Announcements Exam 4 is Monday May 4. Tentatively will cover Chapters 9, 10, 11 & 12 Sample questions will be posted soon Observing Night tomorrow night. Set-up starts at 7:30pm at The Farm. Cancellation notice, if needed, will be posted on by 5:00pm.

Black Holes

The Schwarzschild solution to the equations of general relativity Spherical mass Spherical symmetry gives metric radial (r) dependence only…no dependence on angles around sphere Stationary with no rotation…static: metric should not depend on time Only valid in vacuum outside the mass Karl Schwarzschild: 1873 – Worked out the solution in 1915 while serving as an artillery officer with the German army on the Russian front during WW I

The equation Schwarzschild solved Looks easy enough until you start working the D.E.’s

Schwarzschild’s Metric The shortest space-time interval between two events outside a spherical mass distribution

The Schwarzschild Radius Using the metric becomes

What does it mean? Look at how R S /r affects the terms for various values of r greater than the radius of the object (i.e. R S /r<1) Recall that  s is the proper space-time interval between two events in space-time R S for the S un is 2.95 km, for Earth it is 8.86 mm

“Far” from the mass r>>R S becomes This is the metric for ordinary spherical geometry “Flat” space-time R S /r is very small so

Solution is not “flat” geometry if r~R S  t term gets small while  r term gets large…space and time start to get warped by the mass inside the sphere

Gravitational Time Dilation For clock near object, R S /r is close to 1 so  s is large. The outside observer (far from the black hole) sees the clock for someone near the event horizon (r ≈ R S ) to slow down. At the event horizon it stops completely.

Gravitational Length Contraction R object ≈ R S …”Compact” object Measure length by determining positions of ends simultaneously so  s is zero For r close to R S the  t term is small and the  r term is large. Since 1/(1-(R S /r)) is small,  r must be very large.

What happens when R = R S (at the Schwarzschild radius)? To an observer far from the black hole time for a person falling toward the black hole stops (  t goes to zero) and the distance from the outside observer goes to infinity. In other words, the outside observer never sees the person falling toward the black hole reach the event horizon.

What happens inside the Schwarzschild Radius? Space becomes time-like (positive coefficient) and time becomes space-like (negative coefficient) So, you can move forward or backwards in time? The only direction in space (other that around) is toward the center. You can’t move away from the center, only towards it or around it. r < R S

Inside the Schwarzschild radius, the future is towards the center Near event horizon At event horizon Inside event horizon It is no longer possible to remain stationary. The very fabric of space-time falls towards the event horizon.

At the center is the singularity If you plug r = 0 into the Schwarzschild metric it complete breaks down. This tells us something is very wrong.

The distance at which the orbital speed is the speed of light is the photon sphere The radius of the photon sphere is 3 / 2 the Schwarzschild radius If an object has a mass, its minimum stable circular orbit is 3R S

Black holes have no hair The event horizon of a black hole is perfectly spherical. Only three quantities completely describe any black hole Mass Angular Momentum Electric Charge

The Schwarzschild black hole is a non-rotating black hole The Schwarzschild radius is The Schwarzschild metric is

A rotating black hole is a Kerr black hole Solution developed by Roy Kerr (NZ) while at UT (the REAL UT) in 1963

Rotating black holes: the Kerr Metric

The Kerr black hole has two photon spheres depending on which way the photon is rotating The photon “spheres” are actually ellipsoids of revolution Photons whose orbital axis is the same as the black holes lie on the “spheres”. Those with different orbital axis lie between the “spheres”.

Inside the ergosphere the black hole has two event horizons

The Kerr Black Hole event horizons

Each time an event horizon is crossed the roles of space and time reverse The ring singularity is repulsive!!!

Rotating black holes drag spacetime around with them This rotating space-time is called frame dragging

Hawking Radiation carries away mass and angular momentum

Extracting energy from a rotating black hole

Black Hole Thermodynamics Hawking Radiation has a blackbody characteristic curve which implies that a black hole has a temperature. The temperature is given by h is Planck‘s constant c is the speed of light k is Boltzman’s constant G is the gravitational constant M is the mass of the black hole A one solar mass black hole has a temperature of about 100 nanokelvin

Hawking radiation is a very slow process The time required for a black hole to evaporate is given by