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Relativity 3

Harrison B. Prosper Florida State University YSP 2012. Relativity 3. Topics. Gravity The Global Positioning System. Gravity. Gravity. All objects fall with the same acceleration. G. Galileo 1564 –1642. The Principle of Equivalence. A person falling off a building

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Relativity 3

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  1. Harrison B. Prosper Florida State University YSP 2012 Relativity 3

  2. Topics • Gravity • The Global Positioning System

  3. Gravity

  4. Gravity All objects fall with the same acceleration G. Galileo 1564 –1642

  5. The Principle of Equivalence A person falling off a building experiences no gravity! “The happiest thought of my life” Albert Einstein (1907)

  6. free space view outside view inside view The Principle of Equivalence Gravity is curvedspacetime!

  7. General Relativity (1915) Bending of light Precession of Mercury’s orbit Sir Arthur Eddington Eclipse Expeditions 1919

  8. Metric in Spherical Polar Coordinates z Parameters of spherical polar coordinates (r, θ, φ) Consider the spatial plane θ = 90o AC = rdφ CB = dr AB = dl r θ φ The metric in spacetime is therefore, y Δφ C x B A

  9. Spacetime around a Spherical Star In 1916, Karl Schwarzschild found the first exact solution of Einstein’s equations: Karl Schwarzschild 1873 - 1916 φ r

  10. Spatial Geometry and Proper Distance The proper distance between any two events is their spacetime separation ds when the time difference dt between the events is zero in a given frame of reference. B dt = 0 A

  11. Spatial Geometry and Proper Distance Consider the proper distance along the arc of a circular trajectory, in the Schwarzschild spacetime geometry: dr = 0 dt = 0 ds dφ for a complete circuit. Note: the spacetime separation = proper distance in this case

  12. Spatial Geometry and Proper Distance Now consider a radial trajectory, dφ = 0. Then, the separation ds is given by r Again, in this case, the spacetime separation = proper distance

  13. Spatial Geometry and Proper Distance Problem: Suppose we traveled along a radial trajectory from the Earth to the Sun’s photosphere. How far would we have traveled? That is, what is the proper distance? Wolfram MathematicaOnline Integrator http://integrals.wolfram.com r

  14. Black Holes and wormholes

  15. Black Holes Consider hovering near a black hole at afixedradius r. How would your elapsed time be related to the elapsed time of someone far away? near far away

  16. Visual Distortions near a Black Hole

  17. Wormholes The Morris-Thorne metric (θ = 90o) a = Throat radius Problem 5: Assuming a = 1 km, how far must you walk to get to the center of the wormhole? Assume you start at an “r”-distance that corresponds to a circumference of 2π×10 km. How does the proper distance compare with the “r-distance”?

  18. The global positioning system

  19. The Global Positioning System What is it ? A system of 24 satellites in orbit about Earth that provides accurate world-wide navigation Each satellite contains an atomic clock, accurate to ~ 1 nanosecond Each satellite emits a unique signal giving its position

  20. GPS – Orbits Period: 12 hours Orbital radius: 26,600 km Six orbital planes 60o apart

  21. ct3 ct1 ct2 GPS – Principle 1 2 3 You are here!

  22. GPS – Circular Orbits For circular orbits, r does not change, so dr = 0. Therefore, Now divide by (cdt)2, and noting that v = rdφ/dt, the tangential speed measured by an observer far away, then the elapsed time dτ at any given radius r is given by

  23. GPS Clocks vsatellite vEarth rsatellite t= time far from Earth τ = time at given radius r rS= Schwarzschild radius rEarth Problem 6: How fast (or slow) does the satellite clock run per day relative to the Earth clocks? Give answer in nanoseconds

  24. Summary • The interval between events is invariant. • A timelike interval measures the elapsed time along a worldline. • Gravity is warped spacetime • Time is slowed by gravity

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