10.2. Temperature of horizons
There exists a natural definition of QFT in the original (D + 1)-dimensional space; in particular, we can define a vacuum state for the quantum field on the Z0 = 0 surface, which coincides with the t = 0 surface. By restricting the field modes (or the field configurations in the Schrodinger picture) to depend only on the coordinates in , we will obtain a quantum field theory in in the sense that these modes will satisfy the relevant field equation defined in . In general, this is a complicated problem and it is not easy to have a choice of modes in which will lead to a natural set of modes in . We can, however, take advantage of the arguments given in the last section - that all the interesting physics arises from the (Z0, Z1) plane and the other transverse dimensions are irrelevant near the horizon. In particular, solutions to the wave equation in which depends only on the coordinates Z0 and Z1 will satisfy the wave equation in and will depend only on (t, r). Such modes will define a natural s-wave QFT in . The positive frequency modes of the above kind (varying as exp(- i Z0) with > 0.) will be a specific superposition of negative (varying as ei t) and positive (varying as e-i t) frequency modes in leading to a temperature T = (g / 2) in the 4-dimensional subspace on one side of the horizon. There are several ways of proving this result, all of which depend essentially on the property that under the transformation t t ± (i / g) the two coordinates Z0 and Z1 reverses sign.
Consider a positive frequency mode of the form F(Z0, Z1) exp[- i Z0 + iPZ1] with > 0. These set of modes can be used to expand the quantum field thereby defining the creation and annihilation operators A, A:
The vacuum state defined by A| vac > = 0 corresponds to a globally time symmetric state which will be interpreted as a no particle state by observers using Z0 as the time coordinate. Let us now consider the same mode which can be described in terms of the (t, r) coordinates. Being a scalar, this mode can be expressed in the 4-dimensional sector in the form F(t, r) = F [Z0(t, r), Z1(t, r)]. The Fourier transform of F(t, r) with respect to t will be:
Thus a positive frequency mode in the higher dimension can only be expressed as an integral over with ranging over both positive and negative values. However, using the fact that t t - (i / g) leads to Z0 - Z1, it is easy to show that
This allows us to write the inverse relation to (167) as
The term with K* represents the contribution of negative frequency modes in the the 4-D spacetime to the pure positive frequency mode in the embedding spacetime. A field mode of the embedding spacetime containing creation and annihilation operators (A, A) can now be represented in terms of the creation and annihilation operators (a, a) appropriate to the (t, r) coordinates as
where N is a normalization constant. Identifying a = N(A + e- / g A) and using the conditions [a, a] = 1, [A, A] = 1 etc., we get N = [1 - exp(- 2 / g)]-1/2. It follows that the number of a -particles in the vacuum defined by A| vac > = 0 is given by
This is a Planckian spectrum with temperature T = g / 2. The key role in the derivation is played by equation (168) which, in turn, arises from the analytic properties of the spacetime under Euclidean continuation.