Is 'charged black hole' an oxymoron?

[Dale]

You should have figured it by now - I work it 'backwards'. Setup a gedanken experiment. Check for consistent predictions.

This is a fundamentally impossible approach, for the reasons cited above. You cannot ever either prove or disprove self consistency in this manner.

"In curved spacetime" - my little gedankens are worrying me about the marriage of ME's to such.

OK, now we have something. You doubt that the expression in 196 correctly represents ME in curved spacetime. Do I understand your position correctly?

If you beleived that equation were the correct form of ME in curved spacetime then would you agree that the derivation is correct (ie correct axioms, formally correct derivation, therefore correct conclusions)? If not, then which other axioms are also suspect?

 

[TrickyDicky]

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

Here you are objecting the equivalence principle when it says that the two situations are physically indistinguishable I guess.
Also your example uses a static spacetime too, no time-varying metric there either.

 

[stevendaryl]

Here you are objecting the equivalence principle when it says that the two situations are physically indistinguishable I guess.
Also your example uses a static spacetime too, no time-varying metric there either.

The equivalence principle doesn't say that curved spacetime is indistinguishable from flat spacetime. Of course, they aren't, because the curvature is an observable. The way that it comes into play in GR is just that "gravitational force" is actually a pseudo-force due to a particular choice of coordinates; it can be made to vanish locally by choosing locally inertial coordinates. That's really the only content of the equivalence principle.

As for my examples, yes in both Schwarzschild coordinates and in Rindler coordinates, the metric tensor is time-independent. And in both cases, there is no work done on an object at "rest" in those coordinates. So the idea that it takes work to keep an object in place is not correct.

 

[PeterDonis]

I got 2M in the limit.

Can you elaborate? K goes to 1 in the limit, so the first term in what I obtained is just M, and the other two cancel.

 

[PeterDonis]

My R referred to the shell radius specifically and it's that R that governs the depressed potential experienced by any charge inside said shell.

Ah, ok, I misread. Then I'll defer comment until I've had a chance to analyze the shell case.

 

[PeterDonis]

Interesting result there Peter.

I should note that on comparing this result for Q(r) with the equation I posted early in the thread for the proper acceleration on a test charge due to the charge of the hole, I'm not sure they're consistent. So I need to check things some more.

 

[Q-reeus]

Q-reeus: "You should have figured it by now - I work it 'backwards'. Setup a gedanken experiment. Check for consistent predictions."
This is a fundamentally impossible approach, for the reasons cited above. You cannot ever either prove or disprove self consistency in this manner.

There is a famous line that goes something like "A theory can be proved true a thousand times; but one counterexample and it's dead." I take it you reject that possibility outright - there can be no such thing as a counterexample. So you believe the one and only way to check on a theory is via experimental/observational evidence then?

Q-reeus: ""In curved spacetime" - my little gedankens are worrying me about the marriage of ME's to such."
OK, now we have something. You doubt that the expression in 196 correctly represents ME in curved spacetime. Do I understand your position correctly?

Yes, and not based on any direct analysis of that expression which as you know I can't even properly interpret. I base it on the problems imo raised as per everything previously presented in this thread - #1 and #109 summarise well enough. Do I need to repeat those yet again, or are you now as you should be, thoroughly familiar with exactly where I see inconsistencies appearing?

If you beleived that equation were the correct form of ME in curved spacetime then would you agree that the derivation is correct (ie correct axioms, formally correct derivation, therefore correct conclusions)?

Yes, with possible proviso it's unphysical boundary conditions that have been applied to RN case, and just how Poisson sets his psi = o to get F^tr =Q/r² in (5.22) on p177 of that previously linked article is perhaps under that category, or that best thought of as direct consequence of 'the coupling of EFE's to ME's', I'm not sure.

If not, then which other axioms are also suspect?

None I can think of. Now, with permission from the court I seek temporary leave persuant of legal council with an appointed barrister!

 

[TrickyDicky]

The equivalence principle doesn't say that curved spacetime is indistinguishable from flat spacetime.

I never said the equivalence principle says that.

As for my examples, yes in both Schwarzschild coordinates and in Rindler coordinates, the metric tensor is time-independent.

So the "big difference in particular" that you claimed in your previous post between the gravitational field situation and the SR one is not a difference.

 

[stevendaryl]

So the "big difference in particular" that you claimed in your previous post between the gravitational field situation and the SR one is not a difference.

I was talking on the one hand about freefalling observers, and on the other hand about accelerated observers.

What I said was (and I quote)

"an observer in freefall in flat spacetime sees an unchanging metric, while an observer in freefall in the gravitational field of the Earth sees a time-varying metric"

That's true.

We have 4 cases:

1. Accelerated observer "at rest" in flat spacetime using Rindler coordinates.
2. Observer in "freefall" in flat spacetime using inertial coordinates.
3. Accelerated observer "at rest" in Schwarzschild spacetime using Schwarzschild coordinates.
4. Observer in "freefall" in Schwarzschild spacetime using locally inertial coordinates.

In all cases except the last, the metric is time-independent. In the last case, the metric is time-varying (as he gets closer to the center of the source of gravity, the curvature becomes stronger).

 

[TrickyDicky]

I was talking on the one hand about freefalling observers, and on the other hand about accelerated observers.

What I said was (and I quote)

"an observer in freefall in flat spacetime sees an unchanging metric, while an observer in freefall in the gravitational field of the Earth sees a time-varying metric"

That's true.

We have 4 cases:

1. Accelerated observer "at rest" in flat spacetime using Rindler coordinates.
2. Observer in "freefall" in flat spacetime using inertial coordinates.
3. Accelerated observer "at rest" in Schwarzschild spacetime using Schwarzschild coordinates.
4. Observer in "freefall" in Schwarzschild spacetime using locally inertial coordinates.

In all cases except the last, the metric is time-independent. In the last case, the metric is time-varying (as he gets closer to the center of the source of gravity, the curvature becomes stronger).

Basic facts of GR and differential geometry: Schwarzschild is an static spacetime, changes in coordinates cannot alter the physics nor the metric from time-independent to time-dependent. Do you agree with these facts or not?

 

[stevendaryl]

Basic facts of GR and differential geometry: Schwarzschild is an static spacetime, changes in coordinates cannot alter the physics nor the metric from time-independent to time-dependent. Do you agree with these facts or not?

I agree with the first, in the sense that if you express the physics in covariant form, then the physics is the same in all coordinate systems. But the description in terms of "gravitational potential energy" is NOT a covariant way of describing things. You can only use that description in special coordinates

Your second statement is completely wrong. Changes in coordinates can certainly change a time-independent metric into a time-varying one. If you are changing from one set of coordinates $X_{\mu}$ to another set of coordinates $X'_{\alpha}$, the metric tensor changes as follows:

$g'_{\alpha\beta} = \partial_{\alpha}X^{\mu} \partial_{\beta}X^{\nu} g_{\mu\nu}$

If the quantity $\partial_{\alpha}X^{\mu}$ is time-dependent, then $g'_{\alpha\beta}$ can be time-dependent, even if $g_{\mu\nu}$ is not.

For example, start with Rindler coordinates (in 2D spacetime, for simplicity) the coordinates are $X$ and $T$, and the metric components are:

$g_{TT} = X^{2}$
$g_{XX} = -1$

The metric components are time-independent. Now, switch to new coordinates $x$ and $t$ related to $X$ and $T$
$X = x + vt$
$T = t$

where $v$ is some constant. Then the metric in the new coordinates looks like this:

$g_{tt} = (x+vt)^2 - 1$
$g_{tx} = -1$
$g_{xx} = -1$

The metric component $g_{tt}$ is time-dependent.

 

[stevendaryl]

Basic facts of GR and differential geometry: Schwarzschild is an static spacetime, changes in coordinates cannot alter the physics nor the metric from time-independent to time-dependent. Do you agree with these facts or not?

I gave a detailed counterexample to your last claim, but really, you should see it immediately: whether or not something is "time-varying" depends on what you choose as your time coordinate. So being "time-independent" is not a coordinate-free notion.

 

[TrickyDicky]

Your second statement is completely wrong. Changes in coordinates can certainly change a time-independent metric into a time-varying one.

There is a recent thread devoted to clarify that changes of coordinates cannot change Killing vector fields.

 

[Dale]

So you believe the one and only way to check on a theory is via experimental/observational evidence then?

No. I also believe that it can be falsified by a rigorous formal proof.

Yes, and not based on any direct analysis of that expression which as you know I can't even properly interpret.

Well then I will endeavour to help you learn to interpret it.

For now, why don't you start with the relevant Wikipedia page for Maxwell's equations in tensor notation. Please let me know what, if anything, you have trouble with, and then I can point you to other appropriate resources:
http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism

This page is just about the notation rather than any new physics. I.e. it is a tensor formulation for arbitrary coordinates in flat spacetime.

Yes, with possible proviso it's unphysical boundary conditions that have been applied to RN case

This I agree to without reservation. We have no direct experimental observation of a black hole let alone a charged black hole. The boundary conditions may very well be unphysical, that is an experimental matter which cannot be decided in a thread, but we shouldn't forget it either.

 

[stevendaryl]

There is a recent thread devoted to clarify that changes of coordinates cannot change Killing vector fields.

That doesn't have anything to do with what I said. What I claimed is that whether the metric components are time-varying (meaning: their derivative with respect to the time component is nonzero) is a coordinate-dependent fact. That's obviously true.

It's also true that if there is a timelike Killing vector field, then the metric is unchanged by translation along the Killing vector. But that doesn't mean that the metric is time-independent UNLESS the basis vector in the time direction happens to be the same as the Killing vector field; which is true of Schwarzschild coordinates and Rindler coordinates and inertial coordinates in flat spacetime.

 

[stevendaryl]

There is a recent thread devoted to clarify that changes of coordinates cannot change Killing vector fields.

This relates to the question of time-varying metric components in the following way:
If there is a time-like Killing vector field, then there exists a coordinate system in which the metric components are independent of time. You seem to be interpreting this as: If there is a time-like Killing vector field, then in EVERY coordinate system, the metric components are independent of time. That's clearly not true.

 

[TrickyDicky]

... That's clearly not true.

You might as well take a look at any definition of KV fields and specifically the fact they are coordinate-independent, you don't need to take my word for it.

 

[Dale]

Hi TrickyDicky, stevendaryl is correct. I think that you are confusing the coordinate independent concept of "static" and/or "stationary" with the coordinate dependent concept of "time varying".

 

[stevendaryl]

You might as well take a look at any definition of KV fields and specifically the fact they are coordinate-independent, you don't need to take my word for it.

For a number of exchanges, I have made a statement X, and you've said, No, Y is true. But Y doesn't mean that X is not true.

Statement X: Whether the components of the metric tensor is time-varying depends on which coordinate system you are using.

Statement Y: Whether there is a timelike Killing vector field does not depend on which coordinate system you are using.

Statement X is true AND statement Y is true. They don't contradict each other.

Now, to be fair, it's possible that some people use "static metric" to mean "there exists a timelike Killing vector field". But I explicitly said that I was talking about whether the components of the metric tensor are independent of the time coordinate. Those are two different things, and you act as if you don't understand the distinction. The first is a coordinate-independent notion, and the second is a coordinate-dependent notion.

 

[stevendaryl]

Hi TrickyDicky, stevendaryl is correct. I think that you are confusing the coordinate independent concept of "static" and/or "stationary" with the coordinate dependent concept of "time varying".

Thanks, Dale.

 

[TrickyDicky]

Hi TrickyDicky, stevendaryl is correct. I think that you are confusing the coordinate independent concept of "static" and/or "stationary" with the coordinate dependent concept of "time varying".

Nope, I never used the concept "time varying". Stevendaryl did.
He is saying a static spacetime can have metric components not time-independent and I was merely reminding him that the timelike KV of static spacetimes preserves the metric. Are you confused about this?

 

[TrickyDicky]

I was talking about whether the components of the metric tensor are independent of the time coordinate. Those are two different things, and you act as if you don't understand the distinction. The first is a coordinate-independent notion, and the second is a coordinate-dependent notion.

Both notions are the same notion.
A metric is called stationary if its components are time-independent. And all static spacetimes are stationary.

 

[TrickyDicky]

And just to be clearer, it is of course possible to use coordinates that don't reflect the time-independence of the spacetime but here we are trying to ellucidate the physics of the problem, not coordinate artifacts. I guess I took that for granted.