Does Curving Space by an Approaching Mass Do Work?

[lesaid]

TL;DR 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 traveling 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 traveling 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?

 

[Buzz Bloom]

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.

Hi lesaid:

Your thought experiment is quite interesting, but I think you are confused about the source of the energy that becomes the energy in the stretched elastics. It has nothing to do with the kinetic energy of the mass moving through the center of the rings along the axis perpendicular to the plane of the rings. The source of the elastics' energy is the change in the gravitational field (or the gavitationally distorted geometry) as the mass approaches the rings. This is similar to the changes of the kinetic energy of an small object in a non-circular orbit around a large mass. The potential energy of the small object is greater when the object is further from the mass. As the mass moves closer mass the reduction of its potential energy becomes an increase in its kinetic energy.

BTW, the two rings will also not stay in the same plane as the mass gets closer since the mass will have a greater effect on the inner ring since it is closer to the mass than the outer ring. I am thinking that after the mass has passed through the center, the planes of the two rings will oscillate along the axis under the influence of the elastics. This is the same effect as with a ball vertically attached by an elastic to a fixed point will oscillate vertically if the is ball is positioned so that the elastic is stretched, and then it is let go.

Regards,
Buzz

 

[PeterDonis]

When the mass is still a long way off, the spokes tethering the inner and outer rings together are under an even, gentle tension.

Why? The equilibrium state of the rings and spokes considered as an isolated system is for the spokes to be under zero tension.

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).

No, that's not what will happen. Space is not "stretched" radially by the mass. Space is not Euclidean around the mass; that's not the same thing. In fact, the non-Euclidean-ness of space around the mass has no effect on the stresses in the ring-and-spoke system; the only thing it affects is the proper radial distance from the inner ring to the center, which will be larger with the mass present than it was with the mass absent.

Since you have specified that things happen slowly enough that equilibrium is re-established at each step, we only have to consider what the static equilibrium of the ring-and-spoke system looks like with the mass at the center. The obvious difference between this case and the case where the mass is far away is that, with the mass at the center, the ring-and-spoke system has to support its own weight (whereas it has zero weight when the mass is far away). So there will be a hoop stress in each ring, and the spokes will be, if anything, slightly compressed (not stretched) by the weight of the upper ring (although the hoop stress in the upper ring will also be supporting that ring's weight so the spokes might not have to support very much).

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.

Whether or not the rings are "captured" by the mass depends entirely on the initial conditions you choose.

When the mass is at the centre, the stretched spokes have acquired (strain) potential energy

Actually, per the above, most of the stress induced in the ring-and-spoke system will be hoop stress in the rings.

which can only have come from the kinetic energy of the moving mass as it came closer

No, it comes from the ring-and-spoke system being inside a gravity well--the ring-and-spoke system has "fallen" into the gravity well, but instead of the potential energy due to the "fall" being converted into kinetic energy, it gets converted into strain in the rings (and, to a small extent, as above, in the spokes)--or, in relativistic terms, into a slightly increased rest mass of the ring-and-spoke system.

Note, btw, that this view is frame-dependent--it assumes that we are working in the rest frame of the ring-and-spoke system. In the rest frame of the mass, the ring-and-spoke system does actually fall into the gravity well, and converts its potential energy into kinetic energy--but some of that energy gain shows up as increased rest mass (as just above), rather than increased velocity.

 

[lesaid]

Thanks for the replies!

I think you are confused about the source of the energy that becomes the energy in the stretched elastics.

I am indeed! But before getting into that, I'd like to check my understanding of the geometry of the system with and without the central mass!

Why? The equilibrium state of the rings and spokes considered as an isolated system is for the spokes to be under zero tension.

If we remove the 'heavy mass' for a minute and just think about the rings and spokes - I was envisaging that all the spokes would be under gentle tension, such that the inner ring was held gently in place, with the forces from the individual spokes balancing each other, in static equilibrium. A bit like the spokes under tension on a bicycle wheel (even though they aren't radial) where the two 'rings' would be the hub circumference and the inner part of the rim. I didn't specify it but I was assuming cylindrical symmetry around the axis. Maybe instead of elastic spokes, I should be considering a circular elastic strip joining the two rings together.

The reason for assuming tension at all times was just to make the thought experiment a little simpler, so that the special cases of zero tension or slack/compressed spokes could be ignored.

In fact, the non-Euclidean-ness of space around the mass has no effect on the stresses in the ring-and-spoke system; the only thing it affects is the proper radial distance from the inner ring to the center, which will be larger with the mass present than it was with the mass absent.

This confuses me. I understand that the proper radial distance from the inner ring to the centre will be larger with the mass present. Why would the same not be true of the outer ring, such that the radial distance between the two rings would also be larger. With no mass, if the inner ring had, say, a radius of 1 km and a circumference of about 6.3 km while the outer ring had radius 2km and circumference about 13 km, then the radial distance between them would be around 1 km. A heavy mass in the centre will surely increase the proper radius of each ring, also increasing the radial distance between the two rings?

I demonstrated this to myself by doing a calculation - see end of post. That is where this question (and my confusion about the elastic spokes) was born.

But the mass should not affect the circumference of either ring (assuming negligible compression along the circumference of the ring) or the coordinate radius of the rings. So the endpoints of the elastic spokes are fixed at unchanging coordinate radii, while the proper distance between the endpoints is increased?

I wanted to construct a thought experiment where a piece of elastic had fixed 'coordinate' endpoints that would not be influenced by the central mass, while the proper distance between the endpoints was dependent on the mass. And concentric rings seemed to be the way to do it.

the ring-and-spoke system has to support its own weight (whereas it has zero weight when the mass is far away). So there will be a hoop stress in each ring, and the spokes will be, if anything, slightly compressed (not stretched) by the weight of the upper ring (although the hoop stress in the upper ring will also be supporting that ring's weight so the spokes might not have to support very much).

I understand that. I have been presuming that the rings can be considered (for the purposes of a thought experiment) sufficiently rigid and lightweight that whatever compression they undergo will not cause a significant change in radius compared to the change I expected in the proper radial distances. So, as something that deforms the spokes, it could be ignored. This kind of thing is why I was assuming a gentle tension in the 'no mass' state, to avoid the situation where they could become compressed.

Actually, per the above, most of the stress induced in the ring-and-spoke system will be hoop stress in the rings.

That, however, would be present in a Newtonian system. I was trying to isolate what I thought might be the effect on the radial elastic due to GR alone. In the Newtonian world, I don't see why the radial spokes would be affected much at all, as you implied. But IF the mass causes the proper radial distance between the rings to increase, it would surely have to stretch the elastic. This is what I was trying to explore. Unless I've got something fundamentally wrong in my understanding (quite possible!), it seems to me that the proper length that the spokes traverse has to increase (as measured by lining up rulers end to end).

To illustrate what I mean, I did a calculation, assuming rings of 1 km and 2 km radii, and a central mass of $10^{28} kg$. With no central mass, the radial distance between the two is obviously 1 km. With the mass, the radial distance came out to be about 1.0052 km, or about 5 m longer. Doing the same calculation for each ring to the centre (well, to the SW radius of about 14.85m), came to about 1.034 km and 2.039 km respectively - the difference between them is 1005 m which comes back to the same 5 m increase. So it seems to me that the elastic spokes in that system would have to have been stretched by that amount, specifically due to the change in proper distance between the two rings?

EDIT : typo corrected on the integral !

The formula used was from the radial term in the Schwarzschild metric
$$\sigma=\int_{r=r_{inner}}^{r_{outer}}\frac{1}{\sqrt{1-\frac{R_s}{r}}} dr$$
and resulted in (with the aid of Maxima and some algebra)
$$\sigma=R_s \ln{\left\{\frac{(A_0+1)(1-A_1)}{(A_1+1)(1-A_0)}\right\}}+2r_{outer} A_1-2r_{inner} A_0$$
where $r_{outer}>r_{inner}>R_s$ and
$$R_s=\frac{2GM}{c^2} \hspace{1cm} A_0=\sqrt{\frac{r_{inner}-R_s}{r_{inner}}} \hspace{1cm} A_1=\sqrt{\frac{r_{outer}-R_s}{r_{outer}}}$$

 

[PeterDonis]

was envisaging that all the spokes would be under gentle tension, such that the inner ring was held gently in place, with the forces from the individual spokes balancing each other, in static equilibrium.

Why does the inner ring need to be held in place? What would make it move? If there's no mass present, and no gravity, then the whole ring-spoke assembly is just floating freely in empty space, with no forces to move anything.

The reason for assuming tension at all times was just to make the thought experiment a little simpler, so that the special cases of zero tension or slack/compressed spokes could be ignored.

I don't see how ignoring zero tension makes things any simpler; in fact it makes them more complicated, because the unstressed state is the simplest state to analyze, so if that state is not the state of the ring-spoke assembly when the mass is far away, you are making things more complicated.

I understand that the proper radial distance from the inner ring to the centre will be larger with the mass present. Why would the same not be true of the outer ring, such that the radial distance between the two rings would also be larger.

The proper radial distances of both rings from the center is larger. But that does not necessarily mean the difference between those two proper distances, which is the proper distance between the rings, will be larger. See below.

I demonstrated this to myself by doing a calculation

Your calculation assumes that the radial coordinates of both rings remain constant as the mass comes in. You shouldn't assume that, since it won't automatically be the case.

Note that you use the term "radius" to mean two different things, the radial coordinate and the proper radius. You should not do that; the distinction between the two is important. The radial coordinate is usually thought of as the "areal radius", i.e., $\sqrt{A / 4 \pi}$, where $A$ is the area of a particular 2-sphere centered on the mass. However, it's easier to think of it as $C / 2 \pi$, where $C$ is the circumference of the ring whose proper radius you are trying to calculate. The non-Euclideanness of space around the mass then translates to the proper radius of each ring being larger than $C / 2 \pi$, i.e., larger than its radial coordinate.

So you are assuming that the circumference of each ring remains constant as the mass comes in. But in fact the gravity of the mass will exert a force on each ring that tends to reduce its circumference. It is true that the rings themselves will resist having their circumference reduced; I didn't take that into account in my previous post. So a more complete analysis would be that as the mass comes in there are three effects involved:

(1) If the circumferences of the two rings were constant, the proper distance between them would increase (this is the effect you calculated), which would tend to stretch the spokes;

(2) The gravity of the mass pulls on the rings and tends to reduce their circumferences, which tends to counteract the above effect, but the rings themselves resist having their circumferences reduced, which sets up a hoop stress in each ring;

(3) The weight of the upper ring pushes down on the spokes, which tends to compress them.

I don't think it's possible to calculate the net outcome of all of the above without having a detailed material model of the rings and spokes, including things like their elasticity, stiffness, etc. But I agree, taking into account that the rings will resist having their circumference reduced, that it should be possible, for an appropriate choice of conditions (basically stiff enough rings and elastic enough spokes), for the spokes to be stretched when the mass comes in (assuming that they were under zero tension when the mass was far away).