A demonstration of faster-than-light travel you can try
Using just paper and pencil, you too can warp space and time!
For years now, we have been told that traveling faster than light was impossible. Those who questioned that dogma were quickly silenced. The laws of physics, handed down by Einstein, were clear: thou shalt not travel faster than c, the speed of light in a vacuum.
This law is as strict as the conservation of energy. And, as with that law, general relativity provides an exception.
Special relativity, the theory that enforces the speed of light restriction, is a law about inertial motion, motion without forces or acceleration. It tells us how two observers will appear to one another when they try to measure each other’s length, velocity, and clocks.
Special relativity derives from the idea that all observers measure the speed of light in vacuum to be the same value regardless of their state of motion. Thus, if you are traveling away from me at a significant fraction of the speed of light and emit a beam of light towards me, I will measure the velocity of that light to be c, not, as intuition would have it, c minus your velocity.
What will happen to the light, as you accelerate towards the speed of light, is that it will be redshifted to a lower wavelength. The closer you are to the speed of light, the longer those wavelengths will be. If you could reach the speed of light, the redshift would be infinite and the wavelengths infinitely long.
Reaching the speed of light, or pushing it beyond it, is impossible because of how acceleration works in special relativity. Every time you accelerate to a new velocity with respect to an observer, your clocks, as they measure it, appear slower and your mass appears larger, among other things. If you could reach the speed of light, your clocks would stop and your mass would be infinite. In reality, the amount of energy you would have to apply to accelerate to that speed is also infinite, so there is no chance you will ever reach it.
This scenario is regularly played out in particle accelerators. More and more energy is applied to tiny particles, like protons, and they approach closer and closer to the speed of light, but never reach it. Yet, we observe that particles decay more slowly at those speeds and the particles become more massive as well, both clear indications of Einstein’s predictions.
General relativity lets us accelerate faster than the speed of light because it allows us to redefine what the speed of light is.
At any given point in spacetime, the speed of light is always measured to be c. But that value may not be the same between two different points, i.e., a beam of light may move slower or faster through a curved space than through a flat space.
You can imagine spacetime as being like a big flat chess board with hills and dips in it. Stars, planets, galaxies, and black holes create dips in the flatness. These dips expand space and contract time. In a flat spacetime, moving at the speed of light is like moving at a 45 degree angle (like a bishop in chess). Standing still is like moving straight ahead (like a pawn that isn’t attacking). Moving side to side, which would be akin to disappearing at one point and reappearing at another at the same moment in time, is not allowed (at least not without quantum theory which we aren’t going to talk about).
When a massive body like a star makes a dip in the chess board, the speed of light for in falling light is faster. When you have something like a black hole, this increase can be substantial compared to flat spacetime.
This is why an observer falling into a black hole reaches a velocity which in flat space would be the speed of light by the time it hits the event horizon, but by then the speed of light is twice that.
Another effect on light comes from the expansion of the universe itself. In this case, you can think of the entire flat board being kind of loosely bent like a saddle. As your light beam travels northeast, it also going progressively down a slope. This causes its actual progress to look more like a curve to east-northeast, but at no point is it not moving at a 45 degree angle. Rather, it is the geometry that is curving under it.
It is easy to do an experiment with this. Get a piece of graph paper or, if you don’t have any, draw or print out a grid and draw a 45 degree line on it. Label one axis time and the other space. (We only have room for one space dimension.) Here’s what it will look like:
You can see that for every meter the beam travels in space it travels the same distance in time (measured in units of time where the speed of light in vacuum is one). This is how light behaves in special relativity. Let 90 degrees be the time direction and 0 degrees be space. Nothing travels at an angle lower than 45 degrees here. Everything must travel further in time than in space.
Now we are going to warp space time!
Squeeze the time dimension to simulate space contracting then expanding from bottom to top:
Look at the top right. See how there the light beam now appears to be less than 45 degrees? We have changed the speed of light globally but not locally! Locally, the light beam never changed from 45 degrees, so special relativity still holds within each grid square, but not across many grid squares.
Now squeeze the space dimension to simulate time contracting and expanding from left to right:
Look at the top right again. See how the light beam now appears to be at more than 45 degrees? We have slowed the speed of light down (by changing the rate of time this time), but again, every local observer will measure it as being exactly c.
Of course, spacetime can be warped much more than this. Grid squares can also become larger or smaller too, which is what’s really going on with the expansion of the universe. Space and time can be warped independently as well. Also, warping can be very localized as with black holes. Or it can be very general as with the universe as a whole.
All these effects lead up to the conclusion that, while special relativity can describe what happens at one point in spacetime, where locally beams of light move at 45 degrees, it can’t give an overall picture of a lightbeam that traverses through a curved space time. General relativity, on the other hand, allows light beams to speed up and slow down relative to a beam in a flat spacetime depending on how they move through curved spacetime.
An interesting development in the field of general relativity is the discovery of warp drives, which could allow faster-than-light travel to other stars. These work by the same principles, by changing what light speed means. A warp drive works like a traveling wave in space time. So, if you were to put something like the famous Alcubierre drive in our chess board, it would look like a steep trench going from starting point to destination and then a ridge followed by a plateau where the ship goes, then a matching ridge and trench system. (You can see pictures in this paper.) The trench and ridge can be at almost any angle from due north to northeast, east-northeast, to almost due east. (But probably not to the south as you would be going back in time then.)
For an entity inside the warp bubble, in the plateau, they would be stationary and essentially carried along in the bubble. Their time direction would not be north but in the direction the bubble was traveling. And likewise, their spatial dimension would be perpendicular to their time dimension.
A light beam produced from the bubble towards its rear would traverse the plateau and encounter the ridge of contracting space following by expanding space, then come out into ordinary space. The overall effect would be for the light to be redshifted because of the space expanding behind the ship.
All of this works like a mini-version of the expanding universe case. While at any given point, the light beam never deviates from the speed of light, space expands or contracts to change its velocity relative to a beam of light that is traversing ordinary flat space. (You would measure this as redshift or blueshift in the light.)
As with the folded paper experiment, light never, locally, changes from 45 degrees, but, if that warped spacetime were replaced with a flat one, the distances traveled would be different.
And this is the big point, in special relativity it is easy to define what we mean by speed and faster than light. In a curved space time it turns out to be really hard. After all, every observer measures the speed of light to be the same in general relativity but how long it takes a beam to get from point A to B in the universe is heavily dependent on the curvature of spacetime in between. Thus when we say “faster than light” we are really saying “in less time than a beam of light would take if spacetime in between were flat.” It is a completely different definition.
But on the other hand, when it comes to traveling to the stars that is the definition we really care about.