Spacetime equation derivation

Brussels Sprout

I'm currently reading "Why does E=mc^2" by Cox & Forshaw. In chapter 4 the authors explain Spacetime as an invariant quantity that takes account of both space and time. They then come up with two assumptions in an effort to quantify this relationship, namely:

1. The space part of Spacetime is Euclidean
2. Spacetime is unchanging and the same everywhere

This is followed by the line

Once we make these two simplifying assumptions, we are left with only two possible choices as to how to calculate distances in Spacetime, s^2=(ct)^2+x^2 or s^2=(ct)^2-x^2. There is no other option.

They don't explain how they arrived at these two equations presumably because they didn't want to get bogged down in maths (they aim to go through the entire book without using any maths more complicated than Pythagorean theorem). I'm fairly comfortable with mathematical derivations so I'd like to see how they arrived at these values.


in the above, c: speed which is so far unknown and is used to equate distanceand time via the formuala distance=speed x time
t: time
x: distance
s: spacetime distance

dlouth15

Are you sure the first formula is as it is in the book and that there's no minus sign? I've only ever seen something like: [latex]s^2=-\left(ct\right)^2+x^2[/latex] or [latex]s^{2}=\left(ct\right)^{2}-x^{2}[/latex].

Brussels Sprout

No what I've written is definitely as it's written in the book as they refer to it as being similar to Pythagoras's equation. They then go on to prove that that equation doesn't represent spacetime because it breaks the condition of causality.

dlouth15

The first equation just treats time ([latex]ct[/latex]) as if it were another spacial dimension. In 3-d euclidean space, [latex]\Delta l^{2}=\Delta x^{2}+\Delta y^{2}+\Delta z^{2}[/latex]. If we add a spacial dimension and treat it the same way we get

[latex]$$\Delta s^{2}=\left(c\Delta t\right)^{2}+\Delta x^{2}+\Delta y^{2}+\Delta z^{2}$$[/latex].

With only one spacial dimension and one time dimension it becomes

[latex]$$\Delta s^{2}=\left(c\Delta t\right)^{2}+\Delta x^{2}$$[/latex]

which is what you have in the book. I'm not sure of the usefulness of it in special relativity.

The second is a variation on this but with the [latex]x^2[/latex] bit subtracted. In four dimensions it would be something like:

[latex]$$\Delta s^{2}=\left(c\Delta t\right)^{2}-\Delta x^{2}-\Delta y^{2}-\Delta z^{2}$$[/latex]

or

[latex]$$\Delta s^{2}=\left(c\Delta t\right)^{2}-\Delta l^{2}$$[/latex]

where [latex]\Delta l^{2}=\Delta x^{2}+\Delta y^{2}+\Delta z^{2}[/latex], (i.e. euclidean length in three dimensions). This is called the spacetime separation of events.

In one spacial dimension [latex]x[/latex] and one time dimension, it is

[latex]$$\Delta s^{2}=\left(c\Delta t\right)^{2}-\Delta x^{2}$$[/latex]

which is what is in the book. I've used deltas because it is the difference between the respective components of the two events that are used.

I don't think it is derived as such but rather motivated by its similarity to Euclidean distance. The textbook I have simply states it and then goes on show how it is invariant under the Lorentz transformation and how it is used to determine whether two events have space-like ([latex]\Delta s^{2}>0[/latex]), time-like ([latex]\Delta s^{2}<0[/latex]) or light-like ([latex]\Delta s^{2}=0[/latex]) separation. Proper time and proper length are also closely related to this measure.