Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VII, Chapter 5

QFT in Curved Spacetime

Quantum fields on curved backgrounds: Hawking radiation and the thermodynamics of horizons

Why Study QFT in Curved Spacetime?

In curved spacetime, quantum fields interact with gravity through the background metric gμν. This is not full quantum gravity (which remains unsolved), but rather quantum fields on classical curved backgrounds.

This framework reveals profound connections between gravity, thermodynamics, and quantum mechanics: black holes radiate like blackbodies, accelerating observers see a thermal bath, and spacetime horizons have entropy!

5.1 Scalar Field in Curved Spacetime

The action for a scalar field in curved spacetime is:

$$S = \int d^4x \sqrt{-g} \left[-\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - \frac{1}{2}m^2\phi^2 - \xi R\phi^2\right]$$

where g = det(gμν) is the metric determinant, R is the Ricci scalar, and ξ is thecoupling to curvature.

The Klein-Gordon equation becomes:

$$\left(\Box - m^2 - \xi R\right)\phi = 0$$

where □ = g-1/2∂μ(√-g gμν∂ν) is thecovariant d'Alembertian.

Curvature Coupling

  • Minimal coupling: ξ = 0 (no direct coupling to R)
  • Conformal coupling: ξ = 1/6 (scale invariant in massless limit)
  • General coupling: ξ arbitrary

The choice affects particle creation in expanding universes!

5.2 The Particle Concept Problem

In curved spacetime, there is no unique definition of particles!

The Problem

In flat spacetime, we decompose the field into positive/negative frequency modes using time translation symmetry. In curved spacetime:

  • No global timelike Killing vector → no preferred time coordinate
  • Different observers define different "positive frequency" modes
  • What one observer calls "vacuum," another sees as containing particles!

This is not a problem—it's a feature! It leads to particle creation by gravitational fields.

5.3 Bogoliubov Transformations

Different observers use different mode expansions:

\begin{align*} \phi &= \sum_i (a_i u_i + a_i^\dagger u_i^*) \quad \text{(observer A)} \\ \phi &= \sum_j (b_j v_j + b_j^\dagger v_j^*) \quad \text{(observer B)} \end{align*}

The two sets of modes are related by a Bogoliubov transformation:

\begin{align*} b_j &= \sum_i (\alpha_{ji} a_i + \beta_{ji} a_i^\dagger) \\ b_j^\dagger &= \sum_i (\alpha_{ji}^* a_i^\dagger + \beta_{ji}^* a_i) \end{align*}

If β ≠ 0, the two vacua are different! Observer A's vacuum |0⟩A contains particles as seen by observer B:

$$\langle 0_A | b_j^\dagger b_j | 0_A \rangle = \sum_i |\beta_{ji}|^2 \neq 0$$

5.4 The Unruh Effect

The Unruh effect (1976) shows that an accelerating observerin flat spacetime Minkowski vacuum sees a thermal bath of particles!

Setup

  • Inertial observer (Alice): sees Minkowski vacuum |0M⟩, no particles
  • Uniformly accelerating observer (Bob): acceleration a in x-direction
  • Bob's trajectory: x² - t² = 1/a² (hyperbola in spacetime)
  • Bob defines particles using his proper time τ

Computing the Bogoliubov coefficients, Bob finds the Minkowski vacuum contains a thermal distribution of particles at the Unruh temperature:

$$\boxed{T_{\text{Unruh}} = \frac{a}{2\pi k_B}}$$

in natural units (ℏ = c = 1). In SI units:

$$T_{\text{Unruh}} = \frac{\hbar a}{2\pi c k_B} \approx 4 \times 10^{-23} \text{ K} \times \left(\frac{a}{1 \text{ m/s}^2}\right)$$

Physical Interpretation

The accelerating observer has a Rindler horizon behind them—a boundary beyond which they cannot receive signals. The horizon has associated thermodynamics, just like a black hole!

Unruh temperature is tiny for realistic accelerations: even a = 10⁶ m/s² gives T ~ 4×10-17 K. Extraordinarily difficult to measure!

5.5 Schwarzschild Black Hole

The Schwarzschild metric for a non-rotating black hole of mass M:

$$ds^2 = -\left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1}dr^2 + r^2d\Omega^2$$

The event horizon is at the Schwarzschild radius:

$$r_s = 2GM = \frac{2GM}{c^2} \approx 3 \text{ km} \times \left(\frac{M}{M_\odot}\right)$$

Near the horizon, the metric looks like Rindler spacetime (accelerated observer). This suggests the horizon should have a temperature!

5.6 Hawking Radiation

Stephen Hawking's groundbreaking 1974 discovery: black holes radiate!

Hawking's Calculation

Consider a black hole forming from gravitational collapse:

  1. Define vacuum state in distant past (before collapse): |0in⟩
  2. Follow quantum field modes through collapse
  3. Modes near horizon get redshifted → Bogoliubov transformation
  4. Outgoing modes at late times: |0in⟩ ≠ |0out⟩
  5. Distant observer sees thermal radiation!

The black hole emits radiation with a thermal spectrum at theHawking temperature:

$$\boxed{T_H = \frac{\hbar c^3}{8\pi G M k_B} = \frac{1}{8\pi GM}}$$

in natural units (ℏ = c = kB = 1).

$$T_H \approx 6 \times 10^{-8} \text{ K} \times \left(\frac{M_\odot}{M}\right)$$

For a solar-mass black hole: TH ~ 60 nanokelvin—far below the cosmic microwave background (2.7 K), so stellar black holes absorb more than they emit!

5.7 Physical Picture of Hawking Radiation

Heuristic Explanation

Near the horizon, quantum fluctuations create virtual particle-antiparticle pairs:

  • Normally, the pair annihilates quickly (Heisenberg uncertainty: ΔE·Δt ~ ℏ)
  • Near horizon: one particle can fall in, the other escapes
  • The escaping particle is real (observable radiation)
  • The infalling particle has negative energy relative to infinity
  • Result: black hole loses mass, outgoing radiation!

Note: This is a heuristic picture. The rigorous derivation uses Bogoliubov transformations between early-time and late-time vacuum states.

5.8 Bekenstein-Hawking Entropy

Hawking radiation implies black holes have entropy! Using thermodynamics (dE = TdS):

$$\boxed{S_{BH} = \frac{k_B A}{4\ell_P^2} = \frac{k_B c^3 A}{4G\hbar}}$$

where A = 4πrs² = 16πG²M² is the horizon area and ℓP = √(Gℏ/c³) is the Planck length.

$$S_{BH} = \frac{A}{4} \quad \text{(Planck units)}$$

Why This is Profound

  • Black hole entropy is huge: S ~ M² (not extensive!)
  • Proportional to area, not volume → holographic principle
  • Entropy = (area)/(4ℓP²) suggests one degree of freedom per 4 Planck areas
  • Points to underlying quantum structure of spacetime
  • Key insight for quantum gravity (string theory, loop quantum gravity)

5.9 Black Hole Evaporation

Hawking radiation carries away energy, so the black hole evaporates:

$$\frac{dM}{dt} = -\frac{\hbar c^4}{15360 \pi G^2 M^2} \propto -\frac{1}{M^2}$$

Smaller black holes evaporate faster! The evaporation timescale is:

$$t_{\text{evap}} \sim \frac{G^2 M^3}{\hbar c^4} \approx 10^{67} \text{ years} \times \left(\frac{M}{M_\odot}\right)^3$$

Implications

  • Stellar black holes: evaporation time ≫ age of universe (1010 years)
  • Primordial black holes (M ~ 1011 kg): could be evaporating now!
  • Final stages: M → 0, T → ∞ → explosive endpoint?
  • Information paradox: what happens to information that fell in?

5.10 The Black Hole Information Paradox

Hawking radiation appears to be thermal (no information). But quantum mechanics is unitary—information cannot be destroyed!

The Paradox

Suppose you throw a quantum state |ψ⟩ into a black hole:

  • Information about |ψ⟩ appears lost to the exterior
  • Black hole eventually evaporates via Hawking radiation
  • Radiation is thermal → carries no information about |ψ⟩
  • Pure state |ψ⟩ evolved to mixed state → unitarity violated!

Proposed Resolutions

1. Information encoded in radiation:

Subtle correlations in Hawking radiation carry information (Page curve, AdS/CFT calculations)

2. Remnants:

Black hole leaves behind Planck-scale remnant containing all information

3. Firewalls:

Horizon is not smooth but has high-energy "firewall" (violates equivalence principle)

4. Holography:

Information never truly enters—encoded on horizon (AdS/CFT correspondence)

The paradox remains one of the deepest problems in theoretical physics, pointing to gaps in our understanding of quantum gravity.

5.11 Particle Creation in Cosmology

In an expanding universe (FRW metric), quantum fields can create particles even without black holes!

$$ds^2 = -dt^2 + a(t)^2 d\mathbf{x}^2$$

where a(t) is the scale factor. During inflation (exponential expansion), vacuum fluctuations get stretched to macroscopic scales, seeding cosmic structure!

Key Takeaways

  • QFT in curved spacetime: quantum fields on classical gravitational backgrounds
  • Particle definition is observer-dependent → Bogoliubov transformations
  • Unruh effect: acceleration a creates thermal bath at T = a/2π
  • Hawking radiation: black holes emit thermal radiation at TH = 1/(8πGM)
  • Bekenstein-Hawking entropy: SBH = A/(4ℓP²)
  • Black holes evaporate: tevap ~ M³
  • Information paradox: deep puzzle about unitarity vs. thermality
  • Cosmological particle creation: inflation amplifies vacuum fluctuations
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