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General Relativity: the Notes* Preprint typeset in JHEP style - HYPER VERSION General Relativity: the Notes* C.P. Burgess Department of Physics & Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4M1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5. Abstract: These notes present a brief introduction to Einstein's General Theory of Relativity, prepared for the course Physics 3A03. *©Cliff Burgess, March 2009 Contents 1. Elements of Differential Geometry 2 1.1 The Geometry of Surfaces 3 1.2 More General Curved Space 17 2. Special Relativity and Flat Spacetime 24 2.1 Minkowski Spacetime 25 2.2 Inertial Particle Motion 28 2.3 Non-inertial Motion 31 2.4 Conserved Quantities 37 3. Weak Gravitational Fields 44 3.1 Newtonian Gravity 44 3.2 Gravity as Geometry 48 3.3 Relativistic Effects in the Solar System 53 4. Field Equations for Curved Space 70 4.1 Gravity as curvature 70 4.2 Einstein's Field Equations 71 4.3 Rotationally Invariant Solutions 73 5. Compact Stars and Black Holes 76 5.1 Orbits 77 5.2 Radial geodesics 80 5.3 Singularities of the solution 82 5.4 Black Holes and Event Horizons 84 5.5 Quantum Effects Near Black Holes 86 5.6 Rotating Black Holes 91 6. Other Astrophysical Applications 95 6.1 Stellar interiors 95 6.2 Gravitational Lensing 102 6.3 Gravitational Waves 108 6.4 Binary pulsars 110 6.5 Astrophysical Black Holes 116 { 1 { 7. Cosmology 121 7.1 Kinematics of an Expanding Universe 121 7.2 Distance vs Redshift 129 7.3 Dynamics of an Expanding Universe 137 7.4 The Present-Day Energy Content 148 7.5 Earlier Epochs 155 7.6 Hot Big Bang Cosmology 159 1. Elements of Differential Geometry The essence of general relativity is that gravity is described by the geometry of spacetime, and so this first section pauses to summarize some of the mathematics used to describe non-Euclidean geometries. Before doing so, a brief reminder about Euclidean geometry. Euclidean Geometry Euclid founded his study of plane (i.e. 2-dimensional) geometry on the following five axioms: 1. Any two points can be joined by a straight line. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. All of these seem to be obviously true, given the standard notions of what a point, straight line, circle, right angle and congruence mean. Among the consequences of these axioms are many familiar statements like: the ratio of a circle's circumference, C, to its radius, r, is a universal number: C=r = 2π; the ratio of a circle's area, A, to the square of its radius is also a universal number A=r2 = π; the sum of the interior angles of a triangle sum to 180 degrees, and so on. We are used to taking { 2 { these consequences for granted when understanding the relations amongst objects in physical space. The rest of this section is devoted to describing simple situations where they do not all apply. Once this is done, it becomes an experimental issue whether or not the Euclidean axioms are properties of the space in which we find ourselves situated. The goal of this section is to develop the tools for this, by setting up a precise characterization of these new geometries, and the ways they can differ from Euclidean space. 1.1 The Geometry of Surfaces The non-Euclidean geometries that are easiest to visualize are those of two-dimensional surfaces, such as planes, spheres or hyperbolae. These are easy to picture since we can envision these surfaces embedded in 3-dimensional space. To this end consider the 3-dimensional vector space, IR3, whose vectors, r, de- scribe the distance from an (arbitrary) origin, O, to the various points in space. It is convenient to describe such a vector by its components referred to a `rectangular' basis of unit vectors, fex; ey; ezg, oriented in a fixed but arbitrary direction, so that r = x ex + y ey + z ez i = x ei ; (1.1) where the three coordinates, (x; y; z), each can run from −∞ to 1. Some important notation is introduced in the second equality of eq. (1.1), which 1 2 3 writes x = x, x = y, x = z, and e1 = ex, e2 = ey and e3 = ez. There is also an implied sum from 1 to 3 over the repeated index `i', or any other repeated index taken from the middle of the Latin alphabet for that matter. (Indices taken from the beginning of the Latin alphabet are encountered later, where they run over a; b = 1; 2; and indices from the Greek alphabet also come up, and will be summed from µ, ν = 0; 1; 2; 3.) This rule for summing over repeated indices is called the Einstein summation convention, and in terms of it the dot product of two vectors i i i j with components a = a ei and b = b ei can be written a · b = δij a b , where the Kronecker-δ symbol has the property that δij = 1 if i = j and δij = 0 otherwise. We take the distance, s(r1; r2), between any two points, r1 and r2, in IR3 is given in terms of their rectangular coordinates by the usual Pythagorean rule p s(r1; r2) = jr1 − r2j = (r1 − r2) · (r1 − r2) q i i j j = δij(x1 − x2)(x1 − x2) (1.2) p 2 2 2 = (x1 − x2) + (y1 − y2) + (z1 − z2) ; { 3 { where the middle line again uses the Einstein summation convention. This definition has the important property that it does not depend at all on the origin, O, and orientation of the axes, ei = fex; ey; ezg, that are required to define the coordinates i x = fx; y; zg describing r1 and r2. Curves in Space Before describing two-dimensional surfaces in IR3, it is worth briefly digressing to describe the simpler case of one-dimensional curves. A curve in IR3 is defined by the locus of points that are swept out as a single parameter varies: r(u) = x(u) ex + y(u) ey + z(u) ez i = x (u) ei : (1.3) Here the parameter u labels the points on the curve and our interest is usually in component functions xi(u) = fx(u); y(u); z(u)g that are multiply differentiable with respect to u. For example, straight lines in this picture are described by linear functions, r(u) = a + b u, where a and b are arbitrary constant vectors. When the origin, O, is not on the straight line (i.e. a 6= 0) then the origin together with the line define a plane, which is spanned by the vectors a and b. More generally, a straight line is also given by r(u) = a + b f(u), for any function f(u) that satisfies df=du 6= 0, since this simply represents a relabelling of the points along the curve. By contrast, a curve of the form r(u) = c + a cos u + b sin u traces out a more complicated closed shape, which becomes an ellipse if a and b are perpendicular to one another: a · b = axbx + ayby + azbz = 0. In this case c specifies the position of the ellipse's centre, and its two semi-major axes are p q 2 2 2 p i j a = jaj = a · a = ax + ay + az = δij a a p q 2 2 2 p i j b = jbj = b · b = bx + by + bz = δij b b : (1.4) This ellipse is inscribed on the plane spanned by the vectors a and b, and degenerates into a circle in the special case that a and b have the same length: a = b. The family of vectors that lie tangent to a curve r(u) is found by differentiation, dr dx dy dz dxi t(u) = = e + e + e = e ; (1.5) du du x du y du z du i and a one-parameter family of unit vectors tangent to the curve is found by normal- izing t(u) e (u) = ; (1.6) t jt(u)j { 4 { so et · et = 1 for all u. For a straight line, r(u) = a + bf(u), the tangent dr df t(u) = = b ; (1.7) du du has a constant direction, but a u-dependent length that depends on the precise function f(u) used to parametrize the curve. But for any parametrization the unit tangent vector for a straight line is a constant vector: et = b=jbj. The basis vectors, ei = fex; ey; ezg may themselves be regarded as unit tangent vectors to the curves defining the rectangular coordinate axes themselves: that is, ex is the unit tangent to the curves along which y and z are constant, and similarly for ey and ez. The tangent to the elliptical curve centered at the origin, r(u) = a cos u + b sin u is given by t(u) = −a sin u + b cos u, whose direction changes continuously with u, p with norm jt(u)j = a2 sin2 u + b2 cos2 u (and we use a · b = 0).
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