Are the directions of electric fields lines affected by Gravity?

[Edward Solomo]

I recall that the path of light itself can be altered by gravity, then, being part of the electromagnetic force, then is it safe to assume that the paths of electric fields lines can also be warped?

I would imagine that the consequences would be enormous for electric flux in curved space.

For instance, the flux about a point charge is given by (1/4)(1/pi)(1/Eo)Q/r^2 by Gauss's law INTEGRAL[E * dA] take a sphere so the E field is tanget to the sphere and decreases at a uniform rate from the center, then you get Q/Eo = E(4pir^2), then solve for E.

However, if we're in a region of highly curved space, then the E field no longer retains its uniform rate of change from the center of a point charge. The flux lines would appear something like the picture attached below.

Eventually all of the flux lines would concentrate themselves at a singular point, as if the field of the point charge originated from two or more locations.

Or conversely, in a rapidly expanding area of space, which would be negatively curved, the E-field would then diminish even faster than 1/r^2, much like a dipole, which decreases at a rate of 1/r^3. In a region with a strong enough negative curve, the electric field about a point charge could reach zero over a finite distance (or possibly have reverse effects, an electric field under time reversal? Such as electrons attracting other electrons?)

Of course this sounds like a bunch of nonsense, but the Gaussian laws would suggest this given my current understanding, and thus my understanding must be severely flawed!

 

[Reptillian]

Yep, electric field lines are warped since the space itself is warped. One of the more interesting ideas I've seen is the idea that electrons themselves are a kind of microscopic wormhole. Electric field lines converge on the wormhole giving the sort of point particle behavior we've come to expect. Positrons would be the other end of the wormhole. :D The probelm I see with this view is that you still have a sort of electromagnetic field filling all of spacetime separate from gravity. Which I suppose isn't all that different from quantum field theory in which every fundamental particle has a unique field associated with it that fills the universe. Seems conceptually icky to me. Ideally, the electromagnetic field would itself be understood in terms of geometry.

 

[pervect]

Field lines work fine with General Relativity, and are a fine way to visualize electromagnetism. The tubes of the field lines are representations of anti-symmetric tensors, also known as a "differential forms", of rank 2, referred two as two-forms.

Elelctromagnetism turns out to be represented by a rank 2 anti-symmetric tensor, so it's a perfect match.

See for instance http://125.71.228.222/wlxt/ncourse/DCCYDCB/web/condition/9.pdf , "Teaching Electromagnetic Field Theory Using Differential Forms", IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997. This doesn't specifically talk about the usefulness of two forms with GR, but it does talk about using two forms to teach E&M.

Gravity turns out to be much harder - in general it takes a rank 4 tensor to represent it, and while it has a high degree of symmetry, it's not a differential form.

 

[Muphrid]

It may be easier to talk about EM fields not as "differential forms" but in terms of bivectors. The EM field is a vector field in spacetime but a field of oriented planes. The most natural way to think of this iis to take the electric field lines you know and extend them in the time dimension to draw out sheets. This is the natural object that describes the EM field, and boosting these planes is what gives the relationship between electric and magnetic fields.

This also explains why gravity isn't usually said to have field lines. The Riemann tensor isn't a bivector field like the EM field; rather, it's a linear operator on bivectors.

 

[Q-reeus]

Maybe an occasion to at last get some kind of resolution to what was proposed but never settled here: https://www.physicsforums.com/showthread.php?p=3946413
Seems to be from above entries a general agreement E field will be warped by gravitation. That makes good sense to me, yet how is that squared with the generally accepted as mathematically sound Reissner-Nordstrom charged black hole, where at large r the E field, owing to charge infalling at the infinitely redshifted BH EH (event horizon) is asymptotically close to the flat spacetime expression q/(4πε₀r²)? Does it make any sense to say E field lines could in general be gravitationally bent somehow (as suggested by illustration in #1) yet with absolutely no change in field strength - which the mere existence of a RN BH surely implies? To put this concretely in a very simple and less extreme context, suppose we have a uniformly charged thin spherical mass shell. What does the marriage of EM with GR predict for the dependence of radial E field at large r on the mass M of that shell? Or suppose instead of the surface charge, a 'point' electric dipole is placed at the center of that mass shell. Will the field lines be altered at large r owing to M (assuming, apart from any purely gravitational effects, shell electromagnetic transparency - i.e. ε_r=1)?

 

[PeterDonis]

Does it make any sense to say E field lines could in general be gravitationally bent somehow (as suggested by illustration in #1) yet with absolutely no change in field strength - which the mere existence of a RN BH surely implies?

Hi, Q-reeus, yes, we never really got closure on the questions raised in that other thread. I actually did work through a lot of the computations but never got to a point where I felt ready to post them. If I have time I'll try to go back and look again.

Regarding the specific point in your quote above, in a R-N BH the E field lines are not bent; they all stick straight out, radially, from the BH, all the way out to infinity. That has to be true by spherical symmetry. So that particular case may not be the best one to investigate whether and how gravity can bend EM field lines in a more general scenario that doesn't have that special symmetry.

 

[Q-reeus]

Hi, Q-reeus, yes, we never really got closure on the questions raised in that other thread. I actually did work through a lot of the computations but never got to a point where I felt ready to post them. If I have time I'll try to go back and look again.

Hi Peter. It would be good to see things resolved there (and/or here) if possible (big if!).

Regarding the specific point in your quote above, in a R-N BH the E field lines are not bent; they all stick straight out, radially, from the BH, all the way out to infinity. That has to be true by spherical symmetry. So that particular case may not be the best one to investigate whether and how gravity can bend EM field lines in a more general scenario that doesn't have that special symmetry.

Agreed and I probably could have clarified slightly better that E field bending was not implied in as you say spherically symmetric RN case. However the point there was that imo field warpage of any kind is inconsistent with RN implied total absence of coupling between source charge gravitational potential and far field radial E field strength (radial was specified). Logically *static* EM field (no disputing EM radiation does couple to gravity) is either effected or not by gravity as a package - can't in general admit to directional warpage without logically field strength also being effected. And you know my arguments re finite RN E field.

 

[PeterDonis]

imo field warpage of any kind is inconsistent with RN implied total absence of coupling between source charge gravitational potential and far field radial E field strength (radial was specified).

I'm not sure I would describe such a coupling, if it is present in the spherically symmetric RN case, as "field warpage". But I think by "warpage" you really mean "some observable effect on the field", which seems more general to me and would cover the RN case.

 

[jartsa]

I recall that the path of light itself can be altered by gravity, then, being part of the electromagnetic force, then is it safe to assume that the paths of electric fields lines can also be warped?

I would imagine that the consequences would be enormous for electric flux in curved space.

For instance, the flux about a point charge is given by (1/4)(1/pi)(1/Eo)Q/r^2 by Gauss's law INTEGRAL[E * dA] take a sphere so the E field is tanget to the sphere and decreases at a uniform rate from the center, then you get Q/Eo = E(4pir^2), then solve for E.

However, if we're in a region of highly curved space, then the E field no longer retains its uniform rate of change from the center of a point charge. The flux lines would appear something like the picture attached below.

Eventually all of the flux lines would concentrate themselves at a singular point, as if the field of the point charge originated from two or more locations.

Or conversely, in a rapidly expanding area of space, which would be negatively curved, the E-field would then diminish even faster than 1/r^2, much like a dipole, which decreases at a rate of 1/r^3. In a region with a strong enough negative curve, the electric field about a point charge could reach zero over a finite distance (or possibly have reverse effects, an electric field under time reversal? Such as electrons attracting other electrons?)

Of course this sounds like a bunch of nonsense, but the Gaussian laws would suggest this given my current understanding, and thus my understanding must be severely flawed!

Two identical charges side by side in a gravity field might repel each other in these alternative ways:
1: both experience some lift
2: both experience some down force
3: for some very odd reason the identical charges experience a different vertical force
4: the forces are opposite and horizontal

Well quite obviously number 4 is the only reasonable of these alternatives.

We know that two observers side by side in a gravity field must look upwards, if they want to see each other, but electric charges are felt as being in the direction where they really are.

 

[Q-reeus]

But I think by "warpage" you really mean "some observable effect on the field", which seems more general to me and would cover the RN case.

Yes it just means here 'any departure from flat spacetime field configuration' - field pattern or field strength.

It may be worth reminding at this stage of a finding you made in another thread that isotropic Minkowski spacetime spatial metric components interior to a spherical mass shell are identical to asymptotic exterior Schwarzschild values at infinity. Consequently if one collapses together the plates of a charged parallel plate capacitor lying inside such a shell, it's coordinate determined travel distance is identical to that when done at infinity. Alternately simply discharge the plates at constant separation - the capacitor field effective volumes are identical inside and at infinity. Yet we know that the energy release when operation is performed within the shell is depressed wrt infinity by the redshift factor √-gtt = √(1-2GM/(rc²)).

And that imo inescapable fact presents a rather difficult dilemma for RN supporters (i.e. practically whole of GR community). I raised this scenario several times in the other thread linked to in #5, but worth raising it here again. Cuts right through all the mathematical elegance and sophistication of the standard picture. Which basically wants to have it's cake (RN metric with E field unaffected by infinitely depressed grav redshift) and eat it too (standard grav redshift applies to emitted radiation etc.).

Could splitting between an invariant 'active' charge |E| = q_a/(4πε₀r²) and potential effected 'passive' charge F = q_pE, with q_p = q√-g_tt, allow just that? It certainly works for the collapsing capacitor plates discharge scenario energy-wise. And for the static plates discharge too if we identify gravitationally depressed electrostatic energy density with W = 1/2√-g_ttε₀|E|². Even works if applied to output of a dipole oscillator where field output is reduced solely owing to redshifted frequency - both input and output power drops as -g_tt (radiation field strength drops proportional to frequency and thus as √-g_tt), as required. Alas there is at least one fatal failing.

Suppose we separate two charged spheres apart via a dielectric rod, equal and opposite charges but with one difference - one sphere is more massive than the other. If RN is true, q_a's are identical and hence magnitude of E field induced by one charged sphere on the other. However we had to have q_p dependent on potential, so action and reaction are unequal, with a greater force exerted on the less massive sphere. Newton not happy.

A better approach imo that has no such inconsistency at least for our static scenario was suggested in #248 in the other thread linked in #5. Leave charge as just plain q but with coordinate values for vacuum permittivity and permeability given by: ε,μ = (1/√-g_tt)(ε₀,μ₀). We now have that coordinate determined light speed c, EM energy and power densities, and Newton's 3rd law for electrostatics (and similarly for magnetostatics) all work out good. I do not claim this can be automatically extended to more complex situations involving general motion in non-static, non-symmetric spacetimes, but then it does not fall badly at the first hurdle either!

 

[Q-reeus]

They're going for gold in London, but a different type of gold applies to this thread - as one golden oldie goes 'Silence is golden, golden...' Or another: 'The kiss of death, from Mr Gold Finger...' Gee, hate to think that's my label here. Anyway just in case someone wants to step out of the shadows, here's another thought following on from #10.

Say we have a steady current loop as magnetic dipole of moment m, arranged coaxial with and centred about an electric dipole p of the same energy (as determined by the mutual mm and pp forces of attraction/repulsion between other identical such dipoles). Two such mp pairs, but one pair with opposite relative orientation between m and p, will, provided separations are not too close to invalidate 'point dipole' approximation, experience no net interaction forces or torques. That's in flat spacetime. What about if these pairs are within the spherical mass shell of #10? Still flat spacetime there, but potential depressed by √-g_tt. Locally, non-interaction between dipole pairs must still hold. Which is interesting, since coordinate value of drift speed of magnetic dipole circulating currents is depressed wrt infinity by √-g_tt, and so therefore the magnitude of the magnetic dipole moments m - assuming as per RN metric implies, charge itself as a source of E field is invariant wrt potential.

By RN metric reckoning, the coordinate value for electric dipole moments p are unaffected by the surrounding mass shell (charge magnitude and displacement distance both indifferent to g_tt). Locally then, within the shell, net interaction-free mp pairs, but by coordinate measure there is net interaction owing to dominance by the electric dipoles?! Just doesn't add-up imo. And this coordinate imbalance persists regardless of whether one applies the split into 'active' and 'passive' charge done in #10 or not. Oddly perhaps the force imbalance problem between charged spheres as per #10 cancels out when applied to an equivalent situation involving current loops of differing mass and therefore locally differing √-g_tt factors. Small comfort.

Now try the suggested cure: depressed values for ε,μ = (1/√-g_tt)(ε₀,μ₀). From the Wiki article here: http://en.wikipedia.org/wiki/Dipole, we have expressions for magnetic dipole:

and electric dipole:

At first sight the inverse relative locations of ε₀,μ₀ in the two expressions appears in conflict with having each directly modified by 1/√-g_tt factor - seemingly implying an increased B field in conflict with an expected reduction in E field. Recall however that the magnetic moment m is that owing to circulating charges, which in coordinate measure not only circulate slower by √-g_tt, but with reduced effective charge by the same factor. Hence m by itself is reduced by the factor -g_tt, the field B then further modified by μ = 1/√-g_ttμ₀ whereas electric moment p is unaffected in the above expression for E, it's reduced effective charge appearing outside of p itself - in the single modifier 1/√-g_tt for ε₀. Properly interpreted and applied then, there is imo a harmony here that works. That a well-known 94 yo solution to EFE's is directly under challenge seems to be a matter of sheer indifference here at PF. Or maybe Mr GF just generates too much perplexity/fear/loathing for comfort.

 

[PeterDonis]

It may be worth reminding at this stage of a finding you made in another thread that isotropic Minkowski spacetime spatial metric components interior to a spherical mass shell are identical to asymptotic exterior Schwarzschild values at infinity.

Yes, I did make such a finding. But it doesn't mean what you think it means. The fact that the spatial metric coefficients are the same does not mean you can directly compare distances inside the shell to distances at infinity the way you are trying to. See further comments below.

However, your capacitor scenario is irrelevant to the questions you've raised about RN spacetime, because in the capacitor scenario, unless I'm misunderstanding something, the EM field is supposed to be zero everywhere except between the capacitor plates. So the EM fields of both capacitors are purely local, and a purely local analysis is all that can be applied to them.

discharge the plates at constant separation - the capacitor field effective volumes are identical inside and at infinity.

You're not stating this precisely enough. The correct statement is that the *locally measured* capacitor plate separation is the same inside the shell and at infinity. See below.

Yet we know that the energy release when operation is performed within the shell is depressed wrt infinity by the redshift factor √-gtt = √(1-2GM/(rc²)).

Let's restate this scenario more precisely. We have two capacitors: C1 is at infinity, C2 is inside a spherical mass shell. The proper distance between the plates, measured locally, is D for both C1 and C2. Both capacitors are stipulated to start in the same state of charge, again as measured locally. That means that the same energy E, measured locally, will be released when we discharge each capacitor. So measured locally, the capacitors are identical.

It is true that an observer at infinity will see energy E coming from C1, but energy fE coming from C2, where f < 1 is the "redshift factor" inside the spherical mass shell. What accounts for the difference? Obviously that the energy was redshifted as it climbed out of the gravity well. It doesn't imply any local change in C2 as compared to C1; as we saw above, locally the two capacitors are identical. That means the energy redshifting has nothing to do with the local behavior of the EM field. (If you want, we can transport the energy from C2 to infinity by some means other than EM radiation.) The difference in observed energy received at infinity is purely due to the effects of the intervening spacetime; it has nothing to do with the local properties of C2 vs. C1.

 

[PeterDonis]

That a well-known 94 yo solution to EFE's is directly under challenge seems to be a matter of sheer indifference here at PF.

I've only given your latest scenario a quick read-through, but you are making an awful lot of claims about what GR and the RN metric supposedly say, without backing them up with math. This has happened before. You also continue to insist on focusing on coordinate-dependent quantities instead of invariants, which has also happened before. And you insist on comparing coordinate-dependent quantities in different parts of spacetime (e.g., at infinity vs. inside the shell) and then insisting that these comparisons have some physical meaning. This has happened before as well. All of these are reasons why your proposed challenges are so often received with what you term "sheer indifference".

Furthermore, it's not clear to me exactly what you think you are challenging. Are you trying to claim that the R-N metric is not a solution of the EFE? That's ludicrous; it's easy to check, and thousands of physics undergraduates probably check it every year as a homework assignment. Are you trying to claim that the EFE is wrong? Good luck with that; within its domain of validity it has a lot of experimental support (to put it mildly). Are you trying to claim that the R-N solution, while mathematically correct, is somehow "unphysical"? Well, yes, it is, in at least two ways. One way, which is IMO not that big an issue, is that it assumes exact spherical symmetry; but that's easy to justify as an idealization which is often pretty well approximated by real objects. The other way, which is more of an issue, is that the interior of the R-N solution has a number of features that make it not very reasonable physically. But neither of those issues are relevant to your proposed challenge; you are also assuming perfect spherical symmetry, and your challenge can be applied purely in the exterior region of R-N spacetime. So I'm not sure exactly what you think your challenge is supposed to prove.

 

[Q-reeus]

Yes, I did make such a finding. But it doesn't mean what you think it means. The fact that the spatial metric coefficients are the same does not mean you can directly compare distances inside the shell to distances at infinity the way you are trying to.

Sure it does - as obviously implied in #10 and explicitly stated in that other thread linked to in #5. We can, especially obviously for collapsing plates scenario, use ropes & pulleys, rods & bell cranks etc. and make a 1:1 correspondence that is entirely physically meaningful. Do you challenge that? How so exactly if you do?

However, your capacitor scenario is irrelevant to the questions you've raised about RN spacetime, because in the capacitor scenario, unless I'm misunderstanding something, the EM field is supposed to be zero everywhere except between the capacitor plates. So the EM fields of both capacitors are purely local, and a purely local analysis is all that can be applied to them.

Asserting my examples are irrelevant does not make them so - see my last comments.

Q-reeus: "discharge the plates at constant separation - the capacitor field effective volumes are identical inside and at infinity."
You're not stating this precisely enough. The correct statement is that the *locally measured* capacitor plate separation is the same inside the shell and at infinity.

A trivial true statement which ignores what's going on here. Go back and check context of that entire first main para in #10, esp. preceeding sentence to above. Actually, don't bother; here it is in full:

It may be worth reminding at this stage of a finding you made in another thread that isotropic Minkowski spacetime spatial metric components interior to a spherical mass shell are identical to asymptotic exterior Schwarzschild values at infinity. Consequently if one collapses together the plates of a charged parallel plate capacitor lying inside such a shell, it's coordinate determined travel distance is identical to that when done at infinity. Alternately simply discharge the plates at constant separation - the capacitor field effective volumes are identical inside and at infinity. Yet we know that the energy release when operation is performed within the shell is depressed wrt infinity by the redshift factor √-g_tt = √(1-2GM/(rc²)).

No reasonable excuse for taking it other than how it is clearly meant to be taken. And how it then subsequently leads on.

Q-reeus: "Yet we know that the energy release when operation is performed within the shell is depressed wrt infinity by the redshift factor √-g_tt = √(1-2GM/(rc²))."
Let's restate this scenario more precisely. We have two capacitors: C1 is at infinity, C2 is inside a spherical mass shell. The proper distance between the plates, measured locally, is D for both C1 and C2. Both capacitors are stipulated to start in the same state of charge, again as measured locally. That means that the same energy E, measured locally, will be released when we discharge each capacitor. So measured locally, the capacitors are identical.

Another 'precise' and equally trivially true statement.

It is true that an observer at infinity will see energy E coming from C1, but energy fE coming from C2, where f < 1 is the "redshift factor" inside the spherical mass shell. What accounts for the difference? Obviously that the energy was redshifted as it climbed out of the gravity well.

And via ropes & pulleys etc. we can tie that down to equivalent fields and forces acting 'down there'. As per earlier comments. And it's not just 'absolutes' - this time as per your opening comments in #13, think carefully about the example given in #11 - we have a number of ratios 'paradoxes' of RN making there. If it's all so easy to resolve within standard picture, offer your own full explanation/resolution please, up to your own standards of 'precision'. I maintain logic behind RN metric inevitably leads to paradox.

It doesn't imply any local change in C2 as compared to C1; as we saw above, locally the two capacitors are identical. That means the energy redshifting has nothing to do with the local behavior of the EM field.

Please do not keep repeating that red herring - you aught to know perfectly well it is a false representation of what I have been saying all along. And I don't like going around in circles.

(If you want, we can transport the energy from C2 to infinity by some means other than EM radiation.) The difference in observed energy received at infinity is purely due to the effects of the intervening spacetime; it has nothing to do with the local properties of C2 vs. C1.

Well the intervening spacetime doesn't have any effect on charge as source of E if RN metric is true. Do you disagree? How so if so?
Having read your #13 best I don't respond to it because it's style and tone invites a slanging match we can do without. what would impress in a positive way wold be for you to apply that undoubted prowess with 'the math' to at the very least the simple scenario of #5, and actually commit to a definite prediction. Why was that not done quite some time ago in that other thread? Here I have given an increased choice of alternative configurations - all laughably simple for someone of your grasp of GR surely. The one time you tentatively presented what passed as a proper math solution, not for for charged shell but RN exterior E field was here:https://www.physicsforums.com/showpost.php?p=3969524&postcount=345
My somewhat delicate response was in #348, and you came back, very briefly, in #366 with what looked to be a quiet disowning of the original. Fair enough I suppose, but time to properly settle at minimum the charged shell case, wouldn't you say? Then we might come back to the highly relevant scenarios here and see it all in a new light.

So may I humbly suggest we put off vague accusations of 'imprecision' etc. Once the long awaited kosher GR solution for charged spherical mass shell materializes, discussions of objective substance will be possible. Of course I do not accept there is anything lacking in logical rigor to that already given here in #10 & #11, but I understand the imo underlying Sacred Cow mentality that is denied in word but adhered to in practice. Don't think I'm singling you out on that score - far from it. Must go. :zzz:

 

[PeterDonis]

We can, especially obviously for collapsing plates scenario, use ropes & pulleys, rods & bell cranks etc. and make a 1:1 correspondence that is entirely physically meaningful.

And any such linking mechanism will be affected by the curvature of spacetime in between, just as the energy being transmitted back upward is.

Please do not keep repeating that red herring - you aught to know perfectly well it is a false representation of what I have been saying all along.

I know no such thing. I wasn't making a statement about what you were saying; I was making a statement about what GR says. Did you read the part where I said I was assuming that the EM field is zero everywhere except inside the capacitor plates?

Well the intervening spacetime doesn't have any effect on charge as source of E if RN metric is true.

If you'll remember, in one of those other threads I said I needed to go back and re-evaluate what I said about the charge integral being unchanging, since it didn't seem to be consistent with other "obvious" things about the R-N metric.

Having read your #13 best I don't respond to it because it's style and tone invites a slanging match we can do without. what would impress in a positive way wold be for you to apply that undoubted prowess with 'the math' to at the very least the simple scenario of #5, and actually commit to a definite prediction. Why was that not done quite some time ago in that other thread?

Because such things take time and effort. First, before even constructing a math model, I have to be sure I understand your scenario correctly. Your way of stating scenarios does not always make that easy. Please understand that I'm not saying that as a criticism, and I apologize for coming across as critical in my previous post. But it is a fact that I (and apparently others here on PF) sometimes find it difficult to understand your scenarios. That makes it difficult to model them in the math. Obviously *you* understand your own scenarios, but you can't model them in the math, so there has to be a translation step involved, and often it doesn't appear to work very well.

Second, please understand that I am participating in these discussions because I find them interesting and fun, but that doesn't mean they are at the top of my priority list. (I suspect this goes for others as well who have said things about trying to model these things in math.) I understand that you believe you have a knock-down refutation of GR, and I also understand that nobody here on PF has yet presented a counterargument that you consider valid. That doesn't change the fact that I, and probably others, are applying a heuristic that says that, when presented with a scenario like those you have proposed, which is claimed to refute GR, it is far more likely that the person who is proposing the scenario has made a mistake somewhere, than that it actually refutes GR. That doesn't mean we have actually found the mistake; nor is it a *proof* that there must be a mistake. It's just a heuristic judgment in order to set our priorities for how much time and effort we are willing to spend in trying to either verify your claims, or find your mistake.

I understand that you think it *should* be at the top of all our priority lists (or at least higher than it is now), because you think you are right; you think you *have* found a refutation of GR. You even think it is obvious. But if you are really, really convinced that you have found a refutation of GR, then you should not be depending on us to provide the proof for you. *You* should learn the math yourself; *you* should learn the physics yourself. *You* should be able to write up your arguments and proofs in the standard language of the field. Yes, that's a lot of work; but if you're right, there's probably a Nobel Prize at the end of it. Of course part of the reason I am telling you this is that I think there's a very low probability that you are right; so I think that if you actually do all that work, you will end up just finding out for yourself where you have made mistakes in the scenarios you have proposed here. But I do think the effort is worthwhile quite independently of where you end up with regard to those particular scenarios. Of course that's another heuristic judgment of mine, with which you may disagree.

 

[pervect]

Maybe an occasion to at last get some kind of resolution to what was proposed but never settled here: https://www.physicsforums.com/showthread.php?p=3946413
Seems to be from above entries a general agreement E field will be warped by gravitation. That makes good sense to me, yet how is that squared with the generally accepted as mathematically sound Reissner-Nordstrom charged black hole.

Gauss's law says that we can define the charge enclosed by a surface as the integral of the surface area multiplied by the electric field normal to the surface. There's a very similar theorem in two forms, when the integration process is carried out appropriately.

The easiest way to define the apropriate sense of integration is to use the idea of counting field lines. You can also envision it as a more traditional integral carried out by dividing the whole area into a large number of small pieces, by insisting that you use observers with unit diagonal metrics - i.e. Lorentzian observers - to carry out the integration of each piece.

Consider a space-like sphere surrounding a black hole. This can easily be visualized for any point outside the event horizon by a sphere of constant schwarzschild radius r. The surface area of said 3-sphere will be 4 pi r^2, by he definition of the Schwarzschild r coordinate.

The event horizon isn't space-like , but null. But you can't have a static observer at the event horizon anyway, so it makes no sense to ask what the E-field would be there.

So it is necessary and sufficient that (4/3 pi r^2) * E_normal = constant for charge to be conserved. Which is in fact the prediction of the RN metric when you convert the metric using the usual transformation rules to that of a locally Lorentzian observer.

Note that this is a different behavior than what a local measure of gravitation will give. The integral of local normal force * area (defined in the special way we did above) isn't constant for gravity.

 

[PeterDonis]

The surface area of said 3-sphere will be 4/3 pi r^2, by he definition of the Schwarzschild r coordinate.

This should be "2-sphere" and surface area 4 pi r^2, correct? (Vs. "3-sphere" and 4/3)

So it is necessary and sufficient that (4/3 pi r^2) * E_normal = constant for charge to be conserved. Which is in fact the prediction of the RN metric when you convert the metric using the usual transformation rules to that of a locally Lorentzian observer.

This was my thought, too, in the other thread that Q-reeus linked to; it's why I later said that the calculation in the specific post he linked to didn't look right to me, because it didn't match other things. The calculation was:

Q = \frac{1}{4 \pi} \int{F_{ab} n^{a} u^{b} dS}

where dS is the surface element of a 2-sphere, n^{a} is the outward-pointing normal to the 2-sphere, and u^{b} is the 4-velocity of a static observer--or, more relevant for this calculation, it's the 4-velocity of an observer whose worldline is orthogonal to the spacelike hypersurface in which the 2-sphere lies. Since none of the quantities in the integrand depend on the angular coordinates, the integral just gives a factor of 4 \pi r^2 and we end up with

Q(r) = r^2 F_{ab} n^{a} u^{b}

I inserted u^{b} = (1, 0, 0, 0), n^{a} = (0, \sqrt{1 - 2M / r + Q^2 / r^2}, 0, 0), and F_{10} = Q / r^{2} to obtain

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

which, as I noted in that old post, could be interpreted as the "charge at radius r" being "redshifted" relative to the "charge at infinity". However, by your argument, we should get Q(r) = Q for all r, which would require u^{b} = (1 / \sqrt{1 - 2M /r + Q^2 / r^2}, 0, 0, 0); i.e., the timelike vector would have to be normalized using the metric to be a unit vector. The latter seems more physically reasonable to me, but I haven't been able to convince myself definitely one way or the other, if I don't use the fact that I already know the answer ought to be constant, Q(r) = Q at all r.

 

[PAllen]

Pervect did say:

" You can also envision it as a more traditional integral carried out by dividing the whole area into a large number of small pieces, by insisting that you use observers with unit diagonal metrics - i.e. Lorentzian observers - to carry out the integration of each piece."

This alone would rule out u^b being (1,0,0,0) in RN coordinates. Also, I have never seen this symbol used for anything other than a timelike unit vector.

 

[pervect]

This should be "2-sphere" and surface area 4 pi r^2, correct? (Vs. "3-sphere" and 4/3)

Yes, I wanted the surface of the 3 sphere, which is a 2 sphere, and it's obviously 4 pi r^2

As far as your calculation goes, I haven't looked it over in great detail, but if u^b = (1,0,0,0) g_00 u^b u^b is not equal to 1, so it's not a unit length vector, because g_00 is not unity. At least if you're using the coordinates I think you are (Schwarzschild).

A four-velocity should have unit length, it's only (1,0,0,0) if g_00 = 1.

 

[PeterDonis]

Yes, I wanted the surface of the 3 sphere, which is a 2 sphere, and it's obviously 4 pi r^2

Ok, good.

As far as your calculation goes, I haven't looked it over in great detail, but if u^b = (1,0,0,0) g_00 u^b u^b is not equal to 1, so it's not a unit length vector, because g_00 is not unity. At least if you're using the coordinates I think you are (Schwarzschild).

Yes, I'm using Schwarzschild coordinates. I agree the u^b vector I wrote down is not a timelike unit vector (and so I shouldn't have called it a "4-velocity", as PAllen pointed out); however, it is the timelike Killing vector, correct? So the question is, which is appropriate for the formula I wrote down: the timelike unit vector (in which case the integral gives Q at any radius r), or the timelike Killing vector (in which case the integral gives Q(r), which includes a "redshift factor" at any finite r)?

 

[Q-reeus]

Q-reeus: "We can, especially obviously for collapsing plates scenario, use ropes & pulleys, rods & bell cranks etc. and make a 1:1 correspondence that is entirely physically meaningful."
And any such linking mechanism will be affected by the curvature of spacetime in between, just as the energy being transmitted back upward is.

So you do challenge on that one. We are here dealing with a static spacetime scenario - basically Schwarzschild exterior/Minkowski interior metrics vanishingly perturbed by the presence of what amounts to test charge/currents. Given that, I maintain it's down to just a direct comparison of two spacetime regimes - the one 'down there' (shell interior region) vs the one 'out here' (coordinate observer). The 'intervening spacetime' is an irrelevancy. There is no 'path dependency' muddying things imo - just state dependency. You do I hope acknowledge that our ropes & pulleys etc. are, *as is customary for this kind of gedanken situation*, idealized as light and stiff to the point of not being active participants in the dynamics - they merely convey force, motion, power, from one regime to the other. Given that, your finding of spatial components equal to that at asymptotic infinity most certainly does allow a 1:1 correspondence, just as I claimed. Again, do you disagree, and if so on what substantive basis? Be precise please.

I know no such thing. I wasn't making a statement about what you were saying; I was making a statement about what GR says. Did you read the part where I said I was assuming that the EM field is zero everywhere except inside the capacitor plates?

And you have failed to grasp the point of that cap plate situation - the potential affected energetics directly imply corresponding changes in the field and/or charge response to those fields. Chiming 'locally nothing changes' is another irrelevancy - RN metric is a statement about how charge 'down there' is received 'out here', and my scenarios relate on that same basis. Trivializing things by insisting on a vacuous 'local only' perspective, which reveals nothing, practically by definition. If as I claim consistency requirements imply a coordinate depressed capacitor field, applied then to a charged shell, that situation logically then projects out from 'there' to 'here'. How could it be otherwise?

Because such things take time and effort. First, before even constructing a math model, I have to be sure I understand your scenario correctly. Your way of stating scenarios does not always make that easy. Please understand that I'm not saying that as a criticism, and I apologize for coming across as critical in my previous post. But it is a fact that I (and apparently others here on PF) sometimes find it difficult to understand your scenarios. That makes it difficult to model them in the math. Obviously *you* understand your own scenarios, but you can't model them in the math, so there has to be a translation step involved, and often it doesn't appear to work very well.

Right then, if my scenarios are at all difficult to understand, that criticism aught to be accepted and acted upon. Just can't personally see anything presented in #5, 10, 11, that is at all unclear, hard to grasp, or difficult to evaluate. As you apparently do, kindly point out any and all specific instances that qualifies in your opinion as unclear/difficult. Please - this may help us both.

Second, please understand that I am participating in these discussions because I find them interesting and fun, but that doesn't mean they are at the top of my priority list.

No problem there.

(I suspect this goes for others as well who have said things about trying to model these things in math.) I understand that you believe you have a knock-down refutation of GR,

Let's get one thing clear on that. While you and others are quite aware I have broader misgivings about GR (recall my threads on role of stress, gravity does/does not gravitate etc.), here I have deliberately assumed SM is correct, and am merely challenging, by way of gedanken experiments, the assumptions underlying marriage of ME's and EFE's. Which amounts to challenging the assumption Gauss's law for charge holds in curved spacetime. There is no claim that failure of RN metric undermines all of GR.

and I also understand that nobody here on PF has yet presented a counterargument that you consider valid. That doesn't change the fact that I, and probably others, are applying a heuristic that says that, when presented with a scenario like those you have proposed, which is claimed to refute GR, it is far more likely that the person who is proposing the scenario has made a mistake somewhere, than that it actually refutes GR. That doesn't mean we have actually found the mistake; nor is it a *proof* that there must be a mistake. It's just a heuristic judgment in order to set our priorities for how much time and effort we are willing to spend in trying to either verify your claims, or find your mistake.

And I perfectly understand that attitude. More to say later and in a subsequent thread.

I understand that you think it *should* be at the top of all our priority lists (or at least higher than it is now), because you think you are right; you think you *have* found a refutation of GR. You even think it is obvious. But if you are really, really convinced that you have found a refutation of GR, then you should not be depending on us to provide the proof for you. *You* should learn the math yourself; *you* should learn the physics yourself. *You* should be able to write up your arguments and proofs in the standard language of the field. Yes, that's a lot of work; but if you're right, there's probably a Nobel Prize at the end of it. Of course part of the reason I am telling you this is that I think there's a very low probability that you are right; so I think that if you actually do all that work, you will end up just finding out for yourself where you have made mistakes in the scenarios you have proposed here. But I do think the effort is worthwhile quite independently of where you end up with regard to those particular scenarios. Of course that's another heuristic judgment of mine, with which you may disagree.

And I sure do disagree. Unless, as I have requested above, you think that my scenarios are too hard or unclear to grasp, there really is no excuse but to tackle the specifics of those, and point out 'precisely' where each is iyo manifestly wrong. And this is where it's really hard to understand the problem with you and others avoiding doing just that. Take just the case in #11 of having both an electric and magnetic dipole present within the mass shell. Now either you must claim coordinate values are of no value (and what would that be saying about validity of SM?), or one can quite readily identify the effect of the various metric components on those two entities, and draw valid conclusions. Depends on your response to my first para above no doubt. I reject the notion that at that level it's got to be a drawn out arduous affair re setting up EFE's etc. etc. The relevant basic characteristics are well known (time dilation, spatial metric components). Further, limiting things to weak gravity regime and minimally perturbing test charge/currents is assumed and perfectly adequate here.

 

[Q-reeus]

Gauss's law says that we can define the charge enclosed by a surface as the integral of the surface area multiplied by the electric field normal to the surface. There's a very similar theorem in two forms, when the integration process is carried out appropriately.

The easiest way to define the apropriate sense of integration is to use the idea of counting field lines. You can also envision it as a more traditional integral carried out by dividing the whole area into a large number of small pieces, by insisting that you use observers with unit diagonal metrics - i.e. Lorentzian observers - to carry out the integration of each piece.

But Consider a space-like sphere surrounding a black hole. This can easily be visualized for any point outside the event horizon by a sphere of constant schwarzschild radius r. The surface area of said 3-sphere will be 4 pi r^2, by he definition of the Schwarzschild r coordinate.

The event horizon isn't space-like , but null. But you can't have a static observer at the event horizon anyway, so it makes no sense to ask what the E-field would be there.

So it is necessary and sufficient that (4/3 pi r^2) * E_normal = constant for charge to be conserved. Which is in fact the prediction of the RN metric when you convert the metric using the usual transformation rules to that of a locally Lorentzian observer.

And it's here where I believe there is a flaw. That r is what again? Identically the flat spacetime value when gravity is 'switched off'? Shouldn't we have to in effect do a line integration over dr' = dr√grr to arrive at proper distance r' to q, and one has that r' is always greater than r - infinitely so at the EH. So the source charges 'down there' are really 'further away' than one 'expects'. Seen in that light it seems crazy to expect Gauss's law to hold. I have earlier couched things in terms of potential affected permittivity and permeability, but that is an alternative perspective.

This is all going down a familiar route, and it's best to take a different tack imo. I'm a great believer in properly thought out gedanken experiments. Whether it is taken as arrogant or not, I shall borrow and paraphrase a line from Galileo; ""Gentlemen, please, look through the telescope!" In this case, the 'telescope' is the small collection of scenarios in #10, 11 (and that other thread, but let's not expand too much here). I can only appeal to reason on this - if RN metric etc. is so logically secure, pray tell the harm in actually applying it to those scenarios, and showing how it gives consistency. If there is no interest or willingness to do that, please understand my continued scepticism. Yes this is repeating commentary from the last post - for a reason.

 

[PeterDonis]

We are here dealing with a static spacetime scenario - basically Schwarzschild exterior/Minkowski interior metrics vanishingly perturbed by the presence of what amounts to test charge/currents. Given that, I maintain it's down to just a direct comparison of two spacetime regimes - the one 'down there' (shell interior region) vs the one 'out here' (coordinate observer). The 'intervening spacetime' is an irrelevancy. There is no 'path dependency' muddying things imo - just state dependency.

I agree there is no "path dependency" in the sense that it doesn't matter at what time you make the comparison; you do the comparison now, I do it next year, we both get the same result (whatever that result is). That's what "static" means. (There *is* a "path dependency" in the sense that a purely radial path is different than a path with an angular component--but I've been assuming that we're only talking about purely radial paths here, so that's a non-issue.)

But there is certainly a *radial* dependency; when you lower and raise things from one height to another, *things change*. You can't ignore those changes if you're trying to compare things at different heights. Perhaps this is what you mean by "state dependency"; I'm not sure.

You do I hope acknowledge that our ropes & pulleys etc. are, *as is customary for this kind of gedanken situation*, idealized as light and stiff to the point of not being active participants in the dynamics - they merely convey force, motion, power, from one regime to the other.

Sure, no problem here.

Given that, your finding of spatial components equal to that at asymptotic infinity most certainly does allow a 1:1 correspondence, just as I claimed.

No, it doesn't. You are mistaking the components of the metric for actual physical observables. They're not, not directly. You have to specify *how* you are making the observation, and how what you observe relates to the components of the metric. In this case, any observation you make to compare something inside the shell with something at infinity requires *something* to traverse the intervening spacetime, where the curvature causes things to change. That affects your observations because even though the metric coefficients at the start and end points are the same, they are not the same in between.

Chiming 'locally nothing changes' is another irrelevancy - RN metric is a statement about how charge 'down there' is received 'out here'

Here again you are mistaking metric coefficients for direct physical observables. The "M" and "Q" that appear in the metric are *defined* based on observations made "at infinity". There is no assertion that they *must* correspond to "mass" or "charge" observed at some finite radius r. Also, the metric tells you about physical distances and times; it doesn't, by itself, tell you about other things like charges.

Right then, if my scenarios are at all difficult to understand, that criticism aught to be accepted and acted upon. Just can't personally see anything presented in #5, 10, 11, that is at all unclear, hard to grasp, or difficult to evaluate. As you apparently do, kindly point out any and all specific instances that qualifies in your opinion as unclear/difficult. Please - this may help us both.

I've tried this before and it hasn't worked. Plus, it amounts to the same thing as trying to figure out where you're making a mistake: you're asking *me* to expend time and effort, which I'm going to prioritize like I said.

here I have deliberately assumed SM is correct, and am merely challenging, by way of gedanken experiments, the assumptions underlying marriage of ME's and EFE's. Which amounts to challenging the assumption Gauss's law for charge holds in curved spacetime. There is no claim that failure of RN metric undermines all of GR.

Ah, ok, this helps to clarify the specific point at issue--for this thread, anyway.

Unless, as I have requested above, you think that my scenarios are too hard or unclear to grasp, there really is no excuse but to tackle the specifics of those, and point out 'precisely' where each is iyo manifestly wrong. And this is where it's really hard to understand the problem with you and others avoiding doing just that.

See my comments about heuristic judgment and expending effort. To say "there really is no excuse" is to say that there is a high enough probability that you are right to justify the effort. But I don't believe there is a high enough probability. It's not that I'm "avoiding" making the effort; it's just that it's low on my priority list because of the low expected payoff (from my perspective). If your estimate of the expected payoff is higher, then *you* should be doing the work of learning the math so *you* can go through the derivation of the R-N metric, the EM field tensor that goes with it, the trajectories it predicts for charged objects, etc., etc., and point out where *they* are wrong, inconsistent, physically unreasonable, or whatever. The burden of proof is on you, not me.

Now either you must claim coordinate values are of no value (and what would that be saying about validity of SM?)

It's not that "coordinate values are of no value" (I *think* I understand what you mean by this), it's that they aren't directly observable; they don't have any direct physical interpretation.

or one can quite readily identify the effect of the various metric components on those two entities, and draw valid conclusions.

I understand that you think you can do this, and have done so. I'm not sure I agree, which is why I want to check it with the more accurate math.

 

[PeterDonis]

And it's here where I believe there is a flaw. That r is what again?

The Schwarzschild radial coordinate r, which is *defined* so that the area of a 2-sphere with radial coordinate r is 4 pi r^2.

Identically the flat spacetime value when gravity is 'switched off'?

This has no physical meaning; if gravity is present, there is no way to physically observe anything corresponding to "the flat spacetime value when gravity is switched off".

Shouldn't we have to in effect do a line integration over dr' = dr√grr to arrive at proper distance r' to q

Not if all you're doing is counting field lines passing through a 2-sphere. See above.

 

[Q-reeus]

But there is certainly a *radial* dependency; when you lower and raise things from one height to another, *things change*. You can't ignore those changes if you're trying to compare things at different heights. Perhaps this is what you mean by "state dependency"; I'm not sure.

It was meant as dr₂/dr₁ = (√g_rr)₁/(√g_rr)₂, regardless of what goes on in any amount of intervening static spacetime - assuming our ideal rope & pulleys. Hence a radial tug of x cm out at r₂ produces a radial motion of x(√g_rr)₂/(√g_rr)₁ cm in at r₁. And conveyed inside shell, displacement isotropy applies. Are you saying this is not so?

Q-reeus: "Given that, your finding of spatial components equal to that at asymptotic infinity most certainly does allow a 1:1 correspondence, just as I claimed."

No, it doesn't. You are mistaking the components of the metric for actual physical observables. They're not, not directly. You have to specify *how* you are making the observation, and how what you observe relates to the components of the metric. In this case, any observation you make to compare something inside the shell with something at infinity requires *something* to traverse the intervening spacetime, where the curvature causes things to change. That affects your observations because even though the metric coefficients at the start and end points are the same, they are not the same in between.

There is no confusion here between total traversal time/distance, vs end-point-to-end-point differential motions via an ideally rigid conveyor? I have already made a statement above about this.

Q-reeus: "...RN metric is a statement about how charge 'down there' is received 'out here'"

Here again you are mistaking metric coefficients for direct physical observables. The "M" and "Q" that appear in the metric are *defined* based on observations made "at infinity". There is no assertion that they *must* correspond to "mass" or "charge" observed at some finite radius r. Also, the metric tells you about physical distances and times; it doesn't, by itself, tell you about other things like charges.

How many times must I repeat - that any exterior field exists implies RN 'active' charge has no dependence on metric. There is infinite redshift at EH!

Q-reeus: "or one can quite readily identify the effect of the various metric components on those two entities, and draw valid conclusions."

I understand that you think you can do this, and have done so. I'm not sure I agree, which is why I want to check it with the more accurate math.

When/if that checking gets done, should be interesting comparing notes.

 

[Q-reeus]

The Schwarzschild radial coordinate r, which is *defined* so that the area of a 2-sphere with radial coordinate r is 4 pi r^2

How exactly does one avoid a circular situation - i.e. what then is the measure of area as the 'primary quantity', independent of referring back to r?

Q-reeus: "Identically the flat spacetime value when gravity is 'switched off'?"
This has no physical meaning; if gravity is present, there is no way to physically observe anything corresponding to "the flat spacetime value when gravity is switched off".

Not directly, but to high precision it can be calculated in weak gravity case surely.

Q-reeus: "Shouldn't we have to in effect do a line integration over dr' = dr√grr to arrive at proper distance r' to q"
Not if all you're doing is counting field lines passing through a 2-sphere. See above.

Consider that concept of E field lines may have limited utility in gravity affected regions - or simply accept possibility of 'effective space-charge'. This is evidently true in equivalent gravitating mass case as per last bit in #16. You have no problem accepting that say Poynting theorem fails for power flow between r₂, r₁. Broaden the possibilities.

 

[PeterDonis]

It was meant as dr₂/dr₁ = (√g_rr)₁/(√g_rr)₂, regardless of what goes on in any amount of intervening static spacetime - assuming our ideal rope & pulleys. Hence a radial tug of x cm out at r₂ produces a radial motion of x(√g_rr)₂/(√g_rr)₁ cm in at r₁. And conveyed inside shell, displacement isotropy applies. Are you saying this is not so?

When you say "cm", do you mean "cm of radial coordinate increment" or "cm of actual radial proper distance"? The two are not the same, as you obviously realize. I think you mean the former, but confirmation would be nice. Assuming you do mean the former, I think the constraint of "equal proper distances moved at both ends" is pretty much the *definition* of an "ideally rigid" linkage. So the relationship between *coordinate* distances moved by the ideally rigid linkage would be given by, as you say, the ratio of metric coefficients.

If you want to use the above as your operational definition of "comparing distances", then that's fine. But that, alone, is not enough to derive a contradiction in, for example, your capacitor scenario. That scenario depends on the *potential* difference between the two locations--i.e., on the difference in g_tt, *not* g_rr. Your idealized linkage gives you a way of relating the difference in g_rr to a physical observable, but not g_tt. The difference in g_tt is still what it was before, in terms of observables: it's the actual observed "redshift of energy". In the exterior region, outside the shell, it is true that g_rr = 1 / g_tt, so there is a connection between the two. But the shell breaks that connection, as was discussed ad nauseam in a previous thread. (In the post I hope to make soon regarding the R-N metric and what it says about charge, I'll be restating some of the things I said in that other thread, for context; possibly that will make it a bit easier to see how the shell breaks the connection.)

How many times must I repeat - that any exterior field exists implies RN 'active' charge has no dependence on metric. There is infinite redshift at EH!

Nobody is disputing that the "redshift factor" is infinite (or zero, depending on how you look at it) at the horizon. And in the light of pervect's and PAllen's posts, I think there is agreement that the charge Q, as expressed by the Gauss's law integral over a 2-sphere, is independent of radius. (There is still the open question that I asked, about whether a timelike unit vector or timelike Killing vector is what properly belongs in the integral. But I think I see why it needs to be the unit vector; I'll post separately about that.) I'm not sure if that's what you mean by "active charge has no dependence on metric".

(This is one example of why it is difficult for me, and possibly others, to understand your scenarios; I can't keep track of all the non-standard terminology you keep on using. If you would learn enough of the math and the standard terminology to be able to use it without having to make up your own terms on the fly, it would be a lot easier for me to figure out what you are saying.)

 

[PeterDonis]

How exactly does one avoid a circular situation - i.e. what then is the measure of area as the 'primary quantity', independent of referring back to r?]

You measure distances around the 2-sphere, being careful to stay entirely within the 2-sphere, and compute areas using them. You can do this without ever measuring anything in the radial direction--just as, on an idealized spherical Earth, you can measure distances and areas on its surface without ever having to know its physical radius. Since we are talking about the case of spherical symmetry, we know that we can use *tangential* distances as "primary quantities".

Not directly, but to high precision it can be calculated in weak gravity case surely.

I'm not sure what you mean by this. If gravity is weak enough that curvature is negligible, then our whole discussion is moot; the entire metric is Minkowski to this approximation, and so none of the effects we're arguing about are even observable.

Consider that concept of E field lines may have limited utility in gravity affected regions - or simply accept possibility of 'effective space-charge'.

If you don't like the expression "counting field lines", I can rephrase it the way pervect did in post #16. "Counting field lines" here is just a shorthand for "counting up small elements of surface area times the E field normal to them". In that sense the concept of "field lines" is perfectly applicable, regardless of gravity--remember we are talking about spherical symmetry, so areas on a given 2-sphere are well-defined, as above. In a more general case where the field is not exactly radial, it may be that "field lines" is no longer as good a description; but we aren't talking about that case.

This is evidently true in equivalent gravitating mass case as per last bit in #16.

And pervect explicitly said there that gravity works *different* from charge in this respect. I agree with what he said.

You have no problem accepting that say Poynting theorem fails for power flow between r₂, r₁.

Huh? I have not said anything about the Poynting theorem. As far as I can see, that theorem in a curved spacetime is just a different way of stating local energy conservation: the covariant derivative of the SET due to the EM field (which is the only SET present in R-N spacetime) is zero. I have certainly never said that I would "accept" that theorem failing; quite the opposite.

 

[Q-reeus]

When you say "cm", do you mean "cm of radial coordinate increment" or "cm of actual radial proper distance"? The two are not the same, as you obviously realize. I think you mean the former, but confirmation would be nice. Assuming you do mean the former, I think the constraint of "equal proper distances moved at both ends" is pretty much the *definition* of an "ideally rigid" linkage. So the relationship between *coordinate* distances moved by the ideally rigid linkage would be given by, as you say, the ratio of metric coefficients.

Yes, coordinate ratio was meant, but as per your comments re ideally rigid linkage definition which demands 1:1 locally measured radial displacement both ends is the more physically direct handle here. However that the coordinate ratio is also unity for cap inside shell case admits to a 'direct' comparison of not just the electric forces acting between cap plates - but also the field once we tie down the issue of 'active' vs 'passive' charge - which I claim to have done already in #10. And btw I gave a sufficiently clear and obvious definition of 'active' vs 'passive' back then - re your later remarks below!

If you want to use the above as your operational definition of "comparing distances", then that's fine. But that, alone, is not enough to derive a contradiction in, for example, your capacitor scenario. That scenario depends on the *potential* difference between the two locations--i.e., on the difference in g_tt, *not* g_rr.

Of course, and if you go back and read #10 it will be evident that's what I'm on about. Concentrated on spatial component above because that seemed to be your concern re changes in moving from 'there' to 'here', but that apparently was not so.

Your idealized linkage gives you a way of relating the difference in g_rr to a physical observable, but not g_tt. The difference in g_tt is still what it was before, in terms of observables: it's the actual observed "redshift of energy". In the exterior region, outside the shell, it is true that g_rr = 1 / g_tt, so there is a connection between the two. But the shell breaks that connection, as was discussed ad nauseam in a previous thread. (In the post I hope to make soon regarding the R-N metric and what it says about charge, I'll be restating some of the things I said in that other thread, for context; possibly that will make it a bit easier to see how the shell breaks the connection.)

As per above comments - go back and read #10, 11.

Nobody is disputing that the "redshift factor" is infinite (or zero, depending on how you look at it) at the horizon. And in the light of pervect's and PAllen's posts, I think there is agreement that the charge Q, as expressed by the Gauss's law integral over a 2-sphere, is independent of radius. (There is still the open question that I asked, about whether a timelike unit vector or timelike Killing vector is what properly belongs in the integral. But I think I see why it needs to be the unit vector; I'll post separately about that.) I'm not sure if that's what you mean by "active charge has no dependence on metric".

Yes it is. And as per #10, 11, I maintain it's also nonsensical physically.

 

[Q-reeus]

You measure distances around the 2-sphere, being careful to stay entirely within the 2-sphere, and compute areas using them. You can do this without ever measuring anything in the radial direction--just as, on an idealized spherical Earth, you can measure distances and areas on its surface without ever having to know its physical radius. Since we are talking about the case of spherical symmetry, we know that we can use *tangential* distances as "primary quantities".

Makes sense I guess if one is sure transverse length is unaffected by potential, and SM does claim that. However if r 'draws down' owing to potential, that logically must 'draw down' the shell circumference/area with it to some extent also. Maybe a strain gauge would help sort that one out!

Q-reeus: "You have no problem accepting that say Poynting theorem fails for power flow between r2, r1."

Huh? I have not said anything about the Poynting theorem. As far as I can see, that theorem in a curved spacetime is just a different way of stating local energy conservation: the covariant derivative of the SET due to the EM field (which is the only SET present in R-N spacetime) is zero. I have certainly never said that I would "accept" that theorem failing; quite the opposite.

Sorry hadn't meant to imply you specifically made that statement - just took the liberty you agree with the fact of redshift of both frequency and also power as locally measured at two different potentials. Not so? :zzz:

 

[PeterDonis]

However if r 'draws down' owing to potential, that logically must 'draw down' the shell circumference/area with it to some extent also.

First a correction: the potential does not make r "draw down"; the potential is (related to) g_tt, and the relationship between radial proper distance and radial coordinate increments is governed by g_rr. I have already talked about when g_tt and g_rr are, and are not, related.

Second: even corrected as above, your conclusion above does not follow. There is no necessary connection between the radial proper distance/coordinate increment relation and the tangential one. Another way of saying this is that, with the definition of the radial coordinate r that we are using, the tangential proper distance/coordinate increment relation is *fixed*, by definition--the area of a 2-sphere at radial coordinate r is fixed at 4 pi r^2, since that is what *defines* the radial coordinate r. So variation in g_rr is *equivalent*, in these coordinates, to variation in the radial proper distance//coordinate increment relation, as compared to the tangential one.

just took the liberty you agree with the fact of redshift of both frequency and also power as locally measured at two different potentials. Not so? :zzz:

I agree that "energy redshifts" (which as it stands is pretty vague, but it will do for now since there's already a lot of context in this thread and others helping to pin down what it means). But I don't understand how this relates to the Poynting theorem. As I said, that theorem appears to be just another way of asserting local energy conservation. Any solution to the EFE will obey local energy conservation, because of the Bianchi identities. The R-N metric is a solution to the EFE, so it obeys local energy conservation. The fact that "energy redshifts" doesn't change that.

 

[pervect]

And it's here where I believe there is a flaw. That r is what again?

Spherically symmetric surfaces are defined by the geometry, defining "constant r". You can think of them as surfaces of equal time dilation, or the surfaces of static observers.

The spherical symmetry implies that you'll get the same circumference any direction you measure in. The numerical value of r, by construction of the coordinate system, is the unqiue value of said circumference divided by 2 pi.

The other notable property of "r", besides the above, is that a vector that changes r and only r , which we write formally as \partial / \partial r is a radial vector, it points inwards or outwards and is normal to the surface. The "length" of this vector is not unity, however, so the "distance" between a point on a surface and a surface at r+delta is not equal to delta.

The larger point being made isn't terribly hard to grasp in my opinion. Charge does not get "redshifted", no matter how fast something moves, the charge of the body is still the same. I don[t really follow what your difficulty is, I'm afraid I can't put it any more simply than that.

 

[PAllen]

I will add that it was established as a mathematical proof that GR+Maxwell include charge invariance some time before 1921, as this fact was presented and derived in Pauli's famous relativity exposition in 1921.

Unfortunately, a certain poster here has expressed the point of view that a mathematically based theory can be in conflict with its mathematically based consequences - a concept everyone else finds absurd on its face. The 'method' used is to take specialized results of the theory, or qualitative motivations used pedagogically, as if they are absolute principles and then showing that contradictions result. This whole method is absurd.

 

[pervect]

MTW confirms that E = Q/r^2 e_{\hat{r}}. See pg 841. \hat{r} is the unit length vector in the r direction.

 

[Q-reeus]

First a correction: the potential does not make r "draw down"; the potential is (related to) g_tt, and the relationship between radial proper distance and radial coordinate increments is governed by g_rr. I have already talked about when g_tt and g_rr are, and are not, related.

OK, but it was perhaps something in my subconscious that motivated that - a memory of a past posting that made sense at the time (and still does). Some lengthy digging, and it's probably here that inspiration came from: https://www.physicsforums.com/showpost.php?p=3560246&postcount=229, and your response was in #232! So, it depends on who says it?

Second: even corrected as above, your conclusion above does not follow. There is no necessary connection between the radial proper distance/coordinate increment relation and the tangential one. Another way of saying this is that, with the definition of the radial coordinate r that we are using, the tangential proper distance/coordinate increment relation is *fixed*, by definition--the area of a 2-sphere at radial coordinate r is fixed at 4 pi r^2, since that is what *defines* the radial coordinate r. So variation in g_rr is *equivalent*, in these coordinates, to variation in the radial proper distance//coordinate increment relation, as compared to the tangential one.

This is rather tangential (no pun intended) to our main discussion, but above comments still apply I think. It depends I think on ones point of reference - either comparing radial vs tangential distances in the one gravity affected setting, or on a before/after basis - gravity switched 'on' vs 'off'. It's the latter case that I was thinking of.

I agree that "energy redshifts" (which as it stands is pretty vague, but it will do for now since there's already a lot of context in this thread and others helping to pin down what it means). But I don't understand how this relates to the Poynting theorem. As I said, that theorem appears to be just another way of asserting local energy conservation.

Key word there is *local*. In flat spacetime, Poynting theorem holds 'globally' in that steady-state net power flow across any two nesting closed surfaces is conserved. Not so in GR. And not only the power flow, but the time integrated net energy flow fails - again as locally measured at those two differing potential regions. This is naturally not some 'proof' that charge invariance must similarly fail, but is meant to serve as a reminder that 'common sense' results true in flat spacetime can fail when gravity enters the picture. To me, it seems evident that imperviousness of Gauss's law to gravity is a Sacred Cow tenet incorporated into GR as axiom - manifest as RN metric.

Any solution to the EFE will obey local energy conservation, because of the Bianchi identities. The R-N metric is a solution to the EFE, so it obeys local energy conservation. The fact that "energy redshifts" doesn't change that.

Again, *local* is the key word there. And consider the possibility of analogy with Gauss's law in gravity regime.

 

[Q-reeus]

The larger point being made isn't terribly hard to grasp in my opinion. Charge does not get "redshifted", no matter how fast something moves, the charge of the body is still the same. I don[t really follow what your difficulty is, I'm afraid I can't put it any more simply than that.

Fair enough, but my difficulty on that point is illustrated by example 3: given here:https://www.physicsforums.com/showpost.php?p=3946413&postcount=1 - and the previous 2 examples prep for that 3: You do not get any sense of an issue there?

 

[Q-reeus]

I will add that it was established as a mathematical proof that GR+Maxwell include charge invariance some time before 1921, as this fact was presented and derived in Pauli's famous relativity exposition in 1921.

Why not just refer to original RN solution that began with the R contribution back in 1916.

Unfortunately, a certain poster here has expressed the point of view that a mathematically based theory can be in conflict with its mathematically based consequences - a concept everyone else finds absurd on its face. The 'method' used is to take specialized results of the theory, or qualitative motivations used pedagogically, as if they are absolute principles and then showing that contradictions result. This whole method is absurd.

More deja vu - this mirrors your approach when I raised the matter of stress as source term in another thread. Same attitude then as now - call it absurd to challenge a 'mathematically exact' solution to the EFE's, but simultaneously avoid any actual detailed evaluation of the specific scenarios presented. In other words - "We have no need to peer through the telescope - we know what must be there - 'holy scripture' (read; GR) tells us what will be seen". Well if you think that a tad harsh or unjust, may I suggest a simple redress - actually present your own analysis of those surely simple enough scenarios in #10, 11 - and enlighten me by showing how it all fits nicely with RN metric. I would then and only then accept that the underlying physical assumption of potential invariant Gauss's law makes sense. If you decline as I expect will be the case, don't expect any respect for your criticism of 'a certain poster' here.

 

[PeterDonis]

MTW confirms that E = Q/r^2 e_{\hat{r}}. See pg 841. \hat{r} is the unit length vector in the r direction.

Yes (I called the e_{\hat{r}} vector n^{a} in the equation I wrote down, but it's the same thing).

My question was about the *timelike* vector u^{b} in my equation; should it be a timelike *unit* vector, or the timelike *Killing* vector of the spacetime? As PAllen commented, if the electric field E is going to be measured by local Lorentz observers, then u^{b} must be a timelike *unit* vector. The reason I wondered about whether it should be a timelike Killing vector is that I was thinking about an analogue for charge to the Komar mass integral, which uses a timelike Killing vector; but thinking it over, that sort of thing would only work in a region of nonzero charge density. In the vacuum region of R-N spacetime, the charge density is zero, so a "Komar charge" integral over such a region would just be summing up a lot of zeroes; the "source" of the charge is deeper inside the spacetime, not in the vacuum region.

 

[PeterDonis]

https://www.physicsforums.com/showpost.php?p=3560246&postcount=229, and your response was in #232! So, it depends on who says it?

The discussion in that thread evolved quite a bit from the point you refer to; I'm not sure that either DrGreg's statement or my response in #232 is a good reflection of where things ended up. I seriously doubt DrGreg intended anything that would conflict with what I've been saying here about the area of a 2-sphere at radial coordinate r being 4 pi r^2 (since the tangential line element he gives leads direclty to that area when integrated).

either comparing radial vs tangential distances in the one gravity affected setting, or on a before/after basis - gravity switched 'on' vs 'off'. It's the latter case that I was thinking of.

You can't just switch gravity off, so I don't see how the latter case can be physically realized. You *can* take an object from a region with negligible gravity to a region with non-negligible gravity and compare measurements on it before and after; we've gone into this in gory detail in previous threads, and we've seen that what happens depends on how you do the moving. Or you can try to compare two supposedly "identical" objects, one in a region with negligible gravity and one in a region with non-negligible gravity; then you find that what you get depends on how you do the comparing. In short, there is no unique "standard of comparison" that works the way you are suggesting.

Key word there is *local*. In flat spacetime, Poynting theorem holds 'globally' in that steady-state net power flow across any two nesting closed surfaces is conserved. Not so in GR.

You mean "not so in curved spacetime"; yes, because global "conservation of energy" does not hold in a general curved spacetime.

This is naturally not some 'proof' that charge invariance must similarly fail, but is meant to serve as a reminder that 'common sense' results true in flat spacetime can fail when gravity enters the picture.

Yes, which means that instead of just assuming that the flat spacetime result carries over, you have to *show* that it does, in cases where it does. Pervect showed that it does for the case of charge invariance. Nobody is assuming anything; nobody is claiming that Gauss's Law for charge "must" hold in curved spacetime just because it holds in flat spacetime. We're saying it holds because it has been explicitly *shown* that it holds.

Gauss's law to gravity is a Sacred Cow tenet incorporated into GR as axiom - manifest as RN metric

The R-N metric is a solution to the EFE; that's been proved. The EFE doesn't incorporate Gauss's Law as an axiom. The fact that Gauss's Law for *charge* holds in R-N spacetime is easily shown; pervect did it. Gauss's Law for charge does not have to be assumed or incorporated into GR as an axiom; it is a *theorem*.

Gauss's Law for *gravity* does *not* hold in curved spacetime; as pervect noted, the corresponding integral for gravity is *not* conserved.

 

[PeterDonis]

In the post I hope to make soon regarding the R-N metric and what it says about charge

I want to tie up this loose end (which, as noted earlier in this thread, has actually spanned multiple threads), but it will probably take at least two posts. This first one is partly to review some general properties of static, spherically symmetric spacetimes, and then to make a few comments about the R-N spacetime's geometric properties. I'll defer the specific issues relating to charge to a subsequent post.

(Note: I posted much of the following in a previous thread, here:

https://www.physicsforums.com/showpost.php?p=3843463&postcount=249

However, there I was concentrating specifically on the case of a shell of matter with vacuum both outside and inside, so I didn't comment on some more general properties that are of interest.)

The most general line element for a static, spherically symmetric spacetime can be written as follows (my notation is slightly different from what you'll find in most textbooks, for example MTW):

ds^2 = - J(r) dt^2 + \frac{1}{1 - \frac{2 m(r)}{r}} dr^2 + r^2 d\Omega^2

where J(r) and m(r) are functions of the radius. The function J(r) can be called the "redshift factor", and will be less than or equal to 1; here we will only consider cases where it is positive (i.e., regions outside any horizon that might be present). The function m(r) can be called the "mass inside radius r", and can be defined via its radial derivative as follows:

\frac{dm}{dr} = 4 \pi r^2 \rho (r)

where \rho (r) is the energy density seen locally by a static observer (which is the t-t component of the SET, \rho = T^t_t). This equation for m comes directly from the t-t component of the EFE.

The redshift factor J(r) is governed by the following equation, which comes directly from the r-r component of the EFE:

\frac{1}{2J} \frac{dJ}{dr} = \frac{m(r) + 4 \pi r^3 p(r)}{r \left( r - 2 m(r) \right)}

where the LHS is written this way because it turns out to be more convenient. Here we see one additional function of r, the radial pressure p(r), which is the r-r component of the SET, p = T^r_r. Note that we are *not* assuming isotropic pressure; that is, whatever stress-energy is present need *not* be a perfect fluid. However, the SET must be diagonal (in the chart in which the above line element is expressed), and the tangential components must be equal; we'll see that in a moment.

The radial pressure p(r) is governed by a generalized form of the Tolman-Oppenheimer-Volkoff (TOV) equation which does not assume isotropic pressure; it turns out that that just adds one additional term to the standard TOV equation. The equation can be derived from the tangential component of the EFE, but it turns out to be easier to evaluate the r component of the covariant divergence of the SET, which is equivalent but involves a lot less algebra. The result is:

\frac{dp}{dr} = - \left( \rho(r) + p(r) \right) \frac{1}{2J} \frac{dJ}{dr} - \frac{2}{r} \left( p(r) - s(r) \right)

where we now can see the convenience of writing the J equation as we did above, and where s(r) is the tangential stress, which is the tangential component of the SET, s = T^\theta_\theta = T^\phi_\phi.

I emphasize that all this applies to *any* static, spherically symmetric spacetime; it includes *all* of the cases we have discussed in various threads, including not just vacuum regions, not just the exterior of R-N spacetime, but also interior regions of spherically symmetric bodies such as planets or stars, and interior regions of spherically symmetric shells with vacuum inside (it also applies to the inner vacuum region itself, of course). All we need to do in any specific case is to find appropriate expressions for any two of the five unknowns, rho, p, s, J, m. Then, since we have three equations relating all these unknowns, we can determine the other three from the two we have expressions for.

One other general question we can ask that might be of interest () is, under what circumstances will the redshift factor, J(r), take the following form?

J(r) = f \left( 1 - \frac{2 m(r)}{r} \right)

Notice first of all that there is a constant factor f in front. That is there because, as you can see from the above, we do not have an equation for J or its derivative in isolation; we only have an equation for the *ratio* of dJ/dr to J. That means that, whatever expression we derive for J from the above equations, we can *always* put some constant factor f in front of it and still satisfy the equations. In order to determine that constant factor, we have to look at boundary conditions: for example, in any exterior vacuum region, where J -> 1 as r -> infinity, we must have f = 1; that is, if we have some expression for J that goes to 1 as r goes to infinity, we *cannot* put any constant factor f in front of it and still have a valid solution except f = 1, which is trivial. But there will be cases, as we will see, where we *can* find some f that is not 1 but which satisfies the boundary conditions.

To see what the above condition on J implies, we can simply take its derivative and divide by 2J to obtain:

\frac{1}{2J} \frac{dJ}{dr} = \left( \frac{m(r)}{r^2} - \frac{1}{r} \frac{dm}{dr} \right) \frac{1}{1 - \frac{2 m(r)}{r}}

which quickly simplifies to

\frac{1}{2J} \frac{dJ}{dr} = \frac{m(r) - 4 \pi r^3 \rho (r)}{r \left( r - 2 m(r) \right)}

Comparison with the equation for 1/2J dJ/dr above makes it clear that we must have p(r) = - \rho (r) in order for J(r) to take the special form given above. There are two cases of interest where this condition might be satisfied in a static, spherically symmetric spacetime. Obviously it will be satisfied in any vacuum region, where rho = p = 0. But it is also satisfied, as it happens, by the SET of a static electric field, as in R-N spacetime (I'll give more detail on this in a subsequent post). So in those two cases, we expect to see a relationship between the "redshift factor" and the radial metric coefficient. (The condition itself is also satisfied by "dark energy", such as a cosmological constant, but in cases where that is present the spacetime will not be static, so we won't consider those cases here.)

But the condition p = - rho is obviously *not* going to be satisfied by any kind of normal matter. So in cases like the interior of a shell or the interior of a planet, we do *not* expect to see a relationship between the redshift factor and the radial metric coefficient.

The above should be all the general machinery we will need; in my next post I'll consider special cases corresponding to (at least some of) the scenarios that have been proposed by Q-reeus.

 

[PeterDonis]

The above should be all the general machinery we will need

Actually, there is another thing; we'll need expressions for the proper acceleration of a radially moving test object in a static, spherically symmetric spacetime. Unfortunately these expressions are rather messy and I haven't been able to simplify them very much, but we'll need them for reference if nothing else.

The 4-velocity u^a of a radially moving object has two components, u^t and u^r. However, it will make things look somewhat simpler if we adopt the notation that u^r = w u^t, where w = dr / dt. Also, since the 4-velocity is a timelike unit vector, we must have g_{ab} u^{a} u^{b} = -1. This allows us to derive an expression for u^t in terms of w and other known quantities:

\left( u^t \right)^2 = \frac{r - 2m}{r \left( J - w^2 \right)}

The general formula for the 4-acceleration is

A^b = u^a \nabla_a u^b = u^a \partial_a u^b + u^a \Gamma^b_{ac} u^c

We will have two components to A^b, the t component and the r component (since the motion is purely radial). Expanding out the above formula for each case, we get

A^t = \left( u^t \partial_t + u^r \partial_r \right) u^t + 2 \Gamma^t_{tr} u^t u^r

A^r = \left( u^t \partial_t + u^r \partial_r \right) u^r + \Gamma^r_{tt} \left( u^t \right)^2 + \Gamma^r_{rt} u^t u^r + \Gamma^r_{rr} \left( u^r \right)^2

Substituting for everything we can substitute for, to minimize the number of distinct quantities, we obtain:

A^t = u^t \left( \partial_t + w \partial_r \right) u^t + \frac{w \left( r - 2m \right)}{r J \left( J - w^2 \right)} \frac{dJ}{dr}

A^r = u^t \left( \partial_t + w \partial_r \right) u^r + \left( \frac{\left( r - 2m \right)^2}{r^2} + \frac{w \left( r - 2m \right)}{r} \right) \frac{1}{2 J \left( J - w^2 \right)} \frac{dJ}{dr} + \left( r \frac{dm}{dr} - m \right)\frac{w^2}{r^2 \left( J - w^2 \right)}

As I said, these are messy expressions and I haven't been able to simplify them. But we can at least check one special case: a static observer, for whom w = 0 (i.e., u^r = 0). For this case, all the partial derivatives are zero, so A^t is zero, and we have only

A^r = \left( 1 - \frac{2m}{r} \right)^2 \frac{1}{2 J^2} \frac{dJ}{dr}

Finally, note that the actual measured proper acceleration will be the magnitude of the 4-vector A^b, which in general is

A = \sqrt{ g_{ab} A^a A^b }

For the special case of a static observer, we have

A = \sqrt{ g_{rr} } A^r = \left( 1 - \frac{2m}{r} \right)^{3/2} \frac{1}{2 J^2} \frac{dJ}{dr}

For the further special case where J = 1 - 2m / r (I discussed last post when this special case applies), we obtain

A = \left( \frac{m}{r^2} - \frac{1}{r} \frac{dm}{dr} \right) \frac{1}{\sqrt{1 - \frac{2m}{r}}}

One other note: I think there is another special case of interest and I will probably need to add a post about that before going into details about R-N spacetime and charge. This is the case where A^t = 0, which corresponds to the case where the "total energy" of the test object is conserved.

 

[PeterDonis]

I will probably need to add a post about that before going into details about R-N spacetime and charge.

Still considering some items relating to the acceleration equations I posted, but I'm going to go ahead and post a few things about some special cases, including R-N spacetime.

First special case: Schwarzschild spacetime. This is what you get when you set \rho = p = s = 0, i.e., complete vacuum, no stress-energy anywhere. This immediately gives dm / dr = 0, so the mass m(r) = M is constant. And, as noted previously, this is a case where we know J(r) = f ( 1 - 2 m / r ), and for the exterior vacuum region we know from the boundary condition at infinity that f = 1, so we recover the standard Schwarzschild metric.

Second special case: Schwarzschild exterior, a thin shell of matter, and flat Minkowski interior. (Some of what follows recaps things I have posted in previous threads.) Here we have \rho = p = s = 0 everywhere except inside a thin shell lying between radial coordinates r_{i} (inner) and r_{o} (outer). For simplicity we assume that inside the shell we have constant density \rho. Then we have m(r) = 0 in the interior vacuum region, m(r) = M in the exterior vacuum region (where M is the shell's total mass), and m(r) = 4 / 3 \pi \rho ( r^3 - r_i^3 ) in the region occupied by the shell.

First, what exactly do we mean by saying the interior vacuum region is a "flat Minkowski" region? The fact that m(r) is zero in the interior vacuum region confirms that it is spatially flat (g_rr = 1); and since the pressure p is also zero we know that dJ / dr is zero, so J(r) is constant. However, J(r) is *not* equal to 1 in this region; at least, not with the standard time coordinate we have been using. Instead, J(r) will be equal to its value at r_i, the inner radius of the shell, everywhere in the interior vacuum region. In other words, this is a case where the factor f that I talked about before is not equal to 1. (In fact, we can see that it must be *less* than 1, because the value of J(r) at r_i must be less than 1, since dJ/dr is positive everywhere so J at any finite radius r must be less than its value at infinity, which is 1.)

Of course we can re-scale the time coordinate in the interior vacuum region to make the metric explicitly take the Minkowski form: we just apply the transformation t&#039; = \sqrt{J(r_i)} dt; this is the sense in which the interior vacuum region is indeed a "flat Minkowski" region. Locally, in fact, the transformed coordinates will be the "natural" ones for static observers to use. However, there is still a "gravitational redshift" in this region relative to infinity; for example, if static observers in the interior vacuum region emit light outward, it will be redshifted when it is received by observers at infinity.

How does it happen that the interior region ends up being flat, even though there is still a redshift relative to infinity? As noted before, in the region occupied by the shell, the condition p = - \rho does not hold, so the relationship between g_tt and g_rr that holds in the exterior vacuum region is broken; and when that relationship is "re-established" in the interior vacuum region, it is re-established with a *different* factor f than the one that applied in the exterior region (which, of course, is f = 1).

I won't say any more about this special case; the above should be enough for here. (In other threads I've talked about the behavior of p and s in the region occupied by the shell.)

Third special case: R-N spacetime. Here the parameters are as follows:

\rho (r) = \frac{1}{8 \pi} \frac{Q^2}{r^4}

p (r) = - \rho (r)

s (r) = \rho (r)

m (r) = M - \frac{Q^2}{2r}

Since the condition p = - \rho holds, we know that J = 1 - 2m(r) / r, so we have a complete solution; when written out, of course, we have

J (r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2}

By plugging into other formulas in my past few posts, we can recover the other formulas for R-N spacetime that I posted way back in the early part of this thread.

Some quick observations:

We can now see explicitly that the "mass inside radius r" goes down as r decreases, and that it goes to M as r goes to infinity. (We are limiting ourselves to the region outside the horizon and to Q < M, so m(r) will always be positive; in other words, we are not, strictly speaking, considering an R-N black hole--at least not yet--but rather a charged gravitating body like a planet or star with an appreciable static E field.) This explains, as I noted in a much earlier post, why the "redshift factor" J(r) does not decrease as fast as it does in Schwarzschild spacetime; as you descend, more and more of the mass-energy in the spacetime is above you rather than below you, so it doesn't contribute to the effects of "gravity" that you see. You can also read this off from the equation for dJ / dr, where the negative pressure (p = - rho) makes dJ / dr *smaller*.

We can also see, though, that the "mass inside radius r" behaves *differently* in this respect than the "charge inside radius r", which, as shown in previous posts dealing with the Gauss's Law integral, is *constant*, equal to Q at any radius r. So we have at least one key difference in the behavior of charge vs. the behavior of mass-energy.

 

[PeterDonis]

Actually, there is another thing; we'll need expressions for the proper acceleration of a radially moving test object in a static, spherically symmetric spacetime. Unfortunately these expressions are rather messy and I haven't been able to simplify them very much, but we'll need them for reference if nothing else.

On further review, I'm not sure all of these expressions are right. (I made at least one transcription error, but I also need to check against some other things.) Unfortunately I'm out of the "edit window" now so I can't edit or delete the post, but I'll go over things again and post an update.

Also, there is one more special case I want to make some comments about: the case of a charged gravitating body with a spherical shell some distance outside it. This is basically just a combination of the second and third special cases in my previous post:

We have an exterior vacuum region with the standard R-N geometry, where the mass M that appears in the metric is M_body + M_shell, the mass of the central charged body plus the mass of the shell. So in this region we have m(r) = M_{body} + M_{shell} - Q^2 / 2r, where Q is the charge of the central body (we assume the shell is uncharged), and J(r) = 1 - 2m(r) / r.

We have an interior vacuum region, between the central body and the shell, where the geometry is also R-N, but with two changes from the exterior: the mass M in the metric is now just M_body, the mass of the central charged body, and there is a "rescaled" time coordinate; in other words, the g_tt metric coefficient in this region, written using the "standard" time coordinate, has a factor f in front of it that is less than 1. So in this region we have m(r) = M_{body} - Q^2 / 2r, and J(r) = f \left( 1 - 2m(r) / r \right).

In the shell region, if we assume the shell has constant density, we have m(r) = M_{body} + 4 / 3 \pi \rho_{shell} \left( r^3 - r_i^3 \right) - Q^2 / 2r. We can also see that dJ / dr will still be positive inside the shell, so the value of J at r_i, the shell inner radius (which determines the factor f in the interior vacuum region, see above), will be less than its value at r_o, the shell outer radius. The difference is that there is also a contribution to dJ / dr from the energy and pressure of the static electric field, so the change in J across the shell will be different than in the case of a shell with vacuum inside and outside, assuming identical parameters for the shell (same density and thickness).

 

[Q-reeus]

The discussion in that thread evolved quite a bit from the point you refer to; I'm not sure that either DrGreg's statement or my response in #232 is a good reflection of where things ended up.

First, pardon please delayed response - only partly owing to I assume a lengthy PF server outage. And I must say bravo Peter for a huge and thoughtful campaign of five postings in a row - last three quite lengthy and detailed! Now, re above; understood. Issue there likely only relevant for thick shells or solids where extended radial differential effects could conceivably induce 'appreciable' internal stress/strains, but a side-issue here.

The R-N metric is a solution to the EFE; that's been proved. The EFE doesn't incorporate Gauss's Law as an axiom. The fact that Gauss's Law for *charge* holds in R-N spacetime is easily shown; pervect did it. Gauss's Law for charge does not have to be assumed or incorporated into GR as an axiom; it is a *theorem*.

Read through #16 and #32 again. Not sure proof is the right word. If deciding ab initio that field lines concept is inviolable re E field - lines must begin and end on charge globally, regardless of spacetime curvature or not, what follows - RN solution - will inevitably be made to conform to that position on how nature works. (Realize your next postings do put up a decent argument for that case, but here I am responding as though unaware of those)

Gauss's Law for *gravity* does *not* hold in curved spacetime; as pervect noted, the corresponding integral for gravity is *not* conserved.

Which is interesting and I find it hard to avoid seeing it as a tacit admission gravity really does gravitate; 'lines of g' do indeed terminate in mid-air! Alternately, given the striking parallels between gravitating mass and charge in the almost-Newtonian weak-gravity regime, it begs why and on what basis the assumed departure re 'Gauss's almost law for g', if gravity is rather taken as not gravitating! This is referring here to the restricted case where mass/charge is static in a gravitational potential - thus excluding from consideration the SR velocity/energy dependence for gravitating mass but not charge (though very important when looking at free-infall case).

 

[Q-reeus]

I want to tie up this loose end (which, as noted earlier in this thread, has actually spanned multiple threads), but it will probably take at least two posts. This first one is partly to review some general properties of static, spherically symmetric spacetimes, and then to make a few comments about the R-N spacetime's geometric properties. I'll defer the specific issues relating to charge to a subsequent post.

(Note: I posted much of the following in a previous thread, here:

https://www.physicsforums.com/showpost.php?p=3843463&postcount=249

However, there I was concentrating specifically on the case of a shell of matter with vacuum both outside and inside, so I didn't comment on some more general properties that are of interest.)

The most general line element for a static, spherically symmetric spacetime can be written as follows (my notation is slightly different from what you'll find in most textbooks, for example MTW):

ds^2 = - J(r) dt^2 + \frac{1}{1 - \frac{2 m(r)}{r}} dr^2 + r^2 d\Omega^2...and much following.

There is much that's familiar and also I agree sound and well presented in this and some of the following two entries, but one quite simple and crucial relation has me perplexed. It's in this para:

Comparison with the equation for 1/2J dJ/dr above makes it clear that we must have p(r)=−ρ(r) in order for J(r) to take the special form given above. There are two cases of interest where this condition might be satisfied in a static, spherically symmetric spacetime. Obviously it will be satisfied in any vacuum region, where rho = p = 0. But it is also satisfied, as it happens, by the SET of a static electric field, as in R-N spacetime (I'll give more detail on this in a subsequent post).

The stress components for a magnetostatic field (and E field is perfectly analogous) are given in eq'n (3) here: http://fieldp.com/myblog/2010/theory-and-applications-of-the-maxwell-stress-tensor/ Take x as field direction, then we easily see that for diagonals, |σ_xx| = B²/μ₀, |σ_yy| = |σ_zz| = -1/2|σ_xx|. Convention there is for tension to be positive, but it doesn't really matter as will be seen. When it comes to total stress/pressure contribution, correct me if wrong, but one simply does a scalar sum of the principal stresses - whether stress is owing to a material mechanically/hydrostatically stressed, or that of EM field. And clearly, at least in vacuo, that comes to zero for any EM field! So what's going on here? Is there some justification for excising two components only in EM case but not otherwise? All the rest of your findings is impacted strongly on the resolution of this seemingly straightforward point. Your derivation for that last expression in #42 was though quite helpful, and I will be cataloging these entries as handy reference material - with care. I have no problem with that particular one-way coupling - as already stated in another thread. But two-way coupling still makes more sense to me. Particularly when it gets down to the thorny matter of resolving the difficulties shown up in say #10.

 

[PeterDonis]

And I must say bravo Peter for a huge and thoughtful campaign of five postings in a row - last three quite lengthy and detailed!

Thanks! And more to come...

If deciding ab initio that field lines concept is inviolable re E field - lines must begin and end on charge globally, regardless of spacetime curvature or not

The "field lines concept" is not fundamental; it's just one way of visualizing what the math says. The fundamental concept is the math--in this case, Maxwell's Equations in curved spacetime. Those equations, combined with the EFE, are all that you need to "assume" to derive the R-N metric. The stuff I've been posting all comes out of that derivation. If you're taking the position that Maxwell's Equations or the EFE might not be right, then there's not much point in continuing the discussion. (Even if we wanted to have a discussion along those lines, it's not enough just to say those equations might not be right--you have to put something in their place that we can use to make predictions. What?)

Which is interesting and I find it hard to avoid seeing it as a tacit admission gravity really does gravitate; 'lines of g' do indeed terminate in mid-air!

This is one reason why the "field lines" concept is not fundamental; this interpretation only works in spacetimes where a conserved "energy" can be defined, i.e., in stationary spacetimes. The analogous concept for EM works, to the extent it works, in *any* spacetime, stationary or not.

In a stationary spacetime, as has been discussed ad nauseam in other threads, yes, you can interpret Gauss's Law for gravity not holding as meaning that "gravity gravitates". But you have to be very careful what inferences you draw from that; as those other threads make clear, *lots* of people have real trouble being that careful. This is why I tend not to favor such interpretations, and to prefer concentrating on statements that hold in any spacetime, like local energy conservation--the covariant divergence of the SET is always zero.

 

[PeterDonis]

Take x as field direction, then we easily see that for diagonals, |σ_xx| = B²/μ₀, |σ_yy| = |σ_zz| = -1/2|σ_xx|.

That's right; the radial stress component (in the field direction) is *minus* the tangential components (orthogonal to the field direction). And, when you add in the time component (energy density), you find that the radial stress is *opposite* in sign (but equal in magnitude) to the energy density. I.e., we have radial tension and tangential compression. Also, at least for a static electric field, the tangential stresses are equal in magnitude to the radial stress; there isn't a factor of 1/2. (I've verified that while doing the computations for the R-N metric, since the stress components have to match the components of the Einstein tensor, and they all have the same magnitude--no factors of 1/2.) I'm not sure where you are getting the factor of 1/2 for the magnetic case here.

When it comes to total stress/pressure contribution, correct me if wrong, but one simply does a scalar sum of the principal stresses - whether stress is owing to a material mechanically/hydrostatically stressed, or that of EM field. And clearly, at least in vacuo, that comes to zero for any EM field!

No, it doesn't. At least, not if by "total stress/pressure contribution" you mean what appears in the equation for "attractive gravity"--the initial inward acceleration of a small ball of test particles immersed in whatever "material" we are talking about. That is indeed the sum of *all* the diagonal SET components, or \rho + \sigma_x + \sigma_y + \sigma_z, which in this case comes to 2 \rho--i.e., *not* zero.

Perhaps it's that "in vacuo" that's the issue--an EM field *always* has a positive energy density, so there is never a true "vacuum" when an EM field is present. In "natural" units the energy density is just \left( E^2 + B^2 \right) / 8 \pi.

I have no problem with that particular one-way coupling - as already stated in another thread. But two-way coupling still makes more sense to me. Particularly when it gets down to the thorny matter of resolving the difficulties shown up in say #10.

I'm afraid you lost me here--what is the "one-way" vs. "two-way" coupling referring to?

 

[PeterDonis]

Take x as field direction, then we easily see that for diagonals, |σ_xx| = B²/μ₀, |σ_yy| = |σ_zz| = -1/2|σ_xx|.

On taking a slightly longer look at the page you linked to, it seems to me that for a B field purely in the x direction (i.e., B_x = B, all other components zero), we would have

\mu_0 \sigma_{xx} = B^2 - 1/2 B^2 = 1/2 B^2

\mu_0 \sigma_{yy} = \mu_0 \sigma_{zz} = 0 - 1/2 B^2 = - 1/2 B^2

So the factor of 1/2 is present in *all three* components, and their magnitudes are all the same. And since converting to "natural" units just means replacing \mu_0 with 4 \pi, we have that the magnitude of each stress component in "natural" units is B^2 / 8 \pi, as I said in my last post.

 

[Q-reeus]

46
The "field lines concept" is not fundamental; it's just one way of visualizing what the math says.

Quite the point I have been emphasizing. It was back there in those 'proof' entries I referred to earlier that field line concept were a key part of that 'proof'.

The fundamental concept is the math--in this case, Maxwell's Equations in curved spacetime. Those equations, combined with the EFE, are all that you need to "assume" to derive the R-N metric. The stuff I've been posting all comes out of that derivation. If you're taking the position that Maxwell's Equations or the EFE might not be right, then there's not much point in continuing the discussion.

As said numbers of times, it's over how the two are married. Sooner or later, I [STRIKE]expect[/STRIKE] hope the specific issues in #10, 11, will be faced square-on. They haven't gone away.

(Even if we wanted to have a discussion along those lines, it's not enough just to say those equations might not be right--you have to put something in their place that we can use to make predictions. What?)

Well what did I suggest in #10 - originating actually in thread linked to in #5. It's there - not in sophisticated math, but as key concept.

This is one reason why the "field lines" concept is not fundamental;

Again, can only agree - except I extend that observation to include E, B fields.

 

[Q-reeus]

Q-reeus: "Take x as field direction, then we easily see that for diagonals, |σxx| = B2/μ0, |σyy| = |σzz| = -1/2|σxx|."

That's right; the radial stress component (in the field direction) is *minus* the tangential components (orthogonal to the field direction). And, when you add in the time component (energy density), you find that the radial stress is *opposite* in sign (but equal in magnitude) to the energy density. I.e., we have radial tension and tangential compression. Also, at least for a static electric field, the tangential stresses are equal in magnitude to the radial stress; there isn't a factor of 1/2. (I've verified that while doing the computations for the R-N metric, since the stress components have to match the components of the Einstein tensor, and they all have the same magnitude--no factors of 1/2.) I'm not sure where you are getting the factor of 1/2 for the magnetic case here.

I must admit to getting that wrong - there is no factor 1/2 as you say. [edit out mistaken passage] should have remembered what I argued main para here:https://www.physicsforums.com/showpost.php?p=3840301&postcount=237 .

So when you e.g. write p(r)=−ρ(r) as in #42, I'm supposed to understand that that p is just the radial (-ve sign) component rather than net contribution of p (+ve sign) at r? Dear dear - more precision then!

No, it doesn't. At least, not if by "total stress/pressure contribution" you mean what appears in the equation for "attractive gravity"--the initial inward acceleration of a small ball of test particles immersed in whatever "material" we are talking about. That is indeed the sum of *all* the diagonal SET components, or ρ+σx+σy+σz, which in this case comes to 2ρ--i.e., *not* zero.

Sure, as essentially sorted out above.

Q-reeus: "I have no problem with that particular one-way coupling - as already stated in another thread. But two-way coupling still makes more sense to me. Particularly when it gets down to the thorny matter of resolving the difficulties shown up in say #10."

I'm afraid you lost me here--what is the "one-way" vs. "two-way" coupling referring to?

Raised before in earlier thread - charge via it's E field energy density distorts Schwarzschild geometry (witness your J expression in #42), but as per #34, Mass has no effect on form of E field. Still don't buy it.