A General Relativity Workbook
by Moore
ISBN: 9781891389825 | Copyright 2012
Instructor Requests
General relativity, which lies at the heart of contemporary physics, has recently become the focus of a number of lively theoretical, experimental, and computational research programs. As a result, undergraduates have become increasingly excited to learn about the subject.A General Relativity Workbook is a textbook intended to support a one-semester upper division undergraduate course on general relativity. Through its unique workbook-based design, it enables students to develop a solid mastery of both the physics and the supporting tensor calculus by pushing (and guiding) them to work through the implications. Each chapter, which is designed to correspond to one class session, involves a short overview of the concepts without obscuring derivations or details, followed by a series of boxes that guide students through the process of working things out for themselves.This active-learning approach enables students to develop a more secure mastery of the material than more traditional approaches. More than 350 homework problems support further learning. This book more strongly emphasizes the physics than many of its competitors, and while it provides students a full grounding in the supporting mathematics (unlike certain other competitors), it introduces the mathematics gradually and in a completely physical context.
Published under the University Science Books imprint
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| Front Cover (pg. i) | |
| Contents (pg. viii) | |
| Preface (pg. xvi) | |
| 1. Introduction (pg. 1) | |
| Concept Summary (pg. 1) | |
| Homework Problems (pg. 9) | |
| General Relativity in a Nutshell (pg. 11) | |
| 2. Review of special relativity (pg. 13) | |
| Concept Summary (pg. 13) | |
| Box 2.1 Overlapping IRFs Move with Constant Relative Velocities (pg. 19) | |
| Box 2.2 Unit Conversions Between SI and GR Units (pg. 20) | |
| Box 2.3 One Derivation of the Lorentz Transformation (pg. 21) | |
| Box 2.4 Lorentz Transformations and Rotations (pg. 25) | |
| Box 2.5 Frame-Independence of the Spacetime Interval (pg. 26) | |
| Box 2.6 Frame-Dependence of the Time Order of Events (pg. 26) | |
| Box 2.7 Proper Time Along a Path (pg. 27) | |
| Box 2.8 Length Contraction (pg. 27) | |
| Box 2.9 The Einstein Velocity Transformation (pg. 28) | |
| Homework Problems (pg. 29) | |
| 3. Four-Vector (pg. 31) | |
| Concept Summary (pg. 31) | |
| Box 3.1 The Frame-Independence of the Scalar Product (pg. 36) | |
| Box 3.2 The Invariant Magnitude of the Four-Velocity (pg. 36) | |
| Box 3.3 The Low-Velocity Limit of u (pg. 37) | |
| Box 3.4 Conservation of Momentum or Four-momentum? (pg. 38) | |
| Box 3.5 Example: The GZK Cosmic-Ray Energy Cutoff (pg. 40) | |
| Homework Problems (pg. 42) | |
| 4. Index notation (pg. 43) | |
| Concept Summary (pg. 43) | |
| Box 4.1 Behavior of the Kronecker Delta (pg. 48) | |
| Box 4.2 EM Field Units in the GR Unit System (pg. 48) | |
| Box 4.3 Electromagnetic Equations in Index Notation (pg. 49) | |
| Box 4.4 Identifying Free and Bound Indices (pg. 50) | |
| Box 4.5 Rule Violations (pg. 50) | |
| Box 4.6 Example Derivations (pg. 51) | |
| Homework Problems (pg. 52) | |
| 5. Arbitrary Coordinates (pg. 53) | |
| Concept Summary (pg. 53) | |
| Box 5.1 The Polar Coordinate Basis (pg. 58) | |
| Box 5.2 Proof of the Metric Transformation Law (pg. 59) | |
| Box 5.3 A 2D Example: Parabolic Coordinates (pg. 60) | |
| Box 5.4 The LTEs as an Example General Transformation (pg. 62) | |
| Box 5.5 The Metric Transformation Law in Flat Space (pg. 62) | |
| Box 5.6 A Metric for a Sphere (pg. 63) | |
| Homework Problems (pg. 63) | |
| 6. Tensor Equations (pg. 65) | |
| Concept Summary (pg. 65) | |
| Box 6.1 Example Gradient Covectors (pg. 70) | |
| Box 6.2 Lowering Indices (pg. 71) | |
| Box 6.3 The Inverse Metric (pg. 72) | |
| Box 6.4 The Kronecker Delta Is a Tensor (pg. 73) | |
| Box 6.5 Tensor Operations (pg. 73) | |
| Homework Problems (pg. 75) | |
| 7. Maxwell’s Equations (pg. 77) | |
| Concept Summary (pg. 77) | |
| Box 7.1 Gauss’s Law in Integral and Differential Form (pg. 82) | |
| Box 7.2 The Derivative of m2 (pg. 83) | |
| Box 7.3 Raising and Lowering Indices in Cartesian Coordinates (pg. 83) | |
| Box 7.4 The Tensor Equation for Conservation of Charge (pg. 84) | |
| Box 7.5 The Antisymmetry of F Implies Charge Conservation (pg. 85) | |
| Box 7.6 The Magnetic Potential (pg. 86) | |
| Box 7.7 Proof of the Source-Free Maxwell Equations (pg. 87) | |
| Homework Problems (pg. 88) | |
| 8. Geodesics (pg. 89) | |
| Concept Summary (pg. 89) | |
| Box 8.1 The Worldline of Longest Proper Time in Flat Spacetime (pg. 93) | |
| Box 8.2 Derivation of the Euler-Lagrange Equation (pg. 94) | |
| Box 8.3 Deriving the Second Form of the Geodesic Equation (pg. 95) | |
| Box 8.4 Geodesics for Flat Space in Parabolic Coordinates (pg. 96) | |
| Box 8.5 Geodesics for the Surface of a Sphere (pg. 98) | |
| Box 8.6 The Geodesic Equation Does Not Determine the Scale of x (pg. 100) | |
| Box 8.7 Light Geodesics in Flat Spacetime (pg. 101) | |
| Homework Problems (pg. 102) | |
| 9. The Schwarzschild Metric (pg. 105) | |
| Concept Summary (pg. 105) | |
| Box 9.1 Radial Distance (pg. 110) | |
| Box 9.2 Falling from Rest in Schwarzschild Spacetime (pg. 111) | |
| Box 9.3 GM for the Earth and the Sun (pg. 112) | |
| Box 9.4 The Gravitational Redshift for Weak Fields (pg. 112) | |
| Homework Problems (pg. 114) | |
| 10. Particle orbits (pg. 115) | |
| Concept Summary (pg. 115) | |
| Box 10.1 Schwarzschild Orbits Must Be Planar (pg. 120) | |
| Box 10.2 The Schwarzschild “Conservation of Energy” Equation (pg. 121) | |
| Box 10.3 Deriving Conservation of Newtonian Energy for Orbits (pg. 122) | |
| Box 10.4 The Radii of Circular Orbits (pg. 122) | |
| Box 10.5 Kepler’s Third Law (pg. 124) | |
| Box 10.6 The Innermost Stable Circular Orbit (ISCO) (pg. 125) | |
| Box 10.7 The Energy Radiated by an Inspiraling Particle (pg. 126) | |
| Homework Problems (pg. 127) | |
| 11. Precession of the Perihelion (pg. 129) | |
| Concept Summary (pg. 129) | |
| Box 11.1 Verifying the Orbital Equation for u(z) (pg. 135) | |
| Box 11.2 Verifying the Newtonian Orbital Equation (pg. 135) | |
| Box 11.3 Verifying the Equation for the Orbital “Wobble” (pg. 136) | |
| Box 11.4 Application to Mercury (pg. 136) | |
| Box 11.5 Constructing the Schwarzschild Embedding Diagram (pg. 137) | |
| Box 11.6 Calculating the Wedge Angle d (pg. 138) | |
| Box 11.7 A Computer Model for Schwarzschild Orbits (pg. 138) | |
| Homework Problems (pg. 141) | |
| 12. Photon Orbits (pg. 143) | |
| Concept Summary (pg. 143) | |
| Box 12.1 The Meaning of the Impact Parameter b (pg. 148) | |
| Box 12.2 Derivation of the Equation of Motion for a Photon (pg. 148) | |
| Box 12.3 Features of the Effective Potential Energy Function for Light (pg. 149) | |
| Box 12.4 Photon Motion in Flat Space (pg. 149) | |
| Box 12.5 Evaluating 4-Vector Components in an Observer’s Frame (pg. 150) | |
| Box 12.6 An Orthonormal Basis in Schwarzschild Coordinates (pg. 150) | |
| Box 12.7 Derivation of the Critical Angle for Photon Emission (pg. 151) | |
| Homework Problems (pg. 152) | |
| 13. Deflection of Light (pg. 153) | |
| Concept Summary (pg. 153) | |
| Box 13.1 Checking Equation 13.2 (pg. 159) | |
| Box 13.2 The Differential Equation for the Shape of a Photon Orbit (pg. 160) | |
| Box 13.3 The Differential Equation for the Photon “Wobble” (pg. 160) | |
| Box 13.4 The Solution for u(z) in the Large-r Limit (pg. 161) | |
| Box 13.5 The Maximum Angle of Light Deflection by the Sun (pg. 161) | |
| Box 13.6 The Lens Equation (pg. 162) | |
| Box 13.7 The Ratio of Image Brightness to the Source Brightness (pg. 163) | |
| Homework Problems (pg. 164) | |
| 14. Event Horizon (pg. 167) | |
| Concept Summary (pg. 167) | |
| Box 14.1 Finite Distance to r = 2GM (pg. 172) | |
| Box 14.2 Proper Time for Free Fall from r = R to r = 0 (pg. 174) | |
| Box 14.3 The Future Is Finite Inside the Event Horizon (pg. 175) | |
| Homework Problems (pg. 176) | |
| 15. Alternative Coordinates (pg. 179) | |
| Concept Summary (pg. 179) | |
| Box 15.1 Calculating 2tc/2r (pg. 184) | |
| Box 15.2 The Global Rain Metric (pg. 185) | |
| Box 15.3 The Limits on dr/dtc Inside the Event Horizon (pg. 185) | |
| Box 15.4 Transforming to Kruskal-Szekeres Coordinates (pg. 186) | |
| Homework Problems (pg. 188) | |
| 16. Black Hole Thermodynamics (pg. 189) | |
| Concept Summary (pg. 189) | |
| Box 16.1 Free-Fall Time to the Event Horizon from r = 2GM + f (pg. 194) | |
| Box 16.2 Calculating E3 (pg. 195) | |
| Box 16.3 Evaluating kB, &, and T for a Solar-Mass Black Hole (pg. 196) | |
| Box 16.4 Lifetime of a Black Hole (pg. 197) | |
| Homework Problems (pg. 198) | |
| 17. The Absolute Gradient (pg. 199) | |
| Concept Summary (pg. 199) | |
| Box 17.1 Absolute Gradient of a Vector (pg. 204) | |
| Box 17.2 Absolute Gradient of a Covector (pg. 204) | |
| Box 17.3 Symmetry of the Christoffel Symbols (pg. 205) | |
| Box 17.4 The Christoffel Symbols in Terms of the Metric (pg. 205) | |
| Box 17.5 Checking the Geodesic Equation (pg. 206) | |
| Box 17.6 A Trick for Calculating Christoffel Symbols (pg. 206) | |
| Box 17.7 The Local Flatness Theorem (pg. 207) | |
| Homework Problems (pg. 210) | |
| 18. Geodesic Deviation (pg. 211) | |
| Concept Summary (pg. 211) | |
| Box 18.1 Newtonian Tidal Deviation Near a Spherical Object (pg. 216) | |
| Box 18.2 Proving Equation 18.9 (pg. 217) | |
| Box 18.3 The Absolute Derivative of n (pg. 217) | |
| Box 18.4 Proving Equation 18.14 (pg. 218) | |
| Box 18.5 An Example of Calculating the Riemann Tensor (pg. 218) | |
| Homework Problems (pg. 220) | |
| 19. The Riemann Tensor (pg. 221) | |
| Concept Summary (pg. 221) | |
| Box 19.1 The Riemann Tensor in a Locally Inertial Frame (pg. 224) | |
| Box 19.2 Symmetries of the Riemann Tensor (pg. 225) | |
| Box 19.3 Counting the Riemann Tensor’s Independent Components (pg. 226) | |
| Box 19.4 The Bianchi Identity (pg. 227) | |
| Box 19.5 The Ricci Tensor Is Symmetric (pg. 228) | |
| Box 19.6 The Riemann and Ricci Tensors and R for a Sphere (pg. 228) | |
| Homework Problems (pg. 230) | |
| 20. The stress-energy Tensor (pg. 231) | |
| Concept Summary (pg. 231) | |
| Box 20.1 Why the Source of Gravity Must Be Energy, Not Mass (pg. 236) | |
| Box 20.2 Interpretation of Tij in a Locally Inertial Frame (pg. 236) | |
| Box 20.3 The Stress-Energy Tensor for a Perfect Fluid in Its Rest LIF (pg. 237) | |
| Box 20.4 Equation 20.16 Reduces to Equation 20.15 (pg. 239) | |
| Box 20.5 Fluid Dynamics from Conservation of Four-Momentum (pg. 239) | |
| Homework Problems (pg. 241) | |
| 21. The Einstein Equation (pg. 243) | |
| Concept Summary (pg. 243) | |
| Box 21.1 The Divergence of the Ricci Tensor (pg. 248) | |
| Box 21.2 Finding the Value of b (pg. 249) | |
| Box 21.3 Showing that –R + 4K = lT (pg. 250) | |
| Homework Problems (pg. 251) | |
| 22. Interpreting The Equation (pg. 253) | |
| Concept Summary (pg. 253) | |
| Box 22.1 Conservation of Four-Momentum Implies 0 = do (t0u ) (pg. 258) | |
| Box 22.2 The Inverse Metric in the Weak-Field Limit (pg. 258) | |
| Box 22.3 The Riemann Tensor in the Weak-Field Limit (pg. 259) | |
| Box 22.4 The Ricci Tensor in the Weak-Field Limit (pg. 260) | |
| Box 22.5 The Stress-Energy Sources of the Metric Perturbation (pg. 261) | |
| Box 22.6 The Geodesic Equation for a Slow Particle in a Weak Field (pg. 262) | |
| Homework Problems (pg. 263) | |
| 23. The schwarzschild solution (pg. 265) | |
| Concept Summary (pg. 265) | |
| Box 23.1 Diagonalizing the Spherically Symmetric Metric (pg. 270) | |
| Box 23.2 The Components of the Ricci Tensor (pg. 271) | |
| Box 23.3 Solving for B (pg. 274) | |
| Box 23.4 Solving for a(r) (pg. 275) | |
| Box 23.5 The Christoffel Symbols with t-t as Subscripts (pg. 275) | |
| Homework Problems (pg. 276) | |
| 24. The universe observed (pg. 279) | |
| Concept Summary (pg. 279) | |
| Box 24.1 Measuring Astronomical Distances in the Solar System (pg. 284) | |
| Box 24.2 Determining the Distance to Stellar Clusters (pg. 286) | |
| Box 24.3 How the Doppler Shift Is Connected to Radial Speed (pg. 287) | |
| Box 24.4 Values of the Hubble Constant (pg. 288) | |
| Box 24.5 Every Point Is the Expansion’s “Center” (pg. 288) | |
| Box 24.6 The Evidence for Dark Matter (pg. 289) | |
| Homework Problems (pg. 290) | |
| 25. A metric for the cosmos (pg. 293) | |
| Concept Summary (pg. 293) | |
| Box 25.1 The Universal Ricci Tensor (pg. 298) | |
| Box 25.2 Raising One Index of the Universal Ricci Tensor (pg. 298) | |
| Box 25.3 The Stress-Energy Tensor with One Index Lowered (pg. 298) | |
| Box 25.4 The Einstein Equation with One Index Lowered (pg. 301) | |
| Box 25.5 Verifying the Solutions for q (pg. 302) | |
| Homework Problems (pg. 303) | |
| 26. Evolution of the Universe (pg. 305) | |
| Concept Summary (pg. 305) | |
| Box 26.1 The Other Components of the Einstein Equation (pg. 310) | |
| Box 26.2 Consequences of Local Energy/Momentum Conservation (pg. 311) | |
| Box 26.3 Deriving the Density/Scale Relationship for Radiation (pg. 312) | |
| Box 26.4 Deriving the Friedman Equation (pg. 312) | |
| Box 26.5 The Friedman Equation for the Present Time (pg. 313) | |
| Box 26.6 Deriving the Friedman Equation in Terms of the Omegas (pg. 313) | |
| Box 26.7 The Behavior of a Matter-Dominated Universe (pg. 314) | |
| Homework Problems (pg. 315) | |
| 27. Cosmic Implications (pg. 317) | |
| Concept Summary (pg. 317) | |
| Box 27.1 Connecting the Redshift z to the Hubble Constant (pg. 322) | |
| Box 27.2 Deriving the Hubble Relation in Terms of Redshift z (pg. 322) | |
| Box 27.3 The Luminosity Distance (pg. 323) | |
| Box 27.4 The Differential Equation for a(h) (pg. 323) | |
| Box 27.5 How to Generate a Numerical Solution for Equation 27.18 (pg. 324) | |
| Homework Problems (pg. 325) | |
| 28. The Early Universe (pg. 327) | |
| Concept Summary (pg. 327) | |
| Box 28.1 Single-Component Universes (pg. 332) | |
| Box 28.2 The Transition to Matter Dominance (pg. 333) | |
| Box 28.3 The Time-Temperature Relation (pg. 333) | |
| Box 28.4 Neutrino Decoupling (pg. 335) | |
| Box 28.5 The Number Density of Photons (pg. 337) | |
| Homework Problems (pg. 338) | |
| 29. CMB fluctuations and inflation (pg. 339) | |
| Concept Summary (pg. 339) | |
| Box 29.1 The Angular Width of the Largest CMB Fluctuations (pg. 345) | |
| Box 29.2 The Equation for Xk(t) (pg. 346) | |
| Box 29.3 Cosmic Flatness at the End of Nucleosynthesis (pg. 347) | |
| Box 29.4 The Exponential Inflation Formula (pg. 347) | |
| Box 29.5 Inflation Calculations (pg. 348) | |
| Homework Problems (pg. 349) | |
| 30. Gauge Freedom (pg. 351) | |
| Concept Summary (pg. 351) | |
| Box 30.1 The Weak-Field Einstein Equation in Terms of hno (pg. 355) | |
| Box 30.2 The Trace-Reverse of hno (pg. 356) | |
| Box 30.3 The Weak-Field Einstein Equation in Terms of Hno (pg. 357) | |
| Box 30.4 Gauge Transformations of the Metric Perturbations (pg. 358) | |
| Box 30.5 A Gauge Transformation Does Not Change Rabno (pg. 359) | |
| Box 30.6 Lorenz Gauge (pg. 360) | |
| Box 30.7 Additional Gauge Freedom (pg. 361) | |
| Homework Problems (pg. 361) | |
| 31. Detecting gravitational waves (pg. 363) | |
| Concept Summary (pg. 363) | |
| Box 31.1 Constraints on Our Trial Solution (pg. 368) | |
| Box 31.2 The Transformation to Transverse-Traceless Gauge (pg. 369) | |
| Box 31.3 A Particle at Rest Remains at Rest in TT Coordinates (pg. 371) | |
| Box 31.4 The Effect of a Gravitational Wave on a Ring of Particles (pg. 372) | |
| Homework Problems (pg. 373) | |
| 32. Gravitational Wave energy (pg. 375) | |
| Concept Summary (pg. 375) | |
| Box 32.1 The Ricci Tensor (pg. 379) | |
| Box 32.2 The Averaged Curvature Scalar (pg. 379) | |
| Box 32.3 The General Energy Density of a Gravitational Wave (pg. 379) | |
| Homework Problems (pg. 382) | |
| 33. Generating gravitational waves (pg. 383) | |
| Concept Summary (pg. 383) | |
| Box 33.1 Htn for a Compact Source Whose CM is at Rest (pg. 388) | |
| Box 33.2 A Useful Identity (pg. 388) | |
| Box 33.3 The Transverse-Traceless Components of Ano (pg. 390) | |
| Box 33.4 How to Find ITTjk p for Waves Moving in the nvDirection (pg. 391) | |
| Box 33.5 Flux in Terms of I jk (pg. 393) | |
| Box 33.6 Evaluating the Integrals in the Power Calculation (pg. 394) | |
| Homework Problems (pg. 395) | |
| 34. Gravitational wave astronomy (pg. 397) | |
| Concept Summary (pg. 397) | |
| Box 34.1 The Dumbbell I jk (pg. 402) | |
| Box 34.2 The Power Radiated by a Rotating Dumbbell (pg. 403) | |
| Box 34.3 The Total Energy of an Orbiting Binary Pair (pg. 404) | |
| Box 34.4 The Time-Rate-of-Change of the Orbital Period (pg. 404) | |
| Box 34.5 Characteristics of k Boötis (pg. 405) | |
| Homework Problems (pg. 406) | |
| 35. Gravitomagnetism (pg. 407) | |
| Concept Summary (pg. 407) | |
| Box 35.1 The Lorenz Condition for the Potentials (pg. 412) | |
| Box 35.2 The Maxwell Equations for the Gravitational Field (pg. 413) | |
| Box 35.3 The Gravitational Lorentz Equation (pg. 414) | |
| Box 35.4 The “Gravitomagnetic Moment” of a Spinning Object (pg. 414) | |
| Box 35.5 Angular Speed of Gyroscope Precession (pg. 415) | |
| Homework Problems (pg. 416) | |
| 36. The Kerr Metric (pg. 417) | |
| Concept Summary (pg. 417) | |
| Box 36.1 Expanding R - rv v -1 to First Order in r/R (pg. 421) | |
| Box 36.2 The Integral for htx (pg. 422) | |
| Box 36.3 Why the Other Terms in the Expansion Integrate to Zero (pg. 423) | |
| Box 36.4 Transforming the Weak-Field Solution to Polar Coordinates (pg. 424) | |
| Box 36.5 The Weak-Field Limit of the Kerr Metric (pg. 425) | |
| Homework Problems (pg. 426) | |
| 37. Particle Orbits in kerr spacetime (pg. 427) | |
| Concept Summary (pg. 427) | |
| Box 37.1 Calculating Expressions for dt/dx and dz/dx (pg. 431) | |
| Box 37.2 Verify the Value of [g ] g g 2tz - tt zz (pg. 432) | |
| Box 37.3 The “Energy-Conservation-Like” Equation of Motion (pg. 433) | |
| Box 37.4 Kepler’s Third Law (pg. 434) | |
| Box 37.5 The Radii of ISCOs When a = GM (pg. 435) | |
| Homework Problems (pg. 436) | |
| 38. Ergoregion and Horizon (pg. 437) | |
| Concept Summary (pg. 437) | |
| Box 38.1 The Radii Where gtt = 0 (pg. 441) | |
| Box 38.2 The Angular Speed Range When dr and/or di ≠ 0 (pg. 442) | |
| Box 38.3 Angular-Speed Limits in the Equatorial Plane (pg. 443) | |
| Box 38.4 The Metric of the Event Horizon’s Surface (pg. 444) | |
| Box 38.5 The Area of the Outer Kerr Event Horizon (pg. 445) | |
| Box 38.6 Transformations Preserve the Metric Determinant’s Sign (pg. 445) | |
| Homework Problems (pg. 447) | |
| 39. Negative-Energy Orbits (pg. 449) | |
| Concept Summary (pg. 449) | |
| Box 39.1 Quadratic Form for Conservation of Energy (pg. 454) | |
| Box 39.2 The Square Root Is Zero at the Event Horizon (pg. 454) | |
| Box 39.3 Negative e Is Possible Only in the Ergoregion (pg. 456) | |
| Box 39.4 The Fundamental Limit on dM in Terms of dS (pg. 457) | |
| Box 39.5 dMir ≥ 0 (pg. 458) | |
| Box 39.6 The Spin Energy Contribution to a Black Hole’s Mass (pg. 459) | |
| Homework Problems (pg. 460) | |
| Appendix: A Diagonal Metric Worksheet (pg. 463) | |
| Index (pg. 467) | |
Thomas A. Moore
Thomas A. Moore is a professor in the physics department of Pomona College. He graduated from Carleton College in 1976, and earned an M. Phil. in 1978 and a Ph. D. in 1981 from Yale University. He then taught at Carleton College and Luther College before taking his current position at Pomona College in 1987, where he won a Wig Award for Distinguished Teaching in 1991. He served as an active member of the national Introductory University Physics Project (IUPP), and has published a number of articles about astrophysical sources of gravitational waves, detection of gravitational waves, and new approaches to teaching physics. His previous books include A Traveler's Guide to Spacetime (McGraw-Hill, 1995) on special relativity, and a six-volume introductory calculus-based physics text called Six Ideas That Shaped Physics (McGraw-Hill, 2003).|
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