ASTR2050 Spring In this class we will cover: Hints: Escape Velocity. Relativity and the Equivalence Principle Visualization of Curved Spacetime

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1 ASTR2050 Spring 2005 Lecture 11am 8 March 2005 In this class we will cover: Hints: Escape Velocity Black Holes Relativity and the Equivalence Principle Visualization of Curved Spacetime 1

2 Escape Velocity Conservation of Energy R m v M Set total energy to zero, i.e. object fired upwards from surface escapes with zero velocity at infinity. 1 2 mv2 GMm R = v2 = GM R 2GM v = R 2

3 Examples The Earth: v= m 3 /s 2 kg kg = m/s m A Neutron Star: Homework! v m/s = (2/3)c This is crude (for example, we neglect relativity) but we should expect something odd to happen when R = 2GM c 2 3 Schwarzschild radius

4 Relativity and the Equivalence Principle Some useful references Kutner Chapters 7 and 8 Spacetime Physics, E.F. Taylor and J.A. Wheeler, W.H. Freeman and Company (1992) Exploring Black Holes: Introduction to General Relativity, E.F. Taylor and J.A. Wheeler, Addison Wesley (2000) Visualizing Curved Spacetime, R.M. Jonsson, Amer.Jour.Phys. 73(2005)248 (See our website) 4

5 Special Relativity Moving Clocks Tick More Slowly Proper time dτ is the time between two events according to a moving clock. d! 2 = dt 2 ( dx c ) 2 5

6 Spacetime Diagrams and Worldines Worldline of a particle moving in one dimension Photons move along dx/dt=c. All physical bodies must move inside the yellow area, also known as the light cone. 6

7 The Equivalence Principle Two things that you were taught in introductory physics: F = ma F = G m 1m 2 and r 2 Isn t that amazing!! Mass has two different uses!! In other words, it is not possible to tell an acceleration caused by gravity from any other acceleration. This is the Equivalence Principle. It is the foundation of Einstein s General Theory of relativity, the theory of graviy. Implications: Matter (and energy) bend spacetime, and give rise to accelerations, which we interpret as forces. 7

8 Example: Gravitation from a Static and Spherically Symmetric Mass Distribution The Schwarzschild Solution of the GR Field Equations Takes the form of a metric which defines the curvature. One implication: Gravitational Red Shift r R! 1! 2 M! 1! 2 = Kutner Eq.8.4 [ 1 2GM Rc 2 1 2GM rc 2 8 ] 1/2 Note:! 2 " for R = 2GM c 2

9 Visualization of Curved Spacetime Let motion be in just one space dimension. World line Make plots on warped 2D paper: Space on the X axis, and time on the Y axis. Sprinkle it with tiny spacetime diagrams: The details of how the paper is warped are left to the calculations based on solutions to Einstein s field equations. 9

10 Example: A Radial Line through a Very Dense Star Stationary observers have clock ticks around the circle so a stationary clock runs slower in the star s center! You can also follow a worldline for something moving along the radial line through the center of the star.

11 Example: Mass all inside R Schwarzschild i.e. A Black Hole Clocks run infinitely slowly at the event horizon where the paper pinches off. Inside the event horizon time and space flip and the center is always in the future! 11

12 Stellar Black Holes How can you see a black hole? Stuff falls into it!! After break: End state evolution in close binary systems. Accretion! Example: Cygnus X-1 Optical X-Ray 12