Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 4

Table of contents
- a 4
  - Solution a
- b 4
  - Solution b

a

Solution

For a real free scalar field on a globally hyperbolic spacetime, take

S = - \frac{1}{2} \int d^{4} x - g (\nabla_{a} ϕ \nabla^{a} ϕ + (m^{2} + ξ R) ϕ^{2}) .

The Klein-Gordon equation has a well-posed initial-value problem on every Cauchy hypersurface. Its complex solution space carries the conserved Klein-Gordon inner product. Choose modes with

(u_{i}, u_{j})_{K G} = δ_{ij}, (u_{i}^{*}, u_{j}^{*})_{K G} = - δ_{ij}, (u_{i}, u_{j}^{*})_{K G} = 0,

and expand

ϕ = \sum_{i} (a_{i} u_{i} + a_{i}^{†} u_{i}^{*}), [a_{i}, a_{j}^{†}] = δ_{ij} .

The canonical commutation relations produce the bosonic Fock space, and

a_{i} ∣0 ⟩ = 0

defines the chosen vacuum.

The choice of positive-frequency solutions is extra structure. In a time-dependent geometry there is no preferred time translation, so different choices mix

u_{i}

with

u_{i}^{*}

and give the vacuum ambiguity in a nonstationary spacetime. In a stationary spacetime, a timelike Killing vector field defines positive frequency by

i L_{K} u_{i} = ω_{i} u_{i}

,

ω_{i} > 0

, and hence the preferred vacuum state in a stationary spacetime.

If the spacetime is stationary in the remote past and future, define normalized in- and out-modes there. Their Bogoliubov transformation is

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}) .

The expected out-particle number in the in-vacuum is

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = j \sum ∣ β_{ij} ∣^{2} .

Thus nonzero beta coefficients measure particle creation by the intervening time dependence.

b

Solution

Classically, the laws of black-hole mechanics have the same form as the laws of thermodynamics, but this could have been only a formal analogy. Hawking radiation makes it physical: a stationary black hole emits approximately thermal radiation with

T_{H} = \frac{ℏ κ}{2 π k _{B} c},

so surface gravity is proportional to genuine temperature.

The correspondence is:

constancy of $κ$ on a stationary horizon becomes constancy of equilibrium temperature;
the First law of black-hole mechanics

d M = \frac{κ}{8 π G} d A + Ω_{H} dJ + Φ_{H} d Q

matches the first law of thermodynamics and fixes the Bekenstein-Hawking entropy

S_{B H} = \frac{k _{B} c ^{3} A}{4 G ℏ};

Hawking's area theorem becomes a second law, replaced in the quantum theory by the generalized second law for $S_{B H} + S_{outside}$ ;
the impossibility of reaching $κ = 0$ by a finite process parallels the unattainability of absolute zero.

A real Hawking temperature therefore identifies horizon area as thermodynamic entropy and turns the geometric laws into thermodynamic laws.

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