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WKB-like approach to the Unruh temperature for arbitrary acceleration

Paul M. Alsing

Abstract

In this work we study the Unruh temperature as arising from tunneling through a barrier for an observer in flat Minkowski spacetime with arbitrary acceleration $a(t)$. For the defining case of constant acceleration $a(t) = a_0$, the Unruh temperature (W. Unruh, Phys. Rev. D 14, 870 (1976)) is given by $k_b\,T_U =\tfrac{\hbar\,a_0}{2\,π\,c}$. Extending the work of de Gill et al. (A. de Gill, D. Singleton, V. Akhmedova and T. Pilling, Am. J. Phys. 78, 685 (2010)) we generalize the gravitational WKB approach to derive the Unruh temperature for arbitrary acceleration. We show that the often employed Schwarzschild-like form of the flat metric is not appropriate for the WKB calculation with an arbitrary $a(t)$, and instead derive a generalized Unruh temperature for the generalized Rindler metric where $a_0\to a(t)$. We derive a generalization of the Rindler coordinates appropriate for arbitrary $a(t)$, and stress the importance of the role of the integrated acceleration $χ(t) = \int^t dt'\,a(t')$, which can also act as a temporal coordinate. We explore several non-trivial examples of $a(t)$ and their generalized Unruh temperatures. We additionally develop an approximation to the Unruh temperature for small deviations away from constant acceleration by the standard approach of considering the negative frequency content of a purely positive frequency plane wave of an inertial observer, as measured by the co-moving arbitrarily accelerated observer. Lastly, we develop and explicit coordinate transformation between the arbitrarily accelerated observer and conformal coordinates, where the plane wave structure of the solutions of the wave equation is readily transparent, and analogous to the form for the inertial observer.

WKB-like approach to the Unruh temperature for arbitrary acceleration

Abstract

In this work we study the Unruh temperature as arising from tunneling through a barrier for an observer in flat Minkowski spacetime with arbitrary acceleration . For the defining case of constant acceleration , the Unruh temperature (W. Unruh, Phys. Rev. D 14, 870 (1976)) is given by . Extending the work of de Gill et al. (A. de Gill, D. Singleton, V. Akhmedova and T. Pilling, Am. J. Phys. 78, 685 (2010)) we generalize the gravitational WKB approach to derive the Unruh temperature for arbitrary acceleration. We show that the often employed Schwarzschild-like form of the flat metric is not appropriate for the WKB calculation with an arbitrary , and instead derive a generalized Unruh temperature for the generalized Rindler metric where . We derive a generalization of the Rindler coordinates appropriate for arbitrary , and stress the importance of the role of the integrated acceleration , which can also act as a temporal coordinate. We explore several non-trivial examples of and their generalized Unruh temperatures. We additionally develop an approximation to the Unruh temperature for small deviations away from constant acceleration by the standard approach of considering the negative frequency content of a purely positive frequency plane wave of an inertial observer, as measured by the co-moving arbitrarily accelerated observer. Lastly, we develop and explicit coordinate transformation between the arbitrarily accelerated observer and conformal coordinates, where the plane wave structure of the solutions of the wave equation is readily transparent, and analogous to the form for the inertial observer.
Paper Structure (17 sections, 48 equations, 3 figures)

This paper contains 17 sections, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Rindler metric for arbitrary acceleration
  3. Coordinates for an observer in flat spacetime with arbitrary acceleration $\mathbf{a(t)}$
  4. Coordinate transformation between the inertial and arbitrary acceleration frame: $\mathbf{(T(t,x), X(t,x))}$
  5. Metric $\mathbf{ds_{R,a(t)}^2}$ via flat spacetime Riemann tensor consideration
  6. WKB-type tunneling approach to the Unruh Temperature
  7. WKB tunneling rate for the case of constant acceleration $\mathbf{a_0}$
  8. WKB tunneling rate for the case of arbitrary acceleration $\mathbf{a(t)}$
  9. Examples of Unruh temperature for non-uniform acceleration
  10. $\mathbf{a(t) = \frac{a_0}{\boldsymbol{\cosh}^2(a_0\,t)} }$
  11. $\mathbf{a(t) =a_0\,\boldsymbol{\tanh}(a_0\,t) }$
  12. Negative frequency content of an inertial plane wave
  13. Constant acceleration
  14. $\mathbf{a(t) = a_0\,\boldsymbol{\tanh}(a_0\,t)}$
  15. Arbitrary accleration
  16. ...and 2 more sections

Figures (3)

  • Figure 1: (left) Schematic of surfaces of constant action $S(x)$, (middle) integral for $S_0$, and (right) the contour paths around the pole $x=-1/a_0$ for ingoing and outgoing integration paths.
  • Figure 2: $a(t) = \frac{a_0}{\cosh^2(a_0\,t) }$. (left): (black) $(a_0\,X(t), a_0\,T(t))$ orbit, (black, dashed) $T=\pm X$ asymptotes, (right): (blue) Unruh temperature $T_U(y)$ (Eq.(\ref{['TU:at:line2']})), as a function of $y = (2/\pi)\,\chi(t) = (2/\pi)\,\int dt\, a(t)$ (Eq.(\ref{['Ex:at:inv:coshsqrd:line1']})), for $t\in\{-\infty, \infty\}$. (red) Fractional denominator $\frac{1}{\frac{1}{2}\textrm{Im}[{\mathcal{F}}(y) +1]}$, and (black, dashed) acceleration $a(y)$ as functions of $y\in[-2/\pi, 2/\pi]$ for $\chi = \tanh(a_0\,t)\in[-1,1]$.
  • Figure 3: $a(t) = a_0\,\tanh(a_0\,t)$: (left) (black) $(2\,a_0\,X(t), 2\,a_0\,T(t))$ orbit (for $a_0=1$), (black, dashed) $T=\pm X$ asymptotes, (right) (blue) Unruh temperature $T_U(y)$ (Eq.(\ref{['TU:at:line2']})) , as a function of $y = (2/\pi)\,\chi(t) = (2/\pi)\,\int dt\, a(t)$ (Eq.(\ref{['Ex:at:inv:coshsqrd:line1']})), for $t\in\{-\infty, \infty\}$. (red) Fractional denominator $\frac{1}{\frac{1}{2}\textrm{Im}[{\mathcal{F}}(y) +1]}$, and (black, dashed) acceleration $a(y)$ as functions of $y\in[-10,10]\subset [-\infty, \infty]$ for $\chi \in[-\infty,\infty]$.