General Relativity

1 General RelativityStefanos AretakisJune 30, 20182 Contents1 Special Newtonian Physics .. The Birth of Special Relativity .. The Minkowski SpacetimeR3+1.. Theory .. Observers, Frames of Reference and Isometies .. and Special Covariance .. Mechanics .. Conformal Structure .. Double Null Foliation .. Penrose Diagram .. Electromagnetism and Maxwell Equations .. 232 Lorentzian Causality I .. Null Geometry .. Global Hyperbolicity .. Causality II .. 413 Introduction to General Equivalence Principle .. The Einstein Equations .. The Cauchy Problem .. Gravitational Redshift and Time Dilation .. Applications .. 484 Null Structure The Double Null Foliation .. Connection Coefficients .. Curvature Components.

2 The Algebra Calculus ofS-Tensor Fields .. Null Structure Equations .. The Characteristic Initial Value Problem .. 7234 CONTENTS5 Applications to Null Jacobi Fields and Tidal Forces .. Focal Points .. Causality III .. Trapped Surfaces .. Penrose Incompleteness Theorem .. Killing Horizons .. 886 Christodoulou s Memory The Null InfinityI+.. Tracing gravitational waves .. Peeling and Asymptotic Quantities .. The Memory Effect .. 1017 Black Introduction .. Black Holes and Trapped Surfaces .. Black Hole Mechanics .. Spherical Symmetry .. Setting .. Black Holes .. Kerr Black Holes .. 1088 Lagrangian Theories and the Variational Matter Fields .. The Action Principle .. Derivation of the Energy Momentum Tensor.

3 Application to Linear Waves .. Noether s Theorem .. 1169 Hyperbolic The Energy Method .. A Priori Estimate .. Well-posedness of the Wave Equation .. The Wave Equation on Minkowski spacetime .. 12510 Wave Propagation on Black Introduction .. Pointwise and Energy Boundedness .. Pointwise and Energy Decay .. 138 CONTENTS5 IntroductionGeneral Relativity is the classical theory that describes the evolution of systems underthe effect of gravity. Its history goes back to 1915 when Einstein postulated that the lawsof gravity can be expressed as a system of equations, the so-called Einstein order to formulate his theory, Einstein had to reinterpret fundamental concepts ofour experience (such as time, space, future, simultaneity, etc.) in a purely geometricalframework. The goal of this course is to highlight the geometric character of GeneralRelativity and unveil the fascinating properties of black holes, one of the most celebratedpredictions of mathematical course will start with a self-contained introduction to special Relativity and thenproceed to the more General setting of Lorentzian manifolds.

4 Next the Lagrangian formu-lation of the Einstein equations will be presented. We will formally define the notion ofblack holes and prove the incompleteness theorem of Penrose (also known as singularitytheorem). The topology of General black holes will also be investigated. Finally, we willpresent explicit spacetime solutions of the Einstein equations which contain black holeregions, such as the Schwarzschild, and more generally, the Kerr 1 Special RelativityIn both past and modern viewpoints, the universe is considered to be a continuumcomposed of events, where each event can be thought of as a point in space at an instantof time. We will refer to this continuum as the spacetime. The geometric properties, andin particular the causal structure of spacetimes in Newtonian physics and in the theoryof Relativity greatly differ from each other and lead to radically different perspectives forthe physical world and its begin by listing the key assumptions about spacetime in Newtonian physics andthen proceed by replacing these assumptions with the postulates of special Newtonian PhysicsMain assumptionsThe primary assumptions in Newtonian physics are the following1.

5 There is an absolute notion of time. This implies the notion of simultaneity is The speed of light is finite and observer Observers can travel arbitrarily fast (in particular faster thanc).From the above one can immediately infer the existence of a time coordinatet Rsuchthat all the events of constant timetcompose a 3-dimensional Euclidean space. Thespacetime is topologically equivalent toR4and admits a universal coordinate system(t,x1,x2,x3).Causal structureGiven an eventpoccurring at timetp, the spacetime can be decomposed into thefollowing sets: Future ofp: Set of all events for whicht > tp. Present ofp: Set of all events for whicht= 1. SPECIAL Relativity Past ofp: Set of all events for whicht < generally, we can define the future (past) of a setSto be the union of the futures(pasts) of all points : The Newtonian universe and its causal structureFrom now on we consider geometric units with respect to which the light travels atspeedc= 1 relative to observers at rest.

6 If an observer atpemits a light beam in alldirections of space, then the trajectory of this beam in spacetime will be a null cone withvertex atp. We can complete this cone by considering the trajectory of light beams thatarrive at the : The Newtonian universe and light trajectoriesIt is important to emphasize that in Newtonian theory, in view of the existence of theabsolute timet, one only works by projecting on the Euclidean spaceR3and consideringall quantities as functions of (the space and) s theory gives a very accurate theory for objects moving at slow speeds inabsence of strong gravitational fields. However, in several circumstances difficulties arise:1. A more philosophical issue is that in Newtonian theory an observer is either at restor in motion. But how could one determine if an observerOis (universally) THE BIRTH OF SPECIAL RELATIVITY9rest?

7 Why can t a uniformly moving (relative toO) observerPbe considered atrest sincePis also not affected by any external influence?2. For hundreds of years it has been known that in vacuum light propagates at a veryhigh but constant speed, and no material has been observed to travel faster. If anobserverPis moving at speedc/5 (relative to an observerOat rest) towards alight beam (which is moving at speedcrelative toO), then the light would reachthe observerPat speedc+c/5. However, astronomical observations of doublestars should reveal such fast and slow light, but in fact the speeds turn out to bethe Light is the propagation of an electromagnetic disturbance and electromagneticfields are governed by Maxwell s equations. However, these equations are not well-behaved in Newtonian theory; in particular, in this context, these laws are observerdependent and hence do not take the desired form of universal physical of Einstein s contributions was his persistence that every physical law can beexpressed independently of the choice of coordinates (we will return to this point later).

8 It was this persistence along with his belief that Maxwell s equations are flawless thatled to what is now known as special The Birth of Special RelativityIn 1905 Einstein published a paper titled On the electrodynamics of moving bodies ,where he described algebraic relations governing the motion of uniform observers so thatMaxwell equations have the same form regardless of the observer s frame. In order toachieve his goal, Einstein had to assume the following1. There is no absolute notion of No observer or particle can travel faster than the speed of lightc. The constantcshould be considered as a physical law and hence does not depend on the observerwho measures above immediately change the Newtonian perception for the spacetime, sinceunder Einstein s assumptions the future (past) of an eventpis confined to be the interiorof the future (past) light cone with vertex 1.

9 SPECIAL RELATIVITYIn 1908, Hermann Minkowski showed that Einstein s algebraic laws (and, in partic-ular, the above picture) can be interpreted in a purely geometric way, by introducing anew kind of metric onR4, the so-called Minkowski The Minkowski SpacetimeR3+1 DefinitionA Minkowski metricgon the linear spaceR4is a symmetric non-degenerate bilinearform with signature ( ,+,+,+). In other words, there is a basis{e0,e1,e2,e3}such thatg(e ,e ) =g , , {0,1,2,3},where the matrixg is given byg= 1 0 0 001 0 000 1 000 0 1 .Given such a frame (which for obvious reason will be called orthonormal), one can readilyconstruct a coordinate system (t,x1,x2,x3) ofR4such that at each point we havee0= t, ei= xi, i= 1,2, that from now on in order to emphasize the signature of the metric we will denotethe Minkowski spacetime byR3+1. With respect to the above coordinate system, themetricgcan be expressed as a (0,2) tensor as follows:g= dt2+ (dx1)2+ (dx2)2+ (dx3)2.

10 ( )Note that (for an arbitrary pseudo-Riemannian metric) one can still introduce a Levi Civita connection and therefore define the notion the associated Christoffel symbols andgeodesic curves and that of the Riemann, Ricci and scalar curvature. One can alsodefine the volume form such that ifX , = 0,1,2,3, is an orthonormal frame then (X0,X1,X2,X3) = 1. In the Minkowski case, the curvatures are all zero and thegeodesics are lines with respect to the coordinate system (t,x1,x2,x3). THE MINKOWSKI SPACETIMER3+ Causality TheoryThe fundamental new aspect of this metric is that it is not positive-definite. A vectorX R3+1is defined to , ifg(X,X)>0, , ifg(X,X) = 0, , ifg(X,X)< either timelike or null, then it is calledcausal. IfX= (t,x1,x2,x3) is nullvector atp, thent2= (x1)2+ (x2)2+ (x3)2and henceXlies on cone with vertex atp.