Are the directions of electric fields lines affected by Gravity?

In summary, the conversation discussed the potential for electric field lines to be warped in curved space, similar to the way gravitational fields are warped. The concept of using differential forms to represent electromagnetism and gravity was also mentioned. There was also a discussion about the Reissner-Nordstrom charged black hole and its implications for the relationship between mass and electric field. The conversation ended with the possibility of revisiting these concepts in the future.
  • #1
Edward Solomo
72
1
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!
 

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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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. εr=1)?
 
  • #6
Q-reeus said:
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.
 
  • #7
PeterDonis said:
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. :rolleyes:
 
  • #8
Q-reeus said:
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.
 
  • #9
Edward Solomo said:
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.
 
  • #10
PeterDonis said:
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| = qa/(4πε0r2) and potential effected 'passive' charge F = qpE, with qp = q√-gtt, 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√-gttε0|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 -gtt (radiation field strength drops proportional to frequency and thus as √-gtt), 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, qa's are identical and hence magnitude of E field induced by one charged sphere on the other. However we had to have qp 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/√-gtt)(ε00). 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!
 
  • #11
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 √-gtt. 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 √-gtt, 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 gtt). 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 √-gtt factors. Small comfort.

Now try the suggested cure: depressed values for ε,μ = (1/√-gtt)(ε00). From the Wiki article here: http://en.wikipedia.org/wiki/Dipole, we have expressions for magnetic dipole:
b7d9fcf7464a06bec4fb084517b2927a.png

and electric dipole:
134903c8e2bdf92a4cadf1a7205c57ab.png

At first sight the inverse relative locations of ε00 in the two expressions appears in conflict with having each directly modified by 1/√-gtt 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 √-gtt, but with reduced effective charge by the same factor. Hence m by itself is reduced by the factor -gtt, the field B then further modified by μ = 1/√-gttμ0 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/√-gtt 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. :confused:
 
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  • #12
Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.
 
  • #13
Q-reeus said:
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.
 
  • #14
PeterDonis said:
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 √-gtt = √(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 √-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.
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:
 
  • #15
Q-reeus said:
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.

Q-reeus said:
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?

Q-reeus said:
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.

Q-reeus said:
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.
 
  • #16
Q-reeus said:
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.
 
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  • #17
pervect said:
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)

pervect said:
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:

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

where [itex]dS[/itex] is the surface element of a 2-sphere, [itex]n^{a}[/itex] is the outward-pointing normal to the 2-sphere, and [itex]u^{b}[/itex] 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 [itex]4 \pi r^2[/itex] and we end up with

[tex]Q(r) = r^2 F_{ab} n^{a} u^{b}[/tex]

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

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

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 [itex]Q(r) = Q[/itex] for all r, which would require [itex]u^{b} = (1 / \sqrt{1 - 2M /r + Q^2 / r^2}, 0, 0, 0)[/itex]; 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.
 
  • #18
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 ub being (1,0,0,0) in RN coordinates. Also, I have never seen this symbol used for anything other than a timelike unit vector.
 
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  • #19
PeterDonis said:
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.
 
  • #20
pervect said:
Yes, I wanted the surface of the 3 sphere, which is a 2 sphere, and it's obviously 4 pi r^2

Ok, good.

pervect said:
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)?
 
  • #21
PeterDonis said:
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.
 
  • #22
pervect said:
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.
 
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  • #23
Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus 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. :wink:

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.
 
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  • #24
Q-reeus said:
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.

Q-reeus said:
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".

Q-reeus said:
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.
 
  • #25
PeterDonis said:
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 = (√grr)1/(√grr)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(√grr)2/(√grr)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.
 
  • #26
PeterDonis said:
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.
 
  • #27
Q-reeus said:
It was meant as dr2/dr1 = (√grr)1/(√grr)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(√grr)2/(√grr)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.)

Q-reeus said:
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.)
 
  • #28
Q-reeus said:
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".

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus said:
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.

Q-reeus 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.
 
  • #29
PeterDonis said:
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.
 
  • #30
PeterDonis said:
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:
 
  • #31
Q-reeus said:
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.

Q-reeus said:
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.
 
  • #32
Q-reeus said:
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 [itex]\partial / \partial r[/itex] 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.
 
  • #33
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.
 
  • #34
MTW confirms that [itex]E = Q/r^2 e_{\hat{r}}[/itex]. See pg 841. [itex]\hat{r}[/itex] is the unit length vector in the r direction.
 
  • #35
PeterDonis said:
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.
 
<h2>1. How does gravity affect the direction of electric field lines?</h2><p>Gravity does not directly affect the direction of electric field lines. Electric field lines always point away from positively charged objects and towards negatively charged objects, regardless of the presence of gravity.</p><h2>2. Can gravity change the strength of an electric field?</h2><p>Gravity can indirectly affect the strength of an electric field by influencing the position and movement of charged particles. For example, in a strong gravitational field, charged particles may be pulled towards the source of gravity, causing them to accumulate and create a stronger electric field.</p><h2>3. Do electric field lines curve due to gravity?</h2><p>Yes, electric field lines can curve due to the presence of gravity. This is because gravity can cause the path of charged particles to curve, and electric field lines follow the path of these charged particles.</p><h2>4. Are there any situations where gravity and electric fields interact?</h2><p>Yes, there are several situations where gravity and electric fields can interact. For example, in a gravitational field, charged particles may experience a force due to the electric field, and this force can cause them to move in a curved path.</p><h2>5. Can the direction of electric field lines be affected by other forces besides gravity?</h2><p>Yes, the direction of electric field lines can be affected by other forces, such as magnetic fields or other electric fields. In these cases, the resulting direction of the electric field lines will be a combination of the effects of all the forces acting on the charged particles.</p>

1. How does gravity affect the direction of electric field lines?

Gravity does not directly affect the direction of electric field lines. Electric field lines always point away from positively charged objects and towards negatively charged objects, regardless of the presence of gravity.

2. Can gravity change the strength of an electric field?

Gravity can indirectly affect the strength of an electric field by influencing the position and movement of charged particles. For example, in a strong gravitational field, charged particles may be pulled towards the source of gravity, causing them to accumulate and create a stronger electric field.

3. Do electric field lines curve due to gravity?

Yes, electric field lines can curve due to the presence of gravity. This is because gravity can cause the path of charged particles to curve, and electric field lines follow the path of these charged particles.

4. Are there any situations where gravity and electric fields interact?

Yes, there are several situations where gravity and electric fields can interact. For example, in a gravitational field, charged particles may experience a force due to the electric field, and this force can cause them to move in a curved path.

5. Can the direction of electric field lines be affected by other forces besides gravity?

Yes, the direction of electric field lines can be affected by other forces, such as magnetic fields or other electric fields. In these cases, the resulting direction of the electric field lines will be a combination of the effects of all the forces acting on the charged particles.

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