black hole, one could simply add more mass until a black hole formed, at which point the entropy would go down to , violating the GSL. Thus the entropy must have been less than to begin with. ’t Hooft argued that the inescapable implication of this is that the true space of quantum states in a finite region must be finite dimensional and associated with the two-dimensional boundary of the region rather than the volume.Thus it is not enough even if the system is like a fermion field on a lattice of finite spacing. Rather, the states in the region must be somehow determined by a finite-state system on a boundary lattice! ’t Hooft made the analogy to a hologram, and the idea was dubbed by Susskind the holographic hypothesis. From a classical viewpoint, the holographic hypothesis may correspond to a statement about the phase space of a gravitating system surrounded by a surface of area A that is not inside a black hole. It is not inconceivable that this phase space is compact with a volume that scales as the area. If something like this is true, then the holographic hypothesis could just be a straightforward consequence of quantising a gravitating system. On the other hand, it has been suggested by ‘t Hooft and Susskind that the holographic hypothesis can only be incorporated into physics with a radical change in the foundations of the subject. If so, it provides a tantalising hint as to the nature of that change. There are some suggestions that string theory might be headed in the required direction, or perhaps something very different like a cellular automaton model is correct. For the remainder of this section I will ignore the holographic hint however, and continue to discuss the problem from the point of view of local field theory. 4.2 Formational degeneracy Bekenstein’s original idea was that the entropy of a black hole is the logarithm of the number of ways it could have formed. This is closely related to the Boltzmann definition of entropy as the number of micro-states compatible with the macro-state. A4ℏG= 1AA4A4A4
Page of 1314Reportu6609679Hawking noted that a potential problem arises if one contemplates increasing the number of species of fundamental fields. There would seem to be more ways of forming the black hole, however the entropy is fixed at . Hawking’s resolution of this was that the black hole will also radiate faster because of the extra species, so that there would be less phase space per species available for forming the hole. Presuming these two effects balance each other, the puzzle would be resolved.4.3 Entanglement entropy Another proposal is that the black hole entropy is a measure of the information hidden in correlations between degrees of freedom on either side of the horizon. This entropy is sometimes called entanglement entropy.(It has also been called geometric entropy.) Summary The report can be concluded by saying that black-hole thermodynamics is more general than black holes—that cosmological event horizons also have an entropy and temperature.