Black Holes are Paradoxical!

[Willem F Esterhuyse]

We have that a particle observed from far away takes an infinite time to cross the event horizon of a Black Hole. And the very same particle crosses the event horizon in a finite time (as measured by a clock on the particle).

This causes a clash in my mind.

[swansont]

11 minutes ago, Willem F Esterhuyse said:

We have that a particle observed from far away takes an infinite time to cross the event horizon of a Black Hole. And the very same particle crosses the event horizon in a finite time (as measured by a clock on the particle).

This causes a clash in my mind.

That’s a “you” problem. Not understanding something does not make it a paradox.

In this case it’s an issue of relativity - that the rate of the passage of time depends on the frame of reference of the observer.

[joigus]

Let's say it's puzzling, not paradoxical. Keep in mind inertial observers at spatial infinity are an idealisation. More realistically, far away observers that are locally inertial would have to be falling ever so slowly towards the BH. Times would be very, very long; but not infinite. I'm saying this without doing any calculation, BTW. From intuition.

Static observers, OTOH, would have to be trying to escape to the attraction, in order not to fall. They would have to power up their rockets to stay there.

The closer you get to the BH, the more dramatically the static observer differs from the locally inertial one.

@Markus Hanke and/or @Mordred will probably give a more detailed, and much more rigorous account of it.

[Markus Hanke]

14 hours ago, Willem F Esterhuyse said:

This causes a clash in my mind.

It causes a clash only so long as you tacitly assume that there is only one concept of ‘time’ that is somehow equally valid for every observer anywhere in the universe. But GR tells us that this is not so - there is no global, universal time. There are only local clocks, meaning time is a purely local concept. And because time is local, you cannot use a distant, stationary clock to try and figure out what happens to an in-falling particle; you have to use a clock that is actually local (ie comoving) with that particle. 

That being said, it is important to remember that the distant stationary observer is still right about his own conclusions - but he is only right in his own frame of reference. For that distant observer, the particle really does never reach the horizon. Likewise, the observer attached to the in-falling particle is right about his conclusions, again in his own frame of reference; for him, the particle really does reach (and fall through) the horizon.

So you have a situation where you got two observers who arrive at completely different conclusions, yet they are both right! This flies right in the face of everything we are used to, based on our own direct experience of what the world is like. But the trouble is that our experience is limited to a very specific domain - the classical, low-energy, low-velocity, non-relativistic, Euclidean domain. Within this domain, space and time have the same meaning everywhere, and can be neatly separated - so there is no distinction between “local” and “global” in that sense. But around a black hole, things are very different - time and space are inextricably intertwined, and measurements of time and distances are meaningful only within small local regions. You cannot project some specific observer’s notions of time and space someplace/sometime else, and expect to be able to tell what happens there locally.

Needless to say, once the situation is analysed properly using the appropriate mathematical tools (which aren’t necessarily intuitive), there is no paradox, nor even a contradiction - this is all entirely self-consistent and logical, though counter-intuitive when viewed in light of our own Euclidean-based experience of the world.

[Willem F Esterhuyse]

A god-like observer must conclude the particle falls into and doesn't fall into the event horizon - a contradiction.

[Genady]

5 minutes ago, Willem F Esterhuyse said:

A god-like observer must conclude the particle falls into and doesn't fall into the event horizon - a contradiction.

Not so. A god-like observer must rather conclude that particle falls into the event horizon in one reference frame and doesn't fall into it in another reference frame. No contradiction.

[Markus Hanke]

1 hour ago, Genady said:

Not so. A god-like observer must rather conclude that particle falls into the event horizon in one reference frame and doesn't fall into it in another reference frame. No contradiction.

Precisely! +1

1 hour ago, Willem F Esterhuyse said:

A god-like observer must conclude the particle falls into and doesn't fall into the event horizon - a contradiction.

No such observers physically exist. The closest you can come to it is to describe the situation only in terms of generally covariant quantities, ie quantities that do not depend on the choice of observer at all - meaning all physical observers must necessarily agree on them. That’s pretty much a God’s eye view on the situation. Mathematically, this means tensors (and spinors) of any rank will be used to describe the physics. For the situation at hand, the obvious choice would be the length of the in-falling particle’s world line (which is by definition equal to proper in-fall time) - which is finite and well defined, and as being a rank-0 tensor everyone agrees that it is finite and well defined. 

There’s also another issue - coordinate in-fall time diverging to infinity is valid only in Schwarzschild spacetime, which relies on a number of boundary conditions, most notably asymptotic flatness. In other words, to get Schwarzschild spacetime, you need to assume that the universe is everywhere completely empty so that spacetime far from the black hole is approximately flat. Obviously, in the real world, this only ever holds at most as a rough approximation - in reality, there will be stuff orbiting or falling into the black hole, and other gravitational sources at various distances. If you account for in-falling matter and/or radiation, but retain all other symmetries, then you end up with a different kind of spacetime, called Vaidya spacetime - and here even the coordinate in-fall time as measured by a distant observer is quite finite (though much longer than the proper in-fall time in the particle’s frame).

[Mordred]

Well that's a first I never heard of Vaidya spacetime any recommended literature on it ? 

I spotted this article which seems to have a half decent coverage but if you are familiar with better I would appreciate it

"Geometry of the Vaidya spacetime" by Armand COUDRAY & Jean-Philippe NICOLAS

https://arxiv.org/pdf/2101.06544

Edited December 13, 2022 by Mordred

[Markus Hanke]

13 hours ago, Mordred said:

Well that's a first I never heard of Vaidya spacetime any recommended literature on it ? 

Unfortunately I have yet to find good literature on this topic that is accessible to the general reader. Most papers about this spacetime out there are quite mathematical, including the one you quoted. 

If I can find anything, I’ll post it here on the forum.

Generally speaking, the Vaidya solution is a generalisation of Schwarzschild in that spacetime isn’t assumed to be empty, so it is one of the simplest non-vacuum solutions to the field equations. It’s basically the kind of geometry you get for a spherically symmetric, non-charged, non-rotating body immersed in uniformly ingoing or outgoing matter or radiation. As a result, the parameter M in the metric now depends explicitly on time, so this is can be used as a toy model for a growing or evaporating black hole.

[Mordred]

that would be great, I gathered much of details you described from the mathematics of the article I first found still going through it. Always interested in good resources, I tend to collect good examples of different field treatments so enjoy coming across ones I haven't come across before. They are always handy in model building

[Willem F Esterhuyse]

On 12/12/2022 at 4:42 PM, Markus Hanke said:

No such observers physically exist.

Such an observer spiritually exists. Spiritual beings can also observe physics.

We refer to two times, but we also refer to one and the same particle. The particle falls in and doesn't fall in - a contradiction.

Edited December 29, 2022 by Willem F Esterhuyse

[MigL]

12 minutes ago, Willem F Esterhuyse said:

Such an observer spiritually exists. Spiritual beings can also observe physics.

Science does not address 'spirituality'; nor do we on this Science Forum.

[Willem F Esterhuyse]

3 minutes ago, MigL said:

Science does not address 'spirituality'

Mind Science does address it.

[Genady]

17 minutes ago, Willem F Esterhuyse said:

The particle falls in and doesn't fall in - a contradiction.

A car moves and doesn't move. Is it a contradiction?

[Willem F Esterhuyse]

2 minutes ago, Genady said:

A car moves and doesn't move. Is it a contradiction?

Yes.

[Mordred]

7 minutes ago, Willem F Esterhuyse said:

Mind Science does address it.

nope not really mind science has very little to do with physics.  In particular it certainly doesn't validate some godly observer view point. We don't count spiritual astral projection like states in physics. Now do we count other mental like powers as being involved 

[Genady]

11 minutes ago, Willem F Esterhuyse said:

Yes.

However, it happens all the time. A car moves as seen by a person standing outside, and it does not move as seen by its passenger. No contradiction.

[Mordred]

1 minute ago, Genady said:

However, it happens all the time. A car moves as seen by a person standing outside, and it does not move as seen by its passenger. No contradiction.

I like your approach into the basis of relativity +1

[zapatos]

On 12/11/2022 at 1:10 AM, Markus Hanke said:

That being said, it is important to remember that the distant stationary observer is still right about his own conclusions - but he is only right in his own frame of reference. For that distant observer, the particle really does never reach the horizon. Likewise, the observer attached to the in-falling particle is right about his conclusions, again in his own frame of reference; for him, the particle really does reach (and fall through) the horizon.

 

I seem to remember reading on this forum something along the lines of "people may disagree on the timing of events, but they always agree which events took place".

Your statement above seems at odds with what I remember reading. That is, only one person sees the particle falling through the horizon.

Did I miss something, or perhaps I misremembered what I read?

[Genady]

46 minutes ago, zapatos said:

I seem to remember reading on this forum something along the lines of "people may disagree on the timing of events, but they always agree which events took place".

Your statement above seems at odds with what I remember reading. That is, only one person sees the particle falling through the horizon.

Did I miss something, or perhaps I misremembered what I read?

I think, they all agree that particle falls through the horizon. They disagree only on the timing of this event: for one, it happens in a finite time, for the other - infinite.

[Lorentz Jr]

2 hours ago, Willem F Esterhuyse said:

The particle falls in and doesn't fall in - a contradiction.

The Lorentzian view would be that the particle never falls in (i.e. observers on the outside are "right"), but if it were a person falling into the BH they wouldn't know it, because everything about the person slows down and comes to a stop, including their consciousness and all the atomic and subatomic processes in their body. The relativistic view is that neither observer is "right", but neither metaphysical interpretation of the events has been proved or disproved.

Edited December 29, 2022 by Lorentz Jr

[Markus Hanke]

13 hours ago, zapatos said:

I seem to remember reading on this forum something along the lines of "people may disagree on the timing of events, but they always agree which events took place".

Your statement above seems at odds with what I remember reading. That is, only one person sees the particle falling through the horizon.

Did I miss something, or perhaps I misremembered what I read?

You are correct, in that all observers agree on physical events. In this case, this “event” is the intersection of the world line of the in-falling particle with the event horizon. While observers disagree on the where and when of this, they all agree that the two do in fact intersect - including the distant Schwarzschild observer. The reason why he can’t correlate that intersection with a reading on his own clock is that he shares no concept of simultaneity with a clock that’s actually at the horizon. The issue is simultaneity.

He does, however, fully agree on the length and geometry of the in-fall world line, since these are all geometric quantities that are independent of specific coordinate choices.

So, the fundamental difference between these two observers is that the in-falling one physically measures the length of this world line (since his clock falls along it), whereas the distant Schwarzschild observer does not. Thus it really isn’t a surprise that their clocks disagree. As I have said on many occasions, time becomes a purely local concept once gravity is involved.

13 hours ago, Genady said:

I think, they all agree that particle falls through the horizon. They disagree only on the timing of this event: for one, it happens in a finite time, for the other - infinite.

Exactly +1

14 hours ago, Genady said:

However, it happens all the time. A car moves as seen by a person standing outside, and it does not move as seen by its passenger. No contradiction.

Very nice analogy, I like it +1

[Willem F Esterhuyse]

20 hours ago, Genady said:

and it does not move as seen by its passenger

One can argue that the passenger in the car is mistaken that the car is not moving from a global observer point of view.

[Genady]

7 minutes ago, Willem F Esterhuyse said:

One can argue that the passenger in the car is mistaken that the car is not moving from a global observer point of view.

Not good enough. All other observers disagree about a speed with which the car is moving. An observer standing on the side will see one speed. An observer in another car passing the first one will see a different speed. An observer walking along the street will see yet a different speed. An observer on the Moon, on the Sun, in the center of the Milky Way, in Andromeda galaxy, ... they all see different speeds of the same car. Some very different. For example, the observer in the center of the Milky Way sees the car moving together with the whole Solar system with the speed 820 000 km/h. What is the car's speed "from a global observer point of view"?

[Lorentz Jr]

1 hour ago, Willem F Esterhuyse said:

One can argue that the passenger in the car is mistaken that the car is not moving from a global observer point of view.

Right, but no one has ever observed any experimental evidence to indicate what the correct "global" point of view is. The car may be traveling to the West relative to Earth's surface, but Earth's surface is traveling to the East. The car is probably moving relative to the rest of the visible universe, but for all we know the entire visible universe could be moving relative to some even more universal point of view. It's hard to say for sure who's moving and who isn't.