The Holographic Universe by Leonard Susskind & James Lindesay

Author:Leonard Susskind & James Lindesay [Susskind, Leonard & Lindesay, James]
Language: eng
Format: epub
Published: 2011-02-17T05:00:00+00:00

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88

Black Holes, Information, and the String Theory Revolution VVV

Singularity

VVVVV X+ -X = (MG)2

VV

+

VVVVVVVVVVVVVVVVV

X

A

m

fro

B gathers

e

n

information

and enters

Messag

rizo

Ho

horizon

A

Information

from A

P

B

-

VVVVVVVVVVVVVVVVVVVVVVVVVVV

X

Fig. 9.4

Resolution of Xerox paradox for observers within horizon Now, in the classical theory, there are no limits on how much information can be sent in an arbitrarily small time with arbitrarily small energy.However in quantum mechanics, to send a single bit requires at least one quantum.Since that quantum must be emitted between x− = 0 and x− = M G e−ω∗, its energy must satisfy 1

E >

eω∗

(9.2.10)

M G

From equation 9.2.6 we see that this energy is exponential in the square 2 G

of the black hole mass (in Planck units) E > eM

.In other words, for

M G

observer A to be able to signal observer B before observer B hits the singularity, the energy carried by observer A must be many orders of magnitude larger than the black hole mass.It is obvious that A cannot fit into the horizon, and that the experiment cannot be done.

This example is one of many which show how the constraints of quan-October 25, 2004 15:0

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The Puzzle of Information Conservation in Black Hole Environments 89

tum mechanics combine with those of relativity to forbid violations of the complementarity principle.

9.3

Baryon Number Violation

The conservation of baryon number is the basis for the incredible stability of ordinary matter.Nevertheless, there are powerful reasons to believe that baryon number, unlike electric charge, can at best be an approximate conservation law.The obvious difference between baryon number and electric charge is that baryon number is not the source of a long range gauge field.Thus it can disappear without some flux having to suddenly change at infinity.

Consider a typical black hole of stellar mass.It is formed by the collapse of roughly 1057 nucleons.Its Schwarzschild radius is about 1 kilometer, and its temperature is 10−8 electron volts.It is far too cold to emit anything but very low energy photons and gravitons.As it radiates, its temperature increases, and at some point it is hot enough to emit massive neutrinos and anti-neutrinos, then electrons, muons, and pions.None of these carry off baryon number.It can only begin to radiate baryons when its temperature has increased to about 1 GeV.Using the connection between mass and temperature in equation 4.0.22, the mass of the black hole at this point is about 1010 kilograms.This is a tiny fraction of the original black hole mass, and even if it were to decay into nothing but protons, it could produce only about 1037 of them.Baryon number must be violated by quantum gravitational effects.

In fact, most modern theories predict baryon violation by ordinary quantum field theoretic processes.As a simplified example, let’s suppose there is a heavy scalar particle X which can mediate a transition between an elementary proton and a prositron, as well as between two positrons, as in Figure 9.5. Since the X-boson is described by a real field, it cannot carry any quantum numbers, and the transition evidently violates baryon conservation.The proton could then decay into a positron and an electron-positron pair.

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