A drunk falling into a black hole

[Adel Makram]

According to the holographic principle, It is proposed that the entire information that is kept in a 3D volume of the black hole can be represented in a 2D surface of its horizon. What if we have a drunk falling into a black hole and moves in a random walk in his inertial frame. According to the math of the random walking, the probability that the drunk goes back to the origin from where he starts his motion is 1 in 2D but only 0.34 in 3D when time goes too large. So given the holographic principle of the black hole, where does the difference in information that is resulted from the difference in probabilities of return to the origin go?

 

[Hornbein]

According to the holographic principle, It is proposed that the entire information that is kept in a 3D volume of the black hole can be represented in a 2D surface of its horizon. What if we have a drunk living in a black hole and moves in a random walk. According to the math of the random walking, the probability that the drunk goes back to the origin from where he starts his motion is 1 in 2D but only 0.34 in 3D when time goes too large. So where does the difference in information that is resulted from the difference in probabilities of return to the origin go?

? The information didn't go anywhere.

 

[Adel Makram]

? The information didn't go anywhere.

According to the information theory and Shannon entropy;
$$H=-\sum P(x) log P(x)$$
So if p(x) is different in two situations, the entropy will be also different which means that the black hole hologram may not represent the information in its 3D volume.

 

[Hornbein]

According to the information theory and Shannon entropy;
$$H=-\sum P(x) log P(x)$$
So if p(x) is different in two situations, the entropy will be also different which means that the black hole hologram may not represent the information in its 3D volume.

Sorry, I reacted too quickly. I don't understand how the random walk goes from 2D to 3D.

 

[Chalnoth]

According to the holographic principle, It is proposed that the entire information that is kept in a 3D volume of the black hole can be represented in a 2D surface of its horizon. What if we have a drunk falling into a black hole and moves in a random walk in his inertial frame. According to the math of the random walking, the probability that the drunk goes back to the origin from where he starts his motion is 1 in 2D but only 0.34 in 3D when time goes too large. So given the holographic principle of the black hole, where does the difference in information that is resulted from the difference in probabilities of return to the origin go?

A simple random walk assumes flat space-time. A black hole is highly curved. This modifies the probabilities dramatically.

 

[CalcNerd]

Also, the walk can no longer be random in the vicinity of the BH horizon. All local space is being swallowed up by the BH ie all roads lead to ROME (err BH singularity). There is NO random path once you get too close.
.
At the very least you may have some angular displacement, but that would not be something you would have any random control over, your path would be determined by an initial velocity and distance at the time where your energy of motion would be dwarfed by the holes gravitational field.

 

[Adel Makram]

A simple random walk assumes flat space-time. A black hole is highly curved. This modifies the probabilities dramatically.

What I can see is that a curved 2D space makes it even more easier for the random walker to go back to the origin. Imagine, in extreme case, when the walker starts his first movement, he takes a geodesic which is equal to the equator of the sphere. He will then return to the origin after the first step. So a curved 2D space give the walker the advantage to go back to the origin more frequently than a flat 2D.

 

[Adel Makram]

Also, the walk can no longer be random in the vicinity of the BH horizon. All local space is being swallowed up by the BH ie all roads lead to ROME (err BH singularity). There is NO random path once you get too close.
.
At the very least you may have some angular displacement, but that would not be something you would have any random control over, your path would be determined by an initial velocity and distance at the time where your energy of motion would be dwarfed by the holes gravitational field.

According to Susskind Leonard, the information of a free falling system into a black hole is equally represented on its horizon as seen by an external observer. Because the free falling system is an inertial system, the free falling observer can freely move in a random way and the math of random walking should be applied. Consequently, the total information of the system should be invariant under any representation.

 

[Chalnoth]

What I can see is that a curved 2D space makes it even more easier for the random walker to go back to the origin. Imagine, in extreme case, when the walker starts his first movement, he takes a geodesic which is equal to the equator of the sphere. He will then return to the origin after the first step. So a curved 2D space give the walker the advantage to go back to the origin more frequently than a flat 2D.

That's too simplistic. That's a very simplistic kind if curvature: a constant amount of curvature at every point in space. What's more, the curvature itself has nothing to do with the increased probability: the increased probability simply comes from the fact that this space is finite. You can get an identical increase with a flat toroidal geometry of the same total area.

The black hole complicated matters because the curvature is highly non-uniform.

You could in principle design a spacecraft with thrusters that fire in a random fashion so that the spacecraft moves in a random walk, but its behavior anywhere near a massive object like a black hole will get very complicated (unless it falls in, then it just rapidly falls to the center, until our known physics no longer applies).

 

[Chalnoth]

According to Susskind Leonard, the information of a free falling system into a black hole is equally represented on its horizon as seen by an external observer. Because the free falling system is an inertial system, the free falling observer can freely move in a random way and the math of random walking should be applied. Consequently, the total information of the system should be invariant under any representation.

Once an object enters a black hole, it can't do much of anything. It can in principle* move a little bit in a direction perpendicular to the radius, but it's drawn to the center so quickly that that doesn't really matter.

* In practice it would have a hard time doing even that much because any macroscopic object passing the even horizon gets pulled apart.

 

[Adel Makram]

That's too simplistic. That's a very simplistic kind if curvature: a constant amount of curvature at every point in space. What's more, the curvature itself has nothing to do with the increased probability: the increased probability simply comes from the fact that this space is finite. You can get an identical increase with a flat toroidal geometry of the same total area.

The curved 2D space will not increase the probability of returning to the origin, it only increases the frequency of doing so because in both cases the probability of return to the origin is 1 but only 0.34 in 3D.

 

[jartsa]

1: Hawking radiation does not contain any information about events that happen in a capsule traveling below event horizon. (Information can not travel from below event horizon to above event horizon)

2: Hawking radiation does contain all information that ever was in the event horizon.

From 1 and 2 we can conclude: Event horizon never contains any information about events that happen in a capsule traveling below event horizon.

 

[Adel Makram]

1: Hawking radiation does not contain any information about events that happen in a capsule traveling below event horizon. (Information can not travel from below event horizon to above event horizon)

2: Hawking radiation does contain all information that ever was in the event horizon.

From 1 and 2 we can conclude: Event horizon never contains any information about events that happen in a capsule traveling below event horizon.

The total information about random walking (including steps and direction in 3D) that happens inside a falling capsule, is also encoded fully on the horizon which is a 2D curved space. This is by the holographic principle. So in principle, nothing mentioned about communication by any sort of radiation or waves.

 

[Adel Makram]

What I don`t understand also, is whether encoding information into a space of a lower dimension mandates that the governing laws of physics are running with that space of lower dimension or it would just work like storing 3D information of a video movie into a 2D CD.

 

[jartsa]

The event horizon is exactly the same at time t and one hour later, assuming no disturbances from above.

So no 2D or any other kind of random walking is happening on the event horizon. Problem solved?

 

[Chalnoth]

The curved 2D space will not increase the probability of returning to the origin, it only increases the frequency of doing so because in both cases the probability of return to the origin is 1 but only 0.34 in 3D.

I see this as quibbling over semantics. The amount of time spent at the origin is increased if the space has a smaller volume.

The real difficulty here is that a "random walk" is a highly non-physical process. It involves random displacements at uniform time intervals. But the world doesn't work that way: an object changes its position by accelerating, and then it can only do so a limited number of times as its energy changes. As I mentioned above, you might be able to sort of fake this with an object that has thrusters that fire in a very peculiar random pattern (e.g. fire its thrusters in a random direction at a specific strength for 0.5s, then in the opposite direction at the same strength for 0.5s, repeat every second, but even in this case there will be a constant velocity offset to its motion). The specific movements that result will become extremely complicated when approaching a massive object, and will cease to look much like a random walk at all. Furthermore, as the fuel is expended, the object will lose mass, which complicates the whole system further.

As for entering the black hole, I don't think the random walk is even remotely useful to understand what's going on. A better way to get at what you seem to be trying to say is this:

Imagine that you have a spaceship that falls past the event horizon. After doing this, the spaceship accelerates itself by firing its thrusters. What happens to the information that it has fired its thrusters?

Provided quantum gravity is unitary, the information about whether or not the spaceship would fire its thrusters was encoded on the horizon of the black hole the moment it passed the horizon.

 

[PAllen]

Provided quantum gravity is unitary, the information about whether or not the spaceship would fire its thrusters was encoded on the horizon of the black hole the moment it passed the horizon.

Is that the only solution? I thought in, e.g. the fuzzball model, that such information is preserved inside the horizon, but not lost; and that it comes out again, in principle, as the BH evaporates. In this model, there is no singularity to cause worry about true information loss.

 

[Chalnoth]

Is that the only solution? I thought in, e.g. the fuzzball model, that such information is preserved inside the horizon, but not lost; and that it comes out again, in principle, as the BH evaporates. In this model, there is no singularity to cause worry about true information loss.

My guess is that that's mostly a matter of semantics. I'm pretty sure the information on the horizon can only escape the black hole through Hawking radiation anyway.

 

[PeterDonis]

It is proposed that the entire information that is kept in a 3D volume of the black hole can be represented in a 2D surface of its horizon.

No, that's not what the holographic principle says. It says that the entire information in the 4-D spacetime volume inside the horizon can be represented in the 3-D spacetime surface of its horizon. There is no 2-D spacetime surface involved.

According to the math of the random walking, the probability that the drunk goes back to the origin from where he starts his motion is 1 in 2D but only 0.34 in 3D when time goes too large.

But time cannot "go too large" in this case, because the drunk will reach the singularity at the center of the BH in a finite time, and his worldline ends there. So taking the limit as his time increases without bound is not physically relevant.