Radiation from an inertial mirror horizon

We compute the rapidity $\eta(v)$, by $2\eta(v)\equiv-\ln f^{\prime}(v)$ where the prime is a derivative with respect to the argument, plugging in Eq. (5),

With rapidity, we may easily compute the velocity, $V\equiv\tanh\eta$, plugging in Eq. (7),

and the proper acceleration which follows from $\alpha(v)\equiv e^{\eta(v)}\eta^{\prime}(v)$, using Eq. (7) again, so that

The magnitude of the velocity, Eq. (8), along with the proper acceleration, Eq. (9) are plotted in Figure 3. The speed at the start of time, $v\to-\infty$ is immediately seen to be $|V|=0$, while for $v\to 0$, one recedes at the speed of light, $|V|\to 1$, (or $V\to-1$, the negative sign means the mirror is traveling to the left). The non-monotonic acceleration of Eq. (9) is a different story however. Eq. (9) is both asymptotic inertial, with $\alpha=0$ at the start, $v\to-\infty$ and at the horizon $v\to 0^{-}$. The maximum acceleration is $|\alpha_{\textrm{max}}|=4/(9M)$, which occurs at $v=-M$.

The quantum stress tensor, $\langle T_{\mu\nu}\rangle$, reveals the energy flux, $F\equiv\langle T_{00}\rangle$, emitted by the moving mirror. The quantum stress tensor is the vacuum expectation value of the quantized energy-momentum operator in flat space found by point-splitting. When the mirror has non-uniform acceleration, it is not zero and describes the energy and momentum (not the particles) radiated by the mirror into empty space. The quantum stress tensor enables one to calculate the rate of energy loss of the mirror. Typically, one sees it expressed as Fulling and Davies (1976)

where the energy flux, $F(u)$, is a function of light-cone coordinate retarded time, $u=t-x$, Davies and Fulling (1977); Birrell and Davies (1984) and the brackets define the Schwarzian derivative. However, since we are using advanced time $v$, we write the radiated energy flux using Eq. (5), Good et al. (2017, 2020b)

where the Schwarzian brackets are defined as usual,

which gives, using $f(v)=v+4M^{3}/v^{2}$ of Eq. (5),

It is clear that asymptotically $F(v)\to 0$ for both $v\to(-\infty,0^{-})$. A plot of the energy flux, $F(v)$, as a function of advance time $v$ is given in Figure 4.

Recall that we used $f(v)$ as the trajectory and $v$ as the independent variable because of analytic transcendental tractability. That is, we have not found the equivalent trajectory expressed with independent variables proper time $\tau$, retarded time $u$, or coordinate time $t$. Fortunately, $v$ is sufficient for deriving all the needed and relevant physical observables we are after, i.e. particle production and energy emission. Interestingly, the use of coordinate space $x$ as an independent variable is also readily analytically tractable. It turns out to be illustrative to compute the energy flux as a function of space $x$, in addition to advanced time $v$. To do this, we express the trajectory Eq. (5) as a timespace function $t(x)$:

and also express the energy flux as a function of $x$ as well, $F(x)$, with the primes indicative of derivatives with respect to their argument, $x$,

Then, using Eq. (14) and Eq. (15), one finds:

which is plotted in Figure 5. The benefit of investigating this energy flux in space, is not only an additional mathematical check on the correctness of the results, but an explicit demonstration that negative energy flux is emitted over an infinite distance starting from $x=-2M$ to $x\to-\infty$ in a finite advanced time, $v=-M$ to $v=0$.

The total energy measured by a far away observer at $\mathscr{I}^{+}_{R}$ is Walker (1985b)

where integration occurs over retarded time (it takes the energy time to reach $\mathscr{I}^{+}_{R}$). Since we are using advanced time $v$, we write this with $du=\frac{\operatorname{d}\!{f}}{\operatorname{d}\!{v}}dv$ to get the Jacobian correct,

Plugging in Eq. (5) and Eq. (13) into Eq. (18), a simple analytic form results:

which is finite and positive. Physically, the finite value tells us the evaporation process stops, similar to the ABC’s of extremal black holes (asymptotic uniformly accelerated mirrors Liberati et al. (2000); Good (2020); Good et al. (2020a); Rothman (2000); Foo and Good (2020)), black hole remnants (non-horizon asymptotic coasting mirrors Good et al. (2017, 2019); Good (2018); Myrzakul and Good (2018); Good and Ong (2015); Good (2017)) and complete black hole evaporation (asymptotic static moving mirrors Walker and Davies (1982); Good et al. (2020c, d); Good and Linder (2017); Good et al. (2013); Good and Linder (2018, 2019)). However, it’s clear because of the negative energy flux radiated by the horizon at late times, this trajectory models neither an extremal (no NEF), nor a remnant (no horizon), nor complete evaporation (no horizon). The fact that the total energy is positive is consistent with the quantum interest conjecture Ford and Roman (1999) as derived from quantum inequalities Ford and Roman (1995).

Interestingly, as we have seen from Figures 4 and 5, there is a region of negative energy flux (NEF) at late times or far-left positions. The NEF begins at $v=-M$ or $x=-2M$. The total amount is

which gives an analytic result:

As a percentage of total energy, Eq. (19), it represents

a fairly substantial amount relative to what it appears to be in Figure 4 with light-cone coordinate advanced time $v$. Ultimately, this illusionary discrepancy is revealed as such with the inclusion of the Jacobian, $f^{\prime}(v)$, via integration in Eq. (20).

Let us consider the sum rule of Bianchi-Smerlak Bianchi and Smerlak (2014); Good et al. (2020c), expressed in terms of proper time of the mirror. This sum rule guarantees non-monotonic evaporation via the emission of NEF. The radiation flux $F(u)$ Schwarzian Eq. (10) is more compactly expressed Cong et al. (2020):

demonstrating that jerking toward an observer at $\mathscr{I}^{+}_{R}$, (with positive $+\eta^{\prime\prime}(\tau)$ jerk), yields negative energy flux. From Eq. (23) alone, we know our trajectory Eq. (9) will emit NEF. However, for the sake of argument, let us consider how the sum rule derivation proceeds by integration, which gives

At this point, we can see that any mirror that moves asymptotically with a constant finite rapidity, gives us a zero on the right-hand side because the rapidity $\eta$ as a constant for $\tau\rightarrow\pm\infty$, has derivative zero, and the sum rule is,

Therefore, on the general principle of a universal asymptotic speed limit for the mirror, that it must always remains time-like (as $\tau\rightarrow\infty$, then $\eta\neq\infty$), asymptotic horizonless mirrors will radiate a negative energy flux. Through the information-dynamics relationship (e.g. Chen and Yeom (2017); Bianchi and Smerlak (2014)) $\eta=-6S$, the time-like restriction corresponds to a pure state, i.e. the entanglement entropy, $S$, never diverges and unitary evolution implies negative energy flux; purity $\Rightarrow$ NEF.

However, clearly the trajectory Eq. (5) is a bit of an outlier (see Table 1), where $\eta\to\infty$ and information loss occurs because $|S|\to\infty$, but NEF is present! Non-unitary evolution can result in NEF. This is because even though the mirror is not asymptotically time-like, ($\eta$ is never constant, and in fact diverges, $|\eta|\to\infty$ as the mirror goes light-like in finite advanced time), the acceleration, $\alpha=\eta^{\prime}(\tau)$ still goes to zero at $\tau\to+\infty$. In other words, the asymptotic rapidity, need not asymptote to a finite constant to see NEF radiation. All that is required is that the mirror be asymptotic inertial, which leaves open the strange possibility in the sum rule of Eq. (25) for a light-like horizon to emit NEF. In this case information loss is coupled with negative energy flux.²²2Unitary evolution still implies NEF, regardless. This conclusion from the sum rule, Eq. (25), remains unchanged.

The leading order term in a series expansion for small $M$ or small $\omega$ is

Therefore the leading order term in small $\omega$ for the spectrum is $N_{\omega\omega^{\prime}}=(4\pi^{2}\omega\omega^{\prime})^{-1}$, which is identical to the leading order $\omega$ term for the Schwarzschild black hole spectrum at small $\omega$. However, this similarity between spectrum Eq. (27) and the known spectrum of black holes, Eq. (30), is where the analogy ends.

Looking to late times, which corresponds Hawking’s high-frequency approximation $\omega^{\prime}\gg\omega$ Hawking (1975), the spectrum is expressed as

demonstrating a new form for the late-time spectrum of Hawking radiation emanating from a moving mirror trajectory. Eq. (29) can be compared to the late time spectra of non-extremal and extremal black holes, respectively

Where $\kappa$ is the surface gravity, i.e. $\kappa=1/(4M)$ in the case of Schwarzschild, and/or outer horizon surface gravity for the RN and Kerr non-extremal black holes. Here $\mathcal{A}$ is the extremal parameter, or the asymptotic uniform acceleration Foo and Good (2020) in the case of the mirror system, where $K_{1}$ is the modified Bessel function of the second kind with order $n=1$. If $M\gtrapprox\omega$, then a good approximation for the late-time spectrum Eq. (29) results:

The first salient information gleaned from the spectrum is that the inertial horizon is not thermal, i.e. the particles are not in a Planck distribution and there will be no subsequent associated flattening of the energy flux. Second, we can see that the distribution is rather unusual as given by the Meijer-G function, which in this case does not simplify to a more well-known special function. Such a spectrum empirically measured would indicate particle emission from an inertial horizon and a ‘smoking gun’ of sustained negative energy fluxes.

Moreover, the sustained negative energy flux of the inertial horizon trajectory has an interesting relationship to the second law of thermodynamics Davies (1982). Since there are no problems with the mirror colliding with the observer, there may be a way to direct an oven into the path of oncoming radiation, cooling its temperature, and lowering its entropy. Although the beam would create a fluctuation in the total entropy, the second law of thermodynamics being statistical could permit small fluctuations in line with the constraints on NEF Ford (1991). Globally, the strength and duration of the entropy fluctuations would stay safely within permitted statistical boundaries, but locally, a shutter on the oven could be opened at late-times, permitting negative energy flux to gradually accumulate inside.

The unusual result of the spectrum and rare NEF emission from a horizon emitted to observers at infinity requires checking Eq. (27) for consistency with the total energy, Eq. (19) found from the stress tensor. This is done by quantum summing,

that is, associating a quantum of energy $\omega$ with the particle distribution and integrating over all the colors. The result is

This result is within $0.01\%$ relative error of Eq. (19), for $M=1$ and the numerical integration results in much better accuracy for larger $M$. It is fine to conclude therefore, that yes, the energy is carried by the particles. The beta spectrum Eq. (27) is consistent with the quantum stress tensor, Eq. (13).