Spacetime geometry of a wormhole modiﬁed BTZ black hole

(1)

Spacetime geometry of a wormhole

modified BTZ black hole

David Sundelin Supervisor Bo Sundborg

Stockholm University October 26, 2018

(2)

(3)

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.

(4)

(5)

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 be- ginning 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.

Figure 1: A vizualisation of curved spacetime; in the outskirts spacetime is flat, at the center some heavy object curves it causing the effect of 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 the fusion of the three spatial directions and time into

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

(8)

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.

(9)

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

ds²= g_µνdx^µdx^ν (1)

where g_µν 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 g_µν = η_µν where

η_µν =







−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1







. (2)

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 Einstein tensor¹ 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 T_µν 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

(10)

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 g_µν 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 (T_µν = 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

x²₁+ x²₂+ ... + x²_d+ x²_d+1=

d+1

X

i=1

x²_i = L², (5) 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

−(x⁰)²+

d

X

i=1

(xⁱ)²= L², (6)

where xⁱ are spatial coordinates. Introducting another time coordinate and flipping the sign on the right hand side we obtain AdS spacetime

−(x⁰)²+

d−1

X

i=1

(xⁱ)²− (x^d)²= −L². (7) 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, x² = r sin θ, x³ = ρ sin t.

(10)

(11)

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:

ds²= −(L²+ r²)dt²+ L²

L²+ r²dr²+ r²dθ², (13) 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:

dt = ±L L²+ r²dr.

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

(12)

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

ds²= −r²− M L²

L² dt²+ L²

r²− M L²dr²+ r²dφ², (14) 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

t = v − 2L

√M tanh⁻¹[ r L√

M], gives the line element

ds²= −r²− M L²

L² dv²+ 2dvdr + r²dφ². For a radial light ray this gives

r²− M L²

L² dv = 2dr,

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

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

(13)

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 =√

M L cosh y, (15)

with the modification

sinh²y → sinh²y + λ², (16) gives the wormhole modification line element

ds²= −M (sinh²y + λ²) dt²+ L²dy²+ M L²cosh²y dφ². (17) The event horizon, located at r² = M L², 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 sin- gularities, 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 cosh²y − sinh²y = 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.

(14)

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, Λ = −L⁻²). Given T_µν = 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 d˜t = dt/L, the wormhole modification line element can be written in dimensionsless coordinates

d˜s²= −M (sinh²y + λ²) d˜t²+ dy²+ M cosh²y dφ²,

where d˜s² = ^ds_L2². 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 (sinh²y + λ²) dt²+ L²dy²+ M L²(cosh²y + 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)

κT^φφ = λ²(1 − λ²)

M L⁴cosh²y(sinh²y + λ²)², (20)

(15)

κT^yy= −λ²

L⁴(sinh²y + λ²). (21) 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 ans¨atze.

A dust universe ansatz is given by T^αβ = µu^αu^β, where µ is rest mass density, and u^α is the three velocity. Naturally, we must have u^t 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)u^αu^β+ g^αβp 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 T_µν, either two of the components of the three velocity u^αmust equal zero or p = −ρ. If only one component of u^α is non-zero this will result in either the solution being valid only for a particular (imaginary) value of y given by sinh²y = −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:

κT^tt = λ²(1 + λ²)

M L²(sinh²y + λ²)(cosh²y + λ²)², (22) κT^φφ= λ²(1 − λ²)

M L⁴(cosh²y + λ²)(sinh²y + λ²)², (23) κT^yy= 8λ²(λ²+ cosh 2y)

L⁴(−1 + 8λ⁴+ 8λ²cosh 2y + cosh 4y). (24) 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 ans¨atze. 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

(16)

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

d²x^α

dσ² = −Γ^α_βγdx^β dσ

dx^γ

dσ , (25)

where σ is some affine parameterization of the geodesic and Γ^α_βγ 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

(sinh²y + λ²) dt

dσ = e, (26)

cosh²y dφ

dσ = l, (27)

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, ξ^α = (1, 0, 0), related to the conservation of energy e, and η^α= (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

dσ² + M sinh y cosh y

"

1 L²

dt dσ

2

− dφ dσ

2#

= 0. (28)

(17)

It has not been analytically solved for an expression y = y(σ). What has been obtained is the radial coordinate derivative

dy dσ = ±1

L s

K + M

e²

sinh²y + λ² − L²l² cosh²y

. (29)

Here, K is a constant which equals the norm of a tangent vector u = (_dσ^dt,^dy_dσ,^dφ_dσ), that is being parallel transported along the geodesic:

u · u = gαβu^αu^β = 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

dy dσ →











±_L¹ r

K + M

e²

λ² − R²l²

, y → 0;

±_L¹√

K, y → ∞ ∧ K 6= 0;

±_L² exp(−y)pM(e²− R²l²), y → ∞ ∧ K = 0.

. (31)

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

e²≥ (sinh²y + λ²) ( L²l² cosh²y − K

M). (32)

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 obtained by solving Eq.(28) for ^d_dσ²^y2, which is a function of y and ^dy_dσ. The vector field (^dy_dσ,^d_dσ²^y2) 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).

(18)

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

dσ² +sinh y cosh y sinh²y + λ²

"

dy dσ

2

− K L²

#

= 0. (33)

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 values (y₀, y⁰₀) = (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 itself in _dσ^dy → ±∞ when approaching y = 0.

• For timelike geodesics, _dσ^d²^y₂ 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 with decreasing ^dy_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.

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

(19)

-4 -2 0 2 4 -4

-2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

-4 -2 0 2 4

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.

-4 -2 2 4

σ

-3 -2 -1 1 2 3

y(σ)

-4 -2 2 4 σ

-4 -2 2 4

y(σ)

-4 -2 2 4

σ

-6 -4 -2 2 4 6 8

y(σ)

Figure 3: Numerical solutions to the radial geodesic equation for from left to right time-, light- and spacelike geodesics for λ = 0.5 and initial values (y0, y₀⁰) = (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 insta- bility 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

y = sinh⁻¹ (

±

s M e²

L²(^dy_dσ)²− K − λ² )

. (34)

If ^dy_dσ = 0, which is required for the geodesic to turn around, then the radial turning point equation is given

y = sinh⁻¹ (

±

r−(M e²+ Kλ²) K

)

. (35)

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

(20)

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 real valued radial coordinate, a spacelike geodesic must fulfill L²(^dy_dσ)² > K.

For a lightlike geodesic ^dy_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 geodesic, K < 0, and whenever y → ∞, then _dσ^dy → √

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

dy dσ = ±1

L r

K + M e² 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 lightlike geodesics _dσ^dy → 0 and spacelike geodesics reaching constant radial coordinate derivative, ^dy_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

dt = ± L

√ M

dy psinh²y + 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.

(21)

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

dσ²+sinh y cosh y sinh²y + λ²

"

dy dσ

2

− K L²

#

+ sinh y

(sinh²y + λ²) cosh³y 1 − λ² M l² = 0.

(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).

(22)

-4 -2 0 2 4 -4

-2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

Figure 4: Trajectories for timelike geodesics, for M l² = 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 -2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

Figure 5: Trajectories for lightlike geodesics, for M l² = 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 sinh⁴y = −1

∆ (1 + λ²)∆ + M (e²− L²l²) sinh²y + M (e²− L²l²λ²) + ∆λ² (37) where

∆ = −L² dy dσ

2

+ K.

For a turning point it is required that ^dy_dσ = 0, giving the general turning point equation

sinh⁴y = −1

K (1 + λ²)K + M (e²− L²l²) sinh²y + M (e²− L²l²λ²) + Kλ² . (38)

(23)

-4 -2 0 2 4 -4

-2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

Figure 6: Trajectories for spacelike geodesics, for M l² = 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) = sinh⁴y, g(y) = −1

K (1 + λ²)K + M (e²− L²l²) sinh²y + M (e²− L²l²λ²) + Kλ² , 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, M l² = 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.

(24)

0.5 1.0 1.5 2.0 2.5 3.0y

-3 -2 -1 0 1 2 3 f/g

Figure 7: Graphical solution to the general turning point equation, Eq.(38), for timelike geodesics, λ = 1.5, M l² = 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.

-10 -5 5 10

σ

-2 -1 1 2 y(σ)

-10 -5 5 10

σ

-2 -1 1 2 y(σ)

-10 -5 5 10

σ

-2 -1 1 2 y(σ)

Figure 8: Numerical solutions to the general geodesic equation for timelike geodesics when λ = 1.5, M l²= 4.5 and initial values from left to right (y0, y⁰₀) =(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

M e² = (M L²l²− 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

e² ≥ λ²(L²l²− K M),

(25)

the right hand side of Eq.(38) will always be positive, growing as sinh²y.

Since the left hand side grows as sinh⁴y, 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.

0.5 1.0 1.5 2.0 2.5 3.0y

-3 -2 -1 0 1 2 3 f/g

Figure 9: Graphical solution to the general turning point equation, Eq.(38), for spacelike geodesics, λ = 1.5, M l² = 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.

(26)

-10 -5 5 10 σ

-4 -2 2 4

y(σ)

-10 -5 5 10

σ

-4 -2 2 4 y(σ)

Figure 10: Numerical solutions to the general geodesic equation for spacelike geodesics when λ = 1.5, M l² = 4.5 and initial values (y0, y₀⁰) = (1,0) and (1,1.2).

This is a geodesic in an open, hyperbolic orbit, and a geodesic that passes the wormhole with a slight decrease in ^dy_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

sinh²y

sinh²y + 1

K (1 + λ²)K + M (e²− L²l²)

= 0.

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

y = sinh⁻¹ (

±

rL²l²λ²− e² e²− L²l²

)

. (39)

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.

(27)

1 2 3 4 5 e² 2

4 6 l²

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) gives the value of y for which ^dy_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 one relation between e and l. Expressing this relation as a function e²_l = e²_l(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

(28)

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).

0 1 2 3 4

0 y 5 10 15 20 25 30 e2

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 in Fig.13. Here, the value of e²_l is solved for from the radial coordinate derivative, Eq.(29) and plotted against y and ^dy_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.

(29)

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

y y ′

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

y y ′

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

y y ′

Figure 13: A contourplot of the value of e²_l 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 :

K = L² dy dσ

2

+ M

L²l²

cosh²y − e² sinh²y + λ²

. (40)

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,

e²< L²l²h(y, λ), K > 0. (41) where

h(y, λ) = 1 + λ²− 1 sinh²y + 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 since whenever ^dy_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 geodesics however, Eq.(41) is superfluous for large values of _dσ^dy.

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

(30)

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 (cosh²y + λ²)dφ

dσ = l. (42)

The second modification geodesic equation for y after elimination of the time coordinate becomes

d²y

dσ² +sinh y cosh y sinh²y + λ²

(dy

dσ)²− K L²

+ sinh y cosh y

(sinh²y + λ²)(cosh²y + λ²)²M l²= 0 (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 -2 0 2 4

y y ′

-4 -2 0 2 4

y y ′

-4 -2 0 2 4

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 M l² = 4.5.

The effects of angular motion seen in the wormhole modification has seemingly vanished.

(31)

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

∆ sinh⁴y =

= −(1 + 2λ²)∆ + M (e²− L²l²) sinh²y−(1+λ²)∆λ²−M(1 + λ²)e²− L²l²λ² , (44) where

∆ = K − L²(dy dσ)², and _dσ^dy = 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

e² ≥ (sinh²y + λ²)( L²l²

cosh²y + λ² − K

M). (45)

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 sinh²y = −(1 + λ²)e²− L²l²λ²

e²− L²l² , (46)

which translates to

L²l²λ²

1 + λ² ≤ e²< L²l², 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.

(32)

1 2 3 4 5 e² 5

10 15 20 25 l²

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), gives the value of y for which ^dy_dσ = 0.

0.0 0.5 1.0 1.5 2.0 y

0 2 4 6 8 10 e2

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.

Spacetime geometry of a wormhole modiﬁed BTZ black hole