Black Holes

Lecture #4: Black Holes: 

Lecture #4: Black Holes

Black Hole Definitions: 

Black Hole Definitions Classical Definition (late-1700s): A region from which light cannot escape. Einstein’s Definition (1915-1916): A region that cannot communicate with the external universe (similar to Classical Definition). The boundary of this region is called the Event Horizon, and its existence is predicted in the theory of General Relativity. Observer’s Definition: Any compact object with a mass greater than 3 Msun.

Origins of the Classical Definition: 

Origins of the Classical Definition

Origins of the Classical Definition - 2: 

Origins of the Classical Definition - 2

Escape Velocity: 

Escape Velocity 1 2 R = radius v M = mass The “Escape Velocity” is the velocity that makes E2 = 0. Energy Conservation (E1 = E2): E1 = 0 = ½ mv2 – GMm/R vescape = YDTA From Earth: vescape = 25,000 mi/hr (= 0.004% c) (How can the space shuttle’s speed be 18,000 mi/hr during ascent?) Infinite distance Zero speed

Classical Black Hole: 

Classical Black Hole vescape = (2GM/R)1/2 from previous slide. Definition: A region from which light cannot escape. From these two statements, we can calculate the BH radius: vescape = c R = 2GM/c2 (Same result as General Relativity) Putting in the numbers for G and c gives: R = 3 (M/Msun) km How does this compare to a Neutron Star? R = 3(1.4) km = 4.2 km For the same mass, a BH has a radius ~2.5 times less than a NS. So, its density is higher. However …

Black Hole Scaling: 

Black Hole Scaling

General Relativity and Einstein Definition: 

General Relativity and Einstein Definition

Flat Spacetime: 

Flat Spacetime The Universe we perceive has 4 dimensions: 3 spatial: x, y, z 1 temporal: t This is “spacetime”. The structure of spacetime is defined by the “metric” In flat space, the metric is: ds2 = dt2 - dx2 - dy2 - dz2 ds is the elapsed time as measured by a clock moving through spacetime.

Curved Spacetime: 

Curved Spacetime In General Relativity, gravity is caused by the curvature of spacetime. Orbiting objects move in straight lines, but the space they move through is curved.

Schwarzschild Metric: 

Schwarzschild Metric ds2 = -(1-2GM/c2r) dt2 + (1-2GM/c2r)-1 dr2+ r2 d2 Right after Einstein invented GR, Karl Schwarzschild solved the GR equations and found the metric for spacetime around a non-rotating Black Hole. It is: There are two places where this metric “breaks down” by giving infinite quantities: R = 2GM/c2 (The Event Horizon) and R = 0 (The Singularity)

Observer’s Definition: 

Observer’s Definition (Are there really compact objects that are more massive than 3 Msun?) Radial velocity curve for Cygnus X-1: First observational evidence for a Black Hole (1980s). Using Kepler’s laws of motion to constrain the compact object mass.

Definite Proof of Compact Objects with M>3Msun: 

Definite Proof of Compact Objects with M>3Msun Using Kepler’s laws: f = Porb K3/2G = “mass function” f gives a lower limit to the Black Hole mass. For the X-ray binary GRO J1655-40, f = 3.24 Msun, so M > 3.24 Msun, and this was the first compact object clearly shown to be a BH.

Compact Object Mass Measurements: 

Compact Object Mass Measurements

The Super-Massive Black Hole at the center of our Galaxy: 

The Super-Massive Black Hole at the center of our Galaxy The star that comes the closest to the Black Hole comes within 18 billion km (comparable to the size of the solar system). At this point, the velocity is 11 million miles per hour. Kepler’s laws give a BH mass of 3 million Msun.

The Birth of Black Holes: 

The Birth of Black Holes Astronomer’s see flashes of high energy emission called “Gamma-ray bursts” that may signal the birth of the ~10 Msun Black Holes.

Summary: 

Summary The Classical Definition of a Black Hole, based on the escape velocity of light was developed in the late-1700s and one can use it to obtain the correct radius of the Event Horizon. With the Einstein definition, we obtain the prediction for the Black Hole singularity and a calculation of the structure of spacetime outside the black hole. Observationally, there are compact objects more massive than 3 Msun, and this is currently the best proof that Black Holes exist (note also supermassive BHs). For some Black Holes, Gamma-Ray Bursts may signal their birth.