Length Of A Curve in Spacetime

I'm studying relativity at present and have been trying to follow some of the messages here. I am confused about the way time and clocks are supposed to vary with speed.
I have read that just as the odometers of two cars will read differently if they take different routes between A and B, so will two clocks that take different paths in spacetime. I cannot understand this.

As everyone should know, the length of a curve is given by the equation s = integral(root[1+dy/dx)^2)dx]
Note that dy/dx is a dimensionless ratio.

For a simple spatial plot like this:

Y
| *
| * * *
|* * *
| *
| *
|
__________________________X

where y is a known function of x, the answer is straightforward. Its units are those of length.

However if a similar curve in a 2D plane of spacetime is represented by an X-T graph:

t
| *
| * * *
|* * *
| *
| *
|
__________________________X

we now have the length as: s = integral(root[1+dt/dx)^2)dx]

There is a problem here because the differential term is NOT dimensionless. It has dimensions T/L and so the equation is dimensionally wrong.
The only way around this is to regard TIME as a fourth spatial dimension but to do that would render just about all of modern science and technology useless, which it obviously isn't.
Can somebody please tell me what are the dimensions of a 'length' in spacetime and what its units might be?




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Relevant Pages

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