saderlius said:
what does the sign indicate? You have pointed out how different time is to a spatial dimension, why then should it be treated as a dimension? Perhaps this is a question of semantics, but if it is called a "4th dimension", that implies it is built upon the former 3 dimensions, just as the Y-dimension is only thus in reference to the X- dimension, etc.
The question is: "How are events connected up in space and time?"
Take an infinitesimal interval ds separating two events:
Now the separation in 2 dimensions is given by Pythagoras' theorem:
dx2 + dy2 = ds2
now expand it to 3 dimensions:
dx2 + dy2 + dz2 = ds2
now expand it to 4 dimensions where the fourth dimension is time; do we get
dx
2 + dy
2 + dz
2 + dt
2= ds
2?
Unfortunately this isn't correct, there are two things wrong with it.
First there is a question of units, we have added the squares of 'apples' and oranges'! We need a conversion factor to convert time into distance, such a factor has the dimension of velocity, so call it c, we have to multiply dt
2 by c
2.
Secondly in SR we
do not add the time
2 but
subtract it. This changes the 4D space we are constructing from Euclidean space to Minkowski space. You may ask why do we do this, the first answer is because that is the way the world works, and this approach has been verified in all the experiments that verify SR.
In Minkowski space the maximum velocity is c, massive objects can only approach c asymptotically and massless objects such as photons can only travel
in vacuo at c, so c is the speed of light
in vacuo.
We now have:
dx2 + dy2 + dz2 - c2 dt2= ds2.
This is called the Minkowskian metric and accurately describes the behaviour of objects with clocks and rulers moving relatively to each other at high speed.
The result of this construction of a 'space-time' continuum out of space and time is that time is seen to be a dimension like the other three but with a difference. It bears the same mathematical relationship to them that the Imaginary numbers do the the Real. The fact that if time is a dimension then it is not exactly the same as the others is intuitively self-evident.
I hope this helps.
Garth