Causal mechanism for gravity

[przyk]

You agree that the energy must reside somewhere...but velocity is purely relative.

So is the energy. Like I said, energy is not the same as the full stress-energy tensor. It is just one aspect of it. More precisely, the local energy density is one of the ten components of the stress-energy tensor field if you decompose it in an inertial coordinate system.

Considering the energy and ignoring everything else would be like taking Newtonian mechanics and considering only the $$x$$-component of the force, and then wondering how Newtonian mechanics can respect rotational symmetry when the $$x$$-component of force depends on the coordinate system.

What if the Earth was object A and object B was an interstellar mass from deep space, traveling at a substantial fraction of c? Would the stress-energy tensor reflect this energy in us, particularly prior to our knowledge and measurement of object B?

No, it just depends on the reference frame. In something like the rest frame of object B then yes, Earth has high energy and nonzero momentum. In the rest frame of Earth, it is interstellar object B that has high energy and momentum and Earth just has its normal rest energy. But the full stress-energy tensor field, which is what the gravitational field is coupled to, is covariant and is the same tensor field in either case.

Unfortunately, I did not get to electromagnetism. I took a single Physics course at UNL before I decided that juggling three kids and a mortgage and night classes was too difficult.

Okay. That makes more sense.

I will definitely look at the EM analogy though; my immediate response is that the paradox won't exist in EM because we don't have an analogous blackhole to contend with. Remove the blackhole from GR and any contradictions go away -- the location of energy is irrelevant.

Huh? Even regardless of black holes, the gravitational field is not the same everywhere. It tends to be stronger near where you have matter and it tends to be weaker far away from matter. It also depends on how that matter is distributed and what it is doing and how that is changing over time. E.g., the gravitational field around a rotating mass is not the same as it is around a nonrotating one. The gravitational field around two objects orbiting each other is different still; for example, the objects emit gravitational waves in that case. Obviously GR has to be able to model and predict this. There is nothing special about black holes here.

Object A claims that object B has momentum energy; object B claims that object A has it; the remote observer C could claim that the A-B "system" has such-and-such energy. This would hold true even if A and B were in a tight, mutual orbit, with sufficient angular momentum to traditionally predict the black hole. According to A, they are merely "free-falling" with B buzzing around at terrible speeds. It doesn't really matter because all of the energy effects would be observed as predicted for all parties. Relative energy would have relative energy-related effects that are on a continuous, sliding scale devoid of one-way thresholds.

Again, the gravitational field does not in general depend only on the "energy" or the "momentum" but on the stress-energy tensor field, which is a covariant object. When you see simple thresholds like "you get a gravitational field if the energy is greater than $$M c^{2}$$" it is not a general thing in GR. It is a different type of thing where a human theorist has come along and worked out what happens in a specific situation, under a lot of constraints, often working in a particular reference frame from the very start (so the result is not expressed in a frame-independent way but specifically for whatever frame the theorist is using) and often choosing to introduce simplifying approximations (e.g., pretending a few nearby masses are the same as one aggregate mass, even if that is not exactly true).

Black holes still make no difference here. The gravitational field is described by a manifold, which does not depend on the reference frame, whether it contains a black hole or not. A number of key measures of the gravitational field strength, particularly the curvature tensors, are covariant; they do not depend on the reference frame. So GR predicts the gravitational field in terms of the stress-energy tensor, which includes energy and momentum and so on which are individually frame-dependent, but it needs to do so (and it does) in a way that, in the end, respects the coordinate-independence of the theory. This problem of coordinate-independence and how it is handled and solved in GR has nothing special to do with black holes. It exists anyway.

Anyway, I will accept at face value that the black hole is a valid solution to Einstein's field equations. I haven't really pointed out that the black hole creation has not been shown to be a valid solution.

No, that's also been modelled. E.g., the Tolman–Oppenheimer–Volkoff equation models the process of a sphere of fluid (how cosmologists model stars) collapsing over time and forming a black hole. It is derived by solving the Einstein field equation together with an equation of state for the fluid (this models physical aspects of the fluid, like how its pressure changes when it is compressed), so the gravitational part of it is an exact solution to the Einstein field equation.

 

[river]

No, that's also been modelled. E.g., the Tolman–Oppenheimer–Volkoff equation models the process of a sphere of fluid (how cosmologists model stars) collapsing over time and forming a black hole. It is derived by solving the Einstein field equation together with an equation of state for the fluid (this models physical aspects of the fluid, like how its pressure changes when it is compressed), so the gravitational part of it is an exact solution to the Einstein field equation.

A sphere of fluid , as in a star would never form a black hole .

Because the pressure of gravity is three dimensional . From all sides .

 

[river]

Further any fluid that is pressureized enough becomes a solid .

 

[przyk]

No, that's also been modelled. E.g., the Tolman–Oppenheimer–Volkoff equation models the process of a sphere of fluid (how cosmologists model stars) collapsing over time and forming a black hole. It is derived by solving the Einstein field equation together with an equation of state for the fluid (this models physical aspects of the fluid, like how its pressure changes when it is compressed), so the gravitational part of it is an exact solution to the Einstein field equation.

A correction: I just looked up the Tolman-Oppenheimer-Volkoff equation in more detail and it describes the variation of pressure inside a static spherical-symmetric distribution of fluid. It can be used to show that the internal pressure needed to sustain a sufficiently dense ball of fluid becomes infinite. Hence, a ball of fluid that is too dense becomes unsustainable and must gravitationally collapse. However, since the solution is static, it does not actually model the collapse process. It only shows that a collapse is inevitable beyond a certain point.

What I was thinking of was an actual model of the collapse process, part of which is an exact solution of the Einstein field equation describing the gravitational field in and around a collapsing ball of fluid actually changing and forming a black hole over time. I definitely remember studying this in a GR class (and working through it in detail in my own time afterward) about ten years ago. I don't have my lecture notes (they are probably in a box in another country) so I can't be sure exactly what it was, but it seems like it could be the same thing as is described in section V here: https://arxiv.org/abs/gr-qc/0502040.

In any case, the point here is that exact solutions to the Einstein field equation modelling the formation of a black hole have been derived and do exist. I.e., we do not only have static solutions like the Schwarzschild geometry that describe an eternal black hole.

 

[RJBeery]

So GR predicts the gravitational field in terms of the stress-energy tensor, which includes energy and momentum and so on which are individually frame-dependent, but it needs to do so (and it does) in a way that, in the end, respects the coordinate-independence of the theory.

You do a good job with your explanations. I'm basically complaining about the same issue in a variety of ways, and my issue is what you wrote here. I don't doubt what you've written, I just have a problem accepting that blackholes don't cause a contradiction here. To be clear, I'm not questioning GR at all, but people usually assume that GR and blackholes go hand-in-hand.

In any case, the point here is that exact solutions to the Einstein field equation modelling the formation of a black hole have been derived and do exist. I.e., we do not only have static solutions like the Schwarzschild geometry that describe an eternal black hole.

Is that true, though? I've looked. I'll read the paper you referenced but I notice that it analyzes "massless" creation. There are literally thousands of papers on black holes -- static, spinning, charged, no-hair, evaporating, eternal, etc, yet a curious dearth on the creation of them. That, buttressed by my handful of philosophical problems with black holes, makes my BS-meter go off.

 

[przyk]

You do a good job with your explanations. I'm basically complaining about the same issue in a variety of ways, and my issue is what you wrote here. I don't doubt what you've written, I just have a problem accepting that blackholes don't cause a contradiction here. To be clear, I'm not questioning GR at all, but people usually assume that GR and blackholes go hand-in-hand.

I really don't see what you think is special about black holes here. It seems to be a general issue with the coordinate-independence of GR.

In GR the theory, including particularly the Einstein field equation, is known and mathematically proved to be coordinate-independent. That is, if you have a solution to the Einstein field equation expressed in certain coordinates then you can apply any coordinate transformation to the components of both the stress-energy tensor and gravitational metric, and it is known that the result if you do this will still be a solution to the Einstein field equation.

The implication is that changes to a complete system that are just the active version of a coordinate transformation will not fundamentally change the gravitational field. For example, the gravitational field produced by a single moving mass by itself will not be "stronger" than that produced by one that is the same except it is stationary. If you accept this then it applies to everything. There is no reason black holes should be different.

If you don't accept this, then you're basically doubting that GR is coordinate-independent, which means that you're suspecting that GR might have failed to be the most basic thing that it was meant to be: a relativistic theory. I don't know about you but I would think that would be a much bigger problem than anything to do with black holes.

In either case, fixating on black holes just seems strange here.

Is that true, though? I've looked.

Yes. You would have to consult the research literature for details since it's not something I'm an expert on. It is also getting into theory that is more advanced than is likely to be covered in a textbook.

I'll read the paper you referenced but I notice that it analyzes "massless" creation.

There isn't just one model. There are multiple solutions derived in different cases, some of which are more realistic than others (e.g., fluid with pressure vs. pressureless dust). The paper in question is a pedagogical article that discusses a few models that were mostly already known. (This is the kind of article that the American Journal of Physics typically publishes.) I don't know how comprehensive it is and there could be a few or many other solved cases that it doesn't mention.

One (or two) of the situations considered in the paper is a black hole forming from a collapsing shell of (massless) light. The authors themselves say that this situation is artificial. But in Section V they also model the collapse of a sphere of fluid with nonzero (rest) mass density and pressure. This is closer to describing something realistic like a collapsing star.

There are literally thousands of papers on black holes -- static, spinning, charged, no-hair, evaporating, eternal, etc, yet a curious dearth on the creation of them. That, buttressed by my handful of philosophical problems with black holes, makes my BS-meter go off.

There's a much more reasonable explanation, which is that research on GR is in practice limited by the technical difficulty of actually calculating what the theory predicts. So being able to solve the Einstein field equation even in simple situations (like assuming spherical symmetry) is already impressive, and deriving what happens for even small variations of situations that have been studied before can be a herculean effort that is worth writing a paper about.

In realistic situations the problem is actually even more difficult than my first post in this thread implies, because the gravitational field and the stress-energy tensor both affect each other. For example, gravitational attraction will tend to compress a mass, which increases its internal pressure. But like I've mentioned before, internal pressure is included in the stress-energy tensor and so also affects the gravitational field. So you can't even in general just say what the stress-energy tensor is going to be and solve the Einstein field equation with that fixed, which can already be hard enough. In general you need to solve for both the gravitational field and the stress-energy tensor together as a coupled dynamical system. Now imagine you're a theorist decades ago considering solving this kind of problem. You could understand why a theorist might be tempted to just start with an artificial situation with no pressure and see first if they could already solve for that.

Limitations in our ability to compute predictions is, by the way, not something that is unique to GR. There are situations even in simpler centuries-old theories where it is still difficult or impossible to make accurate predictions. The three-body problem in Newtonian gravity is a famous example of this.

 

[Q-reeus]

...For example, the gravitational field produced by a single moving mass by itself will not be "stronger" than that produced by one that is the same except it is stationary....

Not true. A low density monatomic gas for instance when heated obviously has a higher energy content which is almost entirely due to increased average particle KE. Positive gas pressure and negative container wall stresses, assuming static equilibrium, exactly cancel. Further there is no net momentum flow, so only the T_00 energy density term contributes. It should be obvious then gravitating mass per particle increases as average KE thus average particle speed increases.

The only online article I'm aware of with explicit calculations for a single moving mass (and that needs careful interpretation) is behind a paywall at AJP:
"Measuring the active gravitational mass of a moving object - D.Olson & R.Guarino"

 

[RJBeery]

Not true. A low density monatomic gas for instance when heated obviously has a higher energy content which is almost entirely due to increased average particle KE. Positive gas pressure and negative container wall stresses, assuming static equilibrium, exactly cancel. Further there is no net momentum flow, so only the T_00 energy density term contributes. It should be obvious then gravitating mass per particle increases as average KE thus average particle speed increases.

This is brilliant. Basically, if heat energy contributes to the local gravitational field then linear momentum energy does.

BTW, this is my third time linking to this, but I think its significance is being overlooked: https://authors.library.caltech.edu/1544/

 

[przyk]

...For example, the gravitational field produced by a single moving mass by itself will not be "stronger" than that produced by one that is the same except it is stationary....

Not true. A low density monatomic gas for instance when heated obviously has a higher energy content which is almost entirely due to increased average particle KE.

I think you missed the bit in emphasis. What I said applies to systems that are the same up to a coordinate transformation. So, for example, the gravitational field around a moving planet is not "stronger" than the one around the same planet if it is at rest. On the other hand, a hot gas is not the same as a cold gas described in a different coordinate system, so what I said does not apply there.

The only online article I'm aware of with explicit calculations for a single moving mass (and that needs careful interpretation) is behind a paywall at AJP:
"Measuring the active gravitational mass of a moving object - D.Olson & R.Guarino"

There is no need to derive the gravitational field for a single moving mass. GR is relativistic, so if you know the gravitational field produced by a stationary mass then you can apply a coordinate transformation to describe it from a coordinate system in which it is moving.

The only potential ambiguity is what coordinate system to use. If there's a nontrivial gravitational field then the spacetime is curved which means, basically by definition, that you cannot use an inertial coordinate system to describe it or anything in it. So you would have to use some other coordinate system instead. But assuming you've made a decision about that there's no problem.

 

[przyk]

BTW, this is my third time linking to this, but I think its significance is being overlooked: https://authors.library.caltech.edu/1544/

What exactly do you think is its significance?

Yes, the presence of an electromagnetic field affects the gravitational field in GR. If you were under the impression that this is some kind of obscure discovery then that is not the case. It follows directly from basic GR and is well known to anyone familiar with the theory. In GR, as I've said before, the gravitational field is coupled to the stress-energy tensor. That includes the contribution to the stress-energy tensor of the electromagnetic field.

 

[RJBeery]

...For example, the gravitational field produced by a single moving mass by itself will not be "stronger" than that produced by one that is the same except it is stationary[...]a hot gas is not the same as a cold gas described in a different coordinate system, so what I said does not apply there

I feel like you're getting a little squirrelly here, przyk. We aren't describing the gas in different coordinate systems. We are comparing their relative gravity fields before and after heat is applied. Applied heat is nothing but linear momentum in the gaseous molecules. Reduce the "gas" to a single molecule if you wish. We're clearly dealing with constrained linear molecular momentum.

 

[RJBeery]

What exactly do you think is its significance?

Yes, the presence of an electromagnetic field affects the gravitational field in GR. If you were under the impression that this is some kind of obscure discovery then that is not the case. It follows directly from basic GR and is well known to anyone familiar with the theory. In GR, as I've said before, the gravitational field is coupled to the stress-energy tensor. That includes the contribution to the stress-energy tensor of the electromagnetic field.

Well...you said

Well, the stress-energy tensor plays the analogous role in GR that the charge and current density functions do in the differential form of Maxwell's equations.

Charge and current density are demonstrably affected by relative velocity, but if the tensor is X for one frame then it's X for all frames. This strikes me as a contradiction. I feel like the heated gas highlights the problem.

 

[przyk]

I feel like you're getting a little squirrelly here, przyk.

How so? I was making a point about GR being a relativistic theory. Q-reeus countered with a situation where the relativistic nature of GR by itself does not tell you anything, so was irrelevant to the point I was making.

We aren't describing the gas in different coordinate systems. We are comparing their relative gravity fields before and after heat is applied. Applied heat is nothing but linear momentum in the gaseous molecules.

Applied heat shows up in the stress-energy tensor both as increased energy and local momentum flux, a.k.a. "pressure".

Reduce the "gas" to a single molecule if you wish.

No, that would be wrong. Like I said before, the gravitational field in GR does not obey the superposition principle. The gravitational field produced by a gas is not the sum of the gravitational fields produced by its individual molecules. So understanding the gravitational field produced by a gas with lots of individual molecules moving around past each other at high velocity does not reduce to understanding the gravitational field produced by a single moving molecule.

 

[przyk]

Well, the stress-energy tensor plays the analogous role in GR that the charge and current density functions do in the differential form of Maxwell's equations.

Charge and current density are demonstrably affected by relative velocity, but if the tensor is X for one frame then it's X for all frames. This strikes me as a contradiction.

Well, first, I said "analogue". It is not a contradiction for an analogy to have its limitations and break at some point.

But since you bring it up, the charge and current density are more precisely analogous to the components of the stress-energy tensor, which are also dependent on the coordinate system.

The direct analogue of the stress-energy tensor itself in electromagnetism is the four-current. This is a four-vector whose components are the charge and current densities. A four-vector is a particular kind of tensor (specifically, a tensor of rank one). Its components are frame-dependent but, similar to the stress-energy tensor, the four-current as a whole is a vector in spacetime and can be understood as such independently of any choice of reference frame.

This is partly why I hoped you might have studied electromagnetism earlier. If you had, you would have very likely learned the relativistic formulation of electromagnetism, which includes using the four-current.

 

[Q-reeus]

I think you missed the bit in emphasis....

Not missed. Your wording in natural context makes it clear to me you claimed net gravitating mass of an object with given proper mass is independent of relative speed measured in a given coordinate frame. If you had some totally different word usage in mind, the onus was on you to clearly expound it. You didn't, because my precise expansion above is exactly what you really meant.
Why not just gracefully admit you got it wrong, and move on?

What I said applies to systems that are the same up to a coordinate transformation. So, for example, the gravitational field around a moving planet is not "stronger" than the one around the same planet if it is at rest.

But it is. Not for someone on 'the moving planet' obviously but for someone for which the planet is moving. And THAT latter case is clearly the type of situation under consideration.

On the other hand, a hot gas is not the same as a cold gas described in a different coordinate system, so what I said does not apply there.

???? 'described in a different coordinate system' - what is that but obfuscation? My #87 clearly assumes we stick with one coordinate frame. Where random motions change average speed. Simple.

There is no need to derive the gravitational field for a single moving mass. GR is relativistic, so if you know the gravitational field produced by a stationary mass then you can apply a coordinate transformation to describe it from a coordinate system in which it is moving.

And when that is done, increased gravitational mass is the outcome. I presented the process of elimination in #87 making it real easy to resolve. No call for some full-blown numerical GR attack requiring lots of supercomputer grunt to solve. We are extremely far from merging 'BH's territory.

The only potential ambiguity is what coordinate system to use. If there's a nontrivial gravitational field then the spacetime is curved which means, basically by definition, that you cannot use an inertial coordinate system to describe it or anything in it. So you would have to use some other coordinate system instead. But assuming you've made a decision about that there's no problem.

The counterexample I gave of a dilute gas heated is clearly one of very weak gravity and there is no need at all to resort to curved spacetime metric coordinates. See my expansion of that point next post.

 

[Q-reeus]

How so? I was making a point about GR being a relativistic theory. Q-reeus countered with a situation where the relativistic nature of GR by itself does not tell you anything, so was irrelevant to the point I was making.

I concur with RJB's comment on that, but would use the word weaselly instead.

Applied heat shows up in the stress-energy tensor both as increased energy and local momentum flux, a.k.a. "pressure".

Implying I overlooked role of pressure. Something else here you didn't know evidently. Miss this from my #87?:
"Positive gas pressure and negative container wall stresses, assuming static equilibrium, exactly cancel."
Referring to those contributions to overall system SET obviously. If you wish to dispute that then dispute it also with the following:
http://arxiv.org/abs/gr-qc/0505040
I am here choosing to work strictly within the GR framework where stress in general is for sure a source of gravitation. Exception being pure shear stress. Elsewhere I have posted re my doubts about stress as actual source. It engendered mindless hostility and I only mention this in case someone points to my postings here and down the track accuses me of 'hypocrisy' etc.

No, that would be wrong. Like I said before, the gravitational field in GR does not obey the superposition principle. The gravitational field produced by a gas is not the sum of the gravitational fields produced by its individual molecules. So understanding the gravitational field produced by a gas with lots of individual molecules moving around past each other at high velocity does not reduce to understanding the gravitational field produced by a single moving molecule.

You should well know nonlinearity is totally insignificant for the very weak gravity regime I proposed re heated gas. It's often referred to as 'linearized gravity' and for good reason. One CAN apply superposition there with negligible error. As a rough guide, the maximum fractional error in assuming linear superposition will be ~ r_s/R, where r_s is the system Schwarzschild radius, and R is the characteristic size of the system. Of course it's not always possible to define those values precisely, but assuming a spherical pressure vessel, it can be. I'll leave it to you to do some sums and be confronted with just how misleading your argument here is.
If you insist I supply a specific scenario - take a steel thin shelled (say 1mm wall thickness) spherical pressure vessel. 10cm in inside diameter, filled with helium at STP initially, then heated to say 100 degrees C. There will be lots of zeros after the dot re r_s/R - lots!
Don't try and bounce this back to me re doing the sums - YOU are the one claiming we MUST take into account nonlinearity i.e. need full-blown GR solution. Do the quite easy estimate for given specific case above.

And btw, the relevant system mass re fractional error in assuming linearization will not be the total mass, tiny as it is in gravitational terms, but essentially only the difference in mass between cold and heated. That's because we are considering only the differential in net gravitating mass, something I shouldn't have to remind of.
I was confronted with your kind of tactics a lot back in my PF days, by one or two genuine experts in GR, who imo knew perfectly well they were simply being obstructionists.

 

[foghorn]

I concur with RJB's comment on that, but would use the word weaselly instead.

Q-reeus has arrived on the thread. The hostility begins. Let the ''Nasty'' begin.

 

[Michael 345]

Q-reeus has arrived on the thread. The hostility begins. Let the ''Nasty'' begin.

Iggy is your friend

 

[foghorn]

Iggy is your friend

I do ignore the bulk of his writing after he gets ''nasty''. He likes to think people ignoring him is a sign that he's always correct and the other person has run-away as a result.

 

[Q-reeus]

Q-reeus has arrived on the thread. The hostility begins. Let the ''Nasty'' begin.

I'm curious as to the initial trigger for your nasty obsessive stalking behavior. Want to unload how it all started?
On second thought, given you will have nothing constructive and on topic to share here, just butt out - imo an illegally sanctioned sock of sweetpea.

PS - your skewed commentary in #99 is as expected way off the mark. That SF mods even tolerate your type is a black mark against them as well.