# Length and time for a given metric

1. Jul 2, 2006

### wandering.the.cosmos

I'm looking for recommendations for a good place that discusses at a basic level what physical length, time, simultaneity, etc. mean, for an arbitrary metric.

Landau does discuss this a bit, but in a way that confuses me -- for example he calls

$$\sqrt{g_{00}} dx^0$$

"proper time", but in the SR limit this would just become dt, which isn't proper time, is it? Landau goes on to use this to derive distance in GR

$$\gamma_{ij} dx^i dx^j = \left( -g_{ij} + \frac{ g_{0i} g_{oj} }{g_{00}} \right) dx^i dx^j$$

where my indices are spatial -- they run only from 1 to 3.

I'd like to understand where these expressions come from, and more importantly gain a good understanding of what length and time mean.

2. Jul 2, 2006

### George Jones

Staff Emeritus
Last edited by a moderator: Apr 22, 2017