Is 'charged black hole' an oxymoron?

[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.

 

[Q-reeus]

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.

Not necessarily - I italicized *formally* on purpose. It's already been noted elsewhere recently how local things - conservation/divergence laws, can strictly hold locally yet globally fail in GR.

Q-reeus: "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?

There was discussion over the meaning of Q term as effective spatial SET contribution to M back in e.g. #20 and before/later. I see now from #345 Peter has pointed out that Q has an *effective* spatial variation even within RN model.

Q-reeus: "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 194?

You mean #191. Gets back to my first comments above.

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.

More than a distaste imo. Let's wait a bit for specific predictions for 'charge centred within a shell'. After that we will have more to go on.

 

[Dale]

Not necessarily - I italicized *formally* on purpose.

Do you agree or disagree with the following statement:

"If you use a formally correct derivation from correct axioms then you are guaranteed to get a correct result."

There was discussion over the meaning of Q term as effective spatial SET contribution to M back in e.g. #20 and before/later.

I looked at 20. That seemed to show that acceleration was a function of r, which I certainly agree to. It did not seem to show that M or Q are a function of r.

You mean #191.

Sorry, I meant 196. Please look back at that since you seem to have missed it. It shows the coupling back from the EFE to ME.

More than a distaste imo. Let's wait a bit for specific predictions for 'charge centred within a shell'. After that we will have more to go on.

Why? What is the purpose of the exercise? Is it not to determine self-consistency? We have a correct derivation from correct axioms, therefore it is self-consistent, your personal "more than a distaste" notwithstanding. The thought experiments are simply not relevant in determining self-consistency, for the reasons I pointed out above.

However, if the purpose is to learn some GR by doing some advanced "homework" problems, then I think that the scenarios are worth working through. However, I think that they are pretty advanced for me and most likely also too advanced for you. I am learning a lot by doing the parallel transport exercise.

 

[Q-reeus]

Do you agree or disagree with the following statement:
"If you use a formally correct derivation from correct axioms then you are guaranteed to get a correct result."

"Correct axioms" is imo the suspect link in that chain.

I looked at 20. That seemed to show that acceleration was a function of r, which I certainly agree to. It did not seem to show that M or Q are a function of r. Neither M nor Q are functions of r, so I don't see why their failure to be functions of r should be construed as any kind of a problem.

We are talking effective quantities. Nobody including myself is suggesting locally evident failure of anything. If coordinate acceleration a(r) is properly equated to effective gravitating mass M(r), need I say more?

Sorry, I meant 196. Please look back at that since you seem to have missed it. It shows the coupling back from the EFE to ME.

OK yes I did miss it. Advanced math notation already for me - but really you will surely agree that is a finished statement not a first-principles derivation.

Why? What is the purpose of the exercise? Is it not to determine self-consistency? We have a correct derivation from correct axioms, therefore it is self-consistent, your personal "more than a distaste" notwithstanding. The thought experiments are simply not relevant in determining self-consistency, for the reasons I pointed out above.

Relax DS. My first comment above stands for now. Look at it this way - PF is about educating, right? So if nothing else, if my Quoxotic crusade falls flat as you and hordes others expect, at least a detailed following through of said exercise will do it's little bit in fulfilling the educational PF goal here - yes?

 

[Dale]

"Correct axioms" is imo the suspect link in that chain.

So you agree with the statement, but think that the axioms for deriving the RN spacetime are not correct.

Can you please point out exactly which axiom is incorrect in your view?

We are talking effective quantities. If coordinate acceleration a(r) is properly equated to effective gravitating mass M(r), need I say more?

OK, I don't know the definition of "effective mass" or "effective charge". What is the equation for them to be "properly equated"?

OK yes I did miss it. Advanced math notation already for me - but really you will surely agree that is a finished statement not a first-principles derivation.

It is simply the source-free Maxwell's equations in curved spacetime. I consider that first principles. Do you object to Maxwell's equations as an axiom?

 

[Q-reeus]

So you agree with the statement, but think that the axioms for deriving the RN spacetime are not correct.
Can you please point out exactly which axiom is incorrect in your view?

You should have figured it by now - I work it 'backwards'. Setup a gedanken experiment. Check for consistent predictions. If lacking, trial various possible fixes. Key example here is my #248. Don't ask me to press that into a new axiom. I don't pretend to be a university educated boffin. Just try at being a logical thinker.

OK, I don't know the definition of "effective mass" or "effective charge". What is the equation for them to be "properly equated"?

My take: In #14, first 'a=' RN equation, imo whatever terms modify that expression from purely Schwarzschild solution involving just neutral M. Hence everything there with a Q expresses the effective mass contribution at r owing to distributed E field energy of Q up to that r (or the 'subtraction' from larger than r version). As for effective charge, perhaps you can give your own interpretation on what last expression in #345 is suggesting. My own idea is there in #248. Nothing rigorous, but it seems to be adding up better than elsewhere.

It is simply the source-free Maxwell's equations in curved spacetime. I consider that first principles. Do you object to Maxwell's equations as an axiom?

"In curved spacetime" - my little gedankens are worrying me about the marriage of ME's to such. Feeling a bit feisty at the moment btw?

 

[stevendaryl]

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?

Not EVERYTHING is the same when comparing constant acceleration in flat spacetime with being at rest in a gravitational field. In particular, there is a big difference in that an observer in freefall in flat spacetime sees an unchanging metric, while an observer in freefall in the gravitational field of the Earth sees a time-varying metric.

Here's a way to sort of make sense of work and energy in curved spacetime.

Let's assume a special case to make things simple: In terms of Schwarzschild coordinates, we have a weak nongravitational force corresponding to a static electric potential \Phi. We'll assume that the force is so weak that its contribution to the metric is negligible, so we can use the Schwarzschild metric. Then the following quantity is conserved for a particle of mass m and charge q:

E = mc²√A/√(1-v²/(c²A²)) + q \Phi

where A = 1 - 2GM/(c²r) = the Schwarzschild factor. Notice that in the limit as GM → 0, A → 1 and we get the SR limit:

E = mc²/√(1-v²/c²) + q\Phi

When M is nonzero, the expression for E doesn't cleanly split up into "potential energy" and "kinetic energy"; the first term includes both velocity-dependent and position-dependent quantities. However, we can approximate E in the case where v/c << 1, and GM/(c²r) << 1. In that case, this expression simplifies to

E = mc² - GMm/r + 1/2 mv² + q \Phi

So in this limit, we can see that any work done by the electric field on the particle goes into either increasing the kinetic energy 1/2 mv² or in increasing the gravitational potential energy - GMm/r, just as in Newtonian physics. It's still the case that if r is constant, then no work is being done to the particle by the electric field.

In flat spacetime with an accelerating rocket, we can look at things from two different points of view: from the point of view of an inertial observer, the only energy is kinetic energy, and work must be done in order to keep the rocket accelerating.

From the point of view of an accelerating observer (Rindler coordinates), the rocket is stationary, and no work is being done on the rocket. So where does the energy used up by the rocket go? It goes into throwing exhaust gases backwards. Those exhaust gases are NOT stationary, and so work is done to get them moving backwards.

 

[stevendaryl]

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

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.

You aren't starting with what General Relativity says about these things. You're making stuff up, and then reasoning from it in a sloppy, non-logical way. If you end up with a contradiction, you can't blame it on GR, because you're not using GR.

You object to using complicated mathematics, but mathematics is nothing more than being precise. By not using mathematics, you're not being precise, you're being sloppy. Sloppiness generates apparent inconsistencies where none exist.

 

[Q-reeus]

You aren't starting with what General Relativity says about these things. You're making stuff up, and then reasoning from it in a sloppy, non-logical way. If you end up with a contradiction, you can't blame it on GR, because you're not using GR.

You object to using complicated mathematics, but mathematics is nothing more than being precise. By not using mathematics, you're not being precise, you're being sloppy. Sloppiness generates apparent inconsistencies where none exist.

We've been here before haven't we? I won't debate that kind of polemic.

 

[stevendaryl]

We've been here before haven't we? I won't debate that kind of polemic.

It wasn't polemic--it was, I believe, an accurate description of what you are doing.

 

[stevendaryl]

It wasn't polemic--it was, I believe, an accurate description of what you are doing.

It's actually an explanation for why you are getting inconsistencies--they are your own creation, they are not from GR. That's harsh, but it's accurate.