Crossing the Event Horizon of a Black Hole

[DrGreg]

DiamondGeezer,

By your logic, it's impossible for anyone to travel through the Earth's North Pole because the latitude/longitude coordinate system has a discontinuity there.

All the issues you raise can be resolved if you make the effort the understand the analogous set of disjoint (T,R) coordinate systems for the Rindler rocket (follow the links in post #50), which also exhibit discontinuities at the horizon, where R is time and T is space behind the horizon, where the metric equation expressed in these coordinates makes no sense exactly at the horizon. And yet this is just a weird set of coordinate systems that, between them, describe the flat, continuous, gravity-free Minkowski spacetime of special relativity.

I've explained this the best I can. If you're not getting it, I'm not sure there's much more I can say. If anyone else is still reading this thread and would like to contribute, maybe they can find some different way of explaining it that will get the point across.

 

[DiamondGeezer]

DiamondGeezer,

By your logic, it's impossible for anyone to travel through the Earth's North Pole because the latitude/longitude coordinate system has a discontinuity there.

Nope. The lat/long coordinate system begins and ends at zero and not 200 miles above or below it.

All the issues you raise can be resolved if you make the effort the understand the analogous set of disjoint (T,R) coordinate systems for the Rindler rocket (follow the links in post #50), which also exhibit discontinuities at the horizon, where R is time and T is space behind the horizon, where the metric equation expressed in these coordinates makes no sense exactly at the horizon. And yet this is just a weird set of coordinate systems that, between them, describe the flat, continuous, gravity-free Minkowski spacetime of special relativity.

You keep repeating the Rindler metric without ever dealing with the Schwarzschild Metric. Am I meant to be impressed? One is a coordinate based singularity which is unreachable, the other is not only reachable but inevitable for all infalling particles leading to impossible speeds and impossible energies.

I've explained this the best I can. If you're not getting it, I'm not sure there's much more I can say. If anyone else is still reading this thread and would like to contribute, maybe they can find some different way of explaining it that will get the point across.

Actually you haven't. You've run out of excuses for the bizarre mathematical behaviour of the Schwarzschild Metric. You haven't bothered to explain how tidal acceleration can be measured inside the EH - you simply asserted this as an unassailable fact. You haven't explained how reversing the sign of the Schwarzschild Metric inside the EH is mathematically allowable, nor explained why Schwarzschilds coordinates and any other derived coordinate system such as Kruskal's become imaginary when r<2M.

I'm tired.

 

[Ich]

If anyone else is still reading this thread and would like to contribute, maybe they can find some different way of explaining it that will get the point across.

Some dozen posts ago, I considered jumping in, but thought better of it.

I'm tired.

As this thread comes to an end, maybe it's good to know for you, DiamondGeezer, that I'm backing DrGreg: you obviously suffer from some basic misunderstandings concerning the meaning of coordinates and the metric. Try to do some calculations, like proper time during infall or such, that show you that the singularity at the EH is removable.

 

[Dmitry67]

Some dozen posts ago, I considered jumping in, but thought better of it.

My first thought was "considered jumping in the Black Hole" to prove that we are right :)

 

[Ich]

My first thought was "considered jumping in the Black Hole" to prove that we are right :)

Next time there is one around, I'll give it a thought. Maybe we can persuade the colleagues from the LHC to save one for us. ;)

 

[atyy]

Schwarzschild coordinates have a coordinate singularity. If you start with it, you will always have the singularity. So do not look for an explanation of why Kruskal-Szekeres coordinates are good through the event horizon by starting from Schwarzschild coordinates. Start from the Kruskal-Szekeres metric, verify that it is a vacuum solution of Einstein's equations, and by calculating geometric invariants show that there is an event horizon, and that spacetime is normal all the way until the curvature singularity. Schwarzschild coordinates only cover bits of the Kruskal-Szekeres coordinates.

 

[JesseM]

Schwarzschild coordinates have a coordinate singularity. If you start with it, you will always have the singularity. So do not look for an explanation of why Kruskal-Szekeres coordinates are good through the event horizon by starting from Schwarzschild coordinates. Start from the Kruskal-Szekeres metric, verify that it is a vacuum solution of Einstein's equations, and by calculating geometric invariants show that there is an event horizon, and that spacetime is normal all the way until the curvature singularity. Schwarzschild coordinates only cover bits of the Kruskal-Szekeres coordinates.

Yeah, I think this may get to the heart of the problem--DiamondGeezer is probably starting from Schwarzschild coordinates (as a lot of textbooks do, I think) and seeing other systems as just weird and vaguely suspicious transformations of Schwarzschild coordinates. But you can just as easily start from some other system which has no coordinate singularity at the horizon, like Kruskal-Szekeres coordinates or free-fall coordinates (the latter has a nice physical definition of coordinate time in terms of proper time read by a set of infalling clocks, see the middle of this page, and the discussion on this thread), and derive Schwarzschild coordinates as a transformation of those.

 

[DiamondGeezer]

Schwarzschild coordinates have a coordinate singularity. If you start with it, you will always have the singularity. So do not look for an explanation of why Kruskal-Szekeres coordinates are good through the event horizon by starting from Schwarzschild coordinates. Start from the Kruskal-Szekeres metric, verify that it is a vacuum solution of Einstein's equations, and by calculating geometric invariants show that there is an event horizon, and that spacetime is normal all the way until the curvature singularity. Schwarzschild coordinates only cover bits of the Kruskal-Szekeres coordinates.

I can confirm that the Kruskal-Szekeres metric is a vacuum solution of Einstein's equations. Just like the Schwarzschild Metric. Unfortunately spacetime isn't normal all the way to the curvature singularity - it becomes imaginary.

Yeah, I think this may get to the heart of the problem--DiamondGeezer is probably starting from Schwarzschild coordinates (as a lot of textbooks do, I think) and seeing other systems as just weird and vaguely suspicious transformations of Schwarzschild coordinates. But you can just as easily start from some other system which has no coordinate singularity at the horizon, like Kruskal-Szekeres coordinates or free-fall coordinates (the latter has a nice physical definition of coordinate time in terms of proper time read by a set of infalling clocks, see the middle of this page, and the discussion on this thread), and derive Schwarzschild coordinates as a transformation of those.

The Kruskal-Szekeres coordinates and the free-fall coordinates display the same behavior as the Schwarzschild Metric coordinates - the line integral is continuous to $r=0$ but the spatial or time separations become complex when $r$<$2M$.

The line integral is continuous because it goes through zero at the event horizon regardless of the transformation of coordinate systems used.

All of this should be telling you something important, but unfortunately I've had intelligent people tell me that the square root of a negative number is the same as the negative root of a positive number and I shouldn't worry about this mathematical non sequitur.

 

[DrGreg]

Nope. The lat/long coordinate system begins and ends at zero and not 200 miles above or below it.

The point is when you pass through the North Pole, your longitude "magically" jumps through 180° and your coordinate velocity reverses. Apparently bizarre behaviour if you just look at the coordinates without understanding the bigger picture.

You keep repeating the Rindler metric without ever dealing with the Schwarzschild Metric. Am I meant to be impressed?

I keep mentioning the Rindler case because it's easier to understand than black hole. If you make the effort to understand that example and see just how similar the two cases are, you ought to find the black hole easier to understand too. I don't expect you to just take my word for it, but to investigate the maths further and take some time to make sense of it. But I can't get you to understand black holes until you first understand the Rindler example

One is a coordinate based singularity which is unreachable, the other is not only reachable but inevitable for all infalling particles leading to impossible speeds and impossible energies.

No, in both cases there's a coordinate based singularity in one coordinate system which isn't a singularity in another system. In both cases, the location is physically reachable, as can be seen in one system, but lies outside the other coordinate system and can't be reached in a finite coordinate time in the wrong system. You only get impossible speeds and impossible energies, in both cases, by taking a physically impossible mathematical limit.

I'm tired.

Me too. Merry Christmas!

 

[Dmitry67]

Diamond, Could you explain why time is imaginary inside the BH?
Light cone (for the falling observer) looks normal and oriented vertically.
Check the attachment

 

[atyy]

KS coordinates are something like

dss ~ (1/r)exp(-r/2GM)dtt + ...

For r>0, r/2GM changes from greater than one to less than one as r goes from r>2GM to r<2GM, but there is no change in sign of dss.

As long as a curve remains timelike, a single definition of proper time above and below the event horizon is fine. For a spacelike curve, then we do run into trouble, and have to put in a minus sign, but this is true even in flat spacetime, and is one of the peculiarities of pseudo-Riemannian geometry.

 

[DiamondGeezer]

The point is when you pass through the North Pole, your longitude "magically" jumps through 180° and your coordinate velocity reverses. Apparently bizarre behaviour if you just look at the coordinates without understanding the bigger picture.

But the spacetime separations between two points on the polar coordinate surface remain resolutely real and positive. Quite unlike the Schwarzschild solution below the EH.

I keep mentioning the Rindler case because it's easier to understand than black hole. If you make the effort to understand that example and see just how similar the two cases are, you ought to find the black hole easier to understand too. I don't expect you to just take my word for it, but to investigate the maths further and take some time to make sense of it. But I can't get you to understand black holes until you first understand the Rindler example

I keep mentioning the Schwarzschild Metric because its tougher and because its a static solution.

No, in both cases there's a coordinate based singularity in one coordinate system which isn't a singularity in another system. In both cases, the location is physically reachable, as can be seen in one system, but lies outside the other coordinate system and can't be reached in a finite coordinate time in the wrong system. You only get impossible speeds and impossible energies, in both cases, by taking a physically impossible mathematical limit.

Ah yes, when the Universe divides by zero. But the physically impossible mathematical limit happens regardless of the coordinate system used, and the "inner Schwarzschild metric" isn't continuous with the outside. Nor is the "inner Kruskal-Szekeres" continuous with the outer.

You'd think somebody might be concerned about this sort of problem but apparently its Christmas when the laws of mathematics get suspended for the holidays.

Merry Christmas!

Merry Christmas to you. And a Christmas present: http://www.springerlink.com/content/dl8mu550u7736567/" - its a hint.

 

[Dmitry67]

Here is my present:

http://en.wikipedia.org/wiki/Black_hole
Oppenheimer and his co-authors used Schwarzschild's system of coordinates (the only coordinates available in 1939), which produced mathematical singularities at the Schwarzschild radius, in other words some of the terms in the equations became infinite at the Schwarzschild radius. This was interpreted as indicating that the Schwarzschild radius was the boundary of a bubble in which time stopped. This is a valid point of view for external observers, but not for infalling observers.

In 1958, David Finkelstein introduced the concept of the event horizon by presenting Eddington-Finkelstein coordinates, which enabled him to show that "The Schwarzschild surface r = 2 m is not a singularity, but that it acts as a perfect unidirectional membrane: causal influences can cross it in only one direction

 

[Dmitry67]

... also

The apparent singularity at r = rs is an illusion; it is an example of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates or coordinate conditions. By choosing another set of suitable coordinates one can show that the metric is well-defined at the Schwarzschild radius. See, for example, Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates or Novikov coordinates.

 

[DiamondGeezer]

... also

The apparent singularity at r = rs is an illusion; it is an example of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates or coordinate conditions. By choosing another set of suitable coordinates one can show that the metric is well-defined at the Schwarzschild radius. See, for example, Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates or Novikov coordinates.

Have you actually read this thread or just trying to bore me into submission?

1. All of the above coordinate transformations show that space or time separations below $r=2M$ are imaginary.
2. I have never once claimed that the apparent surface at $r=2M$ is anything other than a coordinate singularity.

What astonishes me is the sheer inability to get to grips with a very simple set of arguments. Instead I have got repeated "proofs by assertion" combined with statements impugning my intelligence. The arguments given so far have been mathematically fallacious.

And please try not to quote Wikipedia on this forum unless you want to be serially ignored.

 

[Dmitry67]

Have you actually read this thread or just trying to bore me into submission? All of the above coordinate transformations show that space or time separations below $r=2M$ are imaginary.

In schwarsheild metrics? of course.
This is why you need to use OTHER metrics for the interior of the BH where these purely mathematical oddities do not exist.

In the metrics from the list (Eddington-Finkelstein are my favourite) it is clear that for the falling observer space and time look normal (even they might looc 'imaginary' in the coordinate system of free far observer)

 

[DiamondGeezer]

In schwarsheild metrics? of course.
This is why you need to use OTHER metrics for the interior of the BH where these purely mathematical oddities do not exist.

In the metrics from the list (Eddington-Finkelstein are my favourite) it is clear that for the falling observer space and time look normal (even they might looc 'imaginary' in the coordinate system of free far observer)

It's clear that changing coordinate systems like that are mathematically invalid.

 

[Dmitry67]

BTW I suggest using the Einstein approach
Forget about the metrics.
Lets talk about what observers will observer.

Say, we are falling inside the spaceship (without the winsows) into the big enough BH so tidal forces are low.
Do you agree that under the horizon austranauts will continue see each other normally?

 

[DiamondGeezer]

BTW I suggest using the Einstein approach
Forget about the metrics.
Lets talk about what observers will observer.

Say, we are falling inside the spaceship (without the winsows) into the big enough BH so tidal forces are low.
Do you agree that under the horizon austranauts will contanue see each other normally?

No. They won't see anything at all. In order to "see" something light has to travel from somewhere outside to the retina and then the electrical impulses travel to your brain.

Inside the EH, all light cones are timelike and nothing (not even light) will travel backwards to hit the retinas.

 

[Dmitry67]

I had drawn a diargam for you (sorry, drawing in notepad using a mouse is terrible). Also, angles are incorrect. So,

Black vertical line is singularity
Vertical red line is horizon
Blue and Green lines are Bob and Alice, they are sitting at the opposite sides of the spaceship.
I had drawn Alice's lightcones and where they intersect Bob's worldline.
You can do the the same for Bob and check that Alice can see Bob as well.
They see each other up to the moment when lightcones become too narrow in these coordinates (they interpret it is the increase of the tidal forces). Spaceship breaks apart. Soon Bob and Alice lose each other behind their apparent horizons (close to the singularity)

Note that lightcones are in fact timelike inside :)

 

[Dmitry67]

And yet another example.
Look at our world from the SUPERLUMINAL rest frame.
You see the same - imaginary time etc.
But it does not affect how we observe things, right?

 

[Ich]

I found a weird metric with imaginary time and space:
ds² = -dx² + dt² +dy² + dz²
Obvoiusly such spacetimes can't exist, as you can't see your nose there.

 

[stevebd1]

No. They won't see anything at all. In order to "see" something light has to travel from somewhere outside to the retina and then the electrical impulses travel to your brain.

Inside the EH, all light cones are timelike and nothing (not even light) will travel backwards to hit the retinas.

I recommend you take a look at http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates" which can be expressed in two forms, free-fall rain frame and global rain frame $(r_s=2M)$-

Free-fall rain frame-

$$c^2d\tau^2=c^2dt_r^2 - dr_r^2 - r^2\left(d\theta^2 + \sin^2\theta \, d\phi^2\right)$$

Global rain frame-

$$c^2d\tau^2=\left(1-\frac{r_s}{r}\right)c^2dt_r^2 - 2\sqrt{\frac{r_s}{r}}\ cdt_rdr - dr^2 - r^2\left(d\theta^2 + \sin^2\theta \, d\phi^2\right)$$

where

$$\begin{flalign} &dt_r=dt-\beta\gamma^2dr\\[6mm] &dr_r=\gamma^2dr-\beta dt \end{flalign}$$

and

$$\begin{flalign} &\gamma=\frac{1}{\sqrt{1-\beta^2}}\\[6mm] &\beta=-\sqrt{\frac{r_s}{r}} \end{flalign}$$

where $\gamma$ is the Lorentz factor and $\beta$ is the velocity of the rain frame relative to the shell frame.

The principle behind the form is-

$$\gamma=\frac{1}{\sqrt{1-(v/c)^2}}\equiv \frac{1}{\sqrt{1-(r_s/r)}}$$

Basically length contraction induced by velocity (which in turn is induced by curvature) balances out the length expansion induced by gravity. This is identical to 'Metric for the Rain Frame' shown on page B-13 of Exploring Black Holes by Taylor & Wheeler.

The free-fall rain frame proper time remains time-like all the way to the singularity, representing the frame of the in-falling observer while the global rain-frame proper time becomes negative at r<2M but with no geometric singularity at the event horizon (dr remains 1).

 

[JesseM]

1. All of the above coordinate transformations show that space or time separations below $r=2M$ are imaginary.

Do you claim that the time separation of events along a timelike worldline is imaginary in Kruskal-Szkeres coordinates? If so I'm pretty certain your wrong, perhaps you'd care to address atyy's post #81 above. As atyy noted, it's true you'll get an imaginary value if you try to calculate the proper time along a spacelike curve, but this is always true in relativity, even in flat SR spacetime (and in KS coordinates it should be just as true outside the horizon as inside it).

 

[DiamondGeezer]

Do you claim that the time separation of events along a timelike worldline is imaginary in Kruskal-Szkeres coordinates? If so I'm pretty certain your wrong, perhaps you'd care to address atyy's post #81 above. As atyy noted, it's true you'll get an imaginary value if you try to calculate the proper time along a spacelike curve, but this is always true in relativity, even in flat SR spacetime (and in KS coordinates it should be just as true outside the horizon as inside it).

No I don't. If you read carefully you'll find I talk about the traditional spacetime separations between events. I know what the difference is between timelike and spacelike and its irrelevant to the arguments.

What I point out (tediously repeating myself I know) is that "inside" a black hole event horizon, spacelike and timelike separations are always imaginary. The various cited coordinate transformations end up with the same result, and curiously nobody seems to be perturbed by the mathematical gyrations which appear to explain what happens when a particle reaches the event horizon in order to allow it to reach the center of curvature.

All of this should tell you lots about the Schwarzschild Metric, but unfortunately nobody's paying any attention.

Nothing I have heard on this thread has shown any insight into the question of black holes or event horizons. Instead I read repeatedly the same arguments which are ipso facto mathematically and physically absurd when examined.

It's not arrogance on my part to say that the mathematical treatment which gives rise to the theory of black holes is badly flawed.

 

[Dmitry67]

DiamondGreezer, did you check my diagram?
Do you agree with it or not?

 

[DiamondGeezer]

DiamondGreezer, did you check my diagram?
Do you agree with it or not?

Yes.

No.

 

[Dmitry67]

Yes.

No.

Then put your version of events on
http://www.valdostamuseum.org/hamsmith/DFblackIn.gif

(Bob and Alice are freely falling into BH looking at each other)

 

[DrGreg]

I know what the difference is between timelike and spacelike and its irrelevant to the arguments.

What I point out (tediously repeating myself I know) is that "inside" a black hole event horizon, spacelike and timelike separations are always imaginary.

If you really did know the difference between timelike and spacelike then you would understand that "imaginary spacelike" means "real timelike" and "imaginary timelike" means "real spacelike".

It's not arrogance on my part to say that the mathematical treatment which gives rise to the theory of black holes is badly flawed.

Just because you don't understand something means you are right and the many, many thousands of relativists who have studied this, some of them eminent physicists and mathematicians, are all wrong?

 

[DiamondGeezer]

If you really did know the difference between timelike and spacelike then you would understand that "imaginary spacelike" means "real timelike" and "imaginary timelike" means "real spacelike".

No. It's irritating that I have to keep repeating myself to people who really should know better.

viz.,

(the square root of a negative quantity)$\neq$(the negative square root of a positive quantity)

Just because you don't understand something means you are right and the many, many thousands of relativists who have studied this, some of them eminent physicists and mathematicians, are all wrong?

If science was really decided by majority voting then relativity would have been rejected more than 100 years ago, when only Einstein understood it. You choose a really poor subject to make such a majoritarian fallacy.

It wasn't so long ago that the majority of experts knew that there were WMDs in Iraq - completely wrong of course.

I don't compare myself to Einstein - he was a genius. Nor do I criticize General Relativity in general. I don't claim to understanding of all of General Relativity.

But on this one small point, the majority on this particular issue are making a mathematical case which is false - and I believe I do have an answer but first there has to be an acknowledgment of a problem.

It does take a certain amount of arrogance to say that "I am right and everyone else is wrong" but there are so many examples of an outsider pointing out something that has somehow escaped the experts in the field that its no longer seen as weird.

I don't know of a single scientist who doesn't have a least 10 hypotheses that goes against what "the majority of experts in the field" believe to be true.

I am perfectly prepared to be wrong (and I may well be). But the argumentation on the issue of black holes and event horizons is mathematically weak and physically contrived - in my humble opinion.

 

[George Jones]

Suppose the components of the metric with respect to a particular coordinate system are given by

$$ds^2 = -dt^2 - dx^2 + dy^2 - dz^2.$$

What do the coordinates $t$, $x$, $y$, and $z$ represent?

 

[DiamondGeezer]

Suppose the components of the metric with respect to a particular coordinate system are given by

$$ds^2 = -dt^2 - dx^2 + dy^2 - dz^2.$$

What do the coordinates $t$, $x$, $y$, and $z$ represent?

George.

Please patronize someone else.

 

[hamster143]

DiamondGeezer, do you need the email of the president of physics?

 

[Ich]

Suppose the components of the metric with respect to a particular coordinate system are given by

ds^2 = -dt^2 - dx^2 + dy^2 - dz^2.

Hey, I claim the https://www.physicsforums.com/showthread.php?p=2497321#post2497321"for this example.

 

[DiamondGeezer]

DiamondGeezer, do you need the email of the president of physics?

No. I have seen the original cartoon though...

Unfortunately I'm not proposing some grandiose thought experiment which undermines relativity...I'm actually interested in the interpretations of the mathematics of black holes, which appear not to make sense from the known laws of mathematics.

Dull, but true.

ETA: Did you ever wonder how many PhDs in Phlogiston Theory there were when the atomic theory was first established? Did majorities work then, Greg?