I just watched this video:
http://www.youtube.com/watch?v=GkCWywO93b8#t=27
and there Mr. Cox states that because of the minus sign in the Minkowski metric nothing can move backwards in time. It's the first time I hear this argument and does anyone know how to show this?
To be precise: I'm looking for an explicit computation that shows why the minus sign in the Minkowski metric means that we can't rotate into a frame where $t \rightarrow -t$
No, because it isn't true. Nothing can move backwards in time because nothing can move forwards in time. Moving in or through time is just a figure of speech. So is “the flow of time”. If you open up a clock you don’t see time flowing through it like it’s some cosmic gas meter. You see little cogs and things, moving. Clocks always feature some kind of regular cyclical local motion, and they clock this up to show you some kind of cumulative display that you call “the time”.
You must have seen some movie where somebody has a device that can stop time. Did you notice that the gizmo doesn’t so much stop time as stop motion? It’s the same kind thing with bullet time. It’s not time going slower. It’s bullets going slower. And maybe you've read Stephen Hawking saying you can “travel” to the future on a superfast train? Imagine you’re on it. The train moves fast. Through space. And because it does, the local rate of motion in your body brain and clocks has to reduce. We call it time dilation, and it occurs because the maximum rate of motion is the speed of light, because of the wave nature of matter. But it isn't time travel. I could watch you every step of the way. All you're doing is travelling through space, and you don't end up in the middle of next week.
Gravitational time dilation is similar. Clocks go slower when they're lower, and so does your local motion. This can be idealised via the stasis box, which is like the ultimate refrigerator. Like time travel it's science fiction, but the important concept is that no motion of any kind occurs inside it. So when I put you inside, electromagnetic phenomena don’t propagate, and you can’t see, hear, or think. Hence when I open the door a week later, to you it’s like I opened the door just as soon as I closed it. And get this: you “travelled” to the future by not moving at all. Instead everything else moved. And all this motion wasn’t through time, or spacetime, it was through space. Don’t forget, we can freeze embryos now. In the future maybe we’ll be able to freeze an adult. Then you could “travel” to the future by stepping into a glorified freezer. But you aren’t really travelling to the future. You aren’t moving. Instead everything else is.
So, the real reason you can't travel to the past isn't because of some minus sign in an equation. It's because there is no such thing as negative motion. And because there is no way you can move such that the motion of everything else is miraculously undone. You can read more about this in A World Without Time: The Forgotten Legacy of Godel and Einstein. It doesn’t say time does not exist, it’s more like time exists like heat exists. And just as you can’t literally climb to a higher temperature, you can’t literally travel forward in time. So you can't travel backwards either.
To understand Brian Cox's actual comment: The minus sign in the metric actually creates a discrepancy which does not exist in Newtonian mechanics: there is a space between light cones. It's a little easier to talk about this in the (+ − − −) convention where particle trajectories have real (not imaginary) lengths. So let the length of a 4-vector $[w, x, y, z]$ be $w^2 - x^2 - y^2 - z^2$, with $w = c t$. Now the condition that nothing goes faster than light means that any 4-vector which describes a particle's displacement has a positive norm.
But, due to these minus signs, there are also parts of the space with negative norms. There are two easy ways to visualize this. The first one is "light bubbles": imagine a supernova explosion: sudden, violent, casting a lot of light around. Since it's sudden, there is a "bubble" of light that "expands outward" at speed $c$, informing the rest of the universe that this supernova has happened. Another way to think about this bubble is, since a massive traveler can move any speed up to the speed of light, the "interior" of the bubble is not just events in spacetime which "know about" the supernova, but also events in spacetime which can be visited by a massive traveler-in-a-spaceship who was at the supernova when it occurred. Outside the bubble, no spaceships: inside the bubble, possible spaceships.
Now imagine two such "bubbles": there are three basic topological organizations to consider. One is that one bubble is inside the other. Since they both expand at the same speed, it turns out that they'll never after intersect. One is objectively before the other: because the light from the one was objectively seen by the other when it started. Similarly, there is no objective distance between them: because there is an inertial spaceship which visits both without accelerating, and in that spaceship's reference frame both of them happen "right here". The other major topology is for the two bubbles to intersect, either at a point (null-separation, 0 metric) or on an expanding circle (space-separation, negative metric). It turns out that the space-separated events are not objectively time-separated: consider a spaceship on the "circle" of intersection: they are seeing both supernovas at this moment. It turns out that there is always a Lorentz boost such that they trace the same distance to the origin of both spheres, so that in this reference frame both spheres have the same size and were emitted at the same instant. But they're objectively space-separated because we know that there is no spaceship which could have visited both events.
The other way to visualize this is to kill one of the dimensions and make the bubbles into expanding circles: since they linearly expand from one point, they describe a cone shape called a "light cone." If you use space to plot time, then either one cone is inside the other one or the two cones intersect on some hyperbola-looking line.
Now take all of the things which the supernova has seen and add it to the mix: this is a cone going in precisely the other direction from the supernova's emitted-light cone.
The space inside the emitted-light cone is the supernova's objective future. The space inside the absorbed-light cone is the supernova's objective past. The space between these two cones is all the events which are spacelike-separated from the supernova. The "minus sign" creates this space-between-the-cones.
Now how does that block "going backwards in time"? In many ways, it doesn't -- wormholes, for example. But it blocks continuous trajectory deformations from going backwards in time, because they have to discontinuously jump across this space between light cones. If they do not, then if you choose a trajectory "in the middle" you'll get superluminal travel. In fact, if you can superluminally travel in two different reference frames you can generally use this to time-travel in relativity: you can do something, then use one reference frame to "jump out" of your future light cone and then the other one will let you "jump in" to your past light cone, before you did that thing.
So the mathematics is not as simple as a calculation; it is the observation that any continuous deformation of a future-pointing timelike vector into a past-pointing timelike vector must either go through 0 or a spacelike vector. If you assume that particle trajectories aren't spacelike then you "discover" that there's no way to go from future-pointing to past-pointing.
