The Unruh vacuum and the in-vacuum in the Reissner-Nordstr¨om spacetime ∗

The Unruh vacuum is widely used as quantum state to describe black hole evaporation since near the horizon it reproduces the physical state of a quantum field, the so called in-vacuum, in the case the black hole is formed by gravitational collapse. We examine the relation between these two quantum states in the background spacetime of a Reissner-Nordstr¨om black hole (both extremal and not) highlighting similarities and striking differences.


I. INTRODUCTION
In Quantum Field Theory in black hole (BH) spacetimes particular relevance is given to the so called 'Unruh vacuum' [1].This quantum state is defined on the maximal analytic extension of the BH spacetime by expanding the quantum field operators in ingoing modes modes that are positive frequency with respect to the asymptotic Minkowski time, while the outgoing modes, emerging from the past horizon, are chosen to be positive frequency with respect to Kruskal time.This state is expected to describe, at late retarded time, the quantum state of a field in the spacetime of a collapsing body forming a BH.
Let us make an explicit example in the simple setting of a two-dimensional Schwarzschild spacetime whose metric reads where m is the BH mass and are, respectively, the retarded and advanced Eddington-Finkelstein coordinates.r * is Regge-Wheeler 'tortoise' coordinate Kruskal's null coordinates are defined as [2] U = ±e −κu , (1.5) where κ = 1/4m is the surface gravity of the BH horizon located at 2m.
The maximal analytic extension of the Schwarzschild metric is depicted in the Penrose diagram (see for example [3]) of Fig. (1) describing the eternal Schwarzschild BH.In eq.
(1.5) the sign − holds in the asymptotically flat region R and in the white hole region W H; while − in W H and L. We shall limit our discussion to the physically interesting regions R and BH.
FIG. 1: Penrose diagram for the Schwarzschild geometry.H + is the future event horizon, H − the past one.I − is past null infinity, I + future null infinity.
In this spacetime we consider a massless scalar quantum field φ minimally coupled to gravity, whose field equation is where ∇ µ ∇ µ is the covariant D'Alembertian.
To get the Unruh vacuum one expands the field as The first term in eq.(1.8) represents the ingoing part, while the second the outgoing one.
Note that U is locally inertial on H − .The â's are annihilation operators and the Unruh vacuum |U ⟩ is defined as for every ω, ω K .

Now consider a
where again but now In this case we have two horizons, an event horizon at r + and an inner one at r − where and two corresponding surface gravities We have two possible ways to implement the construction of Kruskal coordinates, namely which are regular on r + but not on r − where they diverge, or [6] U The Unruh vacuum is usually constructed out of U (+) , namely expanding the field as .11)and the Unruh vacuum |U (+) ⟩ is defined by for every ω, ω K .
We shall focus on the renormalized expectation values of Tµν ( φ) [7][8][9] in this state which read [10] (2.13) (2.14) where and a prime " ′ " indicates the derivative with respect to r. { , } means Schwarzian derivative, and we get This last term in eq.(2.15) describes the Hawking radiation at where The tensor ∆ + µν is conserved and describes radiation propagating along constant u.Mathematically one can define also a Unruh vacuum associated to the Kruskal coordinate U (−) of eq.(2.9), i.e. expanding the field in outgoing modes of the form e −iω K U (−) √ 4πω K .The resulting vacuum state, call it |U (−) ⟩, has the following expectation values of Tµν where as before but now (2.25)

VACUUM
Let us now suppose that the RN BH is formed by gravitational collapse of a charged body.We simplify the discussion by considering the collapse of a charged null shell.One can generalize to an arbitrary body [12] .
The corresponding Penrose diagram is given in Fig. (4).
The spacetime metric of this model can be given as following.The shell is located at v = v 0 .For v < v 0 we have Minkowski spacetime where are, respectively, the null outgoing and ingoing directions.From eqs. (3.2), (3.3) we have For v > v 0 we have the RN metric where f (r) is given in eq.(2.16) and u, v and r * in eqs.(2.2), (2.3) and (2.4).Matching the metric across v 0 we can relate the null outgoing coordinates u and u in [13] We can extend the u in coordinate in the RN region using the above relation.The event horizon (r = r + , u = +∞) is located at The inner horizon outgoing sheet (r = r − , u = −∞) corresponds to Note that this null surface, unlike the previous case of the maximal analytic extension of the RN spacetime (see Fig.  4)).This latter is indeed shifted inside the BH and is located at (using the reflection condition at r = 0, see eq. (3.4)) Note that near i.e. u in behaves as Kruskal U (+) of eq.(2.7) up to a shift.Choosing for example v 0 = 2r + they coincide.On the other side, near the outgoing sheet of the inner horizon (i.e. for u → −∞) we have This means that there u in behaves like U (−) (see eq. (2.9)).So u in , unlike U (±) , is a regular null outgoing coordinate both at (r + , u = +∞) and at (r − , u = −∞).However this coordinate is singular on the Cauchy horizon c o , as can be seen from where from eq. (3.6) which vanishes like (u in − v 0 ) 2 for u in → v 0 .Note that C(r) du du in is nonvanishing neither on H + nor on the outgoing sheet of r − making the (u in , v) coordinates regular there as stated before.
Note that past null infinity I − (see Fig ( 4)) is a Cauchy surface for our null field for the spacetime region within the Cauchy horizon c o ∪ c i .Initial conditions on our field φ on I − determine its evolution in the above region.We choose the quantum state for our field to be Minkowski vacuum on I − .This is achieved by choosing the ingoing modes to be of the form where the last term in eq.(3.17) is the Schwarzian derivative between u in and u to be calculated from eq. (3.6) where U in = u in − v 0 + 2r + .The following limiting behaviours are interesting.For u → +∞ (i.e.u in = v 0 − 2r + , U in = 0) we have (see eq. (3.10))We begin with r + .A coordinate system regular there is the Kruskal one (U (+) , V (+) ) of eqs, (2.7),(2.8).We have to require finiteness of ⟨U (+) | Tµν |U (+) ⟩ as r → r + when expressed in these Kruskal coordinates.This implies that on the event horizon H + (r = r + , U (+) = 0) we need for the regularity of |U (+) ⟩ (see [14]) that i) ii) iii) Conditions ii) and iii) are easily seen to be satisfied using eq.( 2 ii) From eqs. (2.14) and (2.15) we see that ii) and iii) are satisfied, but not i).From eq. ( 2. 13) we see that lim and hence condition i) is not satisfied as f → 0. Hence |U (+) ⟩ is not regular on H − .
We now come to the inner horizon, where a coordinate system regular there can be given |U (+) ⟩ is also singular on c i [15,16] (see also [17]).
A similar analysis can be performed on the state |U (−) ⟩ (see eqs. is extremal?Extremal BHs are very interesting objects: they appear also in supergravity theories [18] and represent in some sense a stable ground state for these theories.For extremal BHs the surface gravity vanishes (see eq. (2.6) for m 2 = Q 2 ) and so they do not emit Hawking radiation.They can be considered as the end state of the evaporation of a non extremal BH when the U (1) charge Q is conserved during the process.
We shall now discuss how we can extend the results of our previous investigation to the extremal BH case.The extremal RN BH is characterized by a metric where as usual but now ( hence κ + = 0, we see that the Schwarzian derivative term (2.17) vanishes, so the Unruh state coincides with the Boulware one leading to Beside regularity issues, one can question if the limiting procedure we used to derive eqs. ( Looking at the (u, u) component, we immediately see that it has an extra contribution with respect to the static vacuum polarization ⟨B| Tuu |B⟩ of eq.(5.7).This term represents transient radiation created by the time dependent collapse of the shell decaying at late retarded time (u → +∞, u in = v 0 − 2m) as (u in − v 0 + 2m) 3 .But near the horizon we have , so the two terms in eq.(5.14) vanish with the same power law and they are opposite in sign.This makes the crucial limit for the regularity condition on H + , i.e. eq.(4.1), satisfied [22] ⟨in| Tuu |in⟩ So |in⟩ is regular on H + .We conclude that |B⟩ cannot represent the state of a quantum field emerging at late retarded time for a collapse forming an extremal RN BH.
Note that in order to get the nonvanishing term coming from the Schwarzian derivative it is necessary to keep also the subleading logarithmic term in eq.(5.11) in the limit u in → v 0 − 2m.Omission of it would result in a vanishing Schwarzian derivative, missing therefore the correct result. 1inally, one can verify that |in⟩ is not regular on the ingoing sheet of the Cauchy horizon and also on the outgoing one (u in = v 0 ), where the Schwarzian derivative diverges like (u in − v 0 ) −6 .
diverging as the shell radius goes to zero: we have a thunderbolt null singularity [25,26].
Concerning the inner sheet of the Cauchy horizon, located for both spacetimes at (r = r − , v = +∞), we have that |U (+) ⟩ and |in⟩ are both not regular making the full Cauchy horizon singular.One expects large backreaction effects through the semiclassical Einstein equations to occur there leading to the formation of a spacetime singularity as predicted by the mass inflation mechanism [27], [28].
We finally analysed the case of an extreme BH.It is often said that the Unruh vacuum

FIG. 2 :
FIG. 2: Penrose diagram of the spacetime of a collapsing star The shadowed region represents the star.

. 10 )
FIG. 3: Penrose diagram of the maximal analytic extension of the RN metric.H + is the future event horizon, H − the past one; c o and c i are, respectively, the outgoing and ingoing sheets of the Cauchy horizon located at r − .

. 18 )
The other terms in eqs.(2.13)-(2.15)represent the vacuum polarization contribution and correspond to the expectation values calculated in the so called Boulware vacuum |B⟩ [11], obtained by expanding the field φ in modes that are positive and negative frequency with respect to t, namely ( e −iωu √ 4πω , e −iωv √ 4πω ).We can formally write eqs.(2.13)-(2.15)as

)FIG. 4 :
FIG. 4: Penrose diagram of the charged BH resulting from the collapse of a null shell.
(3)), does not correspond anymore to the outgoing sheet of the Cauchy horizon (see Fig. (

⟨in|
while for u → −∞ (i.e.u in = v 0 − 2r − , U in = 2(r + − r − )), see eq.(3.11), eqs.(3.15)-(3.17)with the corresponding ones (2.13)-(2.15)and (2.23) we can conclude that in the RN region lim u→+∞ ⟨in| Tµν |in⟩ = lim u→+∞ ⟨U (+) | Tµν |U (+) Tµν |in⟩ = lim u→−∞ ⟨U (−) | Tµν |U (−) ⟩ .(3.22)So the Unruh vacuum |U (+) ⟩ reproduces |in⟩ at late (u → +∞) retarded time, while this does not happen at the inner horizon.There the |in⟩ vacuum is well approximated by |U (−) ⟩.IV.REGULARITY We shall consider a quantum state to be "regular" if the expectation values of Tµν in that given state are finite when expressed in a regular coordinate system.Support to this comes from the fact that the expectation values of Tµν in the semiclassical Einstein equations represent the source which drives the backreaction of the quantum fields on the spacetime metric.Let us start with the Unruh vacuum |U (+) ⟩.The expectation values of Tµν in this state are given by eqs.(2.13)-(2.15)and one sees immediately that they diverge on the physical singularity located at r = 0. However the Eddington-Finkelstein coordinates (u, v) used in eqs.(2.13)-(2.15)are singular on the horizons r ± .So care must be taken when considering these regions.
vacuum polarization induced by the BH and this seems consistent with the fact that extremal BHs do not Hawking radiate.It is easy to see that ⟨B| Tµν |B⟩ is not regular on the horizon r + = m[19].Requiring regularity in the free falling frame across the future horizon we get conditions identical to eqs.(4.1)-(4.3)and we see that (4.1) is not satisfied, since now the denominator there vanishes as (r − m)4 while the numerator only as (r − m) 3 (see eq. (5.7)).Similarly, one finds a singular behaviour on the past horizon and on the Cauchy horizon.It is not yet clear if this singular behaviour is an artifact of the two-dimensional theory[20,21].

(
and also the Hartle-Hawking-Israel one[29,30]) coincide with the Boulware one since the surface gravity of these BHs horizons is vanishing.One may further support this idea by remarking the fact that the Unruh vacuum is constructed by outgoing modes which are positive frequency with respect to a null coordinate (U (+) for a non extremal BH) which is locally inertial on the past horizon.For an extremal RN BH, Eddington-Finkelstein u is locally inertial on the past horizon and the outgoing modes of the Boulware vacuum are indeed positive frequency with respect to u.However, we have clearly shown that this Boulware vacuum for an extremal BH does not correctly describe at large u the physical state of a quantum field if the extreme RN BH is formed by the collapse of a star.So its physical significance is quite unclear and the results for extremal BHs obtained by simply taking the κ → 0 limit of expressions calculated for the Unruh or Hartle-Hawking-Israel state in non extremal BHs should be regarded with suspicion.