Is 'charged black hole' an oxymoron?

[stevendaryl]

Let's consider a 1 cm thick clock hand. Can we say the under side of this clock hand always runs at the same rate as the upper side?
Can this much be said?

As I said, if two clocks A and B are SPATIALLY SEPARATED, then the question of which clock has a faster rate is coordinate-dependent. If they are at the same location, then the question is not coordinate-dependent.

Do you understand that for two clocks A and B that are spatially separated, and for two different coordinate systems (with time coordinates t and T), you can have

d$\tau$_A/dt > d$\tau$_B/dt

and also have

d$\tau$_A/dT < d$\tau$_B/dT

If A and B are at the same location, that is not possible.

 

[stevendaryl]

Yes, and space is filled with these potential fields, and we can take any two objects in space, and say which is the upper object.

Pushing is not absolute work. Pushing upwards, also known as lifting, just absolutely is work.

This discussion is completely full of misunderstandings about General Relativity. The big one is that there is no such thing as a "gravitational potential" in General Relativity. Gravity is not a force in GR, and it does not have a potential. It has a metric, which determines the distance between two points that are spacelike separated and also determines the proper time between points that are timelike separated.

Now, it is possible to choose a coordinate system (the Schwarzschild coordinates) such that the time component of the metric tensor, g_tt is related to the Newtonian gravitational potential:

g_tt = 1 - 2GM/(c² r)

But g is NOT a potential in GR, and that form of the time component is coordinate-dependent.

 

[Dale]

Maybe you think gravitational time dilation is relative. But it's absolute.

I disagree in general, but so what? What does the label "absolute" or "relative" have to do with the question of whether the light's energy increases as it goes down?

If you want to label the energy change "absolute" rather than "relative" it doesn't bother me. I think you are wrong to do so, but it is beside the point which is that it changes.

 

[jartsa]

As I said, if two clocks A and B are SPATIALLY SEPARATED, then the question of which clock has a faster rate is coordinate-dependent. If they are at the same location, then the question is not coordinate-dependent.

Do you understand that for two clocks A and B that are spatially separated, and for two different coordinate systems (with time coordinates t and T), you can have

d$\tau$_A/dt > d$\tau$_B/dt

and also have

d$\tau$_A/dT < d$\tau$_B/dT

If A and B are at the same location, that is not possible.

Gravitational time dilation is different to velocity time dilation. It's much more simple: Upper clock runs faster. You can not find a cordinate system were lower clock runs faster.

Actually, you can find a cordinate system where lower clock runs faster, if lower and upper clock are in relative motion to each other.

 

[PeterDonis]

Classical electrodynamics has this curious thing about charges and Newton's third law, they are not supposed to folllow 3rd law by themselves (not considering the field)

I see what you mean, but I'm not trying to apply Newton's Third Law in this way. Momentum stored in the EM field doesn't play a role in this problem, because we're talking about balancing the Lorentz force on the charge q with the mechanical force exerted on q by the arm, for a net force on q of zero. Momentum stored in the field would only come into play if q was moving freely, with the EM force the only force on it. Then, yes, you would have to keep track of momentum stored in the field for things to balance; if you only looked at the charge q itself it would seem like it was gaining momentum (and energy) "from nowhere". But in this problem q is not gaining momentum and energy at all, because it's not moving at all in the x direction and its motion in the y direction (in situation #2) is at a constant speed.

 

[Dale]

The big one is that there is no such thing as a "gravitational potential" in General Relativity. Gravity is not a force in GR, and it does not have a potential.

You are correct, in general. However, just to be clear, in a static spacetime you can define a potential, and in stationary spacetimes you can define a pair of potentials. But I agree with you that you shouldn't rely on reasoning based on potentials too heavily in GR. It will necessarily be limited in application at best.

 

[PeterDonis]

Now, it is possible to choose a coordinate system (the Schwarzschild coordinates) such that the time component of the metric tensor, g_tt is related to the Newtonian gravitational potential:

g_tt = 1 - 2GM/(c² r)

But g is NOT a potential in GR, and that form of the time component is coordinate-dependent.

It's true that g_tt is coordinate dependent, but it's not correct to state flatly that g_tt is not a potential, or that there is never a potential in GR. "Potential" is a matter of interpretation. In a generic spacetime, you are correct, there is not in general going to be anything that can be interpreted as a potential. But in this case, the spacetime is static and asymptotically flat, so you can indeed define a "potential" (it's basically defined the way the Newtonian potential is defined, but with relativistic corrections) and show that the "acceleration due to gravity" is the gradient of that potential, which is true of the GR version just as it's true of the Newtonian version in the Newtonian approximation. These statements can be formulated in a coordinate-independent way, btw, so even though g_tt itself is coordinate dependent, it's not correct that the potential itself is just a coordinate-dependent thing with no physical meaning, in this particular spacetime. You have to be careful to understand the limitations of viewing things this way, but within its limitations, it's a useful view.

 

[stevendaryl]

Gravitational time dilation is different to velocity time dilation. It's much more simple: Upper clock runs faster. You can not find a coordinate system were lower clock runs faster.

First of all, I didn't say anything about velocities. I was talking about coordinate transformations. Coordinate transformations change velocities, and clock rates.

Second, no, gravitational time dilation is not different from velocity time dilation--not in any absolute sense, anyway. What looks like gravitational time dilation in one coordinate system will look like velocity dependent time dilation in another coordinate system. "Gravitational time dilation" doesn't have anything specifically to do with gravity, it has to do with using curvilinear coordinates. If you use inertial coordinates, there is no gravitational time dilation.

Third, what you said is just not true. You can certainly find a coordinate system in which, at one particular time, the lower clock runs faster than the upper clock.

Let (x,t) be the Schwarzschild coordinates in which the two clocks are at rest. Then, according to those coordinates, Rate_{upper clock}/Rate_{lower clock} = 1+ gh/c². Now, switch coordinates to coordinates (X,T) defined by:

X = x
T = (1+Ax)t

Then in the coordinate system (X,T), the ratio of the clock rates will be given by:

Rate'_{upper clock}/Rate'_{lower clock} = (1+gh/c²)/ (1+Ah)

If A > g/c², then in this coordinate system, the lower clock runs faster than the upper clock.

Clock rates are a coordinate-dependent quantity. There is no absolute sense in which the upper clock always runs faster than the lower clock.

 

[stevendaryl]

It's true that g_tt is coordinate dependent, but it's not correct to state flatly that g_tt is not a potential, or that there is never a potential in GR.

Not according to GR, it's not. GR does not describe gravity in terms of potentials.

"Potential" is a matter of interpretation. In a generic spacetime, you are correct, there is not in general going to be anything that can be interpreted as a potential. But in this case, the spacetime is static and asymptotically flat, so you can indeed define a "potential" (it's basically defined the way the Newtonian potential is defined, but with relativistic corrections) and show that the "acceleration due to gravity" is the gradient of that potential,

But "acceleration" is a coordinate-dependent quantity. An object that is accelerating under gravity in one coordinate system is not accelerating at all in another coordinate system.

which is true of the GR version just as it's true of the Newtonian version in the Newtonian approximation. These statements can be formulated in a coordinate-independent way,

Hmm. I would have to see what you mean by that. I suppose you could define a "stationary observer" to be one for which the local spacetime curvature is unchanging as a function of time. Then you could define "the acceleration due to gravity" to be the relative acceleration between a freefalling object initially at "rest" and a nearby "stationary" observer. But I don't see how those definitions involve potentials.

btw, so even though g_tt itself is coordinate dependent, it's not correct that the potential itself is just a coordinate-dependent thing with no physical meaning, in this particular spacetime. You have to be careful to understand the limitations of viewing things this way, but within its limitations, it's a useful view.

I would say that definitely "gravitational potential" is a coordinate-dependent thing. In the case of the Schwarzschild geometry, there is a set of "preferred" coordinate systems, which are the ones for which the metric is spherically symmetric and time-independent.

 

[stevendaryl]

You are correct, in general. However, just to be clear, in a static spacetime you can define a potential, and in stationary spacetimes you can define a pair of potentials. But I agree with you that you shouldn't rely on reasoning based on potentials too heavily in GR. It will necessarily be limited in application at best.

Maybe it's a matter of terminology. In the Schwarzschild spacetime (and is that the only one?) you can pick a coordinate system in which the metric tensor is time-independent and spherically symmetric. For that particular coordinate system, you can solve for the trajectories of test particles in a way that looks a lot like solving a problem in Newtonian physics with a gravitational potential. But I would not call it a potential.

 

[George Jones]

For good or for bad, "effective potential" is a term used in most GR texts (e.g, Carroll, Wald, Taylor and Wheeler, Hartle, Schutz) in their treatments of geodesics in Schwarzschild.

 

[PeterDonis]

Maybe it's a matter of terminology. In the Schwarzschild spacetime (and is that the only one?) you can pick a coordinate system in which the metric tensor is time-independent and spherically symmetric.

Schwarzschild is not the only one; you can do it in Reissner-Nordstrom spacetime as well. But you can also express the fact that the metric is time-independent and spherically symmetric in a coordinate-free manner, in terms of the Killing vector fields of the spacetime. And you can define the "potential" in terms of the Killing vector fields, so that it has a coordinate-free definition as well.

 

[Q-reeus]

To simplify things somewhat from the "beams inside a synchrotron" scenario, suppose we have a single object of charge q inside a sort of circular capacitor, which I'll call the "ring" for brevity: a device that can create an electric field directed radially. We put the charged object at the end of a massless arm that runs on a track on the inside surface of the ring. The arm holds the charged object at a particular radius r, which is between the inner radius r_i and the outer radius r_o of the ring. The whole thing is floating in free space (flat spacetime).

Our two observables are, again, a scale under the track (O) and a strain gauge on the arm (O').

First, we consider the case where the q+arm assembly is at rest relative to the ring. We turn on the capacitor and find that the scale and strain gauge show nonzero readings. Call this set of readings A. We expect the readings to be equal: O_A = O'_A.

Now we set the q+arm assembly to rotating around the ring with constant angular velocity omega, and therefore constant relative speed v relative to the ring (but with changing direction). At a given instant, we can set up two local inertial frames (MCIFs): one in which the ring+track is at rest and the q+arm assembly is moving at speed v in a direction orthogonal to the E field generated by the ring; and one in which the q+arm assembly is at rest and the ring+track is moving at -v in a direction orthogonal to the E field generated by the ring.

It seems to me that, locally, these two inertial frames are the same in almost all respects (I'll elucidate that "almost all" in a moment) as the two frames in the straight line case: the rest frame of the charge q, and the rest frame of the capacitor. There is one key complication, though; in the rotating case, there is no longer an exact "force balance" in both frames, because the motions are no longer unaccelerated; the instantaneous relative velocity of the two is orthogonal, but the instantaneous acceleration of q+arm is not zero, it has an inward radial component.

This, to me, means we can't draw an exact analogy between the two scenarios, because we no longer have an exact force balance in the direction of the E field. The reasoning that led to O = O' depended on there being an exact force balance. So I don't see the two cases as analogous.

Centripetal acceleration introduced here is of no fundamental consequence. Covered that in the similar and better spinning carousel example back in #92; simply increase radii of your ring capacitor until u²/r acceleration is negligible, or just specify we are concerned only with EM interaction component. The carousel example was better in a number of respects. Centripetal acceleration and electric forces owing to an axially uniform applied E are there orthogonal not co-radial as above, making for a clean separation. But it's your previous response, maintaining there will be an increased axial net force on carousel charges (as measured in non-spinning frame) owing to rotation speed that has an obvious and easily checkable corollary. A necessarily equal and opposite force back on the stationary capacitor charges - assuming one accepts overall force balance must apply.

That no such speed dependent increase in carousel axial forces occurs, just note that it is experimental fact, consistent with charge invariance strictly holding in SR, there is no electrostatic field generated by a current flowing in an electrically neutral circuit. Only an exterior Biot-Savart B field - explained in the usual SR way as differential charge densities seen in a frame moving wrt the circuit. Thus an exterior charge q_e stationary wrt the circuit feels nothing (we assume for simplification here there is no screening charges on the circuit's conducting wire). In the frame of a moving conduction charge q_c, the transverse E field component of q_e is greater for q_c than experienced by a fixed lattice charge q_f, by the drift-velocity gamma factor. Your logic maintains that larger proper 3-force acting on q_c should persist unchanged as measured in the circuit rest frame, where conduction and lattice charge number-densities are equal. Integrating over all conduction and lattice charges, it follows a net force would be exerted by field of q_e on the circuit, despite no such force being exerted by the circuit on q_e. That is the logical consequence of your position. And I've said as much on previous occasions. Force balance is naturally satisfied though by applying the usual 3-force transformation rule - requiring transverse components of both forces and fields measure differently in different frames.

 

[Q-reeus]

I found the original paper, but it was in German, so that wasn't too helpful for me. So I looked around and found a lot of proofs that were just too brief for me to really grasp, until this one by Eric Poisson:
http://books.google.com/books?id=bk...qySBA&ved=0CD4Q6AEwAA#v=onepage&q=176&f=false
Just click any of the hyperlinks to the proof on the bottom of p 176 and all of p 177. Seems pretty solid to me.

Thanks for reference, but apart from p 176 being unavailable I got the impression of a derivation following formal rules without really explaining the guiding philosophy. What raises my eyebrows is a passage down on p 180:
"It therefore appears that RN metric describes more than just a single black hole. Indeed it describes an infinite lattice of asymptotically flat universes connected by black-hole tunnels. Such a fantastic spacetime structure...", 'fantastic' being the key word - derivation from 'fantasy'.

Found this article which only asks more questions imo: https://facultystaff.richmond.edu/~ebunn/ajpans/ajpans.html
I come back to asking how coupling together the implications in #109 together with case 3: in #1 can be dismissed on logical grounds.

 

[Q-reeus]

I don't see how this would work at all. First of all, there is no "fulcrum point", or at least I don't see one; that would require the track to only be supported at one spot between the scale under the track and the location of the arm. That makes no sense, and anyway, we could just change the scenario to have a support directly underneath the scale, so its distance to any "fulcrum" would be zero. Would that change your prediction of the relative forces?

You're taking my metaphors way too literally! Could you please this time answer what I asked of you in last para of #149. And just to be completely clear, in #102 - that's a thin walled right-circular cylinder subject to a uniform axially directed end-forces via say frictionless pistons. Your take on the axial strain in both non-spinning and spinning cases. And to eliminate distractions like effect of coupling centrifugal stresses to axial strain via Poisson's ratio, we specify force is only applied after spin-up has occurred in spinning case. Or if you want, that frictionless thick outer sleeve absorbs all centrigugal stresses, etc. etc. Your understanding of what happens and why, as seen and interpreted in the two frames. Please.

 

[Dale]

Thanks for reference, but apart from p 176 being unavailable I got the impression of a derivation following formal rules without really explaining the guiding philosophy.

I know that we probably disagree on this, but I fundamentally believe that the universe can be analyzed logically. So the formal rules are important.

The assumptions from the derivation, as far as I can tell are:
1) EFE holds
2) Maxwell's equations hold
3) static, spherical symmetry
4) exterior EM field has no sources
5) exterior metric has only EM field as source

I cannot see any other assumptions, and, as you say, the remainder is simply following the formal rules from those.

I will look at the other points you cited, but can you look at those 5 assumptions and see if any of them seem wrong to you? Or let me know if you think I missed one.

 

[PeterDonis]

Centripetal acceleration introduced here is of no fundamental consequence. Covered that in the similar and better spinning carousel example back in #92; simply increase radii of your ring capacitor until u²/r acceleration is negligible

Yes, I can see that you can reduce the acceleration until its effect is very small, so you are very close to having a force balance.

, or just specify we are concerned only with EM interaction component.

Not sure what good that does; force balance is force balance, it has to include all forces acting.

The carousel example was better in a number of respects. Centripetal acceleration and electric forces owing to an axially uniform applied E are there orthogonal not co-radial as above, making for a clean separation.

Yes, this is true, having the forces orthogonal changes the analysis.

But it's your previous response, maintaining there will be an increased axial net force on carousel charges (as measured in non-spinning frame) owing to rotation speed that has an obvious and easily checkable corollary.

I don't think I've even analyzed the case of axial forces at all, have I? I've only analyzed the linear relative motion case and the case of radial forces in a circular ring.

In any case, you keep on multiplying scenarios when we haven't even got the first one taken care of. See below.

You're taking my metaphors way too literally! Could you please this time answer what I asked of you in last para of #149.

Which referred back to #102...

And just to be completely clear, in #102 - that's a thin walled right-circular cylinder subject to a uniform axially directed end-forces via say frictionless pistons.

Which involves forces in two different directions, axial and radial, if the cylinder is spinning. Can't you stick to one scenario? We were talking about a simple scenario with forces in only one direction. I have asked you, repeatedly, how the arm can possibly be motionless if the forces on the two ends of it are different, which is a necessary implication of your claim that the reading on the scale under the track, O, is different than the reading on the strain gauge on the arm, O', when q+arm is moving relative to capacitor+track. Are you ever going to just answer this question, without dragging in multiple other scenarios that add other elements that aren't in this one, and then asking me to wade through all your arguments that purport to show how they really really are related after all?

Sorry if I sound frustrated, but if you really have a good answer to the question I posed (again) in the last paragraph, you ought to be able to give it in a single post that doesn't refer to any other scenarios. Until you do, I'm not going to talk about any of those other scenarios. They might well be interesting to talk about, but let's get the one simple scenario, with forces in just one direction, taken care of first.

 

[PeterDonis]

if you really have a good answer to the question I posed (again) in the last paragraph, you ought to be able to give it in a single post that doesn't refer to any other scenarios.

In fact, I'll make it even easier by simplifying the scenario to show that the fundamental question about force balance doesn't even involve EM fields at all. (That should also keep up my tradition for provocative disagreement .)

Consider an object "q" on an arm that has a strain gauge attached, which runs on a track with a scale under it. There is some unspecified force acting on q in the positive x direction. It doesn't matter what kind of force it is; the only requirement is that it only acts on q, not on any other objects we're considering. (In the version where q is a charged object, this is ensured by stipulating that the arm with its strain gauge, and the track with its scale under it, are electrically neutral.) The reading on the scale under the track is observable O; the reading on the strain gauge on the arm is O'.

(1) In the case where everything is at rest, no relative motion, the force balance in the x direction implies O = O'.

(2) In the case where "q"+arm+gauge is moving in the y direction, relative to track+scale, the force balance in the x direction still implies O = O'. The actual magnitude of O and O' may be different from what it is in #1 above, but the two will still be the same.

Okay, Q-reeus: you agree with #1 but not with #2. What's the difference? Don't use any specific facts about what kind of force acts on "q". Give a general argument that will hold regardless of what kind of force it is, given what is specified above.

(Edit: Just to be clear, though we've discussed it before, by "readings" O and O' I mean the actual numbers appearing on the scale's or strain gauge's readouts.)

 

[Q-reeus]

I know that we probably disagree on this, but I fundamentally believe that the universe can be analyzed logically. So the formal rules are important.

The assumptions from the derivation, as far as I can tell are:
1) EFE holds
2) Maxwell's equations hold
3) static, spherical symmetry
4) exterior EM field has no sources
5) exterior metric has only EM field as source

I cannot see any other assumptions, and, as you say, the remainder is simply following the formal rules from those.

I will look at the other points you cited, but can you look at those 5 assumptions and see if any of them seem wrong to you? Or let me know if you think I missed one.

That summary is helpful. As I have claimed global failure of charge invariance is implied by examples given earlier, that in turn is questioning how 1) and 2) are made to combine in getting to RN metric. The only coupling of relevance there is afaik that EM field acts as an exterior SET source - of further gravity. But that otherwise static gravity modifies static EM field not. You know now what point in particular I refer to - effective charge is not subject to any 'redshift'. Points 3) to 5) are just boundary conditions so no problem there.

 

[Q-reeus]

Which involves forces in two different directions, axial and radial, if the cylinder is spinning.

You may have noticed though, the uncut version of your quote specified how to eliminate radial stresses - and one further option in that regard is as for your radial capacitor - just increase radius arbitrarily, or limit spin-rate to small value. That way, ratio of gamma factor effecting axial strain, to centrifugul stresses is arbitrarily large. Many ways to tame the 'monster' of two forces here! But if you don't want to, ok no gun at your head.

Can't you stick to one scenario? We were talking about a simple scenario with forces in only one direction. I have asked you, repeatedly, how the arm can possibly be motionless if the forces on the two ends of it are different, which is a necessary implication of your claim that the reading on the scale under the track, O, is different than the reading on the strain gauge on the arm, O', when q+arm is moving relative to capacitor+track. Are you ever going to just answer this question, without dragging in multiple other scenarios that add other elements that aren't in this one, and then asking me to wade through all your arguments that purport to show how they really really are related after all?

Well they are related Peter, and honestly your question has been answered umpteen times but not to your liking. We are dealing with effects of motion in SR context and readings in general will distort from one frame to another. In particular for our discussions, the 3-force transformation rules say so!
That bit in #188 about exterior charge and circuit is a good example of how force measured in one frame must measure differently in another. Only real paradox would be if they didn't vary. NASA would love it to be so - "off to the stars with breakthrough propulsion physics."

Sorry if I sound frustrated, but if you really have a good answer to the question I posed (again) in the last paragraph, you ought to be able to give it in a single post that doesn't refer to any other scenarios. Until you do, I'm not going to talk about any of those other scenarios. They might well be interesting to talk about, but let's get the one simple scenario, with forces in just one direction, taken care of first.

To repeat, there is no paradoxical imbalance because, as the 3-force transformation formula insists, what is measured by strain gauge moving with charge and arm, must be greater by gamma factor from what the stationary-in-lab-frame scales under the tracks measures, in order that arm has no net dp_x/dt. We cannot directly compare readings because relative motion is screwing up an exact correspondence. As you know I have coined it RW - strain gauge (or better for our purposes - spring scales holding charge) is 'weakened' as perceived in lab frame - the higher reading is out of whack as far as lab frame folks are concerned. And vice versa from moving frame perspective.

And now that I've caught your #193, I guess the answer to your specific there is going to be obvious. For any transverse force F'_x acting on q in frame S' - in motion at relative velocity u wrt lab frame S, transverse force F_x read by stationary scales in lab frame is, by simply applying the 3-force transformation formulae, reduced according to F_x = F'_x/γ. And conversely F'_x = γF - ie reciprocity of readings applies. And there is then no actual imbalance along x - dp_x/dt (net) = 0.
In the 4D perspective you are more familiar with, isn't it just a case of 4-invariance demanding certain spatial components must therefore vary to preserve overall 4-invariance?
And yes mutual crankiness does seem to be cranking up - what!

 

[Dale]

That summary is helpful. As I have claimed global failure of charge invariance is implied by examples given earlier, that in turn is questioning how 1) and 2) are made to combine in getting to RN metric. The only coupling of relevance there is afaik that EM field acts as an exterior SET source - of further gravity. But that otherwise static gravity modifies static EM field not. You know now what point in particular I refer to - effective charge is not subject to any 'redshift'. Points 3) to 5) are just boundary conditions so no problem there.

Thanks, that really helps to, I think, pinpoint the concern.

Re: the coupling of the EFE and Maxwell's Equations (ME). It isn't just one-way. In other words, the EFE are impacted by the stress energy tensor of the EM field, but ME in curved spacetime are, in turn, impacted by the metric. So they really are a set of coupled equations, not just an input and an output.

There is a specific line in the derivation that shows this quite clearly, I will post it shortly.

EDIT: Here is the line:
"Maxwell's equations in vacuum are $0=F^{\alpha\beta}_{\;\;\; ;\beta} = |g|^{-1/2}(|g|^{1/2}F^{\alpha\beta})_{,\beta}$."

I am still going through the link you posted, that won't be as "shortly".

 

[PeterDonis]

To repeat, there is no paradoxical imbalance because, as the 3-force transformation formula insists, what is measured by strain gauge moving with charge and arm, must be greater by gamma factor from what the stationary-in-lab-frame scales under the tracks measures, in order that arm has no net dp_x/dt.

I understand the argument you are making; I'm just not sure you are properly representing the actual observables in the math. However, I admit I'm having trouble myself coming up with an invariant description of both observables, O and O', so I can't exactly fault you for not coming up with one either.

We cannot directly compare readings because relative motion is screwing up an exact correspondence.

But consider an implication of this. Suppose that the reading O' does not change when we put "q"+arm in motion. (Again, we're not specifying anything about how the force on "q" is produced; just that it doesn't change when "q"+arm is in relative motion wrt scale+track.) Your argument implies that reading O must *decrease* in this scenario.

And conversely F'_x = γF - ie reciprocity of readings applies. And there is then no actual imbalance along x - dp_x/dt (net) = 0.
In the 4D perspective you are more familiar with, isn't it just a case of 4-invariance demanding certain spatial components must therefore vary to preserve overall 4-invariance?

Again, I see the argument for why the mathematical objects you are talking about should vary in the way they are varying; if we restrict attention to one frame at a time the math equations in each frame appear to describe a balance. But again, I'm not sure we agree on which particular math expressions represent the actual physical observables. However, as I said, I'm having trouble coming up with invariant expressions for the observables, so perhaps we need to table this until I can.

 

[PeterDonis]

In the 4D perspective you are more familiar with, isn't it just a case of 4-invariance demanding certain spatial components must therefore vary to preserve overall 4-invariance?

One other thought on this: in cases where what you are describing happens, the time component varies as well (for example, the obvious case of a Lorentz boost changing both the time and space components of a 4-vector).

Here the time component of dp/dt (p is the 4-momentum here, obviously) is *zero* for all the objects involved; the time component of dp/dt describes a change in energy, and that isn't happening for any of the objects. So whatever is going on with the space components, it isn't a simple case of being part of overall 4-invariance as you're describing. Another way of putting this, as I've said before, is that the "variance" in space components is not between representations of the same geometric object, the same 4-vector, in two different frames (which is where "preserving overall 4-invariance" would apply). The variance here is between the space components of two *different* geometric objects, two different 4-vectors--one describing the arm+gauge, the other describing the scale+track.

 

[TrickyDicky]

I see what you mean, but I'm not trying to apply Newton's Third Law in this way. Momentum stored in the EM field doesn't play a role in this problem, because we're talking about balancing the Lorentz force on the charge q with the mechanical force exerted on q by the arm, for a net force on q of zero. Momentum stored in the field would only come into play if q was moving freely, with the EM force the only force on it. Then, yes, you would have to keep track of momentum stored in the field for things to balance; if you only looked at the charge q itself it would seem like it was gaining momentum (and energy) "from nowhere". But in this problem q is not gaining momentum and energy at all, because it's not moving at all in the x direction and its motion in the y direction (in situation #2) is at a constant speed.

If we were restricting the discussion to SR you'd be right, but in GR there isn't such thing as a purely inertial geodesic.

 

[PeterDonis]

If we were restricting the discussion to SR you'd be right, but in GR there isn't such thing as a purely inertial geodesic.

I don't know what you mean by this. If by "purely inertial geodesic" you mean "freely falling worldline", that's obviously false; there are an infinite number of freely falling worldlines (timelike curves) in any spacetime. I can't come up with any other meaning for "purely inertial geodesic", so if the above is not what you meant, please clarify.

 

[TrickyDicky]

I don't know what you mean by this. If by "purely inertial geodesic" you mean "freely falling worldline", that's obviously false; there are an infinite number of freely falling worldlines (timelike curves) in any spacetime. I can't come up with any other meaning for "purely inertial geodesic", so if the above is not what you meant, please clarify.

I meant Minkowskian geodesic.
In the sense that even though locally (in the neighbourhood of a point) GR spacetime is Minkowskian, clearly a geodesic in flat spacetime (what I call an inertial geodesic) must not be the same as a geodesic in curved spacetime. That was supposed to be the main difference between SR and GR.

 

[PeterDonis]

One other thought on this: in cases where what you are describing happens, the time component varies as well (for example, the obvious case of a Lorentz boost changing both the time and space components of a 4-vector).

Just to add some more to this, suppose I wanted to come up with a 4-vector to describe the reading O' on the strain gauge attached to the arm. The obvious thing to do is to take the 3-vector describing the force F' and make a 4-vector out of it by adding a zero time component. Then the invariant proper length of this 4-vector, which is just F', would give us the number corresponding to the reading on the scale. Since the proper length of a 4-vector (unlike a 3-vector) is invariant, this looks good so far; I can compute the components of this 4-vector in any frame I like and it will give me the same number for reading O'.

The problem, if it is a problem, is that when we transform this 4-vector to the unprimed frame, it looks *the same*. Not just its proper length is invariant (which is always true)--*all* of its components are exactly the same! That is, it transforms from (0, F', 0, 0) to (0, F, 0, 0), which requires F = F' (otherwise the proper length will change). This follows, of course, from the fact that a Lorentz transformation in the y direction can only change the t and y components of a 4-vector; but this 4-vector only has an x component.

This is as close as I can come right now to putting into some kind of formal terms the intuition that's been driving my comments all along--a mathematical expression of "force balance in the x direction is unchanged by relative motion in the y direction". Just to be explicit, here's the chain of logic (with notation changed a bit to help with clarity):

(1) In the primed frame (rest frame of q+arm), the 4-vector describing reading O' is (0, F_O', 0, 0). This is required by the fact that in this frame, the 4-vector must have only an x component, so that component must just be the force registered by O'.

(2) Transforming this 4-vector O' into the unprimed frame leaves it unchanged: (0, F_O', 0, 0). This follows, as I said above, from the fact that a Lorentz boost in the y direction can't affect the x component, and there are no other nonzero components.

(3) In the unprimed frame (rest frame of scale+track), the 4-vector describing reading O is (0, F_O, 0, 0), for the same reasons as we used for O' in the primed frame.

(4) For force balance in the unprimed frame to hold, the 4-vector describing reading O must have the same x component in that frame as the 4-vector describing reading O'. This requires that F_O = F_O'.

(5) Hence, reading O must be the same as reading O'.

 

[PeterDonis]

I meant Minkowskian geodesic.

Ah, I see, you are referring to the fact that I said momentum and energy were constant. Yes, you're right, that reasoning, as I gave it, requires flat spacetime. It would still hold in a local inertial frame in curved spacetime, as you say, but it would not if we had to cover a large enough patch of spacetime for tidal effects to come into play.

 

[jartsa]

Third, what you said is just not true. You can certainly find a coordinate system in which, at one particular time, the lower clock runs faster than the upper clock.

So, I guess the clock rates are changing, and only at some particular times the lower clock runs faster, and on the average a lower clock runs slower?

 

[Dale]

Hi Q-reeus, I got the time to go through the page you cited and the post 109.

Found this article which only asks more questions imo: https://facultystaff.richmond.edu/~ebunn/ajpans/ajpans.html

This is all about the event horizon problem. However, from the rest of the discussions it seems that there are some objections outside the horizon. I think we should work on the outside problems first, e.g. by considering a charged planet rather than a charged BH. If we cannot agree on the physics of a charged planet then there is no sense in worrying about any disagreements on the physics of a charged BH.

Actually, it seems that there are even simpler disagreements about the physics of charges outside an uncharged planet. And those should probably be addressed even before worrying about a charged planet.

I come back to asking how coupling together the implications in #109 together with case 3: in #1 can be dismissed on logical grounds.

I haven't joined much in the discussion of redshifting masses and forces and such because I don't think that either you or PeterDonis are approaching it correctly. The correct way to compare tensors at different points in a manifold is to use parallel transport, IMO.

I have never worked out the parallel transport for e.g. an electron's 4 momentum nor for the four-force on an electron, so I am not sure what would "redshift" and what wouldn't. I can attempt it if you are interested in learning about parallel transport and discussing its implications, but this time I will make no cocky guarantees that I can solve it. But I just am not at all convinced either way by either your or PeterDonis' "thought experiments" and "handwaving" analyses.

 

[stevendaryl]

So, I guess the clock rates are changing, and only at some particular times the lower clock runs faster, and on the average a lower clock runs slower?

Yes. On the average, the lower clock runs slower, but not at every moment. It's the same sort of situation as the twin paradox of SR. One twin travels far away at the speed and comes back. The other twin stays put. You can choose a coordinate system in which the traveling twin's clock runs faster during the outward journey. You can choose a coordinate system in which the traveling twin's clock runs faster during the return journey. But you CAN'T find a coordinate system in which the traveling twin's clock runs faster the whole time; during one leg or the other (or both), the traveling twin's clock must run slower. But there is no objective point at which at everyone agrees: Right now, his clock is running slower.

 

[PeterDonis]

The correct way to compare tensors at different points in a manifold is to use parallel transport, IMO.

Just to be clear about where I'm coming from, I agree with this. I think I said at one point that I would need to get deeper into the math; this is basically what I meant. I should also note, though, that much of Q-reeus' and my discussion in this thread has been about a side issue, not directly about whether or not charge "redshifts" in a gravitational field.

 

[Q-reeus]

However, from the rest of the discussions it seems that there are some objections outside the horizon. I think we should work on the outside problems first, e.g. by considering a charged planet rather than a charged BH. If we cannot agree on the physics of a charged planet then there is no sense in worrying about any disagreements on the physics of a charged BH.

Agreed.

Actually, it seems that there are even simpler disagreements about the physics of charges outside an uncharged planet. And those should probably be addressed even before worrying about a charged planet.

You will need to elaborate - you mean I've somehow been getting it wrong on Maxwell's equations in flat spacetime?

I haven't joined much in the discussion of redshifting masses and forces and such because I don't think that either you or PeterDonis are approaching it correctly. The correct way to compare tensors at different points in a manifold is to use parallel transport, IMO.

I have never worked out the parallel transport for e.g. an electron's 4 momentum nor for the four-force on an electron, so I am not sure what would "redshift" and what wouldn't. I can attempt it if you are interested in learning about parallel transport and discussing its implications, but this time I will make no cocky guarantees that I can solve it. But I just am not at all convinced either way by either your or PeterDonis' "thought experiments" and "handwaving" analyses.

Appreciate you're willing to tackle it, but how much of parallel transport in GR do I need to learn? And to what extent is parallel transport aloof from assumptions about how gravity interacts with EM - charge in particular? In the end, whatever your parallel transport approach concludes, it will have to be applied to those situations I gave and provide a consistent resolution. For instance, there can be no disputing that say the arrangement in #109 demands the EM energy of a system of charges gravitationally redshifts - but e.g. a possible 'split' thereafter between 'active' and 'passive' charge may be conjectured.

So parallel transport will need to explain/refute how EM energy redshift meshes with this:
Neutral, non-free-falling mass/energy's contribution to overall Komar mass M redshifts, whereas the very existence of a charged BH requires total immunity of free-falling charge to E field redshift. Which in turn implies in that BH extreme case [STRIKE]a colossally disproportionate EM energy density contribution from charge that is just not at all locally evident[/STRIKE] (hand-wavy situation 3: in #1).
[That bit reads wrong. Meant that locally as EH is approached, hovering observer sees a vanishingly small q/m ratio. That m is of EM nature, and although locally huge (ultrarelativistic particle energy), 'out there' it reduces to just the rest energy of q - assuming initial free-fall from infinity. So there is this enormous (asymptotically infinite) reduction in gravitating mass ('grav charge') moving from local to coordinate measure, yet apparently none at all for electric charge. And that makes sense?]
Hand-wavy perhaps but imo logically necessary conclusions. And a basic reason for my 'Yes' vote to the thread title. But willing to be convinced otherwise.

 

[Q-reeus]

Just to add some more to this, suppose I wanted to come up with a 4-vector to describe the reading O' on the strain gauge attached to the arm. The obvious thing to do is to take the 3-vector describing the force F' and make a 4-vector out of it by adding a zero time component. Then the invariant proper length of this 4-vector, which is just F', would give us the number corresponding to the reading on the scale. Since the proper length of a 4-vector (unlike a 3-vector) is invariant, this looks good so far; I can compute the components of this 4-vector in any frame I like and it will give me the same number for reading O'.

The problem, if it is a problem, is that when we transform this 4-vector to the unprimed frame, it looks *the same*. Not just its proper length is invariant (which is always true)--*all* of its components are exactly the same! That is, it transforms from (0, F', 0, 0) to (0, F, 0, 0), which requires F = F' (otherwise the proper length will change). This follows, of course, from the fact that a Lorentz transformation in the y direction can only change the t and y components of a 4-vector; but this 4-vector only has an x component.

Yes as such but that's owing to the moving-on-rails constraint imposing exact balance of opposing transverse forces as evaluated in any given frame. To see how each of those opposed forces transform we need to consider the situation of instantaneous free-fall of q along x (at the instant u_x = 0) and then we have the compact derivation of dp/dt yielding Lorentz force expression as per bottom of p18 here: "yannis.web.cern.ch/yannis/teaching/relativity.pdf"
And that expression then demands that with rails constraint in place, transverse dp/dt (applying to each opposed electric and mechanical restraint force separately) is independent of longitudinal velocity applying in a given frame - in particular here the lab frame S. Indisputably the transverse E field and thus proper 'active' force qE_x measured in q's proper frame S' is by Lorentz force expression a factor gamma greater than as measured in lab frame S. If you still feel not, I invite you to solve that paradox of #188!

(5) Hence, reading O must be the same as reading O'.

You're being too 'naughty' again.

 

[Dale]

Just to be clear about where I'm coming from, I agree with this. I think I said at one point that I would need to get deeper into the math; this is basically what I meant.

That is good to know. I could see the possibility that components of tensors would redshift, although the conclusion that they do is not foregone in my mind. We have several that might be of interest here:
charge density, the timelike component of four-current
relativistic mass, the timelike component of four-momentum
force, the spacelike component of four-force

Obviously, by the nature of parallel transport, the invariant mass (norm of the four-momentum) does not redshift, but that is about the only thing that I can say for sure. For everything else I would have to work out the parallel transport equation, but in my mind it is not clear what paths to use for the various cases that have been discussed.