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I’m reading M.Pettini: Introduction to Cosmology - Lecture 3

He starts by talking about spacetime:

The appearance of objects at cosmological distances is affected by the curvature of spacetime through which light travels on its way to Earth.

Then he equates spacetime with the metric:

In GR, the fundamental quantity is the metric which describes the geometry of spacetime.

Then he gives the formula (3.3) connecting two events in flat spacetime.

But then at the end of the page, he stops talking about spacetime and mentions space:

In a homogeneous and isotropic universe, although the curvature of space may change with time, it must have the same value everywhere at a given time since the Big Bang.

Is this a typo? Does he mean: “...although the curvature of [spacetime] may change...”

I don’t understand why he switched to space while he was talking about spacetime. Can you clarify? Thanks.

Qmechanic
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zeynel
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3 Answers3

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He is being precise. He does not mean the curvature of spacetime.

In his homogeneous and isotropic universe there is a globally defined $t$ coordinate "time" that foliates the space-time manifold into $t=$ constant leaves. These leaves are homogeneous dimension-$3$ spaces, and it is the curvature of these spaces that may be different at different $t$.

mike stone
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A sphere consists of two poles and a whole lot of circles of latitude. If I talk about the curvature of the sphere, am I forbidden to talk about the curvature of the circles?

(Note that one does not determine the other: The circles are flat but the sphere is not.)

WillO
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He's talking about the large scale shape of space in the universe, which, I suppose, is a constant. Yes it expands with time, but is it overall: flat, closed-like-a-sphere, or hyperbolic.

JEB
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