Black Holes
Underlying principles of General
Relativity
The Equivalence
Principle
No difference between a
steady acceleration and
a gravitational field
Gravity and Acceleration cannot be distinguished
V = a h/c
h
Equivalence principle – this situation should be the same
h
Gravitational
field
Eddington tests General
Relativity and spacetime
curvature
GR predicts lightbending of order 1
arcsecond near the
limb of the Sun
Principe May 1919
Lensing of distant galaxies by a foreground cluster
QSO 2237+0305
The Einstein cross
Curved Space: A 2-dimensional analogy
Flat space
Radius r
Angles of a triangle add up
to 180 degrees
Circumference of a circle is 2πr
Positive and Negative Curvature
Triangle angles >180 degrees
Triangle angles <180 degrees
Circle circumference < 2πr
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.
Time
Timelike
Curved geodesic caused by
acceleration OR gravity
Spacelike
Matter tells space(time)
how to curve
Spacetime curvature tells
matter how to move
Space
** This is actually the path that takes the maximum “proper” time.
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
V = (2ah)1/2
h
Special Relativity
A moving clock runs slow
Observer ON TRAIN
Observer BY TRACKSIDE
Train
speed v
Width of
carriage
s
d
Is d meters
vt/2
t’ = 2d / c
t = 2s / c
So t’ is smaller than t
Observers don’t agree!
Speed of light is c=300,000 km/s
Smaller by a factor g
Where g2 = 1/(1 - v2/c2)
V = (2ah)1/2
h
Special Relativity
A moving ruler is shorter
According to the equivalence principle, this is the same as
h
Gravitational
field
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
Progenitor < 8 M
Planetary Nebula
Remnant < 1.4 M
The Chandrasekhar
limit
A cooling C/O core,
supported by quantum
mechanics! Electron
degeneracy pressure.
Cools forever – gravity loses!
White Dwarf
Progenitor > 8 M
Remnant < 2.5 M
Supernova
Remnant > 2.5 M
20 km
Neutron star, supported by
quantum mechanics! Neutron
degeneracy pressure.
Cools forever – gravity loses!
Black Hole – gravity wins!
Black Holes in binary systems
Cygnus X-1
M3 sin3i = 0.25 (M + m)2 Period = 6 days
M > 5 Msun
Ellipsoidal light curve variations
Depend on mass ratio and orbit inclination
Combine ellipsoidal model with radial velocity curve
BH
mass
Black hole mass 10 –15 x Msun
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