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The In�uence of Black Holes on Light Ray
Trajectories
Conrad Brown
April 26, 2011
1
Declaration of Authorship
This piece of work is a result of my own work except where it forms anassessment based on group project work. In the case of a group project, thework has been prepared in collaboration with other members of the group.
Material from the work of others not involved in the project has beenacknowledged and quotations and paraphrases suitably indicated.
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Abstract
A discussion of light trajectories in Schwarzschild and Kerr geometry withparticular emphasis placed on black holes. These fall into three categories;Plunge, Scatter and unstable circular trajectories. The type of trajectory
depends on the impact parameter (b) of the light ray, the mass (M) and theangular momentum (J) of the black hole. Only trajectories external to the
event horizon, and starting a large distance from the black hole are considered.
3
Acknowledgments
Firstly I would like to thank my supervisors Prof. Ruth Gregory and Prof.Simon Ross for their invaluable advice and guidance with both the subjectmatter and the technicalities in producing an academic report. I would alsolike to thank Mr Chris Brown for being my test third year maths student.Finally I would like to thank my family for o�ering proofreading help.
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Contents
1 Introduction 7
1.1 A Note on Notation . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Non-Euclidean Geometries 8
2.1 The Metric and Line Element . . . . . . . . . . . . . . . . . . . . 82.2 Scalar Products in Non-Euclidean Geometries . . . . . . . . . . . 9
3 Special Relativity Extended 10
3.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.1 Timelike Geodesics . . . . . . . . . . . . . . . . . . . . . 113.3.2 Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Schwarzschild Geometry 14
4.1 Finding the Schwarzschild Metric . . . . . . . . . . . . . . . . . . 144.2 Singularities and The Event Horizon . . . . . . . . . . . . . . . . 164.3 Null Geodesics In Schwarzschild Geometry . . . . . . . . . . . . . 174.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.5 E�ective potential for Schwarzschild Black Holes . . . . . . . . . 20
4.5.1 Interpreting b . . . . . . . . . . . . . . . . . . . . . . . . . 214.5.2 Plotting Weff . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.3 Plunge Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.4 Scatter Orbits . . . . . . . . . . . . . . . . . . . . . . . . 244.5.5 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Gravitational Lensing 27
5.1 Calculating the Angle of De�ection . . . . . . . . . . . . . . . . . 305.1.1 Calculating Small De�ections . . . . . . . . . . . . . . . . 315.1.2 Light De�ection by the Sun . . . . . . . . . . . . . . . . . 325.1.3 The Thin Lens Approximation . . . . . . . . . . . . . . . 33
6 Light Orbits in the Equatorial Plane of Kerr Black Holes 34
6.1 Kerr Metric and Line Element . . . . . . . . . . . . . . . . . . . 356.2 Boyer-Lindquist Coordinates . . . . . . . . . . . . . . . . . . . . 356.3 Singularities in Kerr Geometry . . . . . . . . . . . . . . . . . . . 366.4 Symmetries of Kerr Geometry . . . . . . . . . . . . . . . . . . . . 376.5 E�ective Potential for Kerr Black Holes . . . . . . . . . . . . . . 39
6.5.1 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . 416.5.2 Prograde vs. Retrograde . . . . . . . . . . . . . . . . . . 436.5.3 Variation in Angular Momentum per Unit Mass . . . . . 44
7 Conclusion 46
A Geodesic Equation Solver 47
5
B Light Orbits in Schwarzschild Geometry 49
C Light Trajectories In Kerr Geometry 53
List of Figures
1 2-D Manifold sat in R3, and Tangent Plane . . . . . . . . . . . . 92 Impact Parameter b . . . . . . . . . . . . . . . . . . . . . . . . . 223 Weff in Schwarzschild Geometry . . . . . . . . . . . . . . . . . . 234 Weff Graph for Plunge Orbit in Schwarzschild Geometry . . . . 235 Plunge Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Weff Graph for Scatter Orbit in Schwarzschild Geometry . . . . 257 Scatter Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Weff Graph for Circular Orbit in Schwarzschild Geometry . . . . 269 Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710 Gravitational Lens . . . . . . . . . . . . . . . . . . . . . . . . . . 2811 Gravitational Lensing on 2-D Manifold sat in R3 . . . . . . . . . 2812 Cross-section of �rst and second Einstein Rings . . . . . . . . . . 2913 Einstein Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014 Thin Lens Approximation . . . . . . . . . . . . . . . . . . . . . . 3315 Weff in Kerr Geometry . . . . . . . . . . . . . . . . . . . . . . . 4016 Circular orbits in Kerr Geometry . . . . . . . . . . . . . . . . . . 4317 Plunge orbits in Kerr Geometry . . . . . . . . . . . . . . . . . . . 4318 Scatter orbits in Kerr Geometry . . . . . . . . . . . . . . . . . . 4419 Scatter and Plunge orbits in Kerr Geometry . . . . . . . . . . . . 4420 Distance from the Event Horizon to the Photon Sphere . . . . . 45
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1 Introduction
In Newtonian mechanics light is not a�ected by the in�uence of gravity movingonly in straight lines1. One of the most interesting, and most readily testable,facets of Einstein's General Relativity theory is that massive objects cause lighttrajectories to curve. This report explores this phenomenon with particularattention applied to the most extreme case: black holes.
In the �rst two sections we set ourselves up with the mathematical toolsrequired to study non-Euclidean geometries, and review some important aspectof special relativity and their generalizations. We go on to look, in Section 4, atthe simplest, relevant non-Euclidean geometry which is that surrounding non-rotating, changeless, spherically symmetric masses. This geometry is namedafter Karl Schwarzschild (1873-1916)2 who devised the �rst set of solutions tothe Einstein �eld equations while serving on the eastern front during the FirstWorld War. These solutions describe the geometry that is his namesake.
In Section 4.5 we study the di�erent types of trajectory that light can take inSchwarzschild geometry. We note that these fall into three qualitatively distinctcategories; plunge, scatter and circular orbits. We examine some properties ofthese orbits before going on to discuss, in Section 5, gravitational lensing as aconsequence of scatter orbits. We use this phenomena to produce an approxi-mate equation that relates the mass of the lens to various measurable distancesand angles.
Finally we look at rotating black holes3, called Kerr black holes, which arenamed after Roy Kerr (1934-present)4. In order to maintain as much symmetryas we can, in the Kerr case, we focus on equatorial orbits where the classi�cationsystem we applied to light trajectories in Schwarzschild geometry still appliesand is still exhaustive for orbits from in�nity.
1.1 A Note on Notation
Throughout this document we use notations and conventions broadly the sameas those found in Hartle [1], as he forms the primary source for much of mypreliminary work. Where Hartle does not form part of the source material Ihave altered notation to be in line with his where a precedent has already beenset in the document, and adopted the notation of other authors in the fewoccasions when no such precedent existed.
1This is in its traditional formulation. There are formulations in which light curves, nor-mally by assigning mass to light 'corpuscles'. The curvature produced in general relativity isgreater than in these Newtonian formulations and does not require the non-physical assump-tion of massive light.
2See Weisstein [13].3An interesting consequence of complete gravitational collapse is that much of the in-
formation about the state of matter is lost, leaving only three attributes which completelydetermine the properties of the black hole. These are mass, charge and angular momentum.See Bekenstein [10].
4See the Encyclopaedia Britannica's website [14].
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2 Non-Euclidean Geometries
2.1 The Metric and Line Element
In General Relativity spacetime is curved by massive objects. To understandcurved spaces we need a non-Euclidean geometry. Such a geometry will havesome structure to it and we will need some way to describe how this structurechanges, with respect to spatial translations. We use the line element to do thisas it gives us function of how the space changes locally. Thus to �nd globalchanges we integrate across the relevant domain. It is clear that this requiresus to deal with di�erentiable spaces and for this reason we do not discuss whathappens at singularities. In this Section we discuss material similar to thatfound in Hartle [1] - Section 7.2.
A geometry in coordinates xα is described by the line element:
ds2 = gαβ (x) dxαdxβ (1)
where gαβ(x) is a symmetric, position dependent, matrix called the metric5. Aswe are dealing with four-dimensional spaces, with coordinates xα = (x0, x1, x2, x3),the general form is:
gαβ (x) =
g00(xα) g01(xα) g02(xα) g03(xα)g01(xα) g11(xα) g12(xα) g13(xα)g02(xα) g12(xα) g22(xα) g23(xα)g03(xα) g13(xα) g23(xα) g33(xα)
.For example the metric for four-dimensional Euclidean space, in Cartesian co-ordinates xα = (w, x, y, z), is:
1 0 0 00 1 0 00 0 1 00 0 0 1
. (2)
The metric for the �at (Minkowski) spacetime (ηαβ) of inertial frames, in Carte-sian coordinates xα = (t, x, y, z), is:
ηαβ =
−1 0 0 00 1 0 00 0 1 00 0 0 1
. (3)
We will meet some other metrics that represent curved spacetimes later.The distance ∆s between two points a and b in the geometry is given by
∆s =
ˆ b
a
ds (4)
5Here and elsewhere we make use of the Einstein summation convention.
8
Figure 1: Here we have a two-dimensional manifold sat inthree-dimensional Euclidean space. The tangent space Tp tothe manifold M at P is indicated and is the set of all vectorstangential to any curve γ ∈ M at P . We can increase thenumber of dimensions as required.Source: Carroll (1997) [15]
2.2 Scalar Products in Non-Euclidean Geometries
To perform calculations in non-Euclidean geometries we need some notion of ascalar product. To do this we consider the geometry to be that of the surface ofa manifold in some higher dimensional Euclidean space. It is possible to do thismore generally, without the need for embedding in Euclidean space, howeverthe result is the same and this discussion is more intuitive. The information inthis Section was sourced from Hobson et al. [3] - Chapter 3. For a more detaileddiscussion of manifolds see O'Neill [6] - Chapter 1.
The tangent space Tp to a manifold M at any point P is the set of all vectorstangential to any curve γ ∈ M at P . So any local vector v in M at P lies inTp and Tp is the set of such vectors. This is illustrated in Figure 1 as a two-dimensional manifold sat in three-dimensional Euclidean space. We can expandto greater dimensions as required.
The tangent plane is independent of the coordinate system used to de�nepoints on the manifold, however we may de�ne at each point P a set of basisvectors eα for Tp allowing us to express any vector v at P as a linear combinationof these basis vectors. This allows us to express the local vector �eld v(x), ateach point, in terms of the basis vectors like so:
v(x ) = vα
(x )eα(x ). (5)
If we choose our local tangent space basis vectors eα to be tangential to ourmanifold coordinates xα (whatever choice we have already made for xα) theseare called the coordinate basis, and are de�ned as:
eα = limδxα→0
δs
δxα(6)
where δs is an in�nitesimal displacement vector between the point P and a
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point in the neighborhood of P that is a distance δxα along the coordinatecurve xα. The set eα are linearly independent since the xα are. There are thesame number of eα as there are xα and therefore there are the same number ofeα as dimension of the tangent space. From this we conclude that eα forms abasis of the tangent space. We can now see from equation (6) that:
ds = eα(x )dxα. (7)
Using equation (7), in combination with equation (1), we get:
gαβ (x) dxαdxβ = ds2 = ds · ds= eα (x ) dxα · eβ (x ) dxβ = (eα (x ) · eβ (x )) dxαdxβ
⇒ eα (x) · eβ (x) = gαβ (x) . (8)
Thus we have:
a · b= gαβaαbβ (9)
3 Special Relativity Extended
This Section discusses a few properties of Special Relativity that are relevant toour line of reasoning. It then extends these to the general case.
3.1 Units
It is to be remembered that, as in Special Relativity, we simplify our calcu-lations by measuring time in metres and setting the speed of light c = 1 viathe transformation t(in m) = ct(in s). In General Relativity we also measuremass in metres and set the gravitational constant G = 1 via the transformationM(in m) = MG/c2(in kg). This system is called geometrized units and, unlessotherwise stated, we make use of it. For a more detailed account of this seeHartle [1] - Section 4.6.
3.2 World Lines
The world line of an object is the path it takes through spacetime. This Sectionintroduces a classi�cation system of di�erent types of world lines, basing thediscussion on Hartle [1] - Section 4.3.
A property of a world line that we are interested in is the spacetime distancealong it, which we can calculate using equation (4).
It is worth noting that the �at spacetime metric ηαβ - see equation (3) - isnot the same as the Euclidean metric (2). This di�erence means that in generalds2 ≯ 0. In particular it allows us to draw the distinction between timelike,lightlike and spacelike separated points in spacetime. As summarized below:
(∆s)2 > 0⇒ spacelike
10
(∆s)2 = 0⇒ lightlike
(∆s)2 < 0⇒ timelike. (10)
This classi�cation system still applies in general relativity.
3.3 Geodesics
3.3.1 Timelike Geodesics
It is known that in special relativity the variation principle, de�ned below,summarizes the equations of motion for objects unin�uenced by any force. Nowwe assume that this principle holds in the general case, and use it to constructequations of motion in curved spacetimes. Though we are concerning ourselveswith the trajectories of light it is helpful to look �rst at the timelike geodesicsof massive particles, using information sourced from Hartle [1] - Section 8.1. Inthe following section we look at the lightlike case which is based on Section 8.3of the same.
Variational principle: the world line of a free test particle between two timelikeseparated points extremizes the proper time between them.
We assume throughout the discussion that the curvature of spacetime causedby these particles is negligible, and it is for this reason that we refer to testparticles rather than particles. The Extrema of a function of n variables iswhere all the partial derivatives vanish. The particle being free means that theonly thing in�uencing its trajectory is the curvature of spacetime and its initialconditions, this is distinct from how we use the term free in special relativity.In special relativity the variational principle holds when all forces, includinggravity, are absent. However in general relativity gravity is not a force, but theresult of spacetime curvature induced by the presence of mass. For this reasongravity's in�uence is not excluded for free particles in general relativity.
From equation (1), and the de�nition of proper time dτ2 ≡ −ds2, we cal-culate the proper time interval along the world line of a timelike test particlemoving from a point A to a point B via:
τAB =
B
A
dτ =
B
A
(−gαβ (x) dxαdxβ
)1/2. (11)
If we now parametrize xα by a parameter σ, such that σ = 0 at A and σ = 1 atB, we get:
τAB =
1ˆ
0
dσ
(−gαβ (x)
dxα
dσ
dxβ
dσ
)1/2
(12)
A world line adheres to the variation principle if and only if it satisfy La-grange's equation:
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− d
dσ
(∂L
∂(dxα
dσ
))+∂L
∂xα= 0. (13)
For the Lagrangian:
L =
(gαβ (x)
dxα
dσ
dxβ
dσ
)1/2
. (14)
Plugging equation (14) into equation (13) we get6:
− d
dσ
[(−gγδ
dxγ
dσ
dxδ
dσ
)−1/2(gαβ
dxβ
dσ
)]+
1
2
(−gγδ
dxγ
dσ
dxδ
dσ
)−1/2∂gεβ∂xα
dxε
dσ
dxβ
dσ= 0.
(15)We can now use equation (12) to get:
dτ
dσ=
(−gαβ (x)
dxα
dσ
dxβ
dσ
)1/2
. (16)
Plugging this into equation (15) multiplied by dσ/dτ we �nd that:
− d
dτ
[gαβ (x)
dxβ
dτ
]+
1
2
∂gβγ∂xα
dxβ
dτ
dxγ
dτ= 0
⇒ −gαβd2xβ
dτ− dgαβ
dτ
dxβ
dτ+
1
2
∂gβγ∂xα
dxβ
dτ
dxγ
dτ= 0
which, by the chain rule, implies:
−gαβd2xβ
dτ2+
(−∂gαβ∂xγ
+1
2
∂gβγ∂xα
)dxβ
dτ
dxγ
dτ= 0
⇒ gαδd2xδ
dτ2= −1
2
(∂gαβ∂xγ
+∂gαγ∂xβ
− ∂gβγ∂xα
)dxβ
dτ
dxγ
dτ. (17)
We now introduce the inverse metric gαβ , so gαβgβγ = δαγ where δαγ is the
Kronecker delta. Multiplying equation (17) by gαβ we get:
d2xδ
dτ2= −1
2gδα
(∂gαβ∂xγ
+∂gαγ∂xβ
− ∂gβγ∂xα
)dxβ
dτ
dxγ
dτ. (18)
We simplify this expression by introducing the Christo�el symbols Γαβγwhichare calculated via
Γδβγ ≡1
2gδα
(∂gαβ∂xγ
+∂gαγ∂xβ
− ∂gβγ∂xα
)(19)
6At this point Hartle defers the rest of the derivation of the Geodesic equation (20) to anattachment on his website [16], which may be found in the web supplements Section and isentitled Chapter 8: Derivation of the General Geodesic Equation.
12
and thus equation (18) becomes
d2xα
dτ2= −Γαβγ
dxβ
dτ
dxγ
dτ. (20)
Equation (20) is called the geodesic equation. A geodesic is a generalization ofthe notion of a straight line in �at spacetime. Equivalently a path between 2points, in some geometry, is a geodesic if it is locally the shortest path betweenthose points, in that geometry. A curve is a geodesic if and only if it adheres tothe geodesic equation
We have shown that trajectories that adhere to the variation follow geodesicsin a curved spacetime, there is a large amount of evidence that real worldtrajectories do adhere to the variation principle and indeed follow geodesics.
The geodesic equation consists of coupled, second order ODEs. As we aredealing with four-dimensional spacetime there are four of them. However itis worth noting that Γαβγ is a rank three tensor and therefore has 43 = 64components. For this reason calculating its components is arduous and timeconsuming to perform explicitly so we will use Mathematica to perform thecalculations for us.
3.3.2 Null Geodesics
The geodesic equation, as we have constructed it, restricts the motion of timelikeparticles however we are interested in the motion of light. We need a similarequation that tells us about the motion of lightlike objects which we will �ndin this Section. The information found herein was sourced from Hartle [1] -Section 8.3.
A null geodesic is similar to those discussed above however it is lightlikeand so it describes the trajectories of photons (and other massless particles).However τ ceases to be a useful parameter since the proper time experienced bylight is always zero. For this reason we use another parameter which we denoteλ. However we cannot just use any parameter without it changing the form ofthe Geodesic equation (20), so to keep things simple we stipulate that λ is ana�ne parameter. That the parameter is a�ne means that it belongs to the classof parameters that do not change the form of the geodesic equation7. Thus wehave:
d2xα
dλ2= −Γαβγ
dxβ
dλ
dxγ
dλ. (21)
The actual derivation of equation (21) involves generalizing the laws of elec-tromagnetism to operating in curved spacetimes, which is beyond the scope ofthis project. However there are two things we know that we can use to motivatethe result. Firstly we know from special relativity and Newtonian mechanicsthat in �at spacetime the equation of the motion of light rays is:
7See Hartle [1] - Section 5.5.
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d2xα
dλ2= 0.
This is the statement that light moves in a straight line with constant velocity.Therefore we know that the general case must reduce to this in a local iner-tial frame. We further know that the equation must be of the same form forall coordinate systems, because we have used an arbitrary coordinate systemthroughout. From our discussion in Section 3.3.1 it is clear that equation (20)satis�es this second property, since no speci�c coordinate system was introducedin that discussion either, and so by analogy does equation (21). The �rst prop-erty follows since plugging the metric for �at spacetime ηαβ into equation (19)we get:
Γδβγ =1
2ηδα
(∂ηαβ∂xγ
+∂ηαγ∂xβ
− ∂ηβγ∂xα
)= 0
since all the partial derivatives vanish. This is not surprising since our derivationof the geodesic equation (20) was a process of generalizing unaccelerated motionto curved spaces.
Now if we let xα(λ) be the path of a light ray parametrized by some a�neparameter λ and uα ≡ dxα/dλ be its tangent vector. From equations (1), (9)and the lightlike case in (10) we get:
u · u= gαβ (x )dxα
dλ
dxβ
dλ= 0 . (22)
Which, along with equation (21), gives us all the equations we need to calculatelight trajectories in some geometry.
4 Schwarzschild Geometry
Schwarzschild geometry is the geometry of spacetime outside a spherically sym-metric mass at rest. The Schwarzschild black hole is the simplest model of ablack hole and it is where we assume that the gravitationally collapsing bodyand spacetime outside it are spherically symmetric. We will now �nd the metricthat describes such a geometry.
4.1 Finding the Schwarzschild Metric
To begin with we wish to construct the line element ds2 of Schwarzschild geom-etry and, by equation (1), �nd the components of the metric gαβ . To do this Ihave synthesized information found in Hobson et al. [3] - Section 9.1 and Chow[2] - Section 4.1.
The symmetries in the construction of Schwarzschild geometry tell us twothings, which are;
• The metric gαβ will be independent of the time coordinate t, since theonly mass in�uencing the geometry is at rest.
14
• The metric gαβ can only depend on quantities that are invariant underrotation. This is the condition that ds depends solely on rotationallyinvariant quantities and their derivatives, i.e ds is isotropic.
Rotational invariants of spacelike Euclidean coordinates x = (x, y, z) canonly depend on the scaler quantities:
x · x, dx · dx, dx · x. (23)
Which in polar coordinates are:
x · x = r2, dx · dx = dr2 + r2(dθ2 + sin2 θdφ2), dx · x = rdr. (24)
Now we construct ds2 in the most general possible way such that every term:
• contains the product of precisely two derivatives;
• is independent of time;
• is isotropic.
This gives:
ds2 = −F (r)dt2 +G(r)dt(dx · x) +H(r)(dx · x)2 + I(r)dx · dx
= −F (r)dt2 +G(r)rdtdr +H(r)(rdr)2 + I(r)(dr2 + r2(dθ2 + sin2 θdφ2))
for F , G, H and I general functions of r. This allows us, without loss ofgenerality, to transform F , G, H and I by any function of r and thus we canequate terms and absorb factors of r giving8:
−F (r)dt2 +G(r)dtdr +H(r)(dr)2 + I(r)(dθ2 + sin2 θdφ2).
We can now de�ne a new radial coordinate r via r2 = I(r) yielding9:
−F (r)dt2 +G(r)dtdr +H(r)(dr)2 + r2(dθ2 + sin2 θdφ2).
Since the geometry is at rest we know that the line element should be in-variant under the transformation t→ −t, however currently this is not the caseunless G(r) = 0, which gives us:
ds2 = −F (r)dt2 +H(r)dr2 + r2(dθ2 + sin2 θdφ2
). (25)
We know that as r →∞, ds2 → ηαβdxαdxβ . Thus:
limr→∞
(F ) = limr→∞
(H) = 1. (26)
Equation (26) gives us some restrictions on the functions F and H. However to�nally narrow down F and H to speci�c function requires a diversion into areas
8To keep things simple we neglect to change the symbols though the functions are changing.9We again neglect to change the notation.
15
of tensor calculus that are beyond the scope of this project so I will simply quotethe relevant result10.
The line element which encapsulates Schwarzschild geometry is11:
ds2 = −(
1− 2M
r
)dt2 +
(1− 2M
r
)−1
dr2 + r2(dθ2 + sin2 θdφ2
)(27)
which clearly has the metric:−(1− 2M
r
)0 0 0
0(1− 2M
r
)−10 0
0 0 r2 00 0 0 r2 sin θ
. (28)
Where t is the time as measured at in�nity; r is de�ned so that the surfacearea of nested spheres is 4πr2 like in Euclidean space. However it only gives aprecise measure of radial distances in the �at space limit r → ∞; θ and φ areangular coordinates as usually de�ned12.
4.2 Singularities and The Event Horizon
In the coordinate system we have been using it appears that something strange ishappening at the points r = 0 and r = 2M since at these points the componentsg00 and g11 blow up respectively. When this happens it is called a singularity.To complete our understanding of all possible light trajectories we would needto know what was happening at those points. It only a�ects our understandingof plunge trajectories because both the other kind of trajectory always haver ≥ 3M > 2M as we see later in Section 4.5. In this Section we explore thesesingularities using information sourced from Hartle [1] - Sections 7.1 and 12.1
Singularities
There are two types of singularity that we are interested in: coordinate singular-ities and spacetime singularities. A coordinate singularity is merely a propertyof the coordinate system chosen and does not necessarily correspond to any-thing physically interesting. Coordinate singularities occur because the chosencoordinate system fails at some points to give a unique label to these points.For example in two-dimensional polars the point (0, θ1) = (0, θ2) for any anglesθ1 and θ2. To overcome this problem we use a patchwork of di�erent coordinatesystems to cover the region of spacetime we are interested in. Ensuring thatthe coordinate system we use in any one region does not have any coordinatesingularities in that region.
10For why this is see Hobson et al. [3] - Section 9.2.11See Hartle [1] - page 189.12For a more detailed discussion of interpretation of the coordinates see Hobson et al. [3] -
Sections 11.1 and 11.2.
16
To show that a singularity is a coordinate singularity we need to �nd acoordinate system in which the metric is not singular at the relevant points. Itis beyond the scope of this project to look at this in any more detail but anunderstanding of how this is done in Schwarzschild geometry via Eddington-Finkelstein coordinates can be gleaned from, amongst others, Hobson et al. [3]- Section 11.5
A spacetime singularity is a point at which some physical property, i.e. aproperty that is invariant under coordinate transformations, blows up. It canbe shown that tidal e�ects approach in�nity as r approaches zero, showing thatthe singularity at r = 0 is genuine13.
The Event Horizon
Using Eddington-Finkelstein coordinates we can show that it is impossible forlight to move from r < 2M to r > 2M14. This implies that the outside iscausally separated from the inside (though not the other way round) and for thisreason it is called the event horizon15. If we look at the dt2 term in equation (27)we can see that the time as measured from in�nity blows up as r approaches 2Mfrom the positive side. This means that if we were to observe someone crossingthe event horizon we would see them slow asymptotically to a stand still! Inthis project we only concern ourselves with motion outside of the event horizon.
4.3 Null Geodesics In Schwarzschild Geometry
As we wish to understand light trajectories in Schwarzschild geometry we needto work out what restrictions the geodesic equation places on the motion, andalso stipulate that the four-velocity is null. This is done in this Section whichis based on Chow [2] - Section 4.3 and in�uenced by Hartle [1] - Chapter 9.
Using Mathematica, or otherwise, from the Geodesic equation (21) with theSchwarzschild metric (28) we calculate the equations of motion for free testparticles in a Schwarzschild spacetime as16:
t =Mtr
2Mr − r2
r =Mt2(2M − r)
r3− Mr2
2Mr − r2− φ2(2M − r) sin θ − θ2(2M − r)
θ =1
2φ2 cos θ − 2θr
r
φ = − φ(rθ cot θ + 2r)
r13See Hobson et al. [3] - Section 11.2.14See Hartle [1] - Section 12.1.15Though actually when quantum mechanics is taken into account this is not entirely the
case and black holes very slowly lose energy and emit particles that in�uence the outside.16For the Mathematica notebook used see Appendix A. This is an adaptation of a notebook
supplied on Hartles website [16].
17
where xα = dxα
dλ and xα = d2xα
dλ2 .We can simplify this immediately by noticing that if we assume that at
λ = 0, θ = π/2 and θ = 0, the motion now lies in the plane θ = π/2. We cando this, without loss of generality, since Schwarzschild geometry is sphericallysymmetric, ergo we can always rotate our frame of reference so this is the case.Thus the equations become:
t =2Mtr
2Mr − r2(29)
r =Mt2(2M − r)
r3− Mr2
2Mr − r2− φ2(2M − r) (30)
θ = 0 (31)
φ = −2φr
r. (32)
If we next focus on equation (32) we can see that it implies:
φ
φ= −2Mr
r⇒ d
dλ
(log φ
)= −2
d
dλ(log r)⇒ log φ = log
(hr−2
)⇒ φ =
h
r2
(33)for some constant of integration h. Similarly from equation (29) we have:
t
t=
2Mr
(2M − r)r⇒ d
dλ
(log t
)=
d
dλ
(log
(1− 2M
r
)−1)⇒ t = e
(1− 2M
r
)−1
(34)for some constant of integration e. If the results of equations (31), (33) and (34)are now plugged into (22) the result becomes:
0 =−e2(
1− 2Mr
) +r2(
1− 2Mr
) +h2
r2(35)
⇒ r2 = e2 − h2 (r − 2M)
r3(36)
4.4 Symmetry
Unfortunately the coupled second order di�erential equations given by the Geod-esic equations (20) and (21) tend to be intractable in all but the simplest ofcases, meaning that it is nigh on impossible to solve them analytically usingthe standard methods of integration. Though we have managed to do it forSchwarzschild geometry in Section 4.3 this was only because the Schwarzschildcase is much simpler than most as there is a high degree of symmetry in itsconstruction. We can use our knowledge of how symmetries a�ect geodesicsto give us additional information that will allow us to construct our �rst orderequations more easily. We can do this since a symmetry implies that somequantity is conserved and we can work out what this quantity is via a 'Killing
18
vector' named after the German mathematician Wilhelm Killing (1847-1923)17.In this section we explore why this is, having sourced information from Hartle[1] - Section 8.2. We then go on to apply this to the Schwarzschild case in afashion similar to that found in Hartle [1] - Sections 9.3 and 9.4
Killing Vectors
If we apply equation (13) to a Lagrangian that is independent of a coordinatewhich, without loss of generality, we say is x0 we get:
d
dσ
(∂L
∂(dx0
dσ
)) = 0 (37)
for:
L =
(g0β(x)
dx0
dσ
dxβ
dσ
)1/2
. (38)
This, along with the a�ne parameter version of equation (12), implies that:
∂L
∂(dx0
dσ
) =1
2Lg0β(x)
dxβ
dσ=
1
2g0β(x)
dxβ
dλ= constant. (39)
Now if we introduce a vector ξα = (1, 0, 0, 0), from the scalar product equation(9) and the above we get:
ξ · u = gαβξαuβ = g0β(x)
dxβ
dλ= constant. (40)
We call ξ a Killing vector and it has the associated conserved quantityg0β(x)dxβ/dλ. We can have more complicated Killing vectors, which relateto coordinates of which the Lagrangian is not independent, however this is suf-�cient for our purposes.
Symmetries in Schwarzschild Geometry
We recall that we constructed the metric (28) so that it had the following twoproperties.
• Time independence. As the metric does not depend on time the geome-try is symmetric with regard to translations in time, with the associatedKilling vector ξα = (1, 0, 0, 0). This is useful as it means the quantitye = −ξ · u = (1− 2M/r) dt/dλ is conserved. For su�ciently large valuesof r, e becomes energy per unit mass so in the �at space limit this becomesthe conservation of energy.
17See J. J. O'Conner et al. [17].
19
• Spherical symmetry. The metric is clearly independent of φ and so invari-ant under rotations about the z-axis, this has the associated Killing vectorηα = (0, 0, 0, 1). Other rotational symmetries are less clear but there areassociated Killing vectors, however we will not need them. Similarly tothe above, this means that h = η · u = r2 sin2 θdφ/dλ is conserved. Thisis the conservation of angular momentum.
Thus we have two conservation laws:
e = −ξ · u =
(1− 2M
r
)dt
dλ(41)
h = η · u = r2 sin2 θdφ
dλ= r2 dφ
dλ. (42)
Note that e and h are the constants of integration that appeared when weconstructed the equations of motion directly in Section 4.3. Further note thath ≥ 0 for dφ/dλ ≥ 0, as we can rotate the frame and it makes no di�erence wecan de�ne h ≥ 0. We also chose our a�ne parameter λ so that t > 0, since weare dealing with events outside the event horizon (see Section 4.2), this meansr > 2M and thus e > 0 .
We can now continue, as we did at the end of Section 4.3, to produce equation(36) which rearranges to:
⇒ −(
1− 2M
r
)−1
e2 +
(1− 2M
r
)−1(dr
dλ
)2
+h2
r2= 0.
Later on we use these symmetries and their associated conservation laws tostudy the more complicated case of rotating black holes.
4.5 E�ective potential for Schwarzschild Black Holes
The actual trajectories can only be worked out by integrating the equations ofmotion - (30) and (32) - numerically, which is how the graphs in this Sectionhave been produced18. However we can analyze some of the properties usingthe quantity Weff that is analogous to the e�ective potential Veff used tounderstand the trajectories of massive particles in Schwarzschild geometry19,and other orbital problems. The discussion that follows is based largely onHartle [1] - Section 9.4 and Hobson et al. [3] - Section 9.10
It is clear that equation (36) may be rearranged to:
1
b2−Weff (r) =
1
h2
(dr
dλ
)2
(43)
18The Mathematica notebook used can be found in Appendix B and was based on oneprovided on the accompanying website to Hartle [16] which I adapted from the massive particleto the light case.
19See Hartle [1] - Section 9.3.
20
where:
Weff (r) ≡ 1
r2
(1− 2M
r
), and b ≡ h
e(44)
We also have:1
h2
(dr
dλ
)2
≥ 0⇒ 1
b2≥Weff (r).
This tells us that there is a turning point in r at 1/b2 = Weff (r). We will usethis fact to determine the qualitative properties of light ray trajectories. Weonly consider trajectories that come from r = ∞. Obviously r can't actually=∞ in a literal sense, what we mean is that the beginning of the trajectory issu�ciently far away from the mass that the space is approximately �at. Thus wecan extend its source arbitrarily further away in a strait line tangential to, andin the opposite direction to, its initial trajectory without it signi�cantly a�ectingthe result, i.e. extend it back to in�nity. The circular orbits, which we will seelater, can be thought of as coming from in�nity before getting trapped into theorbit. Nonetheless, for aesthetic reasons, only the circular part of the orbit isshown in the trajectory graphs. However before we do this it is informative forus to �nd a physical interpretation of the quantity b introduced earlier in thissection. Wel do this in the next section taking guidance from Hobson et al. [3]- Section 9.13.
4.5.1 Interpreting b
From equations (36), (42) and our de�nition of b we have:
dφ
dr=φ
r=
b
r2(1− b2
r2
(1− 2M
r
))1/2⇒ r2 dφ
dr=
b(1− b2
r2
(1− 2M
r
))1/2Thus
limr→∞
r2 dφ
dr= ±b
Spherical symmetry means we are free to stipulate, without loss of generality,that:
limr→∞
φ = 0.
Thus we have, at r =∞:b = r sinφ.
We are now in a position to o�er an interpretation of b. It is the distancebetween the origin (the location of the mass) and what would have been theclosest part of the trajectory were the space �at everywhere i.e b is an impactparameter. As shown in Figure 2.
21
Figure 2: represents the impact parameter b as the distancebetween the origin and the point of closest approach of theline tangential to the lights initial trajectory.
4.5.2 Plotting Weff
We next wish to plot Weff against r so that we can compare it with b. Di�er-entiating equation (44) we get:
d
drWeff (r) =
2
r3
(3M
r− 1
).
This implies that the only turning point of Weff is at r = 3M . We also have:
Weff (3M) =1
27M2,
d2
dr2Weff (3M) = − 2
82M4< 0
⇒ maxWeff (r) =1
27M2at r = 3M.
If we now take the limits:
limr→∞
(Weff ) = 0 and limr→0
(Weff ) = −∞
we can see that we get something that looks like Figure 3. Though in fact herewe plot M2Weff against r/M so that it is scaled correctly. It is possible toremove the explicit h dependence in equation (43) by transforming the a�neparameter λ which is why we neglect to scale by it in later graphs20.
4.5.3 Plunge Orbits
Let us �rst consider a light ray with b < 3√
3M . By plotting 1/b2 on to ourWeff graph we get Figure 4.
If we imagine a photon is approaching the central mass from r = ∞, alongthe red arrow that represents the value of b, we know that ∃ λ0 such that:
dr(λ0)
dλ< 0.
We can see from equation (43) that this implies dr/dλ < 0 ∀ λ, since dr/dλ iscontinuous and dr/dλ 6= 0 ∀λ. Thus we get the plunge orbit shown in Figure 5.
20See Hobson et al. [3] - Page 220
22
Figure 3: Here we see how Weff varies with r. To ensurethat it is scaled correctly we actually plot M2Weff againstr/M . Note that the peak, at r = 3M , is the only turningpoint.
Figure 4: Here we see the graph ofWeff in Figure 3 with thevalue of 1/b2 indicated (red arrow). We can see that, sinceWeff and 1/b2 are never equal, there are no turning pointsin r and so a plunge orbit shown in Figure 5 is inevitable.
23
Figure 5: Here we have a plunge orbit indicative of an im-pact parameter such that b < 3
√3M . The actual value of
b used was 5M as indicated in Figure 4. As with previ-ous graphs the axes have been scaled by a factor of 1/M toremove their dependence on M
4.5.4 Scatter Orbits
We now consider the case where b is such that b > 3√
3M . Plotting this asbefore we get Figure 6.
Since b > 3√
3M we have 1/b2 = Weff (r(λ1)) for some λ1. Thus we have:
dr(λ1)
dλ= 0. (45)
By plugging equations (33), (34) and (36) into (30) and simplifying, we produce:
d2r
dλ2=h2(3M − r)
r3. (46)
This implies that r(λ1) is a local minimum since we know that r < 0 as r > 3M .We know r then tends back to ∞, along the black line, as there are no moreturning points in r. This scattering orbit, illustrated in Figure 7, is the e�ectthat leads to gravitational lensing discussed in Section 5.
4.5.5 Circular Orbits
The only possible value of the impact parameter not already considered is b =3√
3M which gives Figure 8.
24
Figure 6: Here we see the graph of Weff in Figure 3 with anew value of 1/b2 indicated (red/black arrows). We can seethat Weff and 1/b2 are equal at a single point thus thereis a minima of r at that point. Consequently we get thescatter orbit shown in Figure 7
Figure 7: For b > 3√
3M we get an orbit similar to thescatter orbit shown above. The actual value of b used was5.65M , as indicated in Figure 6. As with previous graphsthe axes have been scaled by a factor of 1/M to remove theirdependence on M
25
Figure 8: Here we have the limiting case between the scatterand plunge orbits. We see that since Weff = 1/b2 every-where we get the circular orbit shown in Figure 9
Since dr/dλ0 = 0 and d2r/dλ20 = 0 ⇒ r = 3M ∀λ and therefore the tra-
jectory is circular, as in Figure 9. However we can see in Figure 8 that this isunstable, since a small change in one of the parameters will cause the trajectoryto become either a plunge or a scatter.
The set of all such circular orbits is called the photon sphere. Since thecircular orbits only occur at r = 3M the photon sphere is a property that canonly belong to bodies with a radius less than 3M . If we calculate the radius ofthe Sun we get21:
- Solar mass ≈ 1.989× 1030kg≈ 6.673×10−11×1.989×1030
2.9982×1016 m≈ 1.477km- Solar radius ≈ 695500km� 3× 1.477km.
Showing that our star is far too big to have a photon sphere. If we look atthe typical values for a Neutron Star, which is the densest type of star otherthan a black hole, we get something like:
- Mass ≈ 2km- Radius ≈ 12km> 3× 2km
which again is too big22. Having studied the literature it seems there is lit-tle consensus as to whether or not a neutron star of su�cient density to possessa photon sphere is possible in nature. It is certain that all neutron stars haver > 2M , as if one didn't it would have an event horizon and so by de�nitionwould be a black hole. However a detailed discussion of high density physics isbeyond the scope of this project, if you wish to pursue it further see Leung etal. [12]. We can however be certain that photon spheres are not possible for any
21Numerical values found using Wolfram Alpha and, though all are given to four signi�cant�gures, the calculations have been done with eight signi�cant �gures.
22Numerical values sourced from Nemiro� [11] - Page 12.
26
Figure 9: This is a representation of the unstable circularorbit produced by b = 3
√3M . Again the axes have been
scaled
other type of star as white dwarfs are insu�ciently dense and all other typesless dense still.
5 Gravitational Lensing
Light rays, with scattered trajectories (see Section 4.5.4), make their sourceappear to be in a di�erent part of the sky to where it actually is. This sectionwill explore this phenomenon primarily using information found in Hartle [1]-Sections 9.4 and 11.1 and Hobson et al. [3] - Section 10.2. The discussionhas also been informed by Cohen [5] which o�ers a fairly non-mathematicaldiscussion.
The scattering e�ect led to one of the earliest tests of general relativity.In 1919 Arthur Edington measured the apparent position of a star close tothe Sun during a solar eclipse. He compared this to its relative position sixmonths earlier, and found the di�erence to be within the margin of error of thepredictions made using the Schwarzschild solutions to Einstein's equations.
Though the de�ections made by the Sun were tiny, bodies such as black holesor galaxies cause much larger distortions. In fact you can see several images ofthe same star going round large gravitating bodies in di�erent directions, asis shown in Figure 10. We can get a feeling for why this is by consideringFigure 11. This Figure represents our four-dimensional spacetime as a two-dimensional manifold that is curved in the third Euclidean dimension; thuscausing the light to bend both ways around the galaxy to the observer. In
27
Figure 10: This Figure illustrates gravitational lensing. Wecan see how scatter orbits make objects appear to be in adi�erent place to where they actually are. This means thatlight from a single distant object can travel around a mass inmultiple directions to the observer, thus producing multipleimages. This e�ect is known as gravitational lensing and isillustrated aboveSource: Smith [18]
Figure 11: In this Figure our four-dimensional spacetime isrepresented as a two-dimensional manifold which is curvedin the third Euclidean dimension thus causing the light tobend both ways around the galaxy to the observer.Source: Washington University of St. Louis [19]
28
Figure 12: In this �gure we see the trajectories taken bylight that forms a cross-section of the �rst (blue) and second(red) Einstein rings. The apparent position of the source isindicated with dotted lines. We can see that the secondring lies inside the �rst, and by the same process we canshow that each successive ring appears within the other butexternal to the photon sphere.
the diagram there are only two distinct images of the distant galaxy as we areworking with a two-dimensional space. In reality, since we have three spatialdimensions, the light can bend round the mass in all directions and so canproduce several images. In fact if the object and observer are perfectly alignedwith a suitably large mass then they produce a continuous circular image ofthe object known as an Einstein ring. Inside this �rst Einstein ring there willbe another ring corresponding to light that has completely orbited the massbefore traveling on to the observer. Inside this there will be another whichconsists of of photons that have completed two orbits, and so on ad in�nitum23.The limiting angular radius of the rings is that of the photon sphere as thetrajectories move ever closer to it. These rings become progressively less bright.This is because the required precision, such that the photons trajectory �nishesat the observers eye, increases. A cross-section of the �rst two rings is given inFigure 12. Experimental evidence of Einstein rings has been found, thus addingfurther credence to General Relativity theory24. The image in Figure 13 is fromthe Hubble telescope and is believed to be an example of an Einstein ring.
23For a more detailed discussion of these inner rings, as well as other optical properties ofextreme gravity, see Nemiro� [11].
24See Warren et al. [8].
29
Figure 13: This image is of a candidate Einstein ring wastaken by the Hubble telescope [20]
5.1 Calculating the Angle of De�ection
We are interested in knowing how much a system will de�ect the trajectory oflight, or, more usefully, calculating properties of the system based on our mea-surements of the de�ection of light25. To calculate this we start by rearrangingequation (43) to:
dr
dλ= ±h
√1
b2−Weff (r) (47)
as can been seen in Hartle [1] - Section 9.4 [The de�ection of light] which thisSection follows closely. Using the aforementioned equation, θ = π/2, equation(42) and the chain rule we get:
dφ
dr=dφ
dλ/dr
dλ=
±1
r2√
1b2 −Weff (r)
. (48)
Since we are not interested in the direction of travel we are free to drop the ±in the above expression. So the total change in φ (∆φ) is given by:
∆φ =
ˆ ∞−∞
dr
r2√
1b2 −Weff (r)
, (49)
25For a discussion of some of the astronomical applications of gravitational lensing see Ehlerset al. [7] - Section 4.
30
which, by symmetry, is equal to:
2
ˆ ∞r0
dr
r2√
1b2 −Weff (r)
(50)
forWeff (r0) = 1/b2, i.e. r0 is the minimum value of r. We can now see that ∆φonly depends on M/b since the only other parameter in equation (50) is r0 andthat also solely depends on M/b. For large M/b we can use numerical methodsto solve equation (50). However if M/b is small then we can use perturbationtheory to get a good approximation. We do this in the following Section usinginformation sourced from Section 10.2 of Hobson et al. [3]
5.1.1 Calculating Small De�ections
We can rearrange the positive part of equation (48) to:
dr
dφ= r2
√1
b2− 1
r2
(1− 2M
r
). (51)
If we then do the substitution r = 1u we get:
du
dφ=
√1
b2− u2 (1− 2Mu). (52)
This di�erentiates to:d2u
dφ2+ u = 3Mu2. (53)
The vacuum solution (M = 0) to equation (53) gives:
u = A sinφ+B cosφ.
Now plugging in our initial conditions for a photon moving parallel to the x axiswith an impact parameter b we get:
y = r sinφ = b⇒ u =sinφ
b.
We expect a small mass to give a small perturbation δu to the zeroth-ordermassless case and are free to ignore the higher order terms. So we write:
u =sinφ
b+ δu. (54)
Plugging this into equation (53) and dropping higher order terms we get:
d2u
dφ2+u = − sinφ
b+d2δu
dφ2+
sinφ
b+ δu = 3M
(sinφ
b+ δu
)2
≈ 3M
b2sin2 φ (55)
31
which implies:d2δu
dφ2+ δu ≈ 3M
b2sin2 φ. (56)
Solving this we get :
δu ≈ M
b2(3 + cos 2φ) . (57)
Plugging this into equation (54) and taking the limit as r →∞ we get:
limr→∞
sinφ
b+M
b2(3 + cos 2φ) =
φ
b+
4M
b2≈ 0 = lim
r→∞u. (58)
This gives:
φ ≈ −4M
b. (59)
Thus the angle of de�ection, in standard SI units, for M � b is given by:
∆φ ≈ 4GM
c2b. (60)
5.1.2 Light De�ection by the Sun
We now apply this to the Sun, as Arthur Eddington did for his famous exper-iment of 1919, using the values given in Section 4.5.5. For a light ray passingas close as possible to the Sun we take the minimum value of r (r0) to be thesolar radius 695500km. We take the solar mass, as before, to be 1.477km. Viathe equation Weff (r0) = 1/b2, we calculate the value of b to be b ≈ 695500kmand so we get26:
∆φ ≈ 4× 1.477
695500≈ 8.493× 10−6radians ≈ 1.752".
The angle of de�ection is very small and so it created quite a challenge forEddington to measure. The values he got from his two separate sets of mea-surement were ∆φ = 1.98" ±0.16" and 1.61" ±0.427. These where consideredto be su�ciently consistent with the theory to provide the �rst con�rmationof a prediction made using Einstein's General Relativity theory. We would notexpect the measured value to exactly equal the predicted value, even if we couldremove experimental error and equation (60) was exact, since the Sun is a non-isolated, rotating body which is only approximately symmetric, and thereforethe geometry is only approximated by Schwarzschild. However these measure-ments, and much more accurate ones made since, con�rm that it is a reasonablyaccurate model. We are next going to use our de�ection angle approximation toconstruct an equation that allows us to derive the approximate mass of celestialbodies.
26Numerical values are all given to four signi�cant �gures and the calculations have beendone with eight signi�cant �gures as before.
27Numerical values found in Hobson et al. [3] - top of page 235.
32
Figure 14: In this diagram the observer is placed at theorigin, the lens at L and the light source at S. We cansee the two images that the light source produces for theobserver at S1 and S2. The distances and angles betweenthese points are also indicated. This Figure is an adaptationof one found in Wambsganss [22]
5.1.3 The Thin Lens Approximation
To simplify things we are only going to consider gravitational lensing in a geom-etry with a single spherically symmetric source of curvature, i.e. Schwarzschildgeometry. We will assume the mass to be at a point and all of the light de�ec-tion to occur instantaneously in the plane of normals to the line of site of thelens, at the lens (in Figure 14 this is the plane that would result from rotatingξ about the 0− L axis and extending to in�nity). Our �nal assumption is thatthe de�ection angle is given exactly by equation (60). This approximation isaccurate when the b � M and when the distances between the observer, lensand source are much greater than the size of the lens itself. This last require-ment is almost always true when dealing with observable astronomical objects.For a more detailed discussion of some of the properties of gravitational lensesand how they relate to real world observations see Hartle [1] - Section 11.1, thematerial at the beginning of which informed this Section.
In Figure 14 the observer is placed at the origin, the lens at L and the lightsource at S. We can see the two images that the light source produces for theobserver at S1 and S2. The distances and angles between these points are also
33
indicated. To simplify our notation we have replace ∆φ with α in equation (60).Our assumptions imply that Ds � η + ζ so we have, to a good approxima-
tion:η ≈ βDS , ζ ≈ αDLS , η + ζ ≈ θDS , b ≈ ξ ≈ θDL
We assume this approximation holds to su�cient accuracy for our purposes andwe get the lens equation:
θDS = βDS + αDLS . (61)
Plugging in our value for α from equation (60) we get:
θDS = βDS + 4M
bDLS = βDS + 4
M
θDLDLS .
We now rearrange to get:
θ = β + 4MDLS
θDLDS. (62)
It is easy to see that the above equation allows us to calculate the mass (M) ofthe lens simply by measuring the angles β and θ when the values of DL and DS
have been calculated via other means.We can also use equation (62) to glean an understanding of Einstein rings
discussed in the introduction of this Section. Though in some cases our approx-imation is not very accurate. Clearly if the source, lens and observer are alignedthen the angle β = 0 giving:
θ2 = 4MDLS
DLDS⇒ θ = ±2
√M
DLS
DLDS
Thus in this plane images appear in the two positions that are an angle2√MDLS/DLDS from the observer, lens, source axis. The symmetry of the
situation is such that this is true for all planes that are rotations of our startingplane about this axis. Thus we get a ring of images around the lensing masswith a radius of approximately 2
√MDSDLS/DL.
6 Light Orbits in the Equatorial Plane of Kerr
Black Holes
Rotating black holes, known as Kerr black holes, have two parameters thatcompletely determine their properties. These are the black hole's angular mo-mentum (J) and, as with the Schwarzschild case, mass (M). This Section willdiscuss how rotation a�ects light trajectories in the equatorial plane of Kerrblack holes. We start with a discussion of the Kerr line element and metricwhich is based on Hartle [1] - Section 15.2 and Hobson et al. [3] - Section13.5.
34
6.1 Kerr Metric and Line Element
For a suitable set of coordinates, called Boyer-Lindquist Coordinates (see Section6.2), the line element that describes the geometry outside rotating masses is28:
ds2 = −
(1− 2Mr
r2 + J2
M2 cos2 θ
)dt2 − 4Jr sin2 θ
r2 + J2
M2 cos2 θdφdt+
r2 + J2
M2 cos2 θ
r2 − 2Mr + J2
M2
dr2
+
(r2 +
J2
M2cos2 θ
)dθ2 +
(r2 +
J2
M2+
2J2rM sin2 θ
r2 + J2
M2 cos2 θ
)sin2 θdφ2 (63)
which, as one would expect, reduces to the Schwarzschild line element (27) forthe case J = 0.
To keep things simple we are only going to look at motion restricted to theequatorial plane, i.e. θ = π/2 and dθ/dλ = 0. It so happens that, in general,geodesics in Kerr geometry do not lie in a plane, however equatorial orbits do29.
To simplify our notation we introduce the angular momentum per unit massa ≡ J/M and the quantity ∆ ≡ r2 − 2Mr + a2. Applying these simpli�cationsto the line element (63) it reduces to:
ds2 = −(
1− 2M
r
)dt2 − 4aM
rdφdt+
r2
∆dr2 +
(r2 + a2 +
2a2M
r
)dφ2. (64)
With the corresponding metric:−(1− 2M
r
)0 0 − 2aM
r
0 r2
∆ 0 00 0 0 0
− 2aMr 0 0
(r2 + a2 + 2a2M
r
) (65)
6.2 Boyer-Lindquist Coordinates
The coordinates we are using are not the same as Schwarzschild coordinates usedin earlier Sections. Unfortunately the coordinates we use, called Boyer-LindquistCoordinates, do not lend themselves so easily to geometrical interpretation asSchwarzschild coordinates. Nonetheless, in this Section, we will look at someproperties of this coordinate system. This discussion of Boyer-Lindquist coordi-nates is based on Hobson et al. [3] - Section 13.6. The original paper describingthis coordinate system can be found in Boyer et al. [9] which is more completeand more technically demanding. This paper has also informed this Section.
If we now consider the limit of the Kerr line element (63), as M → 0, we seethat we get:
28For a discussion of how this line element can be derived see Hobson et al. [3] - Sections13.1, 13.5 and 13.7.
29See Hobson et al. [3] - Section 13.10.
35
ds2 = −dt2 +r2 + a2 cos2 θ
r2 + a2dr2 +
(r2 + a2 cos2 θ
)dθ2 +
(r2 + a2
)sin2 θdφ2
Now we expect this to be Minkowski spacetime. And indeed one can showwithout too much di�culty that applying the transformation below gives the�at line element.
t = t
x =√r2 + a2 sin θ cosφ
y =√r2 + a2 sin θ sinφ
z = r cos θ
where (t, x, y, z) are ordinary Cartesian coordinates. As we are only going to bedealing with the equatorial plane this becomes:
t = t
x =√r2 + a2 cosφ
y =√r2 + a2 sinφ
z = 0
and we can see that for a = 0 these are just ordinary polar coordinates. Theabove interpretation can be misleading for M 6= 0 however it does allow us tohighlight a key way in which Kerr spacetime di�ers from the Schwarzschild casewhich we do in the next Section.
6.3 Singularities in Kerr Geometry
Inspection of the line element (63) reveals that, in Boyer-Lindquist coordinates,it is singular at r2 + a2 cos2 θ = 0 and ∆ = 0. This section discusses thesesingularities, basing the account on Hobson et al. [3] - Sections 13.4, 13.6 and13.8, Boyer et al. [9] and, to a lesser extent, Hartle [1] - Section 15.3.
The Singularity at r2 +a2 cos2 θ = 0 is real and occurs at r = 0 and θ = π/2(i.e. in the equatorial plane). Following the earlier precedent of this documentwe will not discuss in any detail what happens inside the event horizon(s) ofthe black hole and so we will not look at this singularity in any detail. Howeverwe note that, from our discussion in Section 6.2, this singularity forms a ring ofradius a rather than a point.
The second set of singularities, the solutions to ∆ = 0, are coordinate sin-gularities. They are the solutions to r2 − 2Mr + a2 = 0 which clearly are:
r± = M ±√M2 − a2. (66)
It can be shown that both of these solutions correspond to event horizons30. Itis also worth noting that there are no real solutions to ∆ = 0 for a2 > M2.
30To see how this is done see Hobson et al. [3] - Section 13.4.
36
If this situation is possible it would mean that the ring singularity would benaked, i.e trajectories could come in�nitesimally close to it before returning to∞. However there are reasons to believe that this situation is impossible innature, and for this reason we ignore it31.
Thus we have seen that the structure of Kerr black holes is signi�cantlydi�erent to that of Schwarzschild, having a ring singularity and 2 event horizons.It can be further shown that the Kerr case has another boundary called theErgosphere, which lies at r = 2M in the equatorial plane. In the region boundby the Ergosphere it is impossible for an object to remain at rest with respectto ∞, we did not need to draw the distinction between the event horizon andErgosphere in our Schwarzschild discussion since for a = 0 both occur at r = 2Mand r = 0.
6.4 Symmetries of Kerr Geometry
Since the metric (65) does not depend on the coordinates t and φ we have twoKilling vectors and their associated conservation laws. This section derives a�rst order, ODE in r from these conservation laws, basing the discussion onHobson et al. [3] - Section 13.10 and Boyer et al. [9].
Our Killing vectors and associated conservation laws are:
ξ = (1, 0, 0, 0), ξ · u = g0β (x)dxβ
dλ= −
(1− 2M
r
)dt
dλ− 2aM
r
dφ
dλ= −e
η = (0, 0, 0, 1), η · u = g4β (x)dxβ
dλ= −2aM
r
dt
dλ+
(r2 + a2 +
2a2M
r
)dφ
dλ= h.
Giving the matrix equation:[−(1− 2M
r
)− 2aM
r
− 2aMr
(r2 + a2 + 2a2M
r
) ][ dtdλdφdλ
]=
[−eh
]. (67)
Solving (67) we get:[dtdλdφdλ
]=−1
∆
[ (r2 + a2 + 2a2M
r
)2aMr
2aMr −
(1− 2M
r
) ] [ −eh]
(68)
i.e.dt
dλ=
1
∆
((r2 + a2 +
2a2M
r
)e− 2aM
rh
)(69)
dφ
dλ=
1
∆
(2aM
re+
(1− 2M
r
)h
). (70)
As we are still interested in the trajectories of light, we are still dealing withnull geodesics, therefore equation (22) still applies. Plugging our conservationlaws and the equatorial Kerr metric (65) into equation (22) we get:
31For a discussion of this conjecture, known as cosmic censorship, see Hartle [1] - Section15.1.
37
u · u= gαβ (x )dxα
dλ
dxβ
dλ=
−(
1− 2M
r
)(dt
dλ
)2
−4aM
r
dφ
dλ
dt
dλ+r2
∆
(dr
dλ
)2
+
(r2 + a2 +
2a2M
r
)(dφ
dλ
)2
=
−(
1− 2M
r
)(1
∆
((r2 + a2 +
2a2M
r
)e− 2aM
rh
))2
−4aM
r
1
∆2
(2aM
re+
(1− 2M
r
)h
)((r2 + a2 +
2a2M
r
)e− 2aM
rh
)
+r2
∆
(dr
dλ
)2
+
(r2 + a2 +
2a2M
r
)(1
∆
(2aM
re+
(1− 2M
r
)h
))2
= 0.
As we are dealing with r > r+ we are free to multiply by −∆/(rh)2 and takethe dr/dλ term to the other side, giving:
1
∆r2
(1− 2M
r
)((r2 + a2 +
2a2M
r
)σ
b− 2aM
r
)2
+4aM
r
1
∆r2
(2aM
r
σ
b+
(1− 2M
r
))((r2 + a2 +
2a2M
r
)σ
b− 2aM
r
)
− 1
∆r2
(r2 + a2 +
2a2M
r
)(2aM
r
σ
b+
(1− 2M
r
))2
=1
h2
(dr
dλ
)2
for σ = ha/‖ha‖ restoring the condition h > 0, and giving the condition a > 032,i.e. σ is one if the orbit is in the same direction as the rotation of the mass(corotating/prograde) or minus one if it is rotating in the opposite direction(counterrotating/retrograde). This simpli�es to:
1
h2
(dr
dλ
)2
=1
b2−Weff (r, b, σ) (71)
for:
Weff (r, b, σ) =1
r2
[1−
(ab
)2
− 2M
r
(1− σa
b
)2]. (72)
We can now perform an analysis similar to that performed in Section 4.5 todetermine the shape of our orbits. However now we have two parameters, (b, σ)or, if you like, (h, e) which determine the trajectory around a Kerr black hole,of given mass and angular momentum. We can also see that for a = 0 equation(72) reduces to (44) as required.
32Clearly we can only stipulate that h and a are positive once we have determined σ.
38
6.5 E�ective Potential for Kerr Black Holes
As we saw in Section 4.5, the actual trajectories can only be calculated numeri-cally33. In order that we may continue an analytical discussion we will study theturning points of r, as given by equation (71), to give us some understandingof the trajectories. To do this we base our discussion on Hobson et al. [3] -Sections 13.15, 13.16, 13.17 and 13.18. This section has also been informed byO'Neill [6] - Section 4.14.
From equation (72) we get:
d
drWeff (r) =
2
r3
(3M
r
(1− σa
b
)2
+(ab
)2
− 1
).
This implies that the value of r for which Weff is maximal (rc) is given by:
d
drWeff (rc) = 0
⇒ rc =3M
(1− σ ab
)21−
(ab
)2 =3M (b− σa)
b+ σa. (73)
At this point Weff takes the value:
maxWeff =(b+ σa)
3
27M2b2 (b− σa).
We are only interested in the 0 ≤ a < b. This is because if this condition isviolated Weff < 0 for all r and b > 0 therefore a plunge orbit is inevitable. Inorder to understand how variations in the values of the constants a�ect Weff
we note that:
limb→σa
(maxWeff ) =∞ and limb→−σa
(maxWeff ) = 0
limb→σa
(rc) = 0 and limb→−σa
(rc) =∞.
We still have:
limr→∞
(Weff ) = 0 and limr→0
(Weff ) = −∞.
So plotting Weff against r and σa/b we get the plot in Figure 15. In this �gurethe cross-section taken at the required value of σa/b gives you the analog of theWeff graphs given in Section 4.5. Again we have scaled the axis by powers ofM to remove the M dependence.
It is worth noting the following properties of the graph in Figure 15;
33A Mathematica notebook that does this can be found in Appendix C this was adaptedfrom Appendix B which was adapted from a notebook provided on Hartle's website [16].
39
Figure 15: This plot indicates how Weff varies with r, a, band σ.The cross-section taken at the required value of σa/bgives you the analog of theWeff graphs given in Section 4.5.Notice how the peak value of Weff increases, and occurs ata smaller value of r, for greater values of σa/b
• For any cross-section (ie any speci�c values of b, a, σ) there is only a sin-gle local maxima and no local minima. This means that the qualitativeclassi�cation system, of plunge scatter and circular orbits, we used in theSchwarzschild case still applies.
• The shape of the Weff graph now depends on b which was not true inthe Schwarzschild case. This means that care needs to be taken wheninterpreting Weff as the e�ective potential. Equation (72) shows thatthis does not a�ect our ability to determine turning points in r howeverit could in principle mean that even though the only turning point inWeff is maximal, circular orbits are still stable. In reality this is not thecase and circular orbits are unstable however showing this would requirea digression that is beyond our scope34.
• The more the black hole is rotating with the light ray the smaller the radiusof circular orbits. Furthermore the terning point in r of scatter orbits isallowed to be smaller. This begs questions about the event horizon thatwill be discussed in Section 6.5.3. For retrograde orbits the result is thereverse.
34For how this may be done see Hobson et al. [3] - Section 13.18.
40
6.5.1 Circular Orbits
In order to have a circular orbit, as in the Schwarzschild case, we need:
maxWeff =1
b2.
This gives us the equation:
(b+ σa)3
= 27M2 (b− σa)
Substituting b+ σa = v + 9M2/v into the above and rearranging we get35:(v +
9M2
v
)3
= 27M2
(v +
9M2
v− 2σa
)
⇒ v3 + 3× 9M2v +3× 92M4
v+
93M6
v3= 27M2v +
27× 9M4
v− 2× 27M2σa
⇒ v6 + 54M2σav3 + 93M6 = 0.
Using the quadratic equation we the �nd that:
v = 3M
(−σaM± i√
1−( aM
)2)1/3
.
We can now undo our substitution and we get:
b = 3M
(−σaM± i√
1−( aM
)2)1/3
+3M(
−σaM ± i√
1−(aM
)2)1/3− σa
Multiplying top and bottom by the complex conjugate of the denominator ofthe second term gives:
b = 3M
(−σaM∓ i√
1−( aM
)2)1/3
(−σaM± i√
1−( aM
)2)2/3
+ 1
− σaThis simpli�es too:
b = 3M
(−σaM± i√
1−( aM
)2)1/3
+
(−σaM∓ i√
1−( aM
)2)1/3
− σa⇒ b = 3M
((ei arccos −σa
M
)1/3
+(e−i arccos −σa
M
)1/3)− σa
35For the source of this substitution see Weisstein [21].
41
⇒ b = 6M cos
(1
3arccos
−σaM
)− σa.
We can now use the half angle formula to get:
b = 6M
(1
2
[1 + cos
(2
3arccos
[−σaM
])])1/2
− σa.
Thus circular orbits occur when:
b = 3
(2M2
[1 + cos
(2
3arccos
[−σaM
])])1/2
− σa (74)
and have a radius of:
rc = 2M
[1 + cos
(2
3arccos
[−σaM
])]. (75)
Giving:b = 3
√Mrc − σa. (76)
Note that for a Kerr black hole of given angular momentum (J) and mass(M) in general there will be 2 possible circular orbits, one prograde and oneretrograde. For this reason we, henceforth, denote radii of these orbits rc+(prograde) and rc− (retrograde) and their corresponding impact parameters,bc+ and bc−. Thus we have:
rc± = 2M
[1 + cos
(2
3arccos
[∓ a
M
])]and:
bc± = 3√Mrc± ∓ σa.
Do not be misled by the signs attributed to the radii as these correspond tothe value of σ and not their magnitude. Figure 15 indicates that:
rc+ ≤ rc− and bc+ ≤ bc−.
It is also clear from this �gure that these two orbits only coincide when a = 0,i.e. the Schwarzschild case, and move progressively further apart for larger a.
In the graphs that make up Figure 16 we show these two circular orbits fora = 0.5. The green graphs are for σ = 1. The blue graphs are for σ = −1. Thedirection of rotation of the black hole is indicated in yellow and the values of1/b2c± in red. Care must be taken in interpreting the coordinates (x, y) in thetrajectory graphs for reasons discussed in Section 6.2.
42
Figure 16: In this �gure we see the twopossible circular orbits of a Kerr blackhole with a = 0.5. The left graph showsthe values of b required for these orbits,and why these orbits are circular.
6.5.2 Prograde vs. Retrograde
In the graphs of Figures 17, 18 and 19 corotating orbits are shown in greenand counterrotating in blue. As ever the value 1/b2 is in red. To highlight thedi�erence sigma makes the orbits are shown to be coming from the same place,though in fact they are coming from opposite sides. The qualitative shape ofeach trajectory still relies only on the relationship between b and maxWeff soit would be super�uous to analyze the turning points of each graph individually,as this has been done in Section 4.5, and the result is very similar.
Figure 17: Both the prograde (green)and retrograde (blue) photons followplunge orbits. The images are shown su-perimposed on each other, though oneof them has been re�ected about they/M axis.
43
Figure 18: Both the prograde (green)and retrograde (blue) photons followscatter orbits. The images are shownsuperimposed on each other, as before.
Figure 19: In this Figure the prograde(green) photons follow a scatter orbitand the retrograde (blue) photons aplunge orbit. Again the images areshown superimposed on each other.
The other factors that a�ect the trajectory are b and a. The discussion ofvariation in b is essentially the same as in the Schwarzschild case so a furtherdiscussion is unnecessary. That leaves a discussion of the e�ect of a which willtake place in the next section.
6.5.3 Variation in Angular Momentum per Unit Mass
Figure 15 shows that, for σ = 1, as a increases the radius of circular orbitsdecreases. Correspondingly the minimum value of r (r0) attainable by scatteringorbits decreases. Clearly this can not go on inde�nitely as otherwise it wouldallow scatter orbit to pass within the outer event horizon which produces acontradiction. If we take the positive part of equation (66), i.e. the equationfor the union of outer event horizon and the equatorial plane, we have:
r+ = M +√M2 − a2.
44
Figure 20:
We also have that:
limb→+bc+
r0 = rc+ = 2M
[1 + cos
(2
3arccos
[− a
M
])]since the closest a scatter orbit can get to a black hole is to just miss the innerphoton sphere36. The distance from the outer event horizon to the photonsphere (Deh↔ps) is given by:
Deh↔ps = rc+ − r+ = 2M
[1 + cos
(2
3arccos
[− a
M
])]−M −
√M2 − a2.
⇒ rc+ − r+
M= 1 + 2 cos
(2
3arccos
[− a
M
])−√
1−( aM
)2
.
It is not immediately obvious that this is greater than 0. However plottingDeh↔ps/M against σa/M we get Figure 20. This shows that for non nakedsingularities, i.e. a2 < M2, this distance is always greater than 0. It furthershows that as a → M the radial distance between the photon sphere and theouter event horizon tends to 0. This means that prograde scatter orbits can getcloser to Kerr black holes the faster they are rotating.
36Or it can become part of the photon sphere for a time before scattering due to instability.
45
7 Conclusion
It is not possible for us to perform a completely precise analysis of light-raytrajectories. However we can use numerical methods to give us an arbitrarilyaccurate approximation of a given system37. The appendices that follow provideMathematica notebooks that do this. Despite this restriction, as this reporthas sought to demonstrate mathematically, we can use analytic techniques todetermine the qualitative properties of the trajectories.
Light trajectories that start from r = ∞, lie in the equatorial plane of theblack hole38, and are solely in�uenced by an isolated black hole, which we place,without loss of generality, at the origin, fall into three qualitatively distinctcategories. These are plunge, scatter and unstable circular orbits. For a blackhole of given mass (M), and angular momentum per unit mass (a), the categoryinto which the orbit falls is determined solely by the impact parameter b andwhether the orbit is retrograde or prograde (σ) via the following relations.
b = 3
(2M2
[1 + cos
(2
3arccos
[−σaM
])])1/2
− σa⇒ Circular orbit
b > 3
(2M2
[1 + cos
(2
3arccos
[−σaM
])])1/2
− σa⇒ Scatter orbit
b < 3
(2M2
[1 + cos
(2
3arccos
[−σaM
])])1/2
− σa⇒ Plunge orbit.
We can use our understanding of scatter orbits to analyze gravitational lens-ing. By making a few approximations we can use this e�ect to come up withequations that help us calculate properties of the system. Most notably by us-ing equation (62) to work out the Mass of the source of curvature. This is animportant tool in astronomy and we have shown how it is derived from GeneralRelativity theory.
37This isn't strictly speaking true in that there is a fundamental limit on the number ofcomputations it is possible to make in any given period of time with a given computer. Usingthe algorithm of maximum e�ciency this limits the accuracy that is achievable in a period oftime. If the universe is closed then this means there is a fundamental limit on the accuracyof numerical calculations that may be achieved during the life of the universe. If the universeis open then the heat death predicted by the second law of thermodynamics would put anend to the possibility to further calculations. However it seems safe to assume that in anypractical situation other constraints would apply �rst.
38All orbits of Schwarzschild black holes are equatorial
46
Appendix AGeodesic Equation Solver
In[1]:= Clear@coord, metric, inversemetric, affine, r, Θ, Φ, tDIn[2]:= n = 4
Out[2]= 4
In[3]:= coord = 8t, r, Θ, Φ<Out[3]= 8t, r, Θ, Φ<
In[4]:= metric =
::- 1 - 2M
r, 0, 0, 0>, :0, 1 - 2
M
r
-1
, 0, 0>, 90, 0, r2, 0=, 90, 0, 0, r2 Sin@ΘD=>
Out[4]=
2 M
r- 1 0 0 0
0 1
1-2 M
r
0 0
0 0 r2 0
0 0 0 r2 sinHΘL
In[5]:= inversemetric = Simplify@Inverse@metricDD
Out[5]=
r
2 M-r0 0 0
0 1 -2 M
r0 0
0 0 1
r20
0 0 0 cscHΘLr2
In[6]:= affine := affine = Simplify@Table@1 � 2 Sum@Hinversemetric@@i, sDDL * HD@metric@@s, jDD, coord@@kDDD +
D@metric@@s, kDD, coord@@jDDD - D@metric@@j, kDD, coord@@sDDDL,
8s, 1, n<D, 8i, 1, n<, 8j, 1, n<, 8k, 1, n<DDIn[7]:= listaffine := Table@If@UnsameQ@affine@@i, j, kDD, 0D,
8ToString@G@i, j, kDD, affine@@i, j, kDD<D, 8i, 1, n<, 8j, 1, n<, 8k, 1, j<D
47Printed by Mathematica for Students
In[8]:= TableForm@Partition@DeleteCases@Flatten@listaffineD, NullD, 2D,
TableSpacing ® 82, 2<DOut[8]//TableForm=
G@1, 2, 1D -M
2 M r-r2
G@2, 1, 1D M Hr-2 MLr3
G@2, 2, 2D M
2 M r-r2
G@2, 3, 3D 2 M - r
G@2, 4, 4D sinHΘL H2 M - rL
G@3, 3, 2D 1
r
G@3, 4, 4D -cosHΘL
2
G@4, 4, 2D 1
r
G@4, 4, 3D cotHΘL2
In[9]:= geodesic := geodesic = Simplify@Table@-Sum@affine@@i, j, kDD u@jD u@kD, 8j, 1, n<,
8k, 1, n<D, 8i, 1, n<DDIn[10]:= listgeodesic := Table@8"d�dΛ" ToString@u@iDD, "=", geodesic@@iDD<, 8i, 1, n<DIn[11]:= TableForm@listgeodesic, TableSpacing ® 82<D
Out[11]//TableForm=
d�dΛ u@1D =2 M uH1L uH2L
2 M r-r2
d�dΛ u@2D =M uH1L2 H2 M-rL
r3-
M uH2L2
2 M r-r2- uH4L2 sinHΘL H2 M - rL - uH3L2 H2 M - rL
d�dΛ u@3D =1
2uH4L2 cosHΘL -
2 uH2L uH3Lr
d�dΛ u@4D = -uH4L Hr uH3L cotHΘL+2 uH2LL
r
In[12]:= Clear @listgeodesicDIn[13]:= listgeodesic := Table@8"d�dΛ" ToString@u@iDD, "=", geodesic@@iDD<, 8i, 1, n<DIn[14]:= TableForm@listgeodesic, TableSpacing ® 82<D
Out[14]//TableForm=
d�dΛ u@1D =2 M uH1L uH2L
2 M r-r2
d�dΛ u@2D =M uH1L2 H2 M-rL
r3-
M uH2L2
2 M r-r2- uH4L2 sinHΘL H2 M - rL - uH3L2 H2 M - rL
d�dΛ u@3D =1
2uH4L2 cosHΘL -
2 uH2L uH3Lr
d�dΛ u@4D = -uH4L Hr uH3L cotHΘL+2 uH2LL
r
This notebook is an adaptation of one provided by Hartle [16], in the Mathematica Notebooks Section, entiteled Christoffel.
48Printed by Mathematica for Students
Appendix BLight Orbits in Schwarzschild Geometry
The only parameter that needs to be entered is b; for plunge orbits enter b< 27 , for scatter orbits enter b> 27 and for
circular orbits enter b= 27 .
In[1]:= Clear@"Global`*"DIn[2]:= b = 5
Out[2]= 5
In[3]:= V@u_D := u^2 - 2 u^3
In[4]:= Weff = PlotAV@1 � rD, 8r, 0, 9<, PlotRange ® 0.05,
AxesLabel -> 9"r�M", "M2Weff"=, AxesOrigin ® 80, 0<, Filling ® BottomE
Out[4]=
2 4 6 8r�M
-0.04
-0.02
0.02
0.04
M 2Weff
In[5]:= dV := D@V@uD, uDIn[6]:= max = 1 � 3
Out[6]=1
3
In[7]:= wmax = V@1 � 3D
Out[7]=1
27
In[8]:= sce := PlotA1 � b2, 8r, 0, 9<, DisplayFunction ® IdentityE
In[9]:= Show@Weff, sce, DisplayFunction ® $DisplayFunctionD
49Printed by Mathematica for Students
Out[9]=
2 4 6 8r�M
-0.04
-0.02
0.02
0.04
M 2Weff
In[11]:= soln = NSolve@V@tp D � 1 � b^2, tpDOut[11]= 88tp ® -0.172458<, 8tp ® 0.336229 - 0.0540431 ä<, 8tp ® 0.336229 + 0.0540431 ä<<
In[12]:= tp1 = tp �. soln@@1DDOut[12]= -0.172458
In[13]:= tp2 = tp �. soln@@2DDOut[13]= 0.336229 - 0.0540431 ä
In[14]:= tp3 = tp �. soln@@3DDOut[14]= 0.336229 + 0.0540431 ä
In[15]:= rst = 20
Out[15]= 20
In[16]:= ust = 1 � rst
Out[16]=1
20
In[17]:= eps = 0.00000001
Out[17]= 1.´10-8
In[18]:= orbittype :=
If@H1 � b^2L == wmax, "circle", If@H1 � b^2L < wmax, "scatter", "plunge"DDIn[19]:= orbittype
Out[19]= plunge
In[20]:= norbit := If@orbittype == "plunge", 0.5, If@orbittype � "circle", 2 Π, 1DDIn[21]:= u1 := If@orbittype � "circle", tp2 H1 + epsL, ustDIn[22]:= u2 := If@orbittype � "plunge", 0.5 H1 - epsL, tp2 H1 - epsLDIn[23]:= norbit
Out[23]= 0.5
50Printed by Mathematica for Students
In[24]:= u1
Out[24]=1
20
In[25]:= u2
Out[25]= 0.5
In[26]:= theta@u_, b_, u1_D := NIntegrate@H1 � b^2 - V@wDL^H-1 � 2L, 8w, u1, u<DIn[27]:= delphi = If@orbittype � "circle", 0, theta@u2, b, u1DD
Out[27]= 4.37462
In[28]:= n@z_D := IntegerPart@zDIn[29]:= zf@z_D := FractionalPart@zDIn[30]:= ua@z_D := u1 H1 - 2 zf@zDL + u2 2 zf@zDIn[31]:= ub@z_D := u1 H2 zf@zD - 1L + 2 u2 H1 - zf@zDLIn[32]:= u@z_D := If@zf@zD < .5, ua@zD, ub@zDDIn[33]:= phia@z_D := 2 Hn @zDL delphi + theta@u@zD, b, u1DIn[34]:= phib@z_D := 2 Hn @zD + 1L delphi - theta@u@zD, b, u1DIn[35]:= accphi@z_D := If@orbittype � "circle", z, If@zf@zD < .5, phia@zD, phib@zDDDIn[36]:= x@z_D := Cos@accphi@zDD � u@zDIn[37]:= y@z_D := Sin@accphi@zDD � u@zDIn[38]:= graph = ParametricPlot@8x@tD, y@tD<, 8t, 0, norbit<,
Method -> 8MaxBend -> .001, PlotDivision ® 500<,AspectRatio ® Automatic, PlotRange ® 5, AxesLabel -> 8"x�M", "y�M"<D
Out[38]=
-4 -2 2 4x�M
-4
-2
2
4
y�M
51Printed by Mathematica for Students
In[39]:= eh = ParametricPlot@8r Sin@kD, r Cos@kD<, 8k, 0, 2 Π<,8r, 0, 2<, PlotRange ® 5, AxesLabel -> 8"x�M", "y�M"<, Mesh ® FalseD
Out[39]=
-4 -2 0 2 4
-4
-2
0
2
4
x�M
y�M
In[40]:= Show@eh, graph, DisplayFunction ® $DisplayFunctionD
Out[40]=
-4 -2 0 2 4
-4
-2
0
2
4
x�M
y�M
This notebook is an adaptation of one provided on Hartle’s website [16], in the Mathematica Notebooks Section, entiteledSchwarzorbits.
52Printed by Mathematica for Students
Appendix CLight Trajectories In Kerr GeometryThree parameters need to be entered; Σ, a and b. Σ needs to be chosen as +1 for prograde or -1 for retrograde orbits, a
should be chosen such that a<1. b should be chosen suchg that b>a. For circular motion enter b=Bc, for scatter orbits enter
b>Bc and for plunge orbits enter b<Bc.
In[1]:= Clear@"Global`*"DIn[2]:= a = 0.9
Out[2]= 0.9
In[3]:= Σ = 1
Out[3]= 1
In[4]:= co@B_D = H1 - Ha � BL^2L^3 � H27 H1 - Σ Ha � BLL^4L - 1 � B^2
Out[4]=
J1 -0.81
B2N3
27 I1 -0.9
BM4
-1
B2
In[5]:= CO = Solve@co@BD == 0, B, RealsDOut[5]= 88B ® -6.83232<, 8B ® 0.9<, 8B ® 0.9<, 8B ® 0.9<, 8B ® 1.2879<, 8B ® 2.84442<<
In[6]:= B �. %
Out[6]= 8-6.83232, 0.9, 0.9, 0.9, 1.2879, 2.84442<
In[7]:= Bc = Max@%DOut[7]= 2.84442
In[8]:= b = 3.5
Out[8]= 3.5
In[9]:= W@u_D := H1 - Ha � bL^2L u^2 - 2 H1 - Σ ´ Ha � bLL^2 u^3
53Printed by Mathematica for Students
In[10]:= Weff = PlotAW@1 � rD, 8r, 0, 9<, PlotRange ® 0.14,
AxesLabel -> 9"r�M", "M2Weff"=, AxesOrigin ® 80, 0<, Filling ® BottomE
Out[10]=
2 4 6 8r�M
-0.10
-0.05
0.05
0.10
M 2Weff
In[11]:= dW@f_D := D@W@fD, fDIn[12]:= dW@fD
Out[12]= 1.86776 f - 3.31102 f 2
In[13]:= max = NSolve@dW@exD == 0, exDOut[13]= 88ex ® 0.<, 8ex ® 0.564103<<
In[14]:= ufwm = ex �. max@@2DDOut[14]= 0.564103
In[15]:= Wmax = W@ufwmDOut[15]= 0.0990569
In[16]:= sce := PlotA1 � b2, 8r, 0, 9<, DisplayFunction ® IdentityE
In[17]:= Show@Weff, sce, DisplayFunction ® $DisplayFunctionD
Out[17]=
2 4 6 8r�M
-0.10
-0.05
0.05
0.10
M 2Weff
In[19]:= soln = NSolve@W@tp D � 1 � b^2, tpDOut[19]= 88tp ® -0.258733<, 8tp ® 0.413438<, 8tp ® 0.69145<<
54Printed by Mathematica for Students
In[20]:= tp1 = tp �. soln@@1DDOut[20]= -0.258733
In[21]:= tp2 = tp �. soln@@2DDOut[21]= 0.413438
In[22]:= tp3 = tp �. soln@@3DDOut[22]= 0.69145
In[23]:= rst = 20
Out[23]= 20
In[24]:= ust = 1 � rst
Out[24]=1
20
In[25]:= eps = 0.00000001
Out[25]= 1.´10-8
In[26]:= orbittype :=
If@H1 � b^2L � Wmax, "circle", If@H1 � b^2L < Wmax, "scatter", "plunge"DDIn[27]:= orbittype
Out[27]= scatter
In[28]:= norbit := If@orbittype == "plunge", 0.5, If@orbittype � "circle", 2 Π, 1DDIn[29]:= u1 := If@orbittype � "circle", tp2 H1 + epsL, ustDIn[30]:= u2 := If@orbittype � "plunge", H1 - epsL � H1 + H1 - a^2L^0.5L, tp2 H1 - epsLDIn[31]:= norbit
Out[31]= 1
In[32]:= u1
Out[32]=1
20
In[33]:= u2
Out[33]= 0.413438
In[34]:= theta@u_, b_, u1_D := NIntegrate@H1 � b^2 - W@wDL^H-1 � 2L, 8w, u1, u<DIn[35]:= delphi = If@orbittype � "circle", 0, theta@u2, b, u1DD
Out[35]= 2.51897
In[36]:= n@z_D := IntegerPart@zDIn[37]:= zf@z_D := FractionalPart@zDIn[38]:= ua@z_D := u1 H1 - 2 zf@zDL + u2 2 zf@zDIn[39]:= ub@z_D := u1 H2 zf@zD - 1L + 2 u2 H1 - zf@zDLIn[40]:= u@z_D := If@orbittype � "circle",
H1 - Ha � bL^2L � H3 H1 - Σ Ha � bLLL, If@zf@zD < .5, ua@zD, ub@zDDD
55Printed by Mathematica for Students
In[41]:= phia@z_D := 2 Hn @zDL delphi + theta@u@zD, b, u1DIn[42]:= phib@z_D := 2 Hn @zD + 1L delphi - theta@u@zD, b, u1DIn[43]:= accphi@z_D := If@orbittype � "circle", z, If@zf@zD < .5, phia@zD, phib@zDDDIn[44]:= x@z_D := Cos@accphi@zDD � u@zDIn[45]:= y@z_D := Sin@accphi@zDD � u@zDIn[46]:= graph = ParametricPlot@8x@tD, y@tD<, 8t, 0, norbit<,
Method -> 8MaxBend -> .001, PlotDivision ® 5000<,
AspectRatio ® Automatic, PlotRange ® 7, AxesLabel -> 8"x�M", "y�M"<D
Out[46]=
-6 -4 -2 2 4 6x�M
-6
-4
-2
2
4
6
y�M
In[47]:=
56Printed by Mathematica for Students
In[48]:= eh = ParametricPlot@8r Cos@tD, r Sin@tD<, 8t, 0, 2 Π<, 8r, 0, 1 + H1 - a^2L^0.5<,
PlotRange ® 7, AxesLabel -> 8"x�M", "y�M"<, Mesh ® FalseD
Out[48]=
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x�M
y�M
In[49]:= Show@eh, graph, DisplayFunction ® $DisplayFunctionD
Out[49]=
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x�M
y�M
This notebook is an adaptation of one provided by Hartle [16], in the Mathematica Notebooks Section, entiteledSchwarzorbits.
57Printed by Mathematica for Students
References
[1] James B Hartle, Gravity an Introduction to Einstein's General Relativity,Addison Wesley, San Francisco, 2003
[2] Tai L. Chow, Gravity, Black Holes, and the Very Early Universe, SpringerScience+Business Media, New York, 233 Springer Street, 2008
[3] M. P. Hobson, G. Efstathiou, A. N. Lasenby, General Relativity An Intro-duction for Physicists, Cambridge University Press, The Edinburgh Build-ing, CB2 2RU, 2006
[4] Robert M. Wald, General Relativity, University of Chicago Press, London,1984
[5] Nathan Cohen, Gravity's Lens, John Wiley and Sons, Chichester, 1988
[6] Barrett O'Neill, The Geometry of Kerr Black Holes, A K Peters, Wellesley,Massachusetts, 1995
[7] J. Ehlers and P. Schneider, Gravitational Lensing, Edited By R J Gleiser,C N Kozameh, O M Moreschi, General Relativity and Gravitation: Pro-ceedings of the 13th International Conference on General Relativity andGravitation, Institute of Physics Publishing, Techno House, Redcli�e Way,Bristol, BS1 6NX, 1992
[8] Warren, S. J., Hewett, P. C., Lewis, G. F., Moller, P., Iovino, A., andShaver, A candidate optical Einstein ring, Monthly Notices of the RoyalAstronomical Society, Volume 278, Issue 1, pp. 139-145, 1996
[9] R. H. Boyer and R. W. Lindquist, Maximal Analytic Extension of the KerrMetric, Journal of Mathematical Physics, Volume 8, Number 2, pp. 265-282, 1976
[10] J. D. Bekenstein, Black Holes and The Second Law, Letere Al Nuovo Ci-mento, Volume 4, Number 15, pp. 737-740, 1972
[11] Robert J. Nemiro�, Visual Distortions Near a Neutron Star and BlackHole, American Journal of Physics, Volume 61, pp. 619-631, 1993
[12] Y. C. Leung C. G. WangMinimum Mass of a Neutron Star, Nature PhysicalScience, Volume 240, pp. 132-133, 1972
[13] Eric W. Weisstein, Schwarzschild, Karl (1873-1916), Wolfram Research,http://scienceworld.wolfram.com/biography/Schwarzschild.html,written: 1996, viewed: 2011
[14] Authorship unknown, Roy P. Kerr., Encyclopaedia Britannica, http:
//www.britannica.com/EBchecked/topic/315523/Roy-P-Kerr, written:Unknown, viewed: 2011
58
[15] Sean M. Carroll, Lecture Notes on General Relativity, Enrico Fermi Insti-tute University of Chicago, http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html, written: 1997, viewed: 2011
[16] James B. Hartle, Gravity an Introduction to Einstein's General Rela-tivity, Addison Wesley, San Francisco, http://wps.aw.com/aw_hartle_gravity_1/7/2001/512494.cw/index.html, written: 2003, viewed: 2011
[17] J. J. O'Connor and E. F. Robertson,Wilhelm Karl Joseph Killing, School ofMathematics and Statistics - University of St Andrews, Scotland, http://www-history.mcs.st-and.ac.uk/Biographies/Killing.html, written:2005, viewed: 2011
[18] Gene Smith, Gravitational Lenses, University of California, San DiegoCenter for Astrophysics & Space Sciences, http://casswww.ucsd.edu/
public/tutorial/GLens.html, created: 1999, viewed: 2011
[19] Author Unknown, Gravitational Lens001, Washington University inSt. Louis, http://www.epsc.wustl.edu/classwork/classwork_210a/
transparencies/bernard/gravitational_lens001.jpgl, created: 2008,viewed: 2011
[20] Hubble Telescope, Picture Album: Einstein Ring Gravitational Lens,NASA, http://hubblesite.org/gallery/album/entire/pr2005032g/
web/, taken: unclear, viewed: 2011
[21] Eric Weisstein, Cubic Formula, Wolfram MathWorld, http://mathworld.wolfram.com/CubicFormula.html, written: 1999, viewed: 2011
[22] Joachim Wambsganss, Gravitational Lensing in Astronomy, Astro-physikalisches Institut Potsdam, http://hermes.aei.mpg.de/lrr/1998/12/article.xhtml, written: 1998, viewed: 2011
59
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