Sven Zschocke Lohrmann
email: sven.zschocke@tu-dresden.de

A generalized lens equation for small light-deflection angles

Keywords: numbers: 95.10.Jk, 95.10.Ce, 95.30.Sf, 04.25.Nx, 04.80.Cc

A generalized lens equation in Schwarzschild metric and valid for small lightdeflection angles is suggested. The magnitude of neglected terms in the generalized lens equation is estimated to be smaller or equal than 15 π 4

, where m is the Schwarzschild radius of masive body and d is Chandrasekhar's impact vector. The main applications of this generalized lens equation are extreme astrometrical configurations, where Standard post-Newtonian approach as well as Classical lens equation cannot be applied. It is shown that in the appropriate limits the proposed lens equation yields the known post-Newtonian terms, 'enhanced' post-post-Newtonian terms and the Classical lens equation, thus provides a link between these both essential approaches for determining the light-deflection.

Introduction

Todays astrometry necessitates theoretical predictions of light-deflection by massive bodies on microarcsecond (µas) level, e.g. NASA mission SIM or ESA mission GAIA. In principle, an astrometric precision on microarcsecond level can be achieved by numerical integration of geodesic equations of light-propagation. On the other side, modern astrometric missions like GAIA determine the positions and proper motions of approximately one billion objects, each of which is observed about one hundred times. The data reduction of such huge amount of observations implies the need of analytical solutions, because the numerical investigation of geodesic equations is by far too time-consuming. The metric of a massive body is usually expanded in terms of multipoles, i.e. monopole term, quadrupole term and higher multipoles. Numerically, the largest contributions of light-deflection originates from the spherically symmetric part (Schwarzschild) of the massive body under consideration. The exact analytical solution of light-propagation in Schwarzschild metric [START_REF] Chandrasekhar | The mathematical Theory of Black Holes[END_REF] inherits elliptic integrals, those evaluation becomes comparable with the time effort needed for a numerical integration of geodesic equation. Thus, approximative analytical solutions valid on microarcsecond level of accuracy are indispensible for a highly time-efficient data reduction.

Basically, two essential approximative approaches are known. The first one is the Standard parameterized post-Newtonian approach (PPN) which is of the order O (m). During the last decades, it has been the common understanding that the higher order terms O (m 2 ) are negligible even on microarcsecond level, except for observations in the vicinity of the Sun. However, recent investigations [17,1,12,18] have revealed that the post-post-Newtonian approximation [START_REF] Brumberg | Kinematica i physika nebesnykh tel[END_REF][START_REF] Brumberg | Essential Relativistic Celestial Mechanics[END_REF], which is of the order O (m 2 ), is needed for such high accuracy. Both approximations are applicable for d m, where d being the impact vector.

The second one is the Classical lens equation. Soon after the light-deflection by the Sun has been measured in 1919 [7], the gravitational lens effect at stars was discussed [15,[START_REF] Chwolson | [END_REF]8]. In 1937 Fritz Zwicky hypothesized not only could stars act as lenses but galaxy clusters could as well. Later in 1963, by detailed calculations it has been recognized that quasars as pointlike sources are ideal in order to determine the gravitational lensing effect [14,16,[START_REF] Yu | [END_REF]. One decisive advantage of classical lens equation is it's validity for arbitrarily small values of impact vector d. The classical lens equation is valid for astrometrical configurations where source and observer are far ernough from the lens, especially in case of a d and b d, where a = k • x 1 and b = -k • x 0 , where x 0 and x 1 are the three-vectors from the center of the massive body to the source and observer, respectively, and k is the unit vector from the source to the observer.

However, there are astrometric configurations where neither the post-Newtonian approach nor the classical lens equation are applicable. In order to investigate the lightdeflection in such systems a link between these both approaches is needed. Such a link can be provided by a Generalized lens equation which, in the appropriate limits, coincides with Standard post-Newtonian approach and Classical lens equation. Accordingly, the aim of our investigation is an analytical expression for the generalized lens equation having a form very similar to the classical lens equation. We formulate the following conditions under which our generlized lens equation should be applicable (see Figure 1):

1. valid for d = 0 , a = x 1 m , b = x 0 m, 2. valid for a = 0 , d m , b = 0, 3. valid for b = 0 , d m , a = 0.
These conditions imply that the light-path is always far enough from the lens, thus inherit small light deflection angles. Our paper is organized as follows: In Section 2 we present the standard post-Newtonian approach. In Section 3 we show the steps of postpost-Newtonian approach relevant for our investigation, and we will briefly summarize the main results of our paper [12]. The classical lens equation is derived in Section 4. The generalized lens equation is obtained in Section 5 and discussed in Section 6. A summary is given in Section 7.

We will use fairly standard notations:

• G is the Newtonian constant of gravitation.

• c is the velocity of light.

• β and γ are the parameters of the Parametrized Post-Newtonian (PPN) formalism which characterize possible deviation of the physical reality from general relativity theory (β = γ = 1 in general relativity).

• Lower case Latin indices i, j, . . . take values 1, 2, 3.

• Lower case Greek indices µ, ν, . . . take values 0, 1, 2, 3.

• A dot over any quantity designates the total derivative with respect to the coordinate time of the corresponding reference system: e.g. ȧ = da dt .

• The 3-dimensional coordinate quantities ("3-vectors") referred to the spatial axes of the corresponding reference system are set in boldface: a = a i .

• The absolute value (Euclidean norm) of a "3-vector" a is denoted as |a| or, simply, a and can be computed as a = |a| = (a 1 a 1 + a 2 a 2 + a 3 a 3 ) 1/2 .

• The scalar product of any two "3-vectors" a and b with respect to the Euclidean metric δ ij is denoted by a • b and can be computed as a

• b = δ ij a i b j = a i b i .
• The vector product of any two "3-vectors" a and b is designated by a × b and can be computed as (a × b) i = ε ijk a j b k , where

ε ijk = (i -j)(j -k)(k -i)/2
is the fully antisymmetric Levi-Civita symbol.

• For any two vectors a and b, the angle between them is designated as δ(a, b). Throughout the paper, we work in harmonic gauge, where the components of metric tensor of the Schwarzschild solution is given by g 00 = -1 -a 1 + a , g i0 = 0,

g ij = (1 + a) 2 δ ij + a 2 x 2 1 + a 1 -a x i x j , where a = m x , m = G M c 2
, and M is the mass of massive body.

Standard post-Newtonian solution

Let us consider the trajectory of a light-signal in post-Newtonian Schwarzschild metric:

g 00 = -1 + 2 a + O (c -4 ), g i0 = 0, g ij = δ ij + 2 γ a δ ij + O (c -4
). The light-ray is being emitted at a position x 0 at time moment t 0 and received at position x 1 at a time moment t 1 , see Figure 1. Light propagation is governed by geodesic equation, in post-Newtonian order given by ẍ

= -(1 + γ) c 2 a x x 2 + 2 (1 + γ) a ẋ ( ẋ • x) x 2 + O(c -2 ) . (1)
The unit tangent vector n at the point of observation is defined by . Furthermore, we define the unit vector

n = ẋ(t 1 ) | ẋ(t 1 )| , (2)
k = R R
, where R = x 1x 0 and the absolute value is R = |R|. In the post-Newtonian standard approach [11], the transformation between n and k reads:

n = k -(1 + γ) m d d d x 0 x 1 -x 0 • x 1 R x 1 + O c -4 , (3)
where d is the impact parameter

d = k × (x 0 × k) = k × (x 1 × k) . (4)
Eq. ( 3) is the standard post-Newtonian formula of light-deflection. By means of sin ϕ = |n × k|, we find the light-deflection angle ϕ = δ(n, k) in post-Newtonian approximation:

ϕ = (1 + γ) m d x 0 x 1 -x 0 • x 1 R x 1 + O c -4 . (5)
A post-post-Newtonian approximation shows that the neglected terms in Eq. ( 3) are of the order O m 2 d 2

x 1 d ; see Eq. (94) in [12]. Due to the fact that x 1 d can be arbitrarily large, the contribution of the neglected terms in Eq. (3) can also be arbitrarily large in general, depending on the concrete astrometrical configuration under consideration.

Post-post Newtonian solution

Now we will consider the trajectory of a light-signal in post-post-Newtonian Schwarzschild metric:

g 00 = -1 + 2 a -2 β a 2 + O (c -6 ), g i0 = 0, g ij = δ ij + 2 γ a δ ij + δ ij + x i x j x 2 a 2 + O c -6
. The geodesic equation of light-propagation in post-post-Newtonian approximation is given by [12]

ẍ = -(1 + γ) c 2 a x x 2 + 2 (1 + γ) a ẋ ( ẋ • x) x 2 + 2 c 2 α (β -+ 2 γ (1 + γ)) a 2 x x 2 + 2 α a 2 x ( ẋ • x) 2 x 4 + 2 α (2(1 -β) + -2 γ 2 ) a 2 ẋ ( ẋ • x) x 2 + O c -4 . (6)
The solution of ( 6) and the transformation between the unit vectors n and k in postpost-Newtonian order has been given in [12]. Here, we will write this transformation in the following form:

n = k -(1 + γ) m d d d x 0 x 1 -x 0 • x 1 R x 1 + (1 + γ) 2 m 2 d 2 d d (x 0 x 1 -x 0 • x 1 ) 2 R 2 d x 1 + n scaling 1 + n ppN 1 + O c -6 , (7)
where the terms n scaling × k = 0, we note the esimate (for a proof see [19,20]):

ω ppN 1 = n ppN 1 × k ≤ 15 π 4 m 2 d 2 . ( 8
)
From ( 7) we obtain

ϕ = (1 + γ) m d x 0 x 1 -x 0 • x 1 R x 1 -(1 + γ) 2 m 2 d 2 (x 0 x 1 -x 0 • x 1 ) 2 R 2 d x 1 -ω ppN 1 + O c -6 . (9)
Accordingly, the post-post-Newtonian solution contains 'regular' terms of order m 2 which have been estimated in (8) and so-called 'enhanced' terms of order m 2 (i.e. the last term in the first line of Eq. ( 7) or second term in ( 9)) which can be much larger than the 'regular' ones. Note, that the 'enhanced' terms of order m 2 contain only the parameter γ, thus these terms come from the post-Newtonian terms in the metric and geodesic equation. Therefore, their origin is the second-order (post-post-Newtonian) solution of the first-order (post-Newtonian) geodesic equation given in (1). This fact has been clarified in [12]. Furthermore, in [12] it has been demonstrated that the 'enhanced' terms result from the use of impact vector (4). This impact vector is convinient for determining the light-propagation when positions of source and observer are given in a concrete reference systems, e.g. the BCRS [11].

The light-deflection formula in (7) can be transformed in a form which resembles the post-Newtonian formula (3) by means of the coordinate-independent impact vector d , cf. Eq. (57) of [12]:

d = lim t→-∞ σ × (x(t) × σ) . (10)
This impact parameter has the property d ≥ d and is identical to Chandrasekhar's impact parameter (see Eq.( 215) in chapter 20 of [START_REF] Chandrasekhar | The mathematical Theory of Black Holes[END_REF]), that means in vectorial form d = L E , where L is the orbital three-momentum and E is the energy of the photon on the light-trajectory. While the impact vector d is a simple expression in terms of x 0 and x 1 , the impact parameter d is uniquely determined by x 0 and x 1 and by the mass of the light deflecting body, but a complicated expression in terms of elliptic integrals. An approximative relation between the impact vectors (4) and [START_REF] Yu | [END_REF] has been given in Eq. (62) in [12] and reads:

d = d d d + (1 + γ) m d x 1 + x 0 R x 0 x 1 -x 0 • x 1 R -(1 + γ)m k x 1 -x 0 + R R + O c -4 , (11)
d = d + (1 + γ) m d x 0 + x 1 R x 0 x 1 -x 0 • x 1 R + O c -4 . (12)
With the aid of the coordinate-independent impact vector Eq. ( 10), the transformation between the vectors n and k in post-post-Newtonian order can be written as follows, cf. Eqs. ( 108) and (109) of [12]:

n = k -(1 + γ) m d d 2 x 0 x 1 -x 0 • x 1 R x 1 + n scaling 2 + n ppN 2 + O c -6 . ( 13
)
The scaling term n scaling 2 and n ppN 2 are given in Appendix A. The expressions ( 7) and ( 13) are identical. While n scaling 2 × k = 0, we have the inequality (for a proof see [21])

ω ppN 2 = n ppN 2 × k ≤ 15 π 4 m 2 d 2 . ( 14
)
From Eq. ( 13) we obtain the expression

ϕ = (1 + γ) m d x 0 x 1 -x 0 • x 1 R x 1 -ω ppN 2 + O c -6 . ( 15
)
The form of Eq. ( 15) resembles the post-Newtonian formula (3).

Classical lens equation

In this Section we consider some basic steps leading to the Classical lens equation.

According to the scheme in Figure 2, we obtain the following geometrical relations,

ϕ + ψ = δ , (16)
a tan ϕ = b tan ψ , (17)
where the angles ϕ = δ(n, k), ψ = δ(k, µ), and the unit tangent vector µ = ẋ(t 0 ) | ẋ(t 0 ) | . x 0

If source and observer are infinitely far from the massive body, then the total lightdeflection angle δ = δ (n, µ) in Schwarzschild metric reads [2]

δ = 2 (1 + γ) m d + O m 2 d 2 + O c -6 , (18)
which is a coordinate independent result. The terms of order O m 2 d 2 can be estimated to be smaller or equal than 15 π 4 m 2 d 2 , see [2]. According to (16) in the limit

x 0 = x 1 → ∞ we should have ϕ = δ 2
, a strict condition which is satisfied by the expression (15) up to the given order.

In classical lens approach, the approximation d d + a tan ϕ is used, see Figure 2. Inserting this relation into (18), by means of geometrical relations ( 16) and ( 17), and using tan ϕ = ϕ + O(ϕ 3 ) and tan ψ = ψ + O(ϕ 3 ), we obtain the quadratic equation

ϕ 2 + d a ϕ -2 (1 + γ) m a b a + b = 0 . ( 19
)
The solution of Eq. ( 19) is the classical lens equation,

ϕ class 1,2 = 1 2   d 2 a 2 + 8(1 + γ) m a b a + b ∓ d a   , ( 20
)
where the solution with the upper (lower) sign is denoted by ϕ 1 (ϕ 2 ). The classical lens equation is valid for arbitrary small values of d. Below we will see that ( 20) is restricted for configurations where a d and b d. This is because relation (18) and the used approximation for d is only valid when source and observer are at infinity. In the indroductionary Section we have formulated the three conditions under which the generalized lens equation should be valid. We first notice that according to d = L E , the impact vector d is finite in the limit d → 0. On the other side, the approximative relation ( 12) is not finite in this limit. However, up to the given order we can rewrite (12) as follows:

Generalized lens equation

d = d + (1 + γ) m d x 0 + x 1 R x 0 x 1 -x 0 • x 1 R + O c -4 , (21)
which is a quadratic equation for Chandrasekhar's impact parameter, having the solution

d = d 2 + d 2 4 + (1 + γ) m x 0 + x 1 R x 0 x 1 -x 0 • x 1 R + O c -4 . (22)
We note, that expression ( 22) is finite for d → 0 as well as in the limit R → 0. A comparison of ( 21) with ( 15) yields the relation

d = d + x 1 ϕ + x 0 + x 1 -R R x 1 ϕ + O c -4 , (23)
where ϕ is given by (15). By inserting (23) into ( 15) we obtain the quadratic equation

ϕ 2 + d x 1 ϕ -(1 + γ) m x 1 x 0 x 1 -x 0 • x 1 R x 1 + δ 1 + δ 2 = O(c -6 ) , (24)
where

δ 1 = d x 1 ω ppN and δ 2 = x 0 + x 1 -R R ϕ 2
, and we note δ 1 ϕ 2 and δ 2 ϕ 2 . The solution of quadratic equation (24) reads

ϕ 1,2 = 1 2 d 2 x 2 1 + 4 (1 + γ) m x 1 x 0 x 1 -x 0 • x 1 R x 1 ∓ d x 1 + Ω ppN + O c -6 , (25)
where ϕ 1 (ϕ 2 ) is the solution with the upper (lower) sign. Eq. ( 25) represents the generalized lens equation, which is valid in all those extreme astrometrical configurations defined in 1. -3. in the introductionary Section. One can show, that expression Ω ppN can be estimated as follows:

Ω ppN = 1 4 (δ 1 + δ 2 ) d 2 x 2 1 + 4 (1 + γ) m x 1 x 0 x 1 -x 0 • x 1 R x 1 -1/2 ≤ 15 π 4 m 2 d 2 . (26)
In the following Section we will show that the formula (25) represents a link between Standard post-Newtonian approach and Classical lens equation.

Discussion of generalized lens equation

Comparison with post-Newtonian and post-post-Newtonian solution

In this Section we compare the generalized lens equation (25) with the standard post-Newtonian and post-post-Newtonian approach of light-deflection. A series expansion of the solution ϕ 1 in Eq. (25) for d m yields

ϕ 1 = ϕ pN + ϕ ppN + Ω ppN + O c -6 , (27)
with

ϕ pN = (1 + γ) m d x 0 x 1 -x 0 • x 1 R x 1 , (28)
ϕ ppN = -(1 + γ) 2 m 2 d 2 (x 0 x 1 -x 0 • x 1 ) 2 R 2 d x 1 . ( 29
)
This expression coincides with (9) up to the given order. That means, a series expansion of the generalized lens equation (25) yields the standard post-Newtonian solution (3) and the 'enhanced' post-post-Newtonian term in (7); cf. Eqs. ( 3) and ( 4) in [21]. A further series expansion of Eq. ( 25) would yield the 'enhanced' terms beyond post-post-Newtonian terms.

Derivation of classical lens equation from generalized lens equation

In order to consider the relation of generalized lens equation ( 25) and classical lens equation (20), we rewrite (25) in terms of a and b and obtain A corresponding series expansion of generalized lens equation (30) in terms of 1 and 2 leads to (see Appendix B)

ϕ 1,2 = 1 2       d 2 a 2 + 2(1 + γ) m a (a + b) 2 -a 1 + 2 1 -b 1 + 2 2 2 (a + b) a ∓ d a       1 1 + 2 1 + Ω ppN + O c -6 , (30)
ϕ 1,2 = ϕ class 1,2 + O m a d a + O m a d b + Ω ppN + O c -6 , (31)
where ϕ class 1,2 is defined by Eq. (20). For astrometry the relevant configurations are such that d a, while in extreme but nevertheless real astrometrical configurations the situation b d is possible, e.g. in binary systems. Therefore, the classical lens equation is not applicable for such configurations.

Finally, we also note that in the limit d → 0, known as Einstein ring solution, we obtain

lim d→0 ϕ 1,2 = 2 (1 + γ) m x 1 x 0 x 0 + x 1 . (32)
In the limit d → 0 the classical lens equation coincides with (32); note, that in this limit a = x 1 and b = x 0 . In the extreme configuration b = 0 (in this limit ϕ 2 does not exist) we obtain the finite result

lim b→0 ϕ 1 = 1 2   d 2 x 2 1 + 4 (1 + γ) m x 1 d a (x 1 + d) x 1 - d x 1   , (33)
while the classical lens equation yields simply ϕ class 1 = 0. Obviously, in the limit a → 0 the expression (33) yields zero as it has to be.

Comparison with exact solution

The accuracy of (25) and the estimate (26) has been confirmed by a comparison with the exact numerical solution. For that, we have solved the geodesic equations in Schwarzschild metric by numerical integrator ODEX [START_REF] Hairer | Solving Ordinary Differential Equations 1[END_REF] for several extreme astrometrical configurations. Using forth and back integration a numerical accuracy of at least 10 -24 in the components of position and velocity of the photon is guaranteed. Thus, the numerical integration can be considered as an exact solution of geodesic equation. This numerical approach has been described in some detail in [12]. In all considered extreme configurations the validity of (25) and the estimate (26) have been confirmed. As a typical example, in Figure 3 we present the results for light-deflection of a grazing ray at Sun and Jupiter. The accuracy of generalized lens equation ( 25) is considerably better than the post-post-Newtonian solution investigated in detail in [19], cf. FIG. 3 (B) with FIG. 2 in [19]. x 0 [AU]

(B)

Summary

Modern astrometry of microarcsecond level, e.g. ESA mission GAIA, needs highly precise approximative solutions for light-deflection. In order to determine the light deflection analytically, there are two essential approaches: Standard post-Newtonian approach valid for m d and Classical lens equation valid for source and observer far enough from the lens, especially when a d and b d. However, there are astrometrical configurations, where none of them are applicable, e.g. in binary systems. In order to investigate the light-deflection in such systems a link between these both approaches is needed. Such a link can be provided by a Generalized lens equation which, in the appropriate limits, coincides with Standard post-Newtonian approach and Classical lens equation. In our study we have suggested a generalized lens equation (25) for Schwarzschild metric which is valid for small light-deflection angles. The derivation is based on the solution of geodesic equation in post-Newtonian metric and Chandrasekhar's coordinate independent impact parameter d [START_REF] Yu | [END_REF]. The neglected terms in (25) can be estimated by Ω ppN ≤ 15 π 4 m 2 d 2 . The accuracy of generalized lens equation ( 25) is considerably better than the post-post-Newtonian solution investigated in some detail in [19].

The generalized lens equation satisfies three conditions formulated in the introductionary Section. Especially, we have shown that in the appropriate limits we obtain the post-Newtonian terms, 'enhanced' post-post-Newtonian terms and the Classical lens equation. Thus, the generalized lens equation (25) provides a link between these both essential approaches to determine the light-deflection. Numerical investigations have confirmed the analytical results obtained. The generalized lens equation (25) will allow an analytical understanding and investigation of light-deflection in extreme astrometric configurations.

  Clearly, for an angle between two vectors one has 0 ≤ δ(a, b) ≤ π. Angle δ(a, b) can be computed in many ways, for example, as δ(a, b) = arccos a • b a b .

Figure 1 .

 1 Figure 1. A geometrical representation of the boundary problem under consideration for a light-propagation from the source to the observer.

1 and n ppN 1 are

 1 given in Appendix A. While n scaling 1

Figure 2 .

 2 Figure 2. A geometrical representation of gravitational lens.

O m 2 d 2 .

 2 There are extreme astrometrical configurations, where b ∼ d or even b d are possible. That means, in general astrometric configurations, neither m d nor d b are small parameters. For instance, in binary systems the situations d = 0 or b = 0 are possible. However, a small parameter it given by m d 1, because for b ≥ 0 the impact parameter d is always larger than the radius of massive body. Accordingly, the aim of our investigation is to derive a generalized lens equation valid up to terms of the order Such a generalized lens equation will provide a link between Standard post-Newtonian approach and Classical lens equation.

  lens equation (20) is valid in case of a, b d.

Figure 3 .

 3 Figure 3. Comparison of generalized lens equation (25) with exact numerical solution. (A) for the case of a grazing ray at Sun (m = 1476.6 m, d = 696.0 × 10 6 m). (B) for the case of a grazing ray at Jupiter (m = 1.40987 m, d = 71.492 × 10 6 m).

Acknowledgments

This work was partially supported by the BMWi grants 50 QG 0601 and 50 QG 0901 awarded by the Deutsche Zentrum für Luft-und Raumfahrt e.V. (DLR). Enlighting discussions with the Professors Sergei A. Klioner, Michael Soffel and Chongming Xu are greatfully acknowledged.

According to Eq. ( 87) in [12], the transformation between n and k in post-post-Newtonain order is given by Eq. ( 13), where the scaling term is given by

Furthermore,

In Eq. ( 13) we have introduced the scaling term

and

In case of general theory of relativity we have β = γ = = 1. The choice α = 0 means that we obtain the solution of post-Newtonian geodesic equation (1).

Appendix B. Series expansion of generaized lens equation

The series expansion of generalized lens equation (25) in the form (30) yields ϕ 1,2 = ϕ class 1,2 -


L C Barbado

C Barceló

L J Garay
email: luis.garay@fis.ucm.es

Hawking radiation as perceived by different observers

Keywords: numbers: 04.20.Gz, 04.62.+v, 04.70.-s, 04.70.Dy, 04.80.Cc Black Holes, Hawking Radiation, quantum field theory in curved spacetime, vacuum states

We use a method recently introduced in Barceló et al, 10.1103/Phys-RevD.83.041501, to analyse Hawking radiation in a Schwarzschild black hole as perceived by different observers in the system. The method is based on the introduction of an "effective temperature" function that varies with time. First we introduce a nonstationary vacuum state for a quantum scalar field, which interpolates between the Boulware vacuum state at early times and the Unruh vacuum state at late times. In this way we mimic the process of switching on Hawking radiation in realistic collapse scenarios. Then, we analyse this vacuum state from the perspective of static observers at different radial positions, observers undergoing a free-fall trajectory from infinity, and observers standing at rest at a radial distance and then released to fall freely towards the horizon. The physical image that emerges from these analyses is rather rich and compelling. Among many other results, we find that generic freely-falling observes do not perceive vacuum when crossing the horizon, but an effective temperature a few times larger than the one that they perceived when started to free-fall. We explain this phenomenon as due to a diverging Doppler effect at horizon crossing.

Introduction

Arguably, the discovery that black holes should evaporate by emitting a Planckian spectrum of particles, is the most important result of combining general relativistic with quantum mechanical notions [START_REF] Hawking | Black hole explosions[END_REF][START_REF] Hawking | Particle creation by black holes[END_REF]. As it is by now well known, Hawking radiation is a kinematic effect (that is, does not directly depend on Einstein's equations), and so, much more general than its incarnation in General Relativity. In any phenomenon Confidential: not for distribution. Submitted to IOP Publishing for peer review 24 March 2011 in nature describable in terms of quantum excitations propagating in a Lorentzian geometry with black-hole-like properties, one expects to have Hawking particle (or quasi-particle) production. In [START_REF] Barceló | Analogue gravity[END_REF], for example, you can find a catalogue of systems and situations outside the General Relativity realm in which Hawking radiation is expected to appear. Here, by black-hole-like properties we mean either the presence of trapping horizons or just the asymptotic approach towards configurations with horizons [START_REF] Barceló | Minimal conditions for the existence of a Hawking-like flux[END_REF][START_REF] Barceló | Hawking-like radiation does not require a trapped region[END_REF][START_REF] Grove | Observations on particle creation by static gravitational fields[END_REF].

When the Lorentzian geometry acquires the form of a stationary black hole, the temperature of Hawking radiation becomes constant and proportional to the surface gravity of the black hole at the horizon [START_REF] Hawking | Particle creation by black holes[END_REF]. Moreover, this temperature only depends on this constant, leaving no trace of the history of the black hole geometry formation or preparation [START_REF] Hawking | Particle creation by black holes[END_REF][START_REF] Brout | A Primer for black hole quantum physics[END_REF]. In the General Relativity realm that we shall adopt in this paper, this surface gravity and so Hawking temperature, happens to be inversely proportional to the mass of the black hole.

When investigating Hawking radiation, it is customary to work on a simple static background geometry, namely the Schwarzschild solution, neglecting any back-reaction on the geometry. In order to have Hawking radiation at infinity, it is necessary to set the quantum field in a particular quantum state: the Unruh vacuum state [START_REF] Unruh | Notes on black hole evaporation[END_REF][START_REF] Fabbri | Modelling Black Hole Evaporation[END_REF]. This state is commonly described as being a vacuum state for observers freely falling at the event horizon of the black hole. It is a stationary state, describing a black hole which has always been and will ever be emitting radiation at Hawking temperature at all times. Therefore, it does not capture the process of the black hole formation in General Relativity, in which Hawking radiation switches on in the last stages of collapse. Instead, in this paper we are going to analyse the characteristics of a different vacuum state, which might be called the collapse vacuum. This vacuum is appropriately chosen so as to mimic the process of switching on of Hawking radiation. Initially, there will be no radiation at infinity, but at some particular time, some radiation will start to be noticeable. After some rather quick transient regime, this radiation will become thermal with Hawking temperature (as in this paper we are not going to consider the effects of the evaporation, it will not be necessary to distinguish between the terms thermal and Planckian, and we will use them indistinctively). Our collapse vacuum interpolates between the Boulware vacuum state [START_REF] Boulware | Quantum Field Theory in Schwarzschild and Rindler Spaces[END_REF] at early times and the Unruh vacuum state at late times.

It is well known that "presence of particles" is an observer dependent notion [START_REF] Unruh | Notes on black hole evaporation[END_REF][START_REF] Birrell | Quantum Fields in Curved Space[END_REF]. In this paper we will be specifically interested in analysing how the collapse vacuum state is perceived by different observers. In particular, we are going to consider (i) static observers at different radial positions of spacetime, (ii) observers undergoing a free-fall trajectory from infinity, passing through a fixed radial position at different times, and (iii) observers standing at rest at a radial distance until a moment at which they are released to fall freely towards the hole. A discussion about the different perceptions of the radiation from a black hole in the Unruh state by freely-falling and stationary observers can be found in [START_REF] Greenwood | Hawking radiation as seen by an infalling observer[END_REF].

In order to analyse this particle perception, we could use Bogoliubov coefficients [START_REF] Birrell | Quantum Fields in Curved Space[END_REF], but these, except for a few particular simple cases, are difficult to calculate. Instead, we could use the functional Schrödinger formalism as in [START_REF] Greenwood | Hawking radiation as seen by an infalling observer[END_REF], but it appears difficult to manipulate for non-stationary vacuum states. In this paper we will adopt a different strategy. Following [START_REF] Barceló | Hawking-like radiation from evolving black holes and compact horizonless objects[END_REF], we shall define and calculate a time dependent effective temperature for each observer along its world line. Then, abusing of the language we will describe the perception of the observer as experiencing a thermal radiation wind with a time dependent temperature. This effective temperature will only be a real temperature in situations in which an additional adiabatic condition is satisfied. We will also calculate when this condition is satisfied and when it is not. Even in the cases in which the adiabatic condition is violated, and therefore the radiation spectrum differs from the Planckian shape, it will be argued that the calculated effective temperature is still useful as an estimator of the amount of particles encountered by the observer at each instant of time. But it is important to remark that in these cases our method cannot give information about the characteristics of the spectrum. In any case one should keep in mind that a precise calculation of the particle perception would require the construction of a particle detector, taking into account its spatial and temporal resolutions. This is, however, beyond the scope of this paper. Nonetheless, we will show that the physical image that emerges from our description is rather rich and compelling. For instance, it will allow us to clearly see that freely-falling observers at the horizon will not perceive vacuum unless (a) they closely follow the collapse or (b) they have instantaneously zero radial velocity at the horizon. We will also show that this result is not an artifact of the chosen vacuum, but that it is precisely a characteristic of the final Unruh state. The paper is structured as follows. In section 2, we will start recalling the basic background ingredients needed for our analysis. In section 3, we will describe and fix the vacuum state for the quantum radiation field of our problem. Then, in section 4, after describing what we mean by effective temperature and adiabatic control functions, we will study in detail the radiation perception associated with the already enumerated different observers. We will devote section 5 to the explanation of one of our most important results: the non-zero perception of radiation by freely-falling observers at the horizon, against the opposite naive expectation. After a brief comparison of the results with those that one would obtain in a pure Unruh state (section 6), we will finally summarize and conclude.

Preliminaries

Geometrical set up

We shall restrict ourselves to the analysis of radiation emission in a Schwarzschild black hole. In units where G = c = 1, the Schwarzschild metric reads

ds 2 = -1 - 2m r dt 2 + 1 - 2m r -1 dr 2 + r 2 dΩ 2 2 , (1)
with m being the mass of the black hole. Moreover, we will only use the (t, r) sector of the geometry. In double null coordinates, the corresponding (1 + 1)-metric can be written as

ds 2 = -1 - 2m r dūdv. (2)
Here,

ū := t -r * , v := t + r * , (3)
with r * being the "tortoise" coordinate

r * := r + 2m ln r 2m -1 . (4)

Quantum field theory

We are going to analyse the behaviour of a quantum field in this spacetime. For simplicity, we will only consider a Klein-Gordon massless scalar field (analyses with other fields would yield qualitatively similar results). In this paper, we will only consider spherically symmetric configurations and ignore any backscattering of the scalar field in the geometry, so that our analysis will be equivalent to that in a (1 + 1)-dimensional conformally invariant theory (here on, we will stick to a pure 1 + 1 analysis). Now, the 1 + 1 Klein-Gordon equation in null coordinates reads

∂ ∂ ū ∂ ∂v φ(ū, v) = 0, (5)
so that its general solution can be written as

φ = f (ū) + g(v). (6)
Owing to the conformal invariance of the 1 + 1 Klein-Gordon equation, any relabelling ū = p vs (U ), v = q vs (V ) (the subscript 'vs' in these functions stands for vacuum state; the reason for this notation will be understood later on) can be associated with a mode decomposition of the quantum field of the form

φ = ∞ 0 dω âU ω φ U ω + âV ω φ V ω + âU † ω (φ U ω ) * + âV † ω (φ V ω ) * , (7)
where the normalized modes in this expression are

φ U ω = 1 √ 4πω e -iω U , φ V ω = 1 √ 4πω e -iω V . ( 8
)
This can easily be seen by using the following form of the Klein-Gordon scalar product

φ 1 , φ 2 := -i -dUφ 1 ↔ ∂ U φ * 2 + dV φ 1 ↔ ∂ V φ * 2 . ( 9
)
Once the functions ū = p vs (U ), v = q vs (V ) have been defined, one can select a vacuum state |0 by requiring that it satisfies

âU ω |0 = 0, âV ω |0 = 0. ( 10
)
For any other choice ū = p ob (u), v = q ob (v), we can perform an analogue mode decomposition and also define a vacuum state,

âu ω |0 = 0, âv ω |0 = 0. ( 11
)
In general, the vacuum states |0 and |0 will be different. Thus, an observer for which the vacuum is |0 might detect particles corresponding to the vacuum |0 .

Bogoliubov coefficients

The particle content of the vacuum |0 as seen by observers with an unprimed vacuum notion can be calculated through Bogoliubov coefficients. In this paper we will only worry about outgoing radiation, so we will forget about the φ v ω modes, which correspond to rays going into the black hole. For each frequency ω, we have (see [START_REF] Birrell | Quantum Fields in Curved Space[END_REF])

0 |N u ω |0 = 0 |a u † ω a u ω |0 = dω |β ωω | 2 , ( 12
)
where β ωω is the Bogoliubov coefficient defined by the scalar product

β ωω := -φ u ω , (φ U ω ) * , (13)
with φ u ω defined analogously to [START_REF] Unruh | Notes on black hole evaporation[END_REF]. Plugging the expressions for these modes into [START_REF] Fabbri | Modelling Black Hole Evaporation[END_REF], one arrives at

β ωω = 1 4π √ ωω ω due -i(ω U +ωu) -ω dU e -i(ω U +ωu) , (14)
or, integrating by parts the second term, at

β ωω = 1 2π ω ω due -iω U (u) e -iωu , (15)
(where we have eliminated an irrelevant boundary term). Therefore, to determine the particle content, one only needs to know the relation U (u).

Vacuum state choice

We are interested in analysing how To impose this vacuum state to the quantum field, we will consider a time-like geodesic observer falling into the Schwarzschild black hole from infinity. We will then calculate how this observer would label in a natural way the different outgoing (ū = const) rays that he encounters in its way towards the horizon. This new label U = p -1 vs (ū) will then be used to define the vacuum state via a positive-frequency mode expansion, as explained in section 2.2 [see [START_REF] Brout | A Primer for black hole quantum physics[END_REF]].

So let us first calculate the radial time-like geodesic trajectories by solving the geodesic equation

d 2 r dτ 2 = -Γ r rr dr dτ 2 -Γ r tt dt dτ 2 . ( 16
)
With this aim we will take into account the relation dt dτ

2 = 1 - 2m r -1 1 + 1 - 2m r -1 dr dτ 2 (17)
that follows from the spacetime interval (equivalently, we could use directly a variational principle for the spacetime interval). Here, τ is the proper time of the trajectory, and Γ r rr and Γ r tt are the only non-vanishing Christoffel symbols relevant for the calculation of radial trajectories,

Γ r rr = - m r 2 1 - 2m r -1 , Γ r tt = m r 2 1 - 2m r . (18)
Replacing them in the radial geodesic equation yields

d 2 r dτ 2 = - m r 2 , (19)
which has a form resembling Newton's gravitational force. Integrating once, we obtain

dr dτ = - 2m r 1 - r r 0 1/2 , ( 20
) dt dτ = 1 - 2m r -1 1 - 2m r 0 1/2 , (21)
in which we use the radius r 0 at which the velocity vanishes as an integration constant.

For the time being, we are interested in describing a free-fall trajectory starting at radial infinity, that is, with r 0 → ∞. Then, integrating the radial equation we obtain

r = 3 √ 2m 2 (τ 0 -τ ) 2/3 , (22)
with τ 0 being an integration constant. Note that τ runs from -∞ to τ 0 , at which time the trajectory reaches r = 0. Before that, at τ 0 -4m/3 the event horizon is crossed. We can also find the behaviour of the time coordinate t with respect to τ . Indeed, dividing equations (20) and (21) (with r 0 → ∞), we obtain

dr dt = - 2m r 1 - 2m r , (23)
which can be integrated to yield

t = t 0 -4m r 2m + 1 3 r 2m 3/2 + 1 2 ln r/(2m) -1 r/(2m) + 1 , (24)
where t 0 is an integration constant. Finally, replacing the relation (22) for r(τ ) in this expression, we obtain

t = t 0 -4m 3(τ 0 -τ ) 4m 1/3 + τ 0 -τ 4m + 1 2 ln (3(τ 0 -τ )/4m) 1/3 -1 (3(τ 0 -τ )/4m) 1/3 + 1 . ( 25
)
Now, having the pair (t, r) as a function of τ in ( 22) and (25), we can also write the quantity ū in (3) as a function of τ ,

ū = t 0 -4m 3(τ 0 -τ ) 4m 1/3 + 1 2 3(τ 0 -τ ) 4m 2/3 + τ 0 -τ 4m + ln 3(τ 0 -τ ) 4m 1/3 -1 . ( 26
)
An observer following this world line can naturally use its proper time τ to label the different rays he encounters in its way towards the horizon, namely, he can make the assignment of the label U (ū) = τ , given by the inverse of (26), to the ray ū = const that hits the observer at proper time τ . Then, defining U H := τ 0 -4m/3 and choosing the two constants t 0 = τ 0 so that the two labels U and ū are synchronized at ū → -∞, in the sense that U = U (ū → -∞) → ū, the final relation between both labels is

ū = p vs (U ) := U H + 4m 3 -4m 3 4m (U H -U ) + 1 1/3 + 1 2 3 4m (U H -U ) + 1 2/3 + 1 3 3 4m (U H -U ) + 1 + ln 3 4m (U H -U ) + 1 1/3 -1 . ( 27
)
Note that, while the range of the original null label is ū ∈ (-∞, +∞), the range of the new null label is U ∈ (-∞, U H ), where U H marks the moment at which the falling observer crosses the event horizon. As explained before, associated with this choice U = p -1 vs (ū), we have a vacuum state, and it is this vacuum state which will be the subject of our analysis.

Radiation perception for different observers

In this section, we will investigate what is the particle content of the vacuum state defined above for different types of observers, for which the notion of vacuum might or might not coincide with this one. More specifically, we are going to define yet another label u = p -1 ob (ū) associated, in each case, with a specific type of observer. Once we have characterized the observer by the null-ray labelling function, we will find the function U = p -1 vs [p ob (u)] := U (u) which will inform us about the (observer dependent) particle content of the chosen vacuum state.

Variable temperature-like estimator

As an alternative to the calculation of the exact Bogoliubov coefficients using [START_REF] Carusotto | Numerical observation of Hawking radiation from acoustic black holes in atomic Bose-Einstein condensates[END_REF], we can estimate the amount of particle content by calculating the function

κ(u) := - d 2 U du 2 dU du -1 . ( 28
)
As described thoroughly in [START_REF] Barceló | Minimal conditions for the existence of a Hawking-like flux[END_REF][START_REF] Barceló | Hawking-like radiation from evolving black holes and compact horizonless objects[END_REF], when this function is constant κ(u) κ(u * ) =: κ * over a sufficiently large interval around a given u * , one can assure that during this same interval the system is producing a Hawking flux of particles with a temperature

k B T = κ * 2π . ( 29
)
If the variation of κ(u) is slow, then one can describe the system as a thermal emitter with a slowly varying temperature. Whether κ(u) varies slowly or not in the surroundings of u * , is controlled by an adiabatic condition which, under mild technical assumptions [START_REF] Barceló | Hawking-like radiation from evolving black holes and compact horizonless objects[END_REF], reads * :=

1 κ 2 * dκ du u * 1. ( 30
)
However, even in the case in which the adiabatic condition is not satisfied, the existence of a non-zero κ(u) is an indication that there is particle emission (in this case the Bogoliubov β coefficients will not be zero). The spectrum of this particle content does not follow a precise Planckian profile anymore, but still we will take κ(u) as an estimator of the amount of particles perceived by the observer. In this work, for short we will refer to κ(u) [or more precisely to κ(u)/(2π)] as a variable temperature, although strictly speaking this term is correct only in the intervals in which the adiabatic condition is satisfied. We will call

(u) := 1 κ 2 dκ du (31)
the adiabatic control function

Static observer at a fixed radius

Let us start by analysing the particle content as seen by a static observer sitting at a radius r s . In order to find a new labelling ū = p so (u), now associated with this static observer, we will repeat the same steps that we did before for the definition of the vacuum state. (Here on we will replace the generic subscript 'ob' by a subscript denoting the particular type of observer; in this case 'so' stands for static observer.) The world line of an observer keeping its radial position can be described as r(τ ) = r s = const and

t = 1 - 2m r s -1/2 (τ -τ 0 ), ( 32
) so that ū = 1 - 2m r s -1/2 (τ -τ 0 ) -r * (r s ). ( 33
)
As before, an observer following this world line can naturally use its proper time τ to label the different rays he encounters while standing in its radial position, namely, he can make the assignment of the label u = p -1 so (ū) = τ given by the inverse of this equation to the ray ū = const that hits the observer at proper time τ . Upon synchronization, we obtain the relation between both labels ū = p so (u

) := 1 - 2m r s -1/2 u. (34)
Note that here one cannot demand u → ū in the past infinity, as we did in section 3.

The synchronization in this case reduces to eliminate an irrelevant additive constant.

At this point, we can already construct the relation U = U (u) we were after from the equation ū = p vs (U ) = p so (u). The complexity of (27) does not allow us to write this relation explicitly, but we can write its inverse. Using ( 27) and (34), we obtain

u = 1 - 2m r s 1/2 U H + 4m 3 -4m 3 4m (U H -U ) + 1 1/3 + 1 2 3 4m (U H -U ) + 1 2/3 + 1 3 3 4m (U H -U ) + 1 + ln 3 4m (U H -U ) + 1 1/3 -1 . (35)
One can study the behaviour of this function at the past and future infinities, u → -∞ and u → +∞, respectively. In the asymptotic past, the linear term on the right-hand side is the most important, and this relation becomes

U ≈ 1 - 2m r s -1/2 u, when u → -∞ (U → -∞). ( 36
)
By looking at the definition of κ(u) in (28), one can easily see that in this regime κ = 0, so that there is not radiation at all. On the other hand, at future infinity the behaviour of U (u) is dominated by the logarithmic term in (35), yielding an approximate expression

U ≈ U H - 4m 3    exp 1 4m (U H -6m) exp -1 - 2m r s -1/2 u 4m + 1 3 -1    ≈ U H -4m exp 1 4m (U H -6m) exp -1 - 2m r s -1/2 u 4m , when u → ∞ (U → U H ). ( 37
)
From this expression, it is easy to recognize the well known asymptotic form [2]

U = U H -A exp(-κ a u), (38)
(where U H , A, κ a are constants), which is associated with a Planckian spectrum of particles with temperature k B T = κ a /(2π). Thus, when the static observer waits long enough, it perceives a Hawking flux with temperature

k B T = 1 - 2m r s -1/2 1 8πm . ( 39
)
This is precisely the asymptotic Hawking temperature k B T H = κ H /(2π) = 1/(8πm), multiplied by a gravitational blue-shift factor which runs from unity for observers close to infinity, to arbitrarily large values for observers standing at positions closer and closer to the event horizon.

Numerical κ(u)

Let us describe now the particle perception of one of these observers throughout its entire life outside the horizon. From the definition of κ(u) in ( 28), it is easy to see that we can also write

κ(u) = d 2 u dU 2 du dU -2 U (u) . ( 40
)
Then, evaluating the derivatives we can obtain a close form for κ(u)

κ(u) = 1 4m 1 - 2m r s -1/2 3 4m [U H -U (u)] + 1 -4/3 . ( 41
)
Solving numerically the relation U (u) from ( 35), we can plot κ(u) for different static observers (figure 1). ---, •

• • • • •, -• -, --)]. We use 2m = 1 units.
In these graphs, it is clearly seen how the vacuum state is such that in the past it does not contain particles, but at some instant of time it starts to heat up reaching asymptotically a final temperature. As we mentioned at the beginning, the vacuum state that we have chosen in a static background mimics what would happen in the formation of a black hole through gravitational collapse. Only once the black hole is very close to formation one enters in the asymptotic regime (37), and Hawking radiation is switched on. We can estimate when this happen, but for that let us first describe the behaviour of the adiabatic control function in (31).

Validity of the adiabatic approximation

From the definition of in (31), we have

(u) = dκ dU κ 2 du dU -1 U (u) = 4 3 4m (U H -U (u)) + 1 1/3 -1 . (42)
Thus, decays from infinity at the asymptotic past, to zero when the observer that fixes the vacuum reaches the event horizon. This late time behaviour is consistent with the fact that then the quantum field is finally switched on, so that the radiation that a static observer sees is perfectly thermal. On the other hand, in the asymptotic past, becomes unbounded, but this does not mean a failure of the adiabatic condition, because κ → 0 there. Again, we can numerically plot (u) for different static positions (figure 2). We can see that its value crushes to nearly zero when the temperature enters its asymptotic regime. Now we can introduce two different referential times. On the one hand, we can define an ignition time of the black hole as the moment at which κ = 0. If one calculates this ignition time in terms of U , it does not depend on the value of r s . This is perfectly reasonable, as the label U corresponds to the single free-fall trajectory used to select the vacuum state. It is not difficult to see that this ignition time corresponds to U ig = U H -(676/1029)m, or equivalently to the time when the free-fall trajectory reaches r ig 2m + 0.61m. On the other hand, we can define a thermality time as the moment from which the outgoing radiation can be assumed to have a Planckian shape. This thermality time can be calculated in terms of U through the condition (U th ) = f , meaning that the adiabatic control function has reached a fiduciary value f , that here we will take equal to 0.01. This specific condition happens at U th U H -0.01m, which corresponds to the time when the free-fall trajectory reaches r th 2m + 0.01m. Let us give you some figures. If a neutron star with 1.5 times the mass of the Sun (Schwarzschild radius equal to 4.43 km) and a radius of 12 km started to undergo a free-fall collapse at Schwarzschild time t ini , it would reach r ig 2m + 1.357 km at a time t ig t ini + 0.13 × 10 -3 s, and r th 2m + 0.022 km at a time t th t ini + 0.20 × 10 -3 s. In this way, one can directly see that the detection of Hawking quanta at infinity does not involve any long delay associated with the freezing of the collapsing structure as it approaches the horizon formation.

Finally, let us comment that for different observers the previous two times are seen to happen at different instants of their proper times (their respective u labels). The closer to the horizon, the later the observer will see Hawking radiation to switch on and to become Planckian.

Observers freely falling from infinity

In this subsection, we will analyse particle creation as seen by observers which are freely falling from infinity. To find the relevant re-labelling ū = p ffio (u) (where 'ffio' stands for falling-from-infinity observer ), we have to use the form of the radial timelike geodesic that starts at spatial infinity with zero velocity. However, recall that the calculation done to fix the vacuum state of the field in section 3 was precisely using these time-like trajectories. In that calculation, we had two constants of integration to play with, namely t 0 and τ 0 . We will now consider trajectories differing from the reference trajectory in section 3 by a temporal delay ∆t 0 . Instead of using t 0 = τ 0 = U H + 4m/3, we will use new values

t 0 = τ 0 = U H + 4m/3 + ∆t 0 = u H + 4m/3, where u H := U H + ∆t 0 . We then define the new relation ū = p ffio (u) by [see (27)] ū = p ffio (u) := p vs (u -∆t 0 ) + ∆t 0 , (43)
which explicitly reads

ū = p ffio (u) = u H + 4m 3 -4m 3 4m (u H -u) + 1 1/3 + 1 2 3 4m (u H -u) + 1 2/3 + 1 3 3 4m (u H -u) + 1 + ln 3 4m (u H -u) + 1 1/3 -1 . (44)
Now, as we did for the static observer in section 4.2, we compare both labellings, U in (27) and u in (44), and obtain an implicit relation U (u) from the equation

p vs (U ) = p ffio (u), (45)
which again explicitly reads

-4m 3 4m (U H -U ) + 1 1/3 + 1 2 3 4m (U H -U ) + 1 2/3 + 1 3 3 4m (U H -U ) + 1 + ln 3 4m (U H -U ) + 1 1/3 -1 = ∆t 0 -4m 3 4m (u H -u) + 1 1/3 + 1 2 3 4m (u H -u) + 1 2/3 + 1 3 3 4m (u H -u) + 1 + ln 3 4m (u H -u) + 1 1/3 -1 . ( 46
)
Solutions to this non-algebraic equation must be found numerically. Nevertheless, we can examine the past infinity and event horizon limits analytically. In the past infinity, we obtain U ≈ u, owing to the synchronization we imposed. Near the event horizon, the logarithmic functions provide the leading contribution and therefore,

U near-hor (u) ≈ U H - 4m 3    e -∆t 0 /(4m) 3 4m (u H -u) + 1 1/3 -1 + 1 3 -1    . (47)
The function κ(u) can be readily obtained from ( 43) and (45). Indeed, taking the derivative of (45) with respect to u, we obtain

dU du = dp vs (u -∆t 0 ) du dp vs (U ) dU U (u) -1 , (48)
and differentiating once more

d 2 U du 2 = d 2 p vs (u -∆t 0 ) du 2 - d 2 p vs (U ) dU 2 U (u) dU du 2 dp vs (U ) dU U (u) -1 , (49)
so that κ(u) can be expressed as

κ(u) = d 2 p vs (U ) dU 2 U (u) dU du 2 - d 2 p vs (u -∆t 0 ) du 2 dp vs (u -∆t 0 ) du -1 . (50)
Therefore, the final result is

κ(u) = 1 4m 3 4m (u H -u) + 1 -1 3 4m (u H -u) + 1 1/3 -1 -1 × (3/4m)(u H -u) + 1 (3/4m)(U H -U (u)) + 1 4/3 -1 . ( 51
)
Note that this exact expression is explicit if we considered it as a function of both u and U (although this does not mean that it is a function of two variables). The dependence on ∆t 0 is hidden in the implicit relation between both variables. We can use the limiting expression of U (u) at the event horizon (47) (which is exact there) to find what temperature would the freely-falling observer characterized by the delay ∆t 0 see when he crosses the horizon. This temperature happens to be

κ HC (∆t 0 ) = 1 m 1 -e -∆t 0 /(4m) . (52)
Therefore, the perceived radiation will run from an initial zero temperature to nearly 1/(2πm), which corresponds to four times the standard Hawking temperature, provided that the waiting time ∆t 0 is several times bigger than 4m. This is an interesting and, at first, puzzling result. How can one interpret it? In section 5, we will offer and explanation of this result, but in order to do that we still need to introduce several additional material.

Numerical κ(u)

We can numerically plot the entire relation κ(u). Results are shown in figure 3. For small values of ∆t 0 (less than approximately 20m) the temperature perceived by the freely-falling observer is nearly zero during most of its trajectory towards the horizon. Only when approaching the horizon, the temperature increases to reach a peak value given by (52). For large enough values of ∆t 0 , before this final increase in the temperature, there appears an intermediate plateau in which the temperature is nearly constant. It is easy to understand this behaviour: if the freelyfalling observer is still far from the horizon when Hawking radiation is switched on, at some moment it will start detecting this radiation, which will stay nearly constant till the observer becomes close to the horizon, where additional physics will come into play (see section 5). The effective temperature value along the plateau is somewhat higher than the Hawking radiation temperature, and slightly increasing. As we will see, this is mostly due to a Doppler blue-shift associated with the radial velocity of the falling observer (see also section 5). It is interesting to look at how good is the functional approximation (47) when plugged into (51), as compared with the exact result. This comparison is shown in figure 4 for a long delay ∆t 0 . By construction, this approximation works perfectly well in the surroundings of the horizon crossing. However, it is remarkable that it also fits very well the plateau region.

On the other hand, this approximation is not useful when ∆t 0 is not large enough (see figure 5). The asymptotic approximation (47) correctly reproduces the horizoncrossing value as should be by construction, but at early times it deviates from the exact curve rather quickly.

Adiabatic approximation validity

Having found an expression for κ in terms of u and U (u), we can use it to obtain an expression for the adiabatic control function (u) as defined in (31). The resulting expression, also explicit when considered as a function Temperature as a function of u for a freely-falling observer from infinity with ∆t 0 = 200m using the exact expression (curve depicted --) and using U near-hor (u) approximation (curve depicted ----). We use 2m = 1 units and u H = 0. Temperature as a function of u for a freely-falling observer from infinity with ∆t 0 = 4m using the exact expression (curve depicted --) and using U near-hor (u) approximation (curve depicted ----). We use 2m = 1 units and u H = 0.

of both u and U , is rather involved, and we will not include it here. But, as we did with κ, we can also replace the relation U near-hor (u) (47) valid in the near-horizon limit to find an explicit expression for (u) valid there; finally, we can take the u → u H limit and find its value at horizon-crossing. The result is

HC (∆t 0 ) = 3 8 + 7 4 [e ∆t 0 /(4m) -1] . ( 53
)
This expression runs from 3/8 when ∆t 0 is several times larger than 4m, growing to infinity when ∆t 0 goes to zero. This last value is comprehensible from the definition of , as it is the temperature itself that goes to zero when the freely-falling observer approaches the referential one. The adiabatic condition 1 is never satisfied at horizon crossing. However, it is remarkable that, for ∆t 0 sufficiently large, it is always still smaller than unity, so that the perceived particle spectrum should not be very different from a Planckian spectrum.

Nonetheless, this is just the horizon crossing value. We can see using the numerically evaluated exact result (figure 6) that, for long waiting times, there is part of the trajectory for which the adiabatic condition is with no doubt valid, as there crushes to nearly zero. This of course coincides with the plateau region (in the cases in which it appears), in which κ takes a nearly constant value. Once the final increase in κ shows up, the adiabatic condition starts to be violated, although in a rather mild manner.

Observers freely falling from a finite radius

The last situation we shall consider in this paper is the particle perception of a freelyfalling observer left to fall from a radius r 0 , with zero initial velocity at some waiting time ∆t 0 (in Schwarzschild time) after the reference trajectory used to define the vacuum state had crossed this radial position. Again, we start by calculating this new trajectory. For that, we have to integrate (20), but now keeping r 0 a finite constant. This yields τ0 -τ = -r 0 r 0 2m

r 0 r -1 1/2 r r 0 + arctan r 0 r -1 1/2 , ( 54
)
where τ0 represents the time at which the observer starts to fall (note that, in all expressions in this subsection, 2m < r ≤ r 0 ). The time of horizon-crossing for this trajectory is

ũH := τ0 + 2m r 0 2m r 0 2m -1 1/2 + r 0 2m arctan r 0 2m -1 1/2 . ( 55
)
By means of (20) and (21), we obtain

dr dt = - 2m r 1 - 2m r 1 - r r 0 1/2 1 - 2m r 0 -1/2 , ( 56
)
which can be readily integrated to yield

t(r) = t0 + r 0 r 0 2m -1 1/2 r 0 r -1 1/2 r r 0 + 1 + 4m r 0 arctan r 0 r -1 1/2 + ln r 2m + 2 1 - 2m r 0 1/2 1 - r r 0 1/2 r 2m - 2r r 0 + 1 × r 2m -1 -1 , ( 57
)
where t0 is again an integration constant, in this case representing the time at which the observer starts to fall from r 0 . If we call t vs (r) the expression in (24), we can define t0 as a function of r 0 , t 0 and the delay time ∆t 0 t0 = t vs (r 0 ) + ∆t 0

= t 0 -4m r 0 2m + 1 3 r 0 2m 3/2 + 1 2 ln r 0 /(2m) -1 r 0 /(2m) + 1 + ∆t 0 . ( 58
)
Substituting in (57) the function r(τ ) which is implicitly defined in (54), we can find the function t(τ ).

Let us now construct the spacetime trajectory of an observer that remains at a fixed position r = r 0 until τ = τ0 , at which time he starts to fall freely towards the black hole. This trajectory (t(τ ), r(τ )) is made out of two pieces jointed together at τ = τ0 : the first piece, for τ < τ0 , is such that the radius is fixed r = r 0 and the Schwarzschild time satisfies

t(τ ) = t0 + 1 - 2m r 0 -1/2 (τ -τ0 ); (59)
the second piece, for τ ≥ τ0 , is constituted by the previously found implicit expressions r(τ ; r 0 , τ0 ) in (54) and t(r(τ ; r 0 , τ0 ); r 0 , t 0 , ∆t 0 ) = t(τ ; r 0 , τ0 , t 0 , ∆t 0 ) as defined in (57); the t 0 and ∆t 0 dependence of this last expression comes from the dependence of t0 on t 0 and ∆t 0 in (58).

As explained in previous sections, substituting τ by u, using an appropriate synchronization between t 0 and τ0 , and substituting τ0 by ũH as defined in (55), we can construct the function ū = p ffro (r) := t(r; r 0 , ũH , ∆t 0 ) -r * (r),

where 'ffro' stands for freely-falling-from-a-radius observer. Finally, the U (u) relation we were looking for can be found from the implicit relation ū = p vs (U ) = p ffro (r(u)).

(61) Our next step is to find the form of κ(u) for these observers.

κ(u) = 1 4m 1 - 2m r 0 -1/2 3 4m (U H -U ) + 1 -4/3 , for u < u 0 , (65)
κ(u) = 1 4m r 2m + 1 + r m 1 - 2m r 0 1/2 2m r 1 - r r 0 1/2 - 2m r 0 × 1 - 2m r 0 1/2 + 2m r 1 - r r 0 1/2 r 2m -1 -1 × 3 4m (U H -U ) + 1 -4/3 - 2m r 2 , for u ≥ u 0 , (66)
where r and U must be understood as functions of u (again, these can only be obtained numerically). This function has a finite jump at u = u 0 (= τ0 ), the moment at which the observer at r 0 is released from its previously fixed position. At early times, κ follows the same pattern described in section 4.2, in (41). It increases from zero and tries to reach the asymptotic value associated with the observer at rest at position r 0 . However, at some moment u 0 the observer is released to fall freely. In this moment, κ undergoes a finite jump. The value of κ just after the jump, which we will denote κ released , is found by evaluating (66) at r = r 0 , which gives

κ released (r 0 , ∆t 0 ) = 1 4m 1 - 2m r 0 -1/2 × 3 4m (U H -U 0 (r 0 , ∆t 0 )) + 1 -4/3 - 2m r 0 -2 . ( 67
)
Here U 0 = U (u 0 ) is the value of the label U associated with u 0 , and depends on the parameter ∆t 0 . A particular case is ∆t 0 = 0, which means that the observer starts to fall just when the observer that fixes the vacuum state passes by its side. This implies of course U 0 = U (r 0 ), as determined by the trajectory in section 3 [it can be obtained by reversing (22), noting that U acts as τ now]. Doing this one can check that κ released is zero, as one should expect. This is because in this case, at the starting point, between the observer that determines the vacuum state and our observer there is no difference but in their velocity. If the former one sees nothing, the latter shall see "nothing times a red-shift factor", that is, nothing at all.

On the other hand, if the observer waits long enough so that the black hole is nearly completely switched on (Unruh state), that is, ∆t 0 → ∞, which amounts to substituting U by U H in equation (67), then κ released takes the value

κ Unruh (r 0 ) := κ released (r 0 )| f ∆t 0 →∞ = 1 4m 1 + 2m r 0 1 - 2m r 0 1/2 . ( 68
)
The limit r 0 → 2m gives zero, and the limit r 0 → ∞ gives the usual Hawking temperature κ = 1/(4m), as happens in the Unruh state. This agrees with the fact that our state reproduces the Unruh state for late times.

Another interesting quantity we can calculate is the value of the jump in κ, ∆κ release , at the instant of releasing. We can directly subtract from (65) the expression in (67), and find the jump ∆κ release ,

∆κ release (r 0 ) = 1 4m 1 - 2m r 0 -1/2 2m r 0 2 = 1 - 2m r 0 -1/2 m r 2 0 . ( 69
)
It is remarkable that it does not depend on U or, in other words, does not depend on the waiting time ∆t 0 . This is reasonable, as by switching off the rockets the observer is just loosing the radiation that comes from its own acceleration, and this acceleration only depends on the radius it was staying at. In fact, in the last expression we can directly spot the surface gravity associated with position r 0 [see ( 19)] multiplied by its corresponding gravitational blue-shift factor. We can conclude then that ∆κ release is nothing but Unruh radiation associated with an accelerating observer [START_REF] Unruh | Notes on black hole evaporation[END_REF][START_REF] Birrell | Quantum Fields in Curved Space[END_REF]. In figure 7, we plot the value of κ around the release point for different waiting times. There, one can directly see that the jump in the temperature is always the same. ---, • • • • • •, -• -, --)]. We use 2m = 1 units and u 0 = 0. Now, let us study the radiation perception at horizon crossing. The procedure is analogue to the one followed in section 4.3: in (61) we identify the diverging terms at the event horizon and, from the resulting equation, isolating U we find an expression U near-hor (r) valid near the event horizon. The expression is quite complex and not really useful by itself, but again, plugging it into (66) we obtain an expression κ(r) valid near the event horizon. Finally, performing the limit r → 2m we find the horizon crossing value κ HC , which in this case is

κ HC (r 0 , ∆t 0 ) = 1 m    1 - 2m r 0 1/2 - r 0 /(2m) -1 r 0 /(2m) + 1 1/2 × exp - 5 6 - r 0 + ∆t 0 4m + r 0 2m + 1 3 r 0 2m 3/2 - r 0 4m + 1 r 0 2m -1 1/2 arctan r 0 2m -1 1/2 . ( 70
)
When the observer waits for a sufficiently long time before being released from its position (mathematically, ∆t 0 → ∞), it simplifies to

κ HC | f ∆t 0 →∞ = 1 m 1 - 2m r 0 1/2 , ( 71
)
which reproduces the result κ = 1/m for an observer freely falling from infinity [see the ∆t 0 → ∞ limit of ( 52)]. It is also interesting to analyse the limit of κ HC (r 0 , ∆t 0 ) when r 0 → ∞. The result is

κ HC | r 0 →∞ = 1 m , (72)
which, surprisingly at first sight, does not depend on ∆t 0 . That is, we cannot reproduce the entire formula (52) by taking the r 0 → ∞ limit. The reason is that the trajectory used to fix the vacuum state does never coincide with the trajectory of a released observer even if the release point r 0 is taking close to infinity. Although the velocity difference of these two trajectories at r 0 tends to zero when r 0 → ∞, they arrive at positions close to the horizon at very different Schwarzschild times. The freely-falling-from-aradius observer for far enough releasing radius arrives to the horizon always so late with respect to the reference trajectory, that it sees the black hole as virtually switched on (in its final Unruh state). Plotting expression (70) (figure 8), we can see that the parameter ∆t 0 plays almost no role in the temperature seen when crossing the horizon.

Numerical κ(u)

Let us now plot some examples of exact numerical resolution of κ(u) for different trajectories. The aim is just to have a visual and qualitative understanding of the global evolution of the temperature, as well as to see how the calculated exact limits and behaviours are reproduced. First, we will plot the value of κ(u) for observers starting to fall from the radius r 0 = 10m with different waiting times (figure 9). Graphs show how κ(u) starts from very different values when the observer is released, but converges to virtually the same final value no matter the waiting time, as the starting radius is far enough.

If the radius r 0 is quite near the event horizon (r 0 = 4m in figure 10), there is a small but appreciable difference in the value of κ(u) when crossing it, depending on the waiting time. We can also see that the jump in κ(u) when releasing the observer is more important in this case, as acceleration to stay at the fixed radius is much bigger here.

On the other hand, if the radius is really far from the event horizon (r 0 = 30m in figure 11), the jump in κ(u) is completely negligible. Different waiting times in this case just determine when you start perceiving the radiation, which can happen before or after the releasing point. The plateau found in section 4.3, and of course the final peak, are again reproduced, as should happen for a large radius. But note that, as we already argued, it is not possible to reproduce the cases in section 4.3 for which ∆t 0 is small. The limit r 0 → ∞ here yields the same limit than ∆t 0 → ∞ in section 4.3, no matter the value of ∆t 0 here.

Validity of the adiabatic approximation

Repeating what we have done for the observers studied so far, here we can find a formula for the adiabatic condition (u), in this case as an explicit function of both U = U (u) and r = r(u). Also, once more we can plug the expression U near-hor (r) valid near the event horizon, and find a function (r) valid there. Finally, we may take the limit r → 2m and obtain the value HC at the horizon crossing. Unfortunately, in this case the approximation U near-hor (r), and the exact expression (U, r) are extremely complex in their functional forms. Thus, it seems neither possible, nor interesting, to find an explicit expression of HC in terms of the parameters r 0 and ∆t 0 . Nonetheless, it is still possible to obtain an expression for (r, r 0 ) (not only in the horizon crossing) when ∆t 0 → ∞,

(r, r 0 )| f ∆t 0 →∞ = 3 - 4r r 0 + r 2m 2 -4 1 - 2m r 0 1/2 1 - r r 0 1/2 r 2m × r 2m 2 -1 -2 . ( 73
)
In this expression, one can now take the limit r → 2m and obtain HC when ∆t 0 → ∞ in terms of r 0 ,

HC (r 0 )| f ∆t 0 →∞ = 1 8 3 + r 0 2m -1 -1 . (74)
This result runs from 3/8 when r 0 2m, to infinity when r 0 → 2m. With the first limit 3/8, we reproduce again the result obtained in section 4.3 for long delays [see the limit of (53)]. The second divergent limit reflects the fact that observers released from positions near the horizon by no means perceive thermal radiation. In figure 12, we plot the function in (74), together with some numerically solved graphs of HC for zero and finite waiting times ∆t 0 . Note that the limit 3/8 when r 0 → ∞ is common for all the graphs. Also, we can find an expression for released at the beginning of the falling part of the path, again for ∆t 0 → ∞. We obtain

released (r 0 )| f ∆t 0 →∞ = r 0 2m 2 -1 -1 . (75)
This yields zero for r 0 2m, and infinity when r 0 → 2m. The zero value appears because, at large radius and for large waiting times, what the observer initially sees is a perfect Hawking radiation. The divergent value for r 0 near the horizon has the same explanation as in the case of HC : for trajectories starting too close to the horizon, the adiabatic approximation is never valid. Neither at the beginning of the fall, nor at horizon crossing.

Finally, we will plot the value of (u) for the same cases we plotted κ(u) in section 4.4.1 (figures 13, 14, and 15). As a consequence of the sudden change in κ at the release time, these graphs should exhibit a Dirac-delta peak in there. This has been avoided in the figures for simplicity. Apart from that, graphs need few comments. They of course reproduce the limit values already found. As happened in section 4.3, when plateaus appear becomes nearly zero, and so the radiation is perfectly thermal, with the temperature determined by κ. For earlier parts of the trajectory, diverges, but as we have already explained, this is only reflecting that the temperature there goes to zero.

= (0, 12m, 24m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)].
We use 2m = 1 units and u 0 = 0. ----, • • • • • •, -• -, --)]. We use 2m = 1 units and u 0 = 0.

= (0, 60m, 140m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]
. We use 2m = 1 units and u 0 = 0.

Final increase of the effective temperature

At this stage, we have all the necessary ingredients to explain the, at first sight puzzling, final increase of κ for freely-falling observers. For sufficiently long waiting times, we are sure the vacuum state is indistinguishable from Unruh state (see also next section). As standard lore in the literature, one can read that this state is a vacuum for observers freely falling at the horizon. However, we have here several of these observers and the value of κ HC is non-zero for almost all of them: recall for instance the value of κ HC for observers freely falling from infinity (52) and for observers freely falling from a finite radius (70). The only exceptions are the observer closely following the reference trajectory and the one with zero instantaneous radial velocity at the horizon.

Indeed, once Hawking radiation is switched on, there is only one effective temperature that does vanish at the horizon: the Unruh temperature κ Unruh /(2π) (68). As we explained, this is the κ associated with an observer in free fall at a radial position r and with an instantaneous zero radial velocity in there (we will call this observer an Unruh observer),

κ Unruh (r) = 1 4m 1 + 2m r 1 - 2m r 1/2 . ( 76
)
Let us now make the reasonable hypothesis that all the physics concerning the behaviour of κ(u) can be understood in terms of local properties of the observer's trajectory: its position, its velocity and its acceleration. We will see that this is indeed the case. Under this hypothesis, the only characteristic that distinguishes the Unruh observer from any of the other freely-falling observers is that the latter have non-zero radial velocity. The difference in radial velocities between the freely-falling observers and the Unruh observer implies that their respective radiation perceptions have to be related by a Doppler shift factor. In fact, the Unruh observer is such that in the horizon limit would travel with the very outgoing null ray. Therefore, to obtain κ(u) for the freely-falling observers from κ Unruh , one has to extract from it a Doppler red-shift factor (going to zero at the horizon), or equivalently multiply it by a Doppler blue-shift factor (its inverse, diverging at the horizon). As we are going to show, this divergence counterbalance the zero value of κ Unruh at the horizon yielding the appropriate finite result for the different κ HC 's. Which is the precise form of this Doppler blue-shift factor? Our guess is

D r 0 (r) := dr l dt - dr r 0 dt 1/2 dr l dt + dr r 0 dt -1/2 . (77)
Here, r l (t) represents the trajectory of a null ray going away from the horizon and r r 0 (t) the freely-falling trajectory that starts with zero velocity at r = r 0 . In addition, we have put a subscript r 0 in D r 0 (r) to distinguish the different Doppler factors associated with freely-falling observers released at different radial distances r 0 . The reason behind this guess is the following. A physical Doppler factor has to compare the velocity of the massive objects with the velocity of light,

D = c -v c + v . (78)
However, note that using (t, r) coordinates the velocity of light is different from unity; that is why we have included dr l /dt. The Doppler factor (77) can be alternatively written as

D r 0 (r) = 1 - dr r 0 dr l 1/2 1 + dr r 0 dr l -1/2 . ( 79
)
In this way, one would compare the real local velocity of the observer with respect to the local velocity of light (which is always equal to one). We can take dr r 0 /dt from (56) and calculate dr l /dt from the outgoing rays equation t -r * (r l ) = const, dr l dt r l =r = dr dr * = 1 -

2m r . (80)
After some manipulation, we finally find

D r 0 (r) = 1 - 2m r 0 1/2 + 2m r 1 - r r 0 1/2 1/2 × 1 - 2m r 0 1/2 - 2m r 1 - r r 0 1/2 -1/2 . ( 81
)
If we compute now

κ Unruh (r)D r 0 (r), (82)
and take the limit r → 2m, we can check that one obtains precisely (71). So, as we advanced, it is really the extraction of a huge Doppler red-shift factor the reason why the freely-falling observers perceived some effective temperature even at horizon crossing.

In other words, for long enough waiting times, a generic freely-falling observer sees a small but non-zero κ when crossing the horizon. Only a hypothetical observer with zero radial velocity at the horizon (the Unruh observer there) would see a zero κ, but because of the huge red-shift factor associated to its instantaneous trajectory (this observer is instantaneously travelling with the null ray). The final increase in κ always present for freely-falling observers is caused by the Doppler shift factor. Take the case of an observer freely falling from infinity. As we explained, the perceived radiation starts at zero temperature. Then, if one waits longenough, a plateau region is formed with an almost constant κ. Finally, when the observer approaches the horizon, κ increases up to the value 1/m, which corresponds to four times the standard Hawking temperature. The value of κ from the appearance of the plateau on is due to the competition between Unruh's switching off of the radiation emission when going towards the horizon, which surprisingly passes through an intermediate increasing phase (see figure 16), and the multiplication by a Doppler blue-shift factor due to the velocity of the freely-falling observer with respect the Unruh observer. The final result of this competition is the exact numerical computations we have shown throughout the paper (considering always sufficiently long waiting times).

For instance, let us plot

κ Unruh (r(u))D r 0 →∞ (r(u)) (83)
against the exact numerical κ(u) of section 4.3. Here r(u) is the position of the observer falling from infinity as a function of the label u. From (22) one obtains r(u)

r(u) = 2m 3 4m (u H -u) + 1 2/3 . ( 84
)
In figure 17, we compare this approximation with the exact result for κ(u) for a long waiting time ∆t 0 = 200m. We can see that the approximation is perfect from the starting of the plateau till horizon crossing. Obviously, the previous approximation fails to capture the switching on of Hawking radiation. Actually, this perfect fit for long waiting times can also be checked analytically, and not only for observers falling from infinity, but for the generic case of freely-falling observers from any radius. For instance, one just has to approximate the κ(U, r) in (66) for ∆t 0 → ∞ (this is equivalent to assuming U U H ), which yields

κ(r) = 1 4m r 2m + 1 + r m 1 - 2m r 0 1/2 2m r 1 - r r 0 1/2 - 2m r 0 × 1 - 2m r 0 1/2 + 2m r 1 - r r 0 1/2 r 2m -1 -1 × 1 - 2m r 2 , (85)
and then see that it is precisely equal to

κ Unruh (r)D r 0 (r) = 1 4m 1 + 2m r 1 - 2m r 1/2 × 1 - 2m r 0 1/2 + 2m r 1 - r r 0 1/2 1/2 × 1 - 2m r 0 1/2 - 2m r 1 - r r 0 1/2 -1/2 . ( 86
)
With hindsight, we could have used this equality to obtain all the previous partial results. But we can go now even further: if we put κ released (r, ∆t 0 ) instead of κ Unruh (r) in (86), then it is possible to see that we reproduce the entire formula for the effective temperature in (66). This is because between two freely-falling observers at the same position the only difference is a Doppler shift, and this fact has nothing to do with being in the Unruh state or not. We checked this first for the Unruh state because we wanted to explain the appearance of the final peak in κ, a peak whose appearance was in apparent contradiction with the vanishing value we expected for that state.

Comparison with the Unruh state

One might worry that some of the results shown in this paper related with the late-time behaviour of the system, are peculiarities of an unusual choice of the vacuum state. However, this is not the case. In this section, we will briefly show that at late times this state is really equivalent to the Unruh vacuum state. Thus, all the properties found in the previous sections that applies for late times, are immediately present also in the Unruh state.

The equivalence between the states for late times can be directly seen from ( 27). In that expression, late times means just U U H . So, as we did in (37), we can reverse the approximate relation to find

U ≈ U H - 4m 3 exp 1 4m (U H -6m) e -ū/(4m) + 1 3 -1 ≈ U H -4m exp 1 4m (U H -6m) e -ū/(4m) . ( 87
)
But this is precisely the relation which defines the coordinate U that appears in the Unruh modes [START_REF] Unruh | Notes on black hole evaporation[END_REF] (apart from the irrelevant origin of the coordinates). Thus, if we are using this coordinate to define the vacuum state, we are in fact choosing the Unruh state for late times. Nonetheless, just as an example, let us perform a specific calculation entirely based in the Unruh state choice. For instance, let us calculate the case of a freely-falling observer from infinity. With this aim, consider a redefinition of U using the usual and simple form

U := -4me -ū/(4m) , (88)
and take (44) as the definition of u. That is, U defines the vacuum and is the coordinate that appears in the Unruh modes; and u is the proper time that a freely-falling observer from infinity uses to label the rays. Replacing (44) in (88) we obtain a relation U (u), from which we can directly compute κ(u). The result is

κ(u) = 1 4m 3 4m (u H -u) + 1 -1/3 + 2 3 4m (u H -u) + 1 -2/3 + 1 . (89)
From here one can check the limits

κ(u → -∞) = 1 4m and κ(u → u H ) = 1 m (90)
with the naked eye. The first limit is the usual Hawking radiation temperature, which in the Unruh state is always present, even at the asymptotic past. It is as if we extended the plateau to the u → -∞ limit (see figure 18). The second limit is equal to the one obtained at the horizon crossing for long enough delays in section 4.

Summary and conclusions

In this paper we have considered a quantum scalar field over a Schwarzschild black hole geometry. As a first step, we have set up the quantum scalar field in a specific vacuum state that interpolates between Boulware vacuum state at early times and Unruh vacuum state at late times. In other words, we have chosen a non-stationary vacuum state so that it simulates the physics of a dynamical collapse scenario. In this scenario, before the black hole forms there is no radiation at infinity but, after its formation, observers at infinity start receiving a Hawking flux of radiation: the black hole Hawking emission switches on at some particular ignition time (of course, it is not an instantaneous event). This vacuum state has been fixed by requiring that a reference observer freely falling from infinity detects no radiation. As a second step, we have analysed the radiation associated with this vacuum state in terms of how it is perceived by different observers: static observers at a fixed radius; freely-falling observers from infinity, with different time delays with respect to the reference observer; and observers maintained at a fix radius for some time, and then, after being surpassed by the reference observer in its way in, released to fall freely towards the horizon at different releasing times.

The method used follows the recent analysis in [START_REF] Barceló | Minimal conditions for the existence of a Hawking-like flux[END_REF][START_REF] Barceló | Hawking-like radiation from evolving black holes and compact horizonless objects[END_REF]. For each observer we define a function κ varying along the observer's trajectory. We also define an adiabatic control function as = | dκ/du |/κ 2 . In the portions of the trajectory in which κ stays almost constant ( 1) one can be sure that the observer detects thermal radiation at a slowly varying temperature k B T = κ/2π. When this is not exactly the case, the radiation perceived is non-thermal, but still, we take the value of κ as an estimate of the amount of perceived radiation and, somewhat abusing of the language, we talk about a time varying effective temperature.

• Static observers at fixed radius. We show that they perceived an effective temperature that starts from zero, at early times, and after a transient period, stabilizes at Hawking's temperature 1/(8πm) multiplied by the gravitational blueshift factor associated with the particular radial position of the observer. This description clearly reflects the ignition associated with the formation of a black hole we were seeking for. We also see that the transient period is quite sort: 0.07×10 -3 s for a collapsing neutron star with 1.5 times the mass of the Sun and a radius of 12 km, when detected from far enough positions. Observers closer to the horizon will perceive this process with a slower pace due to the gravitational slow down of their clocks.

• Observers freely falling from infinity. When their delay with respect to the reference trajectory used to fix the vacuum state is small, they perceive almost no radiation at all. The physical picture is the following: If an observer sitting at the very surface of a star that is collapsing in free fall to form a black hole perceived vacuum at the beginning of the collapse, he would perceive vacuum during his entire collapsing trajectory (this is reasonable, as he has not perceived any gravitational field). On the contrary, when the delay is long enough, the observer has time enough to see that the black hole is emitting Hawking radiation. He first sees the switching on of this radiation. Then, during some time he perceives an almost constant temperature radiation passing by. The temperature at this plateau is essentially the result of multiplying Hawking's temperature by a Doppler shift factor associated with the velocity of the observer. In this region the adiabatic condition is perfectly satisfied, so here to describe the perceived radiation as having a temperature is strictly correct. Finally, in the last stages of his approach to the horizon, and surprisingly at first sight, the effective temperature rises reaching exactly four times Hawking's temperature. The adiabatic condition is only mildly violated in these last stages, so although the perceived radiation is not exactly Planckian, it is not hugely different from Planckian either.

• Observers freely falling from a radius. These observers are standing still at a fix radial distance r 0 from the horizon (supporting themselves against the gravitational pull with some rockets) till a release time (when the rockets are switched off), which happens some waiting time after the reference observer, in free fall, has overtaken them. If this waiting time is long enough, the observers will see Hawking radiation appear and settle down to a constant value, right in the same way as described above for static observers. Once the observer is released, the value of the effective temperature undergoes a jump with a value equal to the surface gravity associated with position r 0 times the gravitational blue-shift factor associated with this same position. We calculated the value of the effective temperature after the release and called it k B T released (r) = κ released (r)/(2π). From the release time on, the effective temperature starts to rise, first slowly and later quicker, so as to reach a finite value at horizon crossing. The precise behaviour of the effective temperature depends on the release radius, and so does the asymptotic value of it, which becomes 4T H when r 0 → ∞.

Once Hawking radiation is switched on, the value of k B T Unruh (r) = κ Unruh (r)/(2π), that is, the effective temperature as perceived by an observer freely falling at r and with a vanishing instantaneous radial speed, interpolates from T H when r → ∞ to zero when r → 2m. This is consistent with the idea that the Unruh state is vacuum for freely falling observers at the horizon. However, in general one has to be careful with this assertion. We have shown that after Hawking radiation is switched on, generic freelyfalling observers at the horizon do not perceive vacuum, but a finite effective temperature higher than their initial Unruh effective temperature. The reason is that the Unruh observer experiences a Doppler red-shift factor with respect to a generic observer (due to their velocity difference), or in other words, a generic observer experiences a Doppler blue-shift factor with the respect to the Unruh observer. This Doppler blue-shift factor diverges at the horizon in such a way that, when multiplied by the vanishing value of Unruh's temperature at the horizon, results in a finite temperature for a generic observer. In fact, we have proved the following completely general result: for any freely-falling observer the exact effective temperature T (r) perceived at position r is precisely the product of the temperature when released there T released (r) by the Doppler blue-shift factor associated with the radial velocity of the observer when passing through that position.

Regarding the observability of Hawking radiation, in pure theoretical terms the discussed results lead to the idea that the closer to the horizon one supports oneself against gravity, the better the chances of detecting its radiation, due to the gravitational blue-shift factor. However, in practical astrophysical scenarios the tiny Hawking temperature of a stellar mass black hole, about a few nano-Kelvins, will be masked by the ∼ 3 Kelvin of the cosmic microwave background. This radiation would also experience a gravitational blue shift if detected by observers close to the horizon, so that going closer to the horizon would not improve in principle the chances of detection. Another matter is the detection of Hawking radiation in analogue models of General Relativity. There exist quantum systems, as Bose-Einstein condensates, in which Hawking temperature is not that much masked by the environment temperature. In those situations one could think that analysing the presence of radiation close to the horizon would improve the detection chances, but this is not that clear. These systems exhibit modified dispersion relations so that when blue shifting the modes, they would enter into the non-relativistic regime, behaving very differently to what has been found and discussed in this paper. In addition, the size of the entire system in the lab will be so small that to control the physics localized close to horizon, a tiny fraction of the total size of the system seems implausible. Other ways of detection that could prove better are the so called correlation measurements between outgoing and ingoing partners of the Hawking radiation [START_REF] Balbinot | Non-local density correlations as signal of Hawking radiation in BEC acoustic black holes[END_REF][START_REF] Carusotto | Numerical observation of Hawking radiation from acoustic black holes in atomic Bose-Einstein condensates[END_REF].

Let us finish the paper by mentioning that the results presented here will also constitute an adequate preparatory background material for a future work dealing with analyses of buoyancy over black hole horizons.

Figure 1 .

 1 Figure 1. Temperature as a function of u for different static observers at r s = (3m, 4m, 10m, 20m) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units.

Figure 2 .

 2 Figure 2. Adiabatic control function as a function of u for different static observers at r s = (3m, 4m, 10m, 20m) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units.

Figure 3 .

 3 Figure 3. Temperature as a function of u for different freely-falling observers from infinity with delays ∆t 0 = (2m, 20m, 100m, 200m) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units and u H = 0.

Figure 4 .

 4 Figure 4. Temperature as a function of u for a freely-falling observer from infinity with ∆t 0 = 200m using the exact expression (curve depicted --) and using U near-hor (u) approximation (curve depicted ----). We use 2m = 1 units and u H = 0.

Figure 5 .

 5 Figure5. Temperature as a function of u for a freely-falling observer from infinity with ∆t 0 = 4m using the exact expression (curve depicted --) and using U near-hor (u) approximation (curve depicted ----). We use 2m = 1 units and u H = 0.

Figure 6 .

 6 Figure 6. Adiabatic control function as a function of u for different freely-falling observers from infinity with delays ∆t 0 = (100m, 200m) [curves depicted respectively as (-• -, --)]. We use 2m = 1 units and u H = 0.

Figure 7 .

 7 Figure 7. Temperature around the releasing point as a function of u for r 0 = 10m and delays ∆t 0 = (0, 12m, 24m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units and u 0 = 0.

Figure 8 .Figure 9 .

 89 Figure 8. Temperature when crossing the horizon as a function of r 0 for delays ∆t 0 = (0, 2m, 5m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units.

Figure 10 .Figure 11 .

 1011 Figure 10. Temperature as a function of u for r 0 = 4m and delays ∆t 0 = (0, 4m, 10m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units and u 0 = 0.

Figure 12 .

 12 Figure 12. Adiabatic control function when crossing the horizon as a function of r 0 for delays ∆t 0 = (0, 2m, 5m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units.

Figure 13 .

 13 Figure 13. Adiabatic control function as a function of u for r 0 = 10m and delays ∆t 0 = (0, 12m, 24m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)].We use 2m = 1 units and u 0 = 0.

Figure 14 .

 14 Figure 14. Adiabatic control function as a function of u for r 0 = 4m and delays ∆t 0 = (0, 4m, 10m, ∞) [curves depicted respectively as (----, •• • • • •, -• -, --)].We use 2m = 1 units and u 0 = 0.

Figure 15 .

 15 Figure 15. Adiabatic control function as a function of u for r 0 = 30m and delays ∆t 0 = (0, 60m, 140m, ∞) [curves depicted respectively as (----, • • • • • •, -• -, --)]. We use 2m = 1 units and u 0 = 0.

Figure 16 .

 16 Figure 16. Temperature for the Unruh observer κ Unruh as a function of r. We use 2m = 1 units.

Figure 17 .

 17 Figure17. Temperature as a function of u for a freely-falling observer from infinity with ∆t 0 = 200m using exact expression (curve depicted --) and using (κ Unruh D r0→∞ ) (u) approximation (curve depicted ----). We use 2m = 1 units and u H = 0.

Figure 18 .

 18 Figure 18. Temperature as a function of u for a freely-falling observer from infinity in the Unruh state. We use 2m = 1 units and u H = 0.

  Hawking radiation is perceived by different observers in spacetime. In order to have Hawking radiation in the first place, we need to select an appropriate vacuum state. Instead of making the obvious choice of the Unruh vacuum, we shall choose as vacuum state a non-stationary state which somehow interpolates between the Boulware state (with no radiation at both past and future asymptotic infinities) at early times and the Unruh state at late times. This dynamical (or nonstationary) state in a Schwarzschild static background will mimic what happens in a collapse process, in which the black hole is generated from an initially (almost) Minkowskian spacetime. As opposed to what happens in theUnruh vacuum, in this

situation Hawking radiation will not be present at early times, but will be switched on at a particular ignition time (this would correspond to the time of the formation of the black hole).

Acknowledgments

The authors want to thank S. Finazzi, S. Liberati, S. Sonego and M. Visser for enlightening comments during the elaboration of this paper. L. C. B. and C. B. have been supported by the Spanish MICINN through the project FIS2008-06078-C03-01 and by the Junta de Andalucía through the projects FQM2288 and FQM219. L. J. G. has been financially supported by the Spanish MCINN through the project FIS2008-06078-C03-03 and the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042).