The In uence of Black Holes on Light RayTrajectoriesConrad BrownApril 26, 20111
Declaration of AuthorshipThis 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.2
AbstractA 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 trajectorydepends on the impact parameter (b) of the light ray, the mass (M ) and theangular momentum (J ) of the black hole. Only trajectories external to theevent horizon, and starting a large distance from the black hole are considered.3
AcknowledgmentsFirstly 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.4
Contents1 Introduction72 Non-Euclidean Geometries81.1 A Note on Notation . . . . . . . . . . . . . . . . . . . . . . . . .2.1 The Metric and Line Element . . . . . . . . . . . . . . . . . . . .2.2 Scalar Products in Non-Euclidean Geometries . . . . . . . . . . .3 Special Relativity Extended3.1 Units . . . . . . . . . . . .3.2 World Lines . . . . . . . .3.3 Geodesics . . . . . . . . .3.3.1 Timelike Geodesics3.3.2 Null Geodesics . . . . . . . . . . . . . . . .Finding the Schwarzschild Metric . . . . . . . . .Singularities and The Event Horizon . . . . . . .Null Geodesics In Schwarzschild Geometry . . . .Symmetry . . . . . . . . . . . . . . . . . . . . . .E ective potential for Schwarzschild Black Holes4.5.1 Interpreting b . . . . . . . . . . . . . . . .4.5.2 Plotting Wef f . . . . . . . . . . . . . . . .4.5.3 Plunge Orbits . . . . . . . . . . . . . . . .4.5.4 Scatter Orbits . . . . . . . . . . . . . . .4.5.5 Circular Orbits . . . . . . . . . . . . . . .4 Schwarzschild Geometry184.108.40.206.44.55 Gravitational Lensing5.1 Calculating the Angle of De ection . .5.1.1 Calculating Small De ections .5.1.2 Light De ection by the Sun . .5.1.3 The Thin Lens Approximation.6 Light Orbits in the Equatorial Plane of Kerr Black Holes220.127.116.11.46.5Kerr Metric and Line Element . . . . . . . . . . . . . .Boyer-Lindquist Coordinates . . . . . . . . . . . . . . .Singularities in Kerr Geometry . . . . . . . . . . . . . .Symmetries of Kerr Geometry . . . . . . . . . . . . . . .E ective Potential for Kerr Black Holes . . . . . . . . .6.5.1 Circular Orbits . . . . . . . . . . . . . . . . . .6.5.2 Prograde vs. Retrograde . . . . . . . . . . . . .6.5.3 Variation in Angular Momentum per Unit 333435353637394143447 Conclusion46A Geodesic Equation Solver475
B Light Orbits in Schwarzschild Geometry49C Light Trajectories In Kerr Geometry53List of Figures12345678910111213141516171819202-D Manifold sat in R3 , and Tangent Plane . . . . . . . . .Impact Parameter b . . . . . . . . . . . . . . . . . . . . . .Wef f in Schwarzschild Geometry . . . . . . . . . . . . . . .Wef f Graph for Plunge Orbit in Schwarzschild Geometry .Plunge Orbit . . . . . . . . . . . . . . . . . . . . . . . . . .Wef f Graph for Scatter Orbit in Schwarzschild Geometry .Scatter Orbit . . . . . . . . . . . . . . . . . . . . . . . . . .Wef f Graph for Circular Orbit in Schwarzschild Geometry .Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . .Gravitational Lens . . . . . . . . . . . . . . . . . . . . . . .Gravitational Lensing on 2-D Manifold sat in R3 . . . . . .Cross-section of rst and second Einstein Rings . . . . . . .Einstein Ring . . . . . . . . . . . . . . . . . . . . . . . . . .Thin Lens Approximation . . . . . . . . . . . . . . . . . . .Wef f in Kerr Geometry . . . . . . . . . . . . . . . . . . . .Circular orbits in Kerr Geometry . . . . . . . . . . . . . . .Plunge orbits in Kerr Geometry . . . . . . . . . . . . . . . .Scatter orbits in Kerr Geometry . . . . . . . . . . . . . . .Scatter and Plunge orbits in Kerr Geometry . . . . . . . . .Distance from the Event Horizon to the Photon Sphere . .6.922232324252526272828293033404343444445
1IntroductionIn 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 nonrotating, 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 approximate 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 NotationThroughout this document we use notations and conventions broadly the sameas those found in Hartle , 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.1 This is in its traditional formulation. There are formulations in which light curves, normally 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 assumption of massive light.2 See Weisstein .3 An interesting consequence of complete gravitational collapse is that much of the information 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 .4 See the Encyclopaedia Britannica's website .7
2Non-Euclidean Geometries2.1 The Metric and Line ElementIn 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  - Section 7.2.A geometry in coordinates xα is described by the line element:(1)ds2 gαβ (x) dxα dxβmetric5 .where gαβ (x) is a symmetric, position dependent, matrix called theAswe are dealing with four-dimensional spaces, with coordinates xα (x0 , x1 , x2 , x3 ),the general form is:g00 (xα ) g01 (xα )gαβ (x) 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 coordinates xα (w, x, y, z), is: 1 0 000100 00 .0 10010(2)The metric for the at (Minkowski) spacetime (ηαβ ) of inertial frames, in Cartesian coordinates xα (t, x, y, z), is: ηαβ 0 0 01 0 0 .0 1 0 0 0 1 1 0 00(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 byˆ s bdsa5 Hereand elsewhere we make use of the Einstein summation convention.8(4)
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) 2.2 Scalar Products in Non-Euclidean GeometriesTo 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.  - Chapter 3. For a more detaileddiscussion of manifolds see O'Neill  - 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 twodimensional 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:αv(x ) v (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 limαδx 0δsδx α(6)where δs is an in nitesimal displacement vector between the point P and a9
point in the neighborhood of P that is a distance δx α 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 )dx α .(7)Using equation (7), in combination with equation (1), we get:gαβ (x) dxα dxβ ds2 ds · d s eα (x ) dx α · eβ (x ) dx β (eα (x ) · eβ (x )) dx α dx β eα (x) · eβ (x) gαβ (x) .(8)a · b gαβ a α b β(9)Thus we have:3Special Relativity ExtendedThis 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 UnitsIt is to be remembered that, as in Special Relativity, we simplify our calculations 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) M G/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  - Section 18.104.22.168 World LinesThe 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  - 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 spacelike10
( s)2 0 lightlike( s)2 0 timelike.(10)This classi cation system still applies in general relativity.3.3 Geodesics3.3.1 Timelike GeodesicsIt 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  - Section 8.1. Inthe following section we look at the lightlike case wh
Finally we look at rotating black holes 3, called Kerr black holes, which are named after Roy Kerr (1934-present) 4. In order to maintain as much symmetry as we can, in the Kerr case, we focus on equatorial orbits where the classi cation system we applied to light trajectories in Schwarzschild geometry still applies and is still exhaustive for orbits from in nit.y 1.1 A Note on Notation .
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