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## Main Question or Discussion Point

In a previous series of articles, I posed the question “Does Gravity Gravitate?” and explained how, depending on how you interpreted the terms “gravity” and “gravitate”, one could answer the question either way, yes or no. This article will treat its title question in a similar fashion. :-)
To be sure, this case is a bit simpler than the previous one, because there is only one term that can be given more than one meaning. The term “pressure” is clear: it means the spatial diagonal components of the stress-energy tensor. (More precisely, it means those components in the rest frame of a fluid, but we won’t delve into such technical details very much in this article.) The ambiguity is in the term “source of gravity”. What does that term mean?
The standard answer to that question in GR is that the source of gravity is the stress-energy tensor, the thing that appears on the RHS of the Einstein Field Equation. So...

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Jonathan Scott
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That's an excellent article as it stands (thanks, Peter), but as some people will know I think the usual story misses a couple of interesting points.

Firstly, the pressure integral (negative normal stresses) for a static system is equal and opposite to the potential energy even in Newtonian gravity. For a trivial example, consider a rod of length ##r## and cross-section ##a## holding apart masses ##m_1## and ##m_2## against gravity. The force everywhere in the rod is ##G m_1 m_2 / r^2## so the pressure is that divided by ##a##, and the integral over the cross-section ##a## and length ##r## of the rod is then simply ##G m_1 m_2 / r##, equal to the potential energy. This means that the Komar mass expression has a Newtonian equivalent; it is equivalent to calculating the potential energy by applying the (negative) Newtonian potential everywhere, which counts both sides of each interacting pair of masses so it counts the potential energy twice, but then adding back in the pressure term, so overall the energy of the static system is reduced by its internal potential energy. However, it is very clear that in the Newtonian situation the pressure is equal to the potential energy but does not contain the potential energy, as for example if the rod is removed, the potential energy is initially unchanged.

Secondly, in a dynamic system, the pressure integral is not a conserved quantity, which suggests that the apparent source strength could vary temporarily, affecting the distant field, which seems implausible as in the same situation the Newtonian potential energy remains constant. This paradox is well known; physicist Richard Tolman commented on it in a 1934 relativity text book. I discussed this with various physicists, including a brief email conversation with Kenneth Nordtvedt, who confirmed that as far as he knows, the fully dynamic situation is still unresolved, but he suggested that changes in the metric mean that there are dynamic changes in the relationship between points in physical space and in coordinate space, which effectively create additional acceleration terms, and these could compensate for the changes in the conventional source terms, although on the other hand if they do not, and the distant field can indeed be modulated by pressure changes, that would be very interesting!

martinbn
The article is not Baez's. Its Baez and Bunn's.

Dale
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I think that the “yes” answer is the only reasonable one. While it is true that in a spherically symmetric static spacetime the pressure part of the source (stress energy tensor) cancels out with other parts, that is best understood as an artifact of that specific configuration rather than support for a general claim of “no”.

Similarly we can find distributions of current or charge where EM field contributions cancel out in certain regions. E.g. a spherically expanding shell of charge has current but the external EM field depends only on the charge and not on the current. That does not imply a general claim that currents are not a source of the EM field. The general claim that the sources of the EM field are charges and currents stems directly from Maxwell’s equations irrespective of peculiarities of specific solutions.

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Dale
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the pressure integral is not a conserved quantity, which suggests that the apparent source strength could be vary temporarily, affecting the distant field,
This is a completely erroneous suggestion. Similarly in EM the current density integral is not a conserved quantity but that in no way implies that “the apparent source strength could be vary temporarily, affecting the distant field”. The field equations prohibit that distant effect at faster than c, regardless of changes in the sources.

Jonathan Scott
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This is a completely erroneous suggestion. Similarly in EM the current density integral is not a conserved quantity but that in no way implies that “the apparent source strength could be vary temporarily, affecting the distant field”. The field equations prohibit that distant effect at faster than c, regardless of changes in the sources.
There's no suggestion of any instant effect, just an effect propagating at ##c## as usual which eventually affects the distant field, for example an orbiting body. This is what J.Ehlers et al say about it in the published paper "Pressure as a source of gravity", PHYSICAL REVIEW D 72, 124003 (2005) (also available as an Arxiv preprint):
It was Richard Tolman [5] who studied universes filled with radiation and began to wonder about the consequences of the 3p-term for gravitational theory. The following scenario is known as Tolman’s paradox: A static spherical box has been filled with a gravitating substance of a given mass. If this substance undergoes an internal transformation (e.g. matter and antimatter turning into radiation) raising the pressure, the active mass in the box would change because of the 3p-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].
The reference [5] is to Tolman's 1934 book "Relativity, Thermodynamics and Cosmology", which is available online, although it does not appear to me that Tolman necessarily expressed the paradox in this form. Ehlers et al then cover a variety static cases, showing that the pressure integral is as expected, but I have not found anything in the literature about solutions for cases involving dynamic changes.

Dale
A common system where potential energy is significant compared to rest mass is a neutron star.
A body emits gravitational waves when it has changing quadrupole or higher moments. A centrally symmetric mass distribution does not emit waves.
What does the distant field of a neutron star look like when it is undergoing centrally symmetric pulsations?

Jonathan Scott
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What does the distant field of a neutron star look like when it is undergoing centrally symmetric pulsations?
Birkhoff's theorem says it is constant.

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The article is not Baez's. Its Baez and Bunn's.
Yes, good point. I have updated the article to credit both authors.

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the pressure integral (negative normal stresses) for a static system is equal and opposite to the potential energy even in Newtonian gravity
Do you know of a reference that gives a general proof of this for the GR case? The only exact solution I have been able to find a proof for is the (highly unrealistic) constant density spherically symmetric case I refer to in the article. But as you note, it should be valid for any stationary spacetime (since those are the spacetimes in which "potential energy" has a well-defined meaning).

Jonathan Scott
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The pressure term is well-behaved for a static or quasi-stationary system. Over any extended time scale, the pressure integral for a stable system (including static, cyclic or slowly changing) is on average equal (but opposite) to the gravitational potential energy of the configuration. This is basically because the overall force (pressure times area) across any plane must be zero on average, so mechanical forces exactly balance gravitational forces in all directions. This means for example that if the pressure inside a balloon increases, then the tension in the skin (giving a negative force) must increase to match.

However, a spherically symmetrical pulsation can involve sudden short-term changes of pressure while the external field remains constant (according to Birkhoff's theorem). Local energy and momentum are conserved, at least over small regions and short time scales, so they cannot change abruptly to compensate for the pressure term.

This illustrates (at least for the spherically symmetrical situation) that the "effective" source strength for a dynamic system must also take into account something else in addition to energy, momentum and pressure, in order for the external field to remain unaffected. Ken Nordtvedt suggested that it might relate to the changing mapping between local physical space and coordinate space in such situations, which seems plausible to me, and I think something similar is already taken into account in calculations of the detailed PPN equations of motion for different gravity theories.

Dale
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the fully dynamic situation is still unresolved
In the sense that we do not have exact analytical solutions for the fully dynamic situation, yes, I certainly agree. But AFAIK, gravitational collapses of different kinds of objects have been numerically simulated, and these simulations have not shown any effects on distant fields due to the sudden pressure changes involved. I'll see if I can find a reasonably recent reference.

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Jonathan Scott
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Do you know of a reference that gives a general proof of this for the GR case? The only exact solution I have been able to find a proof for is the (highly unrealistic) constant density spherically symmetric case I refer to in the article. But as you note, it should be valid for any stationary spacetime (since those are the spacetimes in which "potential energy" has a well-defined meaning).
You mean that a proof that the GR pressure integral is equivalent to the Newtonian potential energy? I've not seen such a thing but once I understood the Newtonian model (adding up force vector through each plane due to each pair of constituent particles) it seemed clear to me that the Komar mass expression is exactly equivalent so it would give the same result for any stationary system at least at the weak approximation level, regardless of any symmetry requirement.

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You mean that a proof that the GR pressure integral is equivalent to the Newtonian potential energy?
No, I mean a treatment of the same general type of scenario but using GR in a general stationary spacetime instead of Newtonian gravity. The Komar mass might be general enough, since it doesn't even require the stress-energy tensor to be that of a perfect fluid.

Jonathan Scott
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In the sense that we do not have exact analytical solutions for the fully dynamic situation, yes, I certainly agree. But AFAIK, gravitational collapses of different kinds of objects have been numerically simulated, and these simulations have not shown any effects on distant fields due to the sudden pressure changes involved. I'll see if I can find a reasonably recent reference.
Yes, this is what I would expect, not only from Birkhoff's theorem but from the more general Newtonian equivalent, where potential energy only depends on the energy configuration and not on pressure. I'm not expecting the field to vary; I'm just assuming that there's some additional bit of maths needed to cover the dynamic situation where the pressure varies.

Dale
Dale
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There's no suggestion of any instant effect, just an effect propagating at ccc as usual which eventually affects the distant field
Thank you for the clarification. That is sensible.

Dale
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This illustrates (at least for the spherically symmetrical situation) that the "effective" source strength for a dynamic system must also take into account something else in addition to energy, momentum and pressure, in order for the external field to remain unaffected. Ken Nordtvedt suggested that it might relate to the changing mapping between local physical space and coordinate space in such situations
Your “must” claim here is very doubtful to me. Please post the Nordtvedt reference that supports this claim.

Personally, I do not see why any other sources than those already in the stress energy tensor should be posited. Again, a spherical distribution of charge displays this same behavior without any inference that additional sources are needed.

Jonathan Scott
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If energy and momentum are unchanged on a time scale where the average pressure changes abruptly (for example shortly after an explosion at the center of a large empty spherical shell, long before the time where the exploding material reaches the shell and establishes a new equilibrium) then I cannot see how it could be possible to attribute the source strength only to the quantities of energy, momentum and pressure, as only one of them has changed but the effective source strength has not. That's why I said "must". If you think there is an alternative, please explain.

What Ken Nordtvedt pointed out to me (in an email) was the obvious (at least to him) fact that when the material starts moving, the metric will start changing, which dynamically changes the mapping between points in the original local static physical space and coordinate space. That creates an effective relative acceleration between points in physical space and coordinate space.

Clearly that all depends on the metric, and the metric depends on the terms in the stress-energy tensor, so there is no outside influence, but it would seem that the effective source strength must include some terms related to accelerating sources or similar. Any temporarily unbalanced pressure will result in net forces, causing mass acceleration, so if the accelerated masses caused a compensating effect for the pressure change there would be no problem. A similar effect is already known in the form of the theory of gravitational linear frame-dragging (the induction effect of an accelerating source, creating an effective additional field equal to ##Gm/rc^2## times the acceleration vector), so this seems quite plausible.

Dale
Dale
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I cannot see how it could be possible to attribute the source strength only to the quantities of energy, momentum and pressure, as only one of them has changed but the effective source strength has not. That's why I said "must".
Your inability to see an alternative does not constitute a proof that something must be true. There are many things that I cannot imagine were false, but I recognize that does not constitute a proof of their truth. This is unacceptable, and you are well aware of this fact.

If you think there is an alternative, please explain.
That is not the way this works. If YOU make a claim here then YOU are expected to have professional scientific sources supporting it. It is not incumbent on the rest of us to disprove YOUR claim and you don’t get to keep spouting it without such proof.

However, in this case I at least have an explanation why I am so skeptical of your claim. This explanation is not a disproof of your claim but rather an explanation why I don’t accept your inability to see an alternative as even remotely convincing:

Everything that you are saying about pressure in the EFE could be said about current in Maxwell’s equations. You could have a spherically symmetric static ball of charge which on a very short time scale had a sudden change in current (e.g. exploding ball of charge). Current is a source of the EM field, but that sudden change in current would not lead to any change in the external field. Yet there is no need for any compensatory effect in the charge, nor is there any need for some mysterious “something else” besides charge and current. It is simply the spherical shape that causes this effect. Each radially moving particle of charge produces a magnetic field transverse to the motion, and that field is canceled out by its neighboring particle’s fields.

Since I can see that the reasoning behind your claim fails for EM it is pretty clear that if your claim is valid for gravitation then it requires a rigorous proof based on the EFE.

What Ken Nordtvedt pointed out to me (in an email)
Personal communications with third parties and personal speculation are not valid sources on PF. Please cease posting such unsubstantiated claims

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the obvious (at least to him) fact that when the material starts moving, the metric will start changing, which dynamically changes the mapping between points in the original local static physical space and coordinate space. That creates an effective relative acceleration between points in physical space and coordinate space.
If it's that obvious, someone would presumably have written a paper on it. If no one has, then maybe Nordtvedt should.

Jonathan Scott
Gold Member
If it's that obvious, someone would presumably have written a paper on it. If no one has, then maybe Nordtvedt should.
I think that is already assumed as part of the general PPN equation of motion.

Dale
Jonathan Scott
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If masses move, the metric changes, which changes local scale factors between coordinate and physical space. Depending on how the coordinate system is defined, a reference point can remain fixed, but other points which were at a fixed proper distance from one another may end up at a different coordinate distance from one another as a result of the moves.

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I think that is already assumed as part of the general PPN equation of motion.
Then there should be a reference on the PPN equation of motion that shows it.

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If masses move, the metric changes, which changes local scale factors between coordinate and physical space. Depending on how the coordinate system is defined, a reference point can remain fixed, but other points which were at a fixed proper distance from one another may end up at a different coordinate distance from one another as a result of the moves.
On its face, this seems to be arguing that coordinates can cause physical effects. Which is obviously wrong.

Or, alternatively, this could be arguing that there is in fact no physical effect at all going on--that the actual physical source of gravity does not change at all in the scenarios you describe--but that coordinate effects make it seem like something is changing, until we correctly take into account other coordinate effects that cancel that seeming change out. Which would be valid reasoning, but would also undermine your original claim.