Time, simultaneity, black holes & entropy

[pet_henry]

I'm a newbie, and I've got some questions that are raised by Leonard Susskind's "The Black Hole War," a fascinating non-technical book about his argument with Stephen Hawking about whether or not information is conserved even if it is dropped into a black hole.
I've read several threads and the FAQs and they have been a help but I am still at sea.

My current hangup seems to be about the idea of "now." If information is conserved, or if entropy doesn't decrease, you've got to be able to count it up. This seems to make sense only for a local area, because what does it mean to have something that is a function of time if time is different at different locations?

Susskind mentions a thought experiment whereby someone on the outside of the event horizon drops a container containing atoms of hot gas (with a lot of entropy) into the black hole. When (oops - what does this word mean in this context?) the container crosses the event horizon, entropy in the universe outside the black hole decreases. Crossing the event horizon is a non-event for the container of gas, but to an outside observer - and I guess that means any place outside the event horizon, even only a micron - it takes an infinite time for the container to cross the horizon.

Locally to the container of gas, it experiences no change and just crosses this point (except what about the part of the container which isn't at the event horizon yet?). To the rest of the universe, it appears this event never actually completes due to time dilation. Does the black hole itself ever gain the added mass / entropy / information? Does the universe outside the black hole ever actually lose it? What does "ever" mean in this context?

HELP!

 

[PeterDonis]

There are actually two separate issues here, and it may be easier to see how they are resolved if you keep them separate.

The first issue is the "when" issue, which is a matter of correctly understanding what the Schwarzschild spacetime geometry is telling you. This issue has nothing to do with entropy; you can ask the central question you are asking--when does an infalling object cross the horizon?--without bringing in entropy at all.

The best way to understand this, IMO, is to look at how Schwarzschild spacetime, and the worldline of an infalling object, look in Kruskal coordinates. Take a look at Figure 9 on this page:

http://www.physics.howard.edu/students/Beth/bh_about.html

The horizon (or at least the portion of it we're interested in) is the 45-degree line going up and to the right between regions I and II of the diagram. The worldline of the infalling object crosses that line, the horizon, at a definite event, which occurs at a definite, finite "time" according to the infalling object's clock. So there is a definite "when" that the object crosses the horizon from the object's own point of view.

Now take a look at this page from Wikipedia, and in particular the diagram in yellow part way down the page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

You will notice regions I and II again, but now two sets of dotted curves have been added. These are curves of constant Schwarzschild time coordinate t (the straight lines radiating out from the center) and constant Schwarzschild time coordinate r (the hyperbolas). Notice that in region I, the lines of constant t approach the horizon but never reach it; there is no finite value of t that corresponds to the actual horizon line (the future horizon, the part we're interested in, can be thought of as t = + infinity).

This means that (speaking somewhat loosely), from the point of view of an observer who remains outside the horizon forever, there is no finite "time" at which the infalling observer crosses the horizon. However, the reason for this is not that there is any mystery or strangeness about that event (the infalling observer crossing the horizon); it is simply that the most "natural" assignment of a time coordinate to events, for the observer outside the horizon, only covers region I; it does not cover region II. From the standpoint of the spacetime as a whole, this "time" coordinate is clearly limited (for one thing, all of the lines of constant Schwarzschild t intersect each other at the center of the diagram!).

The upshot of all this is that the definition of "now" depends on how you choose to draw coordinate lines on the spacetime. A "now" can in principle be any spacelike line (on the Kruskal diagram, this is any line whose slope is always less than 45 degrees, i.e., more horizontal than vertical). But *which* spacelike line (or set of lines) you choose makes a difference: there is no "Schwarzschild now" that passes through the event where the infalling observer crosses the horizon, but there are plenty of other perfectly good "now" lines that do.

I'll follow up with discussion of the second issue, entropy, in a separate post.

 

[Naty1]

[oops..I see Peter posted as I was composing..I'll check it tomorrow]

I,too, have Susskinds book, and liked it a lot.

If you enjoyed it, try Kip Thorne's BLACK HOLES AND TIME WARPS which gives many
complementary descriptions...fills in a lot of pieces.

My current hangup seems to be about the idea of "now." If information is conserved, or if entropy doesn't decrease, you've got to be able to count it up. This seems to make sense only for a local area, because what does it mean to have something that is a function of time if time is different at different locations?

'local area' is not what you think it is either: different local observers can readily measure different effects...Unruh effect is one where an inertial observer sees virtual particles while an accelerating observer seems them as actual [physical] particles...! That can explain why a free falling obsever passes an event horizon without effect (observing virtual paticles)
while an accelerating (hovering, stationary) observer is fried by radiation [actual particles].

When (oops - what does this word mean in this context?) the container crosses the event horizon, entropy in the universe outside the black hole decreases.

There is no universal agreement on local time...both speed and gravitational potential affect the flow of time. Time reversal invariance is not a property of the laws in quantum physics,
but is in classical physics. So time is 'mysterious' to an extent,,,,Just as two distant observers will not in general agree on the passage of time, neither do they agree on who is in motion nor the length of the other's vehicle. In fact, the Einstein train example shows that different observers don't even agree on simultaneity.

If you are a glutton for punishment, try this discussion which ends up trying to understand the Einstein clock synchronization...figuring out how and if distant clocks can be set up for experimental purposes [without gravity] in special relativity.

https://www.physicsforums.com/showthread.php?t=567674&highlight=proper+coordinate+time

[There are some good insights, but use caution; many posts turn out to be contested!

Crossing the event horizon is a non-event for the container of gas, but to an outside observer - and I guess that means any place outside the event horizon, even only a micron - it takes an infinite time for the container to cross the horizon.

The 'infinite' time refers to a 'distant' observer but I haven't found out yet whether that is restricted to an infinite distance in flat spacetime. David Finkelstein is the mathematician who stumbled on the resolution.

[Susskind likely address this later in the book...not positive.]

I just found this today under BLACK HOLES in Wikipedia:

In 1958, David Finkelstein identified the Schwarzschild surface as an event horizon, "a perfect unidirectional membrane: causal influences can cross it in only one direction".[15] This did not strictly contradict Oppenheimer's results, but extended them to include the point of view of infalling observers. Finkelstein's solution extended the Schwarzschild solution for the future of observers falling into a black hole. A complete extension had already been found by Martin Kruskal, who was urged to publish it.[16]

It's allso discussed here, but is a bit 'beyond my paygrade"

C:\Users\Owner\Documents\PHYSICS\RELATIVITY black holes,Gravity\Schwarzschild Geometry.mht

Does the black hole itself ever gain the added mass / entropy / information? Does the universe outside the black hole ever actually lose it? What does "ever" mean in this context?

That IS answered in Susskind's book...Susskind was Correct. Did you finish the book?


Here is a related description from another thread...Susskinds perspective is the second..."but there's another..."

Mitchell Porter posts:
... the idea is that the interior of the black hole has a dual (holographic) description in terms of states on the horizon; a lot like AdS/CFT, with the horizon being the boundary to the interior. So when someone crosses the horizon from outside, there's a description which involves them continuing to fall inwards, until they are torn apart by tidal forces and their degrees of freedom redistributed among the black hole's degrees of freedom, all of which will later leak away via Hawking radiation; but there's another description in which, when you arrive at the horizon, your degrees of freedom get holographically smeared across it, once again mingling with all the black hole's prior degrees of freedom (also located on the horizon), which all eventually leak away as Hawking radiation

Susskind describes, I think in that book, that long strings represent more information...and when a piece breaks from the 'stretched' horizon as a result of quantum fluctuations, bits are released as Hawking radiation...

I don't want to give the complete answer as currently understood unless you finished the book.

 

[PeterDonis]

Armed with a better understanding of what "now" means, from my last post, we can now (no pun intended) look at entropy. Consider again the worldline of an infalling object, referring to the first diagram I referenced in my previous post, and suppose now that that object is carrying some amount of entropy. Let's see how things look when we answer the question of "when" this entropy is "where" (i.e., when it is in which region of spacetime) using what we learned in the last post.

From the standpoint of a "Schwarzschild observer" who remains in region I, the entropy carried by the object remains in region I as well; that observer cannot assign any finite "time" to the event at which the object crosses the horizon, leaving region I and entering region II. Notice also, however, that for this observer, the region covered by his "now lines" (the lines of constant time t) does not include the black hole (the region inside the horizon, region II) at all! So for this observer, there is no issue of whether the black hole ever "receives" the entropy carried by the infalling object; this observer's "view" of the spacetime simply does not include the hole.

From the standpoint of the infalling object, however, the entropy leaves region I and enters region II when the object does. (If the object is extended, then there is no single "when" at which the object crosses the horizon; each individual particle in the object carries its own entropy and crosses at its own instant of "time", its own "when". So we can focus on the case of a point-like object with no extension that crosses the horizon at a single instant.) That means the black hole "gains" the entropy and the region outside "loses" it. But notice that, from this point of view, a single "now" line (a line of constant "time" for the infalling observer) covers both regions, so the total entropy at any given "now" has to include both the portion inside the hole (region II) and the portion outside the hole (region I). The sum of the two remains constant, so you can indeed "count up" the entropy at any given "now" and show that it is all accounted for.

Hopefully this helps; if not, it should at least provide a good framework for further questions.