Schwarzschild example: Two observers orbit around the same central point mass at different radii. They measure that the radial separation between them is greater than their orbital circumference / 2pi. They conclude that there is negative (parabolic) spatial curvature in the radial direction in the space between them, as predicted by the external Schwarzschild metric.(adsbygoogle = window.adsbygoogle || []).push({});

If the point mass then hypothetically increases (this is a thought experiment), the two observers will measure that the space between them has become more negatively curved in the radial direction than it was before.

FRW example: Two fundamental comoving observers are receding away from each other with the Hubble flow in an expanding, dust-filled Lambda=0 FRW model at critical density. They measure that the space between them is flat, as predicted by the flat FRW metric. (E.g., they measure each other's apparent transverse size to be appropriately proportional to the distance between them.)

If the homogeneous cosmic matter density then hypothetically increases, but the Hubble rate does not increase, the two observers will measure that the space between them has become positively curved.

Why does adding Schwarzschild mass make the spatial curvature more negative, while adding FRW mass per unit volume (i.e. density) makes the spatial curvature more positive?

Does the difference in results arise from the difference in reference frames? The Schwarzschild negative curvature is measured by an infinitely distant hypothetical observer in flat space, while the FRW positive curvature is measured by observers embedded in the positively curved space. If the Schwarzschild observer were instead located at the center of the point mass (with a radial tunnel through which to observe orbiting test particles), would he measure the space between himself and the test particles to have positive spatial curvature (per the interior Schwarzschild metric) instead of negative? Presumably space should be more negatively curved at the center than at some radius from the center, which implies that the orbital radius will look positively curved when viewed from the center.

But if that's the way it works, then when the orbiting Schwarzschild observer with the lesser radius measures the orbiting observer at greater radius, shouldn't he measure the space between them to be positively curved rather than negatively curved? That doesn't seem right.

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# Spatial curvature in Schwarzschild & FRW

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