The Unruh effect and its applications

It has been $30\text{years}$ since the discovery of the Unruh effect. It has played a crucial role in our understanding that the particle content of a field theory is observer dependent. This effect is important in its own right and as a way to understand the phenomenon of particle emission from black holes and cosmological horizons. The Unruh effect is reviewed here with particular emphasis on its applications. A number of recent developments are also commented on and some controversies are discussed. Effort is also made to clarify what seem to be common misconceptions.

Figure 1

Part of Feynman’s blackboard at California Institute of Technology at the time of his death in 1988. At the right-hand side one can find accel. temp. as one of the issues to learn.

Figure 2

Histogram depicting the number of citations of 251 over the years.

Figure 3

The regions with $\mid t\mid <z$, $\mid t\mid <-z$, $t>\mid z\mid$, and $t<-\mid z\mid$, denoted RR, LR, EDK, and CDK, respectively, are the left Rindler wedge, right Rindler wedge, expanding degenerate Kasner universe, and contracting degenerate Kasner universe, respectively. The curves with arrows are integral curves of the boost Killing vector $z(\partial ∕\partial t)+t(\partial ∕\partial z)$. The direction of increasing $U=t-z$ and that of increasing $V=t+z$ are also indicated.

Figure 4

The world line of a uniformly accelerated detector moving along the $z$ axis in the Minkowski spacetime covered with Cartesian coordinates.

Figure 5

The world line of a uniformly accelerated detector moving along the $z$ axis in Minkowski spacetime covered with Rindler coordinates.

Figure 6

The probability density $d{\mathcal{P}}_{\omega }∕d{\rho }_d$ for different $\omega ∕m$ ratios, where $Mm=1∕4$. Note that the smaller the $\omega ∕m$ ratio, the closer to the horizon (on average) the particle lies, where the “gravitational potential” is lower.

Figure 7

$⟨{\rho }_d⟩$ is shown to be smaller than ${\rho }_p^{\max }\equiv 4M\omega ∕m$ in the high-frequency regime $\omega >{\left(4M\right)}^{-1}$ (i.e., at the right of the vertical broken line) as expected. We have let $mM=10$ but this agreement is verified for any $mM$.

Figure 8

The proton mean proper lifetime as a function of the proper acceleration $a$ for $G_F={10}^{-12}$. (This value of $G_F$ approximately reproduces the inertial neutron proper lifetime of $900\mathrm{s}$ in this model.) Note that ${\tau }_p\to +\infty$ for inertial protons $(a\to 0)$.

Figure 9

Branching ratios ${\mathrm{BR}}_{\left(\mathit{ii}\right)}$, ${\mathrm{BR}}_{\left(\mathit{iii}\right)}$, and ${\mathrm{BR}}_{\left(\mathit{iv}\right)}$. Process (iv) dominates over processes (ii) and (iii) for small accelerations, while processes (iv) and (iii) dominate over process (iv) for high accelerations.

Figure 10

The frequency-dependent temperature ${\beta }^{-1}∕a$ [see Eq. (4.3)] as a function of $\Delta E∕a$ and compared with the corresponding one given by the Unruh effect: $1∕\left(2\pi \right)$. For $\Delta E⪢a$, one obtains ${\beta }^{-1}∕a=1∕\left(2\sqrt{3}\right)$.

Figure 11

The polarizations $P_1$ (solid line) and $P_2$ (dashed line) given in Eqs. (4.4, 4.5), respectively, as functions of $g$. The two curves show substantial differences although some similarities can be also seen.