Black Holes Underlying principles of General Relativity The

Black Holes

Underlying principles of General Relativity The Equivalence Principle No difference between a steady acceleration and a gravitational field

Gravity and Acceleration cannot be distinguished

V = a h/c h

Equivalence principle – this situation should be the same h Gravitational field

Eddington tests General Relativity and spacetime curvature GR predicts lightbending of order 1 arcsecond near the limb of the Sun Principe May 1919

Lensing of distant galaxies by a foreground cluster

QSO 2237+0305 The Einstein cross

Curved Space: A 2 -dimensional analogy Flat space Radius r Angles of a triangle add up to 180 degrees Circumference of a circle is 2πr

Positive and Negative Curvature Triangles >180 degrees Triangles <180 degrees Circle circumference < 2πr Circle circumference > 2πr

The effects of curvature only become noticeable on scales comparable to the radius of curvature. Locally, space is flat.

A geodesic – the “shortest possible path”** a body can take between two points in spacetime (with no external forces). Particles with mass follow timelike geodesics. Light follows “null” geodesics. Timelike Curved geodesic caused by acceleration OR gravity Spacelike Matter tells space(time) how to curve Spacetime curvature tells matter how to move Space ** This is actually the path that takes the maximum “proper” time.

Mass (and energy, pressure, momentum) tell spacetime how to curve; Curved spacetime tells matter how to move A formidable problem to solve, except in symmetric cases – “chicken and egg”

Curvature of space in spherical symmetry – e. g. around the Sun

V = (2 ah)1/2 h Special Relativity A moving clock runs slow

Observer ON TRAIN Observer BY TRACKSIDE Train speed v Width of carriage s d Is d meters vt/2 t’ = 2 d / c t = 2 s / c So t’ is smaller than t Observers don’t agree! Speed of light is c=300, 000 km/s Smaller by a factor g Where g 2 = 1/(1 - v 2/c 2)

V = (2 ah)1/2 h Special Relativity A moving ruler is shorter

According to the equivalence principle, this is the same as h Gravitational field

Curvature of space in spherical symmetry – e. g. around the Sun

Spacetime curvature near a black hole

A black hole forms when a mass is squashed inside it’s Schwarzschild Radius RS = 3 (M/Msun) km Time dilation factor 1/(1 – RS/r)1/2 Becomes infinite when r=RS

Progenitor < 8 M Planetary Nebula Remnant < 1. 4 M The Chandrasekhar limit A cooling C/O core, supported by quantum mechanics! Electron degeneracy pressure. Cools forever – gravity loses! White Dwarf

Progenitor > 8 M Remnant < 2. 5 M Supernova Remnant > 2. 5 M 20 km Neutron star, supported by quantum mechanics! Neutron degeneracy pressure. Cools forever – gravity loses! Black Hole – gravity wins!