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Thanks in advance.

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- Thread starter Magister
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- #1

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Thanks in advance.

- #2

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- #3

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In an expanding cosmological model, the curvature is inversely proportional to the square of the 'radius'. So it's high at the beginning and is decreasing as the cosmos expands.

- #4

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In an expanding cosmological model, the curvature is inversely proportional to the square of the 'radius'. So it's high at the beginning and is decreasing as the cosmos expands.

But is that because as the universe expand, the curvature in at a point is getting smaller? If we look at the cosmos like a sphere, as the radius expands the curvature in the surface is getting smaller. Is this the interpretation one should give to R?

The thing that confuse me more is the fact that R=0 doesn't imply a flat manifold.

- #5

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Yes, it's not very intuitive. To get a flat spacetime all the Christoffel symbols ( connections) have to disappear.The thing that confuse me more is the fact that R=0 doesn't imply a flat manifold.

- #6

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Yes, it's not very intuitive. To get a flat spacetime all the Christoffel symbols ( connections) have to disappear.

I agree that the issue of interpretation is not very intuitive....

but I hope there is a good interpretation to be found.

Note that: in the plane, the Christoffel symbols for the polar coordinates are all not zero.

A flat space requires the Riemann Tensor to be zero (which can be formed from the Christoffel symbols and their derivatives). The scalar curvature is of course formed from the traces of Riemann.

Possibly interesting reading: http://arxiv.org/abs/gr-qc/0103044

Note also that the curvature scalar is featured in the action for GR.

- #7

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so, both the plane in polar coords(PP), and the Schwarzschild metric have zero R and non-zero Christoffel symbols. How do we tell if either is curved ?

I notice that all the Ricci components are zero for both, but there are non-zero Riemann components for Schwarzschild, but not for PP.

M

- #8

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so, both the plane in polar coords(PP), and the Schwarzschild metric have zero R and non-zero Christoffel symbols. How do we tell if either is curved ?

I notice that all the Ricci components are zero for both, but there are non-zero Riemann components for Schwarzschild, but not for PP.

M

As suggested in my earlier post [and as you have done for your examples], compute the Riemann curvature tensor.

- #9

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OK, so space-time is curved if there are tidal effects. Thanks for clearing that up.

- #10

Stingray

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- #11

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A vanishing Ricci tensor doesn't even imply a flat spacetime. In fact the Ricci tensor vanishes in the vacuum of a Schwarzschild spacetime.Yes, it's not very intuitive. To get a flat spacetime all the Christoffel symbols ( connections) have to disappear.

Pete

- #12

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Exactly. In fact the termsOK, so space-time is curved if there are tidal effects. Thanks for clearing that up.

Pete

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