"Time exists to prevent everything happening all at once; space exists to prevent everything happening at all the same place." (G.B)
Indeed events require a space and a time, for we cannot measure the position of anything except at a particular time and we cannot measure the time anything happens except at a particular place. This insight led Einstein to conceive a model of a four dimensional space-time continuum.
But how are these different dimensions, length, width, height, and time, to be connected? How can you measure the space-time separation or interval between two events?
Think first of two dimensions, the surface of a sheet of paper, for example, of length x and width y.
The shortest distance s between two opposite corners is given by Pythagoras' theorem:
s^2 = x^2 + y^2
If we now go to three dimensions, the shortest distance between two opposite corners of your room, for example, one on the floor and the other in the opposite corner on the ceiling, where the room is x long, y wide and z high, is given by:
s^2 = x^2 + y^2 + z^2.
So what happens if we go to four dimensions, and measure the space-time interval between two camera flashes, for example, one happening at one corner of your room and other happening at the opposite corner but a few seconds t later?
We might think the answer would be:
s^2 = x^2 + y^2 + z^2 + t^2, but we would be wrong. There are two things wrong with it.
First of all Einstein had been working on a problem, how to make Maxwell's equations independent of the observer's frame of reference, which would mean the velocity of light is equal for all observers as discovered by Michelson and Morley. He realized that these problems would be resolved if he adopted an idea of his colleague and lecturer, Minkowski, that the t^2 term should be subtracted, not added.
Secondly we are adding "dollars and euros", the dimensions of the terms in the equation are not right, we need a conversion factor, "a rate of exchange", to convert one into the other before we can add or subtract them. The conversion rate that turns time into distance is a velocity we call it c, so if c is measured in kilometres/second and t is so many seconds, then ct will be so many kilometres.
So the correct equation giving the space-time separation s between two events becomes
s^2 = x^2 + y^2 + z^2 - c^2t^2, and now there is one more refinement to make.
Rather than measuring the distance across an extended interval of space-time, it is important to deal only with the separation of adjacent events separated by a infinitesimal change in the coordinates, dx, dy, dz, dt. This allows the possibility that space-time might be 'curved', just as the surface of the Earth is curved into a sphere, although it looks flat on a small scale, or alternatively like the surface of a saddle. Or perhaps like that of a Popperdom, all 'hills and hollows'. So the correct expression of the infinitesimal separation of two adjacent events is:
ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2, this is called the metric of flat space-time. If we want to include 'curved' space-time then we put a coefficient, not equal to one, in front of each term dx^2 etc. In order to find out the separation between two events such as a distant super nova exploding and the event being observed on Earth you have to add up, or integrate all the infinitesimal ds intervals along the light path.
So you can see that the answer to
