Hawking Radiation as Quantum Tunneling

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University of Amsterdam

Bachelor Thesis

Hawking Radiation as Quantum

Tunneling

Auteur:

Marco Los

10171711

Begeleider:

Prof. Dr. E.P. Verlinde

Tweede beoordelaar:

Dr. J.P. van der Schaar

Universiteit van Amsterdam

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Instituut voor Theoretische Fysica Amsterdam

Verslag van Bachelorproject Natuur- en Sterrenkunde, omvang 15EC. Uitgevoerd tussen 05/02/2014 en 06/05/2014

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Bachelor Thesis Physics and Astronomy, 15EC

Title: Hawking Radiation as Quantum Tunneling Author: Marco Los, 10171711, marcolos@live.nl Supervisor: Prof. Dr. E.P. Verlinde

Second assesor: Dr. J.P. van der Schaar Completed between February 5 and May 6, 2014

Institute for Theoretical Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://iop.uva.nl/itfa

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Abstract

In this thesis an attempt is made to derive an expression for the tunneling rate of virtual particles through the event horizon. This is done in the WKB limit describing a particle tunneling away from the black hole where it becomes real and is interpreted as black hole radiation. A tunneling rate proportional to exp(8πEM

~ (1 − E

2M)) is found. This holds some interesting consequences

which are discussed in the final chapter.

To get to this result a geometry is needed which is well behaved at the event horizon. For this purpose the Eddington-Finkelstein coordinates are introduced and spacetime in this new coor-dinate system is analyzed. An expression for the tunneling rate in the WKB limit is derived.

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Populaire samenvatting [Dutch]

Zwarte gaten zijn plekken waar de zwaartekracht sterker is dan elke andere kracht in het heelal. Het zijn zeer kleine objecten die toch een extreem grote massa hebben samengeperst in een punt. Als iets te dicht bij een zwart gat komt kan het niet meer ontsnappen van de aantrekkingskracht en zal het verdwijnen achter de waarnemingshorizon. Zelfs licht kan niet ontsnappen zodra het deze denkbeeldige lijn heeft gepaseerd.

Rond deze waarnemingshorizon gebeuren ook allerlei vreemde dingen. Omdat niks kan ontsnap-pen aan een zwart gat, dacht men ook dat deze zwarte gaten geen straling uit zouden zenden, vandaar de naam ’zwart’ gat. Maar toen kwam Stephen Hawking met een ontdekking. In 1947 zei hij dat wanneer je quantumtheorie, de theorie van de kleinste deeltjes, toepast op een zwart gat, dat je tot de conclusie moet komen dat een zwart gat toch een klein beetje straling uitzendt. Maar hoe kan dit dan?

Een zwart gat bevindt zich in de ruimte, in het vacu¨um. Je zou denken dat het vacu¨um leeg is, maar dat is niet zo. Er worden continue overal deeltjes en antideeltjes in paren gemaakt. Meestal komen deze deeltjes elkaar weer snel tegen en verdwijnen ze weer. Maar als deze twee deeltjes dicht bij een zwart gat gevormd worden, kan het gebeuren dat een van twee deeltjes te dicht bij komt en opgeslokt wordt door het zwarte gat. Het andere deeltje is dan opeens vrij en kan wel van het zwarte gat vertrekken.

Kan dit beeld van twee deeltjes, waarvan eentje opgeslokt wordt door het zwarte gat en de andere ontsnapt, verklaren waarom zwarte gaten toch een beetje straling uitzenden? Om deze vraag te kunnen beantwoorden moet eerst goed gekeken worden hoe de ruimte-tijd rond een zwart gat er uit ziet maar ook hoe het kan dat een deeltje toch eventjes op plekken kan komen, waar het eigelijk niet zou mogen zijn.

In dit onderzoek wordt berekend hoe groot de kans is dat een deeltje naar buiten kan komen, als het deeltje al het zwarte gat in is. Aan de hand van die ’onstsnappings kans’ kunnen dan voorspellingen gedaan worden aan bepaalde eigenschappen van een zwart gat, onder andere welke temperatuur een zwart gat heeft.

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Introduction

In Newtons theory of gravity the escape velocity v at distance r from an object of mass M is given by:

mv2

2 = GM m

r (1.1)

Many physicist asked themselves what would happen if the mass is so big, that at a certain distance the escape velocity would exceed the speed of light. These objects would then be so massive that they generate such a strong gravitational pull that nothing, not even light, could escape from it. This is what defines a black hole. The surface area ABH at a distance RS from

the center of the black hole is called the event horizon, here the escape speed equals the speed of light:

ABH = 4πRs2 Rs=

2GM

c2 (1.2)

This is a very classical discussion and this can be misleading. To get a better understanding of gravity. one needs to include general relativity. This theory, constructed by Einstein, describes how gravity affects the geometry of space-time. Einstein himself thought that black holes where due to an incomplete physical understanding. But these days black holes are not only accepted, but it is also understood how these objects can form and how particles and fields behave near them.

However, black holes still seem to have some properties which are still not completely understood. It was commonly accepted that whenever something gets too close to a black hole, it is lost forever. But in 1947 Stephen Hawking showed with the use of quantum field theory that black holes emit radiation as if it where a blackbody with a so called Hawking Temperature inversely proportional to the black hole’s mass:

TH= ~c 3

8πGM kB

(1.3)

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Chapter 1. Introduction 2

In the last years different approaches have been suggested to explain this phenomenon. In this thesis the radiation will be explained as the result of particles tunneling through the event hori-zon. The tunneling of massless shells is considered in the WKB approximation and a prediction for the Hawking Temperature is given.

This thesis is build up of three parts. In the first an introduction to general relativity is given in which is discussed how geometries are described. The two main geometries are those of the Minkowski space and of the Schwarzschild solution. In the Schwarzschild solution a coordinate transformation will be performed to ensure that it is suitable to describe event horizon crossing events.

In the second part the transmission probability for a tunneling process in the WKB approx-imation is derived. And the final part will consist of a calculation of tunneling process of a massless shell through a black holes event horizon which will lead to a prediction of a black holes temperature.

As a final note I wish that you enjoy the read. Feel free to contact me for any questions or remarks.

Thesis written by Marco Los

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Chapter 2

General Relativity

2.1 Coordinate Systems and Line Elements

A way to specify points in space systematically is to introduce a coordinate system which assigns a unique label to each of these points. This way differential calculus can be used to specify geometries through line elements. Generally, a line element describing an N-dimensional space is given as the square of the distance ds in terms of infinitesimal displacements in coordinates qi= (q1, q2...., qN):

ds2= gijdqidqj (2.1)

where gijindicates the corresponding element of the metric tensor g which describes the geometry

and a sum over repeated indexes is implied. As an example, consider the three dimensional flat space: in Cartesian coordinates (x, y, x) the metric is:

g =     1 0 0 0 1 0 0 0 1     (2.2)

It is also possible to label these points with spherical coordinates (r, θ, φ) with the metric:

g =     1 0 0 0 r2 ₀ 0 0 r2_sin2_θ     (2.3)

Both of these metrics describe the same geometry but in different coordinate systems. The line elements following these metrics are the following:

ds2= dx2+ dy2+ dz2 (2.4) = dr2+ r2dθ2+ r2sinθdφ2 (2.5)

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Chapter 2. General Relativity 4

It is possible to represent the same geometry with various line elements written in different coordinate systems. Of course the physical results should not depend on the choice of coordi-nates, the coordinates themselves hold no physical meaning. This means that it is allowed to perform coordinate transformations to manipulate line elements, without changing the physical description.

2.2 Minkowski space

Einstein’s theory of special relativity unifies the Euclidean space coordinates with time taking ct as a fourth coordinate. This results in the four-dimensional Minkowski space with coordinates (ct, x, y, z) and the geometry is described by:

ds2= −c2dt2+ dx2+ dy2+ dz2 (2.6) or writing the metric η explicitly:

η =       −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1       (2.7)

The quantaty ds2 _{is an invariant quantity, meaning that it will be the same for observers in}

different inertial frames. This line element describes a non-Euclidean geometry due to the minus sign for the time coordinate. But this sign also makes it possible for ds2to be positive, zero and also negative. The three cases are names as followed:

ds2> 0 Spacelike separation (2.8) ds2= 0 Null separation (2.9) ds2< 0 Timelike separation (2.10) Light rays always travel along world lines for which ds2_{= 0 and these are called null geodesics.}

The light cone of a point is the three dimensional surface in spacetime constructed by light rays moving out from the point.

Of course the Minkowski space can also be expressed in terms of the coordinates (ct, r, θ, φ) which would give:

ds2= −c2dt2+ dr2+ r2dθ2+ r2sinθdφ2 (2.11)

2.2.1 Lightcones

To gain a better understanding of the geometry, the world lines of light rays are studied. A world line is the path which an object follows when moving in spacetime. For simplicity only

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two coordinates are considered, but the same principles apply to higher dimensions. For light rays ds2= 0 and from (2.6):

dx

d(ct) = ±1 (2.12)

so graphically light rays are represented by 45◦ lines and in higher dimensions these form cones: lightcones. See figure 2.1.

Figure 2.1: The lightcone from point P in Minkowski space. Q lies within the lightcone of P and they are therefore timelike separated from each other. R lies outside the lightcone, these

points are spacelike separated and they cannot communicate with each other.

Lightcones give a good and clear description how particles behave in spacetime. Since particles with a non-zero mass move with a velocity smaller than the speed of light, they travel along timelike world lines and their world lines are thus confined by the lightcone. Once it is clear how light behaves, one can easily explain what the world lines of massive particles can be like.

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2.3 Curved Spacetime [1]

In the previous section the Minkowski space was discussed which describes a flat spacetime. From Einsteins equivalence principle it seems to follow that light no longer follows straight lines when it is near a mass but its path tends to curve towards the mass. A more elegant way to describe this, is to say that light still follows straight lines in spacetime, but it is spacetime itself which is curved by the gravitational field.

The metrics to describe this curved spacetime can be derived as a solution from the Einstein equa-tions. But solving these equations is far beyond the goal of this thesis. In 1916, K. Schwarschild found an exact solution to the Einstein equations under the assumptions of spherical symmetry and that spacetime is static which, means that there is no dependence on t in any of the co-ordinates. This is called the Schwarzschild solution and this will be discussed in the following section.

2.3.1 Schwarzschild solution

The geometry of spacetime outside a spherically symmetric mass distribution is described by the Schwarzschild line element:

ds2= − 1 −2GM c2_r (cdt)2+ 1 − 2GM c2_r −1 dr2+ r2dΩ2 (2.13) where dΩ2 _{= dθ}2_{+ sin}2_{θdφ. From here on, the equations will be written in geometrized units}

(G = c = 1). It is easy to convert between the geometrized units and the standard mass-length-time units, see appendix A for a short guide.

In geometrized units the line element reduces to:

ds2= − 1 −2M r dt2+ 1 − 2M r −1 dr2+ r2dΩ2 (2.14) This line shows some interesting properties. First off it shows no dependence on time t or the azimuthal angle φ. This, of course, originates from the assumptions made by Schwarzschild when he was solving the Einstein equations.

When there is no mass generating the gravitational potential M = 0 or when a point is very far away from the mass, r → ∞, this line element reduces to the flat Minkowski space. This is consistent since in either cases the effects of the mass’s presence will be negligible and the geometry for flat spacetime is expected.

There are two singularities, one at r = 0 and one at r = 2M . The first one represent the phys-ical singularity where all mass is compressed. To recognize the second singularity the physphys-ical constants are recovered to find:

r = 2GM

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This is exactly the Schwarzschild radius which was calculated earlier on classical grounds. Gen-eral relativity seems to predicts the exact same radius. When the mass of an object is located inside its Schwarzschild radius, the object will collapse and form a black hole. The spherical surface area with its radius the Schwarzschild radius is also known as the event horizon.

2.3.2 Schwarzschild Lightcones

In this section radial light rays in curved spacetime are studied in more detail. These are called radial null geodesics and for these ds2_{= 0 and dθ = dφ = 0. The line element (2.13) reduces to:}

0 = − 1 − 2M r dt2+ 1 −2M r −1 dr2 (2.16)

This expression can be rearranged to find:

dt dr = ± 1 −2M r −1 (2.17)

Where the plus indicates an outgoing ray and the minus indicates a ingoing light ray. Integrating this equation gives two equations, one for each kind of radial ray:

t = r + 2M ln r 2M − 1 + constant outgoing (2.18) t = −r − 2M ln r 2M − 1 + constant ingoing (2.19)

These curves can be plotted in the (t, r)-plane, see figure 2.2. The lightcones show a strange behaviour at the event horizon. When they get closer they get narrower until they suddenly flip when passing the horizon. This comes from the singularity in the line element and at this point the coordinates t and r change in sign and seem to interchange in their roles.

Figure 2.2: Radial light rays in the Schwarzschild geometry. The lightcones get narrower as they approach the black hole and suddenly ’flip’ at the event horizon r = 2M .

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2.4 Eddington-Finkelstein coordinates [2]

The Schwarzschild geometry in the usual (t, x, y, z) coordinates hold a singularity at the event horizon (r = 2M ) and the light cones show a weird behavior near this point. This is unwanted, this line element can not be used to describe event horizon crossing phenomena. The goal is now to make a coordinate transformation such that this singularity is no longer present.

For this a new coordinate ρ is introduced such that:

ρ = Z 1 −2M r −1 dr = r + 2M ln(r − 2M ) (2.20)

and a new definition for the time coordinate is taken as v = t + ρ. It then follows:

dt2= dv2− 2dvdρ − dρ2 _(2.21)

From (2.160) it is clear that dρ = (1 −2M r )

−1_{dr. Using this in (2.21) and multiplying both sides}

by −(1 −2M_r ): − 1 −2M r dt2= − 1 − 2M r dv2+ 2drdv + 1 − 2M 2 −1 dr2 (2.22) Now (2.22) is used to replace the time coordinate in (2.14) and the line element reads:

ds2= − 1 − 2M r dv2+ 2drdv + r2dΩ2 (2.23) This describes the exact same geometry as (2.14) but it no longer has a singularity at r = 2M . Lightcones now behave well near the event horizon and this line element is a proper choice to describe events that take place while crossing the event horizon.

2.4.1 Lightcones in Eddington-Finkelstein coordinates

In this new coordinate system the radial null geodesics are given by:

0 = − 1 −2M r dv2+ 2drdv (2.24) and light rays move along lines of constant v (dv = 0) or:

dr dv = 1 2 1 −2M r (2.25)

These can be integrated to find:

v = 2r + 2M ln r 2M − 1

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which describes ingoing light rays when r < 2M and outgoing when r > 2M . These can again be plotted in a diagram, this time with y ≡ v − r on the vertical axis.

The geodesics of constant v represent ingoing light rays. Notice how ingoing light rays can be described in two different ways and that when r < 2M there can only be ingoing light rays. This indicates again that once something passes the event horizon it can not go back, see figure 2.3.

Figure 2.3: Radial light rays in the Eddington-Finkelstein coordinates, with y ≡ v −r. Notice how the light cones now gradually tip until after the event horizon not even light can escape

the black holes attraction.

2.5 Overview

The Schwarzschild solution described the curvature of space time outside a spherically symmetric mass distribution. For a black hole, all mass is located in one single point, which can conveniently be chosen to be the origin r = 0. In the standard (t, r, θ, φ) coordinates there exist a non-physical singularity at r = 2M , the event horizon. By introducing a new coordinate this singularity can be removed by a transformation and the line element is written in the new Eddington-Finkelstein coordinates (v, r, θ, φ): ds2= − 1 − 2M r dv2+ 2drdv + r2dΩ2 (2.27) From this line element it follow that radial light rays follow eiher paths described by v = constant or: dr dv = 1 2 1 −2M r (2.28)

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Chapter 3

WKB Approximation [3]

The WKB approximation is a useful approach to describe the behavior of wave functions in a slowly changing potential. This approach also gives a mathematical framework to described the tunneling of particles through classically forbidden regions.

3.1 Schr¨

odinger equation and the WKB approximation

The time independent Schr¨odinger equation:

− ~

2

2m d2_ψ

dx2 = (E − V (x))ψ (3.1)

can be written as:

d2_ψ

dx2 = −

p2

~2ψ (3.2)

where p = p2m(E − V (x)), the classic expression for the particles momentum. For now the assumption is that E > V . The wavefunction can be expressed in terms of its amplitude A(x) and its phase θ(x) as:

ψ(x) = A(x)eiθ(x) (3.3) Inserting this wavefunction into (3.2) one finds:

d2_A dx2 + 2i dA dx dθ dx+ iA d2_θ dx2− A dθ dx 2 = −p 2 ~2 A (3.4)

This gives two equations, since both the imaginary and real part much equate on both sides:

d2_A dx2 − A dθ dx 2 = −p 2 ~2 A (3.5) 2dA dx dθ dx + A d2θ dx2 = 0 (3.6) 10

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Chapter 3. WKB Approximation 11

The imaginary part is easily solved. From (3.6) is follows:

d dx A2dθ dx = 0 (3.7) and thus: A = _qc dθ dx (3.8)

To solve the real part, the central assumption of the WKB approximation comes into play. The assumption is made that d_dx2A2/A is much smaller than either (

dθ dx)

2_or p2

~2, then (3.5) reduces to:

dθ dx 2 = p 2 ~2 (3.9)

and then it follows: ℘θ(x) = ±1

~R p(x)dx(3.10)And the general solution will be of the form:

ψ(x) = A pp(x)e i ~R p(x)dx+ B pp(x)e −i ~R p(x)dx (3.11)

3.2 Tunneling

Above discussion assumed that the energy was greater than the potential, but analogously a derivation can be made for the case when E < V . The only difference then is that the exponential in (3.11) becomes imaginary. The general solution becomes:

ψ(x) = A pp(x)e 1 ~ImR p(x)dx+ B pp(x)e −1 ~ImR p(x)dx (3.12)

Equation (3.12) shows how the wave function behaves in the forbidden region, the region where classically the wave function (or particle) is not allowed to exist.

Consider a rectangular potential V (x) with an arbitrary top located from x = a to x = b, see figure 3.1. How does a wave function scatter from this potential? There are three different regions to define: the region where the waves arrive at the potential x < a, the non classical region a < x < b and the region behind the potential x > b.

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Figure 3.1: Scattering from a rectangular potential with an arbitrary top. Indicated are incident, reflected and transmitted waves as respectively A, R, and T .

The wavefunctions in the first region are just those of free particles:

ψ(x) = Aeiηx+ Re−iηx (3.13) where η = 1

~

√

2mE. In the third region the wavefront also is that of a free particle:

ψ(x) = T eiηx (3.14) For the central region equation (3.12) is applied to find:

ψ(x) = B pp(x)e 1 ~Im Rb ap(x)dx+ C pp(x)e −1 ~Im Rb ap(x)dx (3.15)

the potential’s effect will be to lower the amplitude, so one needs to set B = 0 to obtain a physical acceptable result.

These three equations fully describe how the wavefront behaves near and inside the potential. For the goal of this thesis the transmission probability, Γ, is of interest. This is defined as the amplitudes between the ingoing wave (A) and the transmitted wave (T ) which has tunneled through the barrier:

Γ ≡ |T |

2

|A|2 (3.16)

If there was no barrier A = T and R = 0 and if the barrier was infinitely high or broad T = 0 and A = R. The transmission probability is determined solely by the property of the potential. This means that the relative amplitudes of A and T are calculated through the behavior of the wavefront in the non-classical region:

Γ ∝ e−2~Im

Rb

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The exponential here is simply the action for a particle moving along a certain world line. So this is usually written as:

Γ ∝ e−2~Im S (3.18)

where S is the particles action given by:

S = Z

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Chapter 4

Hawking Radiation as Quantum

Tunneling [4] [5]

In Chapter 2 the geometry of space-time outside a black hole was discussed and an expression for radial null geodesics was found. Here the techniques of the WKB approximation, as discussed in Chapter 3, will be used to explain Hawking Radiation as the consequence of particles tunneling outward through the event horizon. These then which move out to infinity, where an observer would see this as radiation emerging from the black hole.

4.1 Pair Creation

In quantum mechanics there exists the possibility for a pair of particles to be created from fluctuations in the vacuum. These particles usually meet together quickly again and annihilate. However when this pair if formed near a black hole’s event horizon, something interesting can happen.

Consider a pair of particles to be created just inside the event horizon, on carrying a positive energy and the other a negative energy. Classically both particles would fall into the black hole. But in quantum mechanics it is allowed that the positive energy particle can tunnel outward, escaping from the black hole and materializing outside the event horizon. This particle can now move on outward and can be considered black hole radiation. The negative energy particle remains inside and falls into the black hole’s singularity, decreasing the black hole’s mass. Since the black holes mass and radius are directly related this would result in the black hole decreasing in size.

The same applies for pair creation just outside the event horizon, where the negative energy particle can tunnel inwards and the positive energy particle remains outside. Considering the negative energy particles simply as normal particles under time inversion, then it is clear both processes are actually the same. In both cases the process can be described as a single positive

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Chapter 4. Hawking Radiation as Quantum Tunneling 15

energy particle tunneling out through the event horizon.

4.2 Where is the barrier? [4]

There is, however, one big problem with this picture. For quantum tunneling the particle needs to traverse some potential barrier. But there seem to be no barrier at the event horizon, there is no trajectory in spacetime what can clearly be classified as the forbidden region.

There is a subtle explanation to this and it lies in the conservation of energy. As the black hole radiates is losses energy in the form of mass which means that the black hole shrinks. The event horizon r = 2M now reduces to a new radius r0= 2(M − E) which is smaller than before. Here E is the energy of the tunneling particle and the amount by which the event horizon reduces is dependent on the particle itself. It is the particle itself that forms the barrier.

4.3 Tunneling [5]

To describe event which cross the event horizon it is required to have a coordinate system which is well behaved at the horizon. The Schwarzschild geometry in Eddington-Finkelstein coordinates satisfy this need.

The following calculations describe gravitating (spherically symmetric) shells in Hamiltonian gravity characterized by energy E. The total mass is held fixed while the black hole mass is allowed to vary. Krauss and Wilczek found that this is done mathematically by replacing M by M − E in the equations. The line element now reads:

ds2= − 1 − 2(M − E) r dv2+ 2drdv + r2dΩ2 (4.1) Here the results from the WKB approximation are going to be used, but is it justified to do so? The radiation from a black hole typically has a wavelength the size of a black hole. So one could argue that this is not in agreement with the approximations made in the derivation for tunneling rate. But when measured by a local observer far away from the black hole, the wavelength traced back towards the horizon will be greatly blueshifted due to gravitational redshift, see appendix B. This leads to a radial wavenumber approaching infinity near the horizon and then the WKB approximation is satisfied.

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Now the action needs to be evaluated:

S = Z rout rin prdr = Z rout rin Z pr 0 dp0_rdr (4.2) where pr is the canonical impulse for r. Using Hamilton’s equation dHdp0

r

= ˙r this can be written as: S = Z rout rin Z M −E M dH ˙r dr (4.3)

where ˙r =_dvdr and rin= 2M and rrout= 2(M − E) the region through which the tunneling takes

place. The Hamiltonian for the gravitating shell is H = M − E and using equation (2.25) with the new expression for the mass and switching the order of integration this gives:

S = −2 Z E 0 Z rout rin dr 1 − 2(M −E_r 0) dE0 (4.4)

The minus sign arises from the Hamiltonian: dH = −dE. To evaluate this integral, note that the integrand holds a first order pole at r = 2(M − E). Here residue integration is applied by deforming the path to the lower half of the E-plane, to ensure a positive result:

Res 1 1 − 2(M −E_r 0) ! r=2(M −E0₎ = 2(M − E 0₎ r2 r=2(M −E0₎ = 2(M − E0) (4.5) Here residue integration gives 2πi Res|f(x) since the deformation is a semi-circle, and not a closed contour. The final integral can be evaluated through Cauchy’s theorem and gives:

S = 2πi Z E

0

2(M − E0)dE0= 4πiEM (1 − E

2M) (4.6)

which is, as expected, an imaginary number. According to equation (3.18) the tunneling rate for such a process is proportional to the exponential decay in this region:

Γ ∝ e−2~Im S= e− 8πEM

~ (1− E

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4.3.1 Black Hole Temperature

An expression for the tunneling rate has been found and it is shown to be proportional to exp(−8πEM

~ (1 − E

2M)). This is not exactly what is expected for a Planck spectral flux. However

for small energies the correction term can be neglected and that would imply a Boltzmann factor exp(−βE). Here β = 1/kBT with kB the Boltzmann constant. Assuming the black hole radiates

as a blackbody this would imply:

T = ~ 8kBπM

(4.8)

this is of course still in the geometrized units. Converting is back to standard units:

T = ~c

3

8kBπGM

(4.9)

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Chapter 5

Discussion and Conclusion

Starting with the Schwarschild solution to Einsteins field equation the behavior of light near a black hole was studied in this thesis. Demanding a well behaving coordinate system near the event horizon led to the introduction of the Eddington-Finkelstein coordinates and these gave a better understanding of processes near and through the event horizon.

To describe the tunneling of particles through the event horizon the WKB approximation needed to be included and in this approximation a tunneling probability was derived. Applying this to self gravitating, massless shells a tunneling probability was derived:

Γ ∝ eexp(−8πEM~ (1− E

2M)) (5.1)

For small values of E the quadratic term can be neglected and this would imply the black hole’s temperature as:

T = ~c

3

8kBπGM

(5.2)

and this is in total agreement with the predictions done by Stephen Hawking, and many more after him. It seems that this picture, of particles tunneling through the horizon is a plausible explanation for this process.

5.1 Information Loss

A topic of discussions is the loss of information in this process. Since all kinds of particles can go into the black hole a certain amount of energy would go into the system. However if the black hole emits thermal radiation, a big piece of information would be lost. For instance, one cannot tell what particle originally fell into the black hole, based solely on the radiation emitted by the black hole.

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Chapter 5. Discussion and Conclusion 19

But that is only true if the black hole emits thermal radiation, with a Boltzman factor of the form exp(−βE). Our calculation did show this factor, but not exactly. There is an extra term: 1 − _2ME . This means that the radiation from a black hole is not precisely thermal and it still contains information about the original particle. This implies that from this approach, the black hole can radiate but there would be no loss of information.

5.2 Final Remarks

The entire derivation rests on the Schwarschild geometry, which is as we now know quite limited. Black holes can also rotate and have an electrical charge which where not included in this thesis. It would be very interesting to take these quantities into consideration and see whether it still agrees with the current knowledge of black holes.

Finally I would like to thank prof. Erik P. Verlinde for supervising this project. The discussions between us were of great value in the process of executing the project and his personal interest in the subject kept me motivated throughout.

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Appendix A

Geometrized Units

Normally equations are written in mass-length-time units certain physicals constants tend to pop up everywhere. In general relativity these are Newtons gravitational constant G and the speed of light c. All these extra symbols can make the equations rather long and messy to read. It is often very usefull to introduce geometrized units in which G = c = 1 and then convert the final result back to the ordinairy mass-length-time unit system.

In geometrized units all quantaties have the dimension of length and to convert the units back to standard units the following conversions need to be made:

M → GM

c2 (A.1)

t → ct (A.2)

L → L (A.3)

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Appendix B

Gravitational Redshift

From the invariance in time displacements in the Schwarzschild geometry it can be derived that the inner product of Killing Vector ~ζ = (1, 0, 0, 0) and the particles momentum ~p is conserved. A photon which has it’s energy measured by an observer at coordinate R. He will find the photon’s energy to be E = −~p · ~uobs. Since the energy of a photon is related to its frequency, E = ~ω, the

observer will measure a frequency:

~ω = −~p · ~uobs (B.1)

The four velocity of an observer hold the normalization condition [1] ~uobs· ~uobs= −1 and then

for a stationairy observer the four vector is written as:

And the four velocity of a stationary observer can be written as:

~ u = 1 −2GM c2_r −1/2 ~ ζ (B.2)

The frequency of the photon as measured by a stationary observer at R:

~ωR= 1 −2GM c2_r −1/2 (−~ζ · ~p)r=R (B.3)

and for a stationary observer at infinity:

~ω∞= (−~ζ · ~p)r=∞ (B.4)

But the quantity −~ζ · ~p is conserved. Then from (B.3) and (B.4) a relationship is found between both frequencies: ω∞= ωR 1 −2GM c2_r 1/2 (B.5)

Equation (3.5) sais that when a photon is moving away from a source of gravity it looses energy and is undergoing redshift. For greater values of r the frequency gets smaller.

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Bibliography

[1] Hartle, J. B. (2003) Gravity, an introduction to Einstein’s General Relativity, Addison-Wesley, San Fransisco, 2003, Chapters 6, 9 and 12.

[2] Hellekman, B. (2013) Thought Experiments on Black Hole Thermodynamics. Bachelor Thesis. University of Amsterdam, Amsterdam, the Netherlands.

[3] Griffiths, D. J. (2003) Introduction to Quantum Mechanics, Pearson Education, 1995, Lon-don, Chapter 8.

[4] M. K. Parikh, A Secret Tunnel Through The Horizon Gen. Rel. Grav. 36, 2419 (2004, First Award at Gravity Research Foundation).

[5] M.K. Parikh and F. Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85 (2000)

Hawking Radiation as Quantum Tunneling

University of Amsterdam

Bachelor Thesis

Hawking Radiation as Quantum

Tunneling

Auteur:

Marco Los

10171711

Begeleider:

Prof. Dr. E.P. Verlinde

Tweede beoordelaar:

Dr. J.P. van der Schaar

Universiteit van Amsterdam

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Instituut voor Theoretische Fysica Amsterdam

Abstract

Populaire samenvatting [Dutch]

Contents

Chapter 1

Introduction

Chapter 2

General Relativity

2.1

Coordinate Systems and Line Elements

2.2

Minkowski space

2.2.1

Lightcones

2.3

Curved Spacetime [1]

2.3.1

Schwarzschild solution

2.3.2

Schwarzschild Lightcones

2.4

Eddington-Finkelstein coordinates [2]

2.4.1

Lightcones in Eddington-Finkelstein coordinates

2.5

Overview

Chapter 3

WKB Approximation [3]

3.1

Schr¨

odinger equation and the WKB approximation

3.2

Tunneling

Chapter 4

Hawking Radiation as Quantum

Tunneling [4] [5]

4.1

Pair Creation

4.2

Where is the barrier? [4]

4.3

Tunneling [5]

4.3.1

Black Hole Temperature

Chapter 5

Discussion and Conclusion

5.1

Information Loss

5.2

Final Remarks

Appendix A

Geometrized Units

Appendix B

Gravitational Redshift

Bibliography