# GR: FRW Metric relation between the scale factor & curvature

1. Dec 18, 2014

### binbagsss

Mod note: OP warned about not using the homework template.
I have read that 'a(t) determines the value of the constant spatial curvature'..

Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption.

I'm trying to see this from what I know, but I'm struggling...

Here's what I know:

The FRW metric takes the form:

$dt^{2}+a^{2}(t) ds^{2}, where ds^{2}=g_{ij}x^{i}x^{j}$, and $g_{ij}$ is the isotropic and homogenous spatial metric.

The Riemann tensor and metric are related via:
$R_{abcd}=\frac{1}{6}R(g_{ac}g_{bd}-g_{ad}g_{bc})$

I have no idea how I can tie these together , or use anything I know to show that 'a(t) determines the value of the constant spatial curvature'

Any hints or links to sources etc really , really appreciated, ta.

Last edited by a moderator: Dec 18, 2014
2. Dec 23, 2014