- Summary
- Question about elastic tethered between concentric rings stretching as a heavy mass passes through the centre of the system, temporarily increasing proper radial distances. Does this happen? If so, where does the energy to do work on deforming the elastic come from, and where does it go when the elastic relaxes again as the mass passes by and space becomes flatter again?

A thought experiment. Two rigid, concentric rings, joined together by a series of lightweight elastic radial spokes. These rings and a heavy mass are approaching each other, such that in the frame of the rings, the mass is travelling along the central, perpendicular axis and will pass through the centre. Nothing else exists in the universe close enough to influence the system. The mass is travelling slowly enough that at any moment, the system can be considered to be in static equilibrium. But fast enough that after the encounter, the mass and rings will recede from each other at least to some extent before getting pulled back (assuming the rings are in fact captured by the mass).

When the mass is still a long way off, the spokes tethering the inner and outer rings together are under an even, gentle tension. Space is flat. The coordinate and proper radial distance between the rings and hence the length of the spokes are the same (proper radius=circumference/2pi).

When the mass is at the central point, the rings are unaffected. However, space is distorted (stretched) radially, so that the proper radial separation between them is longer the coordinate distance (proper radius>circumference/2pi). The elastic spokes must therefore be stretched, as their endpoints are fixed on the rings.

When the mass has receded again far enough, the rings and spokes will again have returned to their original relaxed state. Or perhaps more likely, the ring system will be captured and pulled back, oscillating back and forth for a while before decaying away.

When the mass is at the centre, the stretched spokes have acquired (strain) potential energy which can only have come from the kinetic energy of the moving mass as it came closer. The implication being that the relative velocity of the mass should have been reducing slightly as it approached, doing 'work' to deform the elastic. Similarly, as it left, the elastic would relax and its potential energy should be released back to the mass, causing it to accelerate slightly as it recedes. Additionally, a proportion of the energy should be lost to heat in the elastic as it deforms, ending up presumably as thermal radiation from the spokes.

I know there will be other effects as well, that will be present in the Newtonian world, including tidal losses in the elastic spokes, which should increase their temperature slightly. There will also be a similar effect because as the mass approaches, the inner ring will be attracted more strongly along the axis than the outer one, again stretching and then relaxing the elastic twice as the mass passes. None of these however affect the overall length of the spokes when the mass is in the centre. I think this GR effect is completely separate and its magnitude will be highly dependent on the elastic properties of the spokes.

Does this happen? If not, how is conservation of energy satisfied as work is done to deform the elastic spokes? Or am I missing something obvious?

If it does happen, what is the mechanism? It seems to imply that the act of changing spatial curvature can transfer energy from one object to another purely by changing the relationship between proper distances and coordinate distances and so deforming one of the objects?

When the mass is still a long way off, the spokes tethering the inner and outer rings together are under an even, gentle tension. Space is flat. The coordinate and proper radial distance between the rings and hence the length of the spokes are the same (proper radius=circumference/2pi).

When the mass is at the central point, the rings are unaffected. However, space is distorted (stretched) radially, so that the proper radial separation between them is longer the coordinate distance (proper radius>circumference/2pi). The elastic spokes must therefore be stretched, as their endpoints are fixed on the rings.

When the mass has receded again far enough, the rings and spokes will again have returned to their original relaxed state. Or perhaps more likely, the ring system will be captured and pulled back, oscillating back and forth for a while before decaying away.

When the mass is at the centre, the stretched spokes have acquired (strain) potential energy which can only have come from the kinetic energy of the moving mass as it came closer. The implication being that the relative velocity of the mass should have been reducing slightly as it approached, doing 'work' to deform the elastic. Similarly, as it left, the elastic would relax and its potential energy should be released back to the mass, causing it to accelerate slightly as it recedes. Additionally, a proportion of the energy should be lost to heat in the elastic as it deforms, ending up presumably as thermal radiation from the spokes.

I know there will be other effects as well, that will be present in the Newtonian world, including tidal losses in the elastic spokes, which should increase their temperature slightly. There will also be a similar effect because as the mass approaches, the inner ring will be attracted more strongly along the axis than the outer one, again stretching and then relaxing the elastic twice as the mass passes. None of these however affect the overall length of the spokes when the mass is in the centre. I think this GR effect is completely separate and its magnitude will be highly dependent on the elastic properties of the spokes.

Does this happen? If not, how is conservation of energy satisfied as work is done to deform the elastic spokes? Or am I missing something obvious?

If it does happen, what is the mechanism? It seems to imply that the act of changing spatial curvature can transfer energy from one object to another purely by changing the relationship between proper distances and coordinate distances and so deforming one of the objects?