General Relativity

1. General Relativity

2. Topics • The Principle of Equivalence • Flat and Curved Space • The Schwarzschild Geometry • The Global Positioning System • Summary

3. The Principle of Equivalence All objects fall with the same acceleration G. Galileo 1564–1642

4. The Principle of Equivalence A person falling off a building experiences no gravity! “The happiest thought of my life” Albert Einstein (1907) http://nssdc.gsfc.nasa.gov

5. free space free fall inside view free fall outside view free fall outside view The Principle of Equivalence

6. The Principle of Equivalence Bending of light Gravity is curved spacetime 1915 Sir Arthur Eddington Eclipse Expeditions 1919

7. Flat Space z Interval in Spherical polar coordinates (r, q, f) ds2 = c2dt2 – ds2 r q Temporal distance Spatial distance O f y df C x B A

8. z q f y Df C x B A Flat Space Interval for flat space in the plane q = 90o ds2 = c2dt2 – ds2 Spatial part AC = r df CB = dr (radial) AB = ds ds2 = dr2 + r2df2 O r

9. z q O f y C x B Curved Space Consider a curved space in the plane q = 90o. What is the radial distance between C and B? For a flat space it is drbut, in general, it is not so for a curved space r dr

10. Curved Space O The circle represents the radial distance r in spherically curved space: C dr dz B R dr2 = dr2 + dz2 But, since R2 = r2 + z2 r dr we can write the radial distance as

11. The Schwarzschild Geometry TheSchwarzschild metric (in plane q = 90o) Event horizon rS Schwarzschild radius Karl Schwarzschild 1873 - 1916

12. The Schwarzschild Geometry Properties Far from r = 0, the Schwarzschild metric approaches that of a flat geometry We can therefore interpret t as the time measured by someone far from the origin

13. The Schwarzschild Geometry Circular Orbits By definition, the radius of a circular orbit does not change, that is, dr = 0. Therefore, becomes

14. The Schwarzschild Geometry Circular Orbits Divide by c2dt2, and write v = r df/dt, the orbital speed measured by a far away observer. The elapsed time dt of an observer at radius r is given by Notice: when the r is very large, we get the time dilation formula of special relativity

15. Relativity in Action

16. 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

17. GPS – Orbits Period: 12 hours Orbital radius: 26,600 km 6 orbital planes: 60o apart

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

19. GPS Clocks vS vE rGPS • dt = elapsed time • measured • far from Earth dt = elapsed time at radius r rE

20. GPS Clocks – Extra Credit vS vE rGPS If left uncorrected, by how many microseconds would the satellite clocks be slow or fast each day relative to the Earth clocks? due Feb 22 rE

21. Morris-Thorne Wormhole TheMorris-Thorne metric(q = 90o) a =Throat radius In this spacetime, time is unwarped, while space is. The radius parameter is defined so that r = C/2p where C is the circumference of a circle around the wormhole

22. Summary • The Principle of Equivalence states that an accelerating frame produces the same effect as a gravitational field • From this, Einstein concluded that matter warps spacetime and, consequently, gravity is simply a manifestation of this warping • The Schwarzschild metric describes the spacetime geometry around a spherically symmetric object