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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.

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Underlying principles of general relativity
Underlying principles of General Relativity

The Equivalence Principle

No difference between a steady acceleration and a gravitational field




Equivalence principle – this situation should be the same

Gravitational field

h


Eddington tests General Relativity and spacetime curvature

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

Principe May 1919



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

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.

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”



V = (2ah) Sun1/2

h

Special Relativity

A moving clock runs slow


Observer ON TRAIN Sun

Observer BY TRACKSIDE

Train speed v

Width of carriage

Is d meters

s

d

vt/2

t’ = 2d / c

t = 2s / c

So t’ is smaller than t

Observers don’t agree!

Smaller by a factor g

Where g2 = 1/(1 - v2/c2)

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


V = (2ah) Sun1/2

h

Special Relativity

A moving ruler is shorter


According to the Sunequivalence principle, this is the same as

Gravitational field

h




A black hole forms when a mass is squashed inside it’s SunSchwarzschild Radius RS = 3 (M/Msun) km

Time dilation factor

1/(1 – RS/r)1/2

Becomes infinite when r=RS


Progenitor < 8 M Sun

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 Sun

Supernova

Remnant < 2.5 M

Remnant > 2.5 M

20 km

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

Cools forever – gravity loses!

Black Hole – gravity wins!



Cygnus X-1 Sun

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

M > 5 Msun


Ellipsoidal light curve variations Sun

Depend on mass ratio and orbit inclination


Combine ellipsoidal model with radial velocity curve Sun

BH mass

Black hole mass 10 –15 x Msun



Spaghettification Sun

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

Force increases as 1/r3


Supermassive Black Holes Sun

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


Supermassive Black Hole in the Galactic Centre Sun

Mass is 4 millions times that of the Sun

Schwarzschild radius 12 million km = 0.08 au



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