Spacetime Geometry of a Wormhole Modified BTZ Black Hole

Spacetime geometry of a wormhole modiﬁed BTZ black hole

David Sundelin Supervisor Bo Sundborg Stockholm University

October 26, 2018 2 Spacetime geometry of a wormhole modiﬁed BTZ black hole

David Sundelin

Abstract

The unitarity problem for the BTZ black hole can possibly be solved by a coordinate transformation in which the event horizon is extended to a wormhole. In a model proposed by S.N. Solodukhin, this is done by the addition of a wormhole parameter λ to the BTZ black hole line element. This thesis studies the changes in the spacetime geometry that comes with the addition of such a parameter. The focus of study are geodesic behaviour and possible bound states of waves. Investigating a possible source of the wormhole, the stress-energy tensor ansatz for a perfect ﬂuid is also tested. The thesis concludes that there are notable changes to the spacetime depending on the presence of either a black hole or a wormhole. These changes includes orbital trajectories of geodesics and localized bound states of waves. The changes are most notable for λ > 1 but also detectable for small parameter corrections. The wormhole spacetime can however not be generated by a simple addition of matter in a perfect ﬂuid form.

3 4 Contents

1 Introduction 7

2 Background 9 2.1 General relativity ...... 9 2.2 Anti-de Sitter spacetime ...... 10 2.3 Unitarity and the AdS/CFT correspondence ...... 11 2.4 Wormhole modiﬁcation of a BTZ black hole ...... 12

3 Spacetime geometry of a wormhole modiﬁed BTZ black hole 14 3.1 Implications for the stress energy tensor ...... 14 3.2 Geodesics ...... 15 3.2.1 Preliminaries ...... 16 3.2.2 Radial geodesics ...... 18 3.2.3 General geodesics ...... 21 3.2.4 Elaboration on conserved quantities ...... 27 3.2.5 Geodesics in the second modiﬁcation ...... 29 3.3 Waves ...... 33 3.3.1 Variable substitution ...... 34 3.4 Summary ...... 39

4 Conclusions 41

5 Appendix 43 5.1 The Einstein tensor ...... 43 5.2 Christoﬀel symbols and the Ricci scalar ...... 43 5.3 Image archive ...... 45

5 6 1 Introduction

In physics, there are four fundamental forces. Three of them, electromagnetism and the atomic weak and strong forces, has, through the process of quantization, already been uniﬁed in a single framework known as Quantum Field Theory. The fourth force, gravity, is as of yet a classical theory. The ﬁrst theory of gravity was put forward by Isaac Newton in his work Principia, published in 1687, and is referred to as Newtonian, or classical, mechanics. It is for many the quintessential picture of physics and is often used as an introductory subject in physics. Classical mechanics, however, is not applicable at certain limits:

• At very small distances, e.g. on an atomic level

• At very high velocities, e.g. close to the speed of light

• At very strong gravitational potentials, e.g. close to black holes

All of these limits eventually gave rise to new theories around the beginning of the 20th century. The first resulted in Quantum Mechanics. The second was Special Relativity, which stipulates that nothing can travel faster than the speed of light with unintuitive things happening close to that limit, e.g. time dilation and length contraction. The final theory was General Rel- ativity, which changed the whole perspective on gravity. In general relativity, gravity is not a force in the classical sense that attracts masses like electromagnetism attracts charges of opposite signs but rather it is a con- squence of the curvature of the spacetime. Spacetime is Figure 1: A vizualisation of curved spacetime; in the fusion of the three spa- the outskirts spacetime is flat, at the center some tial directions and time into heavy object curves it causing the effect of gravity. a single quantity which is then curved by presence of mass and energy often vizualised by a two dimensional soft surface being bent in a third direction by some heavy object in the middle, Fig.1. The governing equations of general relativity is the Einstein field equations which states the relationship between the curvature of the spacetime and the mass/energy that curves it. The equations are solved for the metric which contains all the relevant information about how the geometry of a certain spacetime behaves in time and space. General relativity solved many of the problems of Newtonian mechanics, while presenting new physical phenomena, e.g. black holes and gravitational waves. These phenomena have since become research subjects of their own

7 and observations support their existence. Problems that emerged with general relativity includes

• How does the overall geometry of our universe looks like? Is the universe as a whole curved or ﬂat?

• What is the nature of black holes and gravitiational waves?

• How can the theory be quantized to form a unifying theory of all the four fundamental forces?

In the process of solving these issues, models that don’t nesessarily re- semble our own universe, but rather simpliﬁes calculations are often used. This thesis studies the spacetime geometry resulting from the modiﬁcation of such a model. The model, the BTZ black hole, is of interest in resolving some of the issues that arises when formulating a theory of quantum gravity.

8 2 Background

2.1 General relativity In general relativity, time is a coordinate direction on the same footing as the spatial directions of ordinary Newtonian mechanics, distinguished from the latter by opposite sign. The resulting concept is referred to as spacetime which is described by the distance between points in the geometry, with the distance given by the line element. In a chosen system of coordinates, the line element can be expressed as

2 µ ν ds = gµνdx dx (1) where gµν is the components of the metric tensor, or simply the metric, which is symmetric and coordinate dependent. All the information about the geometry of the spacetime is contained within the line element. The simplest geometry, Minkowski spacetime, is the spacetime analogue of ordinary ﬂat space and is given by the metric gµν = ηµν where

 −1 0 0 0   0 1 0 0  ηµν =   . (2)  0 0 1 0  0 0 0 1

The Minkowski line element then reads

ds2 = −dt2 + dx2 + dy2 + dz2. (3)

The metric gµν is obtained from the Einstein ﬁeld equations

Gµν + gµνΛ = κTµν, (4) which is ten equations for the ten inputs of the metric. The equations state the relation between the curvature of the spacetime, given by the 1 Einstein tensor Gµν and the cosmological constant Λ on the left hand side, and the source of the curvature, namely matter and energy, given by the stress-energy tensor Tµν on the right hand side. Given some input for the stress-energy tensor, the equations are solved for the metric. The cosmological constant was added at a later stage by Einstein since without it the ﬁeld equations do not permit static cosmological solutions which was in contrast to contemporary beliefs that the universe was static. After observations of an expanding universe the constant was kept however for theoretical reasons and it is today related with vacuum energy, which for example would be the cause of inﬂation. The constant on the right hand side, κ = 8πG/c4, contains the gravitational constant G, and the speed of light

1see Appendix, section 5.1

9 c, and is determined by taking the limit of low gravitational potential and velocites v << c and demanding that Eq.(4) then reproduces Newtonian mechanics. Being non-linear, the Einstein ﬁeld equations are diﬃcult to solve and few exact solutions are known, [1, 2].

2.2 Anti-de Sitter spacetime

The metric gµν obtained from the Einstein field equations, describes spacetime. Solutions other than Minkowski spacetime includes de Sitter (dS) and Anti-de Sitter (AdS) spacetime. While Minkowski spacetime is a vacuum solution (Tµν = 0) with a zero cosmological constant, dS and and AdS spacetime are vacuum solutions with a positive (Λ > 0) and negative (Λ < 0) cosmological constant, respectively. Constructing a spacetime can be done much like a surface of dimension d is defined by its embedding in d + 1 dimensions. In general a sphere is then defined as d+1 2 2 2 2 X 2 2 x1 + x2 + ... + xd + xd+1 = xi = L , (5) i=1 where L is the radius of the sphere. The surface is defined as every point that satisfies the defining equation, Eq.(5). By the addition of a time coordinate x0 to the sphere, we obtain dS spacetime d X −(x0)2 + (xi)2 = L2, (6) i=1 where xi are spatial coordinates. Introducting another time coordinate and flipping the sign on the right hand side we obtain AdS spacetime

d−1 X −(x0)2 + (xi)2 − (xd)2 = −L2. (7) i=1 In d = 3 dimensions, the line element becomes ds2 = −(dx0)2 + (dx1)2 + (dx2)2 − (dx3)2 (8) with the deﬁning equation (x0)2 − (x1)2 − (x2)2 + (x3)2 = L2. (9) 2+1 AdS spacetime is then all points satisfying Eq.(9). The two time coordinates are constrained which can be seen through a coordinate transformation  x0 = ρ cos t,   x1 = r cos θ, (10) x2 = r sin θ,   x3 = ρ sin t.

10 The transformation gives the line element

ds2 = −(dρ2 + ρ2dt2) + (dr2 + r2dθ2), (11) and the constraint ρ2 − r2 = L2. (12) Eliminating ρ in the line element yields the AdS line element in 2+1 dimensions: L2 ds2 = −(L2 + r2)dt2 + dr2 + r2dθ2, (13) L2 + r2 with one time coordinate, t and two spatial coordinates, r and θ. One usefulness of AdS spacetime is that it has a boundary. The boundary is of one dimension lower than the original spacetime and is flat, i.e. Minkowskian. It can be visualised by studying a radial light ray that travels towards infinity. From the AdS line element, for ds2 = 0 and dθ = 0: ±L dt = dr. L2 + r2 When integrating, the right hand side will converge and thus the light ray can move an infinite distance in a finite amount of time. This is indicative of a boundary in AdS spacetime. With the boundary, AdS spacetime can be used to enclose a gravitational system, in the same manner that particles are enclosed in a box, or in a finite volume, in other areas of theoretical physics, making it an interesting background for the development of quantum gravity, [3].

2.3 Unitarity and the AdS/CFT correspondence There has been no successful attempts of formulating general relativity within a quantum field theory framework. A major inconsistency between the two theories happens at the event horizon of a black hole. Quantum mechanics tells that all the information of a particle is contained within the particle wave function, known as unitarity. Due to quantum effects near the event horizon it could so happen that when a particle-antiparticle pair is created, one of the particles escapes the black hole while the other is absorbed. This is known as Hawking radiation. If the black hole has no means of obtaining mass, the radiation will eventually lead to evaporation of the black hole and ultimately result in a total loss of information, thus contradicting unitarity. The problem could potentially be solved within string theory, which is an attempt to unify all the existing physical theories within one single framework. In string theory, particles are string-like objects, rather than point-like, which vibrates with each vibrating mode corresponding to different particles. By taking the limit of these theories, quantum field theories

11 are obtained and possibly also quantum gravity. Of particular interest is the so called AdS/CFT correspondence or duality. In physics, a duality suggests an intimate relation between two different physical theories in the sense that information about one system can be translated into information about the other system. The AdS/CFT correspondence then suggests that various string field theories in AdS spacetime is dual to various Conformal Field Theories2 (CFT) at the boundary of AdS. In a similar manner that quantum field theory can be derived by taking the limit of string theories, so can also quantum gravity be obtained, [4, 5].

2.4 Wormhole modification of a BTZ black hole The BTZ black hole introduced by [6] is a vacuum solution to the Einstein field equations in 2+1 dimensions. For a non-rotating object the BTZ black hole line element reads r2 − ML2 L2 ds2 = − dt2 + dr2 + r2dφ2, (14) L2 r2 − ML2 where −∞ < t < ∞, 0 < r < ∞, 0 < φ < 2π. M is a Noether charge to be associated with mass (which is dimensionless) and L is the AdS radius which is related to the cosmological constant by Λ = −L−2. Unlike the Schwarzschild black hole solution that describes the geometry outside some non-rotating heavy object (i.e. a black hole) in 3+1 dimensions and is asymptotically flat, the BTZ black hole line element can be shown to be asymptotically AdS. For this reason, due to the AdS/CFT correspondence, it is an interesting model for 2+1 quantum gravity. Note that there is no coordinate singularity at the origin, however it does have an event horizon. Transforming to Eddington-Finkelstein coordinates 2L r t = v − √ tanh−1[ √ ], M L M gives the line element

r2 − ML2 ds2 = − dv2 + 2dvdr + r2dφ2. L2 For a radial light ray this gives

r2 − ML2 dv = 2dr, L2 assuming dv 6= 0. For r2 < ML2, the light ray can only travel inwards towards the black hole, suggesting an event horizon at r2 = ML2, [7].

2i.e. theories that are invariant under conformal transformations

12 Attempts at solving the unitarity problem for the BTZ black hole has been made by [8]. The black hole is described by correlation functions, a system which should when perturbed, in order to exhibit unitarity, return to its original value in a quasi periodic fashion. Essential for the return is a finite value of the entropy. The entropy, however, is not finite and the system never returns to its original values. It is argued that the reason for this is the presence of an event horizon. This promts the addition of a constant to the line element to transform the black hole to a wormhole. This results in the quasi periodic behaviour of the correlation functions and finite entropy. The objective of this thesis then is to study the changes in the geometry that the wormhole parameter λ induces. In cases where there is no possibility of directly observing of what kind an object is, observations of the surroundings and the effects the object has on them is paramount. To this end, geodesics and waves will be studied in the changed spacetime geometry, concluding how their behaviour depends on the presence of a black hole or a wormhole. Additionally, one possible matter energy distribution will be tested if compatible with the wormhole extension of the BTZ black hole. Transforming the coordinates accordingly √ r = ML cosh y, (15) with the modification sinh2 y → sinh2 y + λ2, (16) gives the wormhole modification line element

ds2 = −M(sinh2 y + λ2) dt2 + L2dy2 + ML2 cosh2 y dφ2. (17)

The event horizon, located at r2 = ML2, or equivalently y = 0, is no longer a horizon. It has not vanished though, but rather the region containing the event horizon is not covered by the coordinate transformation, Eq.(15). Strictly speaking, when comparing cases of λ = 0 and λ 6= 0, this is not a comparison between a black hole and a wormhole but rather between a region outside a black hole and a wormhole spacetime without an event horizon. For simplicity, when comparing λ = 0 and λ 6= 0, the two cases will frequently be referred to as a black hole and a wormhole, respectively. The new coordinates cover the entire spacetime with no coordinate singularities, as opposed to the original BTZ black hole line element. Time- and space coordinates then remain time- and space coordinate everywhere in the geometry and any coordinate symmetry related constants are deﬁned uniformly everywhere. Note that the modiﬁcation is only relevant for the geometry close to the black hole, at large y, λ is negligible. Note furthermore, that since cosh2 y −sinh2 y = 1, the particular value λ = 1 is naturally a special case in these coordinates, which will be evident throughout this thesis. The constant λ originates from string theory.

13 3 Spacetime geometry of a wormhole modiﬁed BTZ black hole

The BTZ black hole line element, Eq.(14), solves the Einstein field equations with a negative cosmological constant (related to the AdS radius, −2 Λ = −L ). Given Tµν = 0, it fulfills Eq.(4) and desribes the geometry outside a black hole in 2+1 dimensions. It is the objective of this thesis to examine the wormhole extension of this line element, the wormhole modification line element, Eq.(17) and the spacetime geometry change that comes with it. In section 3.1 this is done by studying how the the addition of the wormhole parameter λ would affect the non-zero nature of Tµν, assuming no modification is done on how gravity affects the equations. In section 3.2, the trajectories of geodesics are examined and in section 3.3, the wave equation is studied. A summary can be found in section 3.4. A note on units, by the substitution dt˜= dt/L, the wormhole modification line element can be written in dimensionsless coordinates

ds˜2 = −M(sinh2 y + λ2) dt˜2 + dy2 + M cosh2 y dφ2,

2 ds2 where ds˜ = L2 . Any dimensionful quantity can now be expressed in powers of L and so it can be regarded as a scaling factor. In subsequent calculations, the choice L = 1 will frequently be used. This does not result in any loss of generality. Additionally, consistent with [8], the choice M = 1 will be made in several calculations. In general, M appears in products with other parameters that are varied leaving variations of M superfluous. Another modification that will be briefly elaborated on in this thesis is the modification, Eq.(16), performed everywhere in the line element. This results in the second modification line element

ds2 = −M(sinh2 y + λ2) dt2 + L2dy2 + ML2(cosh2 y + λ2) dφ2. (18)

3.1 Implications for the stress energy tensor The wormhole modification line element, Eq.(17), is no vacuum solution to the Einstein field equations. This can be interpreted in two different ways. Either it prompts some modification of the theory of gravity, e.g. redefining how the Einstein tensor Gµν depends on the line element, or it suggests that it simply is a solution for a non-zero stress-energy tensor Tµν 6= 0. This latter possibility will be examined here. Calculating the left hand side of Eq.(4) for the wormhole modification line element gives for the right hand side

κT tt = 0, (19) λ2(1 − λ2) κT φφ = , (20) ML4 cosh2 y(sinh2 y + λ2)2

14 −λ2 κT yy = . (21) L4(sinh2 y + λ2) Even though still T tt = 0, due to Eqs.(20)-(21) this is no longer a vacuum solution. The T tt has some consequences for common stress-energy tensor ansätze. A dust universe ansatz is given by T αβ = µuαuβ, where µ is rest mass density, and uα is the three velocity. Naturally, we must have ut 6= 0, but Eq.(19) then implies µ = 0 which is inconsistent with Eqs.(20)-(21). The more general perfect fluid ansatz is given by T αβ = (ρ + p)uαuβ + gαβp where ρ = ρ(t, y, φ) is energy density and p = p(t, y, φ) is pressure. Since any off-diagonal elements must vanish in the effective Tµν, either two of the components of the three velocity uα must equal zero or p = −ρ. If only one component of uα is non-zero this will result in either the solution being valid only for a particular (imaginary) value of y given by sinh2 y = −1 or some typical requirement like λ = 0 or λ = 1. In the second case, if p = −ρ, then Eq.(19) gives p = 0 and so ρ = 0, a trivial solution. In the case of the second modification line element, Eq.(18), the implications for the stress-energy tensor reads:

λ2(1 + λ2) κT tt = , (22) ML2(sinh2 y + λ2)(cosh2 y + λ2)2

λ2(1 − λ2) κT φφ = , (23) ML4(cosh2 y + λ2)(sinh2 y + λ2)2 8λ2(λ2 + cosh 2y) κT yy = . (24) L4(−1 + 8λ4 + 8λ2 cosh 2y + cosh 4y) By the same arguments as for the wormhole modification, these implications are not compatible with the perfect fluid ansatz. Nor are they compatible with a dust universe model due to non-conforming expressions for the rest mass density µ. The attempt to interpret the wormhole modification line element (17) departing from a vacuum solution as being due to the presence of matter is not compatible with standard stress-energy tensor ansätze. Although the ansatz being simplistic and ideal in nature, the wormhole modification arguably does not fit well with matter in a perfect fluid form. This could be indicative of the fact that it is a model of a black hole, not a physical reality.

3.2 Geodesics In classical mechanics, the trajectory of a particle is described by Newton’s equations of motion. Study objects are often taken to be particles which move on straight lines only aﬀected by a gravitational pull. In general

15 relativity, the notion of a straight line is generalized to a geodesic, that is the trajectory of a particle that is unaffected by any force and influenced only by the curvature of the spacetime in which it travels. The analogue of Newton’s equations of motion in general relativity is the geodesic equation. It reads d2xα dxβ dxγ = −Γα , (25) dσ2 βγ dσ dσ α where σ is some affine parameterization of the geodesic and Γβγ are the Christoffel symbols listed for the wormhole modification line element in the Appendix (section 5.2). In this section the geodesic equation is solved. In section 3.2.1 some general observations and relations are listed which will be useful for the treatment of geodesics, section 3.2.2 are limited to radial geodesics while the most general geodesics are treated in section 3.2.3. Section 3.2.4 contains an elaboration on the conserved quantities from the wormhole modification line element and section 3.2.5 deals with the behaviour of geodesics in the second modification line element.

3.2.1 Preliminaries Solving the geodesic equation for the time and angular coordinates, t and φ, it becomes dt (sinh2 y + λ2) = e, (26) dσ dφ cosh2 y = l, (27) dσ where e and l are conserved quantities. These quantities could have been directly obtained by the observation that the line element, Eq.(17), is in- dependent of t and φ, and so there are two Killing vectors, ξα = (1, 0, 0), related to the conservation of energy e, and ηα = (0, 0, 1), related to the conservation of angular momentum l. Note that as y grows large, the change in the time and angular coordinates, with respect to σ, grows small. In the time coordinate case, this could be viewed in light of the AdS boundary elaborated on earlier, section 2.2. In the angular coordinate case, it is reminiscent of orbital trajectories in classical mechanics. Further note that, since for a given value l, the angular coordinate derivative never changes sign, then for whatever orbit in this coordinate system, the orbit always circulate the origin. For the radial coordinate, y, the geodesic equation reads " # d2y 1 dt 2 dφ2 + M sinh y cosh y − = 0. (28) dσ2 L2 dσ dσ

16 It has not been analytically solved for an expression y = y(σ). What has been obtained is the radial coordinate derivative s dy 1 e2 L2l2 = ± K + M − . (29) dσ L sinh2 y + λ2 cosh2 y

Here, K is a constant which equals the norm of a tangent vector u = dt dy dφ ( dσ , dσ , dσ ), that is being parallel transported along the geodesic:

α β u · u = gαβu u = K, (30) and it determines the geodesic to be timelike (K < 0), lightlike (K = 0) or spacelike (K > 0). As L appeared only as a scale factor in the line element (section 3.2), so is K identiﬁed here as a scale factor of the geodesic parameterization. Thus in subsequent calculations or plots throughout, the choices K = −1 for time-, K = 0 for light- and K = +1 for spacelike geodesics will be made. The radial coordinate derivative can either be obtained solving the geodesic equation, or by inserting Eqs.(26)-(27) into Eq.(30). From it, two other useful relations can be derived. The ﬁrst one is the behaviour at the spatial limits. It is given by

 r  ± 1 K + M e2 − R2l2 , y → 0; dy  L λ2 → √ . (31) ± 1 K, y → ∞ ∧ K 6= 0; dσ  L  2 p 2 2 2 ± L exp(−y) M(e − R l ), y → ∞ ∧ K = 0. The second one comes from the requirement that the square root in Eq.(29) must equal or be larger than zero, thus giving the geodesic energy condition

L2l2 K e2 ≥ (sinh2 y + λ2)( − ). (32) cosh2 y M In effect, the geodesic energy condition will be interpreted as a kind of effective potential. Throughout this section, the use of streamplots will be substantial. In the streamplot, the radial coordinate y is plotted along the horizontal axis against the radial coordinate derivative along the vertical axis. This gives a vivid illustration of the trajectory of the geodesics. The streamplot is ob- d2y dy tained by solving Eq.(28) for dσ2 , which is a function of y and dσ . The vector dy d2y field ( dσ , dσ2 ) then generates the streamplot. The treatment of geodesics will be based on these streamplots with supporting arguments principally com- ing from the radial coordinate derivative, Eq.(29), and the geodesic energy condition, Eq.(32).

17 3.2.2 Radial geodesics The ﬁrst kind studied will be geodesics which only have motion in the radial direction, that is geodesics for which dφ = 0. By eliminating the time component in Eq.(28) by the use of Eq.(30), the radial geodesic equation for y is obtained " # d2y sinh y cosh y dy 2 K + − = 0. (33) dσ2 sinh2 y + λ2 dσ L2

The streamplots of the radial geodesic equation for λ = 0.5 are presented in Fig.2. In Fig.3, numerical solutions are shown for λ = 0.5 and initial 0 values (y0, y0) = (3, 1.1) where prime refers to derivation with respect to the parameter σ. In Fig.2 each geodesic is characterized by their conserved energy, e. In Fig.3, this translates to initial values. Studying Eq.(33) and Figs.2-3 the following notes can be made:

• The mass M does not appear as a parameter for radial geodesics.

• The equation has a singularity at y = 0 when λ = 0. This manifests dy itself in dσ → ±∞ when approaching y = 0.

d2y • For timelike geodesics, dσ2 always has the opposite sign of y. If λ = 0, the geodesic plunges into the black hole. If λ 6= 0, the geodesic will pass through the wormhole and come back again in an oscillating manner.

• Lightlike geodesics will, depending on e, either travel away to inﬁnity dy with decreasing dσ , or plunge into the black hole (λ = 0), or pass through the wormhole (λ 6= 0) and onto inﬁnity on the other side.

• Spacelike geodesics behave as lightlike geodesics with the exception of the horizontal stripe −1 < y0 < 1 where a geodesic seemingly turns back from heading towards the origin. This stripe, however, is not permitted which is demonstrated below.

dy • For geodesics that passes the wormhole, dσ increases with decreasing λ and decreases with increasing λ.

18 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 2: Trajectories for from left to right time-, light- and spacelike geodesics for λ = 0.5. Timelike geodesics oscillates back and forth through the wormhole while light- and spacelike geodesics travels through the wormhole and continues onward to inﬁnity on the other side.

y(σ) y(σ) y(σ)

8 3 4 6 2 4 1 2 2 σ σ σ -4 -2 -1 2 4 -4 -2 2 4 -2 -4 -2 -2 2 4 -2 -4 -3 -4 -6 Figure 3: Numerical solutions to the radial geodesic equation for from left to right 0 time-, light- and spacelike geodesics for λ = 0.5 and initial values (y0, y0) = (3, 1.1). In each case for larger initial values, the steepness of the curve at y = 0 increases, eventually resulting in a numerical breakdown. This however is likely an instability in the numerical solution rather than a physical ocurrence. Notice, at the extremities, the ﬂattening of the lightlike curve, and the linearity of the spacelike curve.

To determine the turning points of the timelike geodesics, Eq.(29) can be rewritten as ( s ) Me2 y = sinh−1 ± − λ2 . (34) 2 dy 2 L ( dσ ) − K

dy If dσ = 0, which is required for the geodesic to turn around, then the radial turning point equation is given ( r ) −(Me2 + Kλ2) y = sinh−1 ± . (35) K

The turning point do depend on the mass M, as well as on energy e and λ. It is apparent that it is not possible to have a turning point for other

19 values than K < 0, that is only timelike geodesics do turn around. Thus the horizontal stripe −1 < y0 < 1 is not a permitted region for a spacelike geodesic. The behaviour within the stripe seen in Fig.2 is likely to be a graphical interpretation in Mathematica of how to connect the two separate regions y0 < 1 and y0 > 1. In general, from Eq.(34) it is seen that for 2 dy 2 real valued radial coordinate, a spacelike geodesic must fulfill L ( dσ ) > K. dy For a lightlike geodesic dσ 6= 0, as would be expected for a radial lightlike trajectory. From Fig.2 it is undetermined whether or not every timelike geodesic oscillates through the wormhole and how far out it can travel. The turning point equation does not have an upper limit for the turning point, however the limits of the radial coordinate derivative, Eq.(31), do. For a timelike dy √ geodesic, K < 0, and whenever y → ∞, then dσ → K, suggesting the distance a timelike geodesic can travel is limited. Numerically solving the radial geodesic equation gives a steeper function at y = 0 for large values of e (i.e. initial values), eventually resulting in a singularity. Analytically there is nothing to suggest such a singularity since the radial coordinate derivative gives when y → 0 r dy 1 Me2 = ± K + . dσ L λ2 Also, nothing suggests that not all timelike geodesics do oscillate through the wormhole. The singularity encountered in numerical solutions then is arguably an artefact of the numerical approximation. Thus, given e that fulfills Eq.(32) and larger than λ, every timelike geodesic can pass through the wormhole. Neither light- nor spacelike geodesics can fullfill the radial turning point equation, Eq.(35), thus neither light- nor spacelike geodesics can have turning points. They do, however, pass through the wormhole. Given that the geodesic energy condition is satisfied, they will travel infinitely far out, for dy lightlike geodesics dσ → 0 and spacelike geodesics reaching constant radial dy √ coordinate derivative, dσ → K. Relating back to AdS spacetime described in section 2.2, a lightlike geodesic will travel to infinity in a finite amount of time. Rewriting the wormhole modification line element for a radial light ray (ds2 = 0) gives

L dy dt = ±√ p . M sinh2 y + λ2

Integrating the right hand side will give a limited contribution when y grows suﬃciently large. The equation can be solved with elliptic functions but it is not presented here.

20 3.2.3 General geodesics In this section, geodesics with both radial and angular (dφ 6= 0) motion will be studied. By the same approach as before, eliminating the time coordinate in Eq.(28) with use of Eq.(30), the general geodesic equation for y is obtained in terms of K and the conserved angular momentum l: " # d2y sinh y cosh y dy 2 K sinh y + − + 1 − λ2 Ml2 = 0. dσ2 sinh2 y + λ2 dσ L2 (sinh2 y + λ2) cosh3 y (36) As noted in the introduction of this chapter, M appears in a product with the conserved angular momentum, l. It will be viewed as a single quantity, in practice this means setting M = 1 and varying l2. A general geodesic is then characterized by its value of e, and l. In Figs.4-6 streamplots for the general geodesic equation is shown. Observations include:

• If λ = 1, the third term vanishes, eﬀectively eliminating the angular dependence. This is an artefact of the coordinates, (section 2.4).

• The sign of the third term is dependent on λ and diﬀers in the two cases λ < 1 and λ > 1.

• In the streamplots, the horizontal distance between curves is larger with increasing l. This is suggestive that the geodesics now has angular motion and the possibility to orbit the origin.

• If 0 < λ < 1,

– all timelike geodesics oscillates through the wormhole similar to the purely radial case, – light- and spacelike geodesics can oscillate through the wormhole given a certain value of e, that is initial values, otherwise they behave as if radial and pass through the wormhole once.

• If λ > 1,

– timelike geodesics can either oscillate through the wormhole as before, or stay in bound orbits around the wormhole, – light- and spacelike geodesics can either pass through the wormhole and travel to inﬁnity on the other side, or turn around before reaching the wormhole and return to where it came from. Although spacelike geodesics now have turning points, the horizontal stripe is still not accurately depicted in the ﬁgures (see below).

21 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 4: Trajectories for timelike geodesics, for Ml2 = 4.5 and from left to right λ = 0, 0.5 and 1.5. For λ < 1 the behaviour is similar to that of radial geodesics but when λ > 1 there exists local trajectories where the geodesic orbits the wormhole rather than oscillating throught it

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 5: Trajectories for lightlike geodesics, for Ml2 = 4.5 and from left to right λ =0, 0.5, and 1.5. Contrary to the purely radial case, lightlike geodesics can now have turning points. Either they oscillate through the wormhole (λ < 1) or they turn around before passing through it (λ > 1) as if in an open, hyperbolic orbit.

To evaluate potential turning points of the geodesics, Eq.(29) is rewritten −1 sinh4 y = (1 + λ2)∆ + M(e2 − L2l2) sinh2 y + M(e2 − L2l2λ2) + ∆λ2 ∆ (37) where dy 2 ∆ = −L2 + K. dσ dy For a turning point it is required that dσ = 0, giving the general turning point equation −1 sinh4 y = (1 + λ2)K + M(e2 − L2l2) sinh2 y + M(e2 − L2l2λ2) + Kλ2 . K (38)

22 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 6: Trajectories for spacelike geodesics, for Ml2 = 4.5, and from left to right λ =0, 0.5 and 1.5. Similar to lightlike geodesics, spacelike geodesics can now have turning points and the horizontal stripe −1 < y0 < 1, a region forbidden for radial spacelike geodesics, is now permitted. The spacelike geodesics either oscillates through the wormhole (λ < 1) or they turn around before passing through it (λ > 1) as if in an open, hyperbolic orbit.

The equation can be written on a quadratic from but a graphic solution is arguably more illustrative. Deﬁning

f(y) = sinh4 y, −1 g(y) = (1 + λ2)K + M(e2 − L2l2) sinh2 y + M(e2 − L2l2λ2) + Kλ2 , K as the left and right hand side of Eq.(38), respectively. For timelike geodesics it is expected that the equation f/g = 1 exhibit up to two solutions on each side of the wormhole. Indeed, given λ = 1.5, Ml2 = 4.5, e can be chosen in such a way that in increasing order there exists none, one, two or again one solution, Fig.7. The interpretation of these intersections being

• zero intersections corresponds to some small value of e, excluded by the geodesic energy condition,

• one intersection corresponds to a geodesic at a ﬁxed value of y, that is a circular orbit,

• two intersections corresponding to a geodesic in an elliptic orbit,

• again one intersection corresponding to the trajectories of radial-like geodesics, that is geodesics that oscillates through the wormhole.

23 f/g 3

0 y 0.5 1.0 1.5 2.0 2.5 3.0

-1

-2

-3 Figure 7: Graphical solution to the general turning point equation, Eq.(38), for timelike geodesics, λ = 1.5, Ml2 = 4.5 and e = 3.2 (solid blue), 3.24 (dotted red), 3.4 (dotdashed green) and 3.6 (dashed purple). The values of e represents in order, some value excluded by the energy condition, Eq.(32), circular orbit, elliptical orbit, and a oscillating orbit. The functions are symmetric around the vertical axis.

y(σ) y(σ) y(σ) 2 2 2

1 1 1

σ σ σ -10 -5 5 10 -10 -5 5 10 -10 -5 5 10 -1 -1 -1

-2 -2 -2 Figure 8: Numerical solutions to the general geodesic equation for timelike geodesics 2 0 when λ = 1.5, Ml = 4.5 and initial values from left to right (y0, y0) =(1,0), (1.4,0) and (1.6,0). These corresponds to a bound, circular orbit, a bound elliptical orbit, and a oscillating orbit (through the wormhole).

The orbits are presented graphically as numerical solutions to the general geodesic equation in Fig.8. The region of bound orbits is overlapping with the region of oscillating orbits. As seen from approximating the turning point equation to first order at y = 0, if Me2 = (ML2l2 − K)λ2, then the turning point will be at the very throat of the wormhole. It is plau- sible since the line element still has a spatial extension at y = 0, existing in the angular direction. On the other end, the geodesics do not travel infinitely far out. Given that they fulfill the energy condition at the wormhole, that is K e2 ≥ λ2(L2l2 − ), M

24 the right hand side of Eq.(38) will always be positive, growing as sinh2 y. Since the left hand side grows as sinh4 y, there will always be an intersection between curves. Consistent with earlier statements then, timelike geodesics are conﬁned within a limited radial region. For spacelike geodesics, the solution to the turning point equation is more intricate. When graphically solving f/g = 1 for λ = 0.5 it is possible to obtain a pair of solutions which would suggest spacelike geodesics have bound orbits similar to timelike geodesics. This is not the case however as in neither of these instances both intersections, with diﬀerent y, would simultaneously satisfy the energy condition, Eq.(32). For λ = 1.5, the found intersections of f/g = 1 conforms with the stream plots, as well as with the energy condition, these are shown in Fig.9. In Fig.10, the trajectory of a geodesic in an open orbit and a geodesic passing through the wormhole are presented as numerical solutions to the general geodesic equation.

f/g 3

0 y 0.5 1.0 1.5 2.0 2.5 3.0

-1

-2

-3 Figure 9: Graphical solution to the general turning point equation, Eq.(38), for spacelike geodesics, λ = 1.5, Ml2 = 4.5, and e = 1 (solid blue), 2.1 (dotted red), 3 (dotdashed green) and 4 (dashed purple). A curve can only have maximum one intersection with f/g = 1 (solid black) which simultaneously fulﬁlls the energy condition, Eq(32). These are the turning points of a geodesic in an open, hyperbolic orbit which never passes the wormhole. The curves in the graph which does not intersect f/g = 1 are geodesics that travels through the wormhole and onto inﬁnity on the other side. The functions are symmetric around the vertical axis.

25 y(σ) y(σ)

4 4

2 2

σ σ -10 -5 5 10 -10 -5 5 10

-2 -2

-4 -4

Figure 10: Numerical solutions to the general geodesic equation for spacelike 2 0 geodesics when λ = 1.5, Ml = 4.5 and initial values (y0, y0) = (1,0) and (1,1.2). This is a geodesic in an open, hyperbolic orbit, and a geodesic that passes the dy wormhole with a slight decrease in dσ at the passage.

In stark contrast to the radial case then, spacelike geodesics with angular motion do have turning points. The horizontal stripe at −1 < y0 < 1, however, is still not representative for that region. Returning to the turning point equation, Eq.(38), and neglecting the last two terms on the right hand side when y grows large, it can be written

1 sinh2 y sinh2 y + (1 + λ2)K + M(e2 − L2l2) = 0. K

The square brackets will not equal zero unless e2 << L2l2. This will eventually require e2 < 0 which must be rejected meaning that the turning points for spacelike geodesics, although existing in the general case, are situated to limited values of y. For the turning points of lightlike geodesics, the radial coordinate derivative, Eq.(29) becomes ( r ) L2l2λ2 − e2 y = sinh−1 ± . (39) e2 − L2l2

This translates to the conditions

L2l2 < e2 ≤ L2l2λ2, λ2 > 1,

or L2l2λ2 ≤ e2 < L2l2, λ2 < 1, Thus, in order for a lightlike geodesic to have a turning point, the geodesic characteristic values of e and l must fulﬁll the above relations. These are shown graphically in Fig.11.

26 l2

e2 1 2 3 4 5 Figure 11: Allowed (e2, l2) pairs for which lightlike geodesics have turning points. λ < 1 in the area dotted blue to solid black and λ > 1 in the area dashed red to solid black. The corresponding pair inserted in the turning point equation, Eq.(37) dy gives the value of y for which dσ = 0.

3.2.4 Elaboration on conserved quantities In sections 3.2.2-3.2.3, geodesics were studied by eliminating the time coordinate, in effect the conserved energy e, in the geodesic equation for y, and studied with K, determining the geodesic to be time-, light- or spacelike, and the conserved angular momentum l as parameters. In this section, possible restrictions on the values of e and l will be explored. The geodesic energy condition, Eq.(32), defined in section 3.2.1, stated 2 one relation between e and l. Expressing this relation as a function el = 2 el (y), it can be interpreted as a kind of effective potential. Studies of such a potential completely agrees with earlier statements about the behaviour of the geodesics. These includes:

• if λ = 0, timelike geodesics are conﬁned to a well without the possibility of escaping the black hole while light- and spacelike geodesic can escape given suﬃciently large value of e,

• if λ < 1, all geodesics can oscillate through the wormhole, again light- and spacelike geodesics can travel to inﬁnity while timelike geodesics are conﬁned to a limited region

• if λ > 1, there are bound orbits, both elliptical and circular for timelike geodesics, while light- and spacelike geodesics can not reach the wormhole at y = 0, i.e. they turn around if e is suﬃciently small.

A comparative plot of the eﬀective potential for the diﬀerent kinds of geodesics are shown in Fig.12. It is clearly seen that for timelike geodesics if

27 e is suﬃciently low, there will be bound orbits separated from the wormhole at y = 0. Likewise for light- and spacelike geodesics, they will not be able to reach the wormhole given low enough value of e, i.e. they will turn around. Furthermore the comparison reveals that there are areas of e where only spacelike geodesics reside, areas where both light- and spacelike geodesics reside but not timelike geodesics, and ﬁnally an area where all three kinds of geodesics reside. The geodesics with the lowest possibe value of e are those that stay in bound orbits, either oscillating through or around the wormhole (timelike geodesics), or in open, hyperbolic, orbits that never passes the wormhole (light- and spacelike geodesics).

e2 30

0 y 0 1 2 3 4 Figure 12: A comparison of the energy condition, Eq.(32), for timelike (dotted blue), lightlike (dotdashed red) and spacelike (dashed green) geodesics. The stable bound state of timelike geodesics are clearly seen while for light- and spacelike geodesics with suﬃciently low e will not be able to reach the wormhole at y = 0. In the lowest shaded green area only spacelike geodesics are allowed, in the middle dark green area both light- and spacelike geodesics are allowed while all of them are allowed in the upper gray area. No geodesics are allowed in the low left white corner region.

Another illustrative example of the geodesic dependence of e is shown 2 in Fig.13. Here, the value of el is solved for from the radial coordinate dy derivative, Eq.(29) and plotted against y and dσ showing the value of e corresponding to the streamplots of Figs.4-6. It shows that the orbits, bound or open, belongs to geodesics of low constant energy.

28 1.0 1.0 1.0

0.5 0.5 0.5 y′ 0.0 y′ 0.0 y′ 0.0

-0.5 -0.5 -0.5

-1.0 -1.0 -1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0

y y y

2 Figure 13: A contourplot of the value of el corresponding to the background of the streamplots, Figs.4-6, for from left to right time-, light- and spacelike geodesics. e2 varies in magnitude from ≈ 0 in darker blue to ≈ 10 in ligher beige. Outside the range of this graph, e2 increases dramatically.

A quantative measurement on the relation between e and l can be found by rewriting Eq.(30) for a general K :

dy 2 L2l2 e2 K = L2 + M − . (40) dσ cosh2 y sinh2 y + λ2 Along the path of a geodesic, K is a constant, negative for timelike, zero for lightlike and positive for spacelike geodesics. Obviously, the sign of K is determined by the expression within the square brackets. Evaluating gives the relation e2 ≥ L2l2h(y, λ),K ≤ 0, (41) e2 < L2l2h(y, λ), K > 0. where λ2 − 1 h(y, λ) = 1 + . sinh2 y + 1 Given some input value of l, the function h gives a quantative limit on e, depending on the radial coordinate y. For K ≤ 0, this is a lower limit dy since whenever dσ 6= 0, the square brackets in Eq.(40) must outweigh the derivative term as well in order for it to hold for every y. For spacelike dy geodesics however, Eq.(41) is superﬂuous for large values of dσ .

3.2.5 Geodesics in the second modification In sections 3.2.2-3.2.4, the behaviour of geodesics in the spacetime of the wormhole modification line element was analyzed. Here, geodesics in the spacetime of the second modification line element, Eq.(18) will be studied. The main focus of this section will be the differences between the two line elements, particularly how the second modification line elements effects what was seen in the wormhole modification line element. Naturally, only

29 geodesics with angular motion are of interest here. The Christoffel symbols and the Ricci scalar for this line element are found in the Appendix (section 5.2). The conserved angular momentum, l, is now defined as dφ (cosh2 y + λ2) = l. (42) dσ The second modification geodesic equation for y after elimination of the time coordinate becomes d2y sinh y cosh y dy K sinh y cosh y + ( )2 − + Ml2 = 0 dσ2 sinh2 y + λ2 dσ L2 (sinh2 y + λ2)(cosh2 y + λ2)2 (43) Streamplots of the second modification geodesic equation are shown in Fig.14. The first notion from these is that the effects of angular motion that the wormhole modification induced has seemingly been eliminated.

• λ no longer determines the sign for the third, angular, term in the geodesic equation.

• There are no longer timelike bound orbits around the wormhole.

• Lightlike geodesics no longer turn away from the wormhole, but they can oscillate through the wormhole for both λ < 1 (as before) and λ > 1.

• Spacelike geodesics behave largely as in the wormhole modiﬁcation.

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 14: Trajectories for from left to right time-, light- and spacelike geodesics in the spacetime of the second modfication line element, for λ = 1.5 and Ml2 = 4.5. The effects of angular motion seen in the wormhole modification has seemingly vanished.

30 Determining turning points in the second modification, the modified radial coordinate derivative, Eq.(29), is solved for y, giving the second modification turning point equation

∆ sinh4 y = = − (1 + 2λ2)∆ + M(e2 − L2l2) sinh2 y−(1+λ2)∆λ2−M (1 + λ2)e2 − L2l2λ2 , (44) where dy ∆ = K − L2( )2, dσ dy and dσ = 0 for a turning point. For timelike geodesics, this has a maximum of one solution which corresponds to the turning point of a geodesic which oscillates through the wormhole. As was the case with the wormhole modiﬁcation, the situation is a bit more complex for spacelike geodesics, as the turning point equation can have pair of solutions of which not both simultaneously can satisfy the modiﬁed geodesic energy condition

L2l2 K e2 ≥ (sinh2 y + λ2)( − ). (45) cosh2 y + λ2 M For λ > 1, however, spacelike geodesics only have single solutions to the turning point equation. For the turning points of a lightlike geodesic, ∆ = 0 and so

(1 + λ2)e2 − L2l2λ2 sinh2 y = − , (46) e2 − L2l2 which translates to L2l2λ2 ≤ e2 < L2l2, 1 + λ2 since 0 ≤ λ2/(1 + λ2) ≤ 1, Fig. 15. The modified energy condition, Eq.(45), graphically illustrated for λ = 1.5 in Fig.16, concur with these conclusions. There are no turning points for time- or lightlike geodesics other than those that suggests oscillation through the wormhole. This means no bound orbits around the wormhole for timelike geodesics and no points at which lightlike geodesics turn away from the wormhole and return to where it came from. In this sense lightlike geodesics are now more similar to timelike geodesics in the second modification rather than to spacelike geodesics as was the case with the wormhole modification. There are no qualitative differences between spacelike geodesics in the second and original modification.

31 l2

e2 1 2 3 4 5 Figure 15: (e2, l2) pairs for which lightlike geodesic have turning points. The corresponding pair in the shaded area inserted in the turning point equation, Eq.(44), dy gives the value of y for which dσ = 0.

e2 10

0 y 0.0 0.5 1.0 1.5 2.0 Figure 16: A comparison of the modiﬁed energy condition, Eq.(45), for timelike (dotted blue), lightlike (dotdashed red) and spacelike (dashed green) geodesics. There are no potential wells at which bound orbits can occur for timelike geodesics and y = 0 is no longer isolated from lightlike geodesics of low conserved energy.

32 3.3 Waves Section 3.2 dealt with the generalization of a free particle in classical mechanics to general relativity. In this section, the concept of waves will be generalized. Waves appear in many different forms, be it waves on water, an oscillating string or electromagentic waves in the form of light. In quantum mechanics, the wave function contains all the information about a certain particle (section 2.3), including the probability of finding the particle at a certain location at a certain time. In general relativity, the waves are influenced by the curvature of the spacetime. Thus ordinary derivatives are replaced by covariant derivatives ∂ → ∇ which takes into account both the change of the particle as well as that of the surrounding geometry in which the wave propagate. The wave equation for a scalar field, with mass m reads

(2 + m2)ψ = 0, (47)

µ where 2 = −∇µ∇ and natural units are employed. For the wormhole modiﬁcation line element, Eq.(17), this becomes

2ψ = µ −∇µ∇ ψ = µν −g ∇µ∇νψ = µν −g ∇µ(∂νψ) = µν γ 2 −g ∂µ∂ν − Γµν∂γ ψ = −m ψ

Due to the diagonality of the metric tensor, there is only contributions to the wave equation whenever µ = ν:

h tt 2 y yy 2 φφ 2 y i g (∂t − Γtt∂y) + g ∂y + g (∂φ − Γφφ∂y) ψ = = m2ψ, or writing out the expressions for the metric and Christoﬀel symbols

" 2 2 # −∂t ∂φ sinh y cosh y sinh y 1 + + + + ∂y ∂y ψ = M(sinh2 y + λ2) ML2 cosh2 y sinh2 y + λ2 cosh y L2 = m2ψ.

The equation separates using the ansatz ψ(t, y, φ) = T (t)Y (y)Φ(φ):

− 1 T 00 1 Φ00 1 sinh y cosh y sinh y d T + Φ + + + Y 0 = M(sinh2 y + λ2) ML2 cosh2 y YL2 sinh2 y + λ2 cosh y dy = m2,

33 where a prime refers to diﬀerentiation with respect to the function’s respec- tive arguments. The wave equation then ends up as three separate equations, the time- and angular coordinate equations 1 Φ00 = −l2, (48) Φ 1 T 00 = −ω2, (49) T and the radial wave equation

Y 00 + f(y)Y 0 + g(y)L2Y = 0, (50) where sinh y cosh y sinh y f(y) = + , sinh2 y + λ2 cosh y and ω2 l2 g(y) = − − m2. M(sinh2 y + λ2) ML2 cosh2 y Eqs. (48)-(49) are standard diﬀerential equations with solutions

Φ(φ) = Aφ cos(lφ) + Bφ sin(lφ), (51)

T (t) = At cos(ωt) + Bt sin(ωt). (52) The radial wave equation has not been analytically solved. A ﬁrst approach to the radial wave equation is the behaviour at the limits. At large and small y it can be approximated to yield a solution

κy −κy Y (y) = Aye + Bye , (53) where κ is determined by the limits y → ∞ or y → 0 of the functions f and g. Close to the origin, depending on the values of ω and l, this can result in an oscillating solution.

3.3.1 Variable substitution One approach to an analytic solution of the radial wave equation, Eq.(50), is variable substitution. The goal is to rewrite it on the familiar ”Schr¨odinger form”, thus revealing a potential to be analyzed. Let y = y(x) such that Y = Y (y(x)), then dY dx dY = , dy dy dx and d2Y d2x dY dx2 d2Y = + . dy2 dy2 dx dy dx2

34 After insertion and rearranging, Eq.(50) becomes

dx2 d2Y d2x dx dY + + f + L2gY = 0. dy dx2 dy2 dy dx

d2x dx x(y) is determined to satisfy dy2 + f dy = 0. This gives  √  2 C1 −1  1 − λ sinh y  x(y) = C2 + √ tan , (54) 2 q 2 − 2λ  −(sinh2 y + λ2) by which y = y(x) is implicitly deﬁned. x(y) is everywhere an imaginary function. A real valued function can be obtained by assuming λ > 1, then √ ( 2 ) C1 −1 λ − 1 sinh y x(y) = −√ tan p . (55) 2λ2 − 2 sinh2 y + λ2

On the other hand, if λ < 1, then √ ( 2 ) C1 −1 1 − λ sinh y x(y) = −√ tanh p . (56) 2 − 2λ2 sinh2 y + λ2

Both solves the determining condition for x(y). The radial wave equation is now of ”Schr¨odingerform”:

d2Y 2 2 + 2 [E − U(y(x))]Y = 0, (57) dx C1 M where E = ω2L2 − λ2(l2 + L2m2M), (58) is energy, and

U(y(x)) = (l2−ω2L2+L2m2M(1+λ2)) sinh2 y(x)+L2m2M sinh4 y(x), (59) is the potential, and √ λ2 tan2[ 2λ2−2 x] 2 C1 sinh y = √ , λ > 1, (60) λ2 − 1 − tan2[ 2λ2−2 x] C1 √ λ2 tanh2[ 2−2λ2 x] 2 C1 sinh y = √ , λ < 1. (61) 1 − λ2 − tanh2[ 2−2λ2 x] C1 Notice that for λ > 1, the entire radial axis in y is compressed within the range −C π C π √ 1 < x < √ 1 . 2 2λ2 − 2 2 2λ2 − 2

35 Close to the origin, the correspondence between x and y is linear but further away, small changes in x results in large changes in y. For λ < 1, x takes on any value on the entire real axis and close to the origin the relation is again linear. But now, conversely, further away from the origin, large changes in x is required to make small changes in y. Eqs.(61)-(60) both has singularities originating in the denominator, for λ > 1 whenever ±C p x = √ 1 tan−1[ λ2 − 1], 2λ2 − 2 and for λ < 1 whenever ±C p x = √ 1 tanh−1[ 1 − λ2]. 2 − 2λ2

These two however, both corresponds to some value y where sinh2 y >> λ2, so since it has already been noted that λ is only important at small y, for purposes here it is suﬃcient to only consider the regions contained between these two singularities. In the subsequent analysis of the potential, only those cases in which the wave energy condition is satisﬁed is considered. The energy condition reads

U(x) − E < 0, (62)

and is a requirement in order to have wave like solutions to the wave equation, Eq.(57). The shape of the potential is similar to that of the ”parabola”, varying the parameters ω, m and l it takes three diﬀerent forms:

• an inverted parabola,

• a regular parabola,

• a parabola with two sunken cavities on each side of the symmetry axis.

U(x)-E U(x)-E

100

50 15 000

x -0.4 -0.2 0.2 0.4 10 000 -50

-100 5000

-150 x -200 -1.0 -0.5 0.5 1.0

-250 -5000 Figure 17: The inverted and regular ”parabola” shape of the potential. In the left ﬁgure, λ > 1 and ω > l > m = 0 and in the right ﬁgure, λ < 1 and ω > m > l.

36 The ﬁrst and second case is shown graphically in Fig.17 while the third case is shown in Fig.18. How the variation of the parameters aﬀects the form of the potential is an intricate matter. Some particular notes can be listed:

• For λ > 1, it is required that ω > l in order to satisfy the energy condition.

• It is possible to have four intersections with the horizontal curve, suggesting the existence of bound, localized waves. This happens for λ > 1 and ω > l > m > 0. For λ < 1 a similar form of the potential can be achieved, but not that it intersects the horizontal axis more than twice.

• The parabola is similar to the harmonic oscillator potential, suggesting global standing waves.

• The inverted parabola form can be interpreted as a potential for some- thing similar to a wave packet, analogous to a free particle in the Schr¨odingerequation. This is consistent with the large y approximation, Eq.(53), which suggested exponential solutions when y → ∞.

U(x)-E U(x)-E

1000 x -1.0 -0.5 0.5 1.0

500 -500

x -0.4 -0.2 0.2 0.4 -1000 -500 Figure 18: The potential U(x) − E where E is energy for the case ω > l > m > 0 and λ > 1 (left) and λ < 1 (right). It is possible in the left case to have locally bound, oscillatory states.

The results indicate that for λ > 1, there can be solutions to the radial wave equation which are localized to certain radial regions of the spacetime, as well as global solutions. For λ < 1, there are only global solutions. A crude comparison with the trajectories of geodesics, explored in section 3.2, is largely consistent. For λ > 1 it was possible to have localized orbits, either bound or open, while in the case of λ < 1 the trajectories of the geodesics were global.

37 For completeness, the corresponding procedure for λ = 0 is recapitulated here. Eq.(54) then becomes

C sinh y x(y) = 1 log[ ], (63) 2 cosh y and the radial wave equation on ”Schr¨odinger form” becomes

d2Y 4 2 + 2 [E − U(x)] = 0, (64) dx C1 M where E = ω2L2, is energy, and

U(y(x)) = (L2m2M + l2 − ω2L2) sinh2 y(x) + L2m2M sinh4 y(x)

is the potential, and −e4x/C1 sinh2 y = . (65) e4x/C1 − 1 There is an inverse relation between x and y in this case. When y → ∞, x → 0 and the position of the black hole is at x = −∞. The potential is asymptotic to the horizontal axis in the negative x direction and singular close to the origin, Fig.19. The singularity could be interpreted as the AdS boundary discussed in section 2.2. The shape of the potential is not compatible with any oscillatory, bound, waves.

U(x)-E U(x)-E 100 100

60 50 40

20 x x -2.0 -1.5 -1.0 -0.5 -2.0 -1.5 -1.0 -0.5 -20 -50 -40

Figure 19: The potential U(x) − E where E is energy for λ = 0. In the left ﬁgure ω ≈ l ≈ m and in the right ﬁgure ω > l > m > 0. It is not possible to adjust the curve to intersect the horizontal axis in any other way. Note that in this graph x = 0 corresponds to y → ∞ and the black hole (y = 0) is situated at x = −∞.

38 3.4 Summary In section 3.1 it was shown that the wormhole modification line element of the BTZ black hole, Eq.(17) inserted in the Einstein field equations did not generate any solutions that were applicable to the standard perfect fluid form of the stress-energy tensor. Neither did the second modification line element, Eq.(18) generate any such solutions. In section 3.2 the geodesic equation was solved for radial and general geodesics as well as for geodesics in the second modification. The results are summarized in Table 1 for particular initial values y > 0 and y0 < 0. If y0 > 0, the results are the same for timelike geodesics however light- and spacelike geodesics will always escape to infinity.

λ Time Light Space 0 Plunge Plunge Plunge Radial 0.5 Oscillating Passing Passing 1.5 Oscillating Passing Passing 0 Plunge Plunge Plunge General 0.5 Oscillating Passing/Oscillating Passing/Oscillating 1.5 Oscillating/Bound Passing/Open Passing/Open 0.5 Oscillating Passing/Oscillating Passing/Oscillating 2ndmod. 1.5 Oscillating Passing/Oscillating Passing

Table 1: A summary of the orbital trajectories of the geodesics with initial values y > 0 and y0 < 0. A plunge orbit refers to when the geodesic travels straight into the black hole (λ = 0), a passing orbit refers to when the geodesic travels through the wormhole λ > 0 and onto infinity on the other side, an oscillating orbit refers to when the geodesic oscillates back and forth through the wormhole; a bound orbit refers to a circular or elliptical orbit around the wormhole and an open orbit refers to a hyperbolic path where the geodesic approaches the wormhole from infinity, rounds it and returns to infinity.

Bound and open orbits are characterized by their state of lower conserved energy in comparison with other orbits. Generally in terms of energy, spacelike geodesics can have lower energy than lightlike geodesics, which in turn can have lower energy than timelike geodesics. For a geodesic of low energy then, it is possible to exclude it as either time- or lightlike. The most significant changes that the second modification induces in comparison with the wormhole modification is that the effects of angular motion that the latter induced are eliminated. Due to lightlike geodesics now oscillating for λ > 1, they are somewhat more similar to timelike geodesics, rather than spacelike geodesics. In section 3.3 the wave equation for a massive scalar was studied. Through

39 a variable substitution it could be written on ”Schr¨odingerform” under which the potential could be studied. The prime result was that it was possible to have bound localized states whenever λ > 1, as well as global states. In the case of λ < 1, only global states were possible. A rough comparison with the trajectories of geodesics could be made, where local trajectories (both bound and open) were possible for λ > 1, but only global trajectories was possible when λ < 1.

40 4 Conclusions

The objective of this thesis was to study how the addition of the wormhole parameter λ affected the spacetime geometry of the BTZ black hole. This was done by examining geodesic and wave behaviour of the wormhole modification line element, Eq.(17). An additional analysis was done when the wormhole parameter was added everywhere in the line element, Eq.(18). Differences were typically arranged according to λ < 1 and λ > 1. Arguably, the most intriguing case would be for a small parameter correction for λ, since the wormhole solution, as departing from a black hole solution, could then be attributed to, for example, quantum mechanical corrections near the event horizon. The most distinguishing feature, however, ocurred for when λ > 1. The typical geodesic trajectory when in the presence of a black hole was plunge orbits. This was true for every timelike geodesic while light- and spacelike geodesics could potentially escape the black hole to infinity given proper initial values. For λ > 1, however, the geodesics had the possibility of orbiting the wormhole. Timelike geodesics could stay in circular or elliptical orbits around the wormhole while light- and spacelike geodesics could stay in open orbits where they approached the wormhole, but turned around before they reached it and disappeared to infinity from where they came before. These kinds of orbits would be fairly easy to recognize as they differ significantly from the black hole plunge orbits. In the case of 0 < λ < 1, the geodesics oscillated, back and forth, through the wormhole. As orbits, these are not as recognizable. Assuming it would not be possible to monitor the geodesic when it is on the other side of the wormhole, recognizing this behaviour, although not impossible, would be an arguably more intricate matter than for larger values of λ. In the case of the second modification the typical, recognizable bound orbits for λ > 1 vanished, meaning that in the context of identifying the presence of a black hole or a wormhole, the original wormhole modification would be preferable. A similar situation ocurred for waves. There was no possibility of bound states when in the situation of a black hole. Any wave would disappear into the black hole. With a wormhole present, it was possible to have bound states with a potential similar to the potential of the harmonic oscillator. Particularly when λ > 1 and the wave parameters were correctly adjusted it was possible to have these bound states localized within certain radial regions separated from the wormhole. The potential would then have the shape of a parabola with two sunken cavities on each side of the symmetry axis. Waves were not explored for the second modification. Neither the wormhole modification, nor the second modification spacetime geometry could be generated by a perfect fluid ansatz of the stress- energy tensor, thus the wormhole universe in AdS space differs somewhat from the most common, simple, real world models of the matter energy

41 distribution that are in use. Contingencies left from this thesis is whether or not the above behaviour is exclusive for the existence of a wormhole. Any limitations on the parameter λ are left undetermined, in particular if there is a lower limit on λ for the above eﬀects to still occur. Possibly this would be a task within a numerically oriented work. A continuation of this thesis would also see the bound states thorougly analyzed, in particular the relation of the bound states between geodesics and waves. It is not unreasonable to expect that there is a connection between the wave parameters ω and l and the geodesic conserved quantitites e and l. A solution of the wave equation for some given value of the wave parameters corresponding to some value of the geodesic parameters for which the behaviour is known could turn out fruitful for the otherwise crude comparison between geodesics and waves done in this thesis. The limit m → 0 is a reasonable beginning as light, as is well known, is waves. In this re- gard, another stress-energy tensor ansatz, that of the scalar ﬁeld, could be of particular interest to test as a source of curvature.

42 5 Appendix

5.1 The Einstein tensor

Starting from a given metric tensor gµν, the Christoffel symbols are defined as gγδ Γγ = (∂ g + ∂ g − ∂ g ) . (66) µν 2 µ δν ν µδ δ µν From the Christoffel symbols the Ricci curvature is obtained

γ γ γ δ γ δ Rµν = ∂γΓµν − ∂νΓµγ + ΓµνΓγδ − ΓµδΓνγ, (67) and by contraction the Ricci scalar

µν R = g Rµν. (68)

The Einstein tensor is then deﬁned 1 G = R − g R. (69) µν µν 2 µν For the wormhole modiﬁcation line element of the BTZ black hole, Eq.(17), the Einstein tensors read: −1 Gtt = , (70) ML2(sinh2 y + λ2)

sinh4 y + 2λ2 sinh2 y + λ2 Gφφ = , (71) ML4 cosh2 y(sinh2 y + λ2)2 sinh2 y Gyy = . (72) L4(sinh2 y + λ2) αβ They are consistent with the contracted Bianchi identity, ∇βG = 0.

5.2 Christoffel symbols and the Ricci scalar Below is the Christoffel symbols for the wormhole modification line element

ds2 = −M(sinh2 y + λ2) dt2 + L2dy2 + ML2 cosh2 y dφ2, listed as well as the Ricci tensors and scalars. The Christoﬀel symbols are

 Γt = Γt = sinh y cosh y ;  ty yt sinh2 y+λ2  Γy = M sinh y cosh y; tt L2 (73) Γy = −M sinh y cosh y;  φφ  φ φ sinh y Γφy = Γyφ = cosh y .

43 The components of the Ricci tensor are

 2 2 R = M 2 sinh2 y + λ cosh y ;  tt L2 sinh2 y+λ2  2 2 2 2 4  R = −1 − λ cosh y+λ sinh y+sinh y ; yy (λ2+sinh2 y)2 (74) M cosh2 y(−1+λ2+cosh 2y)  R = − ;  φφ λ2+sinh2 y   α 6= β → Rαβ = 0.

The Ricci scalar is −9 + 3λ2 − 2λ4 + (3 − 5λ2) cosh(2y) − 3 cosh(4y) R = 4 4 (75) L2(sinh2 y + λ2)2

−6 3 which for λ = 0, is a scalar R = L2 , as it should be given AdS . Furthermore −6 −2 1 it behaves as R → L2 as y → ∞ and R → L2 λ2 + 1 for y → 0. For the second modiﬁcation line element

ds2 = −M(sinh2 y + λ2) dt + L2dy2 + ML2(cosh2 y + λ2) dφ2, one of the Christoﬀel symbols is altered cosh y sinh y Γφ = Γφ = , (76) φy yφ cosh2 y + λ2 and the Ricci scalar becomes

−1 1 1 R = 9 + 208λ4+ 4L2 (−1 + 2λ2 + cosh 2y)2 (1 + 2λ2 + cosh 2y)2 +8λ2(11 + 32λ4) cosh 2y + 4(−3 + 44λ4) cosh 4y + 40λ2 cosh 6y + 3 cosh 8y (77)

−6 −6 which also is a scalar R = L2 for λ = 0, and it goes like R → L2 as y → ∞, −2(1+2λ2) and R → L2λ2(1+λ2) as y → 0.

44 5.3 Image archive In this section, additional plots related to the analysis of geodesics, section 3.2 are shown. Some ﬁgures has already been presented as part of the main thesis but are shown here again for the sake of wholeness. In all ﬁgures, L = 1, M = 1 and K = −1 for timelike, K = 0 for lightlike and K = +1 for spacelike geodesics.

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 20: Trajectories for radial timelike geodesics for λ = 0, 0.5 and 1.5. For λ = 0 the geodesic plunges into y = 0. For λ 6= 0 the geodesics can oscillate back dy and forth through the wormhole. The change in dσ at the passing of the wormhole is less dramatic for the larger value of λ.

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 21: Trajectories for radial lightlike geodesics for λ = 0, 0.5 and 1.5. For λ = 0 the geodesic either plunges into y = 0 or travels to inﬁnity while for λ 6= 0 the geodesic passes through the wormhole and onto inﬁnity on the other side. The dy change in dσ at the passage of the wormhole is less dramatic for a larger value of λ.

45 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 22: Trajectories for radial spacelike geodesics for λ = 0, 0.5 and 1.5. The behaviour is similar to that of lightlike geodesics. Note that the erractic behaviour within the horizontal stripe −1 < y0 < 1 must be attributed to some graphical 2 dy 2 interpretation. There is a lower limit L ( dσ ) < K for radial spacelike geodesics.

y(σ) y(σ)

10 3

2 5 1

σ σ -4 -2 2 4 -4 -2 2 4 -1 -5 -2

-3 -10

Figure 23: Numerical solutions to the radial geodesic equation, Eq.(33) for timelike 0 geodesics, λ = 1.5 and initial values (y0, y0) = (3,1) and (10,1). Notice that the curve is more steep at y = 0 whenever initial value y0 increases. For larger, yet ﬁnite, initial values still, the numerical solution will eventually break down. This, however, should be an instability in the numerical solution rather than a physical singularity.

46 y(σ) y(σ)

4 10

2 5

σ σ -4 -2 2 4 -4 -2 2 4

-2 -5

-4 -10

Figure 24: Numerical solutions to the radial geodesic equation, Eq.(33) for lightlike 0 geodesics, λ = 1.5 and initial values (y0, y0) = (3,1) and (10,1). The curve is more steep at y = 0 whenever initial value y0 increases. Note that the curve ﬂattens at the extremities, lightlike geodesics travels towards inﬁnity with constantly decreasing dy dσ .

y(σ) y(σ)

8 15

6 10

4 5 2 σ σ -4 -2 2 4 -4 -2 2 4 -5 -2

-4 -10

Figure 25: Numerical approximations to the radial geodesic equation, Eq.(33) for 0 spacelike geodesics, λ = 1.5 and initial values (y0, y0) = (3,1.1) and (10,1.1). The curve is more steep at y = 0 whenever initial value y0 increases. Note that the curve is linear at the extremitites, spacelike geodesics travels towards inﬁnity with dy constant dσ .

47 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 26: Trajectories for general timelike geodesics for from left to right λ = 0, 0.5, and 1.5 and from top to bottom l2 = 0.5, 1.5 and 4.5. If λ < 1 the geodesics behaves as if radial but if λ > 1 they can stay in bound orbits around the wormhole.

48 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 27: Trajectories for general lightlike geodesics for from left to right λ = 0, 0.5 and 1.5, and from top to bottom l2 = 0.5, 1.5 and 4.5. Contrary to the radial case, lighlike geodesics can now have turning points, either oscillating back and forth through the wormhole for λ < 1 or turning around before reaching the wormhole and return to where it came from for λ > 1.

49 4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y

4 4 4

2 2 2 y′ 0 y′ 0 y′ 0

-2 -2 -2

-4 -4 -4

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

y y y Figure 28: Trajectories for general spacelike geodesics for from left to right λ = 0, 0.5 and 1.5 and from top to bottom l2 = 0.5, 1.5 and 4.5. Contrary to the radial case, spacelike geodesics can now have turning points, either oscillating back and forth through the wormhole for λ < 1 or turning around before reaching the wormhole and return to where it came from for λ > 1. The horizontal stripe, a forbidden region for radial geodesics, is now permitted.

50 y(σ) y(σ)

4 4

2 3

σ 2 -20 -10 10 20

-2 1

σ -4 -20 -10 10 20 Figure 29: Numerical solutions to the general geodesic equation, Eq.(36) for lightlike 2 0 geodesics, λ = 1.5, l = 4.5 and initial values (y0, y0) = (0.5,0.1) and (0.5,1) which represents a lightlike geodesic that turns away from the wormhole, and one that passes through the wormhole.

e2 e2 e2 10 10 20

8 8 15 6 6 10 4 4 2 2 5 0 y 0 y 0 y 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

e2 e2 e2 5 5 14 4 4 12 10 3 3 8 2 2 6 4 1 1 2 0 y 0 y 0 y 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

e2 e2 e2 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 y 0 y 0 y 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2 2 2 L2l2 K 2 Figure 30: Plot of el ≥ (sinh y +λ )( cosh2 y − M ) for l = 4.5 and from left to right λ = 0, 0.5, and 1.5 and from top to bottom time-, light- and spacelike geodesics. 2 dy The value of e must be in the shaded regions in order for a real valued dσ . These correspond to the third rows in Figs. 26-28.

51 4 4

2 2

y′ 0 y′ 0

-2 -2

-4 -4

-4 -2 0 2 4 -4 -2 0 2 4

y y

4 4

2 2

y′ 0 y′ 0

-2 -2

-4 -4

-4 -2 0 2 4 -4 -2 0 2 4

y y

4 4

2 2

y′ 0 y′ 0

-2 -2

-4 -4

-4 -2 0 2 4 -4 -2 0 2 4

y y Figure 31: Trajectories for the second modification for Ml2 = 4.5, for from left to right λ = 0.5 and 1.5 and from top to bottom time-, light- and spacelike geodesics. The second modification erases the effects of angular motion that the wormhole modificaton induced. There are no bound, non-oscillating, orbits for timelike geodesics around the wormhole and no open, hyperbolic orbits for light- and spacelike geodesics. Lightlike geodesics can now, however, oscillate back and forth through the wormhole when λ > 1 in contrast to the wormhole modification case. 52 e2 e2 10 10

8 8

6 6

4 4

2 2

0 y 0 y 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

e2 e2 10 10

8 8

6 6

4 4

2 2

0 y 0 y 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

e2 e2 10 10

8 8

6 6

4 4

2 2

0 y 0 y 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2 2 2 L2l2 K 2 Figure 32: Plot of el ≥ (sinh y + λ )( cosh2 y+λ2 − M ) for l = 4.5, from left to right λ = 0.5 and 1.5 and from top to bottom time-, light- and spacelike geodesics. 2 dy The value of e must be in the shaded regions in order for a real valued dσ . These correspond to each row in Fig. 31.

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