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Lecture 3: General Relativity

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Lecture 3: General Relativity Spacetime diagrams North • A useful way of making spacetime maps • Similar to normal map (north-south & east-west)... • ...but with one axis being time instead of a direction West Spacetime diagram in the car’s rest frame time space Spacetime diagram in the car’s rest frame time space Spacetime diagram in our rest frame time space Spacetime diagram in our rest frame time “car’s wordline” space time space “light cone” “light time “light cone” space • Light always takes shortest path between two points time • Light travels on straight lines • “Light rays” “light cone” • The shortest connections between points are called “light cone” “geodesics” space • Light always takes shortest path between two points time • Light travels on straight lines • “Light rays” “light cone” • The shortest connections between points are called “light cone” “geodesics” space Star party @ Space Place, Monday, 7:15PM time Ice Cream Social @ Michael’s, Sunday, 7:15PM space Star party @ Space Place, Monday, 7:15PM time Ice Cream Social @ Michael’s, Sunday, 7:15PM space Star party @ Space Place, Monday, 7:15PM time Ice Cream Social @ Michael’s, Sunday, 7:15PM space Star party @ Space • Events outside of the “light Place, Monday, 7:15PM cone” cannot be visited by time us, no matter how fast we travel. • They are separated by a real physical distance from us and we can measure that distance Ice Cream Social @ with a physical ruler. Michael’s, Sunday, 7:15PM • Events inside the “light cone” are in our potential past or future and we can visit them space • Their distance from us a time interval Gravity • Special relativity formulated in inertial frames Must find an inertial frame for SR! For electric force, just use a neutral particle to anchor frame to, that will be unaccelerated. But gravity accelerates everything (remember Gallileo?) • Special relativity does not hold under gravity Stationary Elevator with gravity: Ball is accelerated down Stationary Elevator with gravity: Ball is accelerated down Outside of an accelerated elevator: Ball at rest Outside of an accelerated elevator: Ball at rest Inside of accelerated elevator: ball accelerated down Inside of accelerated elevator: ball accelerated down =elevator with gravity! =elevator with gravity! General relativity • Einstein’s fundamental insight: “Equivalence principle” Gravity accelerates everything ⇒ Gravity must be a property of spacetime Gravity and acceleration are indistinguishable (Galileo) ⇒ Formulate physics in terms of accelerated frames Equivalence principle Elevator at rest Elevator at rest Elevator in uniform motion Elevator in uniform motion Inside the moving elevator Inside the moving elevator Accelerated elevator from outside Accelerated elevator from outside Inside the accelerated elevator Inside the accelerated elevator = In an elevator in a gravitational field = In an elevator in a gravitational field Light bending: • Gravity bends light • Recall: light travels on spacetime geodesics Light bending: • Gravity bends light • Recall: light travels on spacetime geodesics ⇒ In spacetime with gravity, geodesics are curved Light bending: • Gravity bends light • Recall: light travels on spacetime geodesics ⇒ In spacetime with gravity, geodesics are curved • Geodesics are the straightest possible lines ⇒ Gravity curves spacetime Spacetime curvature Spacetime curvature Spacetime curvature time space Spacetime curvature time space Spacetime curvature time space In curved space • Parallel lines don’t stay parallel • Triangles don’t add up to 180° • The straightest possible lines are “geodesics” • The stronger the curvature, the stronger theses effects In curved spacetime • The actual length to a destination is changed (try this yourself!) In curved spacetime • The actual length to a destination is changed (try this yourself!) • The circumference of a circle is no longer 2πR (try this yourself!) In curved spacetime • The actual length to a destination is changed (try this yourself!) • The circumference of a circle is no longer 2πR (try this yourself!) • Sometimes, more than one path is the shortest path (try this yourself!) In curved spacetime • The actual length to a destination is changed (try this yourself!) • The circumference of a circle is no longer 2πR (try this yourself!) • Sometimes, more than one path is the shortest path (try this yourself!) Is space curved? What curves spacetime? • Gravity curves spacetime • We know that mass causes gravity ⇒ Mass curves spacetime What curves spacetime? • Gravity curves spacetime • We know that mass causes gravity ⇒ Mass curves spacetime What curves spacetime? • Einstein’s most fundamental equation relates the curvature to mass: More mass, more curvature More curvature closer to mass • Einstein’s equivalent to Newton’s law of gravity “Field equation” What curves spacetime? • Einstein’s most fundamental equation relates the curvature to mass: More mass, more curvature More curvature closer to mass • Einstein’s equivalent to Newton’s law of gravity “Field equation” What curves spacetime? • Einstein’s most fundamental equation relates the curvature to mass: More mass, more curvature More curvature closer to mass • Einstein’s equivalent to Newton’s law of gravity “Field equation” Light bending (2): Light bending (2): • Heavy objects curve spacetime Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them Light bending (2): • Heavy objects curve spacetime • Galaxy clusters are very heavy: 1000 trillion times more massive than the sun • They should curve spacetime a lot • Light should follow curved path around them.
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