By Seahra S.
We study the quantum box conception of scalar box in non-Minkowski spacetimes. We first boost a version of a uniformly accelerating particle detector and show that it'll notice a thermal spectrum of debris while the sector is in an "empty" nation (according to inertial observers). We then strengthen a formalism for touching on box theories in numerous coordinate platforms (Bogolubov transformations),and use it on examine comoving observers in Minkowski and Rindler spacetimes. Rindler observers are discovered to determine a scorching bathtub of debris within the Minkowski vacuum, which confirms the particle detector end result. The temperature is located to be proportional to the correct acceleration of comoving Rindler observers. this can be generalized to second black gap spacetimes, the place the Minkowski body is said to Kruskal coordinates and the Rindler body is said to traditional (t; r) coordinates. We ensure that once the sector is within the Kruskal (Hartle-Hawking) vacuum, traditional observers will finish that the black gap acts as a blackbody of temperature ·=2pi*kB (kB is Boltzmann's constant). We study this lead to the context of static particle detectors and thermal Green's services derived from the 4D Euclidean continuation of the Schwarzschild manifold. eventually, we givea semi-qualitative 2nd account of the emission of scalar debris from a ball of subject collapsing right into a black gap (the Hawking effect).
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Extra resources for Beyond flat-space quantum field theory
The U and V coordinates are given by U = τ − (r − R0 ), (150) V = τ + (r − R0 ). (151) We find the relation between t and τ by equating the induced metric on the surface of the ball as viewed from the inside with the induced metric as viewed from the outside. From the outside, the induced metric is ds2Σ = C dt2 − R˙ 2 2 dτ , C (152) where R˙ = dR/dτ and we evaluate C at R(τ ). From the inside ds2Σ = A(1 − R˙ 2 ) dτ 2 , (153) where A is evaluated at τ and R(τ ). Setting these expressions equal to each other, we get dt = AC 1 − R˙ 2 + R˙ 2 .
The spacetime we imagine is depicted in figure 6. The matter distribution is assumed to be confined to a ball r < R(τ ) where R(τ ) is the world line of the boundary Σ. Outside the ball we assume a line element of the form 1 dr2 = C(r) du dv, (144) ds2+ = C(r) dt2 − C(r) where the precise form of C(r) is not really important, but is in actuality given by the Schwarzschild factor C(r) = 1 − 2M/r. e. the exterior metric has an event horizon that may occur inside or outside the ball. The u and v coordinates are given by u = t − (r∗ − R0∗ ), ∗ v = t + (r − R0∗ ).
Path-integral derivation of black-hole radiance. Phys. Rev. D, 13:2188. 1976.