# What is the dimension of the spacetime interval?

1. May 7, 2006

### neutrino

Just what the title says. In the book Spacetime Physics, by Taylor and Wheeler, the time coordinate is measured in metres of light-travel time, but that's just a roundabout way of saying that they are using the second...or am I missing the point.

2. May 7, 2006

### neutrino

Ok...it took me sometime to realise that $ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$ that has the dimensions of length . But now another question came up...why length? Isn't spacetime a a union of space and time. Even if we divide the whole expression by $c^2$, we get a dimension of time only.

Last edited: May 7, 2006
3. May 7, 2006

### George Jones

Staff Emeritus
Yes. In the first case, everything can be considered to be measured in units of length, e.g., metres, where 1 metre of time is the time taken for light to travel a distance of 1 metre, and, in the second case, everything can be considered to be measured in units of time, e.g., seconds, where 1 second of distance is the distance traveled by light in 1 second of time.

Most relativity books use the former, but I have seen the latter used. In cosmology the latter is often used, i.e., (light)years and years.

Regards,
George

4. May 7, 2006

### neutrino

Yes, I realise that. I'm just wondering why this quantity (ds²), which says something about the unity of space and time does not have a dimension made up of a combination of length and time.

5. May 7, 2006

### robphy

The quantity involved in ds² that "unifies" space and time (namely, the speed of light) has the dimensions of "length/time".

Similarly, the quantity that "unifies" momentum and energy has the dimensions of "momentum/energy".

6. May 8, 2006

### neutrino

Thanks...now it's all clear.

7. May 8, 2006

### Ich

Isn´t "Spacetime Physics" exactly the book where they start with the example of length in northern direction has different units than lenght eastwards, despite both describe the same thing?

8. May 8, 2006

### neutrino

Yes, that's the one. I especially like their spacetime-first approach.

9. May 8, 2006

### George Jones

Staff Emeritus
This is the book from which I first learned special relativity. Great book.

I recommend also "A Traveler's Guide to Spacetime: An Introduction to Special Relativity, which is the book from which I lifted the accelerometer that I used in the "A falling object" thread.

Regards,
George

10. May 8, 2006

### robphy

11. May 8, 2006

### Ich

Yes, the book is a good one (maybe pete would disagree see #20 and #22).
And their point is: c is simply an arbitrary conversion factor from time units to length units. "meter" and "second" are two units where you need only one. Comparable with inches and meters. Two units for the same thing.
The difference between time and space is then not the units, but the metric (-1 1 1 1 instead of 1 1 1 1).

Last edited: May 8, 2006