8 classes left • Rotating black holes • Gravity Waves • Hawking Radiation • Inside a black hole • Wormholes and time travel Rotating Black Holes Rotating Black Holes • Black holes have only 3 properties: – Mass – Charge – Angular momentum Angular Momentum • Any mass rotating around a fixed point has angular momentum – For example, a ball on the end of a piece of string – A planet orbiting a star Angular Momentum • Any mass rotating around a fixed point has angular momentum – For example, a ball on the end of a piece of string – A planet orbiting a star • Angular momentum J = r × mv Angular Momentum • Angular momentum J = r × mv • Angular momentum is conserved – Pull mass in, rotation rate must increase – Mass moved out, rotation rate goes down Angular Momentum • Angular momentum J = r × mv • Rotating objects also have angular momentum • J = Iω • I = moment of intertia • ω = angular frequency Angular Momentum • Angular momentum J = r × mv • J = Iω • For point mass, I = mr2 , ω = v/r • For solid sphere, I = 2/5 mr2 Rotational Energy • Angular momentum J = r × mv • J = Iω • Momentum = mv , Energy = ½ mv2 • Angular momentum = Iω , rotational energy = ½ Iω2 = ½ Jω Rotational Energy • Angular momentum = Iω =J , rotational energy = ½ Iω2 = ½ Jω • If we shrink something: – J stays constant – I decreases – ω increases • So, rotational energy increases Example: the Sun Rotating Black Holes • As a star collapses to a black hole, or stuff falls into a black hole, angular momentum must be conserved • But, we can’t tell anything about what’s inside a black hole from the outside, so blakc hole must have angular momentum Rotating Black Holes • Schwarzschild black hole – Regular, non-rotating black hole • Event horizon at Schwarzschild radius 2 • rs = 2GM/c Rotating Black Holes 2 • rs = 2GM/c • In a rotating black hole, things are more complicated • Solution to general relativity for a rotating black hole found by Roy Kerr in 1963 – Kerr Metric Frame Dragging • In general relativity, a rotating mass drags space along with it • Light moving in direction of rotation goes around quicker than light moving against rotation • Orbits around a rotating mass will precess – Lense-Thirring effect Frame Dragging • Very small effect around typical object, like Earth • Attempts have been made to test it, but not very convincingly • Gravity Probe B measure this effect to within 17% of expected value Frame Dragging Rotating Black Holes • 2 surfaces: – Inner surface – Event Horizon – Outer surface – Static Limit Rotating Black Holes • 2 surfaces: – Inner surface – Event horizon – Outer surface – Static Limit • Ergosphere – Between the two surfaces – Space is rotating faster than light, everything must rotate with black hole .