Spatial curvature in Schwarzschild & FRW

[nutgeb]

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.

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.

 

[nutgeb]

Doggone it, I made a dumb mistake.

Mass in Schwarzschild creates positive spatial curvature, not negative like I said in the last post. The radius of the orbiting observer is disproportionately large compared to the circumference of the orbit. That's positive curvature.

So now it all fits together. Adding mass to both the Schwarzschild and FRW metrics (without changing velocity) makes the spatial curvature more positive.

The coordinate transformation between the empty Milne model and the vanishingly empty FRW model indicates that without gravity, in the reference frame of an "external" observer at zero density, the spatial curvature will be measured to be flat, but moving particles will appear to be $$\gamma$$ Lorentz contracted and time dilated. But if the reference frame is switched to an internal "comoving" reference frame, the spatial curvature will be negative (hyperbolic), which exactly offsets the $$\gamma$$ Lorentz contraction and time dilation of other comovers. This allows all Milne and FRW comovers regardless of velocity to share the same "flat" cosmological proper time frame.

The shift from an "external" to a "comoving" reference frame requires shifting the background spatial curvature in the negative direction (exactly offsetting the SR Lorentz contraction of fellow comovers) and shifting the "curvature" of the time axis in the contracted direction (exactly offsetting the SR time dilation $$\gamma$$ of fellow comovers).

Adding gravity to the FRW model shifts spatial curvature back in the direction of positive curvature. Such that, if the radial velocity of comovers exactly equals the Newtonian escape velocity of the gravitational mass, the gravitational shift exactly offsets $$\gamma$$, so comovers will measure that the spatial curvature has shifted to flat.

However, adding gravity to the FRW model does not shift the curvature of the time axis in the direction of time dilation: comovers continue to measure that each other's cosmological proper time remains flat. This indicates that no Schwarzschild gravitational time dilation occurs as between comovers in FRW comoving reference frames. That makes sense in an FRW model because of the symmetry: an "external" observer will measure each member of any pair of comovers to be gravitationally time dilated compared to the other member of the pair; but if that's true for both of them then by symmetry it must be true for neither of them.

The definition of a "comover" is an observer in freefall, with no proper velocity other than that caused by the gravity of the mass being considered, and any Hubble expansion velocity. In an expanding model, that means a fundamental comover at rest in the local Hubble flow. In a static Schwarzschild model, a "comover" is an observer who is comoving with the central mass, i.e. plunging radially inward in gravitational freefall, or in freefall hurling outward from the central mass, in both cases at the escape velocity of the central mass.

As with the FRW model, adding Schwarzschild gravity to the empty Milne scenario shifts the spatial curvature in the positive direction. So a comover plunging at the escape velocity of the central mass measures flat spatial curvature. However, in a standalone Schwarzschild scenario with a single central mass, the time curvature is shifted asymmetrically. A comover plunging at escape velocity will measure the central mass to be time dilated, and the central mass will measure the plunging comover to be time contracted by the same amount. But if we insert this Schwarzschild central mass into the homogeneous matter distribution of the FRW model, symmetry returns. Now every comover can consider himself to be surrounded by a gravitational sphere which is exactly the same as that of every other comover. This symmetry requires that no net Schwarzschild gravitational time dilation occur between comovers, restoring the same flat cosmological proper time as between all comovers.

This analysis is reassuring because it is consistent with the result that Birkhoff's Theorem requires: the metric for a zero elapsed time snapshot of the FRW metric must be exactly Schwarzschild. Birkhoff's Theorem predicts that Schwarzschild is the stationary metric for any spherically symmetrical mass distribution, regardless of whether the mass is in motion. Birkhoff also requires us to ignore the gravitational effect of all mass located outside any sphere of arbitrary radius from the central mass, as long as the mass distribution inside the sphere is symmetrical.

The only unresolved discrepancy I see between the two metrics is that Schwarzschild mass creates a parabolic spatial curvature (at least as measured by an "external" observer), while the comoving spatial curvature in FRW and Milne are hyperbolic. Perhaps the Schwarzschild spatial curvature changes from parabolic to hyperbolic when the reference frame is shifted to a comover, but I'm not sure.

 

[Entropia]

The difference in results between the Schwarzschild and FRW examples can be attributed to the different reference frames used to measure the spatial curvature. In the Schwarzschild example, the observers are measuring the curvature from an external observer's perspective, while in the FRW example, the observers are embedded in the curved space itself.

In the Schwarzschild example, the external observer measures the space between the orbiting observers to be negatively curved because the point mass is creating a gravitational pull towards the center, causing the space to curve inwards. As the point mass increases, this gravitational pull becomes stronger, resulting in a more negatively curved space between the observers.

On the other hand, in the FRW example, the observers are measuring the curvature from their own perspective, within the expanding space. As the density of matter increases, there is more mass per unit volume, causing the space to curve outwards. This results in a positively curved space between the observers.

The difference in results also highlights the concept of relative curvature - the curvature of space is dependent on the observer's reference frame and the distribution of matter in that frame. The same space can appear to have different curvatures depending on the reference frame of the observer.

To address the question about the Schwarzschild observer at the center of the point mass, it is important to note that the interior Schwarzschild metric is not applicable in this scenario. This metric is only valid for the exterior of a spherically symmetric mass. Therefore, the observer at the center would not measure the space between himself and the orbiting observers to be positively curved. Rather, the space would appear flat to him.

In conclusion, the difference in results between the Schwarzschild and FRW examples arises from the different reference frames used to measure the spatial curvature. It highlights the concept of relative curvature and the importance of considering the observer's perspective when discussing the curvature of space.