Modern Physics · Advanced Topics

QFT in Curved Spacetime

Quantum field theory on a curved background — without quantizing gravity itself — reveals profound effects: particle creation by expanding universes, Hawking radiation from black holes, and the Unruh effect for accelerating observers. These semiclassical results sit at the intersection of quantum mechanics and general relativity.

Prerequisites

General relativity (Ch. GR) \cdot QFT (Ch. QFT) \cdot Tensor calculus (Ch. TC) \cdot Statistical mecha

nics (Ch. SM)

Learning Goals

Explain how Bogoliubov transformations between mode decompositions lead to particle creation in curved spacetime.
$Derive the Hawking temperature T_{H} = \hbarc^{3}/(8\piGMk_B) and compute it for black holes of vario$ us masses.
Apply the $Stefan-Boltzmann law to Hawking radiation to find the black hole evaporation timescale t_{e}$
Explain the Unruh effect and compute the Unruh temperature for a given proper acceleration.
Describe how quantum fluctuations during inflation seed the CMB power spectrum via the Gibbons-Hawking temperature.

CST.1 Quantum Fields on Curved Backgrounds

Replace the flat Minkowski metric η_μν with a general curved metric g_μν(x). The action for a real scalar field becomes:

S = -\frac{1}{2} \int d^{4}x \sqrt(-g) [g^{\mu\nu} \partial_\mu\phi \partial_\nu\phi + (m^{2} + \xiR) \phi^{2}]

(CST.1)

where ξ is the non-minimal coupling to the Ricci scalar R. Minimal coupling: ξ = 0. Conformal coupling (massless): ξ = 1/6 in 4D. The Klein-Gordon equation becomes: (□ − m² − ξR) φ = 0 where □ = (1/√−g) ∂_μ(√−g g^(μν) ∂_ν).

Bogoliubov transformation: in curved spacetime, there is no unique notion of a vacuum. Two observers use different mode decompositions (u_k) and (ū_k): φ = Σ_k (a_k u_k + a†_k u*_k) = Σ_k (b_k ū_k + b†_k ū*_k). The Bogoliubov coefficients α_kk', β_kk' relate a_k to b_k and b†_k: a_k = Σ_(k') (α_(kk') b_(k') + β*_(kk') b†_(k')). The number of particles in mode k of the (ū) vacuum: ⟨0_u|N̂_k|0_u⟩ = Σ_(k') |β_(kk')|² ≠ 0.

CST.2 Hawking Radiation

Hawking (1974): a Schwarzschild black hole emits thermal radiation at temperature:

T_{H} = \hbarc^{3}/(8\piG M k_{B}) = \hbar\kappa/(2\pik_B c) \qquad (Hawking temperature, \kappa = surface gravity)

(CST.2)

Derivation sketch: in the Schwarzschild geometry, modes near the future horizon experience extreme blueshift tracing back to the past horizon. An ingoing vacuum mode in the far past (Unruh/Hartle-Hawking state) appears as an outgoing thermal state at infinity with Planck spectrum T = T_H. The derivation uses the Bogoliubov transformation between early-time (Minkowski-like) and late-time (Schwarzschild) modes — the β coefficients are non-zero, giving a thermal spectrum.

Theorem CST.1 — Bekenstein-Hawking Entropy

A black hole with area A has thermodynamic entropy: S_{BH} = k_{B} A/(4 ℓ_Pl^{2}) = k_{B} c^{3} A/(4 G

\hbar) Equivalently: S_{BH} = k_{B} \times (area in Planck units)/4. For a solar-mass black hole: S_{BH}

\approx 10^{77} k_{B} — vastly larger than the entropy of the original star (~10^{57} k_{B}). The first la

of black hole mechanics: dM = T_{H} dS_BH + \Omega_H dJ + \Phi_H dQ (rotation and charge included

The f

our laws of black hole mechanics map exactly onto the four laws of thermodynamics, with T_

S_{BH} playing the roles of temperature and entropy

Example CST.1 — Black Hole Evaporation Timescale

$A black hole radiates as a blackbody at T_{H} \approx \hbarc^{3}/(8\piGMk_B). Compute how long it takes a b$ $lack hole of initial mass M_{0} to evaporate completely$ .

Luminosity:

Stefan-Boltzmann: L = \sigma A T_{H}^{4} where A = 4\pir_s^{2} = 16\piG^{2}M^{2}/c^{4} and \sigma = \pi^{2}k_{B}^{4}/(60\hbar^{3}c^{2}). Subs

tituting T_{H}: L = (\hbarc^{6})/(15360\pi G^{2}M^{2}) — decreases as M^{-2}. Note: as M decreases, T increase

s, L increases \to runaway (black hole bomb

Mass loss rate:

dM/dt = -L/c^{2} = -\hbarc^{4}/(15360\pi G^{2}M^{2}). Rearrange: M^{2} dM = -\hbarc^{4}/(15360\pi G^{2}) dt. Integrate: M^{3}(

t) = M_{0}^{3} - 3\hbarc^{4}/(15360\pi G^{2}) \times t

Evaporation time:

Set M(t_{ev}) = 0: t_{ev} = M_{0}^{3} \times (15360\pi G^{2})/(3\hbarc^{4}) = 5120\pi G^{2}M_{0}^{3}/(\hbarc^{4}). Numerically: t_{ev} \approx

5120\pi G^{2}/(\hbarc^{4}) \times M_{0}^{3}. For M_{0} = M_\odot = 2\times10^{30} kg: t_{ev} \approx 6.6\times10^{74} s \approx 2.1\times10^{67} years — vastl

longer than the age of the universe (1.4\times10^{10} yr

Primordial BH:

For t_{ev} = 13.8\times10^{9} yr \approx 4.35\times10^{17} s: M_{0} = (\hbarc^{4} \times t_{ev}/(5120\pi G^{2}))^{1/3} \approx 2.6\times10^{11} kg \approx m

ass of a large asteroid. Primordial BHs with M < 2.6\times10^{11} kg have already evaporated. M ~

10^{12} kg: currently evaporating, producing gamma-ray bursts (searched for but not detected

— constrains primordial BH abundance).

CST.3 Unruh Effect

An accelerating observer (Rindler observer with acceleration a) sees the Minkowski vacuum as a thermal bath at the Unruh temperature:

T_{U} = \hbara/(2\pik_B c) \qquad (Unruh temperature)

(CST.3)

Numerically: T_U = 1 K requires a = 2πk_Bc/ℏ ≈ 2.5×10²⁰ m/s² — 10²⁰ times Earth's gravity. The Unruh effect has not been directly measured (requires enormous accelerations), but it is closely related to Hawking radiation via a coordinate transformation.

Both the Hawking and Unruh temperatures have the same mathematical structure: T = ℏκ/(2πk_B) where κ is the relevant acceleration (surface gravity or proper acceleration). The KMS condition: a thermal state satisfies the Kubo-Martin-Schwinger condition on Green's functions, a universal characterization of thermal equilibrium in QFT.

CST.4 Particle Creation in Cosmology

An expanding universe with scale factor a(t) creates particles. The Friedmann-Robertson-Walker (FRW) metric: ds² = −dt² + a²(t)(dx² + dy² + dz²). The conformal time η: dη = dt/a(t); the metric becomes ds² = a²(η)(−dη² + dx²).

Cosmological particle creation: in de Sitter space (inflationary epoch, H = const), the Bogoliubov coefficients between in and out modes give thermal distribution at the Gibbons-Hawking temperature T_GH = ℏH/(2πk_B) — the de Sitter horizon radiates like a black hole. During inflation, quantum fluctuations of the inflaton field (δφ ~ H/(2π)) freeze out when k = aH — this seeds the CMB temperature anisotropies observed today!

Schwinger effect: a strong electric field E creates electron-positron pairs at rate Γ ∝ exp(−πm²c³/(eEℏ)) — the QED vacuum is unstable above E_Sch = m²c³/(eℏ) ≈ 1.3×10¹⁸ V/m. Analogous to Hawking radiation via the Bogoliubov mechanism (exponential suppression ≡ tunneling).

Definition CST.1 — Common Traps

Particle number can be observer-dependent: curved spacetime changes the meaning of vacuum.
Hawking radiation is quantum field theory on curved geometry: it is not classical radiation from the horizon surface.
Backreaction is hard: treating spacetime as fixed ignores energy carried by quantum fields.
Horizons are causal boundaries: coordinate singularities must be separated from physical singularities.

Exercises — CST.1–CST.4 QFT in Curved Spacetime

Calculate the Ha

wking temperature for (a) a solar-mass black hole, (b) a primordial black hole of mass 10^{1}

the CMB temperature (2.7 K).

Straightforward

Explain physically why a uniformly accelerating observer perceives the Minkowski vacuum a

s a thermal state. What is the Unruh temperature for an electron at the Schwinger field E_

m/s²

Intermediate

3.State the black hole information paradox. What is the Page curve, and how does the island formula (replica wormholes) restore unitarity?

Intermediate

4.Derive the nearly scale-invariant power spe

ctrum of scalar fluctuations during inflation. How does the Unruh/de Sitter temperature T_

temperature anisotropies?

Challenging

Key Takeaways

$No unique vacuum in curved spacetime. Bogoliubov transformation: a_{k} = \sum(\alpha b + \beta* b^\dagger). |\beta|$ ² = particle creation.
$Hawking temperature: T_{H} = \hbarc^{3}/(8\piGMk_B). BH radiates thermally. 1 M_\odot BH: T_{H} = 6\times10^{-8} K$ (unobservable).
$BH entropy: S = A/(4ℓ_Pl^{2}). First law: dM = T_{H} dS + \Omega dJ. Evaporation time ~ M_{0}^{3}$ .
$Unruh effect: accelerating observer sees T_{U} = \hbara/(2\pik_Bc). Same mechanism as Hawking via$ Rindler horizon.
$Inflation: de Sitter Gibbons-Hawking T = H/(2\pi). Quantum fluctuations \delta\phi ~ H/2\pi freeze out$ $\to CMB anisotropies$
Information paradox: island formula / replica wormholes restore Page curve and unitarity via gravitational saddles.