Quantum fields in curved spacetime

Choose a smooth classical globally hyperbolic background, appropriate initial data, and a quantum state for the matter field. Global hyperbolicity supports a well-posed causal initial-value problem for familiar free fields. For a scalar field the covariant Klein–Gordon equation can include a curvature coupling. Set $\hbar=c=1$ in that equation, and define $\Box_g=g^{\mu\nu}\nabla_\mu\nabla_\nu$ with the module’s mostly-minus signature. The metric is prescribed rather than a quantum operator.

Physical predictions require more than an equation of motion. Field operators are distributions, so local products need controlled definitions and renormalization. Hadamard states have an appropriate universal short-distance singularity structure, enabling physically meaningful renormalized local observables. One cannot simply subtract an arbitrary flat-space infinity at every point and assume the resulting stress tensor respects local covariance and conservation. The rigorous theory keeps these requirements visible.

In stationary Minkowski spacetime, a preferred inertial time makes positive-frequency modes easy to select. A general time-dependent geometry may have no global timelike symmetry supplying the same choice. Two valid mode decompositions can mix positive- and negative-frequency parts through Bogoliubov coefficients. The same state may then contain particles relative to one decomposition and not another. This does not mean that every local observable is arbitrary or that detectors lack definite predictions.

An explicitly modeled detector follows a worldline and couples to field correlations in a specified state. Its response depends on that trajectory, switching procedure, and state. An eternally uniformly accelerated ideal detector in the Minkowski vacuum has a thermal response associated with the Unruh temperature. A finite experiment has transients and limitations. Likewise, particle creation in an expanding universe requires a background history and a state choice; the words “curved spacetime” alone do not specify a spectrum.

For a Schwarzschild black hole formed by collapse, a state regular for infalling observers and empty of incoming radiation from the distant past produces a late-time outgoing Hawking flux. Its characteristic temperature at infinity is $T_H=\hbar c^3/(8\pi G M k_B)$. Scattering by the exterior curvature adds frequency-dependent greybody factors, so the observed flux is not simply that of a perfect black surface at every frequency. An eternal thermal equilibrium state is a different physical setup.

The approximation treats a macroscopic black hole as slowly evolving and its local curvature as far below quantum-gravity scales. A more ambitious semiclassical equation sources classical geometry using a renormalized stress expectation value; it still does not quantize the metric. Large backreaction, stress fluctuations, and the final evaporation regime can invalidate the approximation. The calculation therefore provides a robust semiclassical prediction within its assumptions, not a completed explanation of quantum gravity or the final fate of all information.