Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect

TL;DR

This work develops the framework of quantum field theory in curved spacetime, emphasizing how time-dependent backgrounds and horizons induce particle creation via Bogoliubov transformations and squeezed-state structures. It connects cosmological particle production, de Sitter perturbations, and black hole Hawking radiation through a unified formalism, including the Klein-Gordon product, mode decompositions, and stress-energy renormalization. The analysis highlights the roles of adiabatic versus sudden transitions, the significance of horizon thermodynamics, and the back-reaction and information-loss debates, while also addressing the trans-Planckian question with string-theory and lattice-analogue perspectives. The work underscores how near-horizon vacuum structure yields thermal spectra and how UV completions or cuts modify entanglement and energy flux, with broad implications for early-universe cosmology and quantum gravity phenomenology.

Abstract

These notes introduce the subject of quantum field theory in curved spacetime and some of its applications and the questions they raise. Topics include particle creation in time-dependent metrics, quantum origin of primordial perturbations, Hawking effect, the trans-Planckian question, and Hawking radiation on a lattice.

Figures (5)

• Figure 1: Spacetime diagram of black hole formed by collapsing matter. The outgoing wavepacket $P$ splits into the transmitted part $T$ and reflected part $R$ when propagated backwards in time. The two surfaces $\Sigma_{f,i}$ are employed for evaluating the Klein-Gordon inner products between the wavepacket and the field operator. Although $P$, and hence $R$ and $T$, have purely positive Killing frequency, the free-fall observer crossing $T$ just outside the horizon sees both positive and negative frequency components with respect to his proper time.
• Figure 2: Spacetime sketch of phase contours of the transmitted wavepacket $T$ and its flipped version $\widetilde{T}$ on either side of the horizon (dashed line). The upper and lower signs in the exponent of the factor $\exp(\mp \pi\omega/\kappa)$ yield the positive and negative free-fall frequency extensions of $T$.
• Figure 3: Dispersion relation $\omega\delta=\pm 2\sin(k\delta/2)$ plotted vs. $k\delta$. Wavevectors differing by $2\pi/\delta$ are equivalent. Only the Brillouin zone $|k|\le\pi/\delta$ is shown.
• Figure 4: A typical wavepacket evolution on the lattice. The oscillations of the incoming wavepacket are too dense to resolve in the plots.
• Figure 5: The ancestors of a Hawking quantum and its negative energy partner. In standard relativistic field theory the ancestors are trans-Planckian and pass through the collapsing matter at the moment of horizon formation. On the lattice the ancestors ar Planckian and propagate in towards the black hole at late times.