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

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

Recap – When is now?

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

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

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

Distances in spacetime

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

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

Distances in Spacetime O Q ds dl cdt Suppose that O and P are events. How far apart are they in spacetime? First guess ds2 = (cdt)2 + dl2 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

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)2 dl > cdt null ds2 = (cdt)2 – dl2 = 0 dl = cdt

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

Gravity

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

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

The Principle of Equivalence free space view inside view outside view Gravity is curved spacetime!

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

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

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

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

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

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

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 Mathematica Online Integrator http://integrals.wolfram.com r

Black Holes and wormholes

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

Visual Distortions near a Black Hole

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”?

The global positioning system

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

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

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

vsatellite vEarth rsatellite GPS Clocks rEarth 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

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