Stress and pressure as gravitational source

[Jonathan Scott]

Going back to the start of this, for the Newtonian situation, even for the most complex configuration, treating gravity as a force, the volume integral of the pressure over a system in equilibrium is exactly equal and opposite to the potential energy of that configuration. If a light rigid support is moved aside or broken, the pressure in that support drops to zero and if there is no other support to take up the pressure, the other parts start accelerating towards one another. On a time scale where the change has just finished propagating, the location and velocity of each part can be arbitrarily close to its previous value, but the pressure in the support has vanished.

As far as I can see, the GR situation is essentially identical at this high level, at least in a weak approximation case, and the subject of this thread is the fact that although all terms in the SET are involved in the gravitational effect, it seems physically implausible that a temporary sudden change in the pressure should affect the external field. PAllen has helpfully related this to a similar paradox noted by Richard Tolman, which is addressed by the paper he referenced, by Ehlers and others.

It appears that this paper asserts that the pressure terms cancel not to the potential energy (as in the Newtonian case and as is usually assumed in the Komar mass formula) but rather to zero, and there is some other term (which I do not yet understand) which accounts for the potential energy correction to the active mass. I find this very interesting but it seems to conflict with what I had previously been told about pressure and gravity in GR, and I would like to understand what differences this makes in terms of the Newtonian model. This paper unfortunately uses some mathematical terminology that I'm not familiar with, so if anyone can shed any light on its interpretation I'd be grateful.

 

[Dale]

This is not helpful and not very polite. The subject has already been discussed.

I am sorry that you think it is impolite, but you are wrong that it is not helpful. This is the key misconception that you have here. I don't know how you think that you will be able to resolve your confusion without listening to the single key point that has been repeated by multiple experts. If you want to actually learn physics then you must keep at the parts that you don't understand until you overcome the challenge, not simply dismiss it as "already been discussed".

There is no law of conservation of pressure, and I've been familiar with continuity equations for something like 45 years.

So what? Look at EM. There is also no law of conservation of current density and conservation of charge density. And yet those are the sources of the EM field.

By your same flawed reasoning current density could not be a source of the EM field because it can "flicker in and out of existence". Current density, one of the components of the source of EM, has the same relationship to the EM fields as pressure does to the gravitational fields.

It is the continuous quantities that act as the sources of fields, not the conserved quantities. This holds in EM as well as in GR.

 

[Jonathan Scott]

I am sorry that you think it is impolite, but you are wrong that it is not helpful. This is the key misconception that you have here. I don't know how you think that you will be able to resolve your confusion without listening to the single key point that has been repeated by multiple experts. If you want to actually learn physics then you must keep at the parts that you don't understand until you overcome the challenge, not simply dismiss it as "already been discussed".

I don't really want to get side-tracked into this, because I'm comfortable that I do understand this aspect and I'm sure it's not really relevant in this case. However, if you insist that there's something that prevents the pressure from being able to just drop to zero in this case (e.g. when a pair of rods holding two masses apart along the x-axis are unpinned and allowed to start falling past one another), or something that happens elsewhere in the SET, please can you be more specific about what you think it is?

 

[Dale]

if you insist that there's something that prevents the pressure from being able to just drop to zero in this case

Pressure certainly can drop to zero. So what? That in no way prevents it from being a source of gravity.

The same thing can be said of current density. You seem to not understand that it is the continuous quantities which act as sources, not the conserved quantities.

 

[snorkack]

Working through the example of current density: say you have a circular wire carrying a current. Then the current is constant through the circumference of the wire - but current density is not, because it increases where the wire cross-section narrows. The current is a source of magnetic field, but not electrostatic.

Now cut the wire at one point.
The current density at that point instantly becomes zero. The current around the rest of the ring is not at first changed. As the charge builds up at the points of the cut wire, though, the electrostatic field grows, and the current (and its density) decrease with time till they become zero. And then reverse.

However, note that between the points of the cut wire, displacement current will flow. And counting the displacement current, the current is still constant all around circuit, nor is it instantly altered by cutting the circuit.

The magnetic field is generated equally by real current and displacement current.

Does pressure have some similar substitute?

 

[Jonathan Scott]

Imagine that there is a device that can either lock the rods together or unlock them to allow them to move past each other. When the device is locked, the pressure appears, and when it is unlocked, the pressure disappears. If the rods are very rigid, this allows the pressure to be turned on and off rapidly.

As Richard Tolman pointed out in his 1934 paper, which in his case related to a sudden pressure increase (for an explosion inside a star) rather than decrease, it seems implausible that this should be able to affect the external field away from the device, for example changing the orbit of a nearby satellite, and indeed for cases covered by Birkhoff's theorem, it should be impossible. The paper by Ehlers and others, referenced by PAllen, is a response to this paradox. However, their answer very interestingly seems to contradict a well-established idea that the pressure term effectively contributes to the active mass of the system, as part of the effective adjustment to match the Newtonian potential energy demonstrated for static systems in the Komar mass formula.

So I clearly need to unlearn something here, but I want it to be the right something!

 

[Dale]

The magnetic field is generated equally by real current and displacement current.

Does pressure have some similar substitute?

Sure. The displacement current does not show up directly in the covariant formulation of EM. It shows up only when you expand out the field tensor derivatives into its components. Similarly, if you were to expand out the Einstein tensor for the EFE in terms of the metric components you would get lots of terms, some of which would be analogous to the displacement current. Since it is a rank 2 tensor and involves 2nd derivatives of the field, there will be many more such terms for gravity than for EM.

 

[Dale]

If the rods are very rigid, this allows the pressure to be turned on and off rapidly.

You can also turn current density on and off very rapidly.

You are focusing on something that is true but irrelevant. The fact that something can be turned on and off in no way implies that it cannot be a component of the source of a field.

 

[Jonathan Scott]

You can also turn current density on and off very rapidly.

You are focusing on something that is true but irrelevant. The fact that something can be turned on and off in no way implies that it cannot be a source of a field.

Firstly, the very close analogy with Newtonian gravity in the weak approximation makes it implausible that some internal change which does not change the location of energy or momentum should have an immediate effect on the external field.

Secondly, we can set up an approximately spherically symmetrical version of this, where a shell is supported by a layer of struts which is suddenly allowed to collapse, and in that case Birkhoff's theorem appears to say that it is impossible for the external field to change, unless of course this somehow violates the assumptions of that theorem.

The RHS of the EFE is simply the SET (ignoring any cosmological constant) so if one component is changed to zero over part of a structure while all other components for that part and other parts remain unchanged (at least initially, but after any energy and displacement stored in the compression has dissipated) it seems that would necessarily affect the field.

The Ehlers paper seems to find this worth investigating, and suggests a new answer. However, I'm having difficulty both with the maths and with the implications of the answer, which contradict other ideas which seem to be well-established but may well be wrong.

 

[Dale]

Firstly, the very close analogy with Newtonian gravity in the weak approximation makes it implausible that some internal change which does not change the location of energy or momentum should have an immediate effect on the external field.

By this logic the very close analogy with Coulomb's law in the electrostatic approximation makes it implausible that some internal change which does not change the location of charge should have an immediate effect on the external EM field. So according to your logic, it would seem implausible that the EM field should be affected by turning a current density on or off.

You may wish to reconsider your logic. Since you are specifically interested in the gravitational effects of pressure it makes as little sense to rely on Newtonian gravity as it does to rely on Coulomb's law to understand the electromagnetic effects of current density.

if one component is changed to zero over part of a structure while all other components for that part and other parts remain unchanged (at least initially, but after any energy and displacement stored in the compression has dissipated) it seems that would necessarily affect the field.

You seem to be ignoring the continuity equation again. Recall that pressure is part of the space-space part of the SET. So it is variations in pressure over space (not time) which are important to continuity. Your discussion about "that part and other parts" is rather unclear, but I suspect that you are neglecting the continuity. The only way for the pressure to change over space is for the momentum density to change over time or for the shear stress to also change over space in a compensatory fashion.

Pressure can change over time without disturbing the SET continuity, but that in no way implies that it is not a source of gravity any more than similar statements about current density.

 

[Dale]

Hi Jonathan Scott,

I need to tone my responses down. I don't know why this question is irritating to me. It is not like it is a question that comes up often and that I am simply exasperated by. I will try to not respond until I can do so more politely.

I am afraid that I still don't understand what you are concerned about. You seem to agree that pressure is part of the SET and you seem to agree that the SET is the source of gravity in GR. The fact that some components of the SET can be turned on and off seems unsurprising to me. And certainly the same can be said of other fields so even if you consider it surprising it seems common and uncontroversial.

Sources can be turned on and off (subject to their continuity equations). That in no way disqualifies them from being sources. Why would you have different expectations for gravity vs. EM in this regard?

 

[Jonathan Scott]

You seem to be ignoring the continuity equation again. Recall that pressure is part of the space-space part of the SET. So it is variations in pressure over space (not time) which are important to continuity. Your discussion about "that part and other parts" is rather unclear, but I suspect that you are neglecting the continuity. The only way for the pressure to change over space is for the momentum density to change over time or for the shear stress to also change over space in a compensatory fashion.

This is how I see it:

If we ignore the initial effect of the decompression wave, which can be arbitrarily small for an arbitrarily rigid rod, then as far as I can see, there is no change in any other term of the SET apart from the pressure. There is a step change in the rate of change of the momentum with respect to time which occurs at the same time as the step change in the pressure with respect to space passes through the relevant location. These two terms match to preserve continuity.

 

[PAllen]

Hi Jonathan Scott,

I need to tone my responses down. I don't know why this question is irritating to me. It is not like it is a question that comes up often and that I am simply exasperated by. I will try to not respond until I can do so more politely.

I am afraid that I still don't understand what you are concerned about. You seem to agree that pressure is part of the SET and you seem to agree that the SET is the source of gravity in GR. The fact that some components of the SET can be turned on and off seems irrelevant to me, and certainly the same can be said of other fields so even if you consider it relevant it seems unsurprising and uncontroversial.

The confusion is that within a volume element of the Komar mass formula, pressure and total energy density show up explicitly (and nothing else does, with the right coordinate choice). This seems to suggest that if pressure changes rapidly throughout a massive body (no need even to worry about speed of light limit - you could arrange a state change that independently happens everywhere throughout the body at the same time in the coordinates used), and you quickly reach a new equilibrium, that the Komar mass has suddenly changed (because the energy density in each volume element can't suddenly change). This really is a seemingly paradoxical situation that was most prominently raised by Tolman long ago. Note that Komar mass is expected to be what determines orbits of test bodies.

The Ehler's paper argues that while pressure must be formally treated as a source, in (all? most?) equilibrium situations, it ends up having no net contribution as Komar mass is integrated over the body due to other constraints that are required for equilibrium. This is, to me, as well, a somewhat surprising way to resolve the Tolman (and similar) paradoxes.

 

[Dale]

The confusion is that within a volume element of the Komar mass formula, pressure and total energy density show up explicitly (and nothing else does, with the right coordinate choice). This seems to suggest that if pressure changes rapidly throughout a massive body (no need even to worry about speed of light limit - you could arrange a state change that independently happens everywhere throughout the body at the same time in the coordinates used), and you quickly reach a new equilibrium, that the Komar mass has suddenly changed (because the energy density in each volume element can't suddenly change).

OK, but I have a hard time getting worked up about that. The Komar mass is not something fundamental in GR, it is not the source of the gravitational field. It is just a useful parameter for characterizing certain spacetimes.

Also, the Komar mass is specifically useful to characterize stationary spacetimes. If the pressure is changing significantly over time then the spacetime is not stationary by definition. Why worry about some problem with some special-case parameter that doesn't even apply to the case being discussed?

 

[PAllen]

OK, but I have a hard time getting worked up about that. The Komar mass is not something fundamental in GR, it is not the source of the gravitational field. It is just a useful parameter for characterizing certain spacetimes.

Also, the Komar mass is specifically useful to characterize stationary spacetimes. If the pressure is changing significantly over time then the spacetime is not stationary by definition. Why worry about some problem with some special-case parameter that doesn't even apply to the case being discussed?

Because you can go quickly from one equilibrium to another, and the Komar mass should be valid at both equilibria. If it has changed substantially, this would mean the orbit of a test body shifts due to internal changes in a massive body that do not result in any flow of radiation or mass to/from the body.

 

[Dale]

The Komar mass should be valid at either stationary spacetime, but I don't know why you would think that it would be valid in a non stationary spacetime that transitions between the two. The Komar mass simply doesn't exist in the spacetime that you are describing.

My solution: don't use the Komar mass for non-stationary spacetimes since they violate the basic assumptions required for the Komar mass.

 

[Jonathan Scott]

Because you can go quickly from one equilibrium to another, and the Komar mass should be valid at both equilibria. If it has changed substantially, this would mean the orbit of a test body shifts due to internal changes in a massive body that do not result in any flow of radiation or mass to/from the body.

I should point out that I didn't actually have a direct problem with the equilibrium case. I had already worked out that for semi-Newtonian gravity (treating gravity as a force) the integral of the pressure for any equilibrium configuration is always equal and opposite to the potential energy of the configuration. (Is this a widely-known fact? It was something I discovered for myself only after looking for the Newtonian equivalent of the Komar mass pressure term). Assuming the weak field approximation, the effective energy as decreased by its own potential (time dilation) is equal to the local energy of the components minus twice the potential energy of the configuration, so the total of the two parts is exactly equivalent to the Newtonian total energy. What bothered me is what happened in the non-equilibrium case when the pressure was unbalanced.

However, the Ehlers paper suggests that it doesn't work the way I thought!

 

[Dale]

If we ignore the initial effect of the decompression wave, which can be arbitrarily small for an arbitrarily rigid rod, then as far as I can see, there is no change in any other term of the SET apart from the pressure. There is a step change in the rate of change of the momentum with respect to time which occurs at the same time as the step change in the pressure with respect to space passes through the relevant location. These two terms match to preserve continuity.

Yes. I agree. For simplicity if we consider a box of pressurized gas then at, say the -x face of the box there is a sudden change change in pressure as a function of x (dp/dx > 0). If the box is rigid then the continuity of the SET requires that there be a change in the shear stress in the wall of the box along y and z to compensate for the change in the pressure along x. If the box cannot support a shear stress, then there must be a temporal change in the x momentum density to compensate for the spatial change in pressure along x. This is what forces the gas to spew out until pressure is spatially equalized.

Of course, you do have to be careful. As a pressure changes in time, it generally disturbs the geometry in space also. Depending on the details a pressure with $\partial_t$ is likely be associated with alterations to $\partial_x$ also. But I am sure that some clever person could come up with a pressure that has a nonzero $\partial_t$ but a zero $\partial_x$ throughout the spacetime.

The question remains, given all that, what is the problem? Why should any of that alter the fact that the SET is the source of gravity in GR and pressure is a component of the SET?

 

[PAllen]

I should point out that I didn't actually have a direct problem with the equilibrium case. I had already worked out that for semi-Newtonian gravity (treating gravity as a force) the integral of the pressure for any equilibrium configuration is always equal and opposite to the potential energy of the configuration. (Is this a widely-known fact? It was something I discovered for myself only after looking for the Newtonian equivalent of the Komar mass pressure term). Assuming the weak field approximation, the effective energy as decreased by its own potential (time dilation) is equal to the local energy of the components minus twice the potential energy of the configuration, so the total of the two parts is exactly equivalent to the Newtonian total energy. What bothered me is what happened in the non-equilibrium case when the pressure was unbalanced.

However, the Ehlers paper suggests that it doesn't work the way I thought!

Can you describe what you mean in reference to the following scenario:

I have box of hydrogen and oxygen, at some pressure. 'Somehow' spontaneous combustion occurs throughout the box. The box is able to prevent any radiation from escaping, and itself has negligible mass and thickness (though it does need to have stress as needed for the stated behavior). Comparing the two equlibria, total energy density is the same in both, pressure is radically different (only water vaper pressure after). So, how is pressure equal and opposite potential energy in both cases? Do you include chemical potential energy?

Ehler's answer is that even though pressure is a source term in the SET, integrated for both equlibria, the pressure does not contribute externally measured gravitational mass in either case. I agree with Dalespam, that Komar mass simply doesn't apply in between. The Bondi mass formulation (which equals Komar mass for stationary cases) proves (non-locally - by integrating to null infinity in AF spacetime) that the mass doesn't change in between as well. Of course, for spherical case, Birkhoff applies, but we need not assume a sphere - I could have a cubical box. Conservation of Bondi mass when there is no radiation says if we meet the problem statement (no radiation) there will be no mass change, even in radically non-stationary evolution.

Lacking, so far in this discussion, is a quasi-local explanation of constancy in between, in terms of SET components.

 

[Jonathan Scott]

For the Newtonian case, you simply need to include the (negative) pressure in the walls of the container; for equilibrium the overall force (including gravitational) must be zero through any plane. Note that changing the internal configuration can change the potential energy, but if there is no external change in energy, the total is still the same.

 

[Jonathan Scott]

I guess from the fact that anyone is asking, the fact that the volume integral of the pressure in the Newtonian equilibrium case is equal to the potential energy isn't that well known. Just consider the force between any two masses in the configuration and you get the usual $-G m_1 m_2 / r^2$, so for equilibrium the integral of the perpendicular pressure over any plane slice of the system in between the masses balances that force (regardless of any local higher or lower pressure areas) and integrating that over the distance $r$ between them gives $G m_1 m_2 / r$, the potential energy of that pair of masses (but positive, as this is the pressure resisting the force). The forces add up as vectors, and the overall integral is equal and opposite to the potential energy.

 

[Jonathan Scott]

I should probably have made it clear that I'm talking about the three separate pressure terms for each perpendicular plane (like those which appear in the SET).

Edit: That means for example that if the perpendicular pressure in a small volume is p in each direction, the result is 3p times the volume, as usual for the Komar mass.

 

[PAllen]

I guess, for me, the upshot (for now) is as follows:

1) In equilibrium, you can use Komar mass, but as shown by Ehlers, the factors that produce equilibrium against pressure result that the 3p term of Komar mass does not actually add anything to the mass over the body as a whole.

2) Out of equilibrium, even instantly, you just can't use Komar mass formula at all. Bondi mass covers the whole evolution and is equal to Komar mass for equilbrium and conserved up to radiation (including GW) - but only defined for AF spacetimes. This guarantees that if you compute Komar at an equilibrium, and radiation is not significant, then Bondi mass equals that Komar mass during non-equlibrium. During non-equlibrium, you simply can't talk about the 3p term of Komar mass. If you want a uniform approach to computation over the evolution, you have to use Bondi mass throughout.

3) Integrating pressure by itself over a volume and expecting this integral to be a separately detectable component of gravitational mass at a distance is simply too simplistic an expectation.

 

[Jonathan Scott]

Earlier in this thread I explained what happens to SET terms when pressure drops locally to zero. However, I now realize there's a problem with the explanation, in that the gravitational force is treated as external which means that momentum is not conserved, which means that the divergence of the SET can't be zero. I know it's the "covariant" divergence, but I need to understand what that implies in terms of gravitational effects. I'll think about it for a bit and if I can't resolve it I'll start a new thread.

 

[Dale]

Earlier in this thread I explained what happens to SET terms when pressure drops locally to zero. However, I now realize there's a problem with the explanation, in that the gravitational force is treated as external which means that momentum is not conserved, which means that the divergence of the SET can't be zero. I know it's the "covariant" divergence, but I need to understand what that implies in terms of gravitational effects.

For me the easiest way is to look locally at a momentarily comoving free-fall frame.

So, for example, consider a non-rotating spherical fluid planet in static equilibrium. It is a non-flowing fluid, so the shear stress components are always 0. There is no pressure gradient in $\theta$ or $\phi$, but there is a pressure gradient in $r$. Given the no-shear condition, the negative pressure gradient in r would imply that there is an increase in momentum density wrt time, per the continuity. Since it is static, that seems to be a contradiction.

However, as you say, it is a covariant derivative, not an ordinary one. So, looking at a locally free-falling frame we see that the fluid is indeed accelerating in the positive r direction and therefore there is, in fact, an increase in momentum density in the r direction which leads to the 0 covariant divergence.