# Spacetime interval and the metrric

## Main Question or Discussion Point

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!

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Matterwave
Gold Member
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 $ds^2$.

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

bapowell
Thanks! A little confused why you referred to $ds^2$ as a scalar. Isn't it a four-vector?
Nope. It's the scalar product of (differential) four-vectors.

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