Would time freeze if you could travel at the speed of light? I read with interest about Einstein's Theory of Relativity and his proposition about the speed of light being the universal speed limit. 


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*So, if I were to travel in a spacecraft at (practically) the speed of light, would I freeze and stop moving? 

*Would the universe around me freeze and stop moving? 

*Who would the time stop for?
 A: Velocity is relative, so it doesn't matter if you're "travelling" at some speed relative to something, or something is travelling at some velocity relative to you - the effects are the same. Right now you have objects in the universe travelling at a wide range of velocities relative to you. If you decided to change your speed to close to the speed of light compared to what it is now, you will find that there is still the same range of velocities of objects relative to you. That's because objects that were travelling close to c in the direction of your increase will have slowed down, and objects that were travelling in the opposite direction will have increased their velocity.
However, you will also find that as objects increase their speed relative to you, the sequence of events there slow down, and that includes the running of their clocks from your view point, which approaches zero as their speed approaches the speed of light.
A: The time wouldn't freeze. Instead, all events in the world will happen at the same time and place (from the viewpoint of the observer travelling at the light speed).
It would be better to say that the world (i.e. space & time) would collapse into single point.
A: I could answer your 3rd question, the others have been answered already.There is two basic questions to answer here:


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*would your time freeze to an external observer

*would time freeze for you, or what would you experience


The answer to #1 is yes, it would. Them mathematical description is in other answers correctly, so I am not going to repeat it. But to understand it, imagine you have small photon clocks inside your body. (to the best knowledge today, we imagine most of our mass/energy due to massless gluons traveling/oscillating at speed c in some kind of confinement, and quarks, but nobody knows what the structure of a quark would be, we think of it as pointlike). So we will take the small photon clocks as analogy to the gluons. The photon clock has mirrors, and the mirrors reflect the photon, that is a tick. As your body speeds up, the photons would have to catch up to the mirrors, but at speed c, the mirrors would move at speed c too, so the photons would never reach the mirror, no ticks. Your time froze to an external observer. 
To point out, this would also mean that the internal structure of your body would freeze to an external observer, since no information could be sent any more about it, since its pieces would be moving at speed c, and the interactions inside your body, and the photon clocks, would seem frozen.
Now also to point out, according to SR length contraction, your body's size would also be pointlike to an observer.
Now to answer #2 is not so simple. You would still see your own photon clocks tick normal. And your body would therefore act normally. But you would see the whole 4th dim. at a glance. You would see the whole timeline all along its path, the beginning of the universe and the end and all of it inbetween as snapshots. And of course as a 3D viewer can see every bit of a 2D plane at once (without obstacle), you, now a 4D viewer would see every bit of the 3D world without obstacle, folded out, every 3D structure would be folded out so that without obstacle you could see every bit of it.
A: When traveling with the speed of light time does not "exist". Well sort of. If a body is traveling with the speed of light its origin and destination are one and the same. The question has no point because traveling with the speed of light means you have always been and always will be traveling with the speed of light. Of course from the viewpoint of an observer the speed of light is a defined number, but for a photon time is not a "thing". So overall it wouldn't freeze so much as stop existing.
A: This kind of question has a long and honorable history. As a young student, Einstein tried to imagine what an electromagnetic wave would look like from the point of view of a motorcyclist riding alongside it. But we now know, thanks to Einstein himself, that it really doesn't make sense to talk about such observers.
The most straightforward argument is based on the positivist idea that concepts only mean something if you can define how to measure them operationally. If we accept this philosophical stance (which is by no means compatible with every concept we ever discuss in physics), then we need to be able to physically realize this frame in terms of an observer and measuring devices. But we can't. It would take an infinite amount of energy to accelerate Einstein and his motorcycle to the speed of light.
Since arguments from positivism can often kill off perfectly interesting and reasonable concepts, we might ask whether there are other reasons not to allow such frames. There are. One of the most basic geometrical ideas is intersection. In relativity, we expect that even if different observers disagree about many things, they agree about intersections of world-lines. Either the particles collided or they didn't. The arrow either hit the bull's-eye or it didn't. So although general relativity is far more permissive than Newtonian mechanics about changes of coordinates, there is a restriction that they should be smooth, one-to-one functions. If there was something like a Lorentz transformation for v=c, it wouldn't be one-to-one, so it wouldn't be mathematically compatible with the structure of relativity. (An easy way to see that it can't be one-to-one is that the length contraction would reduce a finite distance to a point.)
What if a system of interacting, massless particles was conscious, and could make observations? The argument given in the preceding paragraph proves that this isn't possible, but let's be more explicit. There are two possibilities. The velocity V of the system's center of mass either moves at c, or it doesn't. If V=c, then all the particles are moving along parallel lines, and therefore they aren't interacting, can't perform computations, and can't be conscious. (This is also consistent with the fact that the proper time s of a particle moving at c is constant, ds=0.) If V is less than c, then the observer's frame of reference isn't moving at c. Either way, we don't get an observer moving at c.
A: I disagree with those who dismiss this question. As Ben Crowell reminds us- Einstein himself considered it.
One aspect of relativity that is often overlooked is its reciprocity. If you are moving relative to me then any relativistic effects that apply to you from my perspective (eg the slowing of your clock), apply equally to me from your perspective.
The first thing to remember is that your personal experience of time, known as your proper time, will remain unchanged regardless of the speed at which you are moving relative to other observers. 
If you were moving at close to the speed c relative to other observers, then you would seem from their perspective to be experiencing time at a very reduced rate. The effect would be entirely symmetrical, as from your perspective the other observers will seem to be the ones for whom time has slowed almost to a standstill. 
So the answers to your questions are:
1) From your perspective you would not experience any change in the passage of time on board the spacecraft. When timed by observers moving asymptotically close to c relative to you, events on the spacecraft will seem to be frozen in time.
2) When measured against your frame of reference, the clocks of those observers will seem to you to be frozen. (Here I take 'the Universe' to mean any observers for whom your relative speed is close to c).
3) Time doesn't stop for anyone.
A: You can't travel at the speed of light. So it's a meaningless question.
The reason some people will say that time freezes at the speed of light is that it's possible to take two points on any path going through spacetime at less than the speed of light and calculate the amount of time that a particle would experience as it travels between those points along that path. The calculation is
$$\Delta\tau^2 = \Delta t^2 - \frac{1}{c^2}(\Delta x^2 + \Delta y^2 + \Delta z^2)$$
where $\Delta\tau$ is the amount of time experienced by the traveling particle, and the other $\Delta$'s are the differences in space and time coordinates between the two points as measured by an external observer. If you take this same calculation and blindly apply it to a path which is at the speed of light, you get $\Delta\tau = 0$.
A: Yes, I agree with David. If somehow, you were able to travel at the speed of light, it would seem that 'your time' would not have progressed in comparison to your reference time once you returned to 'normal' speeds. This can be modeled by the Lorentz time dilation equation:
$$T=\frac{T_0}{\sqrt{1 - (v^2 / c^2)}}$$
When traveling at the speed of light ($v=c$), left under the radical you would have 0. This answer would be undefined or infinity if you will (let's go with infinity). The reference time ($T_0$) divided by zero would be infinity; therefore, you could infer that time is 'frozen' to an object traveling at the speed of light.
A: As pointed out you can't travel at the speed of light but you can look at the limits we are tending towards as we approach it.

So, if I were to travel in a spacecraft at the speed of light, would I freeze and stop moving?

From the perspective of a stationary observer if your spacecraft was traveling at close to the speed of light, time on the spacecraft would have slowed down (would be approaching zero or frozen).  What does this mean?  Everything in the spacecraft would be moving really slowly, e.g. the person moving, electrical signals, everything is slowed down by the same amount (as seen by the stationary observer).
From the perspective of a person on the spacecraft time appears to travel at it's normal rate (because if time slows down you don't notice it as everything slows down at the same rate, including your thinking process).  So nothing inside the spaceship seems strange.  However if you observe the stars moving past you will observe some strange effects due to aberration and the doppler shift.
See this link for what a spacecraft would see travelling at relativistic speeds.

Would the universe around me freeze and stop moving?

No, the universe keeps on working as it usually does.  Essentially in the spacecraft time is moving slower than outside in the rest of the universe.  So inside you're aging slower than someone outside.  However you don't notice this (time appears to be running normally from your perspective) and you'll just see the stars outside become blue shifted (due to doppler effect of high speed and shifted towards a point around your direction of travel (due to aberration)).  See the link above for more detail on this.

Who would the time stop for?

Nobody notices time slowing from their perspective.  Instead its only the stationary observer who notes that time is slowing down for the person in the spacecraft.
A: I do not agree with David and I found his answer kind of odd. 
For simplicity let us assume we work in $(1,-1,-1,-1)$ Minkowski space. The photon is travelling in direction along $x$-axis. At time $0$ the photon is at point $(0,0,0,0)$, and at time $1$ at point $(1,c,0,0)$. 
Thus the space-time distance of the two events are
$$
c^2-(c-0)^2=0
$$
As a result the photon is not "moving" in Minkowski space. Since our only way of identifying two distinct events is via the metric, this means the two events are indistinguishable. But this type of phenomenon is rather common in mathematics; the world lines of the photon thus become an element of a closed linear subspace of the Minkowski space. The space-time distance in the quotient space can still be defined, and in particular we may assign zero as a value for the time at origin if we would like to preserve additive law.  
