Acceleration without photon pair creation

Abstract

Whenever an experiment can be described classically, quantum physics must predict the same outcome. Intuitively, there is nothing quantum about an accelerating observer travelling through a vacuum. It is therefore not surprising that many people are puzzled by the Unruh effect, which predicts that the observer encounters photons in a thermal state. This paper employs locality and spatial and time translational symmetries to demonstrate that the assumption of a common vacuum of the quantized electromagnetic field in all inertial and non-inertial reference frames is consistent with the principles of special relativity. A key difference between a resting and an accelerating observer is that they each experience a different zero-point energy density.

Figures (5)

• Figure 1: (a) A spacetime diagram illustrating the Doppler effect. Here, a moving observer (Bob) travels at a constant velocity $v_{\rm B}$ (green line) away from a resting observer (Alice). In addition, we assume that Alice sends light signals to Bob at intervals of $\Delta t_{\rm A}$, which he receives at constant time intervals $\Delta t_{\rm B}$. (b) A spacetime diagram illustrating the Unruh effect. Now Bob moves with a time-dependent velocity $v_{\rm B} (t_{\rm A})< c$ (green line) away from Alice. Notice that Bob's dynamics look the same in both figures during a short time interval $\Delta t_{\rm B}$ as his trajectory can be approximated by a collection of short straight lines at velocity $v_{\rm B}^{(m)}$ with $m=1,2,3,...$ .
• Figure 2: (a) Alice's spacetime diagram, with coordinate axes $x_{\rm A}$ and $t_{\rm A}$, showing Bob's time-like trajectory (green) for the case where Bob moves at constant speed $v_{\rm B}> 0$ away from her. Alice and Bob both met at the origin of their respective coordinate systems at an initial time $t_{\rm A} = t_{\rm B} = 0$. The red line marks the light-like trajectory (red) of a narrow light pulse caused by an explosion at Alice's position at a time $t_{\rm A}^{(1)}$ and arriving at Bob's location at $t_{\rm A}^{(2)}$. (b) Illustration of the same scenario in Bob spacetime diagram with coordinate axes $x_{\rm B}$ and $t_{\rm B}$ showing the trajectory of the light pulse (red) caused by an explosion at time $t_{\rm B}^{(1)}$ and reaching him at $t_{\rm B}^{(2)}$. Alice's trajectory is marked in green. Both figures can be used to derive Eq. (\ref{['sk0']}) and subsequently the transformation between the natural coordinates $\chi_{\rm A}$ and $\chi_{\rm B}$ that correspond to the same event in Eq. (\ref{['S01']}).
• Figure 3: A diagram illustrating the discretisation of Bob's trajectory (green) into short periods of inertial motion. A rapid succession of explosions by Alice sends a train of narrow light pulses to Bob at regular time intervals from her position at $x_{\rm A} = 0$ (vertical axis). Bob receives one pulse at the end of each period. As Bob is accelerating, the light beams due to the explosion reach him at different positions and times. The first number in the time superscript indicates the trigger time by Alice and the ,arrival time at Bob, and the second number indicates the explosion number.
• Figure 4: An illustration of the explosions triggered by Alice from Bob's perspective. The positions and times measured by Bob are different, and from his perspective, Alice is accelerating away from him (green trajectory). The first number in the time superscript indicates the trigger time by Alice and the arrival time at Bob, and the second number indicates the explosion number.
• Figure 5: The figure shows a spacetime of Bob's trajectory in Alice's spacetime diagram with perpendicular axes representing Alice's position $x_{\rm A}$ and time $t_{\rm A}$. The green line shows Bob's hyperbolic trajectory in Alice's reference frame. Right-propagating light pulses, shown in red, only intersect with Bob's trajectory if they have a $\chi_{\rm A}$ coordinate in the range $\chi_{\rm A} \in (-c^2/a, 0]$. Left-propagating light pulses, shown in blue, will only intersect with Bob's trajectory if $\chi_{\rm A} \geq 0$. The dashed light pulse at $\chi_{\rm A} = -a/c^2$ meets Bob at a time $t_{\rm A} \to \infty$ and therefore provides a natural horizon for the $\chi_{\rm A}$ and $\chi_{\rm B}$ coordinate systems.