# Spacetime interval and the metrric

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1. Jan 13, 2015

### jmatt

This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other?

Thanks!

2. Jan 13, 2015

### Matterwave

They aren't exactly equivalent, but they are closely related. The space time interval is $ds^2=g_{ab}dx^a dx^b$. Because the metric $g_{ab}$ is a tensor, contracting it with (infinitesimal) vectors $dx^{a}$ will yield an invariant scalar $ds^2$.

3. Jan 13, 2015

### jmatt

Thanks! A little confused why you referred to $ds^2$ as a scalar. Isn't it a four-vector?

4. Jan 13, 2015

### bapowell

Nope. It's the scalar product of (differential) four-vectors.

5. Jan 13, 2015

### Matterwave

Why would it be a 4 vector? What are the 4 components that you are thinking of? It is a scalar because it's just 1 single number, which does not change under an arbitrary Lorentz transformation.

6. Jan 13, 2015

### jmatt

Got it now. Thanks very much.