University of Groningen Field perturbations in general relativity and infinite derivative gravity Harmsen, Gerhard Erwin

Field perturbations in general relativity and infinite derivative gravity

Harmsen, Gerhard Erwin

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10.33612/diss.99355803

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Harmsen, G. E. (2019). Field perturbations in general relativity and infinite derivative gravity. University of Groningen. https://doi.org/10.33612/diss.99355803

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13

Chapter 2

Introduction to QNMs

In this chapter we will introduce the ideas and tools necessary to study the QNMs of black holes due to spin-3/2 perturbations. We begin by introducing a more math-ematical definition of QNMs, and what the general procedure is for obtaining the numerical values of QNMs. We then introduce spin-3/2 fields, by giving the appro-priate equation of motion describing these types of fields, note that due to the space times we wish to study we will need to modify our covariant derivative. Finally, we provide an overview of the numerical methods we shall use to calculate our QNMs.

2.1

A Mathematical description of QNMs

To build on the intuitive picture described in Ch. 1, we will use a more mathemati-cal formalism to describe QNMs. We begin by considering the formula for standing waves, since as stated previously, QNMs are damped standing waves which, cru-cially, obey the boundary conditions given in Ref. [38]. We can represent standing waves in one dimension as [32]

d2

dr2Ψ(r, t) − d2

dt2Ψ(r, t) −V(r)Ψ(r, t) =0 , (2.1) with Ψ describing some wavelike function, where r and t denote space and time coordinates respectively, and V is some r-dependent potential. We can solve this equation by assuming the form of the standing wave as [68]

Ψ(r, t) =e−iωtφ(r). (2.2)

In the case of an undamped wave ω has a purely real value, in the case of a damped wave, however, the parameter ω complex values, where the real part represents frequency and the imaginary part represents the damping that the wave experiences. Plugging Eq. (2.2) into our wave equation we get

d2φ(r) dr2 −  ω2+V(r)  φ(r) =0 . (2.3)

This is the general form of the equations that we will be using to determine the numerical values of our QN frequencies. Note that we will be solving for the QN frequencies. In the chapters that follow our task will be to take an appropriate equa-tion of moequa-tion and obtain the above equaequa-tion. Once we have this we need to impose the appropriate boundary conditions to ensure that we obtain the complex value for

ω. The boundary conditions that we would need to place on the equation is that the

QNMs are purely ingoing at the event horizon, and purely outgoing at spacial infin-ity. As such the appropriate boundary conditions for the asymptotically flat case are

[38]

φ(r) ∼e±iωr∗ ; r∗ → ±∞ , (2.4)

where r∗ is the tortoise coordinate and is determined as follows

dr∗ = dr

f(r), (2.5)

where f(r)is a function specified by the metric of interest [38]. In order to determine the effective potential such that we obtain the wave equation, we must first deter-mine the equation of motion that defines our field. As we are interested in spin-3/2 fields, we will derive their appropriate equation of motion by firstly considering their Lagrangian.

2.2

Spin-3/2 fields

Before we introduce the spin-3/2 field, we will briefly go over some points on nota-tion. In principal we can use the SU(2)×SU(2) Lorentz group to represent different spin fields. Below is a brief overview of this:

• (0, 0) - Represents the spin-0 field. • (1

2, 0)⊕(0, 12) - Represents the spin-1/2 fields or spinor fields. • (1₂, 1₂) - Represents the spin-1 or vector fields.

• (1, 1)⊕(0, 0) - Represents the spin-2 or tensor field.

Where in the above “⊕” is the tensor summation operator, later we also use the tensor product denoted as “⊗”. Spin-3/2 fields are considered to be a combination of the spinor and vector fields, so are naively called spinor-vector fields. This means they require components from both the spinor and vector fields. The representation of these fields in the Lorentz group representation would be [69]

 1 2, 0  ⊕  0, 1 2  ⊗ 1 2, 1 2  = 1 2, 0  ⊗ 1 2, 1 2  ⊕  0, 1 2  ⊗ 1 2, 1 2  . (2.6) Further decomposition of the field yields that the field can be described as follows in the representation  (1,1 2) ⊕ (0, 1 2)  ⊕  (1 2, 1) ⊕ ( 1 2, 0)  . (2.7)

So this field contains both a spin-1/2 component and a spin-3/2 component, given by(1,1₂). We require a spinor representation to show these types of fields, as this makes the notation easier. We will be using a notation first given by Rarita and Schwinger [70] where spinors are represented asΨµ. Note that spinors are consid-ered to be the simplest mathematical objects that can represent Lorentz transforma-tions, such as boosts or rotations [71]. This makes them ideal for describing fields with spin, especially fields with half integer spin. Weinberg has shown that we can isolate the spin-3/2 component of the fields description by requiring that [69]

2.2. Spin-3/2 fields 15

where γµ _{is the Dirac gamma matrix and ψ}

µis a Majorana type spinor representing the field, and µ is a spatial index running from 0 to one minus the total number of space time dimensions. Spinors can be thought of as vectors which can provide a linear representation of the rotation group of dimension n. With each spinor having 2vcomponents, with n=2v+1 or n=2v for the case of n odd or even respectively [72]. This makes them useful in physics, as they naturally encode within them spin (meaning they are occasionally used interchangeably when referring to fermionic particles). In physics there are three main types of spinors, the Dirac, Majorana and Weyl spinors. Majorana spinors were introduced to solve the Majorana equation, written as

γµ∂µψc+mψ=0, (2.9)

where ψc = iψ∗ is the charge conjugate of the field ψ. However, since the field and its charge conjugate appear in the equation, this equation cannot contain fields that have electric charge, since the charge conjugate would be negative. This means when we construct the spin-3/2 field we must assume it has no electric charge. Due to rotational invariance, Eq. (2.8) tells us that a field of momentum q =0 and spin s in the z-component will satisfy the conditions

h0|ψ0(0)|si =0, (2.10) and 3/2

∑

It then follows that the propagator of the spin-3/2 field should be [69] Pµν ₌ P

µν_(q) q2_+m2

g−ie

, (2.12)

where Pµν_{(q)is a Lorentz-covariant polynomial of the four vector q. From the} con-dition that for q = 0 and q0 _{= m}

g, where mg is the mass of the spin-3/2 field, we have Pij = 1+β 2   δij− 1 3γiγj  , (2.13)

with Pi0 _{= P}0i _{= P}00 _{= 0. The unique covariant function that can describe this} behaviour is given as [69] Pµν_{(q) =} ηµν+ q µ_qν m2 g ! −_i/q+mg
− 1 3  γµ−iq µ mg  i/q+mg
 γν−iq ν mg  . (2.14) As such, a Lagrangian that can correctly describe this type of field must have the form L = −1 2 ψ¯ µ_{Dµν(−i∂)} ψν
, (2.15) where Pµν_(q)D νλ(q) =δ µ λ. (2.16)

Weinberg has shown that the relationship in Eq. (2.16) implies that [69] Dνλ(q) = −eνµκλγ5γµqκ− 1 2mg h γν, γλ i . (2.17)

As such the Lagrangian for the massive spin-3/2 field is written as [69] L = −1 2ie νµκλ _¯ ψνγ5γµ∂κψλ
+1 4mg  ¯ ψν h γν, γλ i ψλ  . (2.18)

In this thesis we work with only the massless form of the fields, in which case we can simplify the Lagrangian to its massless form,

L = −1 2ie

νµκλ _¯

ψνγ5γµ∂κψλ
. (2.19) It can be shown straight forwardly, using the gamma matrix identities, that this La-grangian can be rewritten as

L =_ψ¯_νγµνλ

∂κψλ, (2.20)

where γµνα_{= γ}µ_γν_γα_+γµ_gνα_−γν_gµα_+γα_gµν_{is the anti-symmetric gamma matrix} relation. Using the principal of least action to determine the equations of motion, the result of varying the above Lagrangian gives

δL =δ ¯ψµγµνλ∂κψλ−∂κψ¯µγµνλδψλ. (2.21) as shown in Ref. [69] using the “Majorana-flip property” this can be simplified to

δL =2δ ¯ψµγ µνλ

∂κψλ, (2.22)

which by the principal of least action, δL =0, implies that

γµνλ∇νψλ =0. (2.23)

This is the Rarita-Schwinger equation, as given by Rarita and Schwinger in Ref. [70]. This form of the equation of motion, however, is not necessarily gauge invariant in all space times. If we perform the following gauge transformation

ψ0_λ =ψλ+ ∇λϕ, (2.24)

where ϕ is some spinor, and plug this into Eq. (B.10) and then require that

γµνλ∇ν∇λϕ=0, (2.25)

then our Lagrangian is invariant [73]. In the purely gravitational case this equation can be written as γµνλ∇ν∇λϕ= 1 2γ µνλ_[∇ ν,∇λ]ϕ = 1 8γ µνλ_R νλρσγ ρ γσϕ. (2.26)

Using the Bianchi identity we can show that

γµγνγλRνλρσ = −2γλRρσ, (2.27) furthermore γµ

γνγργσRµνρσ = −2R. Using these identities we have that

γµνλ∇ν∇λϕ= 1 8γ µνλ_R νλρσγργσϕ= 1 4  2γλ_Rµ λ−γ µ_R ϕ. (2.28)

The expression on the far right is only zero if we are working in Ricci flat space times, such as the Schwarzschild space time [74]. This is not the case when introducing

2.3. Super covariant derivative 17

charge or when introducing asymptotic curvature to the space time (as in the case of AdS space times). So for both the AdS space time and the Reissner-Nordstöm black hole we will need to modify this equation. We therefore need to modify the derivative function such that it is applicable in the cases that we are interested in. In the next section we show how we have constructed a so called “super covariant derivative”, since it is a modified covariant derivative for spin-3/2 fields.

2.3

Super covariant derivative

In constructing the “super covariant derivative” we need to ensure that

γλµνDµ,Dν



ϕ=0, (2.29)

as alluded to in Eq. (2.26), where here we have used the “super covariant deriva-tive”Dµinstead of the ordinary covariant derivative. This new covariant derivative takes into account the electrical charge and the curvature, due to the cosmological constant. The most general derivative we can construct is of the form

Dµ=De_µ+a √

Λγµ+bγρ_F

µρ+cγµρσF

ρσ_{, (2.30)}

where eDµ = ∇µ−ieAµ, with Aµthe D dimensional form of the four potential, Fµν is the stress tensor andΛ is the cosmological constant. We are then left to determine the values of a, b and c. Plugging this into Eq. (2.29) we have

0=γλµν h e Dµ, eDν i ϕ+2bγλµν h e Dµ, γρFνρ i ϕ+2cγλµν h e Dµ, γνρσFρσ i ϕ +a2Λγλµν γµ, γν]  ϕ+2ab √ Λγλµν γµ, γρFνρ  +2ac√Λγλµν γµ, γνρσFρσ  ϕ +b2γλµνγαFµα, γρFνρ  ϕ+2bcγλµνγαFµα, γνρσFρσ  +c2γλµν h γµαβFαβ, γνρσFρσ i ϕ. (2.31) We first look at the commutation relation of the differential operator,

γλµνDµ,Dν  ϕ= −1 8γ λµν_(R ab µν [γa, γb])ϕ−ieγ λµν_Fµν ϕ = −1 4γ λµν_(R ab µν (γaγb−gab))ϕ−ieγ λµν_F µνϕ =γµG_µλϕ−ieγλµνFµνϕ, (2.32) Where Gλ

µ is the Einstein tensor. Next consider the commutation relation between the differential operator and the Fµν operator,

γλµνDµ, γ ρ_F νρ  ϕ=γλµνγρDµFνρϕ =γλγµγνγρ−γλgµνγρ+γµgλνγρ−γνgλµγρ DµFνρϕ. (2.33) But γµνρ_∇

µFνρ =0 by the Bianchi identities. Furthermore gµρ∇µFνρ = gµν∇µFνρ =0 since∇_µFµρ = 0. Next gνρ∇µFνρ = 0 since the tensor Fµν is antisymmetric. Finally by using the Bianchi identities we can show that

∇_µFνα+ ∇νFαµ+ ∇αFµν =0,

γµν(∇µFνα+ ∇νFαµ+ ∇αFµν) =0, 2γµν_{(∇µFνα) =}

γµν∇αFµν.

This result implies that γλµν Dµ, γρFνρ  ϕ=γλνDµF µ νϕ− 1 2γ µν_Dλ_F µνϕ. (2.35)

In App. A we continue in this fashion, deriving all the commutation relations as seen in Eq. (2.31), where we also provide a brief overview of the gamma matrices. Using the relations obtained above and those in App.A, we can solve for a, b and c in Eq. (2.30).

Beginning with simplest case D=4, Eq. (2.31) reduces to 8c∇µFµλϕ +γµ  Gλ µ−12a 2_Λgλ µ+8a(b+2c) √ ΛFλ µ −4(b 2_+4c2_)F µνFνλFµνFνλ−2b2gλµFρσFρσ  ϕ +γµν(b+2c)  2gλ µ∇F α ν − ∇ λ_Fµν ϕ +γµρσ  −iegλ µFρσ−4ab √ Λgλ µFρσ+4(b2−4c2)Fρσfµλ  ϕ=0. (2.36) In order to ensure that the γµν _{term is zero we set b+2c = 0. This simplifies the} equation as 8c∇_µFµλ ϕ+γµ  Gλ µ−12a 2_Λgλ µ−32c 2_F µνFνλ−8c2gλµFρσFρσ]  ϕ +γµρσ  −iegλ µFρσ+8ac √ Λgλ µFρσ  ϕ=0. (2.37)

This implies that∇_νFµν _{=0, in which case} Gλ µ−12a 2_Λgλ µ−32c 2_FµνFνλ_−8c2_gλ µFρσF ρσ_{=0, (2.38)} and that −iegλ µFρσ+8ac √ Λgλ µFρσ =0. (2.39)

Eq. (2.38) contains the Einstein tensor, and this implies that we need to take 12a2 = −1

12 and c2= −641, implying that b2 = −161. Next for 5 dimensions Eq. (2.31) reduces to

12c∇_µFµλ ϕ+γµ h Gλ µ−24a 2_Λgλ µ+12a(b+4c) √ ΛFλ µ −4 b 2_+8c2
_F µνF νλ −2 b2−4c2
gλ µFρσF ρσi ϕ+γµν h (b+4c)2gλ µ∇αFνα− ∇ λ_Fµνi ϕ +γµρσ h −iegλ µFρσ−8a(b+c) √ Λgλ µFρσ+4(b+4c) (b−c)FρσF λ µ i ϕ +γλµνρσ−2 b2−4c2
 FµνFρσϕ=0. (2.40) Taking b+4c=0 we simplify the above to

12c∇µFµλ ϕ+γµ h Gλ µ−24a 2_Λgλ µ−96c 2_FµνFνλ_−24c2_gλ µFρσF ρσi ϕ +γµρσ h −iegλ µFρσ+24ac √ Λgλ µFρσ i ϕ +γλµνρσ−24c2 FµνFρσϕ=0. (2.41)

2.4. Eigenvalues on the N-Sphere 19

From the γµ_{term we get the Einstein equation} Gλ µ−24a 2_Λgλ µ−96c 2_F µνFνλ−24c2gλµFρσF ρσ _{=0, (2.42)} and get a= i 2√6 and c= i 8√3, implying b= − i 2√3.

For the general case we assume b= −2(D−3)c, then Eq. (2.31) can be rewritten as 4c(D−2)∇µF µλ ϕ +γµ  Gλ µ−2a 2_{Λ(D−1)(D−2)g}λ µ+16(D−2)(D−3)c 2  FµνFλν₋1 4g λ µFρσF ρσ  ϕ +γµρσ h −iegλ µFρσ+4(D−2)(D−3)ac √ Λgλ µFρσ i ϕ +γλµνρσ(−2c2)(D−1)(D−2)FµνFρσϕ=0. (2.43) The first term gives the electromagnetic field equation of motion

∇µFµλ_{=0. (2.44)}

From the γµ_{term we get} Gλ µ−2a 2_{Λ(D−1)(D−2)g}λ µ+16(D−2)(D−3)c 2  FµνFλν₋1 4g λ µFρσF ρσ  =0 (2.45) By expanding the Einstein tensor Gλ

µ, and matching terms in the above equations we find that

2a2Λ(D−1)(D−2) = −1, 16(D−2)(D−3)c2= −1

2.

(2.46)

This implies that a = √ i

2(D−1)(D−2) and c = i 4√2(D−2)(D−3). So the supercovariant derivative is determined to be Dµ =∇µ−ieAµ+ i√Λ p2(D−1)(D−2)γµ− i 2 s D−3 2(D−2)γνF ν µ + i 4p2(D−2)(D−3)γµρσF ρσ (2.47)

This is the most general form of the super covariant derivative that we will use to determine our effective potential for the cases we will investigate in this thesis.

2.4

Eigenvalues on the N-Sphere

For the QNMs investigation we restrict ourselves to studying spherically symmet-rical space times. As such we will need to determine the eigenvalues and eigen-functions of our spinors and spinor-vectors on the N-sphere, SN. In this regard we use results obtained by Camporesi and Higuchi who in Ref. [36] have shown how to obtain these eigenfunctions and values for both the SN and for hyperbolic sur-faces. This approach to using the spherical symmetric space time has the advantage that going to higher dimensions is made much simpler when compared to using the

Newman-Penrose formalism as has been used in Ref. [75–77].

2.4.1 Eigenmodes of spinors on SN

The line element of the SN _{is given as [36]} dΩ2

N =sin2θNd eΩ2N−1+dθ2N, (2.48) where d eΩN−1is the metric of the SN−1, all tilde terms will represent quantities from the SN−1_{. The Dirac covariant derivative on this space is given as}

γµ∇µψλ =γ

µ

∂µψλ+ωµψλ
, (2.49) where ωµis the spin connection, and ψλis a spinor. The spin connection is defined as ωµ = 1 2ωµabσ ab_{, (2.50)} withΣab= 1₄

γa, γb and ωµabis defined as

ωµab=e

α

a ∂µeαb−Γ ρ

µαeρb
. (2.51)

In the above equation eα

a is an n-bein andΓ ρ

µαare the Christoffel symbols on a sphere with the non-zero components given as

ΓθN θiθj = −sin θNcos θNegθiθj ; Γ θj θiθN =cot θNge θj θi ; Γ θk θiθj =eΓ θk θiθj. (2.52) Before moving on we will better define what an n-bein is. The elements in an n-bein allow us to more easily convert between a curved space and an orthonormal space. They follow the following set of relations

gµν =e a µe b νδ ab_{, e}a µe µ b =δab, e a νe µ a = δµν. (2.53)

Note that Greek letters represent our curved space indices, and the Latin letters rep-resent our orthonormal indices. For the SN the n-beins are

eθN N =1, e θi i = 1 sin θie eθi i . (2.54)

By using Eqs. (2.52) and (2.53) it can be easily shown that γθi ₌ 1 sin θNee

θi

i γi and

γθN _{= γ}N_{, where the orthonormal gamma matrices respect the ordinary Clifford} algebra. If we write the spinors as

ψλ = ψ_λ(1) ψ_λ(2) ! =  Aλ(θN)ψe_λ −iB_λ(θN)ψeλ  , (2.55)

where if we let the spinors on the SN−1have the following eigenvalues

e λµ∇e_µψe_λ =0; eλ= ±  l+ N−1 2  , (2.56)

2.5. Numerical methods 21

then using the Jacobi polynomial we can relate Aλto Bλand show that the eigenval-ues of our spinors on the SN can be written as

iλ= ±i  n+ N 2  , n=0, 1, 2, 3, .... (2.57) 2.4.2 Eigenmodes of spinor-vectors on SN

We construct spinors on SN as ψµ= (ψθ1, ψθ2, ψθ3, ..., ψθ,N). Then starting from the S 2 we can construct two orthogonal spinors which can be written as the linear combini-ation of the basis γµψλand∇µψλ. These two spinors are called the “non-Transverse and Traceless modes” (non-TT modes). In the S2these spinors form the complete set of eigenmodes. In higher dimensions, however we must construct more spinors in order to form the complete set of eigenmodes. For the SN we construct the spinors using linear combinations of those that form the set of spinors on the SN−1.

For instance, on the S3we use the two non-TT modes of the S2and then using a linear combination of these two spinors we construct a third spinor. This new spinor, how-ever, does not satisfy the non-TT mode condition and is a “Transverse and Traceless eigenmode”, specifically this is the TT mode I.

For spheres with N>3 we need to introduce another type of TT mode, which we call the TT mode II. In general, to describe a sphere of N>3 we need one non-TT mode, one TT mode I and N−3 TT mode II’s.

2.5

Numerical methods

Once we have the effective potential, V(r), from an equation of the form d

dr2Ψ(r) +V(r)Ψ(r) =ω

2_{. (2.58)}

We will use known methods for solving the equations of this form, such as the WKB method. Note that this is the same form of the equation that we would have for the quantum wave equation.

2.5.1 WKB Approximation

Note that the WKB method can be used to determine the approximate solution to any second order differential equations of the Schrödinger form, and so is typically employed when solving the Schrödinger wave equation [78]. In Ref. [79] we have given an example of how we use the WKB method to solve second order differential equations, which we present in the example below. To start with we have a general second order equation

e2d

2_y

dx2 =Q(x)y , (2.59)

where e1 and Q(x)is some function dependent on x. We can solve this equation by assuming the following solution

Taking the first and second derivatives of this function we see that y(x, e) = A(x, e)eiu(x)/e,

y0(x, e) = A0(x, e)eiu(x)/e+A(x, e) iu 0_(x) e  eiu(x)/e, y00(x, e) = A00(x, e)eiu(x)/e+A0(x, e) iu 0_(x) e  eiu(x)/e+A0(x, e) iu 0_(x) e  eiu(x)/e −A(x, e) u 0_(x) e 2 eiu(x)/e = A00+2A0iu 0_(x) e +A( iu00(x) e −  u0(x) e 2! eiu(x)/e. (2.61) Plugging Eq. (2.61) into Eq. (2.59) we get

e2 A00+2A0iu 0_(x) e +A( iu00(x) e −  u0(x) e 2! −Q(x)A(x, e) =0 . (2.62)

Then since e is very small, we can perform a series expansion of A around the point

e=0;

A(x, e) = A0(x) +eA₁(x) +e2A2(x) +... A0(x, e) = A0₀(x) +eA₁0(x) +e2A0₂(x) +...

A00(x, e) = A00₀(x) +eA₁00(x) +e2A00₂(x) +...

(2.63)

so that Eq. (2.62) becomes 0=e2  A00₀(x) +eA00₁(x) +e2A00₂(x) +...
+2 A₀0(x) +eA0₁(x) +e2A0₂(x) +...
iu 0_(x) e  +e2  A0(x) +eA1(x) +e2A2(x) +...
(iu 00_(x) e −  u0_(x) e 2 −Q(x)A0(x) +eA1(x) +e2A2(x) +...  . (2.64) Grouping in terms of powers of e we can solve for u(x).

This method of obtaining QNMs for black holes has been employed for many years, where it had previously been used up to 3rd order in the approximation [80]. How-ever, Konoplya has extended the method up to 6th order [37]. This extension to the 6th order has resulted in more stable and accurate solutions for the WKB when applied to black holes, at the expense of computational time. This increase in com-putational time is due to the increased complexity in the functions used to solve for the QNMs, see Ref. [37] for the full form of these functions. As such we wish to use a method which could give us the same level of accuracy as the WKB to 6th order without the large computation time. In this respect we have chosen the improved AIM [39–41]. This method is hoped to produce similar results using less computa-tional power, as compared to the WKB 6th order method. In Ch. 3 we show the comparison of results between the two methods.

2.5. Numerical methods 23

2.5.2 Improved AIM

We begin by showing how the AIM can be used to solve for QNMs, as shown in Ref. [81]. We then show how this method was improved by Cho et al. in Ref. [42]. Just as for the WKB, the AIM is useful in solving second order differential equations of the form

y00 =λ0(x)y0+s0(x)y , (2.65) where λ0 and s0 are elements of C∞(a, b)[81], and y is some function of x, with 0 denoting derivatives in terms of x. Taking n derivatives of this equation we obtain the following result

y(n+2) =λny0+sny , (2.66) where λn= λ0_n−1+sn−1+λ0λn−1and sn=s0_n−1+s0λn−1. We then divide the n+2 iteration with the n+1 iteration to get the following ratio

y(n+2) y(n+1) = d dxln(y (n+1)_{) =} λn(y 0₊ sn λny) λn−1(y0+ _λsn_n−₋1₁y) . (2.67)

The objective of this method is to use values of n such that

α= sn λn ∼ = sn−1 λn−1 . (2.68)

So that we can rewrite Eq. (2.67) as d dxln(y

(n+1)_{) =} λn

λn−1

. (2.69)

This has the solution

y(n+1)=C1λn−1exp " x Z α+λ0dt # , (2.70)

where C1is some integration constant. Plugging this into Eq. (2.66) we obtain the first order equation

y0+αy=C1exp h x Z α+λ0dt i , (2.71)

which yields the solution y(x) =exp  − Z x αdt   C2+C1 Z x exp Z t (λ0(τ) +2α(τ))dτ  dt  , (2.72) where C2is another integration constant. In our case this gives the solution of the wavefunction. By using the boundary conditions we are able to obtain the allowed QNMs.

In the case of QNMs we need to know α to a very high precision, and so in order to satisfy the condition given in Eq. (2.68) we would need to use large values of n. Taking this iterative approach would result in large computation time. Instead we use the approach given by Cho et al. in Ref. [42], where they have used a Taylor

expansion to determine the values of λnand snas follows: λn(ξ) = ∞

∑

∑

where ξ is value around which the AIM is performed and ci_nand di_nare our Taylor coefficients. Plugging these into the previous expressions for λnand snwe have the following recursion relation:

ci_n= (i+1)c_ni+₋1₁+di_n−1+ i

∑

∑

We calculate the values of our QNMs by solving d0_nc0_n−1−d0_n−1c0_n = 0, where n rep-resents the number of iterations we wish to perform. This approach of using the Taylor expansion considerably speeds up the computation time. As we do not need to keep the full derivative of the function each time, instead keeping only the coeffi-cients of the Taylor series. Where as stated previously this is advantageous since in order to accurately compute the QNMs we need to consider a very high number of iterations, in fact we use 200 iterations in the following work.

2.6

Absorption probabilities of a black hole

In 1974 Steven Hawking proposed that by applying quantum effects to a black hole one could show that the black hole would evaporate over time [82]. His argument was that due to the quantum fluctuations on the surface of the black hole we could expect the black hole to emit particles in the same way that a hot object radiates heat. This radiation of particles from the surface of the black hole would lead to the black hole losing mass over time, and hence would have a finite lifetime, as it would eventually evaporate away all of its mass. Furthermore Hawking showed that a black hole would have a lifetime of the order 1071(M/M)−3s, where M

is the solar mass and M is the mass of the black hole. So a black hole with a mass of the sun would exist for a very long time indeed. However, for very small black holes the life time could be much shorter. So we may be able to put astrophysical constraints on the evaporation of these types of black holes. Unruh pointed out that an important parameter for determining the likely hood of the quantum evaporation of these small black hole is to know what the absorption probability associated to the black hole is for the various possible fields that it could emit [83]. In our case we wish to calculate the absorption probabilities as this allows us to determine the likelihood of a field with a particular quantum state being formed at the surface of the black hole, and this would give us an indication of the likely QNMs we would detect from the black hole. In Unruh’s paper he provides a methodology for calculating the absorption probabilities associated with scalar fields near Schwarzschild black holes [83]. This method is only valid in the low energy regime. Instead, in this thesis, we use a method developed by Iyer and Will to obtain absorption probabilities for more general cases [80].

2.6. Absorption probabilities of a black hole 25

For the Iyer and Will approach to finding the absorption probabilities we start from Eq. (2.58) and perform a change of variables, namely x = ωr, this is purely for

convenience. Such that we have the following second order differential equation  d2 dx2 ∗ +Q  Φ=0, (2.75)

where Q(x) = −ω2−V(x). The absorption probability is then determined to be [84]

 A_j(ω)   2 =exp  −2 x2 Z x1 dx q Q(x)  , (2.76)

where x1and x2are turning points in Q, namely Q(x1,2) =0 for the given energy ω. This method, however, is only valid for ω2 _{V. In the case of ω}2_{∼V the} exponen-tial will go to infinity and so the method is not valid. However, Iyer and Will have shown that by taking the third order WKB approximation we can obtain solutions that are valid for all energy regimes [80]. In this case the absorption probabilities are given as  A_j(ω)   2 = 1 1+e2S(ω), (2.77) with S(ω) =πk1/2 1 2z 2 0+  15 64b 2 3− 3 16b4  z4₀  +πk1/2 1155 2048b 4 3− 315 256b 2 3b4+ 35 128b 2 4+ 35 64b3b5− 5 32b6  z6₀ +πk−1/2 3 16b4− 7 64b 2 3  −πk−1/2 1365 2048b 4 3− 525 256b 2 3b4+ 85 128b 2 4+ 95 64b3b5− 25 32b6  z2₀. (2.78)

Note that z2₀, b0 and k are determined by the Taylor expansion of Q around its peak x0, and they are determined as follows [80]

Q=Q0+ 1 2Q (2) 0 z2+

∑

∑

Note that in the above equations the subscript zero denotes maximal values obtained when plugging in x0The notations and techniques developed in this chapter shall now be applied to obtain the QNMs for Reissner-Nordström type black holes in Ch.3, and the AdS space time in Ch.4.