# Spatial curvature - effect on objects

1. Sep 3, 2009

### nutgeb

My understanding is that a physical object moving through a spatial curvature gradient (as distinguished from spacetime curvature gradient) will not automatically experience an internally stress-free change in its physical dimensions consistent with the changing background spatial geometry. But the changing spatial geometry can introduce irresistible internal stresses in the object.

Consider a VERY large, simple wagon wheel (with rim, spokes and hub) in free fall inward toward the event horizon of a BH. The wagon wheel was originally constructed very far from the BH. It's a supermassive BH, so the tidal effects are not important near the horizon. (Also, assume that the wheel is free falling at much less than the BH's escape velocity.)

As the radial distance from the wheel to the BH decreases, the spatial curvature progressively increases. Increasing spatial curvature causes the proper length of the wheel's spokes to become longer relative to the circumference of the rim; or it can be thought of as causing the circumference of the rim to decrease relative to the length of the spokes. The circumference increasingly becomes $$< 2\pi r$$.

The wheel's own geometry does not change automatically, in a stress-free way, along with the changing background spatial geometry. However, the changing spatial geometry introduces inexorable internal stresses into the wheel. The wheel was constructed (in essentially flat space) with its circumference equal to $$2 \pi$$ times its proper radius. But such a $$2\pi r$$ planar object cannot exist in space that has significantly positively curved geometry. Therefore, stresses will be introduced that cause the wheel's spokes and hub to deform (bend) out of the plane (causing the wheel to become bowl-shaped), or cause the wheel to fragment (break apart). It's like projecting the surface of a globe's hemisphere onto flat paper -- gaps will appear in the circumference.

Conversely, if the wheel originally was constructed near the BH (in highly curved space), and then is moved away from it, its original circumference was $$< 2 \pi r$$. Therefore the stresses resulting from the curvature gradient will cause the rim of the wheel to deform out of the plane, or will fragment the wheel. In this case it's like trying to wrap a flat sheet of cardboard around the hemispherical surface of a globe -- there will be extra material at the outer edge of the cardboard that can't lie flat without folding.

Is this description correct?

Last edited: Sep 3, 2009
2. Sep 4, 2009

### nutgeb

I should have qualified my examples with the condition that the wheel components are completely inelastic. If on the other hand the wheel components are stretchy/compressible, the wheel's deformation out of the plane, or fragmentation, could be reduced or avoided entirely.

3. Sep 7, 2009

### nutgeb

Any Born Rigidity objections to this scenario can be rendered insignificant by disassembling the very large wheel into small pieces near the BH, and then individually moving them away from the BH. When it is later reassembled far from the BH, it will prove impossible to fit the pieces back into the wheel's original shape, due to the fact that the background spatial curvature has changed while the shapes and sizes of the pieces have not.

4. Oct 18, 2009

### nutgeb

So no one disagrees with my conclusion, particularly as stated in the quote above?