I Gravity changes produced by a compressed and stretched spring

[John K Clark]

If I compress a spring beyond its most relaxed point it will have more energy and thus by General Relativity produce a stronger gravitational field than the uncompressed spring will. But suppose instead I pull in the opposite direction and stretch the spring beyond its most relaxed point it is my understanding that it would produce a weaker gravitational field than the compressed spring even if they both had the same amount of energy because tension, unlike pressure, produces a gravitational repulsion that would cancel out some of the attraction made by the mass/energy. Am I correct?

John K Clark

 

[Drakkith]

But suppose instead I pull in the opposite direction and stretch the spring beyond its most relaxed point it is my understanding that it would produce a weaker gravitational field than the compressed spring even if they both had the same amount of energy because tension, unlike pressure, produces a gravitational repulsion that would cancel out some of the attraction made by the mass/energy. Am I correct?

I don't think so. When you pull on the spring, it stores energy in its bonds, increasing its mass and thus its gravity. If this stored energy is equal to the energy stored in the spring when compressed, then the two should have equal masses. I've never heard of tension causing a repulsion.

 

[timmdeeg]

I think if the work done on the spring is the same in both cases then there shouldn't be any difference. The increase of internal energy of the spring is the same.

 

[PeterDonis]

suppose instead I pull in the opposite direction and stretch the spring beyond its most relaxed point it is my understanding that it would produce a weaker gravitational field than the compressed spring even if they both had the same amount of energy because tension, unlike pressure, produces a gravitational repulsion that would cancel out some of the attraction made by the mass/energy. Am I correct?

No.

The bit about "tension" producing gravitational repulsion is not a general statement in GR; it's a very specific statement about a very specific type of solution, namely that of an isotropic perfect fluid. A spring is not even close to being an isotropic perfect fluid, and the "tension" you produce in a spring when you stretch it is not the kind of "tension" to which that specific statement applies.

The correct intuition here is the one that @Drakkith and @timmdeeg have expressed: when you do work on the spring, either by stretching or compressing it, you increase its total stress-energy and therefore increase the spacetime curvature it produces.

 

[Dale]

The bit about "tension" producing gravitational repulsion is not a general statement in GR; it's a very specific statement about a very specific type of solution, namely that of an isotropic perfect fluid.

I think that @John K Clark is not too far off. There is a paper I remember reading that discussed a container with a perfect fluid. In such a case the container walls can be in tension in a way that "cancels out" the gravity due to the pressure in the fluid. I will try to find the paper

 

[PeterDonis]

There is a paper I remember reading that discussed a container with a perfect fluid. In such a case the container walls can be in tension in a way that "cancels out" the gravity due to the pressure in the fluid.

Hm, yes, good point; the container walls in this case are not perfect fluids, because the stress in them is not isotropic.

However, there's still a key difference from the spring case, because in the spring case work is being done on the system to stretch it. That work should still increase the total stress-energy of the spring, and therefore the spacetime curvature it produces. You can't do work on a system and have it produce weaker gravity afterwards: that would violate local conservation of stress-energy.

I have not tried to model this in more detail mathematically, but I think that might be what would have to be done to resolve the apparent tension (pun intended) between the two statements ("tension cancels out pressure" vs. "doing work adds energy").

 

[John K Clark]

But if the tension wasn't isotropic wouldn't that mean the repulsive gravitational field would also not be isotropic not that the field wouldn't exist? It seems to me that if everything had to be perfect General Relativity would never be applicable in real would situations.

John K Clark

 

[Dale]

You can't do work on a system and have it produce weaker gravity afterwards: that would violate local conservation of stress-energy

Yes, I agree. I think the most you can do is move the gravitational source around a bit. My guess is that if a (static) spring is in tension then something else must be in compression and those two effects cancel.

 

[Dale]

But if the tension wasn't isotropic wouldn't that mean the repulsive gravitational field would also not be isotropic not that the field wouldn't exist? It seems to me that if everything had to be perfect General Relativity would never be applicable in real would situations

Here is the paper I was remembering

https://arxiv.org/abs/gr-qc/0510041

Things get complicated...

 

[PeterDonis]

if the tension wasn't isotropic wouldn't that mean the repulsive gravitational field would also not be isotropic not that the field wouldn't exist?

No. There might be a small negative contribution from tension to the overall gravitational field, but it won't be anywhere near enough to outweigh the other positive contributions. As has already been said, the overall contribution of doing work on the spring has to be positive as far as the gravitational field is concerned; you can't do work on something and have its gravity become weaker.

Remember that, for ordinary objects, stresses are something like 20 orders of magnitude smaller than rest energy density, which of course also acts as a source of gravity (and a positive one). So we're talking about miniscule corrections in any case, way below our ability to actually observe.

It seems to me that if everything had to be perfect General Relativity would never be applicable in real would situations.

Nobody is saying GR is "perfect".

 

[John K Clark]

I know if you put energy into a spring its always going to increase its gravitational field, but I think it matters if that energy is in the form of compression or tension (negative pressure), although the difference would usually be very very small. The reason I got interested in this is I was reading "Black Holes and Time Warps" by Kip Thorne (who won the Nobel Prize for physics just a few days ago) and he indicates that matter/energy density is not the only thing that can make gravity, on page 118 Thorne says to my surprise:

"Mass and pressure warp spacetime... the curvature near a particle is proportional to the density of the mass in the vicinity of the particle (multiplied by the speed of light squared to convert it into the density of energy) plus 3 times the pressure of matter near the particles vicinity". [...] Under most circumstances the pressure of matter is tiny compared to the mass density times the speed of light squared and therefore the pressure is a unimportant contributor to spacetime curvature. Only deep inside Neutron Stars and a few other exotic places is pressure a significant contributor to the warpage." _Kip Thorne

So I know it will be small but I just want to be sure I understand what Thorne is saying hence my question: Will 2 identical springs with identical amounts of stored energy in them produce identical gravitational fields if one spring is under pressure and the other in tension? In my interpretation to what Thorne is saying the answer seems to be no. Am I missing something?

John K Clark

 

[PeterDonis]

matter/energy density is not the only thing that can make gravity

That's correct. The source of gravity is the stress-energy tensor, which includes contributions from pressure and other stresses as well as energy density.

Thorne's specific statement, that the curvature is proportional to the density of energy plus 3 times the pressure, assumes a perfect fluid, i.e., the pressure is positive and the same in all directions. Also, when he says "curvature", he means the spacetime curvature inside the object, at some particular point where we are measuring the energy density and the pressure. That is not the same as the external gravitational field produced by the object.

Will 2 identical springs with identical amounts of stored energy in them produce identical gravitational fields if one spring is under pressure and the other in tension?

It depends on what "identical amounts of stored energy" means. Ordinary language is vague. If it means "identical amounts of work done on the spring to get it from its unstressed to its stressed state", then I think the answer has to be yes, their gravitational fields will be identical. But if "identical amounts of stored energy" means something else, the answer might not be yes.

 

[John K Clark]

You say "identical amounts of work done" but work is just a specific sort of energy and as far as I know heat or any other sort of energy can produce a gravitational field so I don't see why there would be any ambiguity in the term "identical amounts of stored energy". Also if pressure can change the spacetime curvature inside an object I don't understand why that change would suddenly stop at the object's boundary unless there were some sort of spacetime discontinuity there and I thought that only happened at the center point of a Black Hole.

John K Clark

 

[PeterDonis]

You say "identical amounts of work done" but work is just a specific sort of energy and as far as I know heat or any other sort of energy can produce a gravitational field so I don't see why there would be any ambiguity in the term "identical amounts of stored energy"

Because "energy", depending on what you mean by it, can be frame-dependent. But "work done" is an invariant. Mathematically, "work done" must be expressible as a scalar invariant, but "energy" can be just one component of a tensor. They're not necessarily the same thing, because ordinary language is vague. If you told me exactly what you meant by "identical amounts of stored energy" in terms of math, or even in terms of some more precise ordinary language term like "work done" that must be invariant, then there would be no ambiguity. But you didn't do that.

if pressure can change the spacetime curvature inside an object I don't understand why that change would suddenly stop at the object's boundary

You're misunderstanding what I was saying. The Einstein Field Equation, which is what Thorne was describing in his statement that you quoted, is local: it tells you the relationship between the stress-energy tensor and spacetime curvature at a particular point. It doesn't directly tell you the relationship between, say, the energy density and pressure inside a neutron star and the gravitational field observed outside the neutron star. To obtain such a relationship, you need to develop a global solution of the Einstein Field Equation, i.e., a description of stress-energy and spacetime curvature throughout some extended region of spacetime, such that, at each point within that region, the relationship between stress-energy and spacetime curvature at that point satisfies the EFE.

Saying that "pressure can change the spacetime curvature" means that pressure at a particular point in spacetime affects the spacetime curvature at that point in spacetime. That doesn't necessarily mean that, when you develop a global solution, the pressure inside the object will appear in the description of the gravitational field outside the object. As a matter of fact, it can be shown, by an analysis similar to the one given in the paper @Dale linked to, that for the case of a static, spherically symmetric object, the pressure inside the object does not appear in the description of the gravitational field outside the object, i.e., in the externally measured mass of the object. Heuristically, this is because the positive contribution from pressure is exactly cancelled, in equilibrium, by a negative contribution from gravitational potential energy, similar to the way in which the positive contribution from pressure in a gas inside a container is exactly cancelled, in equilibrium, by the negative contribution from tension in the container walls. In both cases, it's not that pressure somehow stops contributing; it's just that its contribution gets canceled by something else.

The bottom line is that you can't just say" pressure contributes to gravity" and expect that to give you an answer for a specific case; you have to actually work out, in a specific case, a solution of the EFE that describes that case, and then see what it tells you about the various contributions to spacetime curvature/gravity. And none of the solutions we have discussed so far in this thread apply to the case of a stretched or compressed spring.

 

[PeterDonis]

My guess is that if a (static) spring is in tension then something else must be in compression and those two effects cancel.

This is quite possible; a common mistake in analyzing situations like this is to leave out a source of gravity, or to put it another way, ignoring the fact that automatic conservation of the source (stress-energy) is built into the Einstein Field Equation. So you can't just say "we'll stretch the spring" in isolation: the energy that produces the work done to stretch the spring has to come from somewhere, and has to be counted in the complete solution of the EFE. So it might, in the end, turn out that it's impossible to model an isolated stretched or compressed spring in GR: you have to model the spring plus whatever it is that stretches or compresses it, and that system as a whole will have an unchanged external gravitational field through the process.

 

[PAllen]

An interesting way to motivate the point in #15 is to consider a simple complete system. Imagine a spring connected to the ends of ratchet, with little rocket engines attached to each end of the ratchet. Ignite the engines. The string stretches, the ratchet prevents the string from relaxing after the fuel is consumed. The spring is under tension, while the ratchet is under compression, after equilibrium.

 

[PeterDonis]

Imagine a spring connected to the ends of ratchet, with little rocket engines attached to each end of the ratchet. Ignite the engines. The string stretches, the ratchet prevents the string from relaxing after the fuel is consumed. The spring is under tension, while the ratchet is under compression, after equilibrium.

As an extension to this, one could consider a second complete system in which the two rockets push the spring into compression, with some ratchet-like mechanism to hold it compressed after the rockets' fuel is expended. Then we could impose the condition that the rockets do the same amount of work on the spring in both cases, and compare the two final states.

 

[Dale]

I think it matters if that energy is in the form of compression or tension (negative pressure),

As @PeterDonis mentioned above, it definitely does matter locally. You have multiple effects going on locally.

The biggest effect is that by placing an object in tension or compression you are changing its shape and density. That primarily effects the "energy density" term of the stress energy tensor. The shape must change, even if the material is perfectly incompressible, and real materials are compressible.

The second, much smaller effect, is the energy density from the external work. This is positive regardless of if the work is done to place the object in compression or tension. It is a small increase to the "energy density" term of the stress energy tensor compared to what you would expect just from the mass density alone.

The third effect, also much smaller than the first, will be the stress itself. This can be positive (compression) or negative (tension), but it is a completely different part of the stress energy tensor than the energy density term. Essentially, it is a different direction in spacetime. Because it is a different direction it is not generally possible to think of it as just increasing or decreasing gravity. It is a change to a tensor, not a scalar.

if pressure can change the spacetime curvature inside an object I don't understand why that change would suddenly stop at the object's boundary

That is precisely the topic studied in the paper I linked to. It is complicated, but if you go through the math that is what you get in certain cases (spherical symmetry and other assumptions described in the paper). Your spring is a more complicated case even than what the paper describes.

 

[jartsa]

Let's say light bounces between mirrors that hang vertically in a gravity field. Mirrors are charged, attractive Coulomb force keeps them at constant distance from each other.

How much does the light attract the planet which is the source of the gravity? Well there is some 'extra' attraction between the planet and the light whenever there is some extra bending of light. There is 'extra' bending of light when light can reach its destination faster by bending an extra amount, and that is the situation when the gravity field is non-uniform and the distance between mirrors is so large that the non-uniformity matters.

I hope I got the 'extra' bending of light correct, I mean the effect where light passes a star and accelerates 200m/s² where normal, slow speed, free falling matter accelerates 100 m/s².

So my point is that there is no 'extra' gravity of pressurized photon gas, when the gas can not detect tidal forces.

 

[pervect]

If I compress a spring beyond its most relaxed point it will have more energy and thus by General Relativity produce a stronger gravitational field than the uncompressed spring will. But suppose instead I pull in the opposite direction and stretch the spring beyond its most relaxed point it is my understanding that it would produce a weaker gravitational field than the compressed spring even if they both had the same amount of energy because tension, unlike pressure, produces a gravitational repulsion that would cancel out some of the attraction made by the mass/energy. Am I correct?

John K Clark

A stumbling block here is that an isolated, static spring in tension or compression can't exist - something needs to hold the ends still. Or the spring needs to be moving. Both present complications.

In the linearized limit, it can be shown that the net effect of tension and compression on an isolated system balances out. So in the linearized system, there won't be any net gain or loss of mass due to tension or compression, it will just "move it around". And "mass" is a concept that's really only applicable to an isolated system, at least with our current conceptions of mass in General relativity - those concepts don't really (and can't as far as I know) localize "mass" at all, they can only give us the total mass, not assign it a definite location.

Something similar happens in the simple non-linear case we can analyze, a pressurized sphere. One can argue that a spherical pressure vessel will, by symmetry, not produce any "force" on an object inside the sphere, so the contribution of the pressure vessel to the reading of the proper acceleration of a static observer inside the pressure vessel will be zero.

Then one changes the pressure inside the pressure vessel by heating up the contents of the sphere. Rather than bring in energy from outside, it's cleaner to have the energy source needed to change the pressure already inside the sphere. We can imagine the pressure containing a bomb. To get detectable effects, I'd suggest a nuclear bomb. Or something better, if available, like a total conversion matter-antimatter bomb. Then the non-pressurized sphere containing the bomb will become a pressurized when we explode the bomb, without importing any external energy. As long as we make it strong enough that the pressure vessel survives the explosion, of course. (That's rather unlikely with any known material, considering the sorts of pressures we need to get measurable effects).

We can then observe the effects on an accelerometer mounted just inside and just outside the surface of the pressure vessel, one that measures the proper acceleration of a static observer. I haven't seen this analyzed in any text, but working it out for myself. I convinced myself that the reading on the external accelerometer does not change when one explodes the bomb, but the reading on the internal accelerometer (just inside the wall of the pressure vessel) does change. (It goes up, the pressure causes more gravity, meaning the accelerometer has a higher reading). This then is a way of demonstrating via a thought experiment how pressure (and tension) affects gravity. If pressure had no effect on gravity, the reading on the interior accelerometer wouldn't change when one exploded the bomb. But even though pressure affects gravity, conservation laws imply that the reading on the external accelerometer doesn't change.

 

[John K Clark]

A stumbling block here is that an isolated, static spring in tension or compression can't exist - something needs to hold the ends still. Or the spring needs to be moving. Both present complications.

In the linearized limit, it can be shown that the net effect of tension and compression on an isolated system balances out

You could weld a bar to the ends of the compressed and tensioned springs to keep the ends still, I suppose the corresponding tension and pressure might cancel out, but if that was always the case I don't understand why pressure and tension are even in the field equations.

Let me propose another thought experiment. I have a spherical metal tank full of high pressure gas and then I add energy in the form of heat to it that increases the pressure of the gas and of course the tension in the metal tank too. I am interested if the change in the gravitational field externally far from the tank is determined only by the increase in the matter/energy due to the energy I added or if the increased pressure must be taken into account too. I also note that if thank tank were to suddenly to rupture the tension would instantly drop to zero but it would take time for the pressure to do so; I guess the two don't always cancel out.

John K Clark

 

[PeterDonis]

@John K Clark, if you are going to quote someone and then respond to the quote, please do both in the same post. There is no need to post the quote, and then your response separately; it just makes it harder to see what you are trying to do.
[Mentor's note: this comment refers to two posts above that have since been properly merged]

 

[PeterDonis]

You could weld a bar to the ends of the compressed and tensioned springs to keep the ends still

In which case there would be stresses in the bar, which would need to be included in the source of the gravitational field.

I have a spherical metal tank full of high pressure gas and then I add energy in the form of heat to it that increases the pressure of the gas and of course the tension in the metal tank too.

If the heat comes from outside the tank (as opposed to the case @pervect considered, where there is a nuclear bomb inside the tank that heats it up), then it comes from somewhere that contains stress-energy, and you have to include that stress-energy in your analysis. Suppose, for example, that you add heat by means of a laser shining on the tank from far away. Then you are transferring stress-energy from the laser source to the tank, which, assuming you can observe the gravitational fields of the two separately, you will see as a decrease in the gravitational field of the laser source and an increase in the gravitational field of the tank.

I am interested if the change in the gravitational field externally far from the tank is determined only by the increase in the matter/energy due to the energy I added or if the increased pressure must be taken into account too.

The answer to this has already been given: if you are only looking at the effects of the external gravitational field, and we assume the tank is allowed to come to equilibrium after being heated up, then the effects of pressure in the fluid in the tank are exactly canceled by the effects of tension in the tank walls.

I also note that if the tank were to suddenly to rupture the tension would instantly drop to zero but it would take time for the pressure to do so; I guess the two don't always cancel out.

You have to be careful not to confuse a local analysis with a global one. The exact cancellation I described just above, between pressure and tension, is global, and assumes that the system is in equilibrium; if the tank wall ruptures, the system is no longer in equilibrium. However, there is a stronger global statement we can make: the externally measured mass of an isolated system doesn't change due to purely internal changes like the tank rupturing; if you are observing it from far away, you won't see any change unless some stress-energy passes you on the way out (such as fluid spraying out of the rupture and flying past you).

Locally, if we look at the stress-energy vs. spacetime curvature at the point where the tank ruptures, the tension does not "instantly" drop to zero; it takes some time. And during that time, fluid goes from being static to being in motion (relative to the tank), as it starts to escape through the rupture. (This is also what causes the pressure to start dropping inside, but the motion of the fluid happens first.) The fluid being in motion means that there is now another nonzero component of the stress-energy tensor, due to the momentum of the fluid (i.e., in addition to the energy density). That is where the stress-energy that was formerly stored in the tension of the tank wall gets transferred to, in order to satisfy the local conservation of stress-energy.

 

[jartsa]

Then one changes the pressure inside the pressure vessel by heating up the contents of the sphere. Rather than bring in energy from outside, it's cleaner to have the energy source needed to change the pressure already inside the sphere. We can imagine the pressure containing a bomb. To get detectable effects, I'd suggest a nuclear bomb. Or something better, if available, like a total conversion matter-antimatter bomb. Then the non-pressurized sphere containing the bomb will become a pressurized when we explode the bomb, without importing any external energy. As long as we make it strong enough that the pressure vessel survives the explosion, of course. (That's rather unlikely with any known material, considering the sorts of pressures we need to get measurable effects).

Let's say we have a large, mostly empty, pressure vessel containing small pressure vessels that contain some stuff.

All the small pressure vessels are opened. What happens? Do we have a pressure increase or decrease?Well I guess first the pressure and the stress of the small vessels cancel out, and after opening the small vessels, the pressure and the stress of the large vessel cancel out.

That was not interesting.

But how about if we have a filled scuba tank and we sink it to the bottom of the sea? Stress on the tank decreases, but I don't see any pressure change anywhere.

 

[pervect]

Here is the paper I was remembering

https://arxiv.org/abs/gr-qc/0510041

Things get complicated...

Nice paper. I haven't gotten all the way through it, but they make the same argument I did, and being peer reviewed rather than something I did on my own is a huge advantage for a discussion. This particular section caught my eye - it's essentially the scenario I analyzed, though in this summary, at least, there's no explicit discussion of the effect of the 3*pressure term on the gravitational acceleration sensor on the inner wall of the box.

The following scenario is known as Tolman’s Paradox: A static spherical box has been filled with a gravitating substance of a given mass. I f this substance
undergoes an internal transformation (e.g. matter and anti-matter turning into radiation) raising the pressure, the active mass in the box would change because of the 3
p-term since the energy is conserved. However, such an internal transformation should not affect the mass measured outside the box, say by an orbiting particle obeying Kepler’s third law. In a spherically symmetric field the particle should be oblivious to all spherically symmetric changes inside its
orbit, a consequence of the vacuum equations known as Birkhoff’s Theorem [6].

Charles Misner and Peter Putnam were intrigued by Tolman’s Paradox [7]. They showed that by increasing the pressure inside the box one has to have stresses in the walls keeping the matter inside confined and the field static [8].

If they're not paywalled, it'd be interesting to track down some of these early references, specifically Misner's and Putnam's analysis of Tolman's Paradox.

 

[Dale]

This particular section caught my eye

Yes, I have also wondered specifically about their reference to Birkhoffs theorem. I wonder how sensitive the final result is to spherical symmetry. If the stress is axisymmetric, do you get a strong "dipole" or does it still mostly cancel out

 

[Nidum]

What might happen in a dynamic case where the spring has mass and where it is caused to oscillate freely in continually repeating cycles of expansion and contraction ?

 

[PAllen]

What might happen in a dynamic case where the spring has mass and where it is caused to oscillate freely in continually repeating cycles of expansion and contraction ?

Gravitational waves, for one thing. In practice, all GR effects for an oscillating spring would be dozens of orders of magitide too small to observe for any conceivable spring or analog thereof.

I am reminded of another theoretical prediction that will likely never be tested. GR predicts that two parallel light beams traveling in the same direction do not attract, while if opposite directions, they will attract and converge. However, one paper on this calculated the direct gravitation influence of stars many light years away is 10 or more order of magnitude larger than mutual attraction of intense antiparallel lasers.

 

[timmdeeg]

In order to avoid stresses on mechanical devices the work done on the spring could be due to tidal forces in curved spacetime. But what stresses would correspond to those on say the bar in this case?

 

[pervect]

You could weld a bar to the ends of the compressed and tensioned springs to keep the ends still, I suppose the corresponding tension and pressure might cancel out, but if that was always the case I don't understand why pressure and tension are even in the field equations.

Let me propose another thought experiment. I have a spherical metal tank full of high pressure gas and then I add energy in the form of heat to it that increases the pressure of the gas and of course the tension in the metal tank too. I am interested if the change in the gravitational field externally far from the tank is determined only by the increase in the matter/energy due to the energy I added or if the increased pressure must be taken into account too. I also note that if thank tank were to suddenly to rupture the tension would instantly drop to zero but it would take time for the pressure to do so; I guess the two don't always cancel out.

John K Clark

The problem with welding a bar to the spring is the bar gravitates too, and since the bar isn't spherically symmetrical, one can't sort out the gravitational effects of the bar from the gravitational effects of the spring. The hollow sphere arrangement canclels out the tension in the enclosing sphere by symmetry arguments, the bar welded to the spring lacks the symmetry to cleanly understand the results.

For the scenario you mentioned, I believe that the far field is only affected by the added energy and not the pressure.

One other important case where the pressure term is importnat is mentioned in the paper that Dale cited. This is the Friedmann equations, which govern the expansion of the universe. One can definitely see the effects of the pressure in the Friedmann cosmology. One can also see the effects of pressure in a spherical tank, even if the effects are not the field at infinity, but rather the internal field, as I argued.

So, the pressure definitely has an effect, just because it's not quite the effect you first imagined doesn't mean that it isn't there.

 

[John K Clark]

Sorry to keep bugging people but I'm still not clear on how pressure and tension fit into General Relativity. It occurs to me that 2 springs with equal energy , a compressed one under pressure and a stretched one under tension (negative pressure) could be both kept static in a arbitrarily light scissor like device and be arbitrarily far apart too, so if the pressure term is positive in one and negative in the other I don't understand how the field equations could say that both springs warp spacetime in identical ways.

John K Clark

 

[PeterDonis]

It occurs to me that 2 springs with equal energy , a compressed one under pressure and a stretched one under tension (negative pressure) could be both kept static in a arbitrarily light scissor like device and be arbitrarily far apart too

No matter how "light" the scissor like device is, or how far apart its arms are, it still has to have stress in it that compensates for the stress in the spring, in order to keep the system as a whole static. There is no way around that; and that is what resolves your question--that you can't just consider the stress in the spring in isolation, you have to consider all of the stresses in the system.

if the pressure term is positive in one and negative in the other I don't understand how the field equations could say that both springs warp spacetime in identical ways

The field equations don't say the springs, in isolation, warp spacetime in identical ways; they say that the static systems, as a whole, warp spacetime in identical ways, because the systems as a whole must include not just the stresses in the spring, but the stresses in whatever it is that is holding the spring static. Those two stresses will be opposite (if the spring is stretched, whatever is holding it will be compressed, and vice versa), so the overall effect is the same either way.

 

[John K Clark]

I understand that far away the effect of tension and pressure would cancel out, but suppose the two springs are far apart and there are 2 observers, one very close to the spring under pressure and the other very close to the spring under tension, would they observe identical spacetime curvature? Are there any circumstances where it would be important to know if the energy in the spring is in the form of pressure or tension?

John K Clark

 

[PeterDonis]

suppose the two springs are far apart

What's holding a particular spring static can't be "far apart" from the spring. So there is no way to just consider a spring under pressure, or a spring under tension, in isolation, if they are static. There has to be something else present that is holding the spring static, that can't be ignored even by an observer close to the spring.

Are there any circumstances where it would be important to know if the energy in the spring is in the form of pressure or tension?

A hypothetical observation made inside one of the springs (i.e., not just close to it but within its internal structure) would obviously see a difference, depending on whether the spring was under pressure or under tension. But under those circumstances you're not talking about the total mass or gravitational field of a static system viewed from the outside.

 

[Dale]

suppose the two springs are far apart and there are 2 observers, one very close to the spring under pressure and the other very close to the spring under tension, would they observe identical spacetime curvature?

The curvature could indeed be different, as mentioned above at least inside the springs it would be different.

In the spherically symmetric case you must measure the curvature inside the system to detect any difference whatsoever. Without the spherical symmetry you probably get some differences very nearby the springs. But the math is very complicated so that is just a guess on my part.

Without doing the math I cannot do more than make a guess about the fields. However, in the above discussion I think that there is one point that we have been glossing over. That is that the energy density and the stress are different components of the stress energy tensor. So it is possible to have two sources with identical energy density but opposite stress. Geometrically that would be akin to having two vectors, one pointing at 89 deg and the other pointing at 91 deg. They wouldn’t be described as “increased” or “decreased”, but they are not “identical” either. The difference is more complicated.

 

[Tomas Vencl]

...However, in the above discussion I think that there is one point that we have been glossing over. That is that the energy density and the stress are different components of the stress energy tensor. So it is possible to have two sources with identical energy density but opposite stress. Geometrically that would be akin to having two vectors, one pointing at 89 deg and the other pointing at 91 deg. They wouldn’t be described as “increased” or “decreased”, but they are not “identical” either. The difference is more complicated.

Sorry for my poor English.
This is an interesting topic generally.
Regarding the mentioned above, let's imagine next example:
We have spherically symmetric thin shell of dust. The dust is collapsing by itself gravity (so we do not have static systém of course, this is a difference against the discussion so far). The observer is inside of the shell and is static and not at the centre of the shell. At some moment the shell is just above the observer. Until this moment the observer is in flat space.
After the shell passes by the observer he feels some gravity field. And the question is, how the gravity field at the observer position will change (if ?, it should not, I think) during next collapsing of the shell, and how the gravity field will change (if) after the dust collides and rising pressure will holds the central compact object static ? After the shell pases the observer, there is no energy flow by the observer so I think, that field keeps static.
I always thought, that the pressure in SET is only another form of energy, but from above discussion I feel, that it is not truth and pressure gives extra gravity beside its energy. Maybe I just do not understand the discussion or I am fully wrong. So the example would help me better understand.

 

[Dale]

how the gravity field at the observer position will change (if ?, it should not, I think)

You are correct. The curvature does not change from that point on, unless some mass or radiation escapes back out past the observer.

 

[1977ub]

There is always an offsetting decrease in the gravity elsewhere, right? You compress the spring with your muscles, and now you have less gravitational attraction?

 

[Dale]

There is always an offsetting decrease in the gravity elsewhere, right? You compress the spring with your muscles, and now you have less gravitational attraction?

Because it is a tensor with multiple components the curvature can be different without being either more or less.

 

[timmdeeg]

The curvature could indeed be different, as mentioned above at least inside the springs it would be different.

Is the increase of invariant mass of the spring the same if the work done to compress resp. to stretch the spring is the same? If so why would the curvature different?

 

[PeterDonis]

Is the increase of invariant mass of the spring the same if the work done to compress resp. to stretch the spring is the same?

If we are assuming that the start and end states are both static, then yes. Remember, as has been pointed out several times in this thread, that for a spring that is stretched or compressed to be static, it cannot be in isolation: there must be something else present that is holding it static, and that something else will have a stress that is opposite to the stress in the spring (stretched if spring is compressed, compressed if spring is stressed). The only meaningful invariant mass will be the invariant mass of the system as a whole, and that will be the same if the same amount of work is done on the system, whether the work stretches the spring (and compresses whatever is holding it static) or compresses the spring (and stretches whatever is holding it static).

If so why would the curvature different?

The spacetime curvature in the spring, considered in isolation, will obviously be different, because that's how you specified the problem. But you cannot deduce the invariant mass of the system from the spacetime curvature in the spring considered in isolation.

 

[timmdeeg]

But you cannot deduce the invariant mass of the system from the spacetime curvature in the spring considered in isolation.

Ok, thanks.

 

[Dale]

Is the increase of invariant mass of the spring the same if the work done to compress resp. to stretch the spring is the same?

Yes (with appropriate caveats)

If so why would the curvature different?

Because the curvature (a rank two tensor) depends on the stress energy tensor (a rank two tensor) and not only on the invariant mass (a scalar)

 

[timmdeeg]

Yes (with appropriate caveats)

Because the curvature (a rank two tensor) depends on the stress energy tensor (a rank two tensor) and not only on the invariant mass (a scalar)

Let me please control with your help, if I see it correctly.

1. Work is done to compress or stretch the spring which increases its stress energy.

2. The holding device is fixed at the spring which increases its stress energy too because it is now under pressure or tension.

3. The mass consisting of spring and device as a system has increased.

4. The curvature due to spring and device as a system doesn’t depend on whether the spring was compressed or stretched. Whereas the curvature inside the spring and inside the device does.

 

[Dale]

1. Work is done to compress or stretch the spring which increases its stress energy.

Not really. The stress energy tensor has many components. So you cannot just talk about it increasing or decreasing. It is a more complicated object.

If you put it in compression then the volume will reduce, so the energy density component will increase both due to the smaller volume and the work. There will also be a separate stress component that will increase.

If you put it in tension then the volume will increase, so the energy density component will decrease due to the larger volume and increase due to the work. Which effect is bigger may depend on the properties of the material but the energy density will remain positive. There will also be a separate stress component that will decrease and may become negative.

2. The holding device is fixed at the spring which increases its stress energy too because it is now under pressure or tension.

Similar comments as above.

3. The mass consisting of spring and device as a system has increased.

Yes, but the source of gravity is the stress energy tensor, not the mass. Note, this assumes that the energy came from outside the system which is not always the case.

4. The curvature due to spring and device as a system doesn’t depend on whether the spring was compressed or stretched. Whereas the curvature inside the spring and inside the device does

I don’t know. The math is beyond me and I don’t know of a reference. I already provided a reference that answers this question for spheres, and your statement is true for spheres. I cannot give a definitive answer for springs, but my intuition is that it is not true for all geometries

 

[pervect]

One of the big issues with generalizing the result to non-spherical geometries is that GR is nonlinear. This means that a spring plus a bar holding it is not quite the same solution as a the sum of a spring solution and a bar solution. So in the strong field, every case has to be analyzed separately, and you can't necessarily find the contribution of pieces of the system, because the system is not represented by the sum of its pieces.

If one introduces some additional assumptions of a weak field, one can get around this problem in principle. I have some guesses as to what hapens, but they're only guesses at this point.

So I'd rather stick with the spherical symmetric example. And I think it illustrates the main principles of what's going on quite well, I don't see the huge difference between it and the less-symmetrical and harder to analyze scenario you're asking about.

[add]
I'm not sure what the goal of the question is, if one is trying to come to grips with GR in the strong field realm, the nonlinearity of the theory is a really important feature, which means that one conceptually can't apply the technique of thinking about the spring separately from whatever is holding it stretched. One needs to analyze the entire system as a single, coherent, piece.

This may or may not be a big issue, depending on one's goals.

 

[timmdeeg]

Not really. The stress energy tensor has many components. So you cannot just talk about it increasing or decreasing. It is a more complicated object.

If you put it in compression then the volume will reduce, so the energy density component will increase both due to the smaller volume and the work. There will also be a separate stress component that will increase.

If you put it in tension then the volume will increase, so the energy density component will decrease due to the larger volume and increase due to the work. Which effect is bigger may depend on the properties of the material but the energy density will remain positive. There will also be a separate stress component that will decrease and may become negative.

I was perhaps naively thinking that stretching respectively compressing the spring changes mainly the shear stress components of the SET and the energy density and pressure components only marginally, because I expected the change of volume to be almost negligible.

Yes, but the source of gravity is the stress energy tensor, not the mass. Note, this assumes that the energy came from outside the system which is not always the case.

Given some stress from outside is put on an isolated mass (no bar etc. has to be considered) then as I understood it only the stress energy increases the invariant mass. Is this stress energy change represented by changes of the shear stress components of the SET? And is the change of the curvature due to changes of the shear stress components plus eventual changes of the energy density and pressure components?

 

[Dale]

I expected the change of volume to be almost negligible.

We are talking about relativistic effects on the gravity of a spring. The whole discussion is negligible. I personally would expect that the change in volume would be the largest of these negligible effects.

Given some stress from outside is put on an isolated mass

I don’t understand the question. If there is stress from outside, then how can it be isolated?

And is the change of the curvature due to changes of the shear stress components plus eventual changes of the energy density and pressure components?

Yes. The stress energy tensor includes components for shear stress, pressure, energy density, and momentum density. All contribute to gravity.

 

[timmdeeg]

I don’t understand the question. If there is stress from outside, then how can it be isolated?

You are right, it can't, I've missed this point.

Thanks for your answers.

 

[John K Clark]

The pressure inside a Neutron Star must be enormous and there is no tension factor to counteract it that I can see, so if the stress energy tensor includes components for shear stress, pressure, energy density, and momentum density and if I want to calculate the gravitational field far from the Neutron Star, does that mean knowing just the mass and spin of the star would be insufficient to make that calculation even if I assume the star is spherically symmetrical? To figure out the spacetime curvature would I need to know the details of the stars interior?

John K Clark