# What is the dimension of the spacetime interval?

by neutrino
Tags: dimension, interval, spacetime
 P: 2,046 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.
 P: 2,046 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.
Mentor
P: 6,245
 Quote by neutrino 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.
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

P: 2,046
What is the dimension of the spacetime interval?

 Quote by George Jones 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.
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.
HW Helper
PF Gold
P: 4,130
 Quote by 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.
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".
P: 2,046
 Quote by 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".
Thanks...now it's all clear.
 Sci Advisor P: 1,909 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?
P: 2,046
 Quote by 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?
Yes, that's the one. I especially like their spacetime-first approach.
Mentor
P: 6,245
 Quote by Ich Isn´t "Spacetime Physics" ...
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