Posted: Mar 01, 2011 3:34 pm
by hackenslash
OK, I just found this video, which doesn't really go into explaining the minus sign, but it does say what it implies:


I'll borrow directly from Cox on this, because the example he gave is quite elegant. Here's how it works:

If we take two events in spacetime, say, getting out of bed and eating breakfast, we can plot them on a spacetime graph. For simplicity, we'll say that the kitchen is 10 metres from your bed and you ate breakfast 1 hour after getting out of bed.

Using the simpler Pythagorean formulation of Twistor's above calculation, we can give the equation for the distance in spacetime as s2=(ct)2+x2, where c is our exchange rate, t is the distance through time and x is the distance through space.

We'll call 'getting out of bed' 'O' and 'eating breakfast' 'A', and we plot them on our spacetime graph thus:


Here. you'll see that s is the distance in spacetime between the two events. On the graph, we have plotted a circle, the radius of which represents all the points in spacetime that are the same distance s from event O. There are also plotted two events, A' and A", which are also the same distance in spacetime from event O. For A', nothing very interesting has happened, but if you look at event A", you can see that something very strange has happened, namely that it occurs in the past of O, which means that yoou ate your breakfast before you got out of bed! Causality is violated.

However, if we use the minus sign version s2=(ct)2-x2, as stated in the video by Cox above, we get a slightly different picture of spacetime, namely a non-Euclidean, hyperbolic spacetime.


The hyperbola still represents all the points that lie the same distance s in spacetime from event O, but you can now see that they all lie within the future light cone of O, thus protecting causality.

So, that's how the minus sign protects the past from the future.

If anybody wants to see this explained in greater detail, I recommend Why Doe E=mc2, by Brian Cox and Jeff Forshaw, from which this example is borrowed, as are the diagrams.

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