Minkowski and Rindler Modes

Rindler modes have positive frequency with respect to boost time in one wedge; Minkowski modes have positive frequency with respect to inertial time across all wedges. The two notions are related by analytic continuation across the acceleration horizons. That continuation forces a mixture of right- and left-wedge modes and produces the factor $\exp(-\pi\omega/a)$ responsible for the Planck spectrum.

Required background. Special Relativity provides boosts, null coordinates, and the four Rindler wedges.

Helpful background. Flat-Spacetime Quantum Field Theory provides Klein–Gordon normalization and positive frequency. Partial Differential Equations and Green Functions supplies mode completeness. Causality and Global Structure develops horizon extensions. Quantum Fields in Curved Spacetime provides the representation and Bogoliubov framework.

Wedge modes and their normalization

In the right wedge,

$$ds^2=-(a\rho)^2d\eta^2+d\rho^2+d\mathbf x_\perp^2.$$

For a scalar of mass $m$, separated positive-boost-frequency modes have the form

$$u^R_{\omega\mathbf k_\perp} =N_{\omega\mathbf k_\perp} K_{i\omega/a}(\mu_\perp\rho) e^{i\mathbf k_\perp\cdot\mathbf x_\perp-i\omega\eta}, \qquad \mu_\perp^2=m^2+\mathbf k_\perp^2,$$

with $\omega>0$ and $\mu_\perp>0$. For the spectral measure $d\omega\,d^2\mathbf k_\perp$ and the transverse plane wave as written,

$$N_{\omega\mathbf k_\perp} =\left(\frac{\sinh(\pi\omega/a)}{4\pi^4 a}\right)^{1/2}, \qquad (u^R_{\omega\mathbf k},u^R_{\omega'\mathbf k'})_{\mathrm{KG}} =\delta(\omega-\omega')\delta^{(2)}(\mathbf k-\mathbf k').$$

The factor $1/(a\rho)$ in the Klein–Gordon surface measure combines with the Bessel orthogonality integral to give this normalization Crispino, Higuchi, and Matsas 2008, arXiv version, §II.D, Eqs. (2.85)–(2.94), pp. 9–10, PDF. The massless, zero-transverse-momentum sector is obtained separately as a chiral limit, rather than substituting zero into a singular Bessel mode. A left-wedge family $u^L$ uses future-directed left-wedge time. Relative phases are conventional, but support, time orientation, and spectral measure must be fixed before comparing coefficients.

The Bessel radial factor is not the origin of thermality. The essential step already appears in a massless chiral sector. With $U=t-x$, the right wedge has $U<0$, and a right-wedge boost mode behaves as

$$(-aU)^{i\omega/a} =e^{i(\omega/a)\ln(a\rho)}e^{-i\omega\eta}.$$

Its branch point at $U=0$ is the horizon.

Minkowski positivity as analyticity

A positive Minkowski-frequency wave packet is the boundary value of a function analytic in the appropriate lower complex-time domain for the convention $e^{-i\Omega t}$. Continuing a wedge mode around the branch point through the corresponding half-plane fixes its value on the other side. One cannot extend a single right-wedge mode by zero and preserve this analyticity.

Two globally positive-Minkowski-frequency Unruh families may be chosen schematically as

$$U^{(1)}_\omega =\frac{ e^{\pi\omega/(2a)}u^R_\omega +e^{-\pi\omega/(2a)}(u^L_\omega)^* }{\sqrt{2\sinh(\pi\omega/a)}},$$ $$U^{(2)}_\omega =\frac{ e^{\pi\omega/(2a)}u^L_\omega +e^{-\pi\omega/(2a)}(u^R_\omega)^* }{\sqrt{2\sinh(\pi\omega/a)}}.$$

Transverse momentum reversal and harmless phases depend on the detailed mode convention. The invariant content is the relative magnitude $e^{-\pi\omega/a}$ and the Klein–Gordon normalization. Crispino, Higuchi, and Matsas 2008, arXiv version, §§II.E–F, Eqs. (2.107)–(2.127), pp. 12–13, PDF derive the mode relations using the opposite metric signature; translating the inner product and time orientation is essential.

Bogoliubov mixing and the Planck factor

Writing a Minkowski-positive annihilator in terms of wedge operators gives a two-mode transformation

$$b^{(1)}_\omega =\alpha_\omega a^R_\omega -\beta_\omega a^{L\dagger}_\omega, \qquad \left|\alpha_\omega\right|^2-\left|\beta_\omega\right|^2=1,$$

with

$$\frac{|\beta_\omega|}{|\alpha_\omega|} =e^{-\pi\omega/a}.$$

Solving the normalization identity yields

$$|\beta_\omega|^2 =\frac{1}{e^{2\pi\omega/a}-1}.$$

Thus the Minkowski vacuum has a Planckian right-wedge occupation with respect to boost frequency. The dimensionless exponent is a check: both $\omega$ and $a$ refer to the chosen boost-time normalization. For an observer at $\rho_0$, converting to proper frequency and proper acceleration gives the same local temperature $a_0/(2\pi)$.

Modewise finiteness is not global unitary implementability. The continuum of modes and infinite wedge volume make the Minkowski and Fulling–Rindler representations inequivalent in the idealized limit. A finite box or wave-packet regulator can provide a two-mode squeezed-vector illustration, but removing the regulator restores the algebraic statement rather than a literal tensor product of two finite systems.

With such a regulator, the vacuum condition $b^{(1)}_\omega|0_M\rangle=0$ yields the formal pair expansion

$$|0_M\rangle_\omega \propto\sum_{n=0}^{\infty} e^{-\pi\omega n/a}|n_\omega^R\rangle|n_\omega^L\rangle.$$

Tracing the regulated illustration over the left oscillator gives geometric right-wedge weights proportional to $e^{-2\pi\omega n/a}$. This reproduces the occupation above and shows why both wedges enter. In the continuum theory, however, local wedge algebras are Type III and there is no canonical wedge density matrix; the KMS property on the algebra is the regulator-independent statement.

The thermal exponential does not depend on the radial Bessel potential or transverse momentum because it comes from horizon analyticity. Mass and transverse momentum still affect the mode functions, normalization, and which wave packets a particular detector samples. Universality of the exponential should therefore not be confused with universality of every response rate.

Checks and limitations

The Klein–Gordon identity $|\alpha|^2-|\beta|^2=1$ checks the sign and normalization. Large $\omega/a$ gives exponential suppression. Here $a$ is a coordinate normalization: rescaling it and $\omega$ together leaves $\omega/a$ unchanged. A physical low-acceleration limit instead holds the proper detector gap fixed while taking the orbit’s acceleration $a_0=1/\rho_0$ to zero. Reversing the analytic half-plane would exchange positive and negative Minkowski frequency and invert the relative exponential, a decisive failure.

These mode formulas assume the Minkowski vacuum, exact wedge geometry, eternal boost evolution, and the free field. They do not say that a finite-time detector has an exactly Planckian spectrum, nor that an accelerated source emits a universal thermal flux. Those are separate observable questions.

Exercise

Given $|\beta_\omega|/|\alpha_\omega|=e^{-\pi\omega/a}$ and the bosonic identity $|\alpha_\omega|^2-|\beta_\omega|^2=1$, derive the wedge occupation and identify the temperature with respect to boost time.

Solution

Set $q=e^{-2\pi\omega/a}$. Then $|\beta|^2=q|\alpha|^2$, while normalization gives $|\alpha|^2(1-q)=1$. Hence $|\beta|^2=q/(1-q)=1/(e^{2\pi\omega/a}-1)$. Comparing with the Bose factor $1/(e^{\beta\omega}-1)$ gives inverse boost temperature $\beta=2\pi/a$. Proper-time temperature requires the additional Killing redshift and is $1/(2\pi\rho_0)$ on the orbit $\rho_0$.

The derivation also checks the analytic continuation: choosing the opposite exponential would make the geometric weights grow with particle number and prevent normalization even in the regulated oscillator model. The decaying branch is therefore fixed by positive Minkowski frequency rather than selected afterward to obtain a thermal answer.

References

• Crispino, Luís C. B., Atsushi Higuchi, and George E. A. Matsas. “The Unruh Effect and Its Applications.” Reviews of Modern Physics 80 (2008): 787–838. DOI. Open PDF.
• Fulling, Stephen A. “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time.” Physical Review D 7 (1973): 2850–2862. DOI.