1 / 28

Relativity

Relativity . part 3. Topics. The Metric The Interval Gravity The Global Positioning System. P. A. O. B. The Metric. dl. The distance between points O and P is given by: OB 2 + BP 2 = OP 2 = OA 2 + AP 2 OP 2 is said to be invariant . The formula dl 2 = dx 2 + dy 2

amy
Download Presentation

Relativity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Relativity part 3

  2. Topics • The Metric • The Interval • Gravity • The Global Positioning System

  3. P A O B The Metric dl The distance between points O and P is given by: OB2 + BP2 = OP2 = OA2 + AP2 OP2 is said to be invariant. The formula dl2 = dx2 + dy2 for computing dl2 is called the metric dy dx In 3-d, this becomes

  4. The Metric z The metric in spherical polar coordinates (r, θ, φ) Consider the spatial planeθ = 90o AC = r dφ CB = dr AB = dl r θ φ y Δφ C x B A

  5. Q O P The Interval ds dt Suppose that O and Q are events. How far apart are they? First guess ds2 = (cdt)2 + dl2 dl Unfortunately, this does not work! In 1908, Hermann Minkowski showed that the correct expression is ds2 = (cdt)2- dl2 ds2is called the interval Hermann Minkowski 1864 -1909

  6. The Interval In general, the interval ds2 between any two events is either timelike ds2 = (cdt)2 – dl2 or spacelike ds2 = dl2 – (cdt)2 or null ds2 = (cdt)2 - dl2 = 0 depending on which difference, temporal cdt, or spatial dl is larger

  7. Proper Distance By Definition: The proper distance is the spatial separation between two simultaneous events D X’ C X A B Example from About Time by Paul Davies

  8. Problem 5 Compute the spacetime distance (ds) between the following events: 1. event 1: solar flare on Sun now (Earth’s now!) event 2: a rainstorm here, 7 minutes later 2. event 1: the fall of Alexandria in 640 AD event 2: Tycho’s supernova seen in 1572 AD(the star was 7,500 ly away from Earth)

  9. Gravity

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

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

  12. free space outside view inside view The Principle of Equivalence (1907) Gravity is curved spacetime!

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

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

  15. Black Holes & Wormholes

  16. Black Holes Consider hovering near a black hole at a fixed radius r. How would your elapsed time Δτ be related to that of someone far away? Far away

  17. Visual Distortions near a Black Hole

  18. Wormholes The Morris-Thorne metric (θ = 90o) a = Throat radius

  19. GPS

  20. 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

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

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

  23. GPS Clocks vsatellite vEarth rsatellite t = time far away from Earth τ = time at 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

  25. Appendix

  26. t t' P x' C Q x O A B The Lorentz Transformation Define Δt = tBC + tCP Δx = xOA + xAB What is the interval between event O and event P? Δx’ = xOQ Δt' = tQP tCP = tQP /γ tBC = v Δx / c2 xOA = xOQ / γ xAB = v Δt

  27. t t' P tCP = tQP / γ x' C Q tBC = v Δx/ c2 x O A B xOA = xOQ / γ xAB = v Δt The Lorentz Transformation We obtain the Lorentz transformation (Δx, Δt) → (Δx', Δt') Δx' = γ (Δx – v Δt) Δt' = γ (Δt – v Δx/c2) What is the interval from O to P? OP2 = (cBP)2 + OB2 No! The correct formula is OP2 = (cBP)2– OB2 = (cQP)2– OQ2

  28. Circular Orbits For circular orbits, r does not change, that is, dr = 0, therefore Dividing by c2dt2, and noting that v = r dφ/dt, the tangential speed measured by an observer far away, we find the elapsed (proper) time dt at radius r to be

More Related