Is 'charged black hole' an oxymoron?

[Dale]

This would be correct only if the table is considered an inertial object (as it is done in most practical physics exercises, there is a lab frame considered to be at rest,it is an idealization that works great for most practical problems), but we know that is not the case in reality, the table is non-inertial and in continuous motion so there is work done. An accelerometer in the surface of the Earth measures proper acceleration.

No, I am not considering the table an inertial object, if the table were inertial then the force on the book would be 0.

You misunderstand how energy works in a non-inertial frame. In a non-inertial frame, like the usual frame attached to the surface of the earth, there is a fictitious force. The fictitious force, in this case, is equal to mg and directed downwards. We know that there is a fictitious force on the book precisely because an accelerometer measures a proper acceleration of g directed upwards and yet the book is not accelerating relative to our frame.

Now, work is f.d, so as an object is moved upwards against this fictitious force it requires an amount of work equal to f.d=mgh. Conversely, as an object is lowered the fictitious force does an amount of work equal to mgh. So, the ficitious force has an associated potential energy. (remember, energy is frame variant)

So, the book, sitting on the table at rest, has no change in KE. It also has no change in PE. No work is being done on it. There is no change to its internal structure or temperature nor anything else where energy could go. There is no change in any energy associated with the book. Despite the fact that there is a force on the book (two forces actually) and the book measures a proper acceleration and is therefore non-inertial.

 

[TrickyDicky]

You misunderstand how energy works in a non-inertial frame. In a non-inertial frame, like the usual frame attached to the surface of the earth, there is a fictitious force. The fictitious force, in this case, is equal to mg and directed downwards. We know that there is a fictitious force on the book precisely because an accelerometer measures a proper acceleration of g directed upwards and yet the book is not accelerating relative to our frame.

(remember, energy is frame variant)

Not exactly, KE is frame dependent, energy is a conserved quantity in Newtonian mechanics (I understand you are restricting this to the Newtonian mechanics POV since you were talking about fictitious forces).

So, the book, sitting on the table at rest, has no change in KE. It also has no change in PE. No work is being done on it. There is no change to its internal structure or temperature nor anything else where energy could go. There is no change in any energy associated with the book. Despite the fact that there is a force on the book (two forces actually) and the book measures a proper acceleration and is therefore non-inertial.

You are forgetting here that work, like KE is a frame dependent quantity. I know the example of the book and the table is considered to have 0 net moment, with net moment defined as change in KE, that is not what I referred to when I said work was being done.
The conventional Newtonian treatment with fictitious forces is fine, I have nothing against it as long as one understands its range of validity.
I was pointing to a more realistic treatment, more like GR's Schwarzschild solution (so energy is not frame dependent) for instance. In which the table would be preventing the book from following its geodesic path. And in order to do that Work in the form of a quantity proportional to the EM force in the table material times distance from the theoretical geodesic path of the book, must be done.
I know is not the conventional way work is referred to in Newtonian physics(wich is the way you were defending it in your post, and as I said I perfectly understand the use of fictitious forces and work in Newtonian physics). But I think by introducing some GR ingredients in the situation it gets closer to reality.

 

[Dale]

Not exactly, KE is frame dependent, energy is a conserved quantity in Newtonian mechanics

All energy is frame dependent, not just KE. And yes, energy is conserved (in SR generally, in GR in static spacetimes). I think you may be mixing up conservation with frame invariance, they are two separate concepts. Energy is conserved, but frame variant.

(I understand you are restricting this to the Newtonian mechanics POV since you were talking about fictitious forces).

No, all of my comments above were wrt GR, not Newtonian physics. What makes you think that GR doesn't have ficitious forces? In fact, GR provides a very easy and consistent mechanism for determining fictitious forces through the Christoffel symbols.

http://en.wikipedia.org/wiki/Curvil...ous_forces_in_general_curvilinear_coordinates
http://www.mathpages.com/home/kmath641/kmath641.htm
http://theoretical-physics.net/dev/src/relativity/relativity.html#inertial-frames

And in order to do that Work in the form of a quantity proportional to the EM force in the table material times distance from the theoretical geodesic path of the book, must be done.

This is an incorrect definition of work. If you believe otherwise then please provide a mainstream scientific reference that defines work as the force times the "distance from the theoretical geodesic path".

I know is not the conventional way work is referred to in Newtonian physics(wich is the way you were defending it in your post, and as I said I perfectly understand the use of fictitious forces and work in Newtonian physics). But I think by introducing some GR ingredients in the situation it gets closer to reality.

No, all of my comments above are GR comments. In GR there are fictitious forces, just like in Newtonian mechanics. See the references above.

The difference is that in Newtonian mechanics gravity is considered a real force and in GR it is considered fictitious. I referred to mg as a fictitious force, so I was definitely discussing from the GR perspective. You are clearly misunderstanding how non-inertial frames and fictitious forces are treated in GR.

 

[TrickyDicky]

Great then, if you are considering gravity as a fictitious force then you indeed are using a GR perspective, sorry I didn't notice it. Yor conclusions made me think you were only considering the Newtonian view. But then you are reinforcing my point, thanks.
From wikipedia page on Fictitious forces:
"Fictitious forces can be considered to do work, provided that they move an object on a trajectory that changes its energy from potential to kinetic."
If the EM forces of the table (and the earth) didn't hold the book, it would spontaneously fly towards its geodesic path.

 

[Dale]

But then you are reinforcing my point, thanks.
From wikipedia page on Fictitious forces:
"Fictitious forces can be considered to do work, provided that they move an object on a trajectory that changes its energy from potential to kinetic."
If the EM forces of the table (and the earth) didn't hold the book, it would spontaneously fly towards its geodesic path.

Far from reinforcing your point, this completely contradicts your point. See the last two paragraphs of 316.

 

[TrickyDicky]

Those two paragraphs look wrong to me. You have the fictitious force of gravity on the book acting in an upward and downward direction at the same time on the same object. That is not possible.

 

[Dale]

Those two paragraphs look wrong to me. You have the fictitious force of gravity on the book acting in an upward and downward direction at the same time on the same object. That is not possible.

Nonsense. Where did I say that?

The fictitious force of gravity always acts downwards. In fact, it is what defines the direction "down". I never said anything to the contrary.

 

[TrickyDicky]

Nonsense. Where did I say that?

Now, work is f.d, so as an object is moved upwards against this fictitious force it requires an amount of work equal to f.d=mgh. Conversely, as an object is lowered the fictitious force does an amount of work equal to mgh.

Despite the fact that there is a force on the book (two forces actually) and the book measures a proper acceleration and is therefore non-inertial.

Here maybe it is your wording but you seem to have the book going up and down in a strange way, and you even transform the fictitious force into ¡two forces¡. Make up your mind, is the fictitious force of gravity one or two forces? is the book going up or down? I'm afraid it can't go up and down at the same time.

 

[Dale]

There are two forces acting on the book. One is the fictitious force of gravity, it points downwards and has a magnitude of mg. The other force is the real force from the table, it points upwards and also has a magnitude of mg. The two forces sum to zero so that there is no acceleration in the non-inertial reference frame.

I never once said that the fictitious force was pointing upwards.

 

[TrickyDicky]

There are two forces acting on the book. One is the fictitious force of gravity, it points downwards and has a magnitude of mg. The other force is the real force from the table, it points upwards and also has a magnitude of mg. The two forces sum to zero so that there is no acceleration in the non-inertial reference frame.

I never once said that the fictitious force was pointing upwards.

Well then you mixed them up to refer to the fictitious force of gravity and its effect on KE and PE.
Do you see now that if you consider gravity as a fictitious force and do not mix it up with the normal force, you have that the potential energy is being changed to kinetic energy (and this is the one that the normal force from the table counters to have an static situation from the Earth's frame, but if the table wasn't there the book would change its path)?

Consider the example in the wikipedia page:
"consider a person in a rotating chair holding a weight in his outstretched arm. If he pulls his arm inward, from the perspective of his rotating reference frame he has done work against centrifugal force. If he now let's go of the weight, from his perspective it spontaneously flies outward, because centrifugal force has done work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from him because it is suddenly allowed to move in a straight line. This illustrates that the work done, like the total potential and kinetic energy of an object, can be different in a non-inertial frame than an inertial one."

 

[Q-reeus]

Others have already commented on this, but I would point out that "logic" requires axioms to start with, and requires a consistent set of propositions to be built up from those axioms. What I'm calling "math" is really a large set of such propositions built up from axioms, whose consistency has been carefully checked. But to carefully check that, you have to have an unambiguous way of expressing propositions, and an unambiguous way of expressing the logical connections between them. "Intuition" doesn't have any of that; it has propositions and logical connections expressed in English, which is not unambiguous, as I've pointed out before, and it certainly doesn't have a large consistent set of propositions whose consistency has been carefully checked.

[First off sorry about not responding earlier to this post but it somehow got missed, perhaps owing to a new page starting last postings of mine, and yours was not up on previous page when I composed last postings. Now to comment on the above and rest...]
Won't argue with the formalities there but it gets down to reasonableness versus unreasonableness. You received high praise from one recent poster for an answer in #287 that you may consider 'precise' but to my mind is just a string of words that asserts a generality without really explaining at all. It is trivially true that everything we experience originates from some 'past light cone(s)'. But where is the actual dynamical connection between exterior field point and inwardly hurtling charged matter asymptotically close to being completely engulfed within an EH? Mere details - as long as we accept it all comes from the past light cone, no problem at all! A matter of opinion perhaps, but I fail to be duly impressed with that.

On the other hand, if I make the point that for there to be any finite exterior BH E field, what most I think reasonably understand by 'gravitational redshift' can have no effect on any EH crossing charge acting as E field source (note this is not an RN situation but neutral BH fed by infall). A rather obvious conclusion I would have thought, but to you it is terribly imprecise and too intuitive to make any sense of. So, in frustration with that line of response, I furnished for your consideration and response numbers of what seems to me are perfectly straight-forward scenarios - examples in #272, #294, and you still manage to make them into 'confused', imprecise' situations somehow impossible to deal with. But let's continue.

As it happens, you go right on to give a good example:
Q-reeus: "Apply then the same 'transmission' approach to field itself. The field 'transmitted' from one place to another suffers zero reduction, regardless of gravitational potential difference, if RN is true."

What does this mean, precisely? What is the "field transmitted from place to place"? What does "suffers zero reduction" mean? Try giving these things precise meanings, before asking us to accept your "intuition" about them. By "precise meanings" I mean actual observables. I've already described several such, and shown how they are all consistent.

I suspect you know very well what was meant, after so much discussion here - E field distribution is unaffected by gravity *if* RN BH is possible. Said on numerous occasions now and no *reasonable* room for doubt as to what I was once again pointing out.

Q-reeus: "And here's a test case: There is a distant star. Also a distant static charge either directly behind or in front of said star wrt our line-of-sight. A massive BH sweeps across our line-of-sight, between us and star/charge. Gravitational lensing uncontroversially distorts the starlight received. What does your 'math' tell you about the field lines of that charge - will they distort or not? You already know my opinion on that one - but I'm asking for yours."

I already know your opinion? This test case is very different from what we have been discussing; what's the connection? Your "intuition" tells you they are connected? Can you give anything more precise than that? And what observable corresponds to the "field lines" of the charge?

You are kidding right? What in essence is different that so stumps you? Bottom line question to you clearly was - will field lines distort at all. And yes from umpteen previous postings you know exactly my view - of course there will be distortion occurring - but logically/intuitively not if one adheres to the RN-is-real view. And you have a problem knowing what 'distort' could possibly mean? Well there is just strength and direction to consider - and apart from using a standard 'test charge' there are lots of specialized detectors out there nowadays. But any excuse will do to avoid a straight answer - one that I suspect you sense will allow me to have you boxed in. A spot of paranoia on my part here is it? I sincerely hope so. Will give one more try at 'breakthrough' later on.

Q-reeus: "You mean GR cannot give us a value for say dipole field strength as function of r,theta, phi, - with and then without a mass M present?"

GR can tell you what the EM field tensor is due to the dipole. Is that what you mean by "dipole field strength"? Or it can tell you what the contraction of the EM field tensor is with some 4-vector. Is that what you mean by "dipole field strength"? Or do you mean something else?

Did I really need to elaborate? How freeking super precise do you need things to be - on certain occasions! Be reasonable please!

Q-reeus: "There is absolutely no experimental/observational support for a RN BH."

This is true. However, there is lots of experimental support for (1) the EFE as applied to spherically symmetric spacetimes; (2) the EM field tensor and the covariant form of Maxwell's equations as applied to electrodynamics under all conditions we have tested. R-N geometry is simply the result of combining the two.

Ah - "simply the result of combining the two". So absolutely no assumptions are made then as to *how* exactly the two are combined? You do recall I have hammered away on just that thing - beginning in #1?

Also, as has been said before, we can just as well consider the R-N geometry exterior to a charged massive object, rather than a BH. That involves only the R-N region exterior to the horizon, which avoids issues with what goes on inside the horizon, and is therefore an unproblematic solution of the EFE combined with Maxwell's equations.

Really. Well let's try it one more time - as best I care to try at being suffciently precise, here's yet another scenario and yet again ask your considered opinion:

There is an essentially 'point' electric dipole of moment p = qd and small mass m_p that initially lies motionless in notionally flat spacetime, and has a field distribution given here:
http://en.wikipedia.org/wiki/Dipole#Field_from_an_electric_dipole
We next assemble a spherical thin shell of mass M>>m_p and effective mean radius R, surrounding and centered on the dipole. Said mass is ideally transparent to dipole E field, and the sole possible effect of M on the dipole is via M's gravitational potential, and/or field external to R. We know the dipole lies in an equipotential region, depressed wrt infinity according to redshift factor (1-2GM/(Rc²)). So, if the dipole were to act as an oscillator, there is I trust no confusion over saying the emitted radiation will be redshifted by factor given above, as received by a distant observer. Now my hopefully crispy clear and concise question is - will the static dipole's field be in any way altered by the presence of M, as determined by a distant observer? That possible alteration is in field strength and/or field direction different to the 'before' case (you know - before we assembled M).
You do of course have a constitutional right to avoid and evade this question - or make it seem altogether too imprecise to handle. I cross fingers and hope for the best.

 

[Dale]

(and this is the one that the normal force from the table counters to have an static situation from the Earth's frame, but if the table wasn't there the book would change its path)?

I already explained exactly this situation in the third paragraph of 316. If an object is in free fall then the fictitious force does work on the object, changing its path (acceleration), gaining KE, and losing PE.

However, because the table is there the book does not accelerate, does not gain KE, and does not lose PE. No work is done on the book.

Consider the example in the wikipedia page:
"consider a person in a rotating chair holding a weight in his outstretched arm. If he pulls his arm inward, from the perspective of his rotating reference frame he has done work against centrifugal force. If he now let's go of the weight, from his perspective it spontaneously flies outward, because centrifugal force has done work on the object, converting its potential energy into kinetic. From an inertial viewpoint, of course, the object flies away from him because it is suddenly allowed to move in a straight line. This illustrates that the work done, like the total potential and kinetic energy of an object, can be different in a non-inertial frame than an inertial one."

Which again directly contradicts your claims and supports my claims.

I don't know how you are getting this stuff so wrong, I have already addressed every point that you are bringing up, and you seem to be misreading everything written either by myself or by anyone else. I think that you should start a new thread on the topic, I feel like we are hijacking Q-reeus' thread on something that is very tangential to his topic.

 

[TrickyDicky]

If an object is in free fall then the fictitious force does work on the object, changing its path (acceleration), gaining KE, and losing PE.

However, because the table is there the book does not accelerate, does not gain KE, and does not lose PE. No work is done on the book.

An object in free-fall doesn't accelerate in the GR perspective.
No work is done on the book in the Earth's frame and considering gravity a real force. But you are mixing the concept of gravity as real force and fictitious force according to your interest.

I won't continue this discussion with you. If you can't admit it when you are wrong is Ok with me I just wanted to clear that up for the benefit of Austin0 and possible other people reading.

 

[PeterDonis]

You received high praise from one recent poster for an answer in #287 that you may consider 'precise' but to my mind is just a string of words that asserts a generality without really explaining at all. It is trivially true that everything we experience originates from some 'past light cone(s)'. But where is the actual dynamical connection between exterior field point and inwardly hurtling charged matter asymptotically close to being completely engulfed within an EH?

The "dynamical connection" ("dynamical" is kind of an odd word to use since the spacetime is static, but I'll go with it here) is given by the EFE and Maxwell's equations. In the case of a neutral massive object collapsing to a Schwarzschild BH, the vacuum EFE determines how the spacetime curvature produced by the stress-energy in the object propagates through the vacuum region to any point. In the case of a charged massive object collapsing to a R-N BH, the EFE with a particular EM field tensor as source determines how the curvature produced by the stress-energy plus charge propagates, and Maxwell's equations determine how the EM field propagates. There is exact math behind all of it, just as I said.

On the other hand, if I make the point that for there to be any finite exterior BH E field, what most I think reasonably understand by 'gravitational redshift' can have no effect on any EH crossing charge acting as E field source (note this is not an RN situation but neutral BH fed by infall). A rather obvious conclusion I would have thought, but to you it is terribly imprecise and too intuitive to make any sense of.

Because I don't see any exact math behind it. I see you reasoning with intuitive concepts without going back to the fundamentals that underlie them. The fundamental laws, the EFE and Maxwell's equations, don't say anything about "redshift". That's a derived concept that can be used to label certain aspects of certain solutions. It may or may not apply in the case you're talking about.

So, in frustration with that line of response, I furnished for your consideration and response numbers of what seems to me are perfectly straight-forward scenarios - examples in #272, #294, and you still manage to make them into 'confused', imprecise' situations somehow impossible to deal with.

Same comment: I don't see the exact math behind them. You don't know any of the exact math, so you can't furnish it; and when I try to match up what you're saying to what I know of the exact math, I don't come up with a match. As I've said before in a discussion like this with you, if I read through one of your scenarios and I can't figure out how it fits into the math, even if what you are saying contains an apparent "paradox" with GR, my conclusion, if you force me to make one, will be that there is some mistake in your scenario that I'm not smart enough to spot, not that GR is wrong. The former is far more likely, IMO, than the latter. It's not that I'm not interested; it's just that if I can't figure out how to translate what you're saying into the fundamentals, the math, I have no way of telling whether it's right, wrong, or not even wrong. So what am I supposed to do?

E field distribution is unaffected by gravity *if* RN BH is possible.

And what does "E field distribution is unaffected by gravity" mean? I know you can't tell me what it means in the math, but can you at least say what specific observation I can make to tell me whether or not the E field distribution is affected? All the ones you've proposed so far have boiled down to an effect of transmitting something through a curved spacetime, which as I and others have said, is obviously due to the spacetime in between, not to the original "source" of what is being transmitted.

Bottom line question to you clearly was - will field lines distort at all.

And how do I tell, by observation/experiment, whether or not the field lines are "distorted"? Give a specific description set in your scenario, not just generalities about test charges and detectors. Why must I always do all the work?

Did I really need to elaborate?

Yes, because all the things I can come up with that "dipole field strength" could mean, GR can calculate for you, as I said, and none of them pose any contradiction. So if that's the best you can do, your example proves nothing.

Ah - "simply the result of combining the two". So absolutely no assumptions are made then as to *how* exactly the two are combined?

Not in the exact math; combining the two is simple and straightforward and requires no "assumptions" beyond the basics necessary to express any physical law in curved spacetime. All the GR textbooks I'm familiar with treat this exact case in some detail. MTW spends several chapters on it.

Well let's try it one more time - as best I care to try at being suffciently precise, here's yet another scenario

This seems to be your standard response in these discussions: when in doubt, pile on another scenario. :sigh:

I'll go ahead and take a look at this, but let's suppose that I come back and say that the radiation emitted by the dipole (which must be time-dependent to radiate, btw, I trust you've taken that into account even though it doesn't appear in your formulation) *is* redshifted when it is seen by an observer far away, as compared to how it is seen by an observer right next to the dipole. What will that prove?

To you, it will prove that somehow the R-N solution is inconsistent. To me, and probably to most others who are participating in this thread, it will mean that the curved spacetime in between the dipole and the observer has an effect on the radiation, just as one would expect, and will be perfectly consistent. What do we do then?

This is why these discussions always go on forever with no resolution.

 

[Dale]

An object in free-fall doesn't accelerate in the GR perspective.

This is only partly correct. In GR there are two different kinds of acceleration: proper acceleration and coordinate acceleration. An object in free fall has a 0 proper acceleration, but in a non inertial coordinate system it may have a non zero coordinate acceleration. That is essentially the defining characteristic of a non inertial frame.

No work is done on the book in the Earth's frame and considering gravity a real force. But you are mixing the concept of gravity as real force and fictitious force according to your interest.

No work is done on the book in the Earth's frame, period. It has nothing whatsoever to do with gravity being real or fictitious.

In non inertial frames fictitious forces can do work and be associated with a potential energy. I am not mixing the concepts at all, you just apparently don't understand how fictitious forces function.

I won't continue this discussion with you. If you can't admit it when you are wrong is Ok with me I just wanted to clear that up for the benefit of Austin0 and possible other people reading.

Why would I admit I am wrong when you haven't actually shown a single mistake in any of my comments? I haven't said anything incorrect and I have provided references to support. You have consistently misunderstood my comments, the references, and the physics.

Again, I recommend opening another thread in order to avoid hijacking Q-reeus' thread.

 

[Q-reeus]

The "dynamical connection" ("dynamical" is kind of an odd word to use since the spacetime is static, but I'll go with it here)...

Is the infalling charge static? But whatever...

And what does "E field distribution is unaffected by gravity" mean? I know you can't tell me what it means in the math, but can you at least say what specific observation I can make to tell me whether or not the E field distribution is affected? All the ones you've proposed so far have boiled down to an effect of transmitting something through a curved spacetime, which as I and others have said, is obviously due to the spacetime in between, not to the original "source" of what is being transmitted.

I meant what was written in caps font there in #306. If that doesn't convey what I mean by 'unaffected by gravity', not much else to say this end.

Q-reeus: "Bottom line question to you clearly was - will field lines distort at all."
And how do I tell, by observation/experiment, whether or not the field lines are "distorted"? Give a specific description set in your scenario, not just generalities about test charges and detectors. Why must I always do all the work?

What work? It's simply down to a qualitative judgement call here - you either allow that, based on your understanding of character of RN metric, that passing neutral BH has the potential to perturb those field lines, or you think there is no such capability. If you think yes, there must 'intuitively' be an in-principle effect - distribution has changed in some manner we need not quantitatively pin down here for the purposes of deciding.

Yes, because all the things I can come up with that "dipole field strength" could mean, GR can calculate for you, as I said, and none of them pose any contradiction. So if that's the best you can do, your example proves nothing.

But I asked you for a simple yes-or-no opinion on what will happen in that scenario - responding with 'there will be no contradiction' might be a very safe play, but is avoiding the question.

Not in the exact math; combining the two is simple and straightforward and requires no "assumptions" beyond the basics necessary to express any physical law in curved spacetime. All the GR textbooks I'm familiar with treat this exact case in some detail. MTW spends several chapters on it.

This is at least saying something definite I guess. So no paradoxical situations should turn up. If you respond to last given scenario with asked-for definiteness, that may be put to the test.

This seems to be your standard response in these discussions: when in doubt, pile on another scenario. :sigh:

Owing to previous ones being side-stepped, and because it's important to check a theory against a simple scenario(s).

I'll go ahead and take a look at this, but let's suppose that I come back and say that the radiation emitted by the dipole (which must be time-dependent to radiate, btw, I trust you've taken that into account even though it doesn't appear in your formulation) *is* redshifted when it is seen by an observer far away, as compared to how it is seen by an observer right next to the dipole. What will that prove?

Well see, straight away there is a big problem - you just haven't got the real issue there at all. The radiation bit was introduced to establish we agree there is some definable meaning to the term 'redshift'. The key matter is how the *static* dipole field is effected - although one might then come back to see how that meshes with radiative case down the track.

To you, it will prove that somehow the R-N solution is inconsistent. To me, and probably to most others who are participating in this thread, it will mean that the curved spacetime in between the dipole and the observer has an effect on the radiation, just as one would expect, and will be perfectly consistent. What do we do then?

This is why these discussions always go on forever with no resolution.

They can certainly drag-on if the basic scenario is misread from the beginning! And honestly I'm about ready to call it quits but let's just see if you are willing to actually commit to a definite in-principle qualitative judgement call on this one at least. :zzz:

 

[PeterDonis]

Q-reeus, a general comment: most people here at PF, when they propose a scenario, understand that it's their scenario; if other people ask for clarification, they give it. You get asked for clarification, and respond by pointing to previous posts and complaining that nobody understands what you're saying. Which may be true, but doesn't help in having a discussion. If you can't state your scenarios in a way that we can respond to, then we can't respond to them. It's as simple as that.

I meant what was written in caps font there in #306. If that doesn't convey what I mean by 'unaffected by gravity', not much else to say this end.

Then there's not much else for me to respond to at my end.

What work? It's simply down to a qualitative judgement call here - you either allow that, based on your understanding of character of RN metric, that passing neutral BH has the potential to perturb those field lines, or you think there is no such capability.

I mean that I have asked you to describe a specific observable that you interpret as "perturbing the field lines", and instead of giving me a specific answer, you say "it's obvious" or something along those lines. I don't have the time to figure out what you mean; if you mean something specific, then just tell me what. If you don't mean anything specific but just have some vague concept of "field lines perturbing", then there's nothing I can really respond to.

But I asked you for a simple yes-or-no opinion on what will happen in that scenario

Without telling me what specific observable corresponds to "dipole field strength", meaning I can't give a definite yes or no. So nothing more for me to add to my previous responses, where I already described how the observables I can come up with act.

Owing to previous ones being side-stepped

No, they haven't been side-stepped; you just haven't been given answers you like. That doesn't make the answers wrong.

and because it's important to check a theory against a simple scenario(s).

Which we already had, but you keep piling on others.

Well see, straight away there is a big problem - you just haven't got the real issue there at all. The radiation bit was introduced to establish we agree there is some definable meaning to the term 'redshift'.

In terms of energy "redshifting", yes, this is ok. But:

The key matter is how the *static* dipole field is effected

Which is either the same as how the radiation field is affected, or it isn't. If it is, I already answered your question, and showed how it doesn't produce any contradiction. If it isn't, then the radiation is irrelevant; you need to specify what observable *does* correspond to "how the static dipole field is affected". You haven't.

They can certainly drag-on if the basic scenario is misread from the beginning!

That's your opinion; my opinion (and I suspect that of others) is that you haven't specified your scenarios precisely enough, and every time we've been able to pin anything specific down, it's turned out to not involve any contradiction.

And honestly I'm about ready to call it quits but let's just see if you are willing to actually commit to a definite in-principle qualitative judgement call on this one at least. :zzz:

I said I'll take a look at it, but I also asked you a question that you haven't really responded to. What will it prove if I find that the radiation *is* redshifted? Or if it isn't? You basically said the radiation isn't the same as the effect on the static dipole field, so why are you asking about the radiation at all? Why can't you describe a specific observable that *does* correspond to how the static dipole field is affected?

If you insist on a "definite in-principle qualitative judgment" without giving me any more information than you already have, here is one: the radiation redshifts. That's my guess about how the math will turn out. Does that help? Notice I haven't said a thing about how the static dipole field is affected by anything, nor have I committed to any judgment about that. You asked about the radiation.

 

[PeterDonis]

given the formula I posted in #262, deriving the result should be pretty straightforward. I'll put that in a separate post.

Just to close this out, here's the derivation of the ADM mass for a Schwarzschild black hole. We want to show that the ADM mass of a Schwarzschild BH equals the M parameter that appears in the metric. To keep the two distinguished, I'll use $M_{ADM}$ for the ADM mass.

The exact version of the formula I posted in #262 is

$$M_{ADM} = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} g^{ij} \left( g_{ik, j} - g_{ij, k} \right) n^{k} d S$$

where I have put back in the constant factor 1/16pi at the front, and I have also changed index notation to make it clear that the metric g is the 3-metric of a slice of constant time, in this case Schwarzschild coordinate time (which I didn't make clear before). If the metric is diagonal, which it will be in the coordinates we will use, we can simplify this a bit to

$$M_{ADM} = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} g^{ii} \left( g_{ik, i} - g_{ii, k} \right) n^{k} d S$$

which reduces the number of terms we have to sum since i and k only range over 3 indexes (and as we'll see, k ends up ranging over only one).

One thing to note about the above formula is that it is not in covariant form; indeed, as the paper I got it from points out, there is no covariant formula involving first derivatives of the metric except the trivial $g_{ab;c} = 0$. So the above formula is only valid for a restricted class of coordinate systems, which the paper calls "asymptotically Euclidean" coordinates. The paper doesn't define exactly what those are, but I suspect that what is meant is really "asymptotically *Cartesian*" coordinates. To see why, let's first try computing the ADM mass of flat Minkowski spacetime in spherical coordinates; the outward normal to a 2-sphere is just $n^{r} = 1$, with other components zero, and we find that only two nonzero terms survive in the integral:

$$M_{ADM} = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \left[ g^{\theta \theta} \left( - g_{\theta \theta, r} \right) + g^{\phi \phi} \left( - g_{\phi \phi, r} \right) \right] n^{r} d S = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \frac{2}{r^{2}} \left( - r^{2} \right)_{,r} dS = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} - \frac{4}{r} dS$$

Taking the integral is now easy since the integrand is a function of r only; the integral just contributes a factor of $4 \pi r^{2}$, the area of the 2-sphere, which gives

$$M_{ADM} = \lim_{S \rightarrow i^{0}} - r$$

The limit is now just a limit as r -> infinity, so we get minus infinity as the ADM mass of flat spacetime in spherical coordinates. Obviously this is not the right answer. We could fix this by using Cartesian coordinates, but the Schwarzschild metric looks really ugly in those coordinates. An easier way is to just evaluate the ADM mass of Schwarzschild spacetime in appropriate spherical coordinates, and then correct the result by "subtracting off" the above value for flat spacetime. This basically corresponds to adjusting the "zero point" of energy to compensate for the behavior of spherical coordinates at infinity.

I said "appropriate spherical coordinates" just now. I'm not sure if standard Schwarzschild coordinates are actually "appropriate" in this sense, since they are not isotropic. In any case, I've done the computation instead in isotropic coordinates, where it is straightforward. The line element is

$$ds^{2} = K(r)^{4} \left( dr^{2} + r^{2} d\theta^{2} + r^{2} sin^{2} \theta d\phi^{2} \right)$$

where $K(r) = \left( 1 + \frac{M}{2r} \right)$. The unit outward normal to a 2-sphere at r is then $n^{r} = 1 / K(r)^{2}$, with all other components zero. Again, only two nonzero terms survive in the sum (the terms with i = r cancel each other out):

$$M_{ADM} = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \left[ g^{\theta \theta} \left( - g_{\theta \theta, r} \right) + g^{\phi \phi} \left( - g_{\phi \phi, r} \right) \right] n^{r} d S = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \frac{2}{K^{4} r^{2}} \left( - K^{4} r^{2} \right)_{,r} \frac{1}{K^{2}} d S = \frac{1}{16 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \left( \frac{4M}{K^{3} r^{2}} - \frac{4}{K^{2} r} \right) d S$$

As above, the integral just contributes a factor of $4 \pi r^{2}$, and we have:

$$M_{ADM} = \lim_{S \rightarrow i^{0}} \left( \frac{M}{K^{3}} - \frac{r}{K^{2}} \right)$$

Again, the limit is now just a limit as r -> infinity, and in this limit K -> 1. As noted above, we have to correct the "zero point" of energy by subtracting off the result we got above for flat spacetime, so we have

$$M_{ADM} = \lim_{r \rightarrow \infty} \left( \frac{M}{K^{3}} - \frac{r}{K^{2}} + r \right)$$

The last two terms cancel in the limit, and we have $M_{ADM} = M$, as desired.

 

[PeterDonis]

There is an essentially 'point' electric dipole of moment p = qd and small mass m_p that initially lies motionless in notionally flat spacetime, and has a field distribution given here:
http://en.wikipedia.org/wiki/Dipole#Field_from_an_electric_dipole
We next assemble a spherical thin shell of mass M>>m_p and effective mean radius R, surrounding and centered on the dipole.

On considering this, I'm not sure it's a workable scenario as it's stated. If the spacetime starts out flat, then where is the mass that forms the shell coming from? If the mass is somewhere to start with, the initial spacetime can't be flat. So I'm not sure how to proceed, since I can't see a consistent solution of the EFE to work with.

There's at least one obvious way to patch the scenario: have an observer at a finite radius r from the dipole, which is at r = 0, and have a spherically symmetric shell of matter start out at some much larger radius and then be moved slowly inward until it is at some smaller radius than the observer. Then we could talk about changes in what the observer observes before and after the shell is moved inward past him.

But in this scenario, an observer "at infinity" would see no change at all, and since the total mass and charge of the spacetime as a whole are defined by flux integrals "at infinity", they would not change, so there would be nothing much to talk about.

So I think I need some further input before I can analyze this scenario.

 

[PeterDonis]

So I think I need some further input before I can analyze this scenario.

Oh, and another item while I'm thinking of it: the field of a dipole is not spherically symmetric, so R-N geometry would not apply. I could go looking for a solution to the EFE with a dipole EM field as the only stress-energy present, but that's more time than I can spend. Since you're really concerned with the effect on a static EM field, I would think that having a point charge Q at the center, r = 0, instead of a dipole, would be sufficient. But I need input on that as well.

 

[Q-reeus]

Q-reeus, a general comment: most people here at PF, when they propose a scenario, understand that it's their scenario; if other people ask for clarification, they give it. You get asked for clarification, and respond by pointing to previous posts and complaining that nobody understands what you're saying. Which may be true, but doesn't help in having a discussion. If you can't state your scenarios in a way that we can respond to, then we can't respond to them. It's as simple as that.

I make a clear distinction between reasonable requests which I do clarify, versus unreasonable obfuscation that seeks to turn the simple and obvious into an extraordinarily complex mess too hard to handle.

I mean that I have asked you to describe a specific observable that you interpret as "perturbing the field lines", and instead of giving me a specific answer, you say "it's obvious" or something along those lines. I don't have the time to figure out what you mean; if you mean something specific, then just tell me what. If you don't mean anything specific but just have some vague concept of "field lines perturbing", then there's nothing I can really respond to.

Rubbish. Here's what I actually wrote on that in #326:

Bottom line question to you clearly was - will field lines distort at all. And yes from umpteen previous postings you know exactly my view - of course there will be distortion occurring - but logically/intuitively not if one adheres to the RN-is-real view. And you have a problem knowing what 'distort' could possibly mean? Well there is just strength and direction to consider - and apart from using a standard 'test charge' there are lots of specialized detectors out there nowadays.

What exactly is unclear about that, given this was to be a *qualitative* call on nature of gravitational-charge interaction?

Without telling me what specific observable corresponds to "dipole field strength", meaning I can't give a definite yes or no. So nothing more for me to add to my previous responses, where I already described how the observables I can come up with act.

Words fail me.

Q-reeus: "Owing to previous ones being side-stepped"
No, they haven't been side-stepped; you just haven't been given answers you like. That doesn't make the answers wrong.

I won't right now embarrass you by quoting from a lengthy list of previous postings - just in this thread, where you have very clearly side-stepped! Challenge me on this - and I will furnish that list!

Q-reeus: "The key matter is how the *static* dipole field is effected"
Which is either the same as how the radiation field is affected, or it isn't. If it is, I already answered your question, and showed how it doesn't produce any contradiction.

Huh? Where did you do that - please quote!

If it isn't, then the radiation is irrelevant; you need to specify what observable *does* correspond to "how the static dipole field is affected". You haven't.

Are you kidding again? I will again quote from what was actually wrote, in #326:

There is an essentially 'point' electric dipole of moment p = qd and small mass mp that initially lies motionless in notionally flat spacetime, and has a field distribution given here:
http://en.wikipedia.org/wiki/Dipole#Field_from_an_electric_dipole
We next assemble a spherical thin shell of mass M>>m_p and effective mean radius R, surrounding and centered on the dipole. Said mass is ideally transparent to dipole E field, and the sole possible effect of M on the dipole is via M's gravitational potential, and/or field external to R. We know the dipole lies in an equipotential region, depressed wrt infinity according to redshift factor (1-2GM/(Rc²)). So, if the dipole were to act as an oscillator, there is I trust no confusion over saying the emitted radiation will be redshifted by factor given above, as received by a distant observer. Now my hopefully crispy clear and concise question is - will the static dipole's field be in any way altered by the presence of M, as determined by a distant observer? That possible alteration is in field strength and/or field direction different to the 'before' case (you know - before we assembled M).

[bold emphasis added here] Again - I make a clear distinction between reasonable requests for clarification, and unreasonable obfuscation!
[on rereading I see there was an error in omitting square-root for redshift expression there. Pardon such an oversight and allow when referring to that bit subsequently.]

That's your opinion; my opinion (and I suspect that of others) is that you haven't specified your scenarios precisely enough, and every time we've been able to pin anything specific down, it's turned out to not involve any contradiction.

See previous comments.

I said I'll take a look at it, but I also asked you a question that you haven't really responded to. What will it prove if I find that the radiation *is* redshifted? Or if it isn't?

Kidding yet again? How can there be any doubt the radiation redshifts precisely as per what I wrote in #326? That's not the issue - not for me, as I made perfectly clear there.

You basically said the radiation isn't the same as the effect on the static dipole field, so why are you asking about the radiation at all?

I said something different - just reread that bit in #326 and stop misquoting and misconstruing what I did say and mean!

Why can't you describe a specific observable that *does* correspond to how the static dipole field is affected?

I have - but a thoroughly unreasonable attitude may not admit to that being so.

If you insist on a "definite in-principle qualitative judgment" without giving me any more information than you already have, here is one: the radiation redshifts. That's my guess about how the math will turn out. Does that help?

As per above, there should have been no doubt whatsoever - no need to guess - redshift occurs *by definition* given the scenario specs.

Notice I haven't said a thing about how the static dipole field is affected by anything, nor have I committed to any judgment about that. You asked about the radiation.

I asked only about the static case - you should have gotten that much perfectly well from the wording in #326, and you certainly know it now!

 

[Q-reeus]

On considering this, I'm not sure it's a workable scenario as it's stated. If the spacetime starts out flat, then where is the mass that forms the shell coming from? If the mass is somewhere to start with, the initial spacetime can't be flat. So I'm not sure how to proceed, since I can't see a consistent solution of the EFE to work with.

There's at least one obvious way to patch the scenario: have an observer at a finite radius r from the dipole, which is at r = 0, and have a spherically symmetric shell of matter start out at some much larger radius and then be moved slowly inward until it is at some smaller radius than the observer. Then we could talk about changes in what the observer observes before and after the shell is moved inward past him.

I disagree it really matters (see below). Nevertheless use that arrangement if it helps - but then we seem to have shifted to yet another scenario as per your #335.

But in this scenario, an observer "at infinity" would see no change at all, and since the total mass and charge of the spacetime as a whole are defined by flux integrals "at infinity", they would not change, so there would be nothing much to talk about.

Why do you make that faulty assumption? The gravitational potential √(1-2GM/(Rc²)) effecting the charge is a function of shell radius R - which we are free to vary as desired. Are you discounting that potential is *the* chief determinant of whether any observed effect will exist "at infinity"?

 

[Q-reeus]

Oh, and another item while I'm thinking of it: the field of a dipole is not spherically symmetric, so R-N geometry would not apply. I could go looking for a solution to the EFE with a dipole EM field as the only stress-energy present, but that's more time than I can spend. Since you're really concerned with the effect on a static EM field, I would think that having a point charge Q at the center, r = 0, instead of a dipole, would be sufficient. But I need input on that as well.

Not comfortable dealing with a static 'point' dipole? OK then, for now - a point charge it is. Go for it!

 

[Dale]

Ah - "simply the result of combining the two". So absolutely no assumptions are made then as to *how* exactly the two are combined? You do recall I have hammered away on just that thing - beginning in #1

I thought I answered your questions about the assumptions and the derivation in combining the two, and you didn't see anything wrong with it. Now that you mention it, that discussion kind of fell away around 196. Perhaps we should resume it.

In 194 you seemed to be under the misconception that the coupling of the equations was one way, which was your only objection to the assumptions of the RN derivation. I corrected that asssumption by pointing out the effect of the metric on Maxwell's equations. So if that was your only objection about the assumptions then it should be resolved.

 

[Q-reeus]

I thought I answered your questions about the assumptions and the derivation in combining the two, and you didn't see anything wrong with it. Now that you mention it, that discussion kind of fell away around 196. Perhaps we should resume it.

In 194 you seemed to be under the misconception that the coupling of the equations was one way, which was your only objection to the assumptions of the RN derivation. I corrected that asssumption by pointing out the effect of the metric on Maxwell's equations. So if that was your only objection about the assumptions then it should be resolved.

I accepted that the derivation given by Poisson seemed formally correct. Usual interpretation of terms in final RN expression has Q contributing to SET as a function of r, and therefore to gravitating mass M = M(r) as a function of radius r. That is the 'one way' coupling part I referred to. Having trouble seeing where M is in turn effecting the value of Q - i.e. where do we have Q = Q(r) showing up? My point all along is RN metric logically can allow no such Q(r) for the reason that predicted mere existence of any E field exterior to an EH, automatically implies infinite reduction by factor √(1-2GM/(rc²)) there - where Q resides - is operative on matter (gravitational charge for our purposes), yet cannot be operative on Q in the least.[I refer you to examples 2: and 3: in #1 for what I mean here] Thus imo it is indeed one-way coupling according to RN. (Aware there are other couplings to EM - e.g. 'light bending', but not relevant to static RN metric case)

My intuitive style is to query the truth of that via the gedanken experiments that have been thrown up - we should reasonably expect a fully self-consistent picture. You will be well aware I claim there are discrepancies thereby revealed - e.g. field energy vs field strength as function of gravitational potential. Those energy/field parameters can be well enough defined for any given scenario. And imo #1 and #248 taken together cover the situation quite well. Hopefully a certain scenario will now be worked through and definite predictions can then be picked over. Not holding my breath though.

 

[stevendaryl]

This would be correct only if the table is considered an inertial object (as it is done in most practical physics exercises, there is a lab frame considered to be at rest,it is an idealization that works great for most practical problems), but we know that is not the case in reality, the table is non-inertial and in continuous motion so there is work done. An accelerometer in the surface of the Earth measures proper acceleration.

I think you're trying to apply concepts that make sense in flat spacetime to curved spacetime. In flat spacetime, if you have a table accelerating upward, and you have a book sitting on the table, then the table (or more accurately, whatever is accelerating the table) is doing work on the book. It takes energy to keep the table accelerating upward, and the presence of the book requires even more energy.

In curved spacetime, it's just not true that the table holding up the book means that anything is doing work. If you have a planet with a solid surface (such as the Earth) and a table is sitting on the Earth, and a book is sitting on the table, then nothing is doing any work on the table, and nothing is doing any work on the book.

 

[stevendaryl]

I accepted that the derivation given by Poisson seemed formally correct. Usual interpretation of terms in final RN expression has Q contributing to SET as a function of r, and therefore to gravitating mass M = M(r) as a function of radius r. That is the 'one way' coupling part I referred to. Having trouble seeing where M is in turn effecting the value of Q - i.e. where do we have Q = Q(r) showing up?

Why would you expect that Q would vary with radius? M varies with radius because an electric field has energy (and therefore mass), but an electric field has no charge, so there is no reason to think Q varies with radius. (I think if you take into account vacuum polarization, then Q actually does vary with distance, but that's going beyond classical GR and electrodynamics).

My point all along is RN metric logically can allow no such Q(r) for the reason that predicted mere existence of any E field exterior to an EH, automatically implies infinite reduction by factor √(1-2GM/(rc²)) there

This is where you are definitely NOT using logic. You say that something implies infinite reduction by a factor of √(1-2GM/(rc²)). What does that mean, and how are you deriving it?

My intuitive style is to query the truth of that via the gedanken experiments that have been thrown up - we should reasonably expect a fully self-consistent picture.

You can't explore the consistency of a set of assumptions unless you know what those assumptions are. This "infinite reduction by a factor of √(1-2GM/(rc²))" is not an assumption of GR. It's something you made up. Now, that factor does come into play in the Schwarzschild geometry. For example, a clock at "rest" at radius r in Schwarzschild cooridnates will show an elapsed time $\tau$ given by:

d$\tau$/dt = √(1-2GM/(rc²))

where t is the Schwarzschild time coordinate.

 

[TrickyDicky]

I think you're trying to apply concepts that make sense in flat spacetime to curved spacetime. In flat spacetime, if you have a table accelerating upward, and you have a book sitting on the table, then the table (or more accurately, whatever is accelerating the table) is doing work on the book. It takes energy to keep the table accelerating upward, and the presence of the book requires even more energy.

In curved spacetime, it's just not true that the table holding up the book means that anything is doing work. If you have a planet with a solid surface (such as the Earth) and a table is sitting on the Earth, and a book is sitting on the table, then nothing is doing any work on the table, and nothing is doing any work on the book.

Well, the thing is that in GR the situation you describe in your first paragraph is made equivalent to the situation you describe in the second paragraph (at least locally but I'm taking the liberty of considering the book a local object) by virtue of the well known Equivalence principle, and that is the only way to consider gravity a fictitious force. See what I mean?

 

[PeterDonis]

I make a clear distinction between reasonable requests which I do clarify, versus unreasonable obfuscation that seeks to turn the simple and obvious into an extraordinarily complex mess too hard to handle.

You *think* you do. Others apparently disagree. I've commented in previous threads that you have a very different idea from me (and others) as to what constitutes a clear and precise statement of a scenario. I doubt we're going to settle that here, so I won't bother commenting further along those lines. The upshot of your #336 is basically that you've given all the information you can about the stuff I asked about. Fair enough; then I'll give all the information I can in response, and that will be it. If it's not enough for you, sorry, it's the best I can do.

Why do you make that faulty assumption? The gravitational potential √(1-2GM/(Rc²)) effecting the charge is a function of shell radius R - which we are free to vary as desired.

The *definition* of the mass M and the charge Q, as observables, depends on flux integrals taken at infinity--more precisely, on taking the limit of a flux integral over a 2-sphere of radius r as r goes to infinity. So in the definition of M and Q, you are *not* free to vary the radius "as desired"--you have to take the limit as r goes to infinity.

Are you discounting that potential is *the* chief determinant of whether any observed effect will exist "at infinity"?

Actually, if you look at the post I made about calculating the ADM mass of a Schwarzschild BH, you will see that the "potential" does not appear at all. The potential is related to the metric coefficient $g_{tt}$. The flux integral for the ADM mass involves spatial metric coefficients only. I'm working on a similar calculation for R-H spacetime to show that M and Q both result from the appropriate flux integrals, using spatial metric coefficients.

It's also worth noting that I did the ADM mass calculation in isotropic coordinates, where $g_{tt}$ is *not* the reciprocal of $g_{rr}$, as it is in Schwarzschild coordinates, so you can't say that the spatial metric coefficients are related to the "potential". As I commented in the other post, I'm not sure the calculation will work in Schwarzschild coordinates; it looks to me like the coordinates have to be isotropic.

Not comfortable dealing with a static 'point' dipole?

Not able to spend the time to work with a more complicated solution to the EFE. You did catch the fact that a dipole is not spherically symmetric, right? That means the R-N geometry does not apply, just as Schwarzschild would not apply to the exterior of a non-spherically-symmetric configuration of mass. I don't know of a handy simple solution to the EFE for the dipole case, and I don't have time to figure one out myself.

 

[PeterDonis]

The *definition* of the mass M and the charge Q, as observables, depends on flux integrals taken at infinity

It occurred to me after writing this that looking at the flux integral for Q might provide a quick way of addressing the "does charge redshift" question.

In post #262 I wrote down an expression for the "ADM charge" of a spacetime; however, I forgot that there needs to be a factor of 1/4 pi in front (at least, in "natural" units for GR; in SI units there would be an $\epsilon_{0}$ somewhere). So the exact integral is

$$Q_{ADM} = \frac{1}{4 \pi} \lim_{S \rightarrow i^{0}} \int_{S} F_{ab} n^{a} u^{b} dS$$

Here there is no weirdness with coordinates since we aren't taking any derivatives of the metric, and we should be able to use the standard Schwarzschild-type coordinates. So we just have the outward-pointing normal $n^{r} = K(r)$, where $K(r) = \sqrt{1 - 2M/ r + Q^{2} / r^{2}}$ is the factor appearing in the denominator of the r-r component of the R-N metric; we have the timelike vector $u^{b} = (1, 0, 0, 0)$, and we have $F_{r0} = Q / r^{2}$ for the EM field tensor. Putting it all together gives

$$Q_{ADM} = \frac{1}{4 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \frac{Q}{r^{2}} K(r) dS = \frac{1}{4 \pi} \lim_{r \rightarrow \infty} 4 \pi r^{2} \frac{Q}{r^{2}} K(r) = \lim_{r \rightarrow \infty} Q K(r) = Q$$

as desired. But what happens if we look at the integral at a finite radius r? We get

$$Q(r) = Q K(r) = Q \sqrt{1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}}}$$

i.e., the "charge seen at radius r" is "redshifted" relative to the "charge at infinity".

Thoughts, Q-reeus?

 

[Q-reeus]

Q-reeus: "I accepted that the derivation given by Poisson seemed formally correct. Usual interpretation of terms in final RN expression has Q contributing to SET as a function of r, and therefore to gravitating mass M = M(r) as a function of radius r. That is the 'one way' coupling part I referred to. Having trouble seeing where M is in turn effecting the value of Q - i.e. where do we have Q = Q(r) showing up?"

Why would you expect that Q would vary with radius? M varies with radius because an electric field has energy (and therefore mass), but an electric field has no charge, so there is no reason to think Q varies with radius.

I refer you to my 'intuitive' resolution in #248 for what that means re Q(r) 'in effect'.

Q-reeus: "My point all along is RN metric logically can allow no such Q(r) for the reason that predicted mere existence of any E field exterior to an EH, automatically implies infinite reduction by factor √(1-2GM/(rc²)) there"
This is where you are definitely NOT using logic. You say that something implies infinite reduction by a factor of √(1-2GM/(rc²)). What does that mean, and how are you deriving it?

Have you gone back as per directive in #340 and looked at the situations there? You have a different slant that actually makes sense? No equations per se there, but please think about it. And there is some supplementary considerations below.

Q-reeus: "My intuitive style is to query the truth of that via the gedanken experiments that have been thrown up - we should reasonably expect a fully self-consistent picture."
You can't explore the consistency of a set of assumptions unless you know what those assumptions are. This "infinite reduction by a factor of √(1-2GM/(rc²))" is not an assumption of GR. It's something you made up. Now, that factor does come into play in the Schwarzschild geometry. For example, a clock at "rest" at radius r in Schwarzschild cooridnates will show an elapsed time τ given by:
dτ/dt = √(1-2GM/(rc²))
where t is the Schwarzschild time coordinate.

Indeed. But not just clock-rate - energy/mass also (being careful to distinguish 'resting' from 'free-fall', and 'coordinate' from 'local' situations). And if the latter can be equated to 'gravitational charge' (T₀₀ part of SET), what is the plain english explanation why one can freely vary as function of 'redshift', but not the other? Just remember here - in standard GR, SET has zero contribution from gravitational field. Which if one follows that through, requires imo rest energy/mass should exhibit a Newtonian 'charge' strictly analogous to electric charge. So if as is the case div'g' = 4πρ_m fails, there is no logical case imo for div E = ρ_e/ε₀ also not to fail equally - i.e. gravity 'redshifts' rest energy and charge to equal degree. There are possibly a number of ways of interpreting that. I gave for charge the effective permittivity/permeability angle in #248, but one might think in terms of 'effective distance' from source perhaps. Personally I think gravitational field energy logically should be a part of SET, but here I have been strictly working from standard GR position it is not.

 

[Q-reeus]

You *think* you do. Others apparently disagree. I've commented in previous threads that you have a very different idea from me (and others) as to what constitutes a clear and precise statement of a scenario. I doubt we're going to settle that here, so I won't bother commenting further along those lines. The upshot of your #336 is basically that you've given all the information you can about the stuff I asked about. Fair enough; then I'll give all the information I can in response, and that will be it. If it's not enough for you, sorry, it's the best I can do.

Fair enough in return.

The *definition* of the mass M and the charge Q, as observables, depends on flux integrals taken at infinity--more precisely, on taking the limit of a flux integral over a 2-sphere of radius r as r goes to infinity. So in the definition of M and Q, you are *not* free to vary the radius "as desired"--you have to take the limit as r goes to infinity.

Understand that in terms of 'at infinity' definitions, but there is it seems a confusion of r with R here. My R referred to the shell radius specifically and it's that R that governs the depressed potential experienced by any charge inside said shell. We are discussing the effect (or not!) of that potential on the distantly observed E field. Thus collapse/expand R and we have a changed E (or not) as measured at r = 'infinity'.

Actually, if you look at the post I made about calculating the ADM mass of a Schwarzschild BH, you will see that the "potential" does not appear at all. The potential is related to the metric coefficient gtt. The flux integral for the ADM mass involves spatial metric coefficients only. I'm working on a similar calculation for R-H spacetime to show that M and Q both result from the appropriate flux integrals, using spatial metric coefficients.

It's also worth noting that I did the ADM mass calculation in isotropic coordinates, where gtt is *not* the reciprocal of grr, as it is in Schwarzschild coordinates, so you can't say that the spatial metric coefficients are related to the "potential". As I commented in the other post, I'm not sure the calculation will work in Schwarzschild coordinates; it looks to me like the coordinates have to be isotropic.

You've lost me a bit here and my non-specialist impression is these are fine distinctions not really important to whether mass effects charge or not in the manner being debated.

Not able to spend the time to work with a more complicated solution to the EFE. You did catch the fact that a dipole is not spherically symmetric, right? That means the R-N geometry does not apply, just as Schwarzschild would not apply to the exterior of a non-spherically-symmetric configuration of mass. I don't know of a handy simple solution to the EFE for the dipole case, and I don't have time to figure one out myself.

Fine, understand. If we can arrive at a definite predictive result for simplest case as per #334 that should suffice.

 

[Q-reeus]

It occurred to me... Putting it all together gives
$$Q_{ADM} = \frac{1}{4 \pi} \lim_{S \rightarrow i^{0}} \int_{S} \frac{Q}{r^{2}} K(r) dS = \frac{1}{4 \pi} \lim_{r \rightarrow \infty} 4 \pi r^{2} \frac{Q}{r^{2}} K(r) = \lim_{r \rightarrow \infty} Q K(r) = Q$$
as desired. But what happens if we look at the integral at a finite radius r? We get
$$Q(r) = Q K(r) = Q \sqrt{1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}}}$$
i.e., the "charge seen at radius r" is "redshifted" relative to the "charge at infinity".
Thoughts, Q-reeus?

Interesting result there Peter. It is of course one derived from standard RN metric. My interpretation is it's describing an effectively *locally depressed* central Q, supplemented progressively by an effective space-charge of the same sign as central Q, yielding a Q at infinity whose magnitude is independent of M.
[Edit: That observation pertains to M completely dominating Q terms, otherwise quite a bit more complex with possible inflexion points etc.]
Heuristic observation, and one more or less opposite to what I proposed in #248. There, locally observed Q is independent of M, and is progressively reduced by an effective dielectric shielding, until at large r we finish with Q' = √(-g_tt)Q.
It would be interesting to work through the contrasting predictions for field energy.

 

[Dale]

I accepted that the derivation given by Poisson seemed formally correct.

Excellent. If you use a formally correct derivation from correct axioms then you are guaranteed to get a correct result. Since the axioms are the EFE and ME, and since those have both been extensively validated (the EFE to the best experimental measurements possible and ME in the classical limit of QED), the axioms seem reasonably correct.

So we have a set of correct axioms, a formally correct derivation from those axioms, and therefore logically must have a correct result.

Usual interpretation of terms in final RN expression has Q contributing to SET as a function of r, and therefore to gravitating mass M = M(r) as a function of radius r. That is the 'one way' coupling part I referred to. Having trouble seeing where M is in turn effecting the value of Q - i.e. where do we have Q = Q(r) showing up?

I don't know what you are talking about here. Neither M nor Q are functions of r, they are constant parameters which describe the entire spacetime. Where are you getting this idea that M is a function of r or that Q should be also?

Thus imo it is indeed one-way coupling according to RN. (Aware there are other couplings to EM - e.g. 'light bending', but not relevant to static RN metric case)

Did you miss the coupling I posted in 196?

My intuitive style is to query the truth of that via the gedanken experiments that have been thrown up - we should reasonably expect a fully self-consistent picture. You will be well aware I claim there are discrepancies thereby revealed - e.g. field energy vs field strength as function of gravitational potential. Those energy/field parameters can be well enough defined for any given scenario. And imo #1 and #248 taken together cover the situation quite well. Hopefully a certain scenario will now be worked through and definite predictions can then be picked over. Not holding my breath though.

When intuition arrives at a different conclusion than logic, then we can be sure that our intuition is not correct. The logical derivation is what ensures that the result is "a fully self-consistent picture", no amount of intuitive gedankens can change that.

You could correctly analyze millions of self-consistent gedankens, and that would not prove that the next gedanken would not be inconsistent. So gedankens cannot prove consistency.

And since it is always possible to make a scenario:
a) difficult enough to analyze that even the most brilliant person would fail
b) sufficiently vague that there is no way to analyze it
c) contain a hidden inconsistency so that the gedanken itself is illogical
so failure to correctly analyze a gedanken can always be attributed to mistakes in specification or skill or the gedanken rather than any real inconsistency in the laws. So gedankens cannot prove inconsistency either.

That is why the logical derivations are so important. The logical derivation has been laid out for you, and I don't see any objections other than a kind of general personal distaste for formal derivations.

 

[Dale]

The last two terms cancel in the limit, and we have $M_{ADM} = M$, as desired.

I got 2M in the limit.