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General Relativity Volker Schlue General Relativity Volker Schlue Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario M5S 2E4, Canada E-mail address: [email protected] Preface I have prepared this manuscript for a course on general relativity that I taught at the University of Toronto in the winter semester 2013. It is based on notes that I took as a student at ETH Z¨urich in a lecture course that Demetrios Christodoulou gave in the fall of 2005. (Chapters that are in significant parts transcribed from these notes will be marked by a .) His teaching and understanding of general relativity made a great impression∗ on me as a student, and I hope that these lecture notes can contribute to a wider circulation of his approach. I am very grateful to Andr´eLisibach, who provided me with most of the exer- cises, as well as complementary notes from a similar course that Prof. Christodoulou gave last fall, which were useful for the preparation of the present manuscript. Toronto, January 2014 V. Schlue iii Contents Preface iii Chapter 1. The equivalence principle and its consequences ∗ 1 1. Classical theory of gravitation 2 2. Special Relativity 5 3. Geodesic correspondence 20 Chapter 2. Einstein's field equations ∗ 47 1. Einstein's field equations in the presence of matter 50 2. Action Principle 66 3. The material manifold 74 4. Cosmological constant 79 Chapter 3. Spherical Symmetry 81 1. Einstein's field equations in spherical symmetry 81 2. Schwarzschild solution 86 3. General properties of the area radius and mass functions 90 4. Spherically symmetric spacetimes with a trapped surface 92 Chapter 4. Dynamical Formulation of the Einstein Equations ∗ 97 1. Decomposition relative to the level sets of a time function 97 2. Slow Motion Approximation 115 3. Gravitational Radiation 125 Bibliography 149 v CHAPTER 1 The equivalence principle and its consequences ∗ There are two motivations for the General Theory of Relativity. (1) Extension of the principle of Special Relativity (invariance of the physical laws under change from one inertial system to another | one such system relative to another is in a state of uniform motion) to all systems of reference in any state of motion whatsoever. (2) Establish a theory of gravitation which is in accordance with Special Relativity. Einstein's Equivalence Principle relates the two motivations. Remark. (1) is sometimes referred to as the requirement that the laws of physics should be valid in any system of spacetime coordinates. (General Covari- ance.) In fact, however, (1) refers to the requirement of invariance of the physical laws under change of physical description, from one relative to one set of observers to another relative to another set of such observers. The history of an observer is represented in relativity by a timelike curve. A family of observers is given by a family of timelike curves that do not intersect each other. Each set of curves is a foliation of a given spacetime region. See Fig. 1. Figure 1. Two families of observers. The history of each observer is a timelike curve. 1 2 1. THE EQUIVALENCE PRINCIPLE Notes on Special Relativity An inertial system is a system of reference relative to which any mass moves on a straight line if no external forces act upon it. In view of Newton's Laws set in Euclidean space, any such system is equivalent to (is not distinguished from) any other system of reference in uniform relative motion, which leads to the principle of Galilean relativity. The corre- sponding transformation laws leave time unchanged (up to translations) and thus entail surfaces of absolute simultaneity (level sets of time) which separate the future from the past. See Fig. 2. Special Relativity is based on the premise that light, which prior to Ein- stein had been assumed to propagate in a medium at rest relative to a distinguished frame of reference, in fact propagates in the absence of any such medium on cocentric spheres with respect to any inertial system when emitted from a point source; in other words the speed of light is finite and the same in any inertial system. Einstein realized this obser- vation can only be reconciled if we give up the notion of absolute simultaneity in favour of a notion of simultaneity relative to the observer such that the propagation of light is independent of the system of reference, c.f. Fig. 2. This is precisely achieved in the theory of Special Relativity according to which space and time are unified in Minkowski space (R3+1; m), a 3 + 1-dimensional linear space endowed with a quadratic form m of index 1. A frame of reference corresponds to the choice of a unit timelike vector e, m(e; e) = 1, − which determines a unique spacelike hyperplane Σe = X : m(e; X) = 0 with positive- definite induced metric m 0, a 3-dimensional Euclideanf space consistingg of all events jΣe ≥ considered simultaneous by observers on Σe at rest relative to the frame defined by e, namely observers whose \world-lines" are straight lines with tangent vector e. The set of null vectors L at a point p, m(L; L) = 0, form a \light cone" consisting of all events reached by light emitted from and received at p. The light cone takes the role of level sets of absolute time in Galilean physics in the sense that it separates the past from the future: The set of time-like vectors T at a point p, m(T;T ) < 0, the interior of the light cone, has two disconnected components referred to as future- and past-directed time-like vec- tors pointing to events that can possibly be influenced by an event at p, or have possibly influenced p, respectively. The exterior of the cone at p, the set of spacelike vectors X at p, m(X; X) > 0, is \causally disconnected" from p, given that no speed of propagation of any physical action whatsoever has ever been observed to surpass the speed of light. The \Principle of special relativity" asserts that all physical phenomena occur in one frame of reference as they do with respect to another, and in principle cannot be used to distinguish a certain frame of reference, i.e. a specific unit time-like vector e. The elements of the orthogonal group with respect to the Minkowski metric m, m(OX; OY ) = m(X; Y ), are called Lorentz transformations, and a \proper" (time-orientation preserving) subgroup mediates in particular the change of one frame of reference to another: e e0 = Oe, m(Oe; Oe) = m(e; e) = 1. Thus the validity of the principle of special relativity7! requires the invariance of all physical− laws under Lorentz transformations. 1. Classical theory of gravitation A postulate of the classical theory (mechanics: motion of bodies in a given force field & gravitation: the gravitational force field of a given distribution of bodies) is the equality (up to a universal constant depending on the choice of physical units) of the passive gravitational mass and the inertial mass. 1. CLASSICAL THEORY OF GRAVITATION 3 Figure 2. Propagation of light in Galilean and Special Relativity. : Newtonian gravitational potential. mG (t; x): Gravitational force acting on a test body of gravitational − r mass mG at time t, the instantaneous position of the body at time t R 3 2 being x R . mG is thus a concept of the classical theory of gravitation. x = x(t): Motion2 of a test body. If a body is subjected to a given force F when at position x and time t, then its acceleration is: d2x 1 (1.1) 2 (t) = F (x; t) dt mI This law (Newton's Second Law of Motion) defines the inertial mass mI . The postulate is (1.2) mG = mI : In the classical framework there is no explanation for this equality. Set (1.3) F = mG (t; x(t)): − r Then mG = mI implies that these cancel in the equations of motion: d2x (1.4) (t) = (t; x(t)) dt2 −∇ That is, the motion of the test body in a given gravitational field is independent of the test body. 1.1. Tidal forces. Consider the gravitational field in a small space-time re- gion, the neighborhood of an event (t0; x0), then a0 = (t0; x0) is the common −∇ acceleration of all test bodies in this region. Let x = x0(t) be the history of the test 0 mass which we take as a reference mass. In the new reference system x = x x0 the origin is translated at each time to the instantaneous position of the reference− mass. (See Fig. 3, 4.) Note that this is an accelerated reference system. 4 1. THE EQUIVALENCE PRINCIPLE t 2 x0 x2 1 x0 x1 x = x0(t) Figure 3. History of a reference mass. t x0 x x0(t) = 00 x2 x1 Figure 4. Reference coordinate system with origin at each time at the instantaneous postion of the reference mass. Now consider the motion of another, arbitrary test body in this new reference system: x0 = x0(t). 2 0 d x 0 (t) = t; x0(t) + x (t) a0(t) dt2 −∇ − (1.5) 0 = t; x0(t) + x (t) + t; x0(t) −∇ r 2 0 0 2 = t; x0(t) x (t) + ( x (t) ) −∇ · O j j In terms of rectangular components, and to linear order in x0(t) this equation reads 2 0i 3 2 d x X @ 0j (1.6) (t) = t; x0(t) x (t) dt2 − @xi@xj · j=1 and is called the tidal equation. 2. SPECIAL RELATIVITY 5 R P 0: Periphery Figure 5.
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