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• Rotating holes • Waves • Hawking • Inside a and travel Rotating Black Holes Rotating Black Holes

• Black holes have only 3 properties: – – Angular

• Any mass rotating around a fixed point has angular momentum – For example, a ball on the end of a piece of string – A orbiting a 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 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

• 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 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-

horizon at

2 • rs = 2GM/c Rotating Black Holes

2 • rs = 2GM/c

• In a rotating black hole, things are more complicated

• Solution to for a rotating black hole found by in 1963 – Frame Dragging

• In general relativity, a rotating mass drags space along with it • 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

• Attempts have been made to test it, but not very convincingly • measure this effect to within 17% of expected value Frame Dragging Rotating Black Holes

• 2 surfaces: – Inner surface – – Outer surface – Static Limit Rotating Black Holes

• 2 surfaces: – Inner surface – Event horizon – Outer surface – Static Limit

– Between the two surfaces – Space is rotating faster than light, everything must rotate with black hole