Black Holes

Underlying principles of General Underlying principles of General RelativityRelativity

The Equivalence Principle

No difference between a steady acceleration and a gravitational field

Gravity and Acceleration cannot be distinguished

h

V = a h/c

h Gravitational field

Equivalence principle – this situation should be the same

Principe May 1919

Eddington tests General Relativity and spacetime curvature

GR predicts light-bending of order 1 arcsecond near the limb of the Sun

Lensing of distant galaxies by a foreground cluster

QSO 2237+0305

The Einstein cross

Curved Space: A 2-dimensional analogy

Flat space

Angles of a triangle add up to 180 degrees

Radius r

Circumference of a circle is 2πr

Positive and Negative Curvature

Triangle angles >180 degrees

Circle circumference < 2πr

Triangle angles <180 degrees

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.

** This is actually the path that takes the maximum “proper” time.

Space

Time

Spacelike

Timelike

Curved geodesic caused by acceleration OR gravity

Matter tells space(time) how to curve

Spacetime curvature tells matter how to move

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

h

V = (2ah)1/2

Special Relativity

A moving clock runs slow

Observer ON TRAIN Observer BY TRACKSIDE

Speed of light is c=300,000 km/s

t’ = 2d / c

Width of carriage

Is d meters

Train speed v

d

vt/2

s

t = 2s / cSo t’ is smaller than t

Observers don’t agree! Smaller by a factor

Where v2/c2)

h

V = (2ah)1/2

Special Relativity

A moving ruler is shorter

h

Gravitational field

According to the equivalence principle, this is the same as

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

Remnant < 1.4 M

Progenitor < 8 M

White Dwarf

Planetary Nebula

A cooling C/O core, supported by quantum mechanics! Electron degeneracy pressure.

The Chandrasekhar limit

Cools forever – gravity loses!

Progenitor > 8 M

Black Hole – gravity wins!

Supernova

Neutron star, supported by quantum mechanics! Neutron degeneracy pressure.

Cools forever – gravity loses!

Remnant < 2.5 M Remnant > 2.5 M

20 km

Black Holes in binary systems

M3 sin3i = 0.25 (M + m)2 Period = 6 days

Cygnus X-1

M > 5 Msun

Ellipsoidal light curve variations

Depend on mass ratio and orbit inclination

BH mass

Black hole mass 10 –15 x Msun

Combine ellipsoidal model with radial velocity curve

Spinning black holes – the Kerr metric

Spaghettification

A 10g stretching force felt at 3700 km (>RS) from a 10 Msun black hole

Force increases as 1/r3

Supermassive Black Holes

Jets propelled by twisted magnetic field lines attached to gas spiralling around a central black hole

Supermassive Black Hole in the Galactic Centre

Mass is 4 millions times that of the Sun

Schwarzschild radius 12 million km = 0.08 au

Falling into a black hole