By Leonard Susskind

During the last decade the physics of black holes has been revolutionized by way of advancements that grew out of Jacob Bekenstein's attention that black holes have entropy. Steven Hawking raised profound matters about the lack of info in black gap evaporation and the consistency of quantum mechanics in a global with gravity. for 2 many years those questions wondered theoretical physicists and finally resulted in a revolution within the method we expect approximately area, time, subject and knowledge. This revolution has culminated in a striking precept referred to as "The Holographic Principle", that's now a massive concentration of consciousness in gravitational learn, quantum box idea and easy particle physics. Leonard Susskind, one of many co-inventors of the Holographic precept in addition to one of many founders of String conception, develops and explains those thoughts.

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11) The eigenvalues ρj can be considered to be probabilities that the system is in the j th state. However, unlike the case of a coherent superposition of states, the relative phases between the states |j are random. There is one special case when the density matrix is indistinguishable from a pure state. This is the case in which only one eigenvalue ρj is Quantum Fields in Rindler Space 35 nonzero. 12) A quantitative measure of the departure from a pure state is provided by the Von Neumann entropy S = −T r ρ log ρ = − j ρj log ρj .

There are several inﬁnities on the Penrose diagram. Future and past time-like inﬁnities (t = ±∞) are the beginnings and ends of time-like trajectories. Space-like inﬁnity (r = ∞) is where all space-like surfaces end. In addition to these there are two other inﬁnities which are called I ± . The Schwarzschild Black Hole 8 t=+ 15 Y +=+1 + t=3 t=2 t=1 r= 8 t=0 r= 3 r=1 r=2 r=0 Y +=Y - Y - =-1 Fig. 6 8 t=- Penrose diagram for Minkowski space They are past and future light-like inﬁnity, and they represent the origin of incoming light rays and the end of outgoing light rays.

The origin of the dimensionless temperature lies in the dimensionless character of the Rindler time variable ω. Nevertheless we should be able to assign to each Fido a conventional temperature that would be recorded by a standard thermometer held at rest at the location of that Fido. We can consider a thermometer to be a localized object with a set of proper energy levels i . The levels i are the ordinary energy levels of the thermometer when it is at rest. The thermometer is assumed to be very weakly coupled to the quantum ﬁelds so that it eventually will come to thermal equilibrium with them.