clear up what might be the Op’s core misunderstanding.
All this business with ‘light cones’ and ‘time cones’ seems to be tripping you up.
The idea of this cone is as follows:
The universe has an ultimate speed limit. This is, in my notation, C. It might also be called ‘Einstein’s constant’. A few posters have called it ‘c’, because it often apperas in the lower case in equations. Whatever. The point is, no matter what the circumstances or the medium, no matter anything at all, nothing, ever can travel faster than it.
So, some physicists cunningly came up with this ‘limitng surface’; this ‘light cone’. Here’s the shape of the cone, in four-dimansional space (the four dimensions are time and the three usual space dimensions):
The farthest anything can have gone in a given time is that time multiplied by the speed limit of the universe. So, if the speed limit of the universe is 300 000 kilometers per second, then one second after you start counting, whatever you’re watching can’t be more than 300 000 km away. After two seconds, anything that started at your point of origin can’t be more than 600 000 km away (2 x 3 = 6).
So the ‘light cone’ limits the distance that anything could possilby be from the origin after a given time. It’s radius increases with time, just like a thief could be anywhere in town an hour after the robbery, but a minute after the robbery, can’t be very far from the bank at all, even if he is the fastest thief ever.
How far can you be from the starting point after a given time? No farther than something traveling at the Ultimate Speed Limit © could have gotten. Not ‘No father than light could have gotten’. No, no. Light is just another traveler. Usually it shoots along at the Ultimate Speed Limit ©, but it’s kind of lackadaisical when it hangs around matter. So we don’t measure the size of the ‘light cone’ by how far light has traveled since the experiment began. We DO NOT do this. Because light doesn’t promise that it will travel at the Ultimate Speed Limit ©. Instead of mucking around with light, we define the limits of that ‘light cone’ to be how far something traveling at the Ultimate Speed Limit © could have made it.
Now, obviously, nothing can have gotten farther away, in the time since we started counting, than something which travels at the Ultimate Speed Limit ©. Why? Because that’s the Ultimate Speed Limit. Nothing can go faster than it.
Now, this whole business of calling it a ‘light cone’ is annoying, because it confuses people sometimes. Since, in a perfect, ideal, not-a-single-speck-of-matter vacuum, light happens to whizz around at The Ultimate Speed Limit ©, you COULD say that the ‘light cone’ is defined by how far light could have gotten, since it’s traveling at the Ultimate Speed Limit. That’s fine, but it becomes rapidly misleading, becuase light tends to slow down and smell the roses anytime except in theses extremely perfect conditions. So, the cool physicists DON’T say that the ‘light cone’ is defined by how far light can get, they say it’s defiend by how far light could have gotten under the super-perfect conditions described above- when it happens to be traveling at the Ultimate Speed Limit ©. So, the COOLEST physicists go even farther, and just define the ‘light cone’ as how far something could have gotten traveling at the Ultimate Speed Limit ©. Now, that’s where the name, and the concept of this ‘cone’ come from.
On to the Water Problem.
In water, how far can something have gotten form it’s origin in the time since it started?
No farther than something which had traveled at the Ultimate Speed Limit ©. (Just like always. This ALWAYS applies, water or not.)
Oh, by the way, in water, ordinary light slows down and window-shops at all the pretty H[sub]2[/sub]O molecules. So, the speed of light slows down to some value less than the Ultimate Speed Limit ©. Let’s called this speed, the speed at which light moseys in water, let’s call it c. Lower-case c.
Now, The size of the ‘light cone’ DOES NOT change in water. Why? Well, what’s the definition of the size of the ‘cone’? It’s how far something could have gotten traveling at the Ultimate Speed Limit ©. It is NOT how far something could have gotten traveling at the lazy, pokey speed that light shuffles along at in water. Just because light gets lazy doesn’t mean more motivated particles can’t travel at the Ultimate Speed Limit ©. But if they do, they WILL be going faster than light.
NOT faster than the Ultimate Speed Limit ©. Definitley NOT so fast that they somehow go back in time. NO. They would be going Faster than light pokes along, but slower than the Universe’s Never-Exceed Speed.
The particles making the Cerenkov rays are not doing anything impossible. They’re just whizzing along at almost the maximum speed of the universe. In the Water meduim, it just so happens that the speed of these particles is greater than the speed at which light sidles along.
So, to sum up:
I think the confusion arises from the common practice of calling the Ultimate Speed Limit of the Universe the ‘Speed of Light’, when we should really call it ‘The Speed at Which Light Happens to Travel If and Only If it Happens To be Traveling in a Complete and Utter Vacuum’.
Cerenkov-ist particles don’t do anything magical. They just play tortoise to light’s hare.
PS- Electrons are apparently interacting with the material in the medium less than photons. Remember that the medium is mostly empty space, so the electrons can just whiz right through, undisturbed. So while the electrons never travel faster than an individual photon, they do pass through the medium faster than the “averaged” speed of the interacting photons.
I see that bryanmcc has added some quite correct, I believe, comments about the propagation of light, which might seem inconsistent with my post. They aren’t.
He says light flies around at C, then bumps into an atom, hnags out, and then whizzes away again at C. This is true. That’s what I meant by the window-shopping analogy. Overall, it’s like a car that does 100 km/h along the highway, and stops at a few red lights along the way. I’ts overall, average speed will end up being slightly less than 100 km/h, Even if it always drives at 100 km/h. (Ignore the fact that cars have to slow down before they stop. That doesn’t really apply to light.)
bryanmcc is giving the very-small-scale mechanism by which my explanation happens.
Ayway, I hope andy_fl can make sense of all this.
Important point which wolfstu pointed out that I think I’ll just emphasize: photons always travel at c. There is no slowing down or speeding up. The instant an electron emits a photon, that photon is whizzing away at c – it doesn’t have to start from zero and speed up. And when a photon crashes into an electron, it instantly winks out of existance – without slowing down – and the electron gains its energy. All within the limits of the uncertainty principle, of course.
bryanmcc Hmmm - I hadn’t considered that ramification of c being a constant. Does that mean that when a photon is emitted from an electron, the photon experiences infinite acceleration ?!
Doesn’t acceleration imply you’re starting from a slow speed? After all, acceleration is change of velocity over time. If the photon doesn’t exist in a ‘slow’ state, the concept of acceleration probably doesn’t apply.
If the photon is emitted in a state of motion, and it’s velocity doesn’t change, then where’s the acceleration?
Oh, it depends on your definition, I guess. But I would say no. The photon doesn’t go from zero to c, it goes from not existing to existing. And when it exists, it has velocity c, so there is no acceleration involved. What it does during that tiny interval between not existing and existing is unknowable, thanks to the Uncertainty Principle. Perhaps someone like Chronos could explain it better…
There’s a misconception that hasn’t been addressed, and that andy_fl has been relying on.
Time travel has nothing to do with light, and nothing to do with communication, or information. Experiments with “exceeding the speed of light” often deal with information, because playing games with spins or phase speed is the only way to “get around” C. Arguments about causality get started by having hypothetical objects travel faster than C. That does not mean, however, that time travel has happened just because an electron wins a race with a photon. After all, if the electron wins, then maybe that was the “speed of communication” in that medium.
I’m a little bewildered by Exapno Mapcase’s rude and condescending attitude towards someone who recognizes that there must be some misconception in their own thinking, since it leads to contradictory results, and is just trying to pin down and correct the misconception. I therefore hesitate to present my own speculation on the origin of andy_fl’s confusion. But here goes…
Might the problem be that the time-traveling effects of traveling faster than C are derived from the assumption that the speed of a light ray traveling through a vacuum is observed to be the same to all observes in inertial reference frames? Could a light ray traveling through a medium appear to have different speeds to observers in different reference frames. If so, then the conclusion that traveling faster than light through a medium leads to time-travel wouldn’t follow.