# Flat spacetime with curved metric? Small change in metric drastically changes geometry? GR

I'm about to be taking an undergrad course in GR. I'm trying to get ahead of the game, and I've hit a problem.

The metric for flat Minkowski space (using the $$-+++$$ signature and $$c=1$$) is:

$$ds^2 = -dt^2+dx^2 +dy^2 +dz^2$$

which in polar coordinates is: $$ds^2 = -dt^2+dr^2 +r^2(d\theta^2 +d\phi^2\sin^2\theta)$$

Firstly, the $$r^2(d\theta^2 +d\phi^2\sin^2\theta)$$ part of the metric is identical to the metric for a 2-sphere, which to me suggests that the Minkowski metric above should signify curved spacetime - why is this part of the metric there if the space the metric describes is flat? In my mind, the metric containing "something curved" should mean that the space it describes is curved.

Also, why does simply doing this:

$$ds^2 = -dt^2+dr^2 +(r^2+b^2)(d\theta^2 +d\phi^2\sin^2\theta)$$

to the metric produce a wormhole geometry ($$b$$ a constant - the size of the wormhole's throat)?

I suppose my questions are summarised by: what does a 2-sphere have to do with flat Minkowski spacetime or the geometry of a wormhole?

This is my first question on stackexchange - sorry if I've not been clear enough!

• A similar situation is a 'flat' FRW spacetime with metric $ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2)$. Here the spatial part is flat but overall spacetime is curved. You can't just look at part of the metric. – jacob1729 Sep 10 at 16:17
• Ordinary Euclidean 3-space contains a sphere, which is curved. Does that mean Euclidean space is curved? – Javier Sep 10 at 16:25
• So just because the metric contains something that looks curved doesn't mean the spacetime is necessarily curved. Thanks! – Ali Chopping Sep 10 at 16:41
• You can’t conclude anything about the curvature of a space by looking at the curvature of one of its lower-dimensional subspaces, as far as I know. – G. Smith Sep 10 at 16:43