On the nature of the infinite fall toward the EH

[Austin0]

Quote by Austin0

Pervect has here given a series of events. Or at least relationships as there seems to be no determinable velocities or explicit spatial coordinates to be derived from this information.

I assumed that the distance to the Tortoise was the spatial coordinate for Achilles, but it is true that we never actually introduced a method to assign spatial coordinates elsewhere. That would require the introduction of a simultaneity convention and a spatial metric elsewhere. It could be done, but would require some more work.

However, since the only thing of interest in the scenario is Achilles I don't see the need. If you want to do more complicated scenarios which are still analogous to the SC horizon then I would recommend going to Rindler coordinates. There the analogy is even closer.

It is not a matter of spatial coordinates not being assigned elsewhere, because there is no means ,with the given information, to assign coordinates to Achilles himself after the initial instant either, is there?

Quote by Austin0

SO we have these times and relative distances and the premise that both Achilles and the tortoise are inertial with which to synthesize a coordinate system and metric.

You have asserted that the Zeno frame is non-inertial so the question is what possible state of motion of that frame could make possible those observed distances between two bodies in uniform motion.

For coordinates non-inertial just means that the metric is not the Minkowski metric, as demonstrated. There is no requirement that a coordinate system correspond with some observer's state of motion.

Please take note of what I actually said . I didn't say coordinates (Which we don't actually have) but distances between the two bodies in inertial motion.which is all that is actually given to work with.
The point was not about coordinates but about inferring a state of motion from the observations.
As far as that goes is there necessarily any rigid constraint besides the signs of the signature and the Pythagorean theorem for a valid inertial metric?
Quote by Austin0

If the observations in the Zeno frame supported a picture of linear decrease in distance between Achilles and the tortoise this would indicate a constant motion of the Zeno frame also , agreed?

Constant motion relative to Achilles, yes. In other words, the coordinate acceleration of Achilles would be 0.

yes and if Achilles is defined as inertial and has zero coordinate acceleration in the Zeno frame then it would follow that the Zeno frame was also inertial (in uniform motion)YES?? Which is what I said.Quote by Austin0

If the observed decrease in distance, itself increased in rate , this would support a conclusion of positive parallel acceleration of the Zeno frame.I.e. Zeno frame increasing it's velocity relative to A and the tortoise.

But according to Pervect the decrease in relative distance between Achilles and the tortoise is decreasing over time non-linearly.
SO what possible motion (acceleration) of the Zeno frame could make this possible?

I am not sure, but it sounds like you want the coordinate acceleration of Achilles, which is easy enough to solve. From post 393 we already found that Achilles' worldline in the Zeno coordinates is given by , so Achilles' coordinate acceleration is the second derivative wrt n which is .

If this is not what you had intended, then could you be more explicit about what you want calculated?

No I am not talking about the coordinate acceleration of Achilles in the Zeno frame which is indeterminable as far as I can see.
If you disagree please explain.
I am talking about what possible motion of the Zeno frame could make the observed relationship between Achilles and the tortoise occur.

Explicitly,, the decreasing rate of the decrease of the distance between them or comparably,, the decrease in the relative velocity between them..

Another perspective is; what possible state of motion of the Zeno frame as charted from the Achilles frame could accomplish this.


It is clear that if the Zeno frame is actually inertial in motion (constant) then an arbitrary clock rate could easily effectuate those observations. Yes?
That this is a state of motion and condition that would be consistent with the Zeno observations. Agreed?

Quote by Austin0

My conclusion is that there is no possible acceleration that could do this alone and therefore the observations in the Zeno frame could only be possible if the Zeno time rate was increasing at a rate not possible through the effects of motion ( Lorentz effects..)

I agree, the same thing happens in SC. The SC coordinate time is increasing at a rate which is not possible through the effects of motion for any local observer. It is only by the use of a simultaneity convention and a distant observer that SC time is related to any observer's proper time. We haven't defined either of those for Zeno coordinates, but we certainly could do so.

Quote by Austin0

Put simply:
Achilles is passing a stream of Zeno clocks and observers. Do you think Achilles sees everything in the Zeno frame speed up exponentially or only the clocks?

Only the coordinate time speeds up exponentially, physical clocks do not. Similarly with a free faller passing a stream of shell observers in SC.

IMO you are incorrect here. In this scenarion we are talking about a system of arbitrarily scaled clocks. Equivalent to the clocks in the GPS system which are artificially calibrated for synch purposes. A physical mechanism.
In the GPS case the artificial rate is constant. In the Zeno case the rate is increasing but the principle is the same.

In the Sc case the static clocks are natural but incrementally decreasing in rate towards the center but that isn't relevant. In the Zeno case we can assume that either all system clocks are identical and exponentially increasing in rate or that the system clocks have increasing rates along the path of Achilles but in either case they must be mechanically operating at different rates , yes?

Quote by Austin0

If you think it is only the clocks, an arbitrary coordinate choice, then you are talking about a mechanism to accomplish this radical increase in rate in actual physical clocks correct?

Clocks measure proper time, not coordinate time. There is no mechanism for coordinates. Coordinates are a mathematical mapping from events in the manifold to R4. They are not physical. That is the whole point.

So you think the calibrated GPS clocks are measuring proper time??

I am not following you here.
If the sole definition of time is that which clocks measure then time has no existence or meaning independent of clocks yes? Coordinates are measured and assigned by physical clocks yes?? All coordinates to events in the manifold are determined and assigned by actual clocks at the actual locations.
All calculations of coordinate times at specific locations are related to actual or hypothetical physical clocks and what they would indicate for proper time at hypothetical events at those locations, yes?
I understand the difference between proper time intervals as measured by a single clock and calculated time intervals between clocks at disparate locations but any such calculated coordinate time interval, in the end corresponds to the times read on physical clocks (even if hypothetical), agreed??

 

[Austin0]

Well you still have that v in the rhs of your equation. What does it represent??

As stevendaryl mentioned, it is just a parameter for the metric. Like M in the Schwarzschild metric. In fact, this is an unintentional similarity.

Well the M in the Sc metric represents specific things yes?? Either mass or a distance. Or is this incorrect?

so if this v represents a velocity , it is the velocity of what relative to the Zeno frame? And how did you arrive at it??

Quote by Austin0 View Post

So in this frame Achilles is a distance behind the turtle. The definition of Zeno time, n, given is .

You have stated that although Achilles and the tortoise are inertial, the Zeno frame is not, so how do you arrive at your identity here to justify your substitution and simplification. The d here in Achilles frame; is not equivalent to the d' here in Zeno's frame; . is it?

Yes, it is the same. Pervect only transformed the time coordinate.

Well I can't speak for Pervect's thought processes but what was actually defined was not a simple transformation of Achilles time.
It was a series of observations which described an obvious similarity to the Sc case.

Zeno time was not a function of Achilles time or Achilles velocity or position. It was limited to a function of the numerical value of the distance between Achilles and the tortoise ,divorced from position.

As I have been trying to make clear these conditions (observations) could be consistent with any number of possible coordinate time/ clock configurations, rates etc in the Zeno frame. Ditto Zeno simultaneity conventions. Obviously these would result in different transformations in each case , yes??
so this would seem to be a classic catch 22. There is not enough info to infer a Zeno metric, Without a defined Zeno metric you cannot derive a valid transformation . Without a valid transformation you cannot derive a Zeno metric.


Quote by Austin0 View Post

Having invoked relativistic principles in this classic scenario how can you now directly equate a distance in one frame with that in another which is not only moving at a relative velocity but which is in non-uniform motion?

It isn't a distance, it is a coordinate. Coordinates and distances are not the same thing. In this case, the coordinate is numerically equal to a distance in an inertial frame, but it is still a coordinate not a distance.

Well in the Achilles frame the distance 100-vt is identical to the tortoise x coordinate it's true but d=100/2^n is clearly a fraction of a distance (a dx') and after the initial instant there is no basis for determining a coordinate location for either Achilles or the tortoise,,yes?SO it can't be a coordinate.
SO then you are claiming equivalence of a coordinate with an interval ,no??
How does whether it is a distance or a coordinate relate to equivalence anyway?.

If x or dx = 100 and x' or dx'= 100 does this mean x =x' or dx=dx' ?

You made similar comments about time and clocks in your previous post. Perhaps this is the root of your problem. In GR time coordinates are not readings on some clock and spatial coordinates are not measurements on some rod. They are mathematical functions which map open subsets of events in spacetime to open subsets of points in R4. They have some mathematical restrictions like being smooth, continuous, and one-to-one, but no physical restrictions. The connection to physical measurements, like clocks and rods, is done through the metric.

How does GR enter into this question. As far as this analogy and discussion is concerned the exercise is taking place in flat spacetime. Otherwise none of the participants are actually inertial are they?

And what does any of this have to do with whether or not 100-vt=100/2^n is a valid equivalence of values between two frames??

Isn't it just making an unsupported assumption of equivalence ,deriving a transformation from that which then makes it equivalent ex post facto?

There seems to be some confusion as to the meaning of an ananolgy. As i understand it , it is taking one situation ( the primary) A , which is simpler or more known and unambiguous and applying inferences or conclusions gained from it to another similar situation B.
In this case from Z to Sc. This is essentially a one way street. If you start taking inferences, principles and conclusions from B and inserting them into A you have simply negated any value as an analogy and made it a tautology.

The point of this analogy as I see it revolves around the question of the finite proper time of the Sc infaller. In the Sc context this seems to be a point of contention for a couple of reasons.

1) Due to the various effects of curved Spacetime the infaller disappears from outside observation even before reaching the EH so there is no possibility of empirical observation of the time at the EH
2) The necessity of integrating proper time and applying the theorem of limits or convergence makes this calculation somewhat less than conclusive in the minds of many.

SO the scenario was recast in the realm of ordinary observable reality where the results by implication could be definitvely determined by observation.Where Achilles would incontrovertibly reach the Tortoise horizon in finite proper time.

So now the finite Achilles proper time is a given and the infinite Zeno time has now become the questionable reality.

So the analogy has been turned upside down. Now Zeno time is analogous to the abstract infinite regression of Achilles motion in the original scenario except it is now ,an equally abstract, infinite progression. And just like Achilles DOES catch up with the tortoise,,,, the Zeno time on the Zeno clocks will in fact register some FINITE time when he does so.

So this is just another abstraction that has no correspondence to the real world. In this case the time is an arbitrary scaling which could not actually occur on real clocks, so are you suggesting that the Sc metric which is based on real natural clocks is equally divorced from reality?
That the Sc metric relating time to clocks is not in correspondence to the real world ?
To me the real point and resolution of Zeno's scenarios is that abstract mathematical and logical ideas do not necessarily work in the actual universe

It has come up on several occasions in this and related threads; the concept of mapping a finite set to an infinite one, with the idea that this was possible and taking place in the Sc question.

Well others may disagree but to me this proposition seems to be self evidently impossible in the real world. It can only occur in the abstract realm of mathematics where virtually anything is possible. But unless you remove any meaning from the word infinity I think the reasoning to accomplish this miracle must always ultimately be incorrect and simply distracting from the obvious mutual exclusivity and contradiction inherent in the premise. Just MHO

I certainly agree that spacetime is a singular continuum with a unique set of events and no coordinate substitution can alter that but I am somewhat confused when you turn around and because you don't like certain events in the Sc coordinates you then change them by switching to SK coordinates. And your proposal that because they do occur is those alternate coordinates they must happen in all coordinate systems ignores the arbitrarity of this choice. I.e . The same reasoning should apply regarding the negative event in the SC coords.
If it doesn't happen there it doesn't happen in any system. Note I am not claiming anything about the reality of events near the horizon but simply commenting on the reasoning behind taking a rigid position either way.

Quote by Austin0 View Post

What about simultaneity?

You are correct, I have not defined a simultaneity convention nor any coordinates off of Achilles' worldline. However, since we are only interested in events on Achilles' worldline it is hard to see why it would matter. If you like, the easiest thing will be to take the standard simultaneity convention of Achilles' inertial frame, however that will make the analogy a bit less direct since Achilles is anlogous to a free-falling local observer and the SC simultaneity convention does not correspond to the standard simultaneity convention of a free-falling local observer.

You could make some remote non-inertial observer and give a simultaneity convention that maps his coordinates to Zeno time. This would make the analogy better, but it seems like a lot of effort.

We can assume the standard simultaneity convention in Achilles (inertial) frame no problem but for the Zeno frame there is not enough information to define one. WHich is a part of my point. In the absense of this information it is just making assumptions without basis. And to redefine the scenario with the relevant parameters explicit would be a completely different case and much of this discussion would have been unnecessary IMO

 

[stevendaryl]

Quote by Austin0

Well i asked a perfectly cogent and relevant question. 1)you deny it is a valid question . then 2) you immediately contradict yourself and present two different possible reasonable answers 3),and 4) but both your answers seem questionable.

I think you're just being argumentative at this point. I didn't deny that what you asked was a valid question, I answered your question as best as I could.

Working from the information defined by pervect it is not possible to derive a velocity for the Achilles frame in the Zeno frame as far as i can see,

Why do you say that? You have one set of coordinates, (x,t), and in those coordinates, the path of Achilles is x = vt

You make a coordinate change to (x',t') given by:

x′=x
t' = log_2(\dfrac{100}{100 - vt}) = log(\dfrac{100}{100 - vt})/log(2)

To compute coordinate velocity in the coordinates (x',t'), you use:

\dfrac{dx'}{dt'} = \dfrac{\dfrac{\partial x'}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial x'}{\partial t}}{\dfrac{\partial t'}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial t'}{\partial t}}For the coordinate change that we're talking about:
\dfrac{\partial x'}{\partial x} = 1

\dfrac{\partial x'}{\partial t} = 0

\dfrac{\partial t'}{\partial x} = 0

\dfrac{\partial t'}{\partial t} = \dfrac{v}{log(2) (100 - vt)}

\dfrac{dx}{dt} = v

So we have:

\dfrac{dx'}{dt'} = log(2) (100 - vt)

Likewise it is not possible to define a velocity for Achilles himself in the Zeno frame.
So again I ask what is the velocity referring to that could be a valid part of the Zeno metric?.

I have no idea what you are talking about. We know Achilles' velocity components in the (x,t) coordinate system, we know how to transform to the (x',t') coordinate system, so we can easily derive Achilles' velocity components in the (x',t') coordinate system. Why do you think there's a problem?

I have no trouble with the idea of an arbitrary non-linear coordinate system.

Then you should understand how to transform velocity components in one coordinate system to velocity components in another.

SO in this case we are given : d=100-vt in the A frame and d=100/2^n in the Z frame.

The word "frame" doesn't mean anything in this case. There are two different
COORDINATE SYSTEMS in use. They aren't two different frames.

Assuming that at A time t =0 Achilles is at x=0 and the tortoise is at x=100 and at Z time n= 0 Achilles is at x'=0 and the tortoise is at x'=100. isn't it axiomatic that if these events are simultaneous in the A frame that they cannot be simultaneous in the Z frame??

No, it's not axiomatic. It's possible that in transforming between two coordinate systems. the synchronization convention changes, but not always.

It follows that the distances , the spatial intervals in the two frames cannot be congruent also Yes??

No. It depends on the transformation. For this particular transformation,
the standard of length is not changed. The standard of simultaneity is not
changed. But the standard for time interval is changed.

in another thread you stated that gravitational time dilation could be eliminated by a coordinate choice remember??

Yes, it's always possible to find a LOCAL coordinate system in which SR is approximately valid (no "gravitational time dilation").

I asked you if you were talking about an arbitrary scaling of clock periodicity and you agreed, correct?
So then we are talking about a physical interpretation of clock rates.
AN artificial mechanical adjustment to the workings of the mechanism.

No, that's not true at all. A time coordinate is just a real number associated with points in spacetime. It's convenient if you have a clock that gives the time, but it's not necessary.

When some switches from rectangular coordinates (x,y) to polar coordinates (r,\theta), do you think it's necessary to perform an artificial mechanical adjustment to all your rulers, so that they can measure r and \theta? No, of course not, because you can compute r and \theta. You don't need a measuring rod that directly measures r and \theta.

Changing the time coordinate is no different. You don't need a clock that computes the new time coordinate, you just need to be able to compute the time coordinate from whatever information you have from normal clocks.

What could be clearer than that??
In this case this means a mechanistic device that exponentially increases the rate at which the hands spin or the LED increments or whatever means that is used to actually indicate the measure of time,,,, CORRECT?

No, that's not correct.

Such artificial scaling is in fact used in the GPS system right?? Those clocks physically increment at a different rate yes??

I don't know for GPS whether the clocks are altered, or whether the times are manipulated afterward. It's conceptually the same, although one or the other might be more convenient.

 

[stevendaryl]

There seems to be some confusion as to the meaning of an ananolgy. As i understand it , it is taking one situation ( the primary) A , which is simpler or more known and unambiguous and applying inferences or conclusions gained from it to another similar situation B.

Yes, and it's clear that you don't understand the simpler case, either, so it was a failed tactic.

The point of the Zeno case is that we have one coordinate system, (x,t) that is used to describe a region of spacetime

0 \leq x \leq 100
0 \leq t \leq 100/v

We have a second coordinate system, (x',t') that describes the SAME region of spacetime as follows:

0 \leq x' \leq 100
0 \leq t' < \infty

The coordinate transformation t' = log_2(\dfrac{100}{100 - vt}) maps a finite interval of time coordinate t to an infinite interval of time coordinate t'.

That's the Zeno case. It's just a change of coordinates. The fact that t' \rightarrow \infty as Achilles approaches the Tortoise does NOT imply that Achilles never reaches the Tortoise, it just implies that that event is not covered by the (x',t') coordinate system.

The Schwarzschild case is ALSO just a change of coordinates. It ALSO maps a finite interval of time in one coordinate system (the freefalling coordinate system) to an infinite interval of time in another coordinate system (the Schwarzschild coordinate system). The fact that Schwarzschild time goes to infinity as an observer approaches the horizon does NOT imply that the observer reaches the horizon. It just implies that that event is not covered by Schwarzschild coordinates.

The analogy is just about perfect.

 

[stevendaryl]

We can assume the standard simultaneity convention in Achilles (inertial) frame no problem but for the Zeno frame there is not enough information to define one.

What are you talking about? For one thing, there is no "Zeno frame". It's an alternative COORDINATE system. It's just like using polar coordinates instead of rectangular coordinates. It doesn't imply that anybody is in a different "frame".

The person wanting to use Zeno coordinates has the exactly the same information as the person wanting to use inertial cartesian coordinates. He can just take the inertial cartesian coordinates and perform a mathematical transformation to get the Zeno coordinates. Why do you believe that there is any problem in doing this?

WHich is a part of my point. In the absense of this information it is just making assumptions without basis.

There are no assumptions being made. They're just using a different coordinate system.

It's really just like polar coordinates. Do you need additional assumptions in order to be able to use polar coordinates? No, you can just COMPUTE r and \theta from x and y, if you know the latter, using the transformations:

r = \sqrt{x^2 + y^2}
\theta = arctan(y/x)

The "Zeno time" is just a mathematical function of the time in the inertial coordinate system. No assumptions are needed to use it.

 

[Dale]

Don't you agree that to assert an equivalence between coordinates or values between two frames in relative motion you need to transform the values from one frame to the other.
If in fact you do not already have the correct transform functions, the T,X,T' and X' in your generalization you cannot simply assume the equivalence between some values in both frames and derive a valid transform from that . There has to be some relevant basis for the equivalence from first principles to justify such an identity and substitution.
Wouldn't you agree??

I don't know what you are talking about here. There are no first principles involved, and I have no idea what you mean by "equivalence" or "relevant basis" in this context. You don't derive the transform, you simply define the transform. There are no "correct" or "incorrect" transforms as long as they are smooth and 1-to-1.

Is the 100 in the A frame equivalent to the 100 in the Z frame?

If you are worried about the 100's then you don't have to simplify, it won't change anything except for making the expressions messier. The limits will still be the same. I leave that as an exercise for you.

isn't it axiomatic that if these events are simultaneous in the A frame that they cannot be simultaneous in the Z frame?? It follows that the distances , the spatial intervals in the two frames cannot be congruent also Yes??

No, it isn't axiomatic. It is derived for a specific class of transforms known as the Lorentz transforms.

how do you justify the equivalence 100-vt=100/2^n over time when the systems are not only in relative motion but one of them is non-linear??

I cannot believe that you actually wrote this. This is by far the silliest objection that you have come up with yet, and it really makes you look extremely unreasonable.

The transitivity of equality is one of the most basic elements of math and you should have learned it in grade school. Since d=100-vt and d=100/2^n then by transitivity of equality 100-vt=100/2^n. You cannot get any stronger justification than that.

It appears to me that to make this assumption of equivalence is unfounded and circular. I.e.,,to determine if these are equivalent requires a valid transformation so to use them to derive a transformation then makes them equivalent circularly.

Again, I don't know what you mean by "equivalence" in this context. Any function can be used as a valid transformation provided that it is smooth and one-to-one. The transform presented meets both of those criteria, so it is a valid transform. See p. 37 here for details: http://arxiv.org/abs/gr-qc/9712019

 

[Dale]

As far as that goes is there necessarily any rigid constraint besides the signs of the signature and the Pythagorean theorem for a valid inertial metric?

The only valid inertial metric is the Minkowski metric.

yes and if Achilles is defined as inertial and has zero coordinate acceleration in the Zeno frame then it would follow that the Zeno frame was also inertial (in uniform motion)YES?? Which is what I said.

Not necessarily. Consider a rotating frame. An object at rest at the center is inertial and has zero coordinate acceleration, but the frame is non-inertial.

No I am not talking about the coordinate acceleration of Achilles in the Zeno frame which is indeterminable as far as I can see.
If you disagree please explain.

It is not indeterminable. I determined it above therefore it is determinable.

I am talking about what possible motion of the Zeno frame could make the observed relationship between Achilles and the tortoise occur.

I don't see the difference between that and the coordinate acceleration of Achilles. Can you express your idea mathematically?

I also don't see the importance of whatever point you are trying to make here. If you want to derive some coordinate or proper accelerations and then try to explain the importance then I would be glad to look into it more, but as it is I feel like I am trying to play a game of "guess what I am thinking" which is only a fun game if your partner is cute and willing .

IMO you are incorrect here. In this scenarion we are talking about a system of arbitrarily scaled clocks. Equivalent to the clocks in the GPS system which are artificially calibrated for synch purposes. A physical mechanism.
In the GPS case the artificial rate is constant. In the Zeno case the rate is increasing but the principle is the same.

OK, I can see your point here. If there is a known mapping between proper time and coordinate time then you can take a clock's measurement of proper time and use that to calculate the coordinate time. I wouldn't call that a "physical mechanism", the part of the clock itself that I would call the "physical mechanism" still measures proper time. Then you can do whatever calculations you want and display any number you wish. But if you want to include the calculation and display as part of what you call the "physical mechanism" then that is just semantics and I am fine with it.

All calculations of coordinate times at specific locations are related to actual or hypothetical physical clocks and what they would indicate for proper time at hypothetical events at those locations, yes?

Certainly not. You have already given a fine counterexample in the GPS where coordinate time is not equal to what the proper time indicates. The proper time has to be scaled in order to get the coordinate time. You may choose to call that scaling part of the "physical mechanism" if you like, but it is NOT proper time.

I understand the difference between proper time intervals as measured by a single clock and calculated time intervals between clocks at disparate locations but any such calculated coordinate time interval, in the end corresponds to the times read on physical clocks (even if hypothetical), agreed??

I really recommend that you read chapter 2 here: http://arxiv.org/abs/gr-qc/9712019

 

[Dale]

Well the M in the Sc metric represents specific things yes?? Either mass or a distance. Or is this incorrect?

The M in the SC metric is just a parameter for the metric. Remember that the SC metric is a vacuum metric, meaning that there is no mass or energy anywhere the metric is valid.

Well I can't speak for Pervect's thought processes but what was actually defined was not a simple transformation of Achilles time.
It was a series of observations which described an obvious similarity to the Sc case.

Zeno time was not a function of Achilles time or Achilles velocity or position. It was limited to a function of the numerical value of the distance between Achilles and the tortoise ,divorced from position.

As I have been trying to make clear these conditions (observations) could be consistent with any number of possible coordinate time/ clock configurations, rates etc in the Zeno frame. Ditto Zeno simultaneity conventions. Obviously these would result in different transformations in each case , yes??
so this would seem to be a classic catch 22. There is not enough info to infer a Zeno metric, Without a defined Zeno metric you cannot derive a valid transformation . Without a valid transformation you cannot derive a Zeno metric.

As described above, the transformation is valid, it is smooth and one-to-one.

Well in the Achilles frame the distance 100-vt is identical to the tortoise x coordinate it's true but d=100/2^n is clearly a fraction of a distance (a dx') and after the initial instant there is no basis for determining a coordinate location for either Achilles or the tortoise,,yes?SO it can't be a coordinate.

Why not? It is smooth and one-to-one.

SO then you are claiming equivalence of a coordinate with an interval ,no??
How does whether it is a distance or a coordinate relate to equivalence anyway?.

I still don't know what you mean by "equivalence" in this context, so I am certainly not knowingly making any claims regarding equivalence.

How does GR enter into this question. As far as this analogy and discussion is concerned the exercise is taking place in flat spacetime. Otherwise none of the participants are actually inertial are they?

True, all of my comments apply to the flat manifold of SR equally as well. GR only enters in as to the SC and the general topic of the thread as a whole, not to pervect's Zeno time example.

The point of this analogy as I see it revolves around the question of the finite proper time of the Sc infaller. In the Sc context this seems to be a point of contention for a couple of reasons.

1) Due to the various effects of curved Spacetime the infaller disappears from outside observation even before reaching the EH so there is no possibility of empirical observation of the time at the EH
2) The necessity of integrating proper time and applying the theorem of limits or convergence makes this calculation somewhat less than conclusive in the minds of many.

The finite proper time of the SC infaller is only a point of contention for one reason: a mistake by those who disagree that it is finite. Unfortunately, those who make that mistake have been reluctant to learn from their mistake.

And just like Achilles DOES catch up with the tortoise,,,, the Zeno time on the Zeno clocks will in fact register some FINITE time when he does so.

No, I already demonstrated that Zeno coordinate time is infinite when Achilles DOES catch up with the tortoise.

So this is just another abstraction that has no correspondence to the real world. In this case the time is an arbitrary scaling which could not actually occur on real clocks, so are you suggesting that the Sc metric which is based on real natural clocks is equally divorced from reality?

The SC metric is based on clocks far away from the horizon. It is not based on clocks near, at, or beyond the horizon. So yes, it is equally divorced from reality in those regions.

That the Sc metric relating time to clocks is not in correspondence to the real world ?

It is certainly not in correspondence to the real world at or beyond the EH. Again, please read ch 2 of http://arxiv.org/abs/gr-qc/9712019

I certainly agree that spacetime is a singular continuum with a unique set of events and no coordinate substitution can alter that but I am somewhat confused when you turn around and because you don't like certain events in the Sc coordinates you then change them by switching to SK coordinates. And your proposal that because they do occur is those alternate coordinates they must happen in all coordinate systems ignores the arbitrarity of this choice. I.e . The same reasoning should apply regarding the negative event in the SC coords.
If it doesn't happen there it doesn't happen in any system. Note I am not claiming anything about the reality of events near the horizon but simply commenting on the reasoning behind taking a rigid position either way.

Your reasoning is wrong. The inference only goes one way. Coordinate systems map open subsets of the manifold to open subsets of R4. So if an event has a coordinate in any chart then it is an element of the manifold. Conversely, the fact that a particular subset of the manifold corresponding to a given chart does not include an event does not imply that the event is not a part of the manifold. C \subset M and E \in C implies E \in M but E \not \in C does not imply E \not \in M

 

[harrylin]

I am not sure which specific topic you are referring to by "this matter", but the whole point of expressing a physical theory in terms of a mathematical framework is precisely in order to ensure that the conclusions/predictions follow logically from the premises/postulates. You just seem to have difficulty with the mathematical framework which enforces the logic. That is a natural part of learning a challenging topic, but it does not in any way indicate a deficit in the logic of the theory.

I elaborated on that topic (which is roughly the topic of this thread) in the preceding sentences and instead of what you think, I suggested that perhaps in neither final version of the theory (either that of Einstein or Finkelstein/xxx) a deficit in logic can be found; but that is what people still seem to be attempting here in vain.

It was only very recently, thanks to the discussions here, that I discovered this subtle issue of interpretation and so I haven't fully made up my mind; however I notice that some others who seem to be smart have been discussing this for at least a year - https://www.physicsforums.com/showthread.php?t=528681&page=18. This makes it unlikely that it is just a matter of understanding the mathematical framework.

 

[Mentz114]

I elaborated on that topic (which is roughly the topic of this thread) in the preceding sentences and instead of what you think, I suggested that perhaps in neither final version of the theory (either that of Einstein or Finkelstein/xxx) a deficit in logic can be found; but that is what people still seem to be attempting here in vain.

You are still asserting that your failure to understand coordinates is a failure of logic in GR. Mathematics is logic and I don't think there are mathematical inconsistencies in GR.

It was only very recently, thanks to the discussions here, that I discovered this subtle issue of interpretation and so I haven't fully made up my mind; however I notice that some others who seem to be smart have been discussing this for at least a year - https://www.physicsforums.com/showthread.php?t=528681&page=18. This makes it unlikely that it is just a matter of understanding the mathematical framework.

There is no subtle issue of interpretation. Why not start a new thread on this 'interpretation' issue.

 

[harrylin]

You are still asserting that your failure to understand coordinates is a failure of logic in GR. Mathematics is logic and I don't think there are mathematical inconsistencies in GR.

I stated that there is likely no failure in either logic; moreover I don't know of anybody here who thinks that there are mathematical inconsistencies in GR (at least, not in modern GR). That shows that you did not understand anything of what I said.

There is no subtle issue of interpretation. Why not start a new thread on this 'interpretation' issue.

That thread already exists, and the discussion was interesting for me.

 

[harrylin]

There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features. [..]
See for instance http://arxiv.org/pdf/0903.1105v1.pdf
[..]

Interesting paper, I quickly browsed through it (and I notice that I have to catch up with not two but three weeks of PF). Can you (or someone else) shortly explain how it supports the claim that according to our reckoning event horizons actually ("now") exist? I cite:

CONCLUSIONS
Recent infrared and mm-VLBI observations imply that if the matter accreting onto Sgr A* comes to rest in a region visible to distant observers, the luminosity associated with the surface emission from this region satisfies Lsurf/Lacc . 0.003. Equivalently, these observations require that 99.6% of the gravitational binding energy liberated during infall is radiated in some form prior to finally settling. These numbers are inconsistent by orders of magnitude with our present understanding of the radiative properties of Sgr A*’s accretion flow specifically and relativistic accretion flows generally. Therefore, it is all but certain that no such surface can be present, i.e., an event horizon must exist.
[..]

In order to draw such a conclusion (experimental evidence for the existence of event horizons) in the context of the discussion in this thread, the predictions of a nearly fully formed EH should be compared with that of a fully formed EH. It is not evident to me that such was done in that paper, but perhaps I overlooked it?

 

[stevendaryl]

I elaborated on that topic (which is roughly the topic of this thread) in the preceding sentences and instead of what you think, I suggested that perhaps in neither final version of the theory (either that of Einstein or Finkelstein/xxx) a deficit in logic can be found; but that is what people still seem to be attempting here in vain.

It was only very recently, thanks to the discussions here, that I discovered this subtle issue of interpretation and so I haven't fully made up my mind; however I notice that some others who seem to be smart have been discussing this for at least a year - https://www.physicsforums.com/showthread.php?t=528681&page=18. This makes it unlikely that it is just a matter of understanding the mathematical framework.

I'm not sure what subtle issue you are talking about, but perhaps it is this:

There are two different views of the Einstein Field Equations: (1) They describe the geometry of curved spacetime. (2) They describe a "universal" force field (a spin-2 field) that unlike that of electromagnetism, doesn't just interact with charged particles, but interacts with all sources of matter or energy. This latter view was explored by various people (Feynmann, and Deser) in the context of understanding the field equations, but also popped up again in string theory.

For weak gravity, there's no difference between the two views, but in the case of event horizons, it seems possible to me that the question of whether anything ever crosses the event horizon might be true for one interpretation and false for the other. At least, it seems conceivable to me that there could be a "force field" interpretation of the field equations that is equivalent to the "curved spacetime" interpretation in weak fields, but not in strong fields. As people have pointed out, from the point of view of general covariance, there is nothing "strong" about gravity right at the event horizon, so if the two interpretations diverge, then it must point to a failure of general covariance.

 

[Dale]

I elaborated on that topic (which is roughly the topic of this thread) in the preceding sentences and instead of what you think, I suggested that perhaps in neither final version of the theory (either that of Einstein or Finkelstein/xxx) a deficit in logic can be found; but that is what people still seem to be attempting here in vain.

It was only very recently, thanks to the discussions here, that I discovered this subtle issue of interpretation and so I haven't fully made up my mind; however I notice that some others who seem to be smart have been discussing this for at least a year - https://www.physicsforums.com/showthread.php?t=528681&page=18. This makes it unlikely that it is just a matter of understanding the mathematical framework.

It is simply a matter of understanding the mathematical framework. I am not sure why you think that smart people discussing it for a year is an indication that it is something other than understanding the mathematical framework. You also seem smart but seem to be making no progress towards understanding the framework, so I would anticipate that it will take more than a year for you also.

If you understood the framework then you would realize that coordinate charts map open subsets of the manifold to open subsets of R4. Therefore, if an event is mapped by a given coordinate chart then that logically implies that it is part of the manifold, but if an event is not mapped by a given coordinate chart then that does not logically imply that it is not part of the manifold. Furthermore, if you understood the framework then you would realize that the portion of the manifold covered by SC is geodesically incomplete, indicating from within SC themselves that the open subset of the manifold covered by SC is not the entire manifold.

You have explicitly stated your opposition to the mathematical framework, so it is unsurprising that you cannot make a logical conclusion about GR. However, your unwillingness to do the math is neither a deficit in nor a logical failing of GR.

 

[Dale]

That thread already exists, and the discussion was interesting for me.

Which thread is the subtle interpretation thread?

 

[harrylin]

I'm not sure what subtle issue you are talking about, but perhaps it is this:

There are two different views of the Einstein Field Equations: (1) They describe the geometry of curved spacetime. (2) They describe a "universal" force field (a spin-2 field) that unlike that of electromagnetism, doesn't just interact with charged particles, but interacts with all sources of matter or energy. This latter view was explored by various people (Feynmann, and Deser) in the context of understanding the field equations, but also popped up again in string theory.

Quite so; Einstein's GR assumes a gravitational field, corresponding to (2). On top of that, I discovered that most people here disagree with Einstein about his EEP. The sum of those subtle differences seems to be without predictive consequence except perhaps on the issue of black holes.

For weak gravity, there's no difference between the two views, but in the case of event horizons, it seems possible to me that the question of whether anything ever crosses the event horizon might be true for one interpretation and false for the other. At least, it seems conceivable to me that there could be a "force field" interpretation of the field equations that is equivalent to the "curved spacetime" interpretation in weak fields, but not in strong fields. As people have pointed out, from the point of view of general covariance, there is nothing "strong" about gravity right at the event horizon, so if the two interpretations diverge, then it must point to a failure of general covariance.

Yes but as I pointed out, that does not necessarily point to such a failure, just as the speed of light does not point to a failure of SR despite the fact that tachyons are proposed.

 

[harrylin]

[..] You have explicitly stated your opposition to the mathematical framework, so it is unsurprising that you cannot make a logical conclusion about GR. [..]

That's nonsense if you refer to Einstein's GR. Evidently you did not understand anything that I brought up; perhaps stevendaryl can explain it better...

Which thread is the subtle interpretation thread?

Already linked in my post #448

 

[Dale]

That's nonsense if you refer to Einstein's GR.

I refer to the mathematical framework of mainstream modern GR, which you have explicitly rejected. I personally don't think that there is any contradiction between "Einstein's GR" and "modern GR", but at a minimum the mathematical framework is the same (pseudo-Riemannian geometry and the EFE). Therefore, since you have rejected Einstein's mathematical framework for GR, you have necessarily rejected "Einstein's GR" also.

Already linked in my post #448

In that thread I proved that charts covering the interior of an EH were indeed legitimate solutions to the EFE, and therefore compatible with GR, and you never disputed that. So I though the subtle interpretation issues were resolved.

 

[stevendaryl]

Quite so; Einstein's GR assumes a gravitational field, corresponding to (2).

I don't agree with that. Einstein's GR is a geometric theory. Einstein himself may not have recognized it, but it is actually implicit in his understanding of gravity in terms of general covariance and the equivalence principle.

On top of that, I discovered that most people here disagree with Einstein about his EEP.

Well, Einstein isn't around to say whether he agrees with you, or not. But in a certain sense, I don't think it is important. General Relativity is a well-understood theory. There is no reason to believe that Einstein had some understanding of it that modern physicists are missing. And it really doesn't matter, anyway. Science is not about personalities and personal preferences, it's about the science. The science of GR is understood much better than it was when Einstein formulated it. He doesn't get any special say about how the theory is to be interpreted by being the inventor of the theory. Inventing a theory in physics is like giving birth to a child: the child goes off to live his own life, independent of the person who gave birth to it.

The sum of those subtle differences seems to be without predictive consequence except perhaps on the issue of black holes.

What do you think that Einstein believed about the equivalence principle that conflicts with what modern physicists believe about it? I should actually ask: what do YOU believe about the equivalence principle that conflicts with what other people say about it? I really do think that what Einstein believed about it is mostly irrelevant.

Yes but as I pointed out, that does not necessarily point to such a failure, just as the speed of light does not point to a failure of SR despite the fact that tachyons are proposed.

I don't think that's correct. In the neighborhood of an event horizon, if one uses local inertial coordinates (or KS coordinates) then gravity doesn't look strong at all. There is nothing unusual happening in the neighborhood of the event horizon. It's only a very specific coordinate system, the Schwarzschild coordinate system, where there is something weird happening at the event horizon. So if you believe that there is something physically significant that is happening at the event horizon, then you are saying that Schwarzschild coordinates somehow count more than other coordinates, which violates the principle of general covariance.

 

[stevendaryl]

That's nonsense if you refer to Einstein's GR.

I do not believe that you are better able to know what Einstein really meant than others are. And, as I said in another post, it's IRRELEVANT what Einstein thought about it. GR is a theory that was not understood when it was first developed, and I believe it is much better understood now. If you prefer a different way of understanding GR, then tell what that different way is; don't expect Einstein to do it for you---he's not around to clarify what he really meant.

 

[stevendaryl]

I refer to the mathematical framework of mainstream modern GR, which you have explicitly rejected. I personally don't think that there is any contradiction between "Einstein's GR" and "modern GR", but at a minimum the mathematical framework is the same (pseudo-Riemannian geometry and the EFE). Therefore, since you have rejected Einstein's mathematical framework for GR, you have necessarily rejected "Einstein's GR" also.

I think it's a waste of time to bring up what Einstein believed about it. The theory is better understood today than it was in Einstein's time, and Einstein's intuitions about it really are not likely to have any relevance today.

 

[Dale]

I think it's a waste of time to bring up what Einstein believed about it. The theory is better understood today than it was in Einstein's time, and Einstein's intuitions about it really are not likely to have any relevance today.

I agree it is a waste of time. But it is an historical fact that Einstein did use the mathematical framework of pseudo-Riemannian geometry for GR, so it is clear that harrylin's rejection of pseudo-Riemannian geometry is also a rejection of Einstein's theory. Therefore, not only is his claim of supporting "Einstein's GR" a waste of time, it is also false.

 

[harrylin]

[..] since you have rejected Einstein's mathematical framework for GR [..]

As you continue to make nonsensical claims about what you think I said, further discussion with you on this topic is useless. Moreover, I found that peterdonis answered my questions (so far) rather well in the thread to which I referred.

 

[harrylin]

[..] There are two different views of the Einstein Field Equations: (1) They describe the geometry of curved spacetime. (2) They describe a "universal" force field (a spin-2 field) that unlike that of electromagnetism, doesn't just interact with charged particles, but interacts with all sources of matter or energy. This latter view was explored by various people (Feynmann, and Deser) in the context of understanding the field equations, but also popped up again in string theory.
[..] it seems possible to me that the question of whether anything ever crosses the event horizon might be true for one interpretation and false for the other. At least, it seems conceivable to me that there could be a "force field" interpretation of the field equations that is equivalent to the "curved spacetime" interpretation in weak fields, but not in strong fields.[..]

[..] Well, Einstein isn't around to say whether he agrees with you, or not. But in a certain sense, I don't think it is important. General Relativity is a well-understood theory. There is no reason to believe that Einstein had some understanding of it that modern physicists are missing. [...]

We already agreed on that. Feynman (note one n) is dead too but evidently understanding no.2 is not "missing" as it has been mentioned here and in other threads. Moreover, the fact that different versions of the EP exist is well known; if you don't care which version originates with whom, then that suffices.

What do you think that Einstein believed about the equivalence principle that conflicts with what modern physicists believe about it?

The EEP is discussed and cited in other threads which are still open for comments. See for example:
https://www.physicsforums.com/showthread.php?p=4180429
https://www.physicsforums.com/showthread.php?t=656240

 

[Dale]

As you continue to make nonsensical claims about what you think I said, further discussion with you on this topic is useless.

Then please clarify your meaning:

Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics.

The referenced section of Carroll's notes were dealing strictly with the mathematical framework of pseudo-Riemannian geometry.

According to what you wrote here it seems clear to me that you disagree with the philosophy of pseudo-Riemannian geometry and consider the standard use of "patchwork" (i.e. different charts to cover a manifold) to be "not good physics". That seems to me to be a clear rejection of pseudo-Riemannian geometry, and hence "Einstein's GR".

I did ask for clarification, which you failed to provide. So if you are not explicitly rejecting pseudo-Reimannian geometry with this quote then please clarify now, because if that is not what you had intended to state then your meaning did not come through.

Btw, it is a pretty poor debate tactic to refuse to clarify your meaning when clarification is specifically requested and then to complain that your views are being misrepresented. I do not believe that I am misrepresenting your views (on pseudo-Riemannian geometry) in the slightest, but the opportunity is yours to clarify.

 

[stevendaryl]

We already agreed on that. Feynman (note one n) is dead too but evidently understanding no.2 is not "missing" as it has been mentioned here and in other threads. Moreover, the fact that different versions of the EP exist is well known; if you don't care which version originates with whom, then that suffices.

The way it seems to me is that the equivalence principle was a HEURISTIC principle that led Einstein to his formulation of General Relativity. It's only use is heuristic, as a way to get an intuitive, non-mathematical idea of what GR predicts. I don't see the point in arguing about what exactly it means, because if the intuitive picture suggested by the equivalence principle isn't precise enough to answer a question, then you can just throw out the equivalence principle and use the actual equations of GR.

 

[stevendaryl]

Anyway, to repeat myself, I think there are two different views of Einstein's Field Equations: (1) As a theory of curved spacetime, and (2) as a spin-2 field theory. I think it's completely backwards to think of (2) as Einstein's view, because Einstein really wasn't coming at GR from the point of view of field theory, at all. He was investigating the nature of motion and generalized coordinate systems. That's a geometric view, not a field-theoretic. From the field-theoretic point of view, the fact that a spin-2 field that couples to stress-energy leads to a generally covariant theory is a nice side-effect, but it isn't the starting point, while with Einstein general covariance was the starting point. So if Einstein is going to be associated with one of the views, it sure seems to me to be the geometric, curved spacetime view. He was the one who formulated the whole thing using Riemannian geometry (adapted for his not-positive-definite metric), and Riemannian geometry is, well, geometry, not field theory.

So I think harrlyin has it exactly backwards about which view is Einstein's view. Okay, maybe not 100%. Einstein's concern was motion and generalized coordinates, not geometry. In hindsight, differential geometry IS the best way to addresses his concerns, but I don't think that Einstein started out thinking that geometry was the heart of what he was doing. But it was certainly closer to geometry than it was to field theory.

 

[harrylin]

[..] I did ask for clarification, which you failed to provide. So if you are not explicitly rejecting pseudo-Reimannian geometry with this quote then please clarify now, because if that is not what you had intended to state then your meaning did not come through.

I considered that whole discussion fruitless which is why I said that I did not participate in it further; and there should be no need to explain that philosophy isn't geometry and that Caroll's philosophy isn't necessarily Riemann's or Einstein's. However I agreed in essence with the comments of stevendaryl who thus correctly understood the kind of interpretation issues that I had in mind; and I already clarified that he did. Moreover I linked to several of the threads in which such issues are discussed. I was becoming riddled why you went on, but now your next sentence clarified it:

Btw, it is a pretty poor debate tactic

Sorry, I think that you mistake a science discussion forum for a place to engage in debates [edit: "debates" in the negative sense that includes the use of tactics]. I dislike debates and refuse to participate in discussions that deteriorate into debates; that is not a "tactic" but my personal policy. Debates are mostly a huge waste of time.

There came no answer to my last question in this thread so I will next unsubscribe from it (if I were a Mentor, I would close it).

 

[harrylin]

[..] Einstein's concern was motion and generalized coordinates, not geometry. In hindsight, differential geometry IS the best way to addresses his concerns, but I don't think that Einstein started out thinking that geometry was the heart of what he was doing. But it was certainly closer to geometry than it was to field theory.

I just found a citation of Einstein in which he disagreed that the gravitational field was somehow more "geometric" than the EM field; but that's again off-topic for this thread. If you are interested, please PM me.

 

[Dale]

I considered that whole discussion fruitless which is why I said that I did not participate in it further; and there should be no need to explain that philosophy isn't geometry and that Caroll's philosophy isn't necessarily Riemann's or Einstein's.

This response is pure evasiveness. There was no philosophy* in that chapter. It was nothing more than an introduction to the mathematical framework of pseudo-Riemannian manifolds.

In any case, despite repeated requests you have not clarified your rejection of "patchwork". I therefore continue to believe (as do others) that you mean covering a manifold with multiple charts. This is a basic and necessary part of pseudo-Riemannian geometry, so I stand my my assertion that you reject pseudo-Riemannian geometry and therefore "Einstein's GR".

You are certainly welcome to unsubscribe or to clarify. The choice is yours, but your rejection of Einstein's math is not a failing of GR.

There came no answer to my last question in this thread so I will next unsubscribe from it

You probably should wait more than one day for pervect to answer the question. As you were not on for 3 weeks recently it shouldn't surprise you that he may not have been on for a day.


*It is possible that you consider all math to be philosophy. In which case the chapter was philosophical, but then it is undisputably an introduction to Riemann's philosophy which was adopted by Einstein.

 

[PeterDonis]

in the case of event horizons, it seems possible to me that the question of whether anything ever crosses the event horizon might be true for one interpretation and false for the other.

It can't work like that, because the Einstein Field Equation for a vacuum spacetime predicts that there *is* an event horizon and that objects *can* cross it going inwards (but not outwards). That's not a matter of "interpretation": it's a unequivocal prediction of the math, which holds regardless of whether you interpret the math as describing geometry or as describing a spin-2 force field. The only ways to avoid that prediction are:

(1) If the stress-energy tensor is not vacuum; however, even this won't avoid the prediction of an EH forming under a fairly wide range of initial conditions;

(2) If the Einstein Field Equation is wrong.

 

[PeterDonis]

Einstein's GR assumes a gravitational field, corresponding to (2).

This is not correct; why do you think this? "Einstein's GR", if by that you mean the theory that Einstein published in 1915, was based on the geometric interpretation of the Field Equation; Einstein never even knew about the spin-2 field interpretation, AFAIK.

Einstein's GR is a geometric theory. Einstein himself may not have recognized it

Einstein certainly *did* recognize it; he spent several years learning Riemannian and pseudo-Riemannian geometry from Marcel Grossman precisely so he could use it as a framework for formulating GR.

 

[stevendaryl]

It can't work like that, because the Einstein Field Equation for a vacuum spacetime predicts that there *is* an event horizon and that objects *can* cross it going inwards (but not outwards). That's not a matter of "interpretation": it's a unequivocal prediction of the math, which holds regardless of whether you interpret the math as describing geometry or as describing a spin-2 force field. The only ways to avoid that prediction are:

(1) If the stress-energy tensor is not vacuum; however, even this won't avoid the prediction of an EH forming under a fairly wide range of initial conditions;

(2) If the Einstein Field Equation is wrong.

I'm specifically talking about case number (2). GR could be wrong in such a way that it gives the right answer in cases of weak gravity, but not for cases of strong gravity. What I said to Harry was that for the event horizon to be such a place where GR breaks down, it means that the notion of "strong gravity" is not a generally covariant notion. The event horizon ISN'T a region of strong gravity in KS coordinates.

 

[stevendaryl]

It can't work like that, because the Einstein Field Equation for a vacuum spacetime predicts that there *is* an event horizon and that objects *can* cross it going inwards (but not outwards). That's not a matter of "interpretation": it's a unequivocal prediction of the math...

I have to disagree a little bit here. The field equations by themselves describe spacetime dynamics within a region of spacetime. They don't say anything about what regions must exist, do they? So in Schwarzschild coordinates, there is a region of spacetime described by Schwarzschild coordinates

2GM/c^2 < r < \infty
- \infty < t < \infty
0 \leq \theta \leq \pi
0 \leq \phi < 2 \pi

The field equations by themselves don't say anything about the existence of other regions. Now, you can argue physically that there should be other regions besides this one, using the principle of geodesic completeness, or by considering how a star collapses, or something. But the field equations themselves don't say what regions of spacetime exist, they only describe how dynamics works within a region. Or at least, it seems that way to me.

 

[stevendaryl]

I just found a citation of Einstein in which he disagreed that the gravitational field was somehow more "geometric" than the EM field; but that's again off-topic for this thread. If you are interested, please PM me.

I've seen that quote, and I don't think it really clarified anything. I'm not into appeal to authority. Einstein is important because he did great things, it's not that they are great because Einstein did them. On philosophical matters about the "real" meaning of Einstein's theory, I actually don't think that Einstein had any more insight than anyone else.

 

[stevendaryl]

As you continue to make nonsensical claims about what you think I said, further discussion with you on this topic is useless.

I would think that the conclusion should not be "further discussion with you on this topic is useless" but rather "I should make more of an effort to clarify what I mean, since people seem confused by it."

 

[PeterDonis]

I'm specifically talking about case number (2).

Ah, ok.

The event horizon ISN'T a region of strong gravity in KS coordinates.

I think this is a somewhat misleading way of stating it. Whether or not gravity is "strong" at the EH seems to me to depend on the mass of the BH; a BH with a small enough mass could indeed have "strong" gravity (in the sense of strong spacetime curvature) at the horizon. But "strong" in this sense *is* a generally covariant notion; curvature invariants are the same regardless of which chart you compute them in. You can even compute them in the SC chart at the horizon if you take limits as r -> 2m.

The way I would put the point I think you're trying to make here is that if you want to claim that GR breaks down at the EH, you have to be relying on some *other* notion than "strong gravity" in the above sense. And since nobody has come up with any such notion that picks out the EH in all cases (i.e., regardless of the mass of the hole) *and* is generally covariant, it seems like any claim that GR always breaks down at the EH must violate general covariance; it must rely on properties of particular coordinate charts (such as the SC chart becoming singular at the EH). Whether or not the (non-generally covariant) notion you pick deserves the name "strong gravity" (in some other sense than the covariant sense I gave above) seems to me to be a side issue.

 

[stevendaryl]

The way I would put the point I think you're trying to make here is that if you want to claim that GR breaks down at the EH, you have to be relying on some *other* notion than "strong gravity" in the above sense. And since nobody has come up with any such notion that picks out the EH in all cases (i.e., regardless of the mass of the hole) *and* is generally covariant, it seems like any claim that GR always breaks down at the EH must violate general covariance; it must rely on properties of particular coordinate charts (such as the SC chart becoming singular at the EH). Whether or not the (non-generally covariant) notion you pick deserves the name "strong gravity" (in some other sense than the covariant sense I gave above) seems to me to be a side issue.

I agree with that paraphrase. Here's a thought experiment about gravity that I think is interesting, even though it might have very little practical use. In quantum scattering, at least in one course I took on the subject years ago, a common mathematical technique is to by-hand add time-dependence to the coupling constants. That is, you imagine that in the distant past, the coupling constant was 0, and that very slowly its strength increased with time to the current value. An example of such a slowly-increasing function might be \lambda = \dfrac{1}{2}\lambda_0 (1+arctanh(kt)) for a very small value of k. The point of having a slowly changing coupling constant is that it (hopefully, anyway) allows you to understand the states of the coupled system as perturbations of the states of the uncoupled system.

Anyway, suppose you tried to do that with gravity. You start off with Minkowsky spacetime and no gravity. Just particles floating around. Then pick a frame (to make this work, you have to choose a frame to serve as your standard for time). Write the field equations in this frame. Then modify the equations as follows: replace the constant G by a function G(t), which starts off at G(-\infty) = 0 and smoothly increases to G(+\infty) = G_0, where G_0 is the current value of G.

Now, these equations are no longer covariant--they have a preferred coordinate system. However, they are still legitimate differential equations. We can still solve them, numerically at least. What I would expect to be the case is that for very large values of t, the solutions would settle down to a solution of the unaltered Einstein Field Equations. However, it's not clear to me that you would ever get the interior of a black hole event horizon. So it would settle down to a solution of the EFE that's missing some regions. Or it seems possible that it would. It's sort of like the case with perturbation theory in physics. Certain solutions (bound states for example) can't be obtained perturbatively.

 

[harrylin]

I would think that the conclusion should not be "further discussion with you on this topic is useless" but rather "I should make more of an effort to clarify what I mean, since people seem confused by it."

Once more, you seem to have understood (and without any confusion) what I think to be a main point of the recent discussions related to black holes and I suggested that perhaps you can explain it better than me (and so you did, although you approached it from a different angle). But IMHO everything that people currently have in mind has already been discussed several times and the last discussions appear to not have helped anyone with anything. So, if you think that something useful can come out of further discussion of the same things, good luck with it.

 

[PAllen]

The way I would put the point I think you're trying to make here is that if you want to claim that GR breaks down at the EH, you have to be relying on some *other* notion than "strong gravity" in the above sense. And since nobody has come up with any such notion that picks out the EH in all cases (i.e., regardless of the mass of the hole) *and* is generally covariant, it seems like any claim that GR always breaks down at the EH must violate general covariance; it must rely on properties of particular coordinate charts (such as the SC chart becoming singular at the EH). Whether or not the (non-generally covariant) notion you pick deserves the name "strong gravity" (in some other sense than the covariant sense I gave above) seems to me to be a side issue.

To serve as an argument that you have to modify GR, I did at some point (I think in this thread) throw out a generally covariant addition to the field equations that I thought accomplished this. I called it, I think, the "universe boundary law". It is:

- closed manifolds are rejected; null infinity must be well defined.
- any points on the manifold not connected to null infinity, or that are part of null infinity, are removed from any solution. Note, an open subset of a manifold is still a valid manifold. It will still everywhere satisfy the EFE, if the 'trial solution' did.

This new law is strictly coordinate independent, thus manifestly generally covariant.

Thus, I think you must add new rules to the EFE to remove horizons and interiors, but it can be done in a generally covariant way. It could be argued that this is in the same spirit as energy conditions that effectively reject mathematically valid Einstein tensors (= stress energy tensor). I, of course, feel that there is no physical basis for an additional law like this - it only serves to violate the equivalence principle.

A less artificial way to change GR is to add evaporation to it in such a way as to guarantee that no event loses connection to null infinity before evaporation completes.

 

[PeterDonis]

- closed manifolds are rejected; null infinity must be well defined.

The problem with this is that it doesn't just rule out closed manifolds; it rules out all manifolds that aren't asymptotically flat. You can only define null infinity in an asymptotically flat manifold. For example, an open FRW manifold such as the one currently used in the "best fit" model for our universe has no null infinity.

A less artificial way to change GR is to add evaporation to it in such a way as to guarantee that no event loses connection to null infinity before evaporation completes.

This doesn't necessarily have to change GR; you could (I believe) construct the classical limit of such an "evaporation" model by using an SET with a sufficiently large negative pressure. This would violate several energy conditions, so such SETs are usually considered "unphysical", but when quantum effects are included it's no longer clear that the energy conditions always have to hold anyway.

 

[PeterDonis]

I, of course, feel that there is no physical basis for an additional law like this - it only serves to violate the equivalence principle.

Not only that, it appears to require that local physics--whatever it is that, locally, prevents an event horizon from forming--must "know" the entire future of the spacetime, so that local events can "know" when they are getting close to losing connection with null infinity (more precisely, with *future* null infinity, which is the point).

 

[PAllen]

The problem with this is that it doesn't just rule out closed manifolds; it rules out all manifolds that aren't asymptotically flat. You can only define null infinity in an asymptotically flat manifold. For example, an open FRW manifold such as the one currently used in the "best fit" model for our universe has no null infinity.

I'm not sure that's right. The technical definition of horizon uses null infinity. I've seen claims in the literature that BH's are only technically undefinable for closed spacetimes. If there is no null infinity, then all singularities are technically naked.

[edit: Here is a reference showing null infinity for De Sitter space: http://www.math.miami.edu/~galloway/papers/cqg7_11_021.pdf

which suggests my comment about 'closed' needs clarification. ]

This doesn't necessarily have to change GR; you could (I believe) construct the classical limit of such an "evaporation" model by using an SET with a sufficiently large negative pressure. This would violate several energy conditions, so such SETs are usually considered "unphysical", but when quantum effects are included it's no longer clear that the energy conditions always have to hold anyway.

But you would need to add a rule that says any SET the produces a horizon is illegal. I call that modifying GR.

 

[Dale]

everything that people currently have in mind has already been discussed several times and the last discussions appear to not have helped anyone with anything.

I agree. There simply is no substitute for actually learning the math. In the end, discussions on internet forums just can't provide a shortcut.

 

[Dale]

I have to disagree a little bit here. The field equations by themselves describe spacetime dynamics within a region of spacetime. They don't say anything about what regions must exist, do they? So in Schwarzschild coordinates, there is a region of spacetime described by Schwarzschild coordinates

2GM/c^2 < r < \infty
- \infty < t < \infty
0 \leq \theta \leq \pi
0 \leq \phi < 2 \pi

The field equations by themselves don't say anything about the existence of other regions. Now, you can argue physically that there should be other regions besides this one, using the principle of geodesic completeness, or by considering how a star collapses, or something. But the field equations themselves don't say what regions of spacetime exist, they only describe how dynamics works within a region. Or at least, it seems that way to me.

Sure, but SC are not the only coordinates, and many of those other coordinates are equally valid solutions of the EFE which do cover regions inside the horizon. Due to the fact that a given chart maps an open subset of the manifold, the existence of any chart covering the interior implies that those events are part of the whole manifold, while the fact that SC doesn't cover them does not imply the opposite.

The only way to get around that is to modify the EFE or impose some sort of ad-hoc restriction to the set of admissible manifolds.

 

[PeterDonis]

The technical definition of horizon uses null infinity.

Yes, I agree. I wasn't disputing your definition of a black hole event horizon; I was only saying that a black hole event horizon can't exist in a spacetime that doesn't have a null infinity. Other kinds of horizons can exist (such as cosmological horizons), but not black hole event horizons.

However, I'm not sure I was right to say that an open (or flat) FRW spacetime doesn't have a future null infinity; I've been trying to find a link to a Penrose diagram of that spacetime but haven't been able to.

my comment about 'closed' needs clarification.

I think the proper term would be "compact", or more precisely "spatially compact"--i.e., any spacelike slice is a compact manifold. Spacelike slices of de Sitter spacetime are, I believe, not compact.

I'm not sure, though, that being spatially compact is equivalent to not having a future null infinity. That's what I think needs further thought.

But you would need to add a rule that says any SET the produces a horizon is illegal. I call that modifying GR.

The rule couldn't be that simple, since a vacuum SET allows a horizon to form. I was actually thinking of something along the lines of: what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET). This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.

 

[PeterDonis]

What I would expect to be the case is that for very large values of t, the solutions would settle down to a solution of the unaltered Einstein Field Equations.

That seems reasonable to me.

However, it's not clear to me that you would ever get the interior of a black hole event horizon. So it would settle down to a solution of the EFE that's missing some regions.

Such a solution would be geodesically incomplete, whereas the initial state (Minkowski spacetime) is geodesically complete. So I'm not sure this would work. What I think would happen instead is that you would not be able to construct a solution with a time-varying G that contained the Schwarzschild exterior. See below.

It's sort of like the case with perturbation theory in physics. Certain solutions (bound states for example) can't be obtained perturbatively.

It's true that the maximally extended Schwarzschild solution to the EFE is something like a soliton; I believe some physicists have actually used that term to describe it. That would mean it's not "reachable" as a perturbation of Minkowski spacetime.

I know that seems weird, since it's obviously possible to express the vacuum exterior region as a perturbation of Minkowski spacetime. But that region is not geodesically complete; so the region we're expressing as a perturbation is not a perturbation of *all* of Minkowski spacetime, it's only a perturbation of a *portion* of Minkowski spacetime; in the simplest case, it's the portion of Minkowski spacetime outside some radius r from a chosen central point.

Which leaves the question of what is the initial condition of the region *inside* that radius? If the region inside radius r starts out in a non-vacuum initial state, then the complete initial state is no longer Minkowski spacetime. But if the region inside radius r starts out as vacuum, then as I said above, I don't think you can construct a solution that turns that vacuum interior into a black hole interior by varying G with time; but you could, perhaps, turn that "vacuum" interior (with particles floating around but no gravity) into a non-vacuum interior with a massive gravitating body in it (if the "particles" have enough mass to form such a body once gravity is "turned on").

 

[zonde]

There simply is no substitute for actually learning the math. In the end, discussions on internet forums just can't provide a shortcut.

It depends on what do you mean by "math" and for what purpose do you need it.

 

[TrickyDicky]

I was actually thinking of something along the lines of: what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET).

I have often considered this very "what if", but you have expressed it in a way clearer than anything I've managed to write myself.

This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.

Agreed. But it is refreshing to see that unlikely as it may be it is at least considered as a possibility something (a plausible way of preventing not only singularities but also event horizons from forming) that was not even admitted as a mathematically and/or physically valid scenario in previous discussions. (Even if it was in the literature as shown by PAllen's references by Krauss et al. in the first posts in this thread).

 

[stevendaryl]

what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET). This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.

It's hard for me to see how that could work, because locally, there is nothing that indicates that you're near an event horizon (in the case of a large enough black hole).