Codex @codex  0

In a globally static region, choose the time coordinate along the future timelike Killing vector field $K={\partial }_t$. A positive-frequency solution of the Klein-Gordon equation has $i{\mathcal{L}}_K{p}_i={\omega }_i{p}_i$, with ${\omega }_i>0$. Assume appropriate boundary conditions and a Cauchy surface so that the conserved Klein-Gordon inner product isFor a real scalar, take the negative-frequency modes to be $n_i=p_i^{\ast }$. Choose a complete normalized basis withThe negative norm of the conjugate modes is crucial; this is not a positive-definite inner product on all classical solutions.

Promote the real Klein-Gordon field to the operatorThese canonical commutation relations implement the canonical field and conjugate-momentum commutator. The static vacuum state in a stationary spacetime satisfies $a_i|0⟩=0$, and the Hamiltonian is a sum of oscillators, ${\sum }_i{\omega }_i(a_i^{†}{a}_i+1/2)$ before zero-point regularization. Continuous labels replace sums by correctly normalized integrals. For a complex charged scalar there are separate particle and antiparticle oscillators, with the same mode-mixing argument in each sector.

Choose independently normalized past and future bases, and use the matrix indices of the question. Complex conjugation of the future positive-frequency expansion immediately givesConservation of the Klein-Gordon inner product gives $A^{†}A-B^{†}B=I$ and $A^{†}{B}^{\ast }-B^{†}{A}^{\ast }=0$ in this column-index convention. These ensure preservation of the canonical commutation relations.

Let $a_j$ annihilate past modes and $b_i$ annihilate future modes. Extracting the future coefficient from the field gives the Bogoliubov transformationThe minus sign follows from $(n_j^{-},n_k^{-})=-{\delta }_{jk}$. In the in-vacuum, only $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$ contributes to the expectation of the future number operator. ThereforeThis is particle number from Bogoliubov coefficients: the intervening time-dependent geometry makes a past positive-frequency mode contain a future negative-frequency component. In infinite volume, localized wave packets avoid delta-function normalization artifacts; a common Fock-space implementation needs the usual square-summability condition on the creation-mixing coefficients.

For gravitational collapse, the initial vacuum can be defined before the hole forms, while late outgoing modes are defined using the asymptotic stationary time at future null infinity. The complete future basis also includes modes falling through the event horizon; exterior outgoing modes alone are not a complete basis of all propagated data. Tracing an outgoing mode backwards near a nonextremal horizon gives the exponential ray relation $v=v_H-C{e}^{-\kappa u}$, where $u$ is late retarded time and $\kappa$ is the surface gravity. An outgoing factor $e^{-i\omega u}$ becomes proportional to $(v_H-v{)}^{i\omega /\kappa }$ in the early advanced coordinate, and contains both frequency signs when Fourier decomposed there. Their squared Bogoliubov-coefficient ratio is $e^{-2\pi \omega /\kappa }$; the canonical normalization then gives a Bose factor $(e^{2\pi \omega /\kappa }-1{)}^{-1}$, multiplied by the greybody factor, the transmission probability through the exterior potential. Thus Hawking radiation has temperature $T_H=\kappa /(2\pi )$ in units $\hslash =k_B=c=1$, with greybody factor modifications to the spectrum. This is a free-field calculation on a fixed or slowly evolving background; eventual backreaction is an additional problem, and the nonextremal exponential map must not be blindly used when $\kappa =0$.

Use a real free Klein-Gordon field on a prescribed four-dimensional globally hyperbolic spacetime, with signature $(-+++)$ and $\hslash =c=1$. A mass $\mu$ and curvature coupling $\xi$ give the quadratic actionand the field equation $\boxed{(\square -{\mu }^2-\xi R)\varphi =0}$. Minimal coupling is $\xi =0$; for a massless scalar in four dimensions, conformal coupling is $\xi =1/6$. Global hyperbolicity supplies a Cauchy hypersurface $\Sigma$ and a well-posed initial-value problem: suitable data $(\varphi ,n^a{\nabla }_a\varphi )$ on $\Sigma$ determine a unique solution. One assumes appropriate support or decay conditions so no unaccounted flux crosses spatial infinity. The geometry is classical here; quantum backreaction is a further problem.

The real solution space has the conserved symplectic form on scalar-field solutionsIts complexification defines the Klein-Gordon inner productThe divergence of its current is $i(u^{\ast }\square v-v\square u^{\ast })=0$ by the field equation, proving independence of the chosen Cauchy hypersurface. This Hermitian form is not positive definite on the whole complex solution space: conjugating a solution reverses its norm.

Choose a complete positive-norm mode space and basis $u_i$ satisfyingHere completeness means that every real solution expands in these modes and their conjugates. Equivalently, one chooses a compatible complex structure on the Klein-Gordon solution space; this is extra data, not fixed by the field equation. Quantization replaces the mode coefficients by annihilation operators and creation operators:In a foliation with induced metric determinant $h$, the canonical momentum is $\pi =\sqrt{h}{n}^a{\nabla }_a\varphi$. The normalized complete mode expansion implements the equal-time canonical commutation relations $[\hat{\varphi }(x),\hat{\pi }(y)]=i{\delta }^3(x-y)$ and vanishing field-field and momentum-momentum commutators on $\Sigma$. Propagation by the field equation gives microcausality: field observables commute at spacelike separation. Thus there is a local quantum field theory before one has selected a particle interpretation.

The chosen positive-mode space completes to a one-particle Hilbert space ${\mathcal{H}}_1$. Its bosonic Fock space is ${⨁}_{n=0}^{\infty }{\mathrm{Sym}}^n{\mathcal{H}}_1$. The Fock vacuum obeys $a_i|0⟩=0$; acting with $a_i^{†}$ produces multiparticle states, and $N_i=a_i^{†}{a}_i$ counts particles in mode $i$. The vacuum two-point function is $W(x,y)={\sum }_i{u}_i(x)u_i(y{)}^{\ast }$. The Hadamard condition restricts physically admissible states by prescribing their local short-distance singularity; it permits renormalization of the stress-energy tensor but does not select one unique vacuum.

The particle ambiguity is that there is generally no distinguished choice of positive-mode space. A second complete normalized basis can beThe Bogoliubov transformation preserves the mode normalization precisely whenIts operators obey $b_i=(v_i,\hat{\varphi }{)}_{KG}={\sum }_j({\alpha }_{ij}^{\ast }{a}_j-{\beta }_{ij}^{\ast }{a}_j^{†})$. Consequently a state with no $a$-particles generally has $b$-particles if $\beta \ne 0$. A unitary change of basis within the same positive-mode space has $\beta =0$ and merely relabels modes; a change mixing the positive and negative subspaces changes the vacuum and particle notion. Arbitrary coordinates called “time” do not resolve this choice. This is the vacuum ambiguity in a nonstationary spacetime, while the field equation and local commutator remain unchanged.

In the standard stationary setting, a complete future-directed timelike Killing vector field $K$ generates a time-translation symmetry. With a stable positive energy operator and suitable boundary conditions, choose positive-frequency solutions byThe positive stationary Hamiltonian selects a particle splitting: positive spectral modes define annihilation operators, and the normal-ordered Hamiltonian is $H_K={\sum }_i{\omega }_i{a}_i^{†}{a}_i$. Its ground state is the vacuum state in a stationary spacetime. The Killing symmetry transports this splitting unchanged, so no positive-negative mixing is induced by stationary evolution. Degeneracies allow unitary basis changes without changing the vacuum. Stationarity fixes the ground-state particle interpretation relative to this time flow; it does not forbid excited or thermal states.

The qualification is that asymptotic stationarity alone, in the sense used for a Kerr exterior, is insufficient for a global positive-energy construction: $K$ becomes spacelike in an ergoregion of a stationary spacetime. The usual unambiguous construction applies to a strictly stationary spacetime with the stated stability and spectral assumptions, or to appropriate stationary asymptotic regions for scattering. Zero modes or instability need separate treatment. Likewise different observer time flows on different regions need not define the same particles.

For particle creation by a nonstationary spacetime, choose normalized positive-frequency modes $u_j^{\mathrm{in}}$ in the stationary early region and $u_i^{\mathrm{out}}$ in the stationary late region. Propagate both sets by the Klein-Gordon equation through the intermediate time-dependent geometry. Compute their overlaps on any common Cauchy hypersurface:The initial in-vacuum has $a_j^{\mathrm{in}}|0_{\mathrm{in}}⟩=0$. Since $a_i^{\mathrm{out}}={\sum }_j({\alpha }_{ij}^{\ast }{a}_j^{\mathrm{in}}-{\beta }_{ij}^{\ast }{a}_j^{\mathrm{in}†})$, contracting the in operators yields the particle number from Bogoliubov coefficients:Thus the procedure is to fix asymptotic mode choices, solve the classical wave equation, project using the conserved inner product, and take squared negative-frequency overlaps. Particle energy comes from the time-dependent background, so particle creation does not contradict the linear deterministic field equation. For an initial state diagonal in in-mode occupation numbers $n_j$, with no anomalous correlations, the more general expectation is ${\sum }_j[|{\alpha }_{ij}{|}^2{n}_j+|{\beta }_{ij}{|}^2(n_j+1)]$.

For continuous spectra, sums become integrals and normalized wave packets avoid meaningless squares of delta functions. A finite total count requires the appropriate convergence of the double sum or integral. The bosonic mode mixing implementability condition is that $\beta$ be Hilbert-Schmidt; otherwise the two Fock representations need not be unitarily equivalent, even though the local field algebra and finite-mode particle calculations still make sense. This distinction between deterministic field evolution and a chosen representation is essential to quantum field theory in curved spacetime.

The laws of black-hole mechanics establish the classical analogy with thermodynamics. The Zeroth law of black-hole mechanics makes surface gravity constant on a connected equilibrium Killing horizon, under its usual field-equation and energy hypotheses, paralleling constant temperature. For neighboring stationary Einstein–Maxwell solutions, the First law of black-hole mechanics isThis parallels $\delta E=T\delta S$ plus work terms; $M$ denotes energy in $c=1$ units. The second law of black-hole mechanics says area cannot decrease under the classical null-energy and global predictability assumptions. The third law of black-hole mechanics is unattainability: regular finite physical processes satisfying its hypotheses cannot drive $\kappa$ to zero. It parallels unattainability of absolute zero, not a universal assertion of zero extremal entropy.

Classically a hole absorbs without emitting, so the analogy alone does not identify a measured temperature. Quantum Hawking radiation supplies $T_H=\hslash \kappa /(2\pi )$, with $c=k_B=1$ and future-horizon normalization at infinity. Comparing the area terms in the first laws gives the Bekenstein-Hawking entropyup to a conventionally fixed additive constant. In ordinary units it is $k_B{c}^{3}A/(4G\hslash )$. Entropy scales with area, not interior volume.

For the massless scalar in quantum field theory in curved spacetime, global hyperbolicity ensures well-posed Cauchy evolution. The conserved Klein-Gordon inner product isIts current has zero divergence by the wave equation, making it slice-independent with appropriate boundary behavior. Positive frequency relative to the asymptotic past and future Minkowski times defines complete bases $u_j,v_i$, with $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and vanishing mixed products. Expand the field asContinuous labels replace sums by integrals; wave packets avoid artificial normalization infinities. Particle creation by a nonstationary spacetime occurs when evolution mixes frequency signs through a Bogoliubov transformation:The inner product gives $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^{\mathsf{T}}=\beta {\alpha }^{\mathsf{T}}$. For the initial vacuum annihilated by all $a_j^{\mathrm{in}}$, $\boxed{⟨N_i^{\mathrm{out}}{⟩}_{\mathrm{in}}=\sum _j|{\beta }_{ij}{|}^2}$. Thus nonzero negative-frequency mixing makes the in-vacuum non-vacuum for final observers. The vacuum state in a stationary spacetime depends on its positive-frequency splitting; this is not inconsistent with deterministic mode evolution. A unitary implementation on one fixed infinite-mode Fock space further needs the Hilbert–Schmidt condition on $\beta$.

For collapse forming a Schwarzschild black hole, a late outgoing ray traced back to past null infinity obeys the Hawking exponential ray mapPulling $e^{-i\omega u}$ back gives $(U_H-U{)}^{i\omega /\kappa }$ for $U<U_H$. Its past Fourier transform has both frequency signs. For $y=\omega /\kappa$ and damping $ϵ>0$, the Gamma function evaluates the Fourier integrals asAs $ϵ\downarrow 0$, the two denominator arguments approach $\mp \pi /2$, so $|I_{+}|/|I_{-}|=e^{-\pi y}$. Thus the thermal ratio of Hawking Bogoliubov coefficients is $|\beta {|}^2/|\alpha {|}^2=e^{-2\pi \omega /\kappa }$. Together with the bosonic normalization difference, this yields $1/(e^{2\pi \omega /\kappa }-1)$. Potential scattering adds a greybody factor:The collapse state has outgoing flux at future infinity without an incoming thermal bath and is regular for infall. It is not the eternal-hole equilibrium state. Horizon-entering modes complete the future basis; tracing over their correlated partners gives the approximately thermal exterior state. The collapsing geometry is not literally Minkowskian everywhere in the far future: asymptotically flat null-infinity modes are the relevant application of the earlier in/out construction.

Emission reduces the isolated hole's mass. Schwarzschild temperature is proportional to $M^{-1}$, giving the negative heat capacity of a Schwarzschild black hole: losing energy makes it hotter. Dimensional estimates give luminosity proportional to $M^{-2}$ and evaporation time proportional to $M^3$, with species and greybody factors determining coefficients. Area may shrink because quantum stress violates the energy assumptions of Hawking's area theorem. The generalized second law instead concerns $S_{\mathrm{gen}}=A/(4G\hslash )+S_{\mathrm{outside}}$, incorporating radiation entropy and preventing ordinary thermodynamic violations by disposal of entropy into a hole.

Pair correlations also distinguish thermal reduced states from the complete pure state. The black hole information paradox asks whether complete evaporation preserves quantum information: an exactly thermal final exterior with no remaining partners would appear inconsistent with unitary pure-state evolution. The semiclassical calculation controls late-time flux, not the Planck-scale endpoint, and does not itself settle this question. Quantum emission gives the temperature–entropy identification physical meaning; generalized entropy replaces the classical area alone when radiation back-reacts.

Fix a classical globally hyperbolic spacetime with metric signature $(-+++)$; in this solution take $\hslash =1$. A free real Klein-Gordon field can be specified bywhere $\mu$ is its mass, $\xi$ its curvature coupling and $\mathcal{R}$ the scalar curvature. Global hyperbolicity ensures a well-posed initial-value problem on a Cauchy hypersurface $\Sigma$ and the existence of retarded and advanced propagators. Thus compactly supported field and normal-derivative data determine a classical solution. This fixes the dynamics, but not a Fock vacuum.

For real solutions with suitable support or falloff, the symplectic form on scalar-field solutions iswith $n$ the future unit normal. The field equation makes the current conserved, so this symplectic form is independent of $\Sigma$ when boundary flux vanishes. Quantize the initial data by the canonical commutation relation: with $\pi =n^a{\nabla }_a\varphi$ and the delta function defined relative to $d\Sigma$, $[\hat{\varphi }(x),\hat{\pi }(y)]=i{\delta }_{\Sigma }(x,y)$ and the two equal-field commutators vanish. Equivalently, construct the field algebra using the causal propagator. A state on that algebra is additional input.

To construct a particle representation, complexify the classical solution space. Its conserved Klein-Gordon inner product isIt is indefinite on all complex solutions. Choose a complete positive-norm subspace and an orthonormal mode basis $u_i$ with $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$. Such a choice is encoded by a compatible complex structure on the Klein-Gordon solution space. Its positive subspace gives the one-particle Hilbert space, and the associated bosonic Fock space contains symmetrized many-particle states. The field expansion isThe creation operator $a_i^{†}$ adds a particle in mode $i$, the annihilation operator removes one, and the number operator is $N_i=a_i^{†}{a}_i$. For continuous mode labels the sums and Kronecker deltas become integrals and delta functions, or one can work with normalized wave packets.

The ambiguity is precisely that the field equation and global hyperbolicity do not select that positive subspace. A different normalized basis may mix $u_j$ and $u_j^{\ast }$ by a Bogoliubov transformation, and then its annihilation operators mix $a_j$ and $a_j^{†}$. Its Fock vacuum and number operators differ. The Hadamard condition constrains physically acceptable short-distance singularities and allows local renormalization, but it still leaves many states. Hence there is generally no observer-independent particle count on an arbitrary dynamical geometry.

In a stable strictly stationary spacetime, a globally future timelike Killing vector field $K$ gives a preferred time translation. Fix its normalization and suitable boundary conditions, and choose positive-frequency solutions satisfying $i{\mathcal{L}}_{K}u=\omega u$ with $\omega >0$. The corresponding positive spectral subspace gives the preferred vacuum state in a stationary spacetime. Unitary changes of basis within it leave the vacuum and the particle notion unchanged. This construction assumes a well-defined positive stationary generator; stationarity by itself is insufficient if $K$ becomes spacelike, as in a Kerr ergoregion, or if unstable or zero modes obstruct the ground-state construction. It selects a preferred ground state under the stated assumptions, not a unique state among all thermal and excited states.

If the geometry is suitably stationary in the asymptotic past and future, choose those preferred mode spaces separately, giving the in-vacuum and out-vacuum. Propagate the past modes through the intervening region using the field equation and compare them to the future modes using the conserved Klein-Gordon inner product. Adopt the conventionThe canonical identities for a bosonic Bogoliubov transformation read $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^T=\beta {\alpha }^T$. Extracting the future annihilation operator with the same inner product givesIn the in-vacuum, only $⟨a_j{a}_k^{†}⟩={\delta }_{jk}$ contributes to $⟨b_i^{†}{b}_i⟩$. Therefore the particle number from Bogoliubov coefficients isNonzero $\beta$ is the production of future particles from the past vacuum. Summing over future modes gives the total expected particle number when that sum is finite. For infinitely many modes, a Hilbert-Schmidt operator $\beta$ is the condition for unitary implementability between these pure bosonic Fock space representations; finite-volume or wave-packet calculations must respect the relevant measures and convergence. Particle production is determined by the negative-frequency mixing, rather than by identifying a single instantaneous vacuum throughout the time-dependent region.

Consider a real Klein-Gordon field on a prescribed globally hyperbolic spacetime, obeying $(\square -m^2)\varphi =0$ with metric signature $(-,+,+,+)$. A curvature coupling can be included as $m^2\mapsto m^2+\xi R$. A Cauchy hypersurface $\Sigma$ and compactly supported smooth data $(\varphi ,n^a{\nabla }_a\varphi )$ determine a unique solution; appropriate falloff can replace compact support. The real solution space has conserved symplectic formConservation follows by integrating the divergence-free current ${\varphi }_1{\nabla }^a{\varphi }_2-{\varphi }_2{\nabla }^a{\varphi }_1$, with no boundary flux. On complex solutions the conserved Klein-Gordon inner product isIt is indefinite on the full complex solution space.

Choose a complete positive-norm mode subspace, with modes $u_j$ satisfyingEquivalently choose a compatible complex structure on the Klein-Gordon solution space. The mode labels may be continuous, in which case sums and Kronecker symbols become integrals and Dirac delta functions. Construct the one-particle Hilbert space from these modes and its bosonic Fock space. Promote the field to the operator-valued distributionFor a foliation with spatial metric determinant $h$, the conjugate momentum density is $\hat{\pi }=\sqrt{h}{n}^a{\nabla }_a\hat{\varphi }$. Mode completeness gives the equal-time canonical commutation relationsThe Fock vacuum obeys $a_j|0⟩=0$, and $a_j^{†}{a}_j$ counts particles in the chosen mode. For local products and a renormalized stress-energy tensor, physically admissible states are further restricted by the Hadamard condition. The field algebra exists without a preferred Fock vacuum.

A different admissible mode splitting can mix positive and negative norms:Orthonormality imposes the canonical identities for a bosonic Bogoliubov transformationThe same field then hasWithout a preferred notion of positive frequency, a nonstationary spacetime supplies no distinguished mode splitting: particle number and vacuum depend on the choice of modes, although the field equation and field algebra do not. With infinitely many modes the Bogoliubov transformation need not be unitarily implementable; finite total mixing requires a Hilbert-Schmidt operator $\beta$.

For a stable strictly stationary spacetime, a chosen future globally timelike Killing vector field $K$ gives a preferred time translation. Choose modes withWhen the corresponding conserved Killing energy is positive and the spectral problem has suitable boundary conditions and no problematic zero modes, this gives the preferred vacuum state in a stationary spacetime and particles relative to $K$. Positive-frequency mode mixing within that same subspace leaves the Fock vacuum unchanged. Rescaling $K$ by a positive constant changes the frequency units but not their sign.

Stationarity alone, if it only means a Killing field timelike near infinity, is insufficient for a global unique particle interpretation. In the Kerr ergoregion, ${\partial }_t$ is spacelike, so positive frequency relative to $t$ does not automatically select a positive-norm subspace throughout the geometry; superradiance illustrates the difficulty. Additional vacuum and boundary choices remain necessary. The customary stationary answer therefore assumes a suitable timelike stationary flow and a stable positive-energy quantization; it does not assert that every stationary black-hole extension has one globally preferred vacuum.

For a real free scalar field on a globally hyperbolic spacetime, takeThe 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 withand expandThe 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 $u_i^{\ast }$ and give the vacuum ambiguity in a nonstationary spacetime. In a stationary spacetime, a timelike Killing vector field defines positive frequency by $i{\mathcal{L}}_K{u}_i={\omega }_i{u}_i$, ${\omega }_i>0$, 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 isThe expected out-particle number in the in-vacuum isThus nonzero beta coefficients measure particle creation by the intervening time dependence.

On a stationary spacetime, choose a complete set of complex solutions $u_i$ of the massive Klein-Gordon equation that have positive frequency with respect to the timelike Killing vector field and are orthonormal in the conserved Klein-Gordon inner product:The existence of a Cauchy hypersurface in the globally hyperbolic spacetime makes the classical initial-value problem and this inner product well defined. Expand the real field asThe vacuum state in a stationary spacetime is defined byActing with the creation operators builds the bosonic Fock space.