Thermality Breakdown in Null-Shifted Rindler Wedges

Abstract: We investigate the behaviour of quantum fields in null-shifted Rindler wedges and analyse the particle spectra perceived by accelerated observers associated with these null deformations. Unlike the standard Unruh effect, our analysis compares two accelerated frames connected by a null displacement. We consider both massive scalar and Dirac fields, constructing their corresponding mode solutions in Rindler coordinates. Using normalised field expansions, we compute the Bogoliubov transformations between modes defined in the two null-shifted wedges. Our results demonstrate a fundamental breakdown of thermality: the presence of mass modifies the mode structure, rendering the characteristic exponential mixing of frequencies absent. This suggests that the massive field remains unexcited on this background, leading to a manifestly nonthermal response. These findings highlight that thermality in accelerated frames depends sensitively on the conformal symmetry of the field, which is broken by the introduction of a mass term.

• The paper shows that Bogoliubov coefficients β₂₁ vanish for massive scalar and Dirac fields, eliminating particle creation and the expected Planck factor in null-shifted wedges.
• The analysis constructs normalized Rindler modes and evaluates null-sector transformations using exact projections and a large-mass stationary-phase approximation for the difficult sector.
• The findings suggest that Unruh thermality depends on conformal invariance, while conclusions remain subject to tests with the Minkowski vacuum, finite mass-to-acceleration ratios, and higher dimensions.

Overview

This paper examines whether the thermal character of the Unruh effect survives when the standard Rindler wedge construction is modified in two ways: the observer's wedge is displaced relative to a reference wedge along a null direction, and the quantum field carries a nonzero mass. Working in $(1+1)$-dimensional Minkowski spacetime, the author constructs mode solutions for massive scalar and Dirac fields in Rindler coordinates, computes Bogoliubov transformations between modes defined on two null-shifted wedges $R_1$ and $R_2$ (with $R_2 \subset R_1 \subset M$), and evaluates the resulting particle spectra. The central claim is that thermality fails: for massive fields, the Bogoliubov coefficient $\beta_{21}(\Omega,\omega)$ vanishes identically, so no particle creation occurs between null-shifted wedges and no Planck factor $e^{-2\pi\omega/a}$ appears. The paper concludes that Unruh-type thermality is not a universal consequence of acceleration but depends on the conformal structure of the field theory, which a mass term breaks.

Geometric setup

The analysis uses conformally flat Rindler coordinates with metric $ds^2 = e^{2ax}(dt^2 - dx^2)$. Two null-shifted wedges are considered. For a shift along the $V$-axis, the Minkowski light-cone coordinates of $R_2$ satisfy $U_M = -e^{-au_2}/a$ and $R_1$0, while $R_1$1 has $R_1$2. Choosing $R_1$3 (argued to be without loss of generality since constant shifts are absorbable into the null-coordinate origin), the wedge relation becomes

$R_1$4

For shifts along the $R_1$5-axis, the complementary relation $R_1$6 holds with $R_1$7. These logarithmic relations generalize the exponential map underlying the standard Unruh derivation. The reference state is taken to be the Rindler vacuum $R_1$8; the paper acknowledges that this state is not globally regular across the horizons but argues it provides a consistent particle interpretation for observers confined to the wedge.

Massive scalar field

Solving the Klein–Gordon equation via separation of variables and the substitution $R_1$9 reduces the radial equation to a modified Bessel equation of imaginary order $R_2$0, giving modes proportional to $R_2$1 with normalization $R_2$2, fixed by the Klein–Gordon inner product and the Macdonald-function orthogonality relation. Unlike the massless case, these modes do not decompose into independent left- and right-movers; only in the near-horizon limit ($R_2$3, small argument of $R_2$4) does the field reduce to a null-mode form, where the two directional sectors can be combined into an effective operator $R_2$5.

Projecting the field onto $R_2$6-modes yields a frequency-diagonal identification $R_2$7, implying $R_2$8 and $R_2$9. The $R_2 \subset R_1 \subset M$0-mode projection is more involved: substituting the logarithmic wedge relation produces an integral over Gamma-function ratios, evaluated by saddle-point approximation (valid for $R_2 \subset R_1 \subset M$1, i.e., $R_2 \subset R_1 \subset M$2). The saddle point sits at $R_2 \subset R_1 \subset M$3, localizing the transformation near a single frequency. Matching onto the Bogoliubov decomposition again gives $R_2 \subset R_1 \subset M$4, with $R_2 \subset R_1 \subset M$5 proportional to $R_2 \subset R_1 \subset M$6. By the symmetry of the conformally flat null coordinates, the $R_2 \subset R_1 \subset M$7-axis analysis is structurally identical. The result is that no positive–negative frequency mixing occurs in either null sector, so the massive scalar spectrum perceived in $R_2 \subset R_1 \subset M$8 is nonthermal — indeed trivially unexcited.

Massive Dirac field

The Dirac analysis follows the same logic. Using a zweibein formalism and a conformal rescaling $R_2 \subset R_1 \subset M$9 that exactly cancels the spin connection, the rescaled Dirac equation separates into coupled first-order equations whose elimination reduces to a modified Bessel equation of order $\beta_{21}(\Omega,\omega)$0. Normalizability at spatial infinity selects $\beta_{21}(\Omega,\omega)$1 solutions, and the spinor norm fixes $\beta_{21}(\Omega,\omega)$2, derived in an appendix through a careful distributional evaluation involving digamma functions and convergence factors.

The Bogoliubov computation mirrors the scalar case: $\beta_{21}(\Omega,\omega)$3-mode projections give $\beta_{21}(\Omega,\omega)$4, $\beta_{21}(\Omega,\omega)$5 directly, while $\beta_{21}(\Omega,\omega)$6-mode projections require the stationary-phase approximation and yield $\beta_{21}(\Omega,\omega)$7 with $\beta_{21}(\Omega,\omega)$8 localized at the saddle frequency. The $\beta_{21}(\Omega,\omega)$9-axis results are identical. The vanishing of $e^{-2\pi\omega/a}$0 in all sectors means the massive Dirac field also exhibits no particle creation under null shifts — a manifestly nonthermal response, consistent with preservation of the canonical anticommutation relations under the diagonal transformation.

Interpretation and significance

The physical picture offered is that massless fields propagate strictly along null directions and their conformal symmetry enforces the exponential frequency mixing responsible for the Planck distribution; a mass term introduces a scale $e^{-2\pi\omega/a}$1 that breaks this symmetry, modifies the mode structure (Bessel rather than exponential modes), and renders the vacuum invariant under the null-shift deformation. Notably, the paper contrasts this with its own prior work on massless fields [Jha:2025tpg], where the analogous null-sector symmetry produced thermal particle creation — establishing that the geometric null-sector symmetry alone is insufficient for thermality. The strong claim here is that thermality in accelerated frames is contingent on conformal invariance rather than acceleration per se, which, if correct, constrains how broadly the Unruh effect can be regarded as a purely kinematic consequence of horizons.

Limitations and open questions

Several caveats bear directly on the strength of the conclusions. First, the key $e^{-2\pi\omega/a}$2-sector results rest on the stationary-phase approximation in the regime $e^{-2\pi\omega/a}$3; the exact transformation away from this high-mass limit is not established, and the numerical constant $e^{-2\pi\omega/a}$4 from the saddle evaluation is left undetermined. Second, the entire analysis is restricted to $e^{-2\pi\omega/a}$5 dimensions, where conformal structure is especially powerful; extension to higher dimensions is explicitly left open. Third, the choice of the Rindler vacuum $e^{-2\pi\omega/a}$6 as reference state — rather than the globally regular Minkowski vacuum used in the standard Unruh derivation — is a substantive assumption, since the Rindler vacuum is singular at the horizons and generally regarded as unphysical as a global state. Whether the nonthermal conclusion persists for the Minkowski vacuum is not addressed. Finally, the paper does not compute an explicit particle spectrum or response function beyond showing $e^{-2\pi\omega/a}$7 within the approximation scheme, leaving open whether subleading corrections to the saddle-point evaluation could generate weak nonthermal excitations.

Conclusion

This paper demonstrates, via explicit mode constructions and Bogoliubov analyses for massive scalar and Dirac fields, that null-shifted Rindler wedges related by logarithmic coordinate maps fail to produce thermal spectra: the $e^{-2\pi\omega/a}$8 coefficients vanish identically within the employed approximations, and the characteristic $e^{-2\pi\omega/a}$9 structure never arises. The result supports the thesis that Unruh thermality in this setting is tied to the conformal symmetry broken by mass, rather than being a generic feature of accelerated observers or horizon geometry. The robustness of this conclusion beyond $ds^2 = e^{2ax}(dt^2 - dx^2)$0 dimensions, beyond the large-mass saddle-point regime, and beyond the Rindler-vacuum reference state remains to be established.