Hawking radiation inside a charged black hole

The Hawking flux perceived by a timelike geodesic observer in a black hole spacetime can be calculated through the use of an effective temperature function

$$\kappa(u)\equiv-\frac{d}{du}\ln\left(\frac{dU}{du}\right)$$

(1)

where the outgoing null coordinate $u$ gives the observer’s position and the null coordinate $U$ gives the position of an emitter that defines the vacuum state [6, 7]. By a slight abuse of notation, the two worldlines labeled by coordinates $U$ and $u$ are connected by a null ray encoded by the function $U(u)$, and as long as $\kappa(u)$ remains approximately constant over a small interval around some point $u_{*}$, it directly implies that the vacuum expectation value of the particle number operator is consistent with that of a Planckian spectrum with temperature

$$T_{H}(u_{*})=\frac{\kappa(u_{*})}{2\pi}.$$

(2)

The constancy condition can be quantified by the adiabatic control function

$$\epsilon(u)\equiv\frac{1}{\kappa^{2}}\left|\frac{d\kappa}{du}\right|,$$

(3)

which must satisfy $\epsilon(u_{*})\ll 1$ in order for a thermal Hawking flux to be detected at $u_{*}$ [8]. However, even if the adiabatic condition is not satisfied, a nonzero $\kappa$ still implies the detection of particles corresponding to a nonzero Bogoliubov coefficient $\beta$; the only difference is that the spectral content will generally be non-Planckian.

Since both the observer and emitter can naturally use their proper times $\tau_{\text{ob}}$ and $\tau_{\text{em}}$ to label the different null rays they encounter throughout their journey, Eq. (1) can be recast in a more intuitive form:

$$\kappa=-\frac{d}{d\tau_{\text{ob}}}\ln\left(\frac{\omega_{\text{ob}}}{\omega_{\text{em}}}\right),$$

(4)

where the frequency $\omega$ (with either subscripts “ob” or “em,” which will be dropped hereafter when either label could apply), defined by

$$\omega\equiv-k^{\mu}\dot{x}_{\mu},$$

(5)

is the temporal component of a null particle’s coordinate 4-velocity ${k^{\mu}\equiv dx^{\mu}/d\lambda}$, measured in the frame of an observer or emitter with coordinate 4-velocity ${\dot{x}^{\mu}\equiv dx^{\mu}/d\tau}$. Eq. (4) makes it apparent that the effective temperature $\kappa$ is nothing more than a measure of the rate of frequency redshifting seen by an observer, an indicator of the exponential peeling of null rays first noted by Hawking as the crucial feature of black hole horizons responsible for particle creation [1, 2].

For black hole spacetimes with a Killing horizon, in the limit as an observer approaches future timelike infinity, the notion of the effective temperature $\kappa(\tau)$ defined above coincides precisely with the notion of the surface gravity $\varkappa$ used to define a black hole’s Hawking temperature [6]. Thus, $\kappa(\tau)$ provides a generalization of the Hawking effect for arbitrary observers around or inside of a black hole.

Instead of performing calculations in a fully dynamical collapse spacetime, it is common to formulate an equivalent problem in an empty, eternal black hole spacetime like the Schwarzschild metric [53]. As a result, the collapsing body must be replaced by appropriate boundary conditions on the past horizon, and these boundary conditions define the quantum field’s vacuum state in that spacetime. Three options are generally discussed in the literature: the Boulware state, in which the quantum field’s modes are defined to be positive frequency with respect to the Killing vector $\partial/\partial t$ on both the past horizon and past null infinity; the Hartle-Hawking state, in which modes are defined to be positive frequency with respect to the past boundaries’ canonical affine coordinates²²2For example, for a Schwarzschild black hole, ${U=-4M\text{e}^{-u/(4M)}}$ is the outgoing Kruskal-Szekeres coordinate, whose vector field ${\partial/\partial U}$ is of Killing type on the past horizon. Positive frequency modes are then defined to be the eigenfunctions of the Lie derivative of the field in the ${\partial/\partial U}$ direction. ${\partial/\partial U}$ and ${\partial/\partial V}$; and the (past) Unruh state [53], in which modes are defined to be positive frequency with respect to ${\partial/\partial U}$ on the past horizon and ${\partial/\partial t}$ at past null infinity. The last of these states is the one that is most physically relevant to the production of a Hawking flux to the future of a collapsing black hole and is the state that will be employed here.

In the effective temperature framework, the vacuum state is specified by the spacetime position and state of motion (the orbital parameters) of the emitter. For example, the Boulware state corresponds to a static emitter maintaining a constant radius $r_{0}$. This state is thus only defined for the exterior portion of the black hole, since an emitter cannot remain static below the event horizon. A freely falling observer measuring in the Boulware state will see diverging stress-energy at the horizon, as a result of the diverging acceleration required for the Boulware emitter to remain static there.

In contrast, the Unruh state is associated with a freely falling emitter, positioned either at the black hole’s horizon or at infinity. The outgoing Unruh modes correspond to the limit ${r_{\text{em}}\to r_{+}}$, so that the observer sees the emitter frozen on the past horizon (one may equivalently take the Unruh emitter’s descent into the black hole to have occurred sufficiently far into the past), and the ingoing Unruh modes correspond to the limit ${r_{\text{em}}\to\infty}$, so that the observer sees the emitter safely resting in the sky above. Since the observer and the Unruh emitter are generally not located at the same spacetime coordinate (as in the Boulware state), their modes must be connected via a null geodesic, since the quantum field under study here is massless.

To see how the choice of vacuum corresponds to the specification of the emitter’s worldline, consider an emitter radially free-falling from rest at infinity³³3The same arguments should hold for any inertial free-faller; here we present the radial, ${E=1}$ case for simplicity. into a static, asymptotically flat, spherically symmetric black hole, which is given by the line element

$$ds^{2}=-\Delta(r)\ dt^{2}+\frac{dr^{2}}{\Delta(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\!\theta d\phi^{2}\right).$$

(6)

The horizon function ${\Delta(r)}$ has the property that it vanishes linearly as $r$ approaches a horizon, and it asymptotes to unity as ${r\to\infty}$.

Such an emitter will have coordinate 4-velocity with nonzero components

	$$\displaystyle\dot{t}\equiv\frac{dt}{d\tau}=\frac{1}{\Delta},$$		(7a)
	$$\displaystyle\dot{r}\equiv\frac{dr}{d\tau}=-\sqrt{1-\Delta}.$$		(7b)

When the emitter is at infinity (${\Delta\to 1}$) sending modes inward, Eq. (7a) implies that the emitter’s proper time $\tau$ will tick at the same proportionate rate as the global timelike Killing coordinate $t$. Thus, $t$ will be the coordinate the emitter uses to define positive frequency, just as expected for ingoing Hawking modes originating from past null infinity.

However, when the emitter reaches a horizon (${\Delta\to 0}$), Eq. (7a) implies that the static Schwarzschild time $t$ will tick at an infinitely faster rate than the emitter’s proper time $\tau$. So heuristically, instead of seeing wave modes of the form ${\exp(-i\omega t)}$, the emitter should end up seeing modes of the form ${\exp[-i\omega\exp(-kt)]}$ (for some constant $k$), so that even when $t$ diverges, the emitter’s proper time will still remain finite. The new time coordinate defined by these modes will be found to coincide with the oft-studied Kruskal-Szekeres coordinate $U$.

To make the above arguments more precise, and to extend the discussion to distinguish ingoing and outgoing modes (which depend on both the emitter’s proper time and the proper distance between wavefronts), consider a set of eikonal waves in the emitter’s locally orthonormal tetrad frame $\{\bm{\gamma}_{0},\bm{\gamma}_{1},\bm{\gamma}_{2},\bm{\gamma}_{3}\}$, whose tangent-space coordinates will be labeled $\xi^{0}$, $\xi^{1}$, $\xi^{2}$, and $\xi^{3}$. This tetrad frame is constructed so that it is continuous across the event horizon and so that the time axis $\bm{\gamma}_{0}$ is always timelike and future-directed, while the radial axis $\bm{\gamma}_{1}$ is always spacelike and outward-directed. In the limit of large frequency $\omega$, to leading order in ${1/\omega}$, the ingoing ($+$) or outgoing ($-$) components of the eikonal wavefront will follow a null geodesic congruence with tetrad-frame 4-momentum (neglecting any normalization factors)

$$k^{\hat{m}}\equiv\frac{d\xi^{\hat{m}}}{d\lambda}=\left(1,\ \pm 1,\ 0,\ 0\right).$$

(8)

The transformation from the emitter’s local tetrad frame to a coordinate frame can be accomplished through the use of the appropriate vierbein. For an external⁴⁴4The case of a free-faller in the black hole interior follows the same line of reasoning as the exterior case presented here, mutatis mutandis. radial free-faller with specific energy $E$ (where ${E=1}$ corresponds to rest at infinity), in the static polar spherical chart this vierbein reads

$$e^{\hat{m}}_{\ \ \mu}=\begin{pmatrix}E&\sqrt{E^{2}-\Delta}&0&0\\ \sqrt{E^{2}-\Delta}&E&0&0\\ 0&0&r&0\\ 0&0&0&r\sin\theta\end{pmatrix}$$

(9)

(where rows label the coordinates $\xi^{0}$, $\xi^{1}$, $\xi^{2}$, $\xi^{3}$ of the emitter’s locally inertial frame, and columns label the global coordinates $t$, $r^{*}$, $\theta$, $\varphi$). Here we define the tortoise coordinate $r^{*}$ by

$$\frac{dr}{dr^{*}}=\Delta.$$

(10)

The coordinate-frame 4-momentum ${k^{\mu}=k^{\hat{m}}e_{\hat{m}}^{\ \ \mu}}$ then follows immediately:

$$k^{\mu}=\left(\frac{E\mp\sqrt{E^{2}-\Delta}}{\Delta},\ \pm\frac{E\mp\sqrt{E^{2}-\Delta}}{\Delta},\ 0,\ 0\right).$$

(11)

If the emitter defines some positive frequency $\omega$ (along with the corresponding wave number $\omega/c$), then their natural choice of ingoing (upper sign) or outgoing (lower sign) modes will take the form ${\exp[-i\omega(\xi^{0}\pm\xi^{1})]}$, which can be written in coordinate form by matching the affine distances of Eqs. (8) and (11):

$$d\xi^{0}\pm d\xi^{1}=\frac{\Delta}{E\mp\sqrt{E^{2}-\Delta}}\left(dt\pm dr^{*}\right).$$

(12)

Asymptotically, as a unit-energy emitter approaches infinity (${\Delta\to 1}$), the fraction in Eq. (12) reduces to unity, so that the proper choice of coordinates to define Unruh modes at infinity is the Eddington-Finkelstein double null coordinate system, defined in both the exterior and interior as

$$u\equiv t-r^{*},\quad v\equiv t+r^{*},$$

(13)

where $u$ here is the same outgoing null coordinate as in Eq. (1).

When the emitter is at a horizon (${\Delta\to 0}$), the mode behavior depends on whether the waves are ingoing or outgoing. For the ingoing modes of a positive-energy free-faller or the outgoing modes of a negative-energy free-faller (neither of which are needed to define an Unruh emitter but will prove useful later to define the natural modes seen by horizon observers), the fraction in Eq. (12) reduces to $2E$, so that the proper modes (after $\omega$ is properly scaled) are once again the Eddington-Finkelstein modes ${\exp[-i\omega(t\pm r^{*})]}$.

But for the outgoing modes of a positive-energy free-faller or the ingoing modes of a negative-energy free-faller at the horizon, the fraction in Eq. (12) vanishes, so a more appropriate coordinate choice must be found. Define a new coordinate $\bar{U}$ such that the outgoing Unruh modes at the horizon will be written as ${\exp[-i\omega\bar{U}]}$. Then Eq. (12) implies that $\bar{U}$ must satisfy

$$\frac{d\bar{U}}{du}\underset{\Delta\to 0}{=}\frac{\Delta}{2E}\approx\frac{r-r_{\pm}}{2E}\frac{d\Delta}{dr}\bigg{|}_{r_{\pm}}$$

(14)

in the near-horizon limit. From this expression one can identify the quantity

$$\varkappa_{\pm}\equiv\frac{1}{2}\frac{d\Delta}{dr}\bigg{|}_{r_{\pm}}$$

(15)

as the outer ($+$) or inner ($-$) horizon’s surface gravity. For an emitter with ${E=1}$, since from Eqs. (8), (10), and (11), the radius $r$ is related to the horizon-limit outgoing proper null coordinate $\bar{U}$ by

$$\frac{dr}{d\bar{U}}=-\frac{1+\sqrt{1-\Delta}}{2}\underset{\Delta\to 0}{=}-1,$$

(16)

then Eq. (14) solves as

$$\bar{U}\propto\exp\left(-\varkappa_{\pm}u\right).$$

(17)

Eq. (17) assumes that $\bar{U}$ is chosen to begin at 0 at the event horizon, when ${u\to\infty}$. This form of the emitter’s proper time (up to an irrelevant normalization factor) is precisely the form of the outgoing Kruskal-Szekeres coordinate $U$ used by Unruh to define positive frequency on the past horizon [53]. Thus, the outgoing modes of the Unruh state correspond to those seen as positive frequency by an emitter in free fall asymptotically close to the past horizon.

In some sense, we have done nothing more than “rederive the obvious” in showing how one may obtain past Unruh null boundary conditions. However, in addition to providing yet another way of understanding the validity of this choice of vacuum state, the generalized derivation above also provides a natural specification of ingoing and outgoing modes for freely falling observers at either horizon, without any reliance on global Killing vector fields or asymptotically Minkowski regimes. We will return to this idea when solving the wave equation in Sec. IV.

As a final comment concerning the choice of vacuum state, an additional family of vacuum states was used by Ref. [8] to mimic the switching on of Hawking radiation as a black hole first forms during a collapse. These “collapse vacua” correspond to emitters in free fall from rest at infinity, each separated from the observer by a time delay $\delta\tau$ (as in the Unruh state), but not necessarily in the limit as they approach the horizon or infinity. However, in the present work, we are not concerned with the initial transient collapse dynamics of a black hole; rather, we will focus on the late-time steady-state behavior once the black hole has settled down into the Unruh state, which should occur only a few light-crossing times after the black hole’s formation.

Consider the line element of Eq. (6), which describes the geometry of a charged, spherically symmetric black hole when the horizon function $\Delta$ takes the form

	$\displaystyle\Delta$	$\displaystyle=\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right),$		(18)
	$\displaystyle r_{\pm}$	$\displaystyle\equiv M\pm\sqrt{M^{2}-Q^{2}}.$		(19)

The black hole modeled by this geometry is known as the Reissner-Nordström black hole, which possesses a mass $M$ and a charge $Q$. The length scales $r_{+}$ and $r_{-}$ are referred to respectively as the outer (event) horizon and the inner (Cauchy) horizon.

The rate of redshift seen by a radially infalling observer has already been calculated for the spacetime of Eq. (6) for arbitrary $\Delta$ (see Appendix B of Ref. [45]), though that analysis was only carried out explicitly for Schwarzschild (${Q/M=0}$). Here we quote the main results and specialize to Reissner-Nordström with a focus on the inner horizon.

The frequency $\omega$ measured in the frame of an observer ($\equiv\omega_{\text{ob}}$) or emitter ($\equiv\omega_{\text{em}}$) with specific energy $E$, normalized to the frequency $\omega_{\infty}$ seen at rest at infinity, is

$$\frac{\omega}{\omega_{\infty}}=\frac{E\pm\sqrt{E^{2}-\Delta}}{\Delta},$$

(20)

where the upper (lower) sign applies to outgoing (ingoing) null rays. The effective temperature $\kappa$ can then be calculated with the help of the chain rule:

	$\displaystyle\kappa$	$\displaystyle=-\frac{d}{d\tau_{\text{ob}}}\ln\left(\frac{\omega_{\text{ob}}}{\omega_{\text{em}}}\right)$
		$\displaystyle=-\omega_{\text{ob}}\left(\frac{\dot{r}_{\text{ob}}}{\omega_{\text{ob}}}\frac{\partial\ln\omega_{\text{ob}}}{\partial r_{\text{ob}}}-\frac{\dot{r}_{\text{em}}}{\omega_{\text{em}}}\frac{\partial\ln\omega_{\text{em}}}{\partial r_{\text{em}}}\right)$
		$\displaystyle=\mp\frac{1}{2}\frac{\omega_{\text{ob}}}{\omega_{\infty}}\left(\frac{d\Delta_{\text{ob}}}{dr_{\text{ob}}}-\frac{d\Delta_{\text{em}}}{dr_{\text{em}}}\right),$		(21)

where an overdot signifies differentiation with respect to the observer’s or emitter’s proper time $\tau$.

For outgoing modes (upper sign), the Unruh emitter must be placed at the event horizon (${r_{\text{em}}\to r_{+}}$), and for ingoing modes (lower sign), the Unruh emitter resides at infinity (${r_{\text{em}}\to\infty}$). The result, for an observer in free fall from rest at infinity (${E_{\text{ob}}=1}$), is the sensation of two independent effective temperatures corresponding to the outgoing ($\kappa^{+}$) and ingoing ($\kappa^{-}$) Hawking modes (throughout the rest of this paper, $\pm$ superscripts will always refer to outgoing/ingoing quantities, while $\pm$ subscripts will always refer to outer/inner horizon quantities):

	$\displaystyle\kappa^{+}$	$\displaystyle=\frac{Mr_{\text{ob}}\left(1-r_{\text{ob}}^{2}/r_{+}^{2}\right)-Q^{2}\left(1-r_{\text{ob}}^{3}/r_{+}^{3}\right)}{r_{\text{ob}}^{2}\left(-r_{\text{ob}}+\sqrt{2Mr_{\text{ob}}-Q^{2}}\right)},$		(22a)
	$\displaystyle\kappa^{-}$	$\displaystyle=\frac{Mr_{\text{ob}}-Q^{2}}{r_{\text{ob}}^{2}\left(r_{\text{ob}}+\sqrt{2Mr_{\text{ob}}-Q^{2}}\right)}.$		(22b)

The rest of this Sec. III.1 will be devoted to exploring the implications of Eqs. (22). As a first comment, because of the square root in the denominator, both temperatures become imaginary when the observer is located close enough to the origin, specifically when ${r_{\text{ob}}<Q^{2}/(2M)}$. However, such values of $r_{\text{ob}}$ are strictly less than the inner horizon radius $r_{-}$ for all choices of $Q$, and the failure of Eqs. (22) in this region coincides with the failure of Gullstrand-Painlevé coordinates in the same region, indicative of the presence of an unphysical negative interior mass $M(r)$ (i.e. this is where an infaller would bounce back due to the effects of the repulsive charged singularity on the spacetime) [54]. Since the region below the inner horizon should be physically disregarded due to the semiclassical singular behavior examined below, that region will not be explored any further here.

Second, it should be noted that for an observer asymptotically far from the black hole, the above formulas reproduce familiar results: the outgoing sector’s temperature asymptotically approaches the black hole’s surface gravity $\varkappa_{+}$ defined by Eq. (15), and the ingoing Hawking sector vanishes:

$$\lim_{r_{\text{ob}}\to\infty}\left\{\kappa^{+},\ \kappa^{-}\right\}=\left\{\frac{r_{+}-r_{-}}{2r_{+}^{2}},\ 0\right\}.$$

(23)

As expected, $\kappa^{+}$ approaches ${1/(4M)}$ in the Schwarzschild ${Q/M=0}$ limit and vanishes in the extremal ${Q/M=1}$ limit. These limits can be seen in the respective panels of Fig. 1, which shows the full behavior of ${\kappa^{+}(r_{\text{ob}})}$ and ${\kappa^{-}(r_{\text{ob}})}$ for different choices of the black hole’s charge-to-mass ratio.

As mentioned in Sec. II.1, the identification of the effective temperature $\kappa$ with a thermal Hawking flux is strictly only valid in conjunction with the adiabatic condition, that $\kappa$ must remain approximately constant over enough e-folds of the arriving modes [6, 7]. This condition is quantified by the adiabatic control function $\epsilon$, which for radial modes in a static, spherically symmetric black hole can be written as

$$\epsilon(r_{\text{ob}})\equiv\left|\frac{\dot{\kappa}}{\kappa^{2}}\right|=\left|\frac{\dot{r}_{\text{ob}}}{\kappa^{2}}\frac{d\kappa}{dr_{\text{ob}}}\right|.$$

(28)

Whenever ${\epsilon\ll 1}$, the adiabatic condition is satisfied and a thermal Hawking spectrum is perceived by the observer.

The exact analytic form of $\epsilon(r_{\text{ob}})$ for the Reissner-Nordström free-faller in the Unruh state is not too illuminating; nonetheless, several key features can be identified. As ${r_{\text{ob}}\to\infty}$, the adiabatic control function for the outgoing modes drops to zero (as anticipated to recover Hawking’s original thermal calculation), except in the extremal case where $\kappa$ itself is already zero and $\epsilon$ therefore diverges. Similar diverging behavior in $\epsilon$ is observed whenever the effective temperature $\kappa$ vanishes, as a result of the $\kappa^{2}$ term in the denominator of Eq. (28), since it is meaningless to define a thermal flux at zero temperature.

Based on the above observations, one might expect that $\epsilon$ would drop to zero whenever $\kappa$ diverges (e.g. when one observes outgoing modes at the inner horizon). However, the adiabatic control function at the inner horizon instead passes through a finite, nonzero value, which nonetheless is still usually smaller than unity for outgoing modes. Specifically, for an ingoing observer,

		$\displaystyle\lim_{r_{\text{ob}}\to r_{-}}\left\{\epsilon^{+},\ \epsilon^{-}\right\}=$
		$\displaystyle\left\{\frac{r_{+}^{2}}{2(2M^{2}-Q^{2})},\ \frac{5Q^{2}+4M\sqrt{M^{2}-Q^{2}}-3M^{2}}{M^{2}-Q^{2}}\right\}.$		(29)

This equality technically only holds when ${Q\neq 0}$; in the Schwarzschild case, instead of approaching unity, both $\epsilon^{+}$ and $\epsilon^{-}$ will asymptotically approach 3 (see the left panel of Fig. 3). But for ${Q>0}$, the value of $\epsilon^{+}$ at the inner horizon is always less than 1, and $\epsilon^{-}$ is always greater than 1. For large enough charge $Q$, Eq. (III.2) thus implies that the outgoing temperature should be approximately thermal for an ingoing observer close enough to the inner horizon. This behavior holds even (and especially) for ${Q/M=1}$, where the inflating negative temperature just above the merged horizons occurs in the black hole’s exterior.

For reference, the behavior of ${\epsilon^{+}(r_{\text{ob}})}$ and ${\epsilon^{-}(r_{\text{ob}})}$ are plotted in Fig. 3 for two of the same values of $Q$ used in Fig. 1. One may observe that for many choices of $r_{\text{ob}}$, $\kappa$ behaves adiabatically and the thermal results fall into place. However, for much of the observer’s trajectory, $\epsilon$ far exceeds unity, and deeper analysis is required, as examined in Sec. IV.

One final technical point related to the discussion of adiabaticity is the comment made by the authors of Ref. [6] that the effective temperature adiabaticity formalism described above is valid “under mild technical assumptions.” These assumptions are related to the more generalized, precise form of the adiabaticity condition, which assumes the existence of a finite quantity

$$D\equiv\underset{n>0}{\text{sup}}\left[\frac{1}{(n+1)!}\frac{|\kappa^{(n)}|}{\kappa^{n+1}}\right]^{1/(n+1)}$$

(30)

such that adiabaticity is implied by the condition ${2D^{2}\ll 1}$ (instead of ${\epsilon\ll 1}$). Usually, the ${n=1}$ term in the definition of Eq. (30) dominates so that the quantity ${2D^{2}}$ is equivalent to the adiabatic control function $\epsilon$ of Eq. (28). But in certain special cases, such as the dip observed in the blue curve in the right panel of Fig. 3 as $\epsilon^{-}$ goes to zero just above the outer horizon, higher-$n$ terms in Eq. (30) dominate. As a result, adiabaticity is not satisfied there, even though $\dot{\kappa}^{-}$ (and therefore $\epsilon^{-}$) vanishes.

The results of Secs. III.1 and III.2 apply to a radial infaller observing modes purely in the radial direction. Since the mass inflation instability involves radial focusing of all null geodesics, one may wonder whether the diverging acceleration seen by an infaller is confined to a single radial point on the sky.

The goal of this section is to provide a generalization of Eq. (21) to account for photons reaching the observer from any direction. The photon’s 4-momentum will now include additional angular terms with the conserved quantity ${b\equiv k_{\vartheta}/k_{t}}$, the photon’s impact parameter, which equals 0 for radial trajectories but in general can take any real value up to infinity. To translate $b$ into a viewing angle on the observer’s sky, it suffices to define a single parameter $\chi$ that measures the angle in the observer’s local tetrad frame between the radial direction and the direction the observer is facing. This viewing angle $\chi$ ranges from 0 degrees (facing radially inward toward the past horizon) to 180 degrees (facing radially outward toward the sky above, at past null infinity). For an observer with specific energy $E_{\text{ob}}$ at radius $r_{\text{ob}}$, the impact parameter $b$ is related to the viewing angle $\chi$ by [45]

$$b=\left|\frac{r_{\text{ob}}\sin\chi}{E_{\text{ob}}-\sqrt{E_{\text{ob}}^{2}-\Delta(r_{\text{ob}})}\cos\chi}\right|.$$

(31)

The frequency $\omega$ measured in the frame of an observer ($\equiv\omega_{\text{ob}}$) or emitter ($\equiv\omega_{\text{em}}$) with specific energy $E$, normalized to the frequency $\omega_{\infty}$ seen at rest at infinity, then generalizes to

$$\frac{\omega}{\omega_{\infty}}=\frac{E\pm\sqrt{\left(E^{2}-\Delta\right)\left(1-b^{2}\Delta/r^{2}\right)}}{\Delta},$$

(32)

where, as in the radial case, the upper (lower) sign applies to outgoing (ingoing) null rays. The calculation of $\kappa$ then follows as in the radial case, though great care must be taken to account for turnaround radii and ensure the correct sign for different viewing angles and observer positions.

Since the perception of particle production is highly dependent on the choice of observer, one must take care to make an appropriate choice depending on the context of the calculation. For example, an observer staring in a fixed direction $\chi$ as they fall inward is not the same as an observer staring at a single infalling emitter, whose position will constantly change in the observer’s field of view. As argued in Ref. [45], the choice of observer that will introduce the least amount of noninertial radiative effects (e.g. from the rotation of the observer’s frame) and will reveal the most “pure” Hawking radiation is an observer staring in a fixed direction $\chi$. Such an observer will see a family of infalling emitters as they fall inward, with each emitter connected to the observer by a null path with the same phase.

If an observer stares at the sky above (corresponding to the ingoing Hawking sector, with a family of Unruh emitters at ${r_{\text{em}}\to\infty}$), the generalization of Eq. (21) to account for the frequency of Eq. (32) seen from any viewing angle $\chi$ is sufficient to satisfy the requirement from the previous paragraph of an inertial observer with fixed $\chi$. However, if the observer stares at the past horizon below them (corresponding to the outgoing Hawking sector, with a family of Unruh emitters at ${r_{\text{em}}\to r_{+}}$), the frequency seen by the emitter or the observer will diverge, as will the affine distance of the null geodesics connecting the two infallers. In order to ensure that the observer is seeing the same emitted in-modes as they follow along a geodesic staring in a fixed direction $\chi$, the emitted affine distance

$$\lambda_{\text{em}}\equiv\omega_{\text{em}}\lambda=\omega_{\text{em}}\int_{r_{\text{em}}}^{r_{\text{ob}}}\frac{dr}{k^{r}}$$

(33)

(where ${k^{r}\equiv dr/d\lambda}$ is the radial component of photon’s coordinate-frame 4-momentum, given by Eq. (80) of Ref. [45]) must be held constant. The resulting effective temperature then takes the form:

	$\displaystyle\kappa$	$\displaystyle=-\frac{\partial}{\partial\tau_{\text{ob}}}\ln\left(\frac{\omega_{\text{ob}}\lambda}{\lambda_{\text{em}}}\right)\bigg{|}_{\chi,\lambda_{\text{em}}}$
		$\displaystyle=-\dot{r}_{\text{ob}}\left(\frac{\partial\ln\omega_{\text{ob}}}{\partial r_{\text{ob}}}\bigg{|}_{\chi}+\frac{\partial\ln\lambda}{\partial r_{\text{ob}}}\bigg{|}_{\chi}\right)-\dot{r}_{\text{em}}\frac{\omega_{\text{ob}}}{\omega_{\text{em}}}\frac{\partial\ln\lambda}{\partial r_{\text{em}}}\bigg{|}_{\chi},$		(34)

where the derivatives of the affine distance (at constant $\chi$) can be expanded with the Leibniz integral rule:

	$\displaystyle\frac{\partial\ln\lambda}{\partial r_{\text{ob}}}\bigg{|}_{\chi}$	$\displaystyle=\frac{1}{\lambda}\left(\frac{1}{k^{r}_{\text{ob}}}+\frac{\partial b}{\partial r_{\text{ob}}}\bigg{|}_{\chi}\int_{r_{\text{em}}}^{r_{\text{ob}}}\!\!\!dr\frac{\partial}{\partial b}\frac{1}{k^{r}}\right),$		(35a)
	$\displaystyle\frac{\partial\ln\lambda}{\partial r_{\text{em}}}\bigg{|}_{\chi}$	$\displaystyle=-\frac{1}{\lambda k^{r}_{\text{em}}}.$		(35b)

The numerical solution to Eq. (III.3) for various values of $r_{\text{ob}}$ and $Q$ is shown in Fig. 4. These plots show similar trends to that found in Ref. [45] for Schwarzschild black holes. First, the outgoing Hawking radiation seen from the past horizon (left two panels) is actually weakest in the radial direction (except when the observer is very close to the inner horizon). As $\chi$ increases from 0^∘ and the observer looks farther away from the center of the black hole’s shadow marking where the past horizon would be, $\kappa^{+}$ increases until it diverges at the edge of the shadow.⁶⁶6This divergence is an artifact of the unphysical metric used; for an astrophysical black hole formed by gravitational collapse a finite time in the past, the Hawking radiation would still exponentially limb brighten but would remain finite before dropping to zero outside of the black hole’s shadow [45]. As the observer falls closer and closer to the inner horizon, the area of sky across which Hawking radiation is visible becomes larger (in conjunction with the growing apparent size of the black hole’s shadow), and the Hawking radiation becomes more and more isotropic across the surface of the shadow. But once the observer falls close enough to the inner horizon, the apparent black hole size begins to decrease as the Hawking area shrinks to a small patch of sky ahead of the observer (this effect is most apparent in the lower left panel of Fig. 4, but even in the upper left panel, additional curves for smaller radii $r_{\text{ob}}$ would begin to shrink since the maximum angle $\chi$ shifts down to 0^∘ as ${r\to r_{-}}$).

When the black hole’s charge $Q$ is nonzero, the main effect on the outgoing effective temperature at arbitrary viewing angle is the same result found in Sec. III.1; namely, an observer close enough to the inner horizon will see a negative $\kappa^{+}$, corresponding to modes that are exponentially blueshifting instead of redshifting. The higher the charge $Q$, the farther out in physical space this blueshifting zone becomes, until it extends beyond the outer horizon and reaches infinity in the extremal case (as already seen in the radial case of Fig. 2).

Similarly, ingoing Hawking radiation seen from an observer looking up at the sky above (right two panels of Fig. 4) reproduces the same behavior found in Ref. [45] for the Schwarzschild case, with minimal modifications when $Q$ is nonzero. The rate of redshifting in the upper hemisphere is strongest when the observer looks straight up to the sky (in the outward radial direction, ${\chi=180^{\circ}}$), and $\kappa^{-}$ changes sign at $90^{\circ}$, reflecting the fact that the infaller is accelerating away from the sky above (so that the upper hemisphere is redshifting) and accelerating toward the black hole below (so that the lower hemisphere is blueshifting).

However, as with the outgoing effective temperature, the ingoing effective temperature changes sign once the observer falls close enough to the inner horizon [seen, e.g., with the dashed pink line at $r_{\text{ob}}={r_{-}+10^{-3}(r_{+}-r_{-})}$ on the right half of the top right panel of Fig. 4], so that the upper hemisphere is blueshifting and the lower hemisphere is redshifting. But unlike the outgoing radiation, the sign change in the ingoing effective temperature is restricted only to infallers within the event horizon, regardless of the value of $Q$.

Aside from the sign reversal in every direction for observers close enough to the inner horizon, the main contribution that an addition of charge has on the angular distribution of Hawking radiation (for both $\kappa^{+}$ and $\kappa^{-}$) is to smooth out the perceived temperature gradients across the sky—the higher the charge $Q$, the less sharp the temperature cutoff is at the black hole shadow’s boundaries, and therefore the less isotropic the temperature is across the observer’s field of view for a given distance above the inner horizon.

Consider the Bogoliubov coefficients between the vacuum state of an Unruh emitter and that of a freely falling observer in a Reissner-Nordström spacetime. In any spacetime with metric $g_{\mu\nu}$, a canonically quantized, massless scalar field $\Phi$ will satisfy the Klein-Gordon wave equation

$$\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^{\mu}}\left(\sqrt{-g}\ g^{\mu\nu}\frac{\partial\Phi}{\partial x^{\nu}}\right)=0.$$

(40)

Motivated by the spacetime’s symmetries, we choose to decompose this field $\Phi$ into a complete set of modes $\phi_{\omega\ell m}$, each accompanied by creation and annihilation operators $a^{\dagger}$ and $a$:

$$\Phi=\int_{0}^{\infty}d\omega\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left(\phi_{\omega\ell m}a_{\omega\ell m}+\phi^{*}_{\omega\ell m}a^{\dagger}_{\omega\ell m}\right).$$

(41)

If these modes are separated as

$$\phi_{\omega\ell m}=\frac{f_{\omega\ell}(t,r)Y_{\ell m}(\theta,\varphi)}{r\sqrt{4\pi\omega}},$$

(42)

then Eq. (40) implies that $Y_{\ell m}$ will satisfy the spherical harmonic equation, while $f_{\omega\ell}$ must satisfy

$$\frac{\partial^{2}f_{\omega\ell}}{\partial r^{*2}}-\frac{\partial^{2}f_{\omega\ell}}{\partial t^{2}}=\Delta\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{1}{r}\frac{d\Delta}{dr}\right]f_{\omega\ell}.$$

(43)

The annihilation operators ${a_{\omega\ell m}}$ of Eq. (41) define the vacuum state of the observer:

$$a_{\text{ob}}|0_{\text{ob}}\rangle=0$$

(44)

(for convenience, the mode indices $\omega$, $\ell$, and $m$ will hereafter be suppressed as needed). However, $\Phi$ could just as easily be decomposed into any other complete set of modes $\bar{\omega}$, $\bar{\ell}$, and $\bar{m}$, so a similar decomposition can be used to define an emitter’s vacuum state as

$$a_{\text{em}}|0_{\text{em}}\rangle=0.$$

(45)

The two vacuum states are related by a Bogoliubov transformation through the coefficients $\alpha^{\omega\ell m}_{\bar{\omega}\bar{\ell}\bar{m}}$ and $\beta^{\omega\ell m}_{\bar{\omega}\bar{\ell}\bar{m}}$ (and note that there should properly be a sum of two integrals for the emitter’s ingoing and outgoing states, which are omitted here for simplicity):

$$a_{\text{ob}}=\int_{0}^{\infty}d\bar{\omega}\sum_{\bar{\ell}=0}^{\infty}\sum_{\bar{m}=-\bar{\ell}}^{\bar{\ell}}\left(\alpha\ a_{\text{em}}+\beta^{*}a^{\dagger}_{\text{em}}\right).$$

(46)

It is then straightforward to show [59] that the vacuum expectation value of the observer’s number operator in the emitter’s vacuum state is related to the Bogoliubov coefficient $\beta$:

	$\displaystyle\langle 0_{\text{em}}|a^{\dagger}_{\text{ob}}$	$\displaystyle a_{\text{ob}}|0_{\text{em}}\rangle=\int_{0}^{\infty}d\bar{\omega}\sum_{\bar{\ell}=0}^{\infty}\sum_{\bar{m}=-\bar{\ell}}^{\bar{\ell}}\left|\beta\right|^{2}$
		$\displaystyle=\int_{0}^{\infty}d\bar{\omega}\sum_{\bar{\ell}=0}^{\infty}\sum_{\bar{m}=-\bar{\ell}}^{\bar{\ell}}\left|\langle\phi_{\text{em}}|\phi^{*}_{\text{ob}}\rangle\right|^{2},$		(47)

where bra-ket notation denotes the Lorentz-invariant Klein-Gordon inner product, which consists of a 3D integral over an arbitrary spacelike Cauchy hypersurface $\Sigma$ that terminates at spacelike infinity and is orthogonal to a future-directed unit vector $n^{\mu}$:

$$\langle\phi_{1}|\phi_{2}\rangle\equiv-i\int_{\Sigma}d\Sigma\ n^{\mu}\sqrt{-g_{\Sigma}}\ \phi_{1}\overset{\leftrightarrow}{\partial}_{\mu}\phi_{2}^{*}.$$

(48)

To determine the expected particle number seen by an observer, one thus needs only to specify the observer’s and emitter’s modes (usually via a set of boundary conditions), propagated through the spacetime via the wave equation so that they coincide on some Cauchy hypersurface.

The Unruh emitter’s ingoing ($-$) and outgoing ($+$) modes are defined with the following boundary conditions at past null infinity $\mathscr{I}^{-}$ and the past horizon $\cal{H}^{+}_{\textit{r}_{+}}\equiv{}^{\text{int}}\cal{H}^{+}_{\textit{r}_{+}}\cup{}^{\text{ext}}\cal{H}^{+}_{\textit{r}_{+}}$ [here $f$ is defined as in Eq. (42), with $\omega\ell$ indices dropped for convenience]:

$$f^{+}_{\text{em}}\to\begin{cases}0,&\mathscr{I}^{-}\\ \text{e}^{-i\omega U},&\cal{H}^{+}_{\textit{r}_{+}}\end{cases},$$

(49)

$$f^{-}_{\text{em}}\to\begin{cases}\text{e}^{-i\omega(t+r^{*})},&\mathscr{I}^{-}\\ 0,&\cal{H}^{+}_{\textit{r}_{+}}\end{cases},$$

(50)

where $U$ is the outgoing Kruskal-Szekeres coordinate, defined in terms of the event horizon’s surface gravity $\varkappa_{+}$ from Eq. (15) by

$$U\equiv\begin{cases}-\text{e}^{-\varkappa_{+}(t-r^{*})}/\varkappa_{+},&r_{+}\leq r<\infty\\ +\text{e}^{-\varkappa_{+}(t-r^{*})}/\varkappa_{+},&r_{-}\leq r<r_{+}\end{cases}.$$

(51)

The relevant surfaces to which these boundary conditions correspond are shown schematically with dotted arrows in the Penrose diagram of Fig. 6. Note that, as shown in the diagram, the outgoing modes can be further split into a pair of substates via $f^{+}\equiv{\left({}^{\text{int}}f^{+}\right)\cup\left({}^{\text{ext}}f^{+}\right)}$, each of whose boundary conditions are zero except on their respective null surfaces. As argued in Sec. II.2, the modes of Eqs. (49) and (50) are precisely those which are positive frequency with respect to the proper time of a freely falling observer skimming asymptotically close to those surfaces. The modes $f^{\pm}_{\text{em}}$ can then be extended to the entire spacetime by solving the wave Eq. (43).

Similarly, the observer’s ingoing ($-$) and outgoing ($+$) modes can be defined via boundary conditions, in this case on the future null hypersurfaces. At future null infinity, the outgoing modes are positive frequency with respect to the outgoing Eddington-Finkelstein coordinate ${u\equiv t-r^{*}}$, since an observer asymptotically close to that surface will define positive frequency with respect to that coordinate (as argued in Sec. II.2). The natural question is then how this vacuum state should be extended to the interior of the black hole. In studies of analogous acoustic black hole systems [60, 61], these interior modes are also defined with respect to the Eddington-Finkelstein coordinates, in part because the inner horizon of those systems is mimicked by a physically infinite asymptotic regime. For the Reissner-Nordström spacetime, an infaller will not reach an asymptotically steady state at the inner horizon; however, they will approach an asymptotic regime (albeit a transient one) where ${\Delta\to 0}$ and the scattering potential of Eq. (43) vanishes. In the regime where this potential vanishes, as shown in Sec. II.2, freely falling observers experience a proper time proportional to the Eddington-Finkelstein coordinates:

$$f^{+}_{\text{ob}}\to\begin{cases}\text{e}^{-i\omega(t-r^{*})},&\mathscr{I}^{+}\\ \text{e}^{-i\omega(r^{*}-t)},&\cal{H}^{+}_{\textit{r}_{-}}\\ 0,&\cal{H}^{-}_{\textit{r}_{-}}\end{cases},$$

(52)

$$f^{-}_{\text{ob}}\to\begin{cases}0,&\mathscr{I}^{+}\cup\cal{H}^{+}_{\textit{r}_{-}}\\ \text{e}^{-i\omega(r^{*}+t)},&\cal{H}^{-}_{\textit{r}_{-}}\end{cases}.$$

(53)

These modes are shown with solid arrows in Fig. 6. They represent the experience of any inertial observer with arbitrary energy $E_{\text{ob}}$ (up to a rescaling of the frequency $\omega$); without loss of generality, an observer with ${E_{\text{ob}}=1}$ is chosen for the left potion of the inner horizon in Fig. 6 ($\cal{H}^{-}_{\textit{r}_{-}}$), while an observer with ${E_{\text{ob}}=-1}$ is chosen for the right portion ($\cal{H}^{-}_{\textit{r}_{-}}$). Also, note that if the observer is placed at the outer horizon instead of the inner horizon, a similar complete set of modes can be defined mutatis mutandis. In what follows, we will present the results for both sets of modes simultaneously, though we will only closely follow the steps of analysis for the inner horizon observers’ set of modes.

Equipped with a complete set of modes for an Unruh emitter and an inertial observer, one may now proceed to calculate the expectation value of the particle number operator seen by the observer in the emitter’s vacuum state via Eq. (IV.1). To do so, consider what will subsequently be referred to as the past null Cauchy hypersurface, consisting of the union of past null infinity with the exterior and interior past horizons ($\mathscr{I}^{-}\cup\ \cal{H}^{+}_{\textit{r}_{+}}$; see Fig. 6). On this surface, the emitter’s modes are given by Eqs. (49) and (50), while the observer’s modes can be found with scattering theory, as described below.

Since the $t$ coordinate used to define the observer’s modes defines a global timelike Killing vector for the spacetime, the field’s modes $f_{\omega\ell}$ can be separated as

$$f_{\omega\ell}(t,r^{*})\equiv\chi_{\omega\ell}(r^{*})\ \text{e}^{\pm i\omega t}.$$

(54)

This separation puts the Klein-Gordon wave Eq. (43) into the form of a 1D scattering equation in $r^{*}$. In the limits as $\Delta$ approaches both 0 and 1, the scattering potential of Eq. (43) vanishes, leading to asymptotic eigenmode solutions of the form ${\exp(\pm i\omega r^{*})}$. As such, the observer’s modes $\chi^{\pm}_{\text{ob}}$ can be backpropagated to the past null Cauchy hypersurface—altogether, for an observer at future null infinity one has

$${}^{\text{ext}}f^{+}_{\text{ob}}\to\text{e}^{-i\omega t}\begin{cases}\text{e}^{i\omega r^{*}}+\mathcal{R}^{+}_{\text{ext}}\text{e}^{-i\omega r^{*}},&r^{*}_{\text{ext}}\to\infty\\ \mathcal{T}^{+}_{\text{ext}}\text{e}^{i\omega r^{*}},&r^{*}_{\text{ext}}\to-\infty\end{cases},$$

(55)

for an outgoing observer at the inner horizon,

$${}^{\text{int}}f^{+}_{\text{ob}}\to\text{e}^{i\omega t}\begin{cases}\text{e}^{-i\omega r^{*}},&r^{*}_{\text{int}}\to\infty\\ \mathcal{T}^{+}_{\text{int}}\text{e}^{-i\omega r^{*}}+\mathcal{R}^{+}_{\text{int}}\text{e}^{i\omega r^{*}},&r^{*}_{\text{int}}\to-\infty\\ \mathcal{R}^{+}_{\text{int}}\mathcal{T}^{-}_{\text{ext}}\text{e}^{i\omega r^{*}},&r^{*}_{\text{ext}}\to\infty\\ \mathcal{R}^{+}_{\text{int}}\left(\text{e}^{i\omega r^{*}}+\mathcal{R}^{-}_{\text{ext}}\text{e}^{-i\omega r^{*}}\right),&r^{*}_{\text{ext}}\to-\infty\end{cases},$$

(56)

and for an ingoing observer at the inner horizon,

$${}^{\text{int}}f^{-}_{\text{ob}}\to\text{e}^{-i\omega t}\begin{cases}\text{e}^{-i\omega r^{*}},&r^{*}_{\text{int}}\to\infty\\ \mathcal{T}^{-}_{\text{int}}\text{e}^{-i\omega r^{*}}+\mathcal{R}^{-}_{\text{int}}\text{e}^{i\omega r^{*}},&r^{*}_{\text{int}}\to-\infty\\ \mathcal{T}^{-}_{\text{int}}\mathcal{T}^{-}_{\text{ext}}\text{e}^{-i\omega r^{*}},&r^{*}_{\text{ext}}\to\infty\\ \mathcal{T}^{-}_{\text{int}}\left(\text{e}^{-i\omega r^{*}}+\mathcal{R}^{-}_{\text{ext}}\text{e}^{i\omega r^{*}}\right),&r^{*}_{\text{ext}}\to-\infty\end{cases},$$

(57)

where $r^{*}_{\text{int}}$ and $r^{*}_{\text{ext}}$ represent the radial tortoise coordinates $r^{*}$ for the black hole’s interior and exterior, respectively. The reflection coefficients $\mathcal{R}^{\pm}_{\text{int,ext}}$ and transmission coefficients $\mathcal{T}^{\pm}_{\text{int,ext}}$, which depend on the observer’s mode numbers $\omega$ and $\ell$, can be computed numerically (or semianalytically with confluent Heun functions) with the above boundary conditions on the wave Eq. (43); see the Appendix for more details.