General Relativity and Quantum Gravity

[...]... equation The gravitational field of a simple Einsteinian model star consists of the exterior and the interior Schwarzschild solutions which are joined together at the surface of the star Their derivation and interpretation will be discussed; in particular the Schwarzschild radius (for the sun ≈3 km) and its relation to the event horizon of the corresponding black hole will be investigated 1.1 Newton’s... supermassive black holes are rather common and probably reside at the center of every galaxy Cosmologically speaking, the supermassive black hole in the Galactic Center is in our backyard, only about 26 000 light years away from us This makes it the best observed candidate for studying all aspects of black hole physics and is an ideal laboratory for black hole physics The theory of black hole physics,... only some of the catchwords, • • • • the gravitational red shift, the gravitational deflection of light (→ gravitational lensing), the general relativistic perihelion and periastron advance, and the time delay of radar pulses (the Shapiro effect) Using additional structure from Einstein’s theory, more predictions can be verified: • • the Hulse–Taylor pulsar: emission of gravitational waves, the Lense–Thirring... (1.18) The summation convention is assumed, i.e summation is understood over repeated indices The values of the components of tensors do change, but only in the specific linear and homogeneous manner indicated here Equations of tensors remain form invariant or covariant, i.e the transformed equations look the same but with the unprimed indices replaced by primed ones If one contracts co- and contravariant... gravitational theory in quasi-field-theoretical form Gravity exists in all bodies universally and is proportional to the quantity of matter in each If two globes gravitate towards each other, and their matter is homogeneous on all sides in regions that are equally distant from their centers, then the weight of either globe towards the other will be inversely as the square of the distance between the centers... invariant but only a limiting case of the wave equation for static e situations The first idea for a Poincar´ -covariant equation for the gravitational potential e would be the obvious generalization by admitting the gravitational potential φ and the source ρ General Relativity and Quantum Gravity General Relativity and Quantum Gravity Bởi: OpenStaxCollege When we talk of black holes or the unification of forces, we are actually discussing aspects of general relativity and quantum gravity We know from Special Relativity that relativity is the study of how different observers measure the same event, particularly if they move relative to one another Einstein’s theory of general relativity describes all types of relative motion including accelerated motion and the effects of gravity General relativity encompasses special relativity and classical relativity in situations where acceleration is zero and relative velocity is small compared with the speed of light Many aspects of general relativity have been verified experimentally, some of which are better than science fiction in that they are bizarre but true Quantum gravity is the theory that deals with particle exchange of gravitons as the mechanism for the force, and with extreme conditions where quantum mechanics and general relativity must both be used A good theory of quantum gravity does not yet exist, but one will be needed to understand how all four forces may be unified If we are successful, the theory of quantum gravity will encompass all others, from classical physics to relativity to quantum mechanics—truly a Theory of Everything (TOE) General Relativity Einstein first considered the case of no observer acceleration when he developed the revolutionary special theory of relativity, publishing his first work on it in 1905 By 1916, he had laid the foundation of general relativity, again almost on his own Much of what Einstein did to develop his ideas was to mentally analyze certain carefully and clearly defined situations—doing this is to perform a thought experiment [link] illustrates a thought experiment like the ones that convinced Einstein that light must fall in a gravitational field Think about what a person feels in an elevator that is accelerated upward It is identical to being in a stationary elevator in a gravitational field The feet of a person are pressed against the floor, and objects released from hand fall with identical accelerations In fact, it is not possible, without looking outside, to know what is happening—acceleration upward or gravity This led Einstein to correctly postulate that acceleration and gravity will produce identical effects in all situations So, if acceleration affects light, then gravity will, too [link] shows the effect of acceleration 1/11 General Relativity and Quantum Gravity on a beam of light shone horizontally at one wall Since the accelerated elevator moves up during the time light travels across the elevator, the beam of light strikes low, seeming to the person to bend down (Normally a tiny effect, since the speed of light is so great.) The same effect must occur due to gravity, Einstein reasoned, since there is no way to tell the effects of gravity acting downward from acceleration of the elevator upward Thus gravity affects the path of light, even though we think of gravity as acting between masses and photons are massless (a) A beam of light emerges from a flashlight in an upward-accelerating elevator Since the elevator moves up during the time the light takes to reach the wall, the beam strikes lower than it would if the elevator were not accelerated (b) Gravity has the same effect on light, since it is not possible to tell whether the elevator is accelerating upward or acted upon by gravity Einstein’s theory of general relativity got its first verification in 1919 when starlight passing near the Sun was observed during a solar eclipse (See [link].) During an eclipse, the sky is darkened and we can briefly see stars Those in a line of sight nearest the Sun should have a shift in their apparent positions Not only was this shift observed, but it agreed with Einstein’s predictions well within experimental uncertainties This discovery created a scientific and public sensation Einstein was now a folk hero as well as a very great scientist The bending of light by matter is equivalent to a bending of space itself, with light following the curve This is another radical change in our concept of space and time It is also another connection that any particle with mass or energy (massless photons) is affected by gravity 2/11 General Relativity and Quantum Gravity There are several current forefront efforts related to general relativity One is the observation and analysis of gravitational lensing of light Another is analysis of the definitive proof of the existence of black holes Direct observation of gravitational waves or moving wrinkles in space is being searched for Theoretical efforts are also being aimed at the possibility of time travel and wormholes into other parts of space due to black holes Gravitational lensingAs you can see in [link], light is bent toward a mass, producing an effect much like a converging lens (large masses are needed to produce observable ...GENERAL RELATIVITY & COSMOLOGY for Undergraduates Professor John W. Norbury Physics Department University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee, WI 53201 1997 Contents 1 NEWTONIAN COSMOLOGY 5 1.1 Introduction 5 1.2 Equation of State 5 1.2.1 Matter 6 1.2.2 Radiation 6 1.3 Velocity and Acceleration Equations 7 1.4 Cosmological Constant 9 1.4.1 Einstein Static Universe 11 2 APPLICATIONS 13 2.1 Conservation laws 13 2.2 Age of the Universe 14 2.3 Inflation 15 2.4 Quantum Cosmology 16 2.4.1 Derivation of the Schr¨odinger equation 16 2.4.2 Wheeler-DeWitt equation 17 2.5 Summary 18 2.6 Problems 19 2.7 Answers 20 2.8 Solutions 21 3 TENSORS 23 3.1 Contravariant and Covariant Vectors 23 3.2 Higher Rank Tensors 26 3.3 Review of Cartesian Tensors 27 3.4 Metric Tensor 28 3.4.1 Special Relativity 30 3.5 Christoffel Symbols 31 1 2 CONTENTS 3.6 Christoffel Symbols and Metric Tensor 36 3.7 Riemann Curvature Tensor 38 3.8 Summary 39 3.9 Problems 40 3.10 Answers 41 3.11 Solutions 42 4 ENERGY-MOMENTUM TENSOR 45 4.1 Euler-Lagrange and Hamilton’s Equations 45 4.2 Classical Field Theory 47 4.2.1 Classical Klein-Gordon Field 48 4.3 Principle of Least Action 49 4.4 Energy-Momentum Tensor for Perfect Fluid 49 4.5 Continuity Equation 51 4.6 Interacting Scalar Field 51 4.7 Cosmology with the Scalar Field 53 4.7.1 Alternative derivation 55 4.7.2 Limiting solutions 56 4.7.3 Exactly Solvable Model of Inflation 59 4.7.4 Variable Cosmological Constant 61 4.7.5 Cosmological constant and Scalar Fields 63 4.7.6 Clarification 64 4.7.7 Generic Inflation and Slow-Roll Approximation 65 4.7.8 Chaotic Inflation in Slow-Roll Approximation 67 4.7.9 Density Fluctuations 72 4.7.10 Equation of State for Variable Cosmological Constant 73 4.7.11 Quantization 77 4.8 Problems 80 5 EINSTEIN FIELD EQUATIONS 83 5.1 Preview of Riemannian Geometry 84 5.1.1 Polar Coordinate 84 5.1.2 Volumes and Change of Coordinates 85 5.1.3 Differential Geometry 88 5.1.4 1-dimesional Curve 89 5.1.5 2-dimensional Surface 92 5.1.6 3-dimensional Hypersurface 96 5.2 Friedmann-Robertson-Walker Metric 99 5.2.1 Christoffel Symbols 101 CONTENTS 3 5.2.2 Ricci Tensor 102 5.2.3 Riemann Scalar and Einstein Tensor 103 5.2.4 Energy-Momentum Tensor 104 5.2.5 Friedmann Equations 104 5.3 Problems 105 6 Einstein Field Equations 107 7 Weak Field Limit 109 8 Lagrangian Methods 111 4 CONTENTS Chapter 1 NEWTONIAN COSMOLOGY 1.1 Introduction Many of the modern ideas in cosmology can be explained without the need to discuss General Relativity. The present chapter represents an attempt to do this based entirely on Newtonian mechanics. The equations describing the velocity (called the Friedmann equation) and acceleration of the universe are derived from Newtonian mechanics and also the cosmological constant is introduced within a Newtonian framework. The equations of state are also derived in a very simple way. Applications such as conservation laws, the age of the universe and the inflation, radiation and matter dominated epochs are discussed. 1.2 Equation of State In what follows the equation of state for non-relativistic matter and radiation will be needed. In particular an expression for the rate of change of density, ˙ρ, will be needed in terms of the density ρ and pressure p. (The definition ˙x ≡ dx dt , where t is time, is being used.) The first law of thermodynamics is dU + dW = dQ (1.1) where U is the internal energy, W is the work and Q is the heat transfer. Ignoring any heat transfer and writing dW = Fdr = pdV where F is the 5 6 CHAPTER 1. NEWTONIAN COSMOLOGY force, r is the distance, p is the pressure and V is the volume, then dU = −pdV. (1.2) Assuming that ρ is a relativistic energy density means that the energy is expressed as U = ρV (1.3) from which it follows that ˙ U =˙ρV + ρ ˙ V = −p ˙ V (1.4) where the term Introduction to Differential Geometry & General Relativity 4th Printing January 2005 Lecture Notes by Stefan Waner with a Special Guest Lecture by Gregory C. Levine Departments of Mathematics and Physics, Hofstra University 2 Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries 3 2. Smooth Manifolds and Scalar Fields 7 3. Tangent Vectors and the Tangent Space 14 4. Contravariant and Covariant Vector Fields 24 5. Tensor Fields 35 6. Riemannian Manifolds 40 7. Locally Minkowskian Manifolds: An Introduction to Relativity 50 8. Covariant Differentiation 61 9. Geodesics and Local Inertial Frames 69 10. The Riemann Curvature Tensor 82 11. A Little More Relativity: Comoving Frames and Proper Time 94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 100 13. Two Basic Premises of General Relativity 109 14. The Einstein Field Equations and Derivation of Newton's Law 114 15. The Schwarzschild Metric and Event Horizons 124 16. White Dwarfs, Neutron Stars and Black Holes, by Gregory C. Levine131 References and Further Reading 138 3 1. Preliminaries Distance and Open Sets Here, we do just enough topology so as to be able to talk about smooth manifolds. We begin with n-dimensional Euclidean space E n = {(y 1 , y 2 , . . . , y n ) | y i é R}. Thus, E 1 is just the real line, E 2 is the Euclidean plane, and E 3 is 3-dimensional Euclidean space. The magnitude, or norm, ||y|| of y = (y 1 , y 2 , . . . , y n ) in E n is defined to be ||y|| = y 1 2 !+!y 2 2 !+!.!.!.!+!y n 2 , which we think of as its distance from the origin. Thus, the distance between two points y = (y 1 , y 2 , . . . , y n ) and z = (z 1 , z 2 , . . . , z n ) in E n is defined as the norm of z - y: Distance Formula Distance between y and z = ||z - y|| = (z 1 !-!y 1 ) 2 !+!(z 2 !-!y 2 ) 2 !+!.!.!.!+!(z n !-!y n ) 2 . Proposition 1.1 (Properties of the norm) The norm satisfies the following: (a) ||y|| ≥ 0, and ||y|| = 0 iff y = 0 (positive definite) (b) ||¬y|| = |¬|||y|| for every ¬ é R and y é E n . (c) ||y + z|| ≤ ||y|| + ||z|| for every y, z é E n (triangle inequality 1) (d) ||y - z|| ≤ ||y - w|| + ||w - z|| for every y, z, w é E n (triangle inequality 2) The proof of Proposition 1.1 is an exercise which may require reference to a linear algebra text (see “inner products”). Definition 1.2 A Subset U of E n is called open if, for every y in U, all points of E n within some positive distance r of y are also in U. (The size of r may depend on the point y chosen. Illustration in class). Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set. Examples 1.3 (a) If a é E n , then the open ball with center a and radius r is the subset B(a, r) = {x é E n | ||x-a|| < r}. 4 Open balls are open sets: If x é B(a, r), then, with s = r - ||x-a||, one has B(x, s) ¯ B(a, r). (b) E n is open. (c) Ø is open. (d) Unions of open sets are open. (e) Open sets are unions of open balls. (Proof in class) Definition 1.4 Now let M ¯ E s . A subset V ¯ M is called open in M (or relatively open) if, for every y in V, all points of M within some positive distance r of y are also in V. Examples 1.5 (a) Open balls in M If M ¯ E s , m é M, and r > 0, define B M (m, r) = {x é M | ||x-m|| < r}. Then B M (m, r) = B(m, r) Ú M, and so B M (m, r) is open in M. (b) M is open in M. (c) Ø is open in M. (d) Unions of open sets in M are open in M. (e) Open sets in M are unions of open balls in M. Parametric Semi-Riemann Geometry and General Relativity Shlomo Sternberg September 24, 2003 2 0.1 Introduction This book represents course notes for a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einstein’s theory of general relativity. The background assumed is a good grounding in linear algebra and in advanced calculus, preferably in the language of differential forms. Chapter I introduces the various curvatures associated to a hypersurface embedded in Euclidean space, motivated by the formula for the volume for the region obtained by thickening the hypersurface on one side. If we thicken the hypersurface by an amount h in the normal direction, this formula is a polynomial in h whose coefficients are integrals over the hypersurface of local expressions. These local expressions are elementary symmetric polynomials in what are known as the principal curvatures. The precise definitions are given in the text.The chapter culminates with Gauss’ Theorema egregium which asserts that if we thicken a two dimensional surface evenly on both sides, then the these integrands depend only on the intrinsic geometry of the surface, and not on how the surface is embedded. We give two proofs of this important theorem. (We give several more later in the book.) The first proof makes use of “normal coor- dinates” which become so important in Riemannian geometry and, as “inertial frames,” in general relativity. It was this theorem of Gauss, and particularly the very notion of “intrinsic geometry”, which inspired Riemann to develop his geometry. Chapter II is a rapid review of the differential and integral calculus on man- ifolds, including differential forms,the d operator, and Stokes’ theorem. Also vector fields and Lie derivatives. At the end of the chapter are a series of sec- tions in exercise form which lead to the notion of parallel transport of a vector along a curve on a embedded surface as being associated with the “rolling of the surface on a plane along the curve”. Chapter III discusses the fundamental notions of linear connections and their curvatures, and also Cartan’s method of calculating curvature using frame fields and differential forms. We show that the geodesics on a Lie group equipped w ith a bi-invariant metric are the translates of the one parameter subgroups. A short exercise set at the end of the chapter uses the Cartan calculus to compute the curvature of the Schwartzschild metric. A second exercise set computes some geodesics in the Schwartzschild metric leading to two of the famous predictions of general relativity: the advance of the perihelion of Mercury and the bending of light by matter. Of course the theoretical basis of these computations, i.e. the theory of general relativity, will come later, in Chapter VII. Chapter IV begins by discussing the bundle of frames which is the modern setting for Cartan’s calculus of “moving frames” and also the jumping off point for the general theory of connections on principal bundles which lie at the base of such modern physical theories as Yang-Mills fields. This chapter seems to present the most difficulty conceptually for the student. Chapter V discusses the general theory of connections on fibe r bundles and then sp e cialize to principal and associated bundles. 0.1. INTRODUCTION 3 Chapter VI returns to Riemannian geometry and discusses Gauss’s lemma which asserts that the radial geodesics emanating from a point are orthogo- nal (in the Riemann metric) to the images under the exponential map of the spheres in the tangent space centered at the origin. From this one concludes that geodesics (defined as self parallel curves) locally minimize arc length in a Riemann manifold. Chapter VII is a rapid review of special GENERAL RELATIVITY & COSMOLOGY for Undergraduates Professor John W. Norbury Physics Department University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee, WI 53201 1997 Contents 1 NEWTONIAN COSMOLOGY 5 1.1 Introduction 5 1.2 Equation of State 5 1.2.1 Matter 6 1.2.2 Radiation 6 1.3 Velocity and Acceleration Equations 7 1.4 Cosmological Constant 9 1.4.1 Einstein Static Universe 11 2 APPLICATIONS 13 2.1 Conservation laws 13 2.2 Age of the Universe 14 2.3 Inflation 15 2.4 Quantum Cosmology 16 2.4.1 Derivation of the Schr¨odinger equation 16 2.4.2 Wheeler-DeWitt equation 17 2.5 Summary 18 2.6 Problems 19 2.7 Answers 20 2.8 Solutions 21 3 TENSORS 23 3.1 Contravariant and Covariant Vectors 23 3.2 Higher Rank Tensors 26 3.3 Review of Cartesian Tensors 27 3.4 Metric Tensor 28 3.4.1 Special Relativity 30 3.5 Christoffel Symbols 31 1 2 CONTENTS 3.6 Christoffel Symbols and Metric Tensor 36 3.7 Riemann Curvature Tensor 38 3.8 Summary 39 3.9 Problems 40 3.10 Answers 41 3.11 Solutions 42 4 ENERGY-MOMENTUM TENSOR 45 4.1 Euler-Lagrange and Hamilton’s Equations 45 4.2 Classical Field Theory 47 4.2.1 Classical Klein-Gordon Field 48 4.3 Principle of Least Action 49 4.4 Energy-Momentum Tensor for Perfect Fluid 49 4.5 Continuity Equation 51 4.6 Interacting Scalar Field 51 4.7 Cosmology with the Scalar Field 53 4.7.1 Alternative derivation 55 4.7.2 Limiting solutions 56 4.7.3 Exactly Solvable Model of Inflation 59 4.7.4 Variable Cosmological Constant 61 4.7.5 Cosmological constant and Scalar Fields 63 4.7.6 Clarification 64 4.7.7 Generic Inflation and Slow-Roll Approximation 65 4.7.8 Chaotic Inflation in Slow-Roll Approximation 67 4.7.9 Density Fluctuations 72 4.7.10 Equation of State for Variable Cosmological Constant 73 4.7.11 Quantization 77 4.8 Problems 80 5 EINSTEIN FIELD EQUATIONS 83 5.1 Preview of Riemannian Geometry 84 5.1.1 Polar Coordinate 84 5.1.2 Volumes and Change of Coordinates 85 5.1.3 Differential Geometry 88 5.1.4 1-dimesional Curve 89 5.1.5 2-dimensional Surface 92 5.1.6 3-dimensional Hypersurface 96 5.2 Friedmann-Robertson-Walker Metric 99 5.2.1 Christoffel Symbols 101 CONTENTS 3 5.2.2 Ricci Tensor 102 5.2.3 Riemann Scalar and Einstein Tensor 103 5.2.4 Energy-Momentum Tensor 104 5.2.5 Friedmann Equations 104 5.3 Problems 105 6 Einstein Field Equations 107 7 Weak Field Limit 109 8 Lagrangian Methods 111 4 CONTENTS Chapter 1 NEWTONIAN COSMOLOGY 1.1 Introduction Many of the modern ideas in cosmology can be explained without the need to discuss General Relativity. The present chapter represents an attempt to do this based entirely on Newtonian mechanics. The equations describing the velocity (called the Friedmann equation) and acceleration of the universe are derived from Newtonian mechanics and also the cosmological constant is introduced within a Newtonian framework. The equations of state are also derived in a very simple way. Applications such as conservation laws, the age of the universe and the inflation, radiation and matter dominated epochs are discussed. 1.2 Equation of State In what follows the equation of state for non-relativistic matter and radiation will be needed. In particular an expression for the rate of change of density, ˙ρ, will be needed in terms of the density ρ and pressure p. (The definition ˙x ≡ dx dt , where t is time, is being used.) The first law of thermodynamics is dU + dW = dQ (1.1) where U is the internal energy, W is the work and Q is the heat transfer. Ignoring any heat transfer and writing dW = Fdr = pdV where F is the 5 6 CHAPTER 1. NEWTONIAN COSMOLOGY force, r is the distance, p is the pressure and V is the volume, then dU = −pdV. (1.2) Assuming that ρ is a relativistic energy density means that the energy is expressed as U = ρV (1.3) from which it follows that ˙ U =˙ρV + ρ ... theory of quantum gravity Hawking is a long-time survivor of ALS and has produced popular books on general relativity, cosmology, and quantum gravity (credit: Lwp Kommunikáció) Gravity and quantum. .. are 6/11 General Relativity and Quantum Gravity another The first significant connection between gravity and quantum effects was made by the Russian physicist Yakov Zel’dovich in 1971, and other... Schwarzschild radius RS and is given by 4/11 General Relativity and Quantum Gravity RS = 2GM c2 , where G is the universal gravitational constant, M is the mass of the body, and c is the speed of