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Relativity 3

Harrison B. Prosper Florida State University YSP. Relativity 3. Topics. Recap – When is now? Distances in spacetime Gravity The Global Positioning System. Recap – When is now?. Line of simultaneity. Δt = t B - t E. t E. E. But events D and E are simultaneous for the starship

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Relativity 3

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

  2. Topics • Recap – When is now? • Distances in spacetime • Gravity • The Global Positioning System

  3. Recap – When is now?

  4. Line of simultaneity Δt = tB - tE tE E But events D and E are simultaneous for the starship so tE = tD / γ When is Now? Events B and D are simultaneous for Earth so tD = tB / γ tB tD B D Line of simultaneity O

  5. When is Now? “Nows” do not coincide Δt = tB - tE Line of simultaneity Line of simultaneity Writing distance between B and D as x = BD the temporal discrepancy is given by B tB D tD tE E O Problem 3: derive Δt Problem 4: estimate Δt between the Milky Way and Andromeda, assuming a relative speed between the galaxies of 120 km/s

  6. ct x O When is Now? ct' x' α B D E worldline of a photon emitted from O x' θ θ

  7. Distances in spacetime

  8. Q A O B Distances in Space dl The distance between points O and Q is given by: OB2 + BQ2= OQ2= OA2 + AQ2 OQ2is said to be invariant. The formula dl2 = dx2 + dy2 for computing dl2 is calledametric dy dx In 3-D, this becomes dl2 = dx2 + dy2 + dz2

  9. ct ct' P x' C Q x O A B Distances in Spacetime What is the “distance” between event O and event P? (x, ct) (x', ct') What is BC? What is AB?

  10. P O Q Distances in Spacetime Suppose that O andPare events. How far apart are they in spacetime? First guess ds2 = (cdt)2 + dl2 ds cdt dl However, this does not work for spacetime! In 1908, Hermann Minkowski showed that the correct expression is ds2 =(cdt)2 – dl2 ds2 is called the interval Hermann Minkowski 1864 -1909

  11. The Interval In general, the interval ds2 between any two events is either timelike ds2 = (cdt)2 – dl2 cdt > dl or spacelike ds2 =dl2 – (cdt)2dl > cdt or null ds2 = (cdt)2– dl2 = 0 dl = cdt

  12. C 1. Which is the longest side and which is the shortest side? A B F ct 2. Which path is longer, D to F or D to E to F? 3 E 3 5 6 D units: light-seconds x from Gravity by James B. Hartle

  13. Gravity

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

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

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

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

  18. 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

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

  20. 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

  21. 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

  22. 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

  23. 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

  24. Black Holes and wormholes

  25. 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

  26. Visual Distortions near a Black Hole

  27. 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”?

  28. The global positioning system

  29. 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

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

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

  32. 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

  33. 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

  34. 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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