Geometrical and Perturbative study of Tidally Deformed Schwarzschild Spacetime

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Master Thesis

Geometrical and Perturbative study of Tidally Deformed Schwarzschild

Spacetime

Division of Theoretical Physics Department of Physics and Astronomy

Uppsala University

Author:

Zeyd Sam

Supervisor:

Asst.Prof. Giuseppe Dibitetto Subject Reader:

Prof. Ulf Danielsson

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Abstract

We undertake a study of the tidal deformation of a Schwarzschild black hole. We formulate the geometry in terms of intrinsic and extrinsic quantities on the event horizon, which is described by a null hypersurface embedded in spacetime, and we do so by formulating the Gauss-Codazzi theory of null hypersurfaces, extending standard results for the spacelike and timelike cases. This formalism is then applied to a solution of the Einstein field equations in vaccum, in order to describe tidal distortions to the black hole horizon due to a small orbiting body. The techniques that we employ are valid for rapid orbits in the strong field of the black hole. This analysis relies on a perturbative approach for bodies with a very large mass ratio. We conduct the study in the frequency domain of black hole perturbation theory, for a small body orbiting in a circular orbit. The results are visualized by embedding the distorted horizon in Euclidean space, showing how the horizon is deformed from an embedded sphere to an ellipsoid when the small body is close to the black hole, and when the orbital separation is increased, showing in both cases the angular offset between the horizon bulge and the body that induces it.

For both orbital separations considered, we provide snapshots of the tidal distortion as the small body orbits around the black hole.

Popul¨arvetenskaplig sammanfattning

Vi undersöker tidvattendeformationen av ett icke-roterande, s˚a kallade Schwarzschild, svarta h˚al. Vi formulerar geometrin i form av inre och yttre geometriska kvantitet p˚a händelsehorisonten, som beskrivs av en hyperyta inbäddad i rumtiden, och vi gör det genom att formulera Gauss- Codazzi teorin om nollhyperytor som sträcker det standard resultat för rumslika och tidslika fallen. Denna formalism tillämpas sedan p˚a en lösning av Einstein-fältekvationerna i vaccum, för att beskriva tidvat- tensförvrängningar till svarth˚alshorisonten p˚a grund av en liten kretsande objekt. De tekniker som vi använder är giltiga för sm˚a objekt som rör sig snabbt i omloppsbanor, i det svarta h˚alets starka gravitationsfält. Denna analys bygger p˚a en perturbativ strategi för tv˚a kroppar med ett mycket stort massförh˚allande - det vill säga, det sm˚a objektet är mycket mindre

¨

an den andra. Vi genomför studien i frekvensdomänen för störningsteori för svart h˚al, för en liten kropp som g˚ar i en cirkulär bana. Resultaten visualiseras genom att bädda in den förvrängda horisonten i det euklidiska rymden, vilket visar hur horisonten deformeras fr˚an en inbäddad sfär till en ellipsoid när den lilla kroppen är nära det svarta h˚alet, och när orbita- lavskiljningen ökar, vilket i b˚ada fallen visar vinkelförskjutning mellan ho- risontbukten och kroppen som producerar den. För b˚ada orbitalskillnader som beaktas tillhandah˚aller vi ögonblicksbilder av tidvattenförvrängnin- gen när den mindre kroppen kretsar runt det svarta h˚alet.

Uploaded Oct 20 for access, Uppsala universitet.

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1 Introduction

1.1 Context and motivation

The astounding first detection of gravitational waves by LIGO (Laser Interfer- ometer Gravitational Wave Observatory) in 2015 [1] not only validated one of the famed predictions of Einstein’s theory of gravitation, the celebrated theory of General Relativity, but it also ushered in a new era in how we observe our Uni- verse - gravitational wave astronomy. No longer are we bound just by ”sight”, but are we are now attuned to the ”sound” of spectacular events unfolding in the cosmos. This first detection came from the orbiting inspiral and merger of two black holes, of 36 and 29 solar masses, and was awarded the 2017 Nobel prize in physics. Unlike orbits in the Newtonian theory of gravity, accelerating bodies in GR produce gravitational waves, which manifest as ripples in the fabric of spacetime, carrying energy and angular momentum and traveling radially away from the source at the speed of light. This loss of orbital energy and angular momentum due to gravitational wave emission causes the bodies to spiral into one another and merge. The two-body problem, as it is known, is thus far more complicated in GR than in Newtonian theory, and the key to understanding the rich information carried by gravitational waves from compact binaries lies in accurately modeling the two-body problem for a range of masses, spins, and orbital configurations. A lot of work is being done on this front, for detection with current and future gravitational wave detectors.

1.2 Two-body problem in GR

As mentioned, a detailed understanding of the dynamics of inspiraling compact objects (black holes and neutron stars) is required in order to understand the information carried by gravitational waves. So how do we go about this?

There exists a collection of exact solutions to the Einstein field equations [2].

However, apart from a handful of solutions, such as the trivial Minkowski spacetime, Schwarzschild Reissner-Nordstr¨om and Kerr black hole solutions, and the Friedmann-Robertson-Walker spacetime encountered in cosmology along with de Sitter and anti-de Sitter spacetimes [3], most of the solutions are algebraic in value and offer little to no physical interpretation. From the physical black hole solutions that we have listed, none can do the job of accurately describing and evolving the dynamics of a pair of coalescing compact objects and the particular gravitational waveform they produce, for a set of initial conditions.

So how does one go about solving the two-body problem in GR? Given a set of initial conditions, we wish to evolve the system forward in time. To do this, we must solve the Einstein field equations (EFE) for this complex system.

This is easier said than done. The EFE are a complicated set of ten coupled and highly nonlinear partial differential equations given by

Gαβ= 8πG

c⁴ Tαβ (1.1)

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where Gαβ is the Einstein tensor, Tαβ is the energy-momentum tensor of the system, G is the universal gravitational constant and c the velocity of light in vacuum. Often one works in natural units in which G = c = 1. Can we get an exact solution to this problem, or do we have to approximate it? We follow [4]

to give an overview.

One method is to use numerical relativity. Using powerful supercomputers and sophisticated computational techniques, one can solve the EFE numerically and propagate the solution forward in time. We can regard numerical relativity as being an almost exact method of solution (if we ignore inevitable numerical errors arising from the computations). This technique is well suited to compact bodies of comparable mass, such as the first event detected by LIGO. In their last moments before merger, the two bodies are orbiting very closely about one another, and at a significant fraction of the speed of light. The spacetime in their vicinity is highly nonlinear, dynamical, warped and nontrivial. Numerical relativity runs into problems however, when the separation of the bodies is too large. In theory, this configuration should be tangible, but the issue is due to a matter of practicality. When the separation is increased, evolving the system over many orbits until it approaches the merging phase takes too long. Re- solving over a long orbiting timescale is therefore too computationally taxing.

Numerical relativity also runs into problems when the mass ratio between the two bodies is large. The computational techniques are suited to comparable mass bodies, but when the masses differ by orders or magnitude, we must re- solve over the two disparate length scales, the large and the small, in the system.

Again, this issue is a matter of practice, as this is also computationally taxing.

As the mass ratio is increased, this becomes computationally prohibitive.

So what can be done when the separation of the bodies is large? In this configuration, the bodies are far apart and the orbital velocity is low. The orbit is yet to reach the rapid, highly relativistic merging phase. We say that this regime is in the weak field, and as such, we can make approximations. The weakness of the gravitational field allows us to decompose the metric into the flat Minkowski spacetime ηαβ with a small perturbation pαβon top of it,

gαβ= ηαβ+ pαβ (1.2)

where |p_αβ| << 1. This is the regime of validity for post-Minkowskian and post- Newtonian theory. We are essentially asking, how much are we deviating from flat, Minkowski spacetime, and/or from slow, Newtonian orbits? In practice, coalescing binary black holes can be described by an expansion scheme that utilizes both regimes; we expand in powers of G (emphasizing the weak field) as we expand in powers of c⁻² (emphasizing slow orbital velocities). This is because in bound gravitating systems, the gravitational potential energy must be equal to the kinetic energy of the system,

G ∼ (v/c)²≈ 1/c² (1.3)

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which necessitates expanding in both parameters, in a mutually consistent man- ner. These perturbative approximations, although limited in the range of orbital separation and velocity, are not limited in the mass ratio.

Finally, we arrive at the remaining regime of the two-body problem. LIGO is not sensitive to the frequency of gravitational waves produced from such binaries, instead we turn to the future space based detector eLISA. The mass ratio is very large, and the separation between the bodies may be small, so we are in the strong field. Given that one body is far more massive than the other, we take the gravitational field of the large body to be the known metric of a Schwarzschild or Kerr black hole solution to the EFE, and we assume the much smaller body to be a perturbation on top of g_αβ⁰ , the known background spacetime, such that

gαβ= g⁰_αβ+ pαβ (1.4)

The small expansion parameter chosen is the mass ratio, or the small mass µ.

This is the domain of black hole perturbation theory. This regime is suited to perturbations in the strong field of the central body, but also not limited in the orbital separation. The techniques employed can be extended to large distances. Again, as a matter of practicality, we may not want to go to very large separations, as the techniques employed for dealing with the strong field specifically, can be substituted for post-Newtonian methods which are more appropriate in that regime. Although black hole perturbation theory can be pushed far towards large separations (within reason for computational practicality), it cannot be pushed towards a comparable mass ratio. The starting assumption of a small body acting as a small perturbation on a highly curved background solution would then no longer be accurate.

Different techniques are therefore employed in their respective domains of validity, but the boundaries between them are not sharply defined. Indeed, the different regimes can be reconciled with each other at the boundaries, and work is actively being done to push one regime closer to the other, and to use results from one domain to inform progress on the other. It should be noted that a further regime, known as effective-one-body (EOB) sets out to encompass arbitrary separations and mass ratios, by combining results from the different domains. This combines analytical and numerical results, and fitting with phenomenological models to produce waveform templates for the inspiral, merger, and finally ringdown phase of coalescing binary compact bodies [4].

1.3 Tidal distortion of Newtonian fluid body

Black holes, singularities in the fabric of spacetime geometry, are remarkable artifacts of Einstein’s celebrated theory of General Relativity. Although black holes do not exist in Newtonian theory, can we draw any parallels with our Newtonian intuition, in our study of tidal deformability of a black hole in GR?

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In Newtonian graivtational theory, we must consider fluid objects. A spherical fluid object will be deformed into an ellipsoid by the tidal force exerted from an external, perturbing companion object [5]. If the central fluid body is non-rotating, and the perturbing body is stationary, the induced bulge (location of greatest distortion) points towards the perturbing body that caused it. This is intuitively sensible.

If however, the perturbing body is an orbiting companion, and we assume the central fluid object to have some viscosity ν, the tidal bulge ”lags” the orbiting body by a time interval

τ ∝ Rν

M (1.5)

where R and M are the fluid object’s average radius and mass, respectively [6].

We note that if the fluid body is instead rotating, the bulge will lag behind the orbiting companion, if the spin frequency is lower than the orbital frequency.

If the spin frequency is higher than the orbital frequency, the bulge will instead

”lead” the orbiting body.

For the purpose of comparison, we will restrict our attention to the case in which the fluid body is non-rotating, and hence the distortion lags behind the orbit. How will the event horizon of a non-rotating, Schwarzschild black hole respond to an orbiting companion?

1.4 Tidal distortion of compact objects

The effect of tides on compact objects is an interesting topic of astrophysical study. This is of particular interest in binary systems, in which the bodies may distort one another as the orbit about each other. Therefore, the goal of detect- ing the tidal imprint on gravitational waves from these extreme gravity objects has been pushing research to uncover the precise dynamics of these systems. In the case that one or both of the members in the binary is a neutron star, the prospect of learning about this objects equation of state is particularly enticing (see e.g. [7]). Among other research, some investigations have been concerned with the tidal distortion of black holes, such as the case of the non-rotating (Schwarzschild) (e.g. [8], [9]) or rotating (Kerr) (e.g. [5], [10], [11]) black holes of general relativity. A topic that is of critical importance in such studies is how one chooses to quantify the distortion, given the choice of coordinates describing the geometry of the system. In the case of non-rotating black holes, a gauge in- dependent measurse, the so-called Love numbers, are vanishing quantities [12].

Therefore, a way of quantifying the distrotion, that does not vanish, or rely on the choice of gauge, has been desired. A resolution to this problem has been presented in [6].

In this project, we investigate how a tidally deformed black hole can be described by the intrinsic and extrinsic geometry of its event horizon by closely following the approach in [6], in particular. In our investigation of the geometry of a deformed horizon, we undertake a mathematical study of the Gauss-Codazzi

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theory of a two-dimensional null hypersurface embedded in the external, four- dimensional spacetime manifold. This is an extension of the well-known description of spacelike or timelike hypersurfaces (e.g [13]). Again, we conduct our study with minimal assumptions as we develop the necessary quantities in differential geometry.

We focus on the tidal distortion of the Schwarzschild spacetime solution to the Einstein field equations of general relativity. The assumption we make is that the tidal distortion is small enough to be described to first (linear) order in perturbation theory. Aside from this requirement, the formalism we study presents no greater restrictions. Following this, we investigate the tidal deformation in the regime of black hole perturbation theory. We will discuss the use of master-functions in a gauge-invariant framework [14] that describe the perturbation in the even and odd-parity sectors, in which the metric perturbation is decomposed into even and odd-parity spherical harmonics, respectively.

By closely following the approach in [5], we will investigate how to express a measure of tidal deformability in a way that is amenable to visualization, so that we may plot our results in Euclidean space, and clearly discern the tidal effect of a small orbiting body on the Schwarzschild black hole.

Central questions that we aim to answer are therefore: What can we learn from a thorough study of the differential geometry describing the event horizon?

Can we derive meaningful measures of tidal deformability? How does the event horizon respond to a small perturbation? How does it compare with our New- tonian intuition? How can we use black hole perturbation theory to understand the tidal distortion? How can we visualize the tidal influence of an orbiting body, on the Schwarzschild black hole? How does the tidal distortion differ for orbits at different separations?

1.5 Outline of project

This project can be seen as divided into these main categories: geometry, perturbation theory, and numerical results. Here, we provide a brief outline of the project.

In Sec. 1, we provided an introduction to tidally deformed black holes, by noting that gravitational wave astronomy gives context and motivation for their study, and that we may study them within the two-body problem in general relativity. We noted some of the aspects of tidal distortion of compact objects in the literature, and provided and overview of what our physical intuition from Newtonian theory provides.

In Sec. 2, having set the stage for our investigation, we begin our study of the geometry of null hypersurfaces embedded in spacetime. We begin by carefully detailing the formalism and then use it to derive the Gauss-Weingarten equations, which also lead us to notions of intrinsic and extrinsic quantities, along

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with notions of curvature. These relations are used to derive the Gauss-Codazzi equations, which are then used to obtain components of the Ricci tensor, which ultimately lead us to several measures of tidal deformability.

In Sec. 3, we introduce the Schwarzschild metric and a small perturbation thereof, which sets the focus for later sections. We set up coordinates that en- able the study of a distorted horizon, leading us to horizon equations for the perturbations. We also choose to focus on particular extrinsic and intrinsic geometrical quantities identified earlier, and study them in the context of linear perturbation theory.

In Sec. 4, we introduce techniques from black hole perturbation theory. We provide an overview of master-functions that are well-known in literature and can be related to our geometrical quantities on the horizon. From a near-horizon analysis, we obtain such a relation for the Ricci scalar curvature. We also provide an overview of the Teukolsky equation for Kerr black holes, which we can apply to the scalar curvature in our Schwarzschild study.

In Sec. 5, we study the tidal influence on the Schwarzschild black hole, by a small body in a circular, equatorial orbit. We discuss how the orbital energy has been reconciled with tidal deformability, and provide details of how to embed the horizon in Euclidean space. We also provide a calculation of the dephasing angle between the tidal bulge and the orbiting body, for two different orbital separations.

In Sec. 6, we describe the Teukolsky code which produces the numerical data which we use. We provide plots of the mode convergence of the perturbed scalar curvature, along with snapshots of the tidal bulge at various angular dis- placements of the orbiting body, for both orbital separations that we consider, allowing us to compare the results.

In Sec. 7, we provide a summary of the project, and outline numerous ways to extend this work, and numerous new avenues for investigation. We then provide some concluding remarks.

In App. A, we investigate the effect of reparameterizing the horizon’s null generators. In App. B, we discuss the scalar, vector, and tensor spherical harmonics. In App. C, we provide and a detailed calculation of the induced metric.

In App. D, we provide a detailed calculation of an extrinsic horizon quantity.

Throughout this work, we use units in which G = c = 1.

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2 Hypersurface geometry

Given a spacetime manifold, a hypersurface is a submanifold. A hypersurface is characterized as being either spacelike, timelike or null, with the latter subject to subtleties and requiring special care in its analysis. This is precisely the case that we are interested in when studying the event horizon of a black hole.

It is therefore worth taking some time to develop the appropriate formalism [15].

We may for instance consider a codimension 1 manifold, that is, a 3-dimensional hypersurface Σ embedded in 4-dimensional spacetime manifold M. This would be the case of a spacelike or timelike hypersurface, which may be described in parametric form as

x^α= x^α(y^a) (2.1)

for spacetime coordinates x^α and coordinates on the hypersurface y^a, and for lower-case Latin indices a = 1, 2, 3, or by a function Φ placing a restriction on the spacetime coordinates for some constant C,

Φ(x^α) = C. (2.2)

Now, to study the geometry of a hypersurface, we need the notion of a unit normal vector to the surface. This (normalized) normal, nα is defined to be

n_α= ± ∂_αΦ

p|g^µν∂_µΦ∂_νΦ| (2.3)

where ± denotes a timelike (+) or spacelike (−) hypersurface. This notion is not applicable for the null case, as the denominator is zero. Instead, we define

k_α= −∂_αΦ (2.4)

to be the normal. Given that the event horizon of a black hole is described by a null hypersurface, this is the case we will pursue further.

2.1 Constructing a null hypersurface

The first steps to be taken, with the goal of formulating the geometry of the event horizon, is to discuss the congruence of null geodesics that generate the null hypersurface Σ_n. A congruence of geodesics is essentially a family of curves that do not intersect. We closely follow the description and notation as presented in [6] to set the formalism to follow.

In the null case, we introduce the parametric equations x^α as

x^α= x^α(λ, αÂ), αÂ= (α, β) (2.5) where λ is a scalar parameter along each generator, and αÂ are for A = 2, 3 labels on the generators that remain constant on each generator, which will

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be more suitable than the form Eq. (2.1). With this, the null vector field introduced previously can be written as

k^α= ∂x^α

∂λ

α^A

(2.6) is both normal and lies tangent to the congruence. The spacelike displacement vectors are similarly defined as

e^α_A= ∂x^α

∂α^A

λ

(2.7) These basis vectors satisfy the orthogonality conditions

kαk^α= 0 , kαe^α_A= 0 (2.8)

Since kαis defined to be the normal to the Σn, the first condition is a statement that it is orthogonal to itself, hence also tangent to Σn. This shows the subtleties involved in the null case. Now, as we are dealing with a null hypersurface, the intrinsic geometry (i.e geometry on the surface) is necessarily described by a degenerate two-dimensional metric, which gives the relation between two spacelike displacement vectors as

γ_AB = g_αβe^α_Ae^β_B (2.9)

where we have contracted over spacetime indices and γ_AB is taken to be this degenerate 2-metric on Σ_n and g_αβ is the spacetime metric. We let ∇_A repre- sent the covariant derivative compatible with the hypersurface metric γAB with connection coefficient given by Γ^C_AB. From the above definitions, the vectors k^α and e^α_Asatisfy the Lie-transport equations

Le_Ak^α= Lke^α_A= 0 (2.10a) LeBe^α_A= L_e_Ae^α_B= 0 (2.10b) which clearly imply that

e^β_A∇βk^α= k^β∇βe^α_A (2.11a) e^β_B∇_βe^α_A= e^β_A∇_βe^α_B (2.11b) where ∇_α, the covariant derivative in spacetime, is taken with respect to a connection that is metric compatible with gαβ.

In order to have a full four-dimensional spacetime basis, we complete this set of basis vectors by introducing an auxiliary null vector N^αsuch that

NαN^α= 0 , Nαk^α= −1 , Nαe^α_A= 0 (2.12) where the second condition is a typical normalization choice, and the third follows for the same reason as for k^α, namely that Nαlies tangent to e^α_A.

Now that we have constructed a complete vector basis on Σn, a natural question to ask now is how does one differentiate a vector A^αon the hypersurface.

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2.2 Derivation of Gauss-Weingarten equations

In order to arrive at the Gauss-Codazzi equations that we seek to study the geometry of the black hole’s event horizon, we first require the Gauss-Weingarten equations. These serve to express the rate of change of the unit normal vector to a given surface using the derivatives of the position vector on the surface.

Given that the null vector Nαis auxiliary, this leaves us seeking such equations for k^β∇βk^α, e^β_A∇βk^αand e^β_B∇βe^α_A.

The first Gauss-Weingarten equation is easily obtained, as it is simply given by the general form of the geodesic equation [15], namely

k^β∇βk^α= κk^α (2.13)

for some scalar κ. An appropriate choice is one that preserves the notion that k^α is null on the hypersurface, so we seek no terms proportional to N^αon the RHS. This leads us to find

κ = −N_αk^β∇_βk^α (2.14)

to be an appropriate choice. The remaining relations require more work to elucidate.

Consider first the case of spacelike or timelike hypersurfaces. The quantity defined as

DbAa := e^α_ae^β_b∇βAα (2.15) for a tangent vector Aa where lower-case Latin indices denote intrinsic coordinates, is the intrinsic covariant derivate. It describes the tangential components of the vector e^β_b∇βA^α on Σ. Given that we are working with Σn, we seek to find an analogue for null hypersurfaces.

To do this, we require the normal component of the vector defined above.

We start by re-writing the vector as

e^β_b∇_βA^α= g^α_µe^β_b∇_βA^µ (2.16) where we can now decompose the metric into its normal and tangential parts, using the metric on Σn, so the spacetime metric for the null case is [15]

g^αβ= −k^αN^β− N^αk^β+ γ^ABe^α_Ae^β_B (2.17) We proceed as follows

e^β_B∇βA^α= g^α_µe^β_B∇βA^µ

= (−k^αNµ− N^αkµ+ γ^AMe^α_AeM µ)e^β_B∇βA^µ

= −(N_µe^β_B∇βA^µ)k^α− (kµe^β_B∇βA^µ)N^α+ γ^AM(e_{M µ}e^β_B∇βA^µ)e^α_A (2.18)

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From the tangent vector fields in our basis, we obtain the remaining Gauss- Weingarten equations in the following, by setting A^α = k^α and A^α = e^α_A, respectively.

e^β_A∇_βk^α= ω_Ak^α+ B_A^Be^α_B (2.19a) e^β_B∇βe^α_A= BABN^α+ KABk^α+ Γ^C_ABe^α_C (2.19b) where, as we have noted previously, the Lie transport equations Eq. (2.11a) and Eq. (2.11b) hold, and we have defined

ωA= −Nαe^β_A∇βk^α (2.20a) BAB= e^α_Ae^β_B∇βkα (2.20b) K_AB= −N_αe^β_B∇βe^α_A (2.20c) Γ_CAB= e_Cαe^β_B∇_βe^α_A (2.20d) where BAB= BBA, KAB = KBAand ΓCAB = ΓCBAare symmetric in AB. To show that BAB is symmetric, for example, we proceed as follows

BAB= e^α_Ae^β_B∇βkα

= e^β_B∇β(kαe^α_A) − e^β_Bkα∇βe^α_A

= −kαe^β_A∇βe^α_B

= −e^β_A∇β(kαe^α_B) + e^β_Ae^α_B∇βkα

= e^α_Be^β_A∇_βk_α

= BBA

(2.21)

where the second line follows from the product rule for the covariant derivative, the third line follows from orthogonality and the Lie transport equation Eq. (2.11b), Eq. (2.8) and the remainder of the calculation proceeds by the same set of arguments. Symmetry of K_AB follows from a very similar set of arguments.

We can see from Eq. (2.20) that the geometric quantities γ_AB, B_AB and Γ^C_AB are related to the intrinsic geometry of Σn, whereas κ, ωA, and KAB in contrast, are related to the extrinsic geometry of Σn, i.e how this hypersurface is embedded in the external four-dimensional spacetime.

2.3 Intrinsic and extrinsic quantities

We give some further consideration to the geometric quantities that we have encountered. Now, we further note that

B_AB= e^α_Ae^β_B∇βk_α

= 1

2e^α_Ae^β_BLkg_αβ (2.22)

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represents the intrinsic curvature of the null hypersurface, and we have used a helpful result that relates the covariant derivative of k^αto the Lie derivative of the metric along k^α[3], [15]. Since k^αis null and tangent to the hypersurface, Lkgαβ is a tangential derivative.

We may then wonder what the quantity K_AB represents. We write KAB= 1

2e^α_Ae^β_BLNgαβ

= 1

2e^α_Ae^β_B(∇_βN_α+ ∇_αN_β)

= −N_αe^β_B∇_βe^α_A

(2.23)

and define this quantity as the transverse curvature of the null hypersurface, using similar techniques as in the aforementioned calculation, and note that it represents the transverse derivative of the metric.

To conclude this section, we note that these quantities, intrinsic or extrinsic, all refer to a specific parameterization choice (λ, α^A) of the horizon’s null generators. A demonstration of the effect of reparamterization is given in App.

A.

2.4 Derivation of Gauss-Codazzi equations

We mentioned that for spacelike or timelike hypersurfaces, the notion of an intrinsic derivative can be defined. From such a derivative, we may further define the Riemann tensor for the hypersurface as

R^c_dabA^d= DbDaA^c− DaDbA^c (2.24) in which this curvature tensor is entirely intrinsic. Returning our attention to the null case at hand, the situation is not as straightforward. We know that the Riemann tensor Rαβγδ describes the curvature of a 4D spacetime manifold.

Given that γAB acts as a metric on Σn, we now wish to consider how the 2D Riemann curvature tensor on Σn can be related to the 4D curvature tensor.

This will ultimately serve to bring us closer to understanding how the geometry of a black hole’s event horizon is described, and how it is then altered by an external influence.

To arrive at the Gauss-Codazzi equations, we follow Sec. 3.5 of [15], and begin by taking the covariant derivative along e^γ_Cof the Gauss-Weingarten equations. Consider the first such equation given by Eq. (2.13). We have

e^γ_C∇γ(k^β∇βk^α) = e^γ_C∇γ(κk^α) (2.25) and we consider each side of the equality separately. Consider first the LHS,

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which gives

e^γ_C∇γ(k^β∇βk^α) = e^γ_C∇γk^β(∇βk^α) + ∇βk^α(e^γ_C∇γk^β)

= e^γ_Ck^β∇γ∇βk^α+ ∇_βk^α(ω_Ck^β+ B_C^De^β_D)

= e^γ_Ck^β∇_γ∇_βk^α+ k^β∇_βk^αω_C+ B_C^De^β_D∇_βk^α

= e^γ_Ck^β∇γ∇βk^α+ k^β∇βk^αωC+ B_C^De^β_D∇βk^α

= e^γ_Ck^β∇γ∇βk^α+ κk^αωC+ B_C^D(ωDk^α+ B_D^Ee^α_E)

(2.26)

where we have used the product rule to obtain two terms, then used the Gauss- Weingarten equation Eq. (2.19a) to rewrite the second term. We then expanded this second term, and used the Gauss-Weingarten equations Eq. (2.13) and Eq.

(2.19a) to rewrite the resulting first and second terms, respectively.

We follow similar steps to expand out the RHS and write the appropriate combinations of terms as Gauss-Weingarten relations,

e^γ_C∇γ(κk^α) = k^αe^γ_C∇γκ + κe^γ_C∇γk^α

= k^αe^γ_C∇γκ + κ(ωCk^α+ B_C^De^α_D) (2.27) With these expansions at hand, we set LHS = RHS and solve for the term e^γ_Ck^β∇_γ∇_βk^α. The reason we are interested in this term will become apparent shortly. We find

e^γ_Ck^β∇γ∇βk^α= −κk^αωC− B_C^D(ωDk^α+ B_D^Ee^α_E) + k^αe^γ_C∇γκ + κ(ωCk^α+ B_C^De^α_D)

= −B_C^D(ωDk^α+ B_D^Ee_E^α − κe^α_D) + k^αe^γ_C∇γκ

(2.28) where on the first line, the first terms cancels out, and we are able to re-factorize.

Recall now the form of the Reimann tensor. If we subtract an analogous equation for k^βe^γ_C∇β∇γk^α, we get precisely −R_µβγ^α k^µk^βe^γ_C. We find that

R^α_µβγk^µk^βe^γ_C= k^βe^γ_C∇β∇γk^α− e^γ_Ck^β∇γ∇βk^α

= B_C^D(ωDk^α+ B_D^Ee_E^α − κe^α_D) − k^αe^γ_C∇γκ + k^αk^β∇βω_C+ e^α_Dk^β∇βB_C^D

(2.29)

What is left to do now is to project along the various basis vectors. Projecting along Nαgives

R^α_µβγk^µk^βe^γ_CN_α= −B_C^Dω_D+ e^γ_C∇γκ − k^β∇βω_C

= −B_C^Dω_D+ ∂_λω_C− ∂Cκ (2.30) where we recall that κ is a scalar, and we have used Eq. (2.6) to write the last covariant derivative as ∂_λ. If we instead project along e_Aα, the resulting

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equation is

R^α_µβγk^µk^βe^γ_CeAα= B_C^DB_D^Ee^α_EeAα− κB_C^De^α_DeAα+ e^α_DeAαk^β∇βB_C^D

= B_C^DB_C^DBDA− κBCA+ ∂λBCA

(2.31) Following similar steps with the remaining Gauss-Weingarten equations, and considering the various projections along our basis vectors, we arrive at a set of equations which define the Gauss-Codazzi relations, listed below.

k^µN^νk^λe^α_ARµνλα= ∂λωA− ∂Aκ + B_A^BωB (2.32a) k^µe^α_Ak^νe^β_BRµανβ= −∂λBAB− κBAB+ B_A^CBCB (2.32b) k^µe^α_AN^νe^β_BRµανβ= −∂λKAB− κKAB+ ∇AωB+ ωAωB+ K_A^CBCB (2.32c) k^µN^νe^α_Ae^β_BR_µναβ= ∇_Aω_B− ∇Bω_A− B_A^CK_CB+ B_B^CK_CA (2.32d) k^µe^α_Ae^β_Be^γ_CR_µαβγ= ∇_CB_AB− ∇_BB_AC− ω_CB_AB+ ω_BB_AC (2.32e) N^µe^α_Ae^β_Be^γ_CRµαβγ= ∇CKAB− ∇BKAC+ ωCKAB− ωBKAC (2.32f) e^α_Ae^β_Be^γ_Ce^δ_DRαβγδ= ¯RABCD+ BACKBD− BADKBC+ KACBBD− KADBBC

(2.32g) These relations are the various projections of the spacetime Riemann tensor that we are able to express in terms of the intrinsic and extrinsic curvature quantities on the null hypersurface. The relation Eq. (2.32g) directly relates the spacetime Riemann tensor to the Riemann tensor ¯RABCDon the null hypersurface Σn.

2.5 Components of Ricci tensor

With this set of Gauss-Codazzi equations, we are now equipped go on to find the components of the spacetime Ricci tensor. We begin by writing

Rµν = g^αβRαµβν

= (−k^αN^β− N^αk^β+ γ^ABe^α_Ae^β_B)Rαµβν

= −R_αµβνk^αN^β− RαµβνN^αk^β+ γ^ABR_αµβνe^α_Ae^β_B

(2.33)

where at the first equality, we have written the Ricci tensor as a the spacetime metric Eq. (2.17) contracted with the Riemann tensor; at the second equality we inserted the form of this metric; and at the last equality we expanded out to write as Riemann tensors projected along two basis vectors. Projecting along a further two basis vectors allows us to find the components of the Ricci scalar.

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The first component we find is

Rµνk^µk^ν = −Rαµβνk^αk^µN^βk^ν− RαµβνN^αk^µk^νk^β+ γ^ABRαµβνe^α_Ak^µe^β_Bk^ν

= γ^ABR_αµβνe^α_Ak^µe^β_Bk^ν

= γ^ABR_µανβk^µe^α_Ak^νe^β_B

= γ^AB(−∂λBAB− κBAB+ B_A^CBCB)

= −∂λΘ + κΘ + BABB^AB

(2.34) where we have used the symmetry in the indices µ, α and α, ν to cancel the first and second terms, respectively; rearranging the indices on the remaining term allows us to relate it directly to Eq. (2.32b); contracting with the two-metric then gives us our result where we defined the scalar trace

Θ := γ^ABBAB (2.35)

which denotes the expansion of the null geodesic congruence. Recall that a congruence of null geodesics generates the event horizon, so this quantity is a useful measure of how the event horizon expands or contracts. We will have more to say about this in a later section.

The second component of the Ricci tensor is found from similar steps, and we find

R_µαk^µe^α_A= −R_µανβk^µe^α_Ak^νN^β− R_µανβk^µe^α_AN^νk^β+ γ^{N B}R_µανβk^µe^α_Ae^ν_Ne^β_B

= −Rµνλαk^µN^νk^λe^α_A+ γ^BCRµαβγk^µe^α_Ae^β_Be^γ_C

= ∂λωA− ∂Aκ + B_A^BωB+ γ^BC(∇CBAB− ∇BBAC− ωCBAB+ ωBBAC)

= ∂_λω_A− ∂_Aκ + B_A^Bω_B+ ∇_BB_A^B− ∇_AΘ − ω_BB_A^B+ ω_BΘ

= ∂λωA− ∂Aκ − ∂AΘ + ωBΘ + ∇BB_A^B

(2.36) The third component requires slightly more work to elucidate. We begin as before, and write this component of the Ricci tensor as the sum of components of the Riemann tensor,

R_αβe^α_Ae^β_B= −R_µανβk^µe^α_AN^νe^β_B− R_µανβN^µe^α_Ak^νe^β_B+ γ^CDR_µανβe^µ_Ce^α_Ae^ν_De^β_B (2.37) where on the first line, we have used the fact that the first term is given by Eq.

(2.32c) and the third term is composed of the induced metric and Eq. (2.32g).

We now use the first Bianchi identity,

Rµ[ανβ]= 0 (2.38)

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to rearrange and rewrite the second term as

R_µανβN^µk^νe^α_Ae^β_B= −R_µνβαN^µk^νe^α_Ae^β_B− R_µβανN^µk^νe^α_Ae^β_B

= −Rνµβαk^µN^νe^α_Ae^β_B− Rνβαµk^µN^νe^α_Ae^β_B

= −Rµναβk^µN^νe^α_Ae^β_B− Rµανβk^µN^νe^α_Ae^β_B

(2.39)

where we have rearranged indices on the Riemann tensors, and thus expressed the second term as a sum of two known Gauss-Codazzi relations, namely Eq.

(2.32d) and Eq. (2.32c). With this, we may now continue calculating the final component of the Ricci tensor

Rαβe^α_Ae^β_B = −2Rµανβk^µe^α_AN^νe^β_B+ Rµναβk^µN^νe^α_Ae^β_B+ γ^CDRµανβe^µ_Ce^α_Ae^ν_De^β_B

= 2(∂λKAB+ κKAB− ∇AωB− ωAωB− K_A^CBCB) + ∇AωB− ∇BωA

− B_A^CKCB+ B_B^CKCA+1 2

Rγ¯ ^AC(γACγBD− γADγBC) + γ^AC(B_ACK_BD− B_ADK_BC+ K_ACB_BD− K_ADB_BC)

= 2(∂_λ+ κ + 1

2)K_AB− (∇Aω_B+ ∇_Bω_A) − 2ω_Aω_B

− 2(K_A^CB_CB+ K_B^CB_CA) + KB_AB+1 2

Rγ¯ _AB

(2.40) where at the second equality, we used a geometric property of the Riemann tensor in 2D [6], namely that it can be written as

R¯ABCD=1 2

R(γ¯ ACγBD− γADγBC) (2.41) where ¯R is the degenerate two-dimensioanl Ricci scalar on Σn. In analogy to Eq. (2.35), we have also defined a trace term for K_AB for convenience, as

K := γ^ABKAB (2.42)

which we used at the third equality. Although, the quantity K presents no similar insight as the analogous scalar Θ.

To summarize the results of these calculations, we have computed the components of the Ricci tensor. They involve the intrinsic and extrinsic quantities we have discussed earlier, and they are given as follows

k^µk^νRµν = −(∂λ− κ)Θ + BABB^AB (2.43a)

k^µe^α_AR_µα= ∂_λω_A− ∂Aκ − (∂_A− ωA)Θ + ∇_BB_A^B (2.43b) e^α_Ae^β_BRαβ= 2(∂λ+ κ + 1

2)KAB− (∇AωB+ ∇BωA) − 2ωAωB

− 2(K_A^CB_CB+ K_B^CB_CA) + KB_AB+1 2

Rγ¯ _AB

(2.43c)

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where in Eq. (2.43a), we have corrected the sign of the final term to have a plus, as compared with the result stated in [6]. Equation Eq. (2.43a) will be particularly useful in revealing physical insight on the behavior of the event horizon.

2.6 Evolution of expansion and shear

We found previously that the geometric quantity BAB is a characteristic of the intrinsic geometry of our null hypersurface Σn. Furthermore we found that it represents the intrinsic curvature of Σn. We now find further interpretation of this quantity, by instead examining its components. What we may do is to split this tensor into components and write

BAB= 1

2ΘγAB+ σAB (2.44)

where the first term is Eq. (2.35) as before and denotes the expansion of the null geodesic congruence, and σAB is the remaining term. As such, it is purely tracefree and instead denotes the shearing of the null congruence. To see how these quantities change, we desire an evolution equation for the rate of expansion and shear. For the expansion Θ, this is precisely given by Raychaudhuri’s equation [15], which we arrive at by making use of the results recently developed.

Rearranging Eq. (2.43a) and substituting in the decomposition Eq. (2.44), the rate of expansion is

∂λΘ = κΘ − BABB^AB− Rαβk^αk^β

= κΘ − (1

2ΘγAB+ σAB)(1

2Θγ^AB+ σ^AB) − Rαβk^αk^β

= κΘ −1

2Θ²− 2ΘγABσ^AB− σABσ^AB− Rαβk^αk^β

(2.45)

where at the final equality, we have made use of γABγ^AB= δ^A_A= 2. We continue the calculation by rearranging Eq. (2.44) and substituting it in for σAB, which enables us to perform a useful cancellation of the term we are taking the trace of at the second equality. This is again due to σAB being purely tracefree.

Continuing the calculation,

∂λΘ = κΘ −1

2Θ²− 2ΘγAB(B^AB−1

2Θγ^AB) − σABσ^AB− Rαβk^αk^β

= κΘ −1

2Θ²− 2ΘγAB(Θ − Θ) − σABσ^AB− Rαβk^αk^β

= κΘ −1

2Θ²− σABσ^AB− Rαβk^αk^β

(2.46)

We have now simplified the evolution equation and it is left in terms of a component of the Ricci tensor. This is a useful relation, however we can do a bit more to obtain a deeper physical interpretation. Therefore, we are now at the

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stage where we wish to introduce the Einstein field equations Eq. (1.1), Gαβ:= Rαβ−1

2Rgαβ= 8πTαβ (2.47)

where we write the Einstein tensor Gαβin terms of quantities we have developed.

Using the Einstein field equations, we may re-express the component of the Ricci tensor in question as

Rαβk^αk^β= (8πTαβ+1

2Rgαβ)k^αk^β

= 8πTαβk^αk^β+1 2R kβk^β

| {z }

=0

(2.48)

where the final term vanishes by virtue of orthogonality Eq. (2.8), so we may now express Raychaudhuri’s equation as

∂λΘ = κΘ −1

2Θ²− σABσ^AB− 8πTαβk^αk^β (2.49) where the term introduced involving the energy-momentum denotes matter moving across Σ_n. This is our evolution equation for the expansion scalar Θ, which describes the expansion of a null geodesic congruence generating a null hypersurface, which in turn describes the black hole event horizon.

Now, if instead we start from Eq. (2.32b), we obtain a different evolution equation. Rearranging yields,

∂λBAB = −κBAB+ B_A^CBCB− Rµανβk^µe^α_Ak^νe^β_B (2.50) which can be written in terms the tracefree component as

∂λσAB= (κ − Θ)BAB+ B_A^CBCB− Rµανβk^µe^α_Ak^νe^β_B (2.51) so we arrive at an evolution equation, analogously to that for Θ, for the shear tensor σ_B^A

∂_λσÂ_B= (κ − Θ)σÂ_B− C_µανβk^µeÂαk^νe^β_B (2.52) where C_µανβis the well known Weyl curvature tensor and is the tracefree contribution of the Riemann tensor, whereas the trace contribution is given by the Ricci tensor.

The effect of the shear tensor in general, is slightly more complicated than that of the expansion scalar [15]. The shear tensor can be thought of as being made up of components, or shear parameters, σ+ and σ_× which dictate the geometry of the shearing effect, which is area preserving. If σ_× = 0, the geodesic congruence, if initially thought of as circular, will become an ellipse with its major axis along an angle of φ = 0 for instance. If instead σ+ = 0, then the initially circular configuration will form an ellipse with major axis in the direction φ = π/4 in comparison. The shearing effects of σ+ and σ_× thus

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amount to deformation of the initial configuration along φ = 0 and φ = π/2 and along φ = π/4 and φ = 3π/4, respectively, in a ”plus” and ”cross” pattern, as the labeling of the shear parameters suggests. This is similar to the effect of the + and × polarizations of gravitational waves on a ring of test particles [3].

Looking ahead, in Sec. 4.4 we investigate how these measures of tidal deformability can be developed further using black hole perturbation theory, and how they apply to the case of a small body external to the black hole, which induces the tidal distortion of the black hole.

Geometrical and Perturbative study of Tidally Deformed Schwarzschild Spacetime

Master Thesis