Causal mechanism for gravity

No. You're once again making assumptions. Angular momentum is handled via the stress-energy tensor just like everything else energy/matter/momentum-related that is coupled to the gravitational field. That is, if you have something like a rotating mass then its local momentum will vary (have different magnitudes and be pointing in different directions) depending on position and distance relative to the axis of rotation, which makes the stress-energy tensor for a rotating system different than for a non-rotating one.
I shouldn't have used the word "contradiction" there when "point of confusion for me" would have been better. I laid out my confusion in the second paragraph of my last post. If momentum of any type is coupled to the gravitational field then why is only one type of momentum (apparently) capable of creating black holes? Now I'm stuck thinking about what would make angular momentum intrinsically different from linear momentum, and so far, I do not appreciate the difference. Having momentum "vary depending on position" seems terribly arbitrary to me when trying to identify a difference. We could easily have a stream of water whose linear momentum increases as we get further from the shore.

Having momentum "vary depending on position" seems terribly arbitrary to me when trying to identify a difference.

Well first, as a simple matter of fact, the momentum density (and the rest of the stress-energy tensor) does vary depending on position. This is because the universe is not filled up with a homogeneous fluid moving everywhere with the same speed in the same direction. Look around you to see that this is not the case.

But more importantly, this comment of yours is exactly my point about assumptions. Where, exactly, is it that you are getting your ideas from about what the gravitational field should or shouldn't depend on? Where are you getting ideas about how sensitive it should or should not be to local variability of momentum density, in any situation, or anything else for that matter? In GR, the only source of knowledge we have about that sort of thing is from the Einstein field equation and the (often approximate) solutions people have been able to derive for it in specific cases over the years. That includes solutions showing that it can make a difference if you have a glob of matter that is rotating v.s. not rotating. This is not in conflict with anything because there simply does not exist some alternative method of reasoning you can follow to derive that the gravitational field should behave any differently. It is just you who seem to be approaching GR with a lot of beliefs about what sort of behaviour the Einstein field equation should predict without having mathematically studied or solved it.

I'd already commented on the complexity of the Einstein field equation a couple of posts ago, but just to emphasise one thing about it here: like I said before, it is a nonlinear differential equation. In more physical terms, that means it does not obey the superposition principle. That means that you absolutely cannot understand the gravitational field produced by a complicated system by reducing it to its parts. GR is not like Newtonian gravity where you can find the gravitational field produced by any mass distribution by adding the gravitational fields produced by all its constituent parts. For example, you can't derive the gravitational field around two nearby masses in space just by adding two Schwarzschild metrics, or anything so simple. You would basically need to solve the Einstein field equation for that specific situation, starting from scratch, if you wanted to do that.

Last edited:
Well first, as a simple matter of fact, the momentum density (and the rest of the stress-energy tensor) does vary depending on position. This is because the universe is not filled up with a homogeneous fluid moving everywhere with the same speed in the same direction. Look around you to see that this is not the case.

But more importantly, this comment of yours is exactly my point about assumptions. Where, exactly, is it that you are getting your ideas from about what the gravitational field should or shouldn't depend on? Where are you getting ideas about how sensitive it should or should not be to local variability of momentum density, in any situation, or anything else for that matter? In GR, the only source of knowledge we have about that sort of thing is from the Einstein field equation and the (often approximate) solutions people have been able to derive for it in specific cases over the years. That includes solutions showing that it can make a difference if you have a glob of matter that is rotating v.s. not rotating. This is not in conflict with anything because there simply does not exist some alternative method of reasoning you can follow to derive that the gravitational field should behave any differently. It is just you who seem to be approaching GR with a lot of beliefs about what sort of behaviour the Einstein field equation should predict without having mathematically studied or solved it.

I'd already commented on the complexity of the Einstein field equation a couple of posts ago, but just to emphasise one thing about it here: like I said before, it is a nonlinear differential equation. In more physical terms, that means it does not obey the superposition principle. That means that you absolutely cannot understand the gravitational field produced by a complicated system by reducing it to its parts. GR is not like Newtonian gravity where you can find the gravitational field produced by any mass distribution by adding the gravitational fields produced by all its constituent parts. For example, you can't derive the gravitational field around two nearby masses in space just by adding two Schwarzschild metrics, or anything so simple. You would basically need to solve the Einstein field equation for that specific situation, starting from scratch, if you wanted to do that.
Thanks for posting this. Very informative.

This is not in conflict with anything because there simply does not exist some alternative method of reasoning you can follow to derive that the gravitational field should behave any differently. It is just you who seem to be approaching GR with a lot of beliefs about what sort of behaviour the Einstein field equation should predict without having mathematically studied or solved it[...]You would basically need to solve the Einstein field equation for that specific situation, starting from scratch, if you wanted to do that.
I disagree with this attitude. I'm simply expressing my points of confusion. You're frustrated that I don't accept "just trust me, it's more complicated than you think and if you don't believe me then go do the math yourself" as an answer. Remember, this all started with Halc saying "well...I don't know but SOMEBODY does" and linked to the Gibbs/Baez explanation. It doesn't take a genius to parse through the Gibb/Baez explanation to see that it had no substance.

I'm not asking a math question, I'm asking a philosophical question -- why would angular momentum be treated differently than linear momentum in GR? Linear momentum energy isn't "local" according to Misner, Thorne and Wheeler but neither is angular momentum energy. Angular momentum energy only exists when we look at the rotating system holistically -- and if we're going to do that then there is nothing preventing us from looking at the "system" of two objects in relative motion and analyzing that system's linear momentum energy.

And saying that solutions to Einstein's field equations are difficult to come by is fine, but that's a lot different from saying that other solutions don't exist. Frankly, I'm not even sure which concept you're defending, or which idea you believe I'm promoting. I'm just basically thinking out loud about why the Physics GR community would treat angular momentum differently from linear momentum in some fundamental way. It doesn't pass the smell test with me. Perhaps it's offensive to you that a layman won't accept something that you have, apparently based on faith?

I disagree with this attitude. I'm simply expressing my points of confusion. You're frustrated that I don't accept "just trust me, it's more complicated than you think and if you don't believe me then go do the math yourself" as an answer. Remember, this all started with Halc saying "well...I don't know but SOMEBODY does" and linked to the Gibbs/Baez explanation. It doesn't take a genius to parse through the Gibb/Baez explanation to see that it had no substance.

I'm not asking a math question, I'm asking a philosophical question -- why would angular momentum be treated differently than linear momentum in GR? Linear momentum energy isn't "local" according to Misner, Thorne and Wheeler but neither is angular momentum energy. Angular momentum energy only exists when we look at the rotating system holistically -- and if we're going to do that then there is nothing preventing us from looking at the "system" of two objects in relative motion and analyzing that system's linear momentum energy.

And saying that solutions to Einstein's field equations are difficult to come by is fine, but that's a lot different from saying that other solutions don't exist. Frankly, I'm not even sure which concept you're defending, or which idea you believe I'm promoting. I'm just basically thinking out loud about why the Physics GR community would treat angular momentum differently from linear momentum in some fundamental way. It doesn't pass the smell test with me. Perhaps it's offensive to you that a layman won't accept something that you have, apparently based on faith?
You seem to be demanding that someone explain GR to you in detail without maths. But, from what I understand, GR is the maths. I don't know GR but I do know some QM and it's the same with that. One can produce analogies, up to a point, but in the end you just have to do the maths to understand properly what is going on. The model is mathematical and there is no way of getting round that.

Your angular momentum issue is way beyond me, but one point of difference between linear and angular momentum is that the latter involves acceleration, does it not? So I am not surprised it behaves differently in GR from linear momentum. But this may be a superficial observation - I don't know.

Your angular momentum issue is way beyond me, but one point of difference between linear and angular momentum is that the latter involves acceleration, does it not? So I am not surprised it behaves differently in GR from linear momentum. But this may be a superficial observation - I don't know.
I don't think that's superficial, I think it's something to consider.

one point of difference between linear and angular momentum is that the latter involves acceleration, does it not?
Angular momentum, like linear momentum, does not require acceleration unless there is to be a change to it. So no, this is not a difference. An external force is required to change the momentum of a system, and likewise an external torque is required to change the angular momentum of a system.

One difference is that angular momentum is absolute, that is, it is not something that can be made zero in any reference frame, not even a rotating one. Linear momentum on the other hand, like other values like kinetic energy, is completely frame dependent.

Angular momentum, like linear momentum, does not require acceleration unless there is to be a change to it. So no, this is not a difference. An external force is required to change the momentum of a system, and likewise an external torque is required to change the angular momentum of a system.
OK, but any rotation about an axis involves a constant centripetal acceleration of off-axis parts of the system towards the axis, doesn't it? So if there is angular momentum, there is acceleration present, isn't there? This is something you can feel , in a way that is indistinguishable from gravitation, just like linear acceleration. Or is that wrong in GR?

OK, but any rotation about an axis involves a constant centripetal acceleration of off-axis parts of the system towards the axis, doesn't it? So if there is angular momentum, there is acceleration present, isn't there?
Well, a system of two rocks in a passing trajectory has angular momentum and no acceleration of anything period, so no, angular momentum does not necessarily require acceleration. You're only thinking of a restricted case of a system of one rotating rigid object, which yes, involves acceleration of any components not at the axis.

Well, a system of two rocks in a passing trajectory has angular momentum and no acceleration of anything period, so no, angular momentum does not necessarily require acceleration. You're only thinking of a restricted case of a system of one rotating rigid object, which yes, involves acceleration of any components not at the axis.
Aha, thanks, yes I can see that. So centripetal acceleration is not intrinsic, then.

Aha, thanks, yes I can see that. So centripetal acceleration is not intrinsic, then.
It makes me question a binary star system. Both stars are in "free fall" as they orbit each other -- we consider them to be accelerated, but do they "feel" acceleration? I suspect the off-the-cuff Physics answer is "no" because they are following a geodesic path, but that the technical answer is "yes" because the gravity field surrounding each body is graded.

It makes me question a binary star system. Both stars are in "free fall" as they orbit each other -- we consider them to be accelerated, but do they "feel" acceleration? I suspect the off-the-cuff Physics answer is "no" because they are following a geodesic path, but that the technical answer is "yes" because the gravity field surrounding each body is graded.
Yes I suppose so.

Did you ever get in contact with Markus Hanke?

Yes I suppose so.

Did you ever get in contact with Markus Hanke?
Yep, he responded but I wanted to form a specific follow-up question before I went back to that forum.

I disagree with this attitude. I'm simply expressing my points of confusion. You're frustrated that I don't accept "just trust me, it's more complicated than you think and if you don't believe me then go do the math yourself" as an answer.

I didn't say "go do the math", but what you're paraphrasing here is a valid criticism. General relativity is a mathematically-defined theory and what we know about what it predicts comes from mathematically investigating it. If you don't learn the math then you are handicapping your own ability to understand how that works and to understand people's answers to your questions.

When people tell you "go do the math" I doubt anyone realistically expects you to actually drop what you're doing and spend three months studying general relativity in detail. But there's perhaps a dim hope you will do that at some point and that it would lead to one of two much better outcomes than what we're seeing here:
• You find out how things actually work in GR and end up solving your own problem.
Imagine if, for example, you studied a derivation of the Lense-Thirring metric and were able to point out to people precisely where, in the modelling, you thought angular momentum was being treated "fundamentally differently" from "linear momentum". That kind of detail could be genuinely useful to people trying to help you. What you've written instead is not useful and makes it seem like you've just ignored what I've said:

why would angular momentum be treated differently than linear momentum in GR?

I've already told you it isn't.

Linear momentum energy isn't "local" according to Misner, Thorne and Wheeler but neither is angular momentum energy. Angular momentum energy only exists when we look at the rotating system holistically -- and if we're going to do that then there is nothing preventing us from looking at the "system" of two objects in relative motion and analyzing that system's linear momentum energy.

This doesn't even make any real sense to me. A lot of this isn't even using terminology I recognise or using it in ways I don't recognise. What are "linear momentum energy" and "angular momentum energy" supposed to be? What do you mean by "local", or that angular momentum only exists "holistically"? Why do you think we need to group things into "system"s?

I can only reiterate what I've already said here. The gravitational field in general depends on the stress-energy tensor. The stress-energy tensor contains information about a lot of things including mass/energy, momentum, pressure, and angular momentum (i.e., if you know the stress-energy tensor then you can calculate the angular momentum with respect to any reference point). So, logically, the gravitational field can be related to any of these things or any combination of them, and figuring out the details beyond this is a math problem. Where do you see any conflict in this?

Last edited:
RJ - getting back to the initial query re why a moving mass cannot gravitationally collapse simply owing to relative velocity. The golden rule is 'evaluate in the center-of-energy frame'. According to GR, a collision of ultra-relativistic particles with equal and opposite momenta in some frame might generate sufficient energy density to then form a 'BH'.
Which according to Hawking would immediately 'evaporate' into a particle shower. The telltale signature has never been observed at LHC, and speculatively relied on existence of 'large hidden higher dimensions' permitting 'BH' formation at much lower collision energies than for conventional 4D GR.

Something roughly analogous is involved in experiments to 'boil the vacuum' to produce real electron-positron pairs via ultra high powered lasers. It requires interference between oppositely directed beam components to produce extremely high intensity standing wave E fields. No matter how powerful the individual beams may be, they could never produce that standing wave condition individually. Moral again - evaluate system in center-of-energy frame.

Your issue with angular momentum imo has confused non-locality of angular momentum density residing in a GR gravitational field region generated by a spinning source, with the accepted fact net angular momentum in a given system is an SR frame-independent quantity.

Last edited:
Screwed around again with EXTREMELY sluggish SF posting behavior - one tries repeatedly but it doesn't improve. Then - voila - 'double post' but not my fault.

Last edited:
It makes me question a binary star system. Both stars are in "free fall" as they orbit each other -- we consider them to be accelerated, but do they "feel" acceleration? I suspect the off-the-cuff Physics answer is "no" because they are following a geodesic path, but that the technical answer is "yes" because the gravity field surrounding each body is graded.
There should be no confusion between zero proper acceleration of each star's center of gravity (not generally coincident to the star's center of mass), and tidal forces acting on matter surrounding each star's center of gravity. The vector sum of those tidal forces must add to zero in accordance with conservation of system linear and angular momentum (ignoring usually negligible GW emissions).

Last edited:
RJBeery said:
why would angular momentum be treated differently than linear momentum in GR?
I've already told you it isn't.
RJBeery said:
Linear momentum energy isn't "local" according to Misner, Thorne and Wheeler but neither is angular momentum energy. Angular momentum energy only exists when we look at the rotating system holistically -- and if we're going to do that then there is nothing preventing us from looking at the "system" of two objects in relative motion and analyzing that system's linear momentum energy.
przyk said:
This doesn't even make any real sense to me. A lot of this isn't even using terminology I recognise or using it in ways I don't recognise. What are "linear momentum energy" and "angular momentum energy" supposed to be? What do you mean by "local", or that angular momentum only exists "holistically"? Why do you think we need to group things into "system"s?

I'm conflicted on where to go from here. Other posters seem to understand my point of confusion. Learning the full-treatment of GR mathematics might work, but that's a tall order when there should be thousands of experts online capable of explaining this to their grandmother.

I'll try again -- when a mass is spinning, that rotation adds to the stress-energy tensor such that a black hole collapse becomes more likely (i.e. less rest mass is required). Where does that black hole collapse become more likely? Next door? Millions of miles away? No, the obvious answer is that this stress-energy tensor is affected by rotation in an area restricted to the immediate vicinity of the rotating mass. I choose to call this stress-energy tensor modification "angular-momentum energy," but you didn't appreciate that, so we'll just call it the stress-energy tensor modification.

Now, when two bodies, A and B, are in relative motion they are said to possess kinetic energy. Kinetic energy is a real phenomenon. Relativistic mass gravitationally affects trajectories, so this "energy" is not illusory. Energy in all forms modifies the stress-energy tensor. The stress-energy tensor is a field, as you said, so it varies by position. WHERE does this kinetic energy reside? Object A, or object B? Does it exist in the "A-B system" (hence the talk of holistic systems)? Does it exist nowhere at all?

You should maybe consider really reading the questions posed in some of these posts. If nothing else it gives us some interesting philosophical points to ponder.

Learning the full-treatment of GR mathematics might work, but that's a tall order

That depends on the person. Most people don't care enough about GR for it to be worth spending months learning about in detail just to know the answer to one or two questions about it.

But in your case, you've been trying to gain a few insights about GR from threads like these for about ten years, while physics students with the right prerequisites usually learn the full theory in a few months. It's just way more effective. Especially if you consider that, in a typical class, you'll have N students learning GR with the help of just one instructor, while you, who are just one "student", seem to want N experts to craft perfect explanations tailored specially just for you.

when there should be thousands of experts online capable of explaining this to their grandmother.

Really, I'm surprised to see someone with an IT background invoke that grandmother thing. A lot of grandmothers really struggle to use a computer. I guess this is because all the supposed "computer experts" in the world actually don't understand computers.

WHERE does this kinetic energy reside? Object A, or object B? Does it exist in the "A-B system" (hence the talk of holistic systems)? Does it exist nowhere at all?

This really makes no sense. The energy resides where it actually resides. Why do you think it should reside somewhere else? For your two hypothetical objects, the answer is that some of the energy is located where object A is and some of it is located where object B is, and this is how you would describe it in the stress-energy tensor.

What's really confusing for me here is that you've said you did a physics degree some years ago. Which means you maybe didn't learn GR but you should certainly have learned electromagnetism. Well, the stress-energy tensor plays the analogous role in GR that the charge and current density functions $$\rho(\boldsymbol{x}, t)$$ and $$\boldsymbol{j}(\boldsymbol{x}, t)$$ do in the differential form of Maxwell's equations. Do you have the same confusions here? If you have two charges A and B, do you have the same question about where the charge and the current reside? When do two charges become "one holistic system"?

Last edited:
This really makes no sense. The energy resides where it actually resides. Why do you think it should reside somewhere else? For your two hypothetical objects, the answer is that some of the energy is located where object A is and some of it is located where object B is, and this is how you would describe it in the stress-energy tensor.
You agree that the energy must reside somewhere...but velocity is purely relative. 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?
przyk said:
What's really confusing for me here is that you've said you did a physics degree some years ago. Which means you maybe didn't learn GR but you should certainly have learned electromagnetism. Well, the stress-energy tensor plays the analogous role in GR that the charge and current density functions $$\rho(\boldsymbol{x}, t)$$ and $$\boldsymbol{j}(\boldsymbol{x}, t)$$ do in the differential form of Maxwell's equations. Do you have the same confusions here? If you have two charges A and B, do you have the same question about where the charge and the current reside? When do two charges become "one holistic system"?
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. 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. 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.

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.

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

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. 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. 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.

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.

That , which is highlighted , last statement , is a most important statement .

The difference between the mathematical and real physical evidence of a black hole is in a physical form .

Last edited: