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# SPEC & GEN RELATIVITY PHZ 6607

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Special and General Relativity Lecture Notes Day 17 110408 Shawn Mitryk Contents 1 O CO Horizons 11 De nitions i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 112 Null Solutions i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 113 Schwarzschilcl Metric i i i i i i i i i i i i i i i i i i i i i i i i i 114 Isotropic Coordinates i i i i i i i i i i i i i i i i i i i i i i i i 1 Comparison to Electrodynamics Next Class 311 Reading i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 6 1 Horizons 11 De nitions 0 Event Horizon global o Apparent Horizon local convenient for Numerical Relativity 0 Killing Horizon Not a 77horizon77 rather a geodesically complete null surface 12 Null Solutions 0 Minkowski X2 7 T2 0 o Schwarzschild R2 7 T2 0 13 Schwarzschild Metric 2M 1 2 7 7 7 7 2 7 2 2 2 d3 7 1 T ldt 17 dr 7quot d9 1 Making the transformation T2 7 R2 7 le we obtain 3 432 e 7dT2dR2r2d92 2 7 ln Mikoswski coordinates there is a killing vector of the form K zapT8 This is much like the angular momentum Pl may 7 you Taking the vector K X7 T7 07 0 We can show KHKquot 7X2 T2 gt 0 for X iT K orbits cover the killing horizon X2 7 T2 0 7 The Killing vector is null on the killing horizon but KHVHK 7HK 0 Taking X T7X7070 and U 171070 olt K Then a 7 7 ltvlmgtltvwgt lt3 n2 7 gw mvwxv w lt4 2 7 1 7 H 7 7 71711 5 H 11 6 ln schwartzchild We consider the killing vector K RVT TV Noting 3 7T WK 32M eW7R2T2 0RiT 7 7 Taking the transformation V TRgtRV7U 8 U TiRgtTVU 9 E 8U 6 8V 6 W WW 1 WW 10 Thus we can de ne the killing vector in the form K RBT Jr T63 11 1 E E 1 E E K V7UWW VU7WW 12 E E K iUw VW 13 Then recalling V e 14 U 76 15 Thus K becomes 6t 6 6T 6 at 6 6T 6 K 7UEEVEE 16 We then want to nd a function such that f d39r iUdV 7 VdU Thus we nd if 71 an 7 And nally f7 iUV 17 f39r eviuilM 18 M 1i 7 11aW 19gt Taking the ratio 76W Differentiating dV V e Vi dU 2Mdt 20 From Which we can read dt 2M 72M W 7 U62 21 dt 2MV 72M W We U 22 Thus we can calculate the killing vector K 72M 6 2M 6 E K 7UTE 4ME 23 K 4M0 0 0 24 From Which we calculate H2 gVMKVV LKV 25 1 n2 7 ltwngtKVgtltvpKUgt 26gt VOKO FgoKO 27 VOKl rgoKO 28 V1K rglKO 29 V1K1 rglKO 30 Finally 1 H2 ElQOOQOMVoKO 900911V0K12911900V1K02911911V1K313 1 2M M 4M Alarm 2 7 777 77 2 7 2 7 7 7 H 7 2 lt1 TgtltT21 gt kn 32gt 16M4 H T4 enmi 33 De ning a 77more useful77 form k a 34 M2 H2 T7 35 M 1 H T7 QT2M m 36 14 Isotropic Coordinates The Schwarzschild metric reads 17 M M d2 1723M 1774M R249 37 s 17 lt 2R gt H Flgute 1 Kluskal Mappmg Conformal Mappmg Ra maps the left Side of the plot to the ught Side For small ltlt1 dsi 717 dt7 1 dll7 Rim 2 Comparison to Electrodynamics Consldet e charge denslty p Q 7dV W 4120 Q emdietnamonw Ed lt38 lt39 lt40 41 But thls has no General Reletmty eqnwelent There exlsts no volume den nty for mass or energy 01 I constant 2M 0 Owl 1 Tn0n0 2M quotM 1 iw0n0n0 T We note the followlng Ieptesentatlons for the Energy 1 1 E 7 i 1 VM ml Konm 41G ADJ z 7 who lt44 lt45 Note that K is a timelike killing vector Ara 1 Another form 1 V V EADM mBz d21v720iajh aihj 46 Where 9 n V hW has a solution if h is of the order Tin falls off fast enough for integral to converge Finally we also note the angular momentum 1 JKoma r 7 d21v 72nHUVV R 47 87rG 32 48 Where R is the rotational killing vector 5 3 Next Class 31 Reading 0 Ch 33 in Gravitation Misner7 Thorne and Wheeler 0 Section 67 and Chapter 7 in Spacetime and Geometry Carroll 0 p 8an02 22 2 A2 2 P22 22 7r a 0 Asin 0ld gt Zdr p 10 where AT272GMTCLZ p2 r2 a2 cos20 JaM Where does gtt go to zero Where does gM go to 00 Where does 9 go to 007 9 goes to zero when r2 a2c0s20 7 QGMT 0 r2 1217 sin20 7 QGMT 0 A2 7a2sin20 0 9M goes to zero at A 0 then Arirrir Tgt7L hr and L are both event horizons z r2 a212 sin0cos gt y r2 a212 sin0sin gt zrcos0 In the neighborhood of 7 it is linear7 need to do a coordinate transfor mation PHZ 6607 Special and General Relativity I Handout 5 Consider the two dimensional metric for at space in cartesian coordinates d52 dx2 d242 and impose the coordinate transformation 3 zcos l ysin l and y inc sin l ycos for which the new metric becomes d82 dab2 dyZ As you know this is a simple example of a rotation which preserves angles and distances and it clearly leaves the form of the at space metric unchanged Now consider the coordinate transfor mation x zcosh6 tsinh l and atsinh l tcosh H for the two dimensional at spacetime metric c132 7dt2 dzZ You will find it straightforward and to your advantage to show that the form of this metric also remains unchanged and you will recognize this coordinate transformation as a simple example of a Lorentz transformation for which vc tanh l and y cosh 6 Thus boosts are to a Lorentzian or pseudo Reimannian geometry with indefinite metric what rotations are to an Euclidean or Reimannian geometry with definite metric In accord with the similarity we have just established between these two operations there is also a Lorentzian analogue of the transformation to polar coordinates z39e x p COShT andt psinhT for which the metric becomes ds2 7p2d r2 dp2 Although you will recognize that this coordinate transformation as given is not defined over all of Minkowski space a similar transformation turns out to be extremely important in the study of black hole spacetimes and we will be making further use of it in that context Another exercise which is very helpful in developing an understanding of black hole spacetimes is to solve the equations of motion for a particle whose Lagrangian is given by l imngl c in which 39 ddA where now the metric is that of four dimensional at spacetime but again expressed in spatial spherical polar coordinates To obtain the full advantage of this exercise it is important to have explicit expressions for the conserved components of linear and angular momentum in terms of the canonical momenta defined from this Lagrangian namely 8 dx39 You should give this exercise some further thought since it will be appearing as a homework problem shortly INTRODUCING VECTORS THE OLD AND THE NEW The concept of a metric can be introduced as a fairly simple generalization of Pythago ras7s theorem But the definitions a vector is a tensor is a manifold is may at first sight seem a little formal Occasionally you will find it useful to refer back to their formal de nitions but for the most part you can use minor extensions of concepts that you may have already encountered but only implicitly An important part of what we shall do is to make those hidden concepts explicit Within certain inexactitudes we will do this by recognizing familiar examples first rather than by discussing the formal definitions A good example with which to start is the idea of a vector which in a schoolboy definition was a directed line segment77 Well our vectors will be that and so much more But other related ideas which we might take for granted will not automatically be applicable For example the idea of a length for a vector rests upon a scheme for measuring length and whereas a vector itself does not represent such a scheme a metric does previously you have probably always implicitly used a metric without specific reference to it Also the apparently fundamental idea of a position vector will be seen to break down in an elementary way with the formal definition we shall work with henceforth The first generalization of the notion of vector we shall introduce is the idea that we can have vectors of different types In fact you will already have used these two types previously without distinction For us this distinction will prove very useful and an advantage which it entails is that the two types can be dotted into each other without the need for a metric What will pass for ordinary vectors we will represent with an index upstairs thus Va The other type we shall denote with its index downstairs Wu and in three dimensions the ordinary vector cross or wedge product will produce such an entity almost As you will recall the vector triple product is a scalar and is obtained by dotting a vector into a cross product that is by something of the form VaWa where we have introduced a convention in which the repeated index is automatically to be summed as is required for a dot product No reference to a metric was necessary to construct this scalar Instead we have used the two different types of vectors to accomplish that In fact the notation above already hides part of the definition we have always used for a vector since the idea of basis vectors is implied but not stated The notation above refers simply to components of the vectors in their respective bases while a further con vention about the dot product between different basis vectors is hidden in the summation convention used above Although often we will restrict our use to what are called coor dinate bases you have already encountered both coordinate and non coordinate bases for example when you have dealt with the vector potential in cartesian and polar coordinate systems in Electromagnetism In particular a coordinate basis may conveniently be used to investigate how coordinate transformations affect the two different types of vectors As you might guess the transformation properties of basis vectors and components are com plementary so that vectors themselves remain unchanged but not their representation 2 INTRODUCING TENSORS THE NEW AND THE NEWER It should have occurred to you as curious by now that the cross product of two vectors of one kind each with an index upstairs should be capable of resulting in a vector of the other type with its index downstairs In this case to understand clearly we have to both introduce the broarder notion of a tensor and relate it to a particular object of previous encounter namely the totally antisymmetric 5 symbol The notion of a tensor itself is fairly easy to introduce Examples of tensors can be obtained from a particular product of vectors in which the indices are all independent and not in general summed for example Tab XaYb would be a tensor of rank two as also would Van2 and WaWb in particular though again of different types In fact I shall later have to point out that there is a deception in this statement 3 PHZ 6607 Special and General Relativity I Handout 1 General Relativity is an exceptionally powerful physical theory Ultimately it is also a theory about geometry the geometry of a four dimensional spacetime In that theory timelike geodesics represent the possible paths of in nitesimal test particles whereby curvature in the spacetime re ects the presence of matter subjecting the test particle to an in uence which we have familiarly come to know as the gravitation force due to that matter But the theory purports more not only is curvature in the spacetime geometry meant to be due to the presence of physical matter but also that curvature in turn determines the motion of all matter present corresponding entirely to its gravitational interaction all other interaction between the matter being determined by the other known forces in physics in the usual way From this point of view geodesic motion for a point in nitesimal test particle which is presumed to have no in uence back on the spacetime itself represents one of the simplest cases of matter motion which might be taken for consideration Why is a theory such as this necessary Firstly Newtonian gravity is not a relativistic theory Secondly Special Relativity which serves perfectly well to account for the behaviour of electro magnetic elds in a at space does not deal in any way with the consequences of gravitational interactions Thirdly Quantum Mechanics makes us acutely aware that particles even photons identified as particles must lose energy when climbing out of a gravitational potential Presum ably then local electromagnetic energy density must depend on exactly where a local region lies within a gravitational potential We have seen that the scalar potential of electrostatics must in general become a vector potential in the full theory of classical electromagnetism We are then forced to ask what generalization becomes necessary for the static Newtonian potential especially to describe situations characterized by very high relative velocities General Relativity is one of the simplest attempts to answer that question which it does in a fully relativistic way and which so far is not in con ict with experiment nor with other direct or indirect physical observation General Relativity is not the only current theory of gravitation Moreover whenever an attempt has been made to construct the corresponding quantum theory serious di iculties have always arisen which became insurmountable suggesting to some that if General Relativity is a useful theory in physics it is only as the classical low energy limit of some more fundamental and very different quantum theory There is so little hard experimental evidence to support this view at the present time but overwhelming evidence within a specific theoretical framework Nevertheless the di iculty of understanding General Relativity fully is currently so great that we shall entirely content ourselves with its direct investigation in this course leaving any more complicated approach to a different study One further observational aside may well be in order given the current climate That part of quantum mechanics which showed how a system could have a ground state and a discrete spectrum of excited states was far easier to accomplish than was solving all details of the Hydrogen atom Even today in the eyes of some Mathematical purists the existence of the photon as a quantum particle state in electromagnetism is still not completely proven though there are probably few Physicists who seriously doubt that proposition The first part of our observation is that the same problem for the propagating degrees of freedom in General Relativity is basically no harder in at spacetime than it is for electromagnetism It is the nonlinearities in strong field regions and the non propagating degrees of freedom which present the most serious problem for a quantum theory of General Relativity On the other hand and this is the second part of our observation the non propagating degrees of freedom are much closer in nature to the particle aspects of the Hydrogen atom problem than they are to eld theory aspects of electromagnetism PHZ 6607 Special and General Relativity I Handout 4 One dif cult result we will have to deal with at this point is the fact that in three space the so called position vector is not a vector under general coordinate transformations although it does behave as a vector under the restricted class of transformations known as rotations However the components of in nitesimal displacement really do form a contravariant vector since they satisfy the appropriate transformation law m 8952 39 dac d Thus some of the most common vectors we use to refer to the motion of a particle in Newtonian physics namely velocity and momentum both behave as contravariant three vectors in that context time is merely a parameter with which we can label points along the path of the particle Before considering acceleration we will need to know how to deal more fully with the differen tiation of a vector The fact that proper four vectors can be formed in a similar way also plays a crucial role in the invariant formulation of special relativity This will be a particularly useful perspective to keep in mind when we come to deal with special relativity again shortly At that time we will also expand on the usual understanding of electromagnetism within the framework of special relativity Current reading A rst course The first two thirds of this book cover the introductory material necessary to be able to write down and make sense of the non linear field equations of General Relativity It also includes an extensive treatment of the familiar topic of special relativity which we will use heavily in a generalization of existing knowledge to the wider context needed for analysis on manifolds and in particular for General Relativity The last third of the book deals with application of the equations to modern topics in astrophysics including black holes and gravitational radiation 52 Tensor algebra in polar coordinates 53 Tensor calculus in polar coordinates 54 Connections Le Christoffel symbols and the metric 55 The non tensorial nature of P36 56 Noncoordinate bases 59 Exercises 4 8a Geometrical methods Much of this book is concerned merely with giving a new mathematical formulation to ideas which are already familiar from physics Thus through it topics such as vectors the inertia tensor inner products special relativity spherical harmonics the rotation group and angular momentum operators conservation laws the theory of intergration gradient divergence and curl Gauss7 and Stokes7 theorems of vector calculus Maxwe117s equations and other gauge theories of physics will all take on a new light based on a deeper and uni ed understanding founded in a geometrical approach to these topics On the other hand a topic such as fiber bundles is not essential to General Relativity and we will not be concerned with it in this course However the last chapter includes many of the concepts and methods which are most heavily used in General Relativity and we will deal with it in some detail Chapter 1 Some Basic Mathematics Exercise 21 PHZ 6607 Special and General Relativity I Handout 6 The velocity vector along a curve is a familiar example of a contravariant vector The gradiant operator acting on a function de ned everywhere on a manifold is also familiar and it creates a covariant vector field which is itself defined at every point on a manifold Their dot product can be defined without any aditional structure on the manifold and it results in something we can actually identify namely the total change of the function along the curve which is a scalar quantity Z Z Z t dt 201 One might perhaps use an anology from quantum mechanics to clarify the distinction between a covariant and a contravariant vector In quantum mechanics one has both bra states QM and ket states We also normally think that we can map from one of these to the other but technically this requires a structure called complex conjugation to be defined with respect to Hilbert space The advantage of having both types of state vectors on Hilbert space is that we can form their their dot or inner product Similarly without a metric covariant and contravariant vectors are completely distinct but we can still form their dot product The advantage of a metric as a defined structure on a manifold anologously to the complex structure on Hilbert space is that it allows to construct a map from contravariant vectors to covariant vectors INTRODUCTION ON DIFFERENTIAL GEOMETRY General relativity is a theory of the geometry of spacetime and of how it responds to the presence of matter Thus our first task will be to come to an understanding of geometry suitable for use un the theory of general relativity Geometry may mean slightly different things in different contexts and even slightly different things to different people The primary elements we will have to deal with are distances and shape We will need invariant ways to refer to distances and to characterize shape we will most frequently use curvature As in real life the distance between two points will generally depend on the path taken between them We will thus actually compute distance by integration along each particular path The basic construct we will use for each infinitessimal element of path length will be a generalization of Pythagoras7s theorem and of the familiar cosine rule This construct will be embodied in the introduction of a metric simply a convenient way of specifying infinitessmal distances An orange has a different shape from a sheet of paper There are both topological and geometrical differences between these two objects and for the most part we will concentrate on the geometrical differences as determined by the curvature We will find that some measures of curvature depend on how we choose to refer distinct points on the objects while othere do not and invariance will be an important part of out investigation Curvature will turn out to be very important for us since it is the quantuty which relates the presence of matter its in uence on the geometry of spacetime So far I have been referring to points and elements of path length as though they were well understood familiar entities While that is true our use of them will eventually go beyond the currently familiar and we will need to introduce one other very important geometrical notion that of a manifold This will be necessary not only from a purely geometric point of view but also to recreate the familiar notion of vector in a suf ciently broad and more geometrical context Vectors and tensors their generalization will become our bread and butter This course is not just about geometry Matter curves geomerty in particular ways and we will especially want to understand and investigate those ways But matter moves very differently in a curved space and we will also need to become familiar with those new and peculiar ways A good springboard for launching into a full awareness of general relativistic effects will be to look at the relatively minor changes which occur when curva ture is everywhere small This too will require a particular language but by then we will be able to use the language already learnt in the discussion of curved lines in a curved manifold PHZ 6607 Special and General Relativity I Handout 2 A manifold can be described simply as a mapping of a set of points into Rquot Typical well known examples are the solid cube the solid sphere and the solid donut as three dimensional objects with their obvious embedding in R3 What might not be at all clear is whether the surfaces of each of these objects can be embedded as two dimensional manifolds into R2 In fact none of them can be embedded globally but a re nement of our de nition allows us to continue to regard all these surfaces as two dimensional manifolds It turns out to be su icient practically to require that the embedding can be done locally in open balls7 which overlap to cover entirely the complete set of all points to be embedded We will nd it necessary to de ne geometric structures on the manifolds with which we shall working We will also nd it very convenient to refer to other objects de ned on manifolds and the most common of these will be tensors of which vectors are a simple and familiar example However perhaps in contrast with your current practice we will use index position to have a meaning and will for a start consider Vi and Vi as inequivalent allowing them to refer to dif ferent tensor types The most distinguishing property of tensors which we need to comprehend is their transformation law under changes of coordinates on the underlying manifold These changes we can recognize from the way partial derivatives change under coordinate transformations Thus z g V 816 V for vectors with their index upstairs which we will refer to as contravarz39ant vectors and li ja a z for vectors with their index downstairs which we will refer to as covariant vectors For tensors of more general type with both upstairs and downstairs indices the transformation law is simply a generalization of the above with one factor for each index the type of factor depending on the type of index upstairs or downstairs Certain operations on tensors can now be de ned without any additional geometric structure being given Perhaps the most obvious are multiplication between tensors to form tensors of higher rank and contraction of a pair of indices to form a tensor of lower rank For example AiAj AiBj and 3133 are all different examples of distinct types of tensors of rank two while AiBi would be an example of contraction to a tensor of rank zero Le a scalar Notice that at present we have no way of de ning the sort of contraction you might expect to correspond to 2 which requires the additional structure known as a metric A perhaps unexpected but familiar example of a second rank tensor might be a square matrix M27 de ned on a manifold is a scalar known as the trace of the matrix We can also see that a matrix can be contracted with a vector to form another vector of the same type eg 14ng and MgBj while the contraction of a matrix with another leads to a third matrix exactly as in the familiar matrix multiplication z39e Pf Mng Two other special tensors at least one of which you may have encountered before are eijk and Eijk each de ned to be 1 71 or 0 in locally cartesian coordinates depending on whether is an even odd or not a permutation of 123 With these we can also de ne DetM eijkM Mijgelmn as well as the familiar vector triple product eijkAinDk Finally another operation which can sometines be used to de ne a new tensor is differentiation Thus 813 7 BjBZ Exterior Airin 7 OiBiAj and AlliB Bizin Lie where 81 88152 are all tensors More general differentiation requires additional structure a connection PHZ 6607 Special and General Relativity I Handout 3 You have just seen that the invariant line element d32 7dt2 dac2 dy2 d22 is preserved under a restricted Lorentz transformation In fact the complete Lorentz group of transformations can be de ned as that which leaves this invariant element unchanged This might appear to be the reverse of the order in which things have been de ned for you in Special Relativity but actually we also de ne other groups of transformations in exactly this way Two examples which we shall encounter again are the Rotations in three space which leave angles and distances unchanged and Conformal transformations which leave angles unchanged but not distances This line element also introduces one of the structures on manifolds which we shall be work ing with continually namely the metric The metric carries the information which determines distances and time separations and is very helpful in establishing a relationship between the manifolds with which we shall be working and the physics on them which we shall be interested in discussing Another structure which will also prove to be essential is a thing called a connection which again can be demonstrated from purely at space considerations Unlike the metric the connection is not a tensor but it will be required in the de nition of the curvature tensor which we shall also be using repeatedly There is a very easy way to witness the emergence of a connection As you will notice from a course textbook the equation for a geodesic as a suitably parameterized curve embedded in a higher dimensional manifold may be given by d2i Z dxj dank d WK where A is a parameter along the curve in three dimensional space it may be taken as time Now geodesics in Euclidean three space R3 are actually straight lines And you also know that free motion in at space takes place in straight lines Suppose then we look at trying to derive this information in spherical polar coordinates instead of the usual cartesian coordinates Then for some chosen orientation of coordinates the non relativistic Lagrangian for this motion would be 1 1 m02 gm f2 r262 1 r2 sin26 12 The Euler Lagrange equations become 77 1 r612 1 7 sin26 2 0 7r26l39 r2 sin l cos l 2 0 and 7r2 sin26 0 where time has indeed assumed the role of the parameter A above From these we are almost immediately able to read off and identify 1 1 Pge 77 Pk 77 sin2 6 P29 1 qu P Lid isin lcos l and Pg cot 6 all other possibilities being zero We shall shortly nd the geometrical reason for these results which in at space correspond to a rather special but physically natural choice for the structure which has been called the connection and is represented here by P PHZ 6607 Special and General Relativity I Handout 4 One dif cult result we will have to deal with at this point is the fact that in three space the so called position vector is not a vector under general coordinate transformations although it does behave as a vector under the restricted class of transformations known as rotations However the components of in nitesimal displacement really do form a contravariant vector since they satisfy the appropriate transformation law a i j 816 die dabi Thus some of the most common vectors we use to refer to the motion of a particle in Newtonian physics namely velocity and momentum both behave as contravariant three vectors in that context time is merely a parameter with which we can label points along the path of the particle Before considering acceleration we will need to know how to deal more fully with the differen tiation of a vector The fact that proper four vectors can be formed in a similar way also plays a crucial role in the invariant formulation of special relativity This will be a particularly useful perspective to keep in mind when we come to deal with special relativity again shortly At that time we will also expand on the usual understanding of electromagnetism within the framework of special relativity Current reading A rst course The first two thirds of this book cover the introductory material necessary to be able to write down and make sense of the non linear field equations of General Relativity It also includes an extensive treatment of the familiar topic of special relativity which we will use heavily in a generalization of existing knowledge to the wider context needed for analysis on manifolds and in particular for General Relativity The last third of the book deals with application of the equations to modern topics in astrophysics including black holes and gravitational radiation 52 Tensor algebra in polar coordinates 53 Tensor calculus in polar coordinates 54 Connections Le Christoffel symbols and the metric 55 The non tensorial nature of P36 56 Noncoordinate bases 59 Exercises 4 8a Geometrical methods Much of this book is concerned merely with giving a new mathematical formulation to ideas which are already familiar from physics Thus through it topics such as vectors the inertia tensor inner products special relativity spherical harmonics the rotation group and angular momentum operators conservation laws the theory of intergration gradient divergence and curl Gauss7 and Stokes7 theorems of vector calculus Maxwe117s equations and other gauge theories of physics will all take on a new light based on a deeper and uni ed understanding founded in a geometrical approach to these topics On the other hand a topic such as fiber bundles is not essential to General Relativity and we will not be concerned with it in this course However the last chapter includes many of the concepts and methods which are most heavily used in General Relativity and we will deal with it in some detail Chapter 1 Some Basic Mathematics Exercise 21 Special and General Relativity Lecture Notes Day 10 100708 Shawn Mitryk Contents 1 Weyl Tensor Revisited 11 De nition i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 12 On a 2Sphere i i i i i i i i i i i i i i i i i i i i i i i i i i i i i N Gravitation i Einstein Equation i i i i i i i i i i i i i i i i i i i i i i i i i i 22 Schwarzschild Metric i i i i i i i i i i i i i i i i i i i i i i i i i 221 De nition i i i i i i i i i i i i i i i i i i i i i i i i i i i 222 Spacetime Boundaries i i i i i i i i i i i i i i i i i i i i 22 Properties i i i i i i i i i i i i i i i i i i i i i i i i i i i The 2body Problem i i i i i i i i i i i i i i i i i i i i i i i i i 231 Problems With nding a solution i i i i i i i i i i i i i i 232 Extreme mass ratio limit i i i i i i i i i i i i i i i i i i 23 Equal mass case i i i i i i i i i i i i i i i i i i i i i i i 2 CAD 3 Schwarzschild Dynamics 31 Equation of Motion and Conserved Quantities i i i i i i i i i i i 32 Light propagation near a massive body i i i i i i i i i i i i i i i 4 Next Class 41 Homework i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 42 Reading i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 Weyl Tensor Revisited 11 De nition 0 puny 2 2 P qu iglgmevhr QumRzMl mg ugypgl n 7 Note 0 CWW is tracefree All traces 0 on 0W is conformally invariant 12 On a 2Sphere Coef cients o Riw sin2 9 0 R99 1 o R sin2t9 For a sphere of radius 77a o R9 9 a2 sin2 9 o R E2 Ricci scalar units of 2 Gravitation 21 Einstein Equation 0 Previously we Viewed Newtonian gravity as a force 0 NOW the geometry of spacetime Will de ne the gravitational affect Einstein7s Equation 0W SWGNTW 2 WOW 0 3 WTW 0 4 Dual effects of spacetime curvature 0 CW gtmatter curves spacetime STrGNTW o STrGNTW gtspace moves matter CW 22 Schwarzschild Metric 221 De nition 2GM dT2 2 2 2 2 2 ds 7517 T62ldtWTDQ 5 Properties 0 Nontrivial metric curved spacetime o Spherically Symmetric 0 Vacuum solution of Einstein Equation 7 GW 0 7 RW 07 R 0 o Non at CWW olt o OWN is wellbehaved at T 2M but 77blows up77 at T 0 3 7 Diverges as T as T 7gt 0 7 Diverges fast enough that you can t pass T 0 7 T 0 is a spacetime boundary 222 Spacetime Boundaries 0 ln at spacetime7 all geodesics extend to ioo af ne parameter 0 ln Schwarzschild spacetime7 geodesics Which reach T 0 stop at that point With a nite af ne parameter 0 With missing points we obtain geodesics Which are singular 223 Properties 0 Vacuum Solution GM 0 TW 0 7 No apparent matter but still curved spacetime 7 Spacetime can curve on its own 0 Schwarzschild is a 77static77 spacetime 7 Timeindependent does not explicitly depend on 77t77 0 Every spherically symmetric static spacetime is Schwarzschild outside of matter 0 This replaces the Newtonian notion of gravitation 23 The 2body Problem Recall for Newtonian gravitation for a 2body problem we can break it up into two parts 0 center of mass motion 0 reduced mass gravitational problem in center of mass This can not be done when working with the Schwarzschild metric The 2 body problem does not have an analytic solution for the Schwarzschild metric Note A rotating black hole can be described by the Kerr metric It is stationary but not static 231 Problems with nding a solution Consider a mass 77m77 rotating around a massive black hole 77M o Timedependant rotation due to gravitational radiation 0 We can obtain a perturbed analytic 77solution77 for a binary system for 70 232 Extreme mass ratio limit 0 For Extreme mass ration perturbation solution 7 This is situation exists in the center of all galaxies 7 Note VMTWW 0 aMp39p39 0 7 but GM SWGNTMMPPJ 0 Close to m7 the external spacetime is a small perturbation 0 Far from m7 the mass7s affect on the spacetime of M is a small perturbation 0 In some overlap region we can connect these solutions to obtain a complete result 233 Equal mass case With a signi cant amount of computing power the approximately equal mass case up to about M 4m can be solved using numerical relativity 3 Schwarzschild Dynamics Consider a 0 in a xed Schwarzschild black hole spacetime 0 Find departure from Newtonian Gravitation problem for massive particles 7 Orbit precesses 0 Find affect on the straight ight path for light massless particle propa gation 7 Light path is bent 31 Equation of Motion and Conserved Quantities Conserved Quantities Pt E 911 6 10 J gasg 7 09 0 999 8 Lac7 Ly7 L are all conserved Equation of Motion 712 Eii 7 9 32 Light propagation near a massive body Solving the Equation of Motion for 57 g Etlligl 10gt Differentiating we obtain d27 J2 3M W lliil 11 7 is the 77bending77 term which would not be present in Newtonian dy narnics Different cases 0 for T 00 the path is straight o for T 3M the path is Circular 0 for T lt 3M the path converges to T 0 photon can not escape Noting dT J2 2M 7 2 7 7 7 7 d E T2 1 T l 12 dab J g 7 2 13 o Figure 1 Bending of Light around a massive body We can solve for all d dr 7 2 12 2M 7 i7 E2 7 7 1 7 7 14 01 J T2 T l In the flatispace limit 0 dr 7 2 12 i i 2 7 i 01 i J E T2 15 If We then de ne u then 77 1amp7192 7 JW 16 Ju f 505W 17 We then obtain the result J E Tcos 5 18 NOW de ning a and 7 0 We can Write d 2M 77 1 7 a2 1 7 7a 19 01 1 To 1 lt gt Differentiating We nd 01 3M W 7 u1 7 To u 20 NOW de ning a uo u1 and E 3734 ltlt 1 We may expand the result to obtain d2 wmo u1 uo ul 5UDU12l 21 a 41 Noting that uo cosq is a solution to the hornogenous equation we nd d2u1 dab u1 cos2 Which has the result 2 ul acosq sinq W Thus to rst order in 5 a 1 8a cos 55 Sims 21 sin2q5 2 Using boundary conditions we nd that a 7 and 0 Thus the nal result for the deViation AT 7 7 7 02 7 GM lsin2q 0 A PW CZ l Next Class Homework 2 23 24 25 o Noting that the armlength of the LlGO detector is 4km7 calculate a pho ton7s deViation distance from a straight line at the center of the detector Given the values of the constants G 667 ioill g 39 TEanh 6378 103km o c 299 96108 0 MEarth 597 1024kg We calculate 0024 44249610 3771 o qbo tan 1TELah 017990 Finally 2 AT 7 iGMu 7 1Sl nl lquot 6542 1010m 62 cos o 42 Reading 0 Finish Chapter 4 Special and General Relativity Lecture Notes lDay 19 111808 Shawn Mitryk and Steve Hoohman 111808 Contents H N Kerr Metric 11 Where does 9quot go to 00 1 Killing Vectors Next Class 211 Reading 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Kerr Metric 11 Where does 9 go to 00 Starting With the Kerr Metric ZGMT 7 2 p2dt92 02 a22 7 a2A sin2 9dq52 2 2 432 in thMAZ www id 1 A T272GMTa2 2 p2 7 2 04260826 3 J aM 4 lnverting the Matrix we can nd 9quot 7 7 2 a2 2 7 aQAsith 9 ltlt 1 gt 5 This goes to 00 at A 0 thus we can solve A T272GMTa20 6 Ti MiM27a2 7 We can also solve for ZGMT 2aA 7 7 2 a2 5 7 7 g 2 p2A p2A 8 A 7 a2 sin2 9 545 9 g pA sin2 9 Despite that all of these go to 00 at A 0 there exists a component that remains nite On the TihOTiZOTL Expanding out 9W 3 31V a a T2a2 7ai2 asin2t9 7i 9W7 7 3 345 322 a 2 10 BI 81 HA p2 s1n 9 Note as a 7gt 0 this becomes the Schwarzschild Solution 12 Killing Vectors 3715 and are killing vectorsi Writing out the components explicitly E 7 T 1 0 0 0 11 at lt gt lt gt E 2M7 74GMTa sin2 9 7 T 7 17 0 0 12 ad tltlt p21 p2 ltgt 2 Calculating the scalar Tl TM THTH 717 2N2 lt 0 p 13 for A 7 a2 sin2 9 gt 0 10 Note that any 1 5 is a killing vector for a and constanti Consider the numerator 7 2 a2 7 a in the diverging component of Eq At a xed r7 this points in the direction of a killing vector but it is not a killing vector unless the coef cients are constan 1 Evaluating this at TZ Ti 712 7 a 23 Tia2 7 at Tia2 8415 a 14 7 Q Tia2 H 15 This 9 can be interpreted as the angular momentum of the horizon If we throw something into a black hole radially from 00 M M6M 16 A 167rM 6M2 17 Thus 6M gt 0 and 6A gt 01 Considering Kerr Black Holes Given AH 47r39ri a2 18 AH 16WGQMgreducablewhere MiM gt 0 19 Plugging in Ti gives AH 2 2 J2 SWG M M 4 7 Thus as we increase M7 AH increases and as we reduce J AH increasesi Finding SAH 20gt SWGa 87r 6AH 7 6M7QH6J gt0 21 Q Qj 7 77 H Penrose considered throwing a mass into the black hole with an angular momentum opposite to that of the spinning black holes This decreases the angular momentum but it increases the Area Gravitational Radiation spin 2 carries away both spin and mass Next we consider whether we can throw radiation entropy into a black hole thus reducing the systems entropy and violate thermodynamicsi Consider 65 gt 0 which combines the entropy derived from QFT with the classical Area and Gravitational constant but still holds true This implies the temperature TBH olt H 87rM 2 Next Class 21 Reading 0 Chapter 7 in Spacetime and Geometry Carroll Lecture Notes for 12208 Benjamin Hall and Hidris Pal December 4 2008 01 Continuing linear perturbation As stated in the last lecture7 we decompose the metric perturbation hW as W ii710175 1 where S is traceless The vector part ml is decomposed further wi wi w i where 9101 0 2 eijkdiwll 0 3 These conditions imply that wi has 2 degrees of freedom and mi if has only one degree of freedom H H H S can be similarly decomposed S ST S Sf where 616 0 4 61678157 0 5 eiikajals k 0 6 Again7 5 has only 1 dof7 S has 27 and S has the remaining 2 This is because we can write 1 SH aiajiivz m I SSH A 00 V As was stated in previous classes7 gravitational waves are spin 2 under spatial rotations traceless tensors the only part of the perturbation match ing this criterion is S 7 speci cally Sf as it is the only spin 2 part of the metric perturbation For the remainder of these notes we will use the traditional notation hijTT E Sf where the TT stands for transverse traceless77 lf expanded in spherical harmonics for example hijTT has only terms with L 2 27 so it doesnt include the gravitational monopole Am7 S 0L 07 the gravitational dipole relating to motion against the background7 S 07 L 1 or the static angular momentum S LL 1 terms that may be present from the source Various choices of gauge can push the gravitational waves to other parts of the perturbation by reducing spin7 but cannot change 02 Gravitational Waves For this section we will consider Sf hijTT Aexpmwt 7 I and also set k 07 07 k1 Under these assumptions we can write 7 11 hm 0 we Mm0 0 0 0 The action of a pure h or hm polarized gravitational wave on a circular array of masses is shown in Figure 1 Note7 this formalism only describes the weak eld limit small sources or large separation between source and detector It is totally insuf cient to describe the generation of gravitational waves close to strong sources such as black holes We can thus write GW77 h 87TTM for weak sources and slow motion We know from conservation that time and spatial derivatives of TM can be related LTm 0 a aOTOO 761T 10 LTM39 0 a aOTOi fagT7 11 After much work7 we can then write the metric perturbation as 7 26 d2 m7 wwwi ma 2 where EM E h 7 lhg w The quantity Ii is the mass quadrupole moment 2 de ned as 155 d3yyquotyquotT00tC 13 evaluated at t t 7 xi 7 yi yi is the distance between a portion of the source and the point where the measurement is being done With care this formulation also works at large distances from even moderate sources This gives us the amplitude but to get the energy density we need to con sider terms quadratic in the perturbation 2nd order perturbations Since the waves are being measured far from the source at in nity to be precise there are no sources which means TW 0 This implies that we can write G77hG0G1G2quot39 14 where each term is of the indicated order in h The energy density can be written as 1 W 7 327TG brackets indicate an average over a suitable region and the power radiated to in nity as t ltahTTthTTgt 15 G d3 de 1377 7 505ltdt3 dt3 Jia 16 where J I 7156 739 For 2 newtonian like ignoring higher order effects masses in circular orbit de ne b Gill21 g 1 P 7M296R4 We know from previous work that Q fl2 for Schwartzchild This implies that P N 155 5 gtgt 1 For solar mass black holes this is for a small time more luminous than stars As the black holes spiral in the amplitude grows as 775 and the frequency increases both with modulation due to the structure of the infalling objects afterwards the frequencies and decay periods are completely characterized by the total mass and total angular momentum of the new black hole The maximum amplitudes expected can be estimated L0 few 17 L0 0L0 L06L xd212h N dx1h 18 hL g 10 1 19 PHZ 6607 Special and General Relativity I Handout 1 General Relativity is an exceptionally powerful physical theory Ultimately it is also a theory about geometry the geometry of a four dimensional spacetime In that theory timelike geodesics represent the possible paths of infinitesimal test particles whereby curvature in the spacetime re ects the presence of matter subjecting the test particle to an in uence which we have familiarly come to know as the gravitation force due to that matter But the theory purports more not only is curvature in the spacetime geometry meant to be due to the presence of physical matter but also that curvature in turn determines the motion of all matter present corresponding entirely to its gravitational interaction all other interaction between the matter being determined by the other known forces in physics in the usual way From this point of view geodesic motion for a point infinitesimal test particle which is presumed to have no in uence back on the spacetime itself represents one of the simplest cases of matter motion which might be taken for consideration Why is a theory such as this necessary Firstly Newtonian gravity is not a relativistic theory Secondly Special Relativity which serves perfectly well to account for the behaviour of electro magnetic elds in a at space does not deal in any way with the consequences of gravitational interactions Thirdly Quantum Mechanics makes us acutely aware that particles even photons identified as particles must lose energy when climbing out of a gravitational potential Presum ably then local electromagnetic energy density must depend on exactly where a local region lies within a gravitational potential We have seen that the scalar potential of electrostatics must in general become a vector potential in the full theory of classical electromagnetism We are then forced to ask what generalization becomes necessary for the static Newtonian potential especially to describe situations characterized by very high relative velocities General Relativity is one of the simplest attempts to answer that question which it does in a fully relativistic way and which so far is not in con ict with experiment nor with other direct or indirect physical observation General Relativity is not the only current theory of gravitation Moreover whenever an attempt has been made to construct the corresponding quantum theory serious di iculties have always arisen which became insurmountable suggesting to some that if General Relativity is a useful theory in physics it is only as the classical low energy limit of some more fundamental and very different quantum theory There is so little hard experimental evidence to support this view at the present time but overwhelming evidence within a specific theoretical framework Nevertheless the dif culty of understanding General Relativity fully is currently so great that we shall entirely content ourselves with its direct investigation in this course leaving any more complicated approach to a different study One further observational aside may well be in order given the current climate That part of quantum mechanics which showed how a system could have a ground state and a discrete spectrum of excited states was far easier to accomplish than was solving all details of the Hydrogen atom Even today in the eyes of some Mathematical purists the existence of the photon as a quantum particle state in electromagnetism is still not completely proven though there are probably few Physicists who seriously doubt that proposition The rst part of our observation is that the same problem for the propagating degrees of freedom in General Relativity is basically no harder in at spacetime than it is for electromagnetism It is the nonlinearities in strong eld regions and the non propagating degrees of freedom which present the most serious problem for a quantum theory of General Relativity On the other hand and this is the second part of our observation the non propagating degrees of freedom are much closer in nature to the particle aspects of the Hydrogen atom problem than they are to eld theory aspects of electromagnetism 1 PHZ 6607 September 18 11 Expanding Universe Energy momentum tensor TW puMuquot pg MWquot and metric dsz 7dt2 a2tdx2 dyz 122 Some solution exists for at that satis es Einstein s Equations VTW needs to be satis ed VTW EMTW PgMT rg w For the spatial part II 2 the connection coef cients are ng 0616 a P30 g5 and H H I I I 1 VJ diT PinT PinTm then 1 8 7 0 1a2p p is spatially constant For 1 07 we have 80T00 PEMTOO PEMTW 3d 1 p p 3 729 a a 3d 0 p p 0 An interesting case for cosmology is when p wp If we take the condition the w constant7 then B 731 w p a if w 07 as is the case for non interacting dust7 then p N 1 3 For radia tion7 w 13 isotropic Then p N f4 If the universe if dominated by radiation like matter then p N f4 Vacuum case7 TW AgW7 if p p 0 and A p Then p 7p and p is constant PHZ6607 Class Notes W Zach Korth 922008 1 Action principle SLw 1 d i A d 7 A 1 i i L Em 1 zd Em r2 r202 r2 sinZ 0a The metric of at space in spherical polar coordinates is d5 drz r2d62 r2 sinZ 6d 2 On the surface of a sphere7 r a7 dr 07 so at 12 d62 sinz was 2 Geodesic equation The geodesic equation is given by dzxi l dxj dzk 7 d2 739de dA where i 1 il jk i9 91 9jlk 9m is known as the a ne connection 21 Paralneterized curves If We consider a parameterized curve r O with parameter A the proper length between points A and B 39s given 3 l 1 MW AB A 77 But What does it mean to integrate over these in nitesimals This is Why We choose a parameter A over which to integrate as lt7 B dzdy IMFA x i dA 5 17 dA dA e note that this form is Tepammeterizatian invammt as any changes in the parameter A 4r x leave the physics unchang 3 3 Acceleration vector We have What We de ned as the tangent nectar 1 dr u dA We now de ne an acce1eration Vector77 Du 9 du M nguiu 31 Relation to geodesic How do we de ne a geodesic o A curve of extremal length77 or i o 153 oltfui Aside For ui ddisi dxidzj d52 2 l i 7 7 u u 9 39M E 1 11 Du2Duuau0 l2 4 Parallel transport Suppose we have a vector eld de ned in 3 D space This eld would have a value wilx along some curve If we parameterize the curve as xj it makes sense to ask how the vector wi changes along the curve For this however we must be sure to use an appropriate derivative a Pljwku not a good vector proper vector 13 We are used to being able to move vectors around at will This cannot be done in curved space and indeed is not trivial even in at space for non cartesian bases In general to move a vector along some curve we must impose a condition I I le dwl dA dA which we term the condition of parallel transport rgjwkuj 0 14 5 Covariant derivatives If a eld wilx as above is de ned on a manifold not just along some curve 9670 azi DA dwi i dxj 310i 7 143w Du 7 3M dA dA 3x7 3x7 TQM uj 15 Here we de ne 3x7 to be the covariant derivative which is necessary to ensure that tensors remain as tensors under differentiation This arises because a true vector is 6 i i Figwk E ijk 16 W wie 17 dw dwiei do6 widei 18 Hence we have two terms in the covariant derivative one for component changes and one for changes in the coordinate bases 6 Christo 39el symbols We can now give a proper de nition of the Christoffel symbols ng E eidkej 19 Note that this is not a tensor but rather a tensor like object Good tensors can be constructed from the Christoffel symbols however for example the torsion tensor 1 I I Tit 3 ng Fi 20 In GR we will be dealing with the Einstein Equations in which manifolds have no torsion thus T2 2 Pg 7 r2 0 a Pg r2 21 J 7 Covariant derivative examples Let us examine the form of what results from a covariant derivative of a scalar 643 l i I 22 v we 6 lt gt a vector avk k k i VjV 6W PijV 23 a oneform de Wm gigm w some higherrank tensor vaik MW rijlk Pij 25 71 Operators Recall some operators gradient scalar 5145 V145 26 divergence vector 1 I I I VlVl 91Vl lidV 27 Noting the indices7 we see that this must contract into a scalar lets look at the connection 1 P21 5911gw 9711 9m 28 The last two terms in the parentheses cancel7 leaving 1 l 1 91 91M 7 WM 2 MM where g is the determinant of the metric Thus7 P21 1 1 mwaW4iltmuweiww m w m m curl vector eijkVZBj This equation works well for 3 D7 but for higher dimensions it is useful to de ne where the last step follows from the fact that our connections Pgk are symmetric in their lower indices 5 the Laplacian with 1 W WW VialMW Vi gijv b 9ij5j PHZ 6607 Special and General Relativity I Handout 2 A manifold can be described simply as a mapping of a set of points into Rquot Typical well known examples are the solid cube the solid sphere and the solid donut as three dimensional objects with their obvious embedding in R3 What might not be at all clear is whether the surfaces of each of these objects can be embedded as two dimensional manifolds into R2 In fact none of them can be embedded globally but a re nement of our de nition allows us to continue to regard all these surfaces as two dimensional manifolds It turns out to be su icient practically to require that the embedding can be done locally in open balls7 which overlap to cover entirely the complete set of all points to be embedded We will nd it necessary to de ne geometric structures on the manifolds with which we shall working We will also nd it very convenient to refer to other objects de ned on manifolds and the most common of these will be tensors of which vectors are a simple and familiar example However perhaps in contrast with your current practice we will use index position to have a meaning and will for a start consider Vi and V as inequivalent allowing them to refer to dif ferent tensor types The most distinguishing property of tensors which we need to comprehend is their transformation law under changes of coordinates on the underlying manifold These changes we can recognize from the way partial derivatives change under coordinate transformations Thus Vi a i j Bacj 7 for vectors with their index upstairs which we will refer to as contravariant vectors and Bacj 7 Vi T Biciv for vectors with their index downstairs which we will refer to as covariant vectors For tensors of more general type with both upstairs and downstairs indices the transformation law is simply a generalization of the above with one factor for each index the type of factor depending on the type of index upstairs or downstairs Certain operations on tensors can now be de ned without any additional geometric structure being given Perhaps the most obvious are multiplication between tensors to form tensors of higher rank and contraction of a pair of indices to form a tensor of lower rank For example AiAj AiBj and 3133 are all different examples of distinct types of tensors of rank two while AiBi would be an example of contraction to a tensor of rank zero ie a scalar Notice that at present we have no way of de ning the sort of contraction you might expect to correspond to 2 which requires the additional structure known as a metric A perhaps unexpected but familiar example of a second rank tensor might be a square matrix Mi de ned on a manifold is a scalar known as the trace of the matrix We can also see that a matrix can be contracted with a vector to form another vector of the same type eg 14ng and M173 while the contraction of a matrix with another leads to a third matrix exactly as in the familiar matrix multiplication ie Pf Mng Two other special tensors at least one of which you may have encountered before are eijk and Eijk each de ned to be 1 71 or 0 in locally cartesian coordinates depending on whether is an even odd or not a permutation of 123 With these we can also de ne DetM Z7quot leMijgelmn as well as the familiar vector triple product eijkAinDk Finally another operation which can sometines be used to de ne a new tensor is differentiation Thus BiBj 7 BjBZ Exterior Airin 7 OiBiAj and AiBiBj Bizin Lie where 81 88152 are all tensors More general differentiation requires additional structure a connection NewmanPenrose Formalism Tetrad formalism Specia cases NP formalism GHP method Appication Summary Tetrad formalism ntroduction At every point in space set up four linearly independent vectors emf a 1723 4 k 900i Qikera emlemi 2 6mm 60 j 61p e b e b 7abeai emu quotM eai 9 139 Where quotammbm 5a Am 3 gran1 AM quot10m Aw Ai em AM 3 3a Aj 3aAj em 39 e a I Am Tetrad formalism Intrinsic Derivative amp Ricci Rotation Coefficients era emfW 9 eui E grail 151quot Ara h ebi Ara ermi 01in I eibiivalemimi I eibiieaini Aheiuik l J39 i i k c Anni 9a Amen 9aiki9bi em A Define Ricci rotation coefficients MW ekfemwewf 39 i Ara b emf Amen in a b A Intrinsic derivative A an inward on1w 3aiAafbebj Tetrad formalism tooth 3 t to to 1b J39taHbtlr z cotemf em em leml 2 th Wow twat 10mm Mono 3 Moo Commutation relations com 9o mewetcr C c CC Ib 7 WHYquot ub Tetrad formalism Ricci amp Bianchi Identities m eaikl eaik Rmiklew a RWWMWWMWHWWW MmmMW MWM I f Vumm m m nnmmWMw Ralbtcdl1f 116 2 WWW 1 WWW nnt n meanmmmWmmanwmm mmn mwmmWmmanmmm h Genera Izatlon 0 Mer WW Coordinate and Tetrad transformation Special Tetrad system Four vectors at each point are in the direction of the coordinate axes that is parallel to the four coordinate differentials elm lm I 9iiglilt base vectors of a Cartesian coordinate system in the local Minkowski system of the point concerned drain dia l lll ll gij u 1lj lquot ill 1 39C quotvl Q i nu vectorsas tetrad vectors 39 Z ll t l ll 3l in Tit in ii ll using this system complex tetrad components can arise 39 Special cases Dina 85 la n 39mai NP tetrad 0 1 0 0 UL7 l 0 0 0 MmZU 0 71 U u 71 u m r m ymbz wU 21mm 7 Zruwzm mamx 1 Null tetrad approach to NP Formalism ntroduction Spin Coefficient in terms of Ricci Rotation Coefficients e1e2n eze1l e3 e4 m and e e3 m e1e2D e2e1A e3 e46 and e4 e36 Ki 311 Pi 314 8 07313i 243 i i zizi 342 ii244 Ti 312 a fiimwm Vi 242 i 241 B ffi mf39i mi 211i 341 f Weyl Ricci and Riemann Tensors in NP formalism R11de C11denacRbdquotbcRadquotudec quotbdRacj mam adnbc Rz P0 C1313 Cpqslpmqlrms l1 1 C1213 Cpqrslpnql39ms W2 Cpq slpmqmrns W3 C1242 Cpqrslpnqmrns 1 1 L C2424 Cpqsn r n r s 00 lt31 1 21 3R24 1 D11 R12 1334 Dm 3B33 1 101 313 1 22 3 122 1 i 112 31353 120 1 110 3R14 H 2 ii NP set of equations commutation relations Ricci Identities eliminant relations Bianchi Identities Ricci Identities Dp 6K p2 aap s KT 3aB 7 Dooa R1314 Dry 3K app3 KT397 a3l3q o R1313 Dr Ax pr7rar7rr K3 1 441301 R1312 Dex Vs ap 2 Ba B Kl Kynspltl10 01358aa Bp 8 Kuv 3a n l 1 R3414 R1214 R1213 R3413 Dy A3 ar7r3r n ygg 30 T7 Wlt W2 Du Aa R1212 R3412 IDA w pxaunna m wlt A3s sltlgt20 R2441 Dp Sn ppal1r7r al3 pg VK I 22A R2431 Dv An unrlnr 7 v3 1 3ltl21 R2421 AA 6v Auu3y y v3aBn r I 4 R2442 Sp VapaB U3a BTP P Kp u I 1 4130 R3143 Sat 6 up Aaaa BB 2a3 P P3H N 2 D11 Aa R1234 R3434 515M vppnu Mual3 Aa 3B I 3q21 R2443 6v M 2MHvvv7r vr 3B a 22 R2423 6y AByr a Bm av sv B7 V Hal 12 R1232 R3432 61 A0 alprrBa U3 KVCD02 R1332 Ap Vr ppalrB a r pvvvK I 22A R1324 Act wvps irl3avM B T I 3 R1242 R3442 Bianchi Identities 6 F0 mg 4a 1r 0 22p8 l 1 3qu2 Ricci 0 R13U3l4 0 6 1 1 D I Z A I o 27 DOWl 30 1 2 21613 RiCCi 0 R13214 0a 5112 mg 2A I 1 3n l 2 28 p 1 3 W4 Rioci 0 R42U3l4 0 5 3 mat 32 221ra 1 3 46 p 1 4 RiCCi 0 R42214 0 A I O5 I 14y u l 0 22r 3 1 1 30 Rioci 0 R13132 0a ADI16 I 2 v 1 02y u 1 1 3t 1 2 20 I 3 Ricci 0 R13432 0 M2 5 113 2v 1 1 34mg 23 r 1 3 an4 Ricci 0 R42132 0 Awa 6W4 3VT2 T Ricci 0 R42432 0 Urquot 2a 2 lt1300 5lt1gt01 MOO 2azcbm 2pclgt11 ac1gt02 y 2y 2yCDOO 21D10 2DA DD215q32o2P 8ltD21 2ul1027zltl11 KD22 2a 23 7clt1gt20 26A A32olt3 1gt212ot rltilgt212vltilgt1oaltlgt22 2LltD11 u2v27lt1gt2o Dclno2 6CD01 21z mm 2min ANDOO 20a p28 23ltD02 M01 5 1302quot392 Do1 2PCD12 V Doo2T 11 DD22 6ltD2121riCIJ21 2 I11 ACDZO21ID12 p 28 2 I22 2AA M1321 6ltI222uy 21 2VDU vd2021l12 r 2a 2Bd322 Spinor calculus Spinors in minkowskian space Isomorphism between Unimodular T and LTs uWwweuW uv0 x0 H554503 5 EVL x H5 5 E x x2 7i27 y 1 39 x34850655 HEW 5uMmu HEV 3ukH x We wam we ww Spinors in minkowskian space A39 w A B quotaquotB B 6 a 8396 X 2 ijxj 5002 3012 5022 3032 1 aoo aol 1 1 a0 0 1 39 39 A A de ne 5AaAB B nAa gnB 3W BH Ha yH L 5 C W n 01 310 in Cquot aAg AnB mlnvanant Spinors in minkowskian space xi 050 CDC 139 l X0x3 x1ix2 H 615039 515139 75 x1lx2 x0x3 i 50039 1 XGX3 X1EX2 rw H 51039 421139 7E X11X2 X0X3 X02X12X22X32 X0X3X0X3 X1iX2X1iX2 2500 51139501 51039 COO50039 1l39 ll39 10 lol5013950139 XAB XAB 9inin SACSB39D39XAB XCD39 39 AB39 AB39 39 X101ABVXAB X 0 iXl General connection between Tensors and Spinors and Spinor Affine Connection i SACBB39D39 Qua AB39O39JCD ij i 1 EF AB CD39 Y kUAB39UCD39Ui Y EF yAB CD EF39 H Yuk AB CD AB CD It Y 5p 0 id in EF39Y39Jk Vin Xi39iHVAB XCD39 XCD39AB39 VAB39 7w XCD AB O CD39 G39AB39Xji 1 v E 0 cog1339 0 Emmi gem r grail 3CDAB 0 2Spinor approach to NP Formalism Dyad mea ism F uo Gum Bum Gum A A z50A0A 1 quot eABoAzB 0011 0110 0313 0quot1A1 A r J 339 C 5a 5AM hm Emmi iv Squot quot quot39 3 0039 1139 01 10quot 09V Fquot H fn 2 7 1 liHo 39 m HoA quot i 1 P mi JAB 1 n r i 0 391 i T m Hz al niHlAIB quotB 75 m n39 v2 mi 1 Spin Coefficients in terms of Spinor Affine Connec on a001 all zA a01quot 5 510395 rabcd39 CaFCD39Cbfgcyc mqbi F CaiFCUJ 8ab39 01 ab 00 or 11 d 39 10 00 K 8 7r rmnbncnd39 1039 P a A 01 a 3 p 1139 r v v GHP l v 39rlquot nquot r ln Ina rewind mquot 2241420 M n xqun l vil m X 1m Wquot X391Xr n n gt X39li ln New and appropriate operations 31quotna 110 232 39 1139 f t1 0 39 C m m m II m pz K Azao i 12 33 5 71 Ia 339 y a 39 PIquot D pp qE39M 11 1 251 1 1m D p6 wan 6 nlt639p 39 a n pr11 2539 11 M 0 p 4339 1 Z1P39 HUIj mad 17115 1 p 1 Va 7 31qnbvaib 31 7 1171139Vanb 1MP Q 71 4 13941 H 11247 Application summary In finding amp analyzing new solutions of Einstein field equations In studying asymptotic properties of radiation fields In particular GHP method turn out to be very effective in 2 surfaces calculations Developing approaches to quantization through the study of complexified space times PACS numbers I PROBLEM 13 DISCUSSION Problem 13 considered a transformation in 2d at space A t a sinh7 a A z a cosh7 a d82 7dt2 dz2 dz u d coshasinha from this we can calculate the acceleration according to the equation previously derived Du du 7 a b 0 DA dA Fbcu lsinhacosha a a2 aaabgab 7 sinh2 cosh2 7 Note that we have a path that has constant acceleration but it is NOT a parabola 7 its a hyperbolai II SCALAR FIELD ACTION PROBLEM DISCUSSION The action of a scalar eld 45 is S gab Vaqb qub d4z We obtain the equations of motion by varying wirit 45 63 53 ga Vaqb vb6 d4z ga M 650 h 7 aw gab va gt6 h 61M 9 W15 M h ignoring the boundary term for now7 we have ab ab 4 657 7 7 7 Va 6 d 0 g lt15 lt15 I beabvaab iO L1 L2 L3 L4 L5 L6 L7 LLL 1L2 1L3 1L4 1L5 116 which are the equations of motion for a scalar eld The boundary term is generally dealt with by the initial conditions III CONTINUATION OF CRISTOFFEL SYMBOLS Recall from last time the de nition of the Cristoffel symbols welll derive the relation shown in class in more detaili 1 Tuba Ega gdcxy gbdc 7 gbcd 11111 1 Paac Egadgdca gauge 7 911001 1112 l 7 i9 er 9m 1113 7 i 1114 7 29 c g 1 7 86 lng 11115 60 11mg 1116 IV SYMMETRIES IS OMETRIES Take for example at space where d32 d12dy21 Rotation in Xy space leaves the space unchanged 7 the metric is form invariant As another example consider 2d Minkovski space d32 7dt2 dz Now let I Hzcoshq tsinhq5 IVil t A tcosh zsinhq IViZ we can easily see that the metric remains unchanged This looks a bit like a rotation in Xt space To clarify this let s attempt the coordinate change I p cosh 739 dz dp cosh 739 psinh TdT IViS y psinh 739 dy dpsinhT pcosh TdT IVA Using this we arrive at d32 dp2 7 p2d73921 This looks like polar coordinates and we see that 739 A 739 6739 leaves the distance unchangedi This change of coordinates only maps 1 gt t However we can be properly clever and adjsut the map eigi by allowing complex 739 to join it with z and 711 We further observe that the proper time is 1 Lines of constant p are hyperbolae and lines of constant 739 are the t 0 lines for Lorentzboosted frames V VECTORS AND DIFFERENTIALS We7ve shown that we can use a differential to describe a tangent vector space df BI E f 7 V11 dz 8A 81 gt d BI E i V12 dA 6A BI In 2space let s consider the vectorial object 1 In polar coordinates this is d E E i 0 i 1 i V13 d6 w 66 l d a 7 0 1 V4 lt d lt 7 gt lt gt VI LOCAL INFINITESIMAL TRANSFORMATIONS Let 1 z 65 Eb Then what is gbza Consider the constant ds2 d3 gab dz dzb V111 gflb dz dz VIZ We expand this in the coordinate transformation above d32 gbdza 6805 dz dzb ead b dzd d8 gbz gtltdz due 4ng dzbaasc M 9 dzaabsc dz 911 1 gab 6chaaiC 911491750 ayab c 0 69a are 595 91195 BIC 9159 650 92590 gab10 7 490218115 gacab c 501291127 v13 v14 v15 V16 v17 V18 v19 vmo and for the metric to be invariant under the transformation that is for there to exist a symmetry the quantity in the brackets must be 0 General Relativity 10 16 2008 ProfWhitings Lecture Note October 237 2008 In the previous class we derived 1 d Goo ewa To 7 a SWGNTOO 87rGNp52 7 1 2 G11 7T7252A17 5 71 87rGNT11 87rGNp62A 1 A G22 r25 1 7 7 gt A 7 7 87rGNT22 87rGNT2p For static and spherically symmetric matter7 we can write the metric as 152 fewdtz ezAdrz 720192 At rest7 ut 1407070707 where 140 5 15 Energy momentum tensor is given by T p FWUV W dia 572 57 2 7p gltp p r2 r2 sin2 9 Now let us consider WTW 0 WTW QMTW rgMTW PZMTW 7 0 For 1 0 80T00 PEMTOO rgMTW 0 For 1 1 81TH P fMTll 1 inW 0 For 1 2 82T22 113T PEWTW 0 For 1 3 93ng rgMT33 1 ngW 0 We need to calculate Pi al and Fi Covariant derivative for Vt is 1 VFLVF QFLVF PgMV QFLVF iaa V Recall the affine connection coefficients 0 1 at Play 59 gm 9490 9mg we need to calculate the following to get the equation for 1 1 11 1 P232 59 91719 1 91117 919ml 1 1 7 P50 9119100 9100 9001 9119001 62 MW 2 1 P11 59119114 A 1 A I z 29119211 7 5 2 7 1 1 3 759119334 6 2A 811120 Therefore for 1 1 2 2 8199 gt A lo 6 2Ap gt 191V 7 y 0 This gives p PP 0 10 11 12 15 16 17 18 Now from G00 87139GNT00 22 we get d dir17 572A 87139Gp7 2 23 7 or d M1 7 a 87err2dr 24 Now de ne pdV E 7717 25 e care 7t 1s oes not e ne amass 1ns1 e r en we can e uce r m r as B fulh d d iid Th dd 02 d d mr pdV pr2sin0drd0d7r 47139 przdr 26 Using 26 and 24 we get 1 717 5 2A 2Gm or 52A W G 1 27 7 i 1 27717 2 7 87rGNp GIV EWG HfH W 28 7 7 From this we get L MGM MN 3 W 29 7 Using 21 gives 7r47rGNP mrr3 19 WU P 30 47 Now we have 4 unknowns and 3 equations7 to go further give pp for example7 p 13p for electromagnetic radiation or pr Let s try a simpli ed assumption pr 007 then mr is 7717 p047rrzdr gwporg 31 19 T47quotP WP00 p 17 7rp0 0 p 32 3 Setting R and M as gn39poR3 M7 we get pilt17 tiger2 7 lt17 gt12i P 317127 7T212 3 2M 1 2M 15 7 7 1277 7 212 5 21 R 21 R3 7 1 A e 17 and also we get the metric 3 2M 1 2M 2 d5 7dt2 517 if 7 517 772 7 1r 7 r2d02 1 1 2 Next class7 we will start from discussing what happens to ligh 77 36 PHZ 6607 Special and General Relativity I Handout 3 You have just seen that the invariant line element d32 7dt2 dac2 dy2 d22 is preserved under a restricted Lorentz transformation In fact the complete Lorentz group of transformations can be de ned as that which leaves this invariant element unchanged This might appear to be the reverse of the order in which things have been de ned for you in Special Relativity but actually we also de ne other groups of transformations in exactly this way Two examples which we shall encounter again are the Rotations in three space which leave angles and distances unchanged and Conformal transformations which leave angles unchanged but not distances This line element also introduces one of the structures on manifolds which we shall be work ing with continually namely the metric The metric carries the information which determines distances and time separations and is very helpful in establishing a relationship between the manifolds with which we shall be working and the physics on them which we shall be interested in discussing Another structure which will also prove to be essential is a thing called a connection which again can be demonstrated from purely at space considerations Unlike the metric the connection is not a tensor but it will be required in the de nition of the curvature tensor which we shall also be using repeatedly There is a very easy way to witness the emergence of a connection As you will notice from a course textbook the equation for a geodesic as a suitably parameterized curve embedded in a higher dimensional manifold may be given by d2i Z dxj dank dv 1quot dA dA where A is a parameter along the curve in three dimensional space it may be taken as time Now geodesics in Euclidean three space R3 are actually straight lines And you also know that free motion in at space takes place in straight lines Suppose then we look at trying to derive this information in spherical polar coordinates instead of the usual cartesian coordinates Then for some chosen orientation of coordinates the non relativistic Lagrangian for this motion would be 771112 77103 r292 r2 sin26 The Euler Lagrange equations become 77 r92 rsin26 52 0 7r26l39 r2 sin l cos 6 2 0 and 7r2 sin26 0 where time has indeed assumed the role of the parameter A above From these we are almost immediately able to read off and identify Pge 77 P2925 77 sin2 6 P29 i I Limb isin cos and P3 cote all other possibilities being zero We shall shortly nd the geometrical reason for these results which in at space correspond to a rather special but physically natural choice for the structure which has been called the connection and is represented here by P

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