By Leonard Susskind

Over the past decade the physics of black holes has been revolutionized by means of advancements that grew out of Jacob Bekenstein’s awareness that black holes have entropy. Steven Hawking raised profound matters in regards to the lack of details in black gap evaporation and the consistency of quantum mechanics in an international with gravity. for 2 many years those questions questioned theoretical physicists and finally ended in a revolution within the manner we predict approximately area, time, subject and data. This revolution has culminated in a outstanding precept known as "The Holographic Principle", that's now an incredible concentration of consciousness in gravitational learn, quantum box thought and basic particle physics. Leonard Susskind, one of many co-inventors of the Holographic precept in addition to one of many founders of String thought, develops and explains those innovations.

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The integral over the ﬁrst wedge is deﬁned by constaining the ﬁelds at θ = 0 and θ = δθ. This deﬁnes a transfer matrix G in the Hilbert space of the ﬁeld conﬁguration χR . The matrix is recognized to be G = (1 − δθ HR ) . 27) In other words, the path integral deﬁning Ψ is computed as a transition matrix element between initial state χR and ﬁnal state χL . The inﬁnitesimal generator which pushes θ surfaces forward is just the Rindler Hamiltonian. Now we are prepared to compute the density matrix ρR .

To compute the total entropy we begin by replacing the inﬁnite transverse X, Y plane by a ﬁnite torus with periodic boundary conditions. This has the eﬀect of discretizing the values of k. Thus kX = 2nX π B where B is the size of the torus. 15) Entropy of the Free Quantum Field in Rindler Space 47 The entropy stored in the ﬁeld χk can be estimated from the entropy density of a 1+1 dimensional massless free boson at temperature T . 16) where T is the temperature. 18, the integral must be cut oﬀ when k > 1 .

The events in the outer region can send light rays to I + and time-like trajectories to t = ∞. The horizon, as we have seen, is a global concept whose location depends on all future events. It is composed of a family of light rays or null geodesics, passing through each space-time point on the horizon. 12. Notice that null geodesics are vertical after the shell crosses the horizon and essentially at 45o prior to that crossing. These light rays are called the generators of the horizon. Singularity forms XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX The Schwarzschild Black Hole 21 Horizon Shell crosses Horizon Shell of radially moving light-rays Horizon Forms Fig.