PWrong wrote:Space and time have different units. There is a concept of "distance" in spacetime, but it doesn't follow Pythagoras' theorem.
s^2 = x^2 + y^2 + z^2 - t^2
Notice that its t squared and not (ct) squared. By unit analysis, t must have units of length in your equation, which is correct in "God given units" which jinydu already stated.
These quantities have the same unit. The speed of light is set to one, the reason being that it appears so much throughout relativity that it seemed natural to absorb it into the units.
The minus sign is the crucial difference that makes relativity relativity and not just some adding of one more dimension. If you just added a 4th dimension in the spirit of Pythagoras, you would not have time dilation, length contraction, black holes or light cones. Relativity IS something different that is not just the addition of another dimension......all because of that minus sign.
An attempt was made with entropy to define time, but it was pointed out that while the Universe must increase in entropy, individual systems are allowed to decrease in entropy. For instance, the evolution of a human was a non-equilibrium process. A magnet is the alignment of electrons that has almost the lowest number of states, ie, the lowest entropy. When you stretch a rubber band, its entropy decreases, but yours increased.
But alas, I think this is the best answer. Based on PWrong's equation there, if the time component is greater than the space component, then s^2 is negative! These are called timelike events, which is what we exist in. If s^2 is positive, then the space component is greater and these spacelike events have motion faster than the speed of light.
Here's something for you people who think they can imagine the 4th dimension, try and imagine this. If s^2 is zero, then there is zero displacement. Massless particles that travel at the speed of light have s^2 = 0. So the light you see, that travels so incredibly fast in 3D, has no displacement in this "4th dimension." It just sits at zero displacement in 4D spacetime, independent of what its displacement or time of travel is in 3D. A motionless point, that travels only in time, is the light you see in relativity.
In conclusion, since we live in a Universe that must have s^2 negative upto and including zero and since zero is the ultimate limit, then we must have as much space as there is time to get that zero result.