Relativity part 3
Topics • Special Relativity: Recap • The Interval • Gravity • The Global Positioning System
Special Relativity Principles • The Principle of Relativity: The laws of physics are the same in all inertial frames of reference • The speed of light, in vacuum, is independent of the motion of its source It follows from the above that the speed of light must be independent of the motion of the observer, also.
Time Dilation and Length Contraction Moving clocks tick more slowly Moving lengths contract T is time in moving clock frame t is time in stationary clock frame D is length in stationary object frame d is length in moving object frame
P A O B The Interval The value of the interval between O and P is independent of the components used to calculate it: OB2 + BP2 = OP2 = OA2 + AP2 OP2 is said to be invariant. The formula for computing it is called the metric. Spacetime interval Herman Minkowski (1908) Hermann Minkowski 1864 -1909
t t' P x' C Q x O A B The Interval – II Define Dt = tBC + tCP Dx = xOA + xAB What is the interval between event O and event P? Dx' = xOQ Dt' = tQP tCP = tQP / g tBC = v Dx / c2 xOA = xOQ / g xAB = v Dt
t t' P tCP = tQP / g x' C Q tBC = v Dx/ c2 x O A B xOA = xOQ / g xAB = v Dt The Interval – III We obtain the Lorentz transformation (Dx, Dt) → (Dx', Dt') Dx' = g (Dx – v Dt) Dt' = g (Dt – v Dx/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 Problem: Prove this
The Interval – IV In general, the interval Ds2 = OP2 between two events is either timelike Ds2 = c2Dt2 – Ds2 or spacelike Ds2 = Ds2 – c2Dt2 or null Ds2 = Ds2 – c2Dt2 = 0 depending on whether the temporal or spatial difference dominates.
The Interval – V z Interval in spherical polar coordinates (r, q, f) Consider the spatial planeq = 90o Ds2 = c2Dt2 – Ds2 r q AC = r Df CB = Dr AB = Ds Ds2 = Dr2 + r2Df2 f y Df C x B A
Gravity All objects fall with the same acceleration G. Galileo 1564–1642
A Happy Thought! A person falling off a building experiences no gravity! “The happiest thought of my life” Albert Einstein (1907) http://nssdc.gsfc.nasa.gov
General Relativity (1915) Bending of light G = 8pT Gravity is warped spacetime Sir Arthur Eddington Eclipse Expeditions 1919
The Spacetime Around A Star Event horizon The Schwarzschild metric (q = 90o) rS Schwarzschild radius Karl Schwarzschild 1873 - 1916
The Schwarzschild Geometry Properties Far from r = 0, the Schwarzschild metric approaches that of special relativity t is the time measured by someone far from r = 0.
The Schwarzschild Geometry – II Circular Orbits Since the radius r does not change, that is, dr = 0 the interval has the form Dividing by c2dt2, and noting that v = r df/dt, the tangential speed measured by a far away observer, we find the elapsed (proper) time dt at radius r to be
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
GPS – Orbits Period: 12 hours Orbital radius: 26,600 km Six orbital planes 60o apart
ct3 ct1 ct2 GPS – Principle 1 2 3 You are here!
GPS Clocks vSatellite vEarth rSatelitte • t = time far away from Earth • = time at radius r rS = Schwarzschild radius rEarth Problem: How fast or slow does the satellite clock run per day relative to the Earth clocks? Give answer in nanoseconds
Morris-Thorne Wormhole The Morris-Thorne metric (q = 90o) a = Throat radius
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
