Einstein for Everyone Lecture 6: Introduction to General Relativity Dr. Erik Curiel Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universit¨at Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry 1 Introduction to General Relativity 2 Newtonian Gravity Kepler's Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of Light 5 Rejection of Absolute Space 6 Euclidean Geometry Deductive Structure Fifth Postulate 7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary 8 Riemannian Geometry Intrinsic vs. Extrinsic Curvature Geodesic Deviation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Why a New Theory of Gravity? Newtonian Gravity Einstein's Special Relativity Incredibly empirically Newtonian ) Minkowski successful spacetime Force of gravity: Space and time observer - depends on spatial dependent, replaced by distance at a single invariant space-time instant of time interval - instantaneous interaction ) absolute simultaneity Challenge New theory of gravity compatible with special relativity? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Responses to the Challenge Einstein's Contemporaries (Poincar´e,Minkowski, Max Abraham, Gustav Mie. ) - Reformulate gravity in Minkowski spacetime - Preserve special relativity, change theory of gravity Einstein's Approach - Relativity as an incomplete revolution - Change both \special relativity" and theory of gravity - Conceptual problems within Newtonian gravity - ) reformulate notion of relativistic spacetime - ) need to generalize notion of \geometry" Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Responses to the Challenge Einstein's Contemporaries (Poincar´e,Minkowski, Max Abraham, Gustav Mie. ) - Reformulate gravity in Minkowski spacetime - Preserve special relativity, change theory of gravity Einstein's Approach - Relativity as an incomplete revolution - Change both \special relativity" and theory of gravity - Conceptual problems within Newtonian gravity - ) reformulate notion of relativistic spacetime - ) need to generalize notion of \geometry" Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Why Geometry? Einstein's \rough and winding road" (1907-1915) 1905 Special relativity 1907 \Happiest thought of my life" (principle of equivalence) Equivalence between gravity and acceleration Need to extend relativity to accelerated frames 1909 Ehrenfest's Rotating Disk Acceleration ) Non-Euclidean Geometry 1912-13 Hole Argument 1915 General theory of relativity Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry 1 Introduction to General Relativity 2 Newtonian Gravity Kepler's Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of Light 5 Rejection of Absolute Space 6 Euclidean Geometry Deductive Structure Fifth Postulate 7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary 8 Riemannian Geometry Intrinsic vs. Extrinsic Curvature Geodesic Deviation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Kepler's Laws Copernican Revolution Ptolemaic Hypothesis Copernican Hypothesis Images from Hevelius, Selenographica (1647) (courtesy of Trinity College, Cambridge) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Kepler's Laws Johannes Kepler (1571-1630) Kepler's Innovations Orbit of Mars: Ellipse Motion of planets caused by sun, analogy with magnetism Kepler's \Laws" 1 Planets move along an ellipse with the sun at one focus. 2 They sweep out equal areas in equal times. 3 The radius of the orbit a is related Kepler's New Astronomy to the period P as P 2 / a3 (1609) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Inertia and Acceleration Types of Motion Inertial Motion Motion in a straight line with uniform velocity (that is, covering equal distances in equal times). Accelerated Motion Change in velocity (speed up or slow down) or direction (e.g., rotation) Based on Newtonian space and time: Spatial Geometry: straight line; distances measured by measuring rods Time: time elapsed, measured by a clock Location over time: distance traveled over time Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Inertia and Acceleration Newton's First Law Law I Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed upon it. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Inertia and Acceleration Newton's Second Law Law II A change in motion is proportional to the motive force impressed and takes place in the direction of the straight line along which that force is impressed. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Inertia and Acceleration Force and Inertial Mass Modern formulation of second law: F = mia Force F Measured by departure Inertial Mass mi from inertial motion Intrinsic property of a Treated abstractly, body quantitatively Measures how much force Examples: impact, is required to accelerate a attraction (magnetism, body gravitation), dissipation Not equivalent to weight (friction), tension (oscillating string), . Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Gravity Newtonian Gravity Attractive force between interacting bodies M; m: Mgmg F = G r2 Dependence on Distance 1 - F / r2 1 - Move bodies twice as far apart, force decreases by 4 Dependence on Masses - Force depends on gravitational masses of both interacting bodies Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Gravity Newton's Argument for Universal Gravitation 1 1 Kepler's Laws ! Force F / r2 - Kepler's laws hold for planets and satellites 2 \Moon Test": this force is gravity! - Compare force on moon to force on falling bodies near Earth's surface 3 Dependence on Mass - Pendulum Experiments 4 Conclusion: universal, mutual attractive force Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Gravity Galileo on Freely Falling Bodies Bodies fall in the same way regardless of composition Two separate concepts of mass 1 Inertial mass: F = mia 2 Gravitational mass: Mg mg F = G r2 If mi = mg, then Galileo's result is true! Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Gravity Objections to Newton's Theory (ca. 1907) 1 Problems due to Special Relativity - Space and time no longer invariant - Instantaneous interaction 2 \Epistemological Defect" in Newton's theory - Why are inertial and gravitational mass equal? - Absolute space 3 Empirical Problems - Motion of Mercury - (Lunar motion, motion of Venus) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry Conservatives vs. Einstein Relativity Meets Gravity Problems due to Special Relativity Spatial distance between two bodies observer-dependent Time at which force acts observer-dependent Conservative Response Reformulate gravity in terms of space-time distance Minkowski, Poincar´e,Abraham, Mie, N¨ordstr¨om:several possibilities, fairly straightforward