Clarifying Black Hole Horizons: An Examination of Observer Perspectives

[Dale]

i find it interesting to understand where the line is o
between what isp rovable and what is in principle not, in the modelling from terrestrial observers

Fair enough. Terrestrial observers would definitely be privileged observers, but for selfish reasons they would also be particularly interesting.

 

[wabbit]

Yes I must admit a parochial interest in that particular class of observers, though I will gladly share my toughts and compare notes with visiting aliens : )

 

[Dale]

(By "non-trivial" I mean excluding obviously contrived cases like only taking the open submanifold of events within some small radius of the worldline, which is a valid open submanifold that contains the entire worldline but obviously excludes almost all of the past light cone of any event on the worldline.)

I think what you want is not "non trivial" submanifold, but rather submanifolds bounded by a null surface. I think many such sub manifolds have the property mentioned by wabbit, if not all of them.

 

[wabbit]

I think what you want is not "non trivial" submanifold, but rather submanifolds bounded by a null surface. I think many such sub manifolds have the property mentioned by wabbit, if not all of them.

Ah that is possible, it would be an interesting characerization of this class of submanifolds. Hmm need to consider.

 

[PeterDonis]

Any open submanifold strictly contained in M_p, but containg the worldline, satifies that prescription.

Hm, yes, my definition of "non-trivial" wasn't restrictive enough, as DaleSpam pointed out. I agree with you that the restriction to manifolds bounded by a null surface, as he suggests, would be an interesting case to consider.

 

[wabbit]

To clarify, the property i mentionned as interesting for E (contains its own past) is stronger than the one quoted by PeterDonis (contains the past of one given worldline) or different if the latter is meant as "contains the past of a given maximally extended M-worldline".

The second just defines submanifolds S that contain M_p(gamma), and is not very special. I doubt that any submanifold containing M_p(gamma) must be bounded by anything special. In fact, take any closed subset of the interior of M\M_p(gamma), bounded by any surface you like, and M minus that subset fits the requirement, and its boundary includes that surface.

Hmm just reread you, not sure this answers it - you want a non trivial example of a submanifold that is bounded by a null surface, contains a maximally extended M-worldline, and doesn't contain the past of that worldline? If so this does seem exotic at first sight, I don't know if it's possible or not.

Here a wordline is not necessarily maximally extended, it can be just a segment.

The first property above says that E contains the past of any event within it, or equivalently E contains the past of any E-worldine. Among these E-wordlines are some maximally extended M-wordlines, and E of course also contains their past.
The interior region I, though bounded by a null surface, does not contain its own past, nor that of I-wordlines. (I don't think it contains any maximally extended M-wordline however)

 

[PAllen]

I don't find this category of manifolds very interesting. Suppose I define the manifold consisting of the past light cone of myself at 3 PM today, minus the null surface (and me at 3 PM). This manifold will contain the past of every event and every world line contained in it. I think most people would consider this an 'egocentric' manifold of little significance.

 

[wabbit]

This is true. Any submanifold defined as you did as the past of something, contains its own past. The converse isn't true I think, though the exterior region is the past of the horizon.

Anyway it's just one property of E that not everyone shares, call it "past-complete" or whatever, it's not the eighth wonder of the world for sure. The fact that I found it interesting as a tourist is no indication that it should be interesting to those better versed in GR.

 

[Haelfix]

It would also preclude a meaningful physical interpretation of things like DeSitter cosmologies in the static patch, where your friend Alice eventually falls off the end of the world.

 

[wabbit]

Not sure what you mean - but then again I have no idea what the static patch of deSitter cosmologies is.

 

[harrylin]

except that the coordinates you label as meaningful for an external observer of a BH correspond precisely to Rindler coordinates for a uniformly accelerating rocket. If you believe the one 'never' you must believe the other if you are logically consistent. The other logically consistent position - accepted by almost all physicists - is that both 'nevers' are coordinate artifacts, not something physically meaningful. [..].

An essential assumption is missing in that argument: you assume that the physical interpretation of the coordinates must be identical. However, that is not the case.

Probably most physicists believe that acceleration and gravitation are equivalent but not identical; for starters gravitation functions without a rocket engine. Consequently the physical interpretation differs. For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates, while the same apparent difference in a gravitational field may be interpreted as physically real. The logically consistent conclusion is then the contrary from the one you advance.

 

[PeterDonis]

Probably most physicists believe that acceleration and gravitation are equivalent but not identical

More precisely, they believe that acceleration in free space and being at rest in a gravitational field are (locally) equivalent (assuming the proper acceleration in both cases is the same).

For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates

No, it isn't. Two astronauts at the rear and front ends of the rocket can exchange repeated round-trip light signals and verify that the rear one's clock rate is slower (less elapsed time between successive signals). This is as "real" as the corresponding experimental result for two people at rest in a gravitational field at slightly different altitudes.

 

[harrylin]

More precisely, they believe that acceleration in free space and being at rest in a gravitational field are (locally) equivalent (assuming the proper acceleration in both cases is the same).

Yes indeed, thanks for the precision :-)

No, it isn't. Two astronauts at the rear and front ends of the rocket can exchange repeated round-trip light signals and verify that the rear one's clock rate is slower (less elapsed time between successive signals). This is as "real" as the corresponding experimental result for two people at rest in a gravitational field at slightly different altitudes.

That is erroneous; but I'm afraid that this is a permanent bug. I'll nevertheless clarify this once more.

Doppler effect and clock rate are different physical concepts; their definitions are unrelated. The elapsed time between successive signals coming from a distant clock is a function of both.

An astronaut in the rocket can measure light signals from a clock in the front with a clock in the rear; accounting for the rocket's acceleration (which she calculates from the thrust of the rocket engine and the rocket's mass) she will conclude that the rear clock rate is approximately the same (less elapsed time between successive signals as predicted by the Doppler effect).

 

[PeterDonis]

An astronaut in the rocket can measure light signals from a clock in the front with a clock in the rear; accounting for the rocket's acceleration (which she calculates from the thrust of the rocket engine and the rocket's mass) she will conclude that the rear clock rate is approximately the same (less elapsed time between successive signals as predicted by the Doppler effect).

What happens when light signals make a round trip? Or repeated round trips? Or does your qualifier "to first order" mean you were ignoring that?

Also, exactly the same logic can be applied in a local inertial frame in a gravitational field, for example to a light signal traveling between two observers at rest in the Earth's gravitational field at slightly different altitudes. When viewed in a local inertial frame, the frequency shift in the light signal can be entirely attributed to the Doppler effect. So if you are only looking at things "to first order", there is no difference between the two scenarios.

 

[PAllen]

An essential assumption is missing in that argument: you assume that the physical interpretation of the coordinates must be identical. However, that is not the case.

Probably most physicists believe that acceleration and gravitation are equivalent but not identical; for starters gravitation functions without a rocket engine. Consequently the physical interpretation differs. For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates, while the same apparent difference in a gravitational field may be interpreted as physically real. The logically consistent conclusion is then the contrary from the one you advance.

I would argue that the local physics is identical. To hover above a large horizon (so tidal effects are not extreme), you would need a steadily firing rocket (or suspension from a more distant steadily firing rocket). Any local measurements you make, including the behavior of signals bounced of objects closer to the horizon, and the fact that you could send to, but not receive messages, from on object that fell through the horizon are identical to the equivalent experiments in a uniformly accelerating rocket [in 'empty space' far away from anything]. If you use any natural procedure for setting up coordinates around this hovering observer, you get coordinates identical (delta second order tidal effects) to the rocket in deep space (Rindler coordinates). Further, in both cases, the redshift between higher and lower altitudes in a lab is purely Doppler in both situations, if expressed in locally inertial coordinates.

Thus, per local physics, as well as mathematics, the coordinates are equivalent and the horizons are equivalent, and any sense of 'never happens' based on two way signal behavior is identical. In both cases, you can choose to stop your physically experienced acceleration and then immediately access the other side of the horizon.

 

[wabbit]

For what it's worth, I agree with that. The equivalence is "local" but valid in a finite region, not just an infinitesimal one. An accelerometer will also tell the same story in both cases.

The only difference I can see is the horizon curvature vs plane Rindler horizon, but with a large enough black hole that won't be noticable.

 

[harrylin]

What happens when light signals make a round trip? Or repeated round trips? Or does your qualifier "to first order" mean you were ignoring that?

Regretfully you did not comment on the correction to the erroneous statement. And while I left the effect of length contraction out of my clarification, I did not ignore anything. Light signals in the other direction don't add information for this question, but the following simple fact does. Satellite clocks are typically slowed down in order to run at the correct clock rate in space around the Earth; Doppler effects can only explain that by pretending that the Earth is exploding at almost 10 m/s². As most people don't believe that, it is generally acknowledged that clock rate is reduced in a gravitational field.

Also, exactly the same logic can be applied in a local inertial frame in a gravitational field, for example to a light signal traveling between two observers at rest in the Earth's gravitational field at slightly different altitudes. When viewed in a local inertial frame, the frequency shift in the light signal can be entirely attributed to the Doppler effect. So if you are only looking at things "to first order", there is no difference between the two scenarios.

As you see above, the interpretation that you advocate does not match all the observations - even not to first order.
Coincidentally I showed in a recent thread that even local considerations discredit that interpretation, as also explained in the literature; I added a simple calculation example to verify the related arguments by Einstein and Okun - #3 , #5 , #48 , #49 , #52 .

As everyone can see, the two of us discussed this already exhaustively and by now I think that this is probably a permanent bug; it won't be useful for us to discuss the same things all over again. I'm open to clarify more to others who may have missed the earlier discussions. In a nutshell, the question is easily solved by looking one level deeper than is done in usual discussions.

 

[harrylin]

For what it's worth, I agree with that. The equivalence is "local" but valid in a finite region, not just an infinitesimal one. An accelerometer will also tell the same story in both cases.

The only difference I can see is the horizon curvature vs plane Rindler horizon, but with a large enough black hole that won't be noticable.

An accelerometer gives very limited information and equations require physical interpretation. The way I look at it, it all depends on one's belief if the physical interpretations of the following scenarios are identical during the time A2:

A.1. A rocket is in "deep space", and the accelerometer reads "0g".
A.2. The rocket engine is fired, at a power to reach "1g" accelerometer reading; the astronaut slowly reduces the fire power in order to keep the accelerometer reading constant.
A.3. The fuel runs out, and the accelerometer reading drops to zero.

B. A rocket is standing on Earth. No rocket engine is fired but the astronaut reads "1g" on the accelerometer. This reading does not change.

 

[PAllen]

An accelerometer gives very limited information and equations require physical interpretation. The way I look at it, it all depends on one's belief if the physical interpretations of the following scenarios are identical during the time A2:

A.1. A rocket is in "deep space", and the accelerometer reads "0g".
A.2. The rocket engine is fired, at a power to reach "1g" accelerometer reading; the astronaut slowly reduces the fire power in order to keep the accelerometer reading constant.
A.3. The fuel runs out, and the accelerometer reading drops to zero.

B. A rocket is standing on Earth. No rocket engine is fired but the astronaut reads "1g" on the accelerometer. This reading does not change.

But that is not at all the scenario I gave. What I gave, if you want to include history before and after uniform acceleration would be:

Rocket falling toward large BH fires rockets, above some target g, then stabilized at g, such that the result is to approach some distance above the horizon, then hover. Then at some point turn off the rocket, at which point the fall into BH is resumed.

This is what compares to your A1. I claim, that up to second order tidal effects that can be made arbitrarily small , the quasi-local physics of the two situations is identical.

 

[wabbit]

Sorry I'm lost now - which A1?

 

[PeterDonis]

Regretfully you did not comment on the correction to the erroneous statement.

I don't know what you are referring to. I commented on the latest post you made prior to mine.

Light signals in the other direction don't add information for this question

Yes, they do. See below.

it is generally acknowledged that clock rate is reduced in a gravitational field.

Fine. Now tell me what "clock rate is reduced in a gravitational field" actually means, in terms of direct observables. See below.

As you see above, the interpretation that you advocate does not match all the observations

You missed the key phrase "local inertial frame". You are not even addressing what I'm actually saying.

Satellite clocks are typically slowed down in order to run at the correct clock rate in space around the Earth; Doppler effects can only explain that by pretending that the Earth is exploding at almost 10 m/s2.

Once again, it doesn't seem like you are reading my posts. I specifically talked about a local inertial frame. The GPS satellite clock scenario (which I assume is what you are referring to) obviously cannot be covered by a single local inertial frame. Also, an orbiting satellite is a bad example because it is not at rest relative to an observer on the Earth's surface; it would be better to talk about an observer at rest on Earth's surface, compared to an observer at rest on, say, a platform high above the first observer.

I think that this is probably a permanent bug

We agree on this; but I don't think we agree on what the bug is. Let me re-state the key points from my perspective.

We want to compare two scenarios:

Scenario #1: A rocket accelerating in flat spacetime. Observer 1R is at the rear of the rocket; observer 1F is at the front.

Scenario #2: At rest in a gravitational field. Observer 2R is at rest at some altitude in the field; observer 2F is at rest at some higher altitude.

We stipulate that observers 1R and 2R feel the same proper acceleration, and observers 1F and 2F feel the same proper acceleration. We stipulate that the proper distance between observers 1R and 1F is the same as the proper distance between observers 2R and 2F, and that both proper distances are unchanging.

We have observer 1R send a light signal to observer 1F, and observer 2R send a light signal to observer 2F. Both light signals are redshifted when they are received. Why is this? We have two ways of analyzing it:

Local Inertial Frame Analysis: Pick a local inertial frame in which the R observer is at rest at the instant the light signal is emitted. Because both the R and F observers are accelerating in this frame, the F observer will be moving away from the light signal when it is received; so there will be a Doppler redshift.

Non-Inertial Frame Analysis: Construct a non-inertial frame in which the observers are at rest. For scenario #1, this will be Rindler coordinates; for scenario #2, it will be Schwarzschild coordinates. In this frame, there will be a "gravitational redshift"--or, if you don't like the term "gravitational" in the flat spacetime case, you can simply look at the timelike Killing vector field with respect to which the R and F observers are both following integral curves--both coordinate charts are adapted to this KVF, so that the KVF corresponds to the timelike basis vector and its integral curves are curves of constant spatial position. The invariant length of the KVF is different for the R and F observers--it is "shorter" for the R observer than for the F observer. This causes observer F to see light signals from observer R as redshifted (the math is simple and applies to any stationary spacetime).

So both analyses give the same answer. The advantage of the second analysis is that it can be extended beyond a single local inertial frame, so if you want to say you prefer it for that reason, that's fine. But that doesn't make the first analysis invalid; it just restricts its scope. If we're talking about the equivalence principle, the scope is restricted to a single LIF anyway.

However, there is another issue. You had claimed, in the post I originally responded to that started this subthread, that

"(to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates"

and I had responded that we can look at repeated round-trip light signals to verify that the difference in clock rates is not an artefact. You apparently still do not understand how that works. The scenario is simple: the R and F observers send repeated round-trip light signals back and forth, and each measures his own elapsed proper time between successive signals. The R observer finds less elapsed proper time than the F observer does from signal to signal. This is a direct observable that shows the difference in clock rates.

It is true that this result is simplest to derive in the non-inertial coordinates of the second analysis above, but that doesn't make it an "artefact" of using accelerating coordinates. Elapsed proper time along a given worldline between two given events is an invariant, independent of coordinate choice. So I am entirely unable to understand how you can justify your claim that I quoted above.

 

[Dale]

Doppler effects can only explain that by pretending that the Earth is exploding at almost 10 m/s2.

This type of language is inflammatory and unhelpful. You are making a straw-man argument by taking a first-order "local" approximation, applying it over a region where it is well-known not to hold (and nobody is claiming that it does hold), and then using emotionally-laden words like "pretending" and "exploding".

As you see above, the interpretation that you advocate does not match all the observations - even not to first order.

To first order a local inertial frame does match the observations of accelerometer readings as well as differences in transmitted and received frequencies and everything else. It is only by setting up a scenario which covers a large enough region of spacetime for the first order approximation to fail that you would get any discrepancy between what is predicted by a local inertial frame and what is observed.

 

[harrylin]

I would argue that the local physics is identical. [..]

I had overlooked that post. My point was that different people argue differently and as a result they reach different conclusions.
And while I won't participate in a long argument between people who may be entrenched in their thinking, here's a brief comment on your arguments. Mostly I already discussed those arguments in my other post, except for one:

To hover above a large horizon (so tidal effects are not extreme), you would need a steadily firing rocket (or suspension from a more distant steadily firing rocket). Any local measurements you make, including the behavior of signals bounced of objects closer to the horizon, and the fact that you could send to, but not receive messages, from on object that fell through the horizon are identical to the equivalent experiments in a uniformly accelerating rocket [in 'empty space' far away from anything]. If you use any natural procedure for setting up coordinates around this hovering observer, you get coordinates identical (delta second order tidal effects) to the rocket in deep space (Rindler coordinates). Further, in both cases, the redshift between higher and lower altitudes in a lab is purely Doppler in both situations, if expressed in locally inertial coordinates.

Thus, per local physics, as well as mathematics, the coordinates are equivalent and the horizons are equivalent, and any sense of 'never happens' based on two way signal behavior is identical. In both cases, you can choose to stop your physically experienced acceleration and then immediately access the other side of the horizon.

- The tidal effects are no indication of different physics if a sound constructive model exists according to which the two effects are physically the same.
- The hovering rockets are somewhat similar to the communication satellites around the Earth which I mentioned earlier; there is even no complication from velocity Doppler. And while I'm not up for discussions about black holes (indeed I only commented on a logical argument), for low field approximations such as on Earth it's not difficult to show that only one interpretation can be correct. If you start a topic on that, I'll be happy to participate as it is easy to understand basic physics.

 

[harrylin]

But that is not at all the scenario I gave. [..]

I did not pretend to discuss your scenario; instead I was talking about the physical interpretation of such cases. Some people may think that such cases show that gravitation is the same in appearance as acceleration but not in essence (so that it quacks like a duck and walks like a duck, but its genetic structure differs); and logically they may then reach a different conclusion as you.

 

[harrylin]

[..] What I gave, if you want to include history before and after uniform acceleration would be:

Rocket falling toward large BH fires rockets, above some target g, then stabilized at g, such that the result is to approach some distance above the horizon, then hover. Then at some point turn off the rocket, at which point the fall into BH is resumed.

This is what compares to your A1. I claim, that up to second order tidal effects that can be made arbitrarily small , the quasi-local physics of the two situations is identical.

Sorry I'm lost now - which A1?

A.1. A rocket is in "deep space", and the accelerometer reads "0g".

PAllen argues that not just the local phenomena but also the physical interpretation of a rocket falling into a black hole is the same as that of an inertial rocket far away from heavy matter.

 

[harrylin]

[..]t we can look at repeated round-trip light signals to verify that the difference in clock rates is not an artefact. You apparently still do not understand how that works. The scenario is simple: the R and F observers send repeated round-trip light signals back and forth, and each measures his own elapsed proper time between successive signals. The R observer finds less elapsed proper time than the F observer does from signal to signal. This is a direct observable that shows the difference in clock rates. [..].

(emphasis mine; see also post #118 )

This is basic SR; for a rocket accelerating from 0 to 10 m/s it's even correct to apply classical Doppler physics. If there is anyone else here who thinks that Doppler plays no role in such an analysis of an accelerating rocket, I will start it as a topic (later, these days I'm busy). That understanding is essential for many discussions on this forum.

 

[PAllen]

...

- The hovering rockets are somewhat similar to the communication satellites around the Earth which I mentioned earlier; there is even no complication from velocity Doppler. And while I'm not up for discussions about black holes (indeed I only commented on a logical argument), for low field approximations such as on Earth it's not difficult to show that only one interpretation can be correct. If you start a topic on that, I'll be happy to participate as it is easy to understand basic physics.

As so often, I don't understand your intent at all here.

Are you claiming similarity between a hovering rocket that is firing thrusters to maintain hovering, with satellites in free fall (orbit) around the earth? If so, I have no idea how such a claim could be justified.

Are you claiming (second sentence) that the redshift observed when an emitter on a tall tower sends to a receiver on another lower floor, it is physically incorrect to accept the interpretation of a free fall frame, in which all the observed red shift (second order effects are many orders of magnitude below detectability) is due to Doppler? Thus, that free fall frames provide invalid physical interpretations?

 

[PeterDonis]

If there is anyone else here who thinks that Doppler plays no role in such an analysis of an accelerating rocket

Sigh. That's not what I said. I gave two analyses; one uses Doppler in a local inertial frame; the other uses a non-inertial frame in which both objects are at rest, so there is no Doppler. I don't see how you can get from that to the claim that I "think that Doppler plays no role". Obviously it plays a role in any frame in which there is motion.

 

[Dale]

It seems like the OP's question has been answered so it seems like it is time to close the thread. Everything else in the recent conversation seems to be an argument about a position that nobody is actually taking.