Relativity
Structure of the Universe: from beginning to end, requires GR
to understand spacetime. Here we shall apply GR to black
holes, the early universe and the present Universe.
1. SR,
GR,
Spacetime,
gravitational redshift
Curvature,
Equivalence,
2. Time dilation, intervals, geodesics, proper time, the
metric for flat spacetime
3. Schwarzschild metric, black holes, wormholes, Hawking
radiation
4. Observations, examples, the confirmation. Mercury’s
perihelion.
5. Cosmological principles, the FRW metric, cosmological
constant, dark matter, dark energy.
6. Critical density, age of the Universe, WMAP
7. Early evolution, last scattering, recombination, distance
measures
Carroll and Ostlie, An Introduction to Modern Astrophysics,
2nd edition, Chapters 17 and 29
Galilean transformations: x’ = x –ut
v’ = v – u
1. Special Relativity
A whirlwind tour of special relativity (SR) and life in flat spacetime.
In the absence (or distant from) massive objects.
Inertial frames: uniform motion or rest sustained in the absence of
a force.
There is no absolute inertial frame.
Einstein’s two postulates:
The principle of relativity: The laws of physics are the same in all
inertial reference frames.
The constancy of the speed of light: Light moves through a
vacuum at a constant speed that is independent of the motion of
the light source.
An event: a single moment in space and time, characterized
uniquely by (t, x, y, z). The spacetime interval between two
events:
Some references, especially field theory books, define the metric
with the opposite sign, so be careful. Or:
Lorentz transformations:
is the Lorentz factor.
Einstein said that all of the consequences of special relativity can
be found from examination of the Lorentz transformations.
These transformations, and hence special relativity, lead to
different physical predictions than Newtonian mechanics when
relative velocities become comparable to the speed of light. The
speed of light is so much larger than anything humans encounter
that some of the effects predicted by relativity are initially counterintuitive.
Essential concept: Spacetime.
– Since space affects time, cannot think of 3-D space
and time separately, but only as a single 4-dimensional
manifold.
– We all live in spacetime.
– If we could not fly, the surface of Earth is a 3-D
spacetime; it is curved.
Special relativity overthrows Newtonian notions of absolute
space and time by stating that distance and time depend on the
observer, and that time and space are perceived differently,
depending on the observer.
It yields the equivalence of matter and energy, as expressed in
the famous equation E=mc2, where c is the speed of light.
Special relativity agrees with Newtonian mechanics in their
common realm of applicability, in experiments in which all
velocities are small compared to the speed of light.
Proper Time & Time Dilation
Clocks that are moving with respect to an inertial system of
observation (the putatively stationary observer) are found to be
running slower. This effect is described precisely by the Lorentz
transformations.
The shortest time interval between two events is found by a
clock at rest relative to the two events. This clock measures the
proper time.
The time dilation effect is reciprocal: as observed from the point
of view of any two clocks which are in motion with respect to each
other, it will be the other party's clock that is time dilated.
(This presumes that the relative motion of both parties is uniform;
that is, they do not accelerate with respect to one another during
the course of the observations.)
The time lapse between two events is NOT invariant from one
observer to another, but is dependent on the relative speeds of the
observers' reference frames (e.g., the twin paradox which
concerns a twin who flies off in a spaceship travelling near the
speed of light and returns to discover that his twin has aged much
more).
Relativity of simultaneity — two events happening in two
different locations that occur simultaneously to one observer, may
occur at different times to another observer (lack of absolute
simultaneity).
Proper Length & Length Contraction
A decrease in length is detected by an observer in objects that
travel at any non-zero velocity relative to that observer.
The proper length is measured in the object’s rest frame.
Note: the ladder paradox involves a long ladder travelling near the
speed of light and being contained within a smaller garage).
Relativistic Doppler shift formula
For motion along the line of sight:
If the observer and the source are moving directly away from each
other with velocity v, the observed frequency is different from the
frequency of the source (observer and emitter):
Defining the redshift:
Inverting this formula:
which is derived from special relativity. You can see by the
presence of the v/c in the denominator of the fraction that as v gets
close to c the redshift becomes increasingly large. (v = c would
yield an infinite redshift). This means the Hubble law at high
redshift becomes
For motion in an arbitrary direction
If, in the reference frame of the observer, the source is moving
away with velocity at an angle relative to the direction from the
observer to the source (at the time when the light is emitted), the
frequency changes as
i.e. there is a transverse Doppler shift (theta = 90o) purely due
to time dilation
2. Consequence of Special Relativity
Superluminal Motion & Beaming
``Superluminal motion'' describes proper motion of source
structure (traditionally mapped at radio wavelengths) that, when
converted to an apparent speed vobs, gives vobs > c.
This phenomenon occurs for emitting regions moving at very high
(but still sub-luminal) speeds at small angles to the line of sight.
Relativistically moving sources ``run after'' the photons they emit,
strongly reducing the time interval separating any two events in the
observer's frame and giving the impression of faster than light
motion.
With the advent of very long baseline interferometry (VLBI), we
obtained very high resolution radio images of jets.




To measure 1 parsec at 100 Mpc requires a resolution of 10-8
radians. This is about 2 milli-arcseconds.
These images show that jets form coherent structures over 5
orders of magnitude in scale (~1 pc -> ~ 100 kpc)
Somewhat disturbingly, knots seen in the jets observed by
VLBI are sometimes moving along the jets at superluminal
velocities (v > c)!
This effect can be explained geometrically if the jet is
directed nearly along the line of sight:
Taking observations in 2 epochs:
The blobs are actually moving towards us at an angle  measured
from the line of sight:
Photon emitted along the line of sight at time t=0, travels a
distance d to us, taking a time t1 to arrive: t1 = d/c.
A second photon is emitted at a time t later, when the blob is a
distance d – vt cos away from us. The second photon arrives
at t2 = t + (d - vt cos)/c.
The observed difference in the time of arrival from photon 1 & 2 is:
tobs = t2 - t1 = t (1 – v cos /c) < t
The apparent transverse velocity is vapp = v t sin / tobs
= v sin / (1 – v cos/ c).
As v approaches c, vapp can appear > c
Superluminal motion, typically 5-10 c
Take  = 1/(1- v2/c2)1/2 : the Lorentz factor.
Then: vapp  v (maximum observed velocity occurs when cos  =
v/c or sin  = 1/).
We will only observe superluminal motion when the jets are
pointed within an angle of 1/ towards the line of sight, but this light
will be beamed and brightened.
Relativisitic Beaming
Beaming results in (1) rapid variability and (2) boosted flux.
(1) To a stationary observer, the “clocks” on the blob run slow by a
factor of  (time dilation). From before:
tobs = t
(1 – v cos  /c)
tobs = te 
(1 – v cos  /c) .
So the observed frequency of the light is increased according to:
obs = e [ (1 – vcos/c)]-1.
Note: v is directed towards the observer here.
Blueshift and Rapid Variability can be explained by beaming!
(2) A light source at rest radiates isotropically. In a frame moving
towards the observed with speed u, a transverse ray of light in the
moving frame is transformed into a ray moving with speed
u
(1-u2/c2)1/2
towards the observer,
transverse to the observer.
When  >>1, half the light is focussed into a narrow cone with
sin = 1/ ~ . This is the headlight effect.
Thus the light will be brightened by a factor of order 22
Explains why we usually only see one relativistic jet, the one
being beamed towards us.
3. General Relativity; Principles
Gravitation binds matter to matter. But how does matter "know"
that the other matter is "out there”?
The insight: gravitation is not due to a force but rather is a
manifestation of curved space and time, with this curvature
being produced by the mass-energy and momentum content of the
space-time.
• The greater the mass, the greater the distortion of
spacetime.
• Gravity is not “action at a distance” because spacetime
curvature is automatically generated by all mass!
One of the defining features of general relativity is the idea that
gravitational 'force' is replaced by geometry.
In general relativity, phenomena that in classical mechanics are
ascribed to the action of the force of gravity (such as free-fall,
orbital motion, and spacecraft trajectories) are taken in general
relativity to represent inertial motion in a curved spacetime.
So what people standing on the surface of the Earth perceive as
the 'force of gravity' is a result of their undergoing a continuous
physical acceleration caused by the mechanical resistance of
the surface on which they are standing.
General relativity is distinguished from other metric theories of
gravitation by its use of the Einstein field equations to relate
space-time content and space-time curvature.
General relativity is currently the most successful gravitational
theory, being almost universally accepted and well confirmed by
observations. The first success of general relativity was in
explaining the anomalous perihelion precession of Mercury.
Then in 1919, Sir Arthur Eddington announced that observations of
stars near the eclipsed Sun confirmed general relativity's prediction
that massive objects bend light.
Since then, many other observations and experiments have
confirmed many of the predictions of general relativity, including
gravitational time dilation, the gravitational redshift of light,
signal delay, and gravitational radiation.
The Principle of Equivalence
Weak Equivalence Principle or WEP. The WEP states that the
"inertial mass" and "gravitational mass" of any object are equal.
Every object falls at the same rate in a gravitational field,
independent of the composition of the object
The equivalence principle dictates that free-falling observers
are the ones in inertial motion. A consequence of this insight is that
inertial observers can accelerate with respect to each other.
(Think of two balls falling on opposite sides of the Earth, for
example.)
This redefinition is incompatible with Newton's first law of motion,
and cannot be accounted for in the Euclidean geometry of special
relativity. To quote Einstein himself:
"If all accelerated systems are equivalent, then Euclidean
geometry cannot hold in all of them."
["In small enough regions of spacetime, the laws of physics reduce
to those of special relativity; it is impossible to detect the existence
of a gravitational field."
Locally inertial frames: those which follow the motion of freely
falling particles in small enough regions of spacetime.]
The Principle of Equivalence: all local freely-falling, nonrotating laboratories are fully equivalent for the performance
of all physical experiments.
Thus the equivalence principle led Einstein to search for a
gravitational theory which involves curved space-times.
Fundamental principles
General relativity is based on the following set of fundamental
principles which guided its development.






The general principle of relativity: The laws of physics must
be the same for all observers (accelerated or not).
The principle of general covariance: The laws of physics
must take the same form in all coordinate systems.
The principle that inertial motion is geodesic (straightest
possible worldline) motion: The worldlines of particles
unaffected by physical forces are timelike or null geodesics
of spacetime.
The principle of local Lorentz invariance: The laws of special
relativity apply locally for all inertial observers.
Spacetime is curved: This permits gravitational effects such
as freefall to be described as a form of inertial motion.
Spacetime curvature is created by stress-energy within the
spacetime. This is described in general relativity by the
Einstein field equations.
Causality and prohibition of motion faster than light
Figure: Light cone
The interval AB is 'time-like'; i.e., there is a frame of reference in which event
A and event B occur at the same location in space, separated only by
occurring at different times.
If A precedes B in that frame, then A precedes B in all frames. It is
hypothetically possible for matter (or information) to travel from A to B, so
there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; i.e., there is a frame of
reference in which event A and event C occur simultaneously, separated
only in space.
However there are also frames in which A precedes C (as shown) and
frames in which C precedes A. If it was possible for a cause-and-effect
relationship to exist between events A and C, then logical paradoxes would
result. For example, if A was the cause, and C the effect, then there would be
frames of reference in which the effect preceded the cause.
The logical situation is not completely clear in general relativity, so it is
an open question whether or not there is some fundamental principle that
preserves causality (and therefore prevents motion faster than light) in
general relativity.
.
Bending of light
Take a freely falling lab/lift. Let a photon of light be directed
horizontally across the lab from one wall. GR states that the
photon hits the opposite wall, moving in a straight horizontal line.
The ground-based observer will observe a curved path because
the lab has accelerated (path is curved as if attracted to the earth).
This curved path is the quickest route in the curved spacetime.
The deflection is tiny, of course.
Gravitational Redshift
Now send a photon from the floor of the accelerating lab upwards.
Its frequency is unaltered when it reaches the ceiling. (by principle
of GR)
However, the ground-based observed would expect to record a
higher frequency since the ceiling is observed to be moving
towards the photon. To reconcile with what the ceiling records, the
light must be gravitationally redshifted.
The exact result for gravitational redshift valid for a strong
gravitational field is:
   2GM 
 1 

o 
ro c 2 
1/ 2
for the frequency at infinity. The redshift is:
 2GM 
z 1 

ro c 2 

1 / 2
1
Time passes more slowly as the surrounding spacetime
becomes more curved. t(infinity) > t(proper)
t o  2GM 
 1 

t  
ro c 2 
1/ 2
Clocks will run slower at lower gravitational potentials (deeper
within a gravity well).
Interlude:
EFEs, Intervals and Geodesics
Relativity calls for gravitation to be expressed as a rank-two tensor.
The Einstein field equations (EFE) describe how stress-energy causes curvature of
spacetime and are usually written as
where
Gab is the Einstein tensor,
Tab is the stress-energy tensor and
kappa is a constant.
The tensors Gab and Tab are both rank 2 symmetric tensors, that is, they can each be
thought of as 4×4 matrices, each of which contains 10 independent terms.
The EFE reduce to Newton's law of gravity in the limiting cases of a weak
gravitational field and slow speed relative to the speed of light.
In fact, the value of kappa in the EFE is determined to be
by making these two approximations. Being non-linear differential equations, the
EFE often defy attempts to obtain an exact solution; however, many such solutions
are known.
The geometry of space-time
SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time.
This space, however, is very similar to the standard 3 dimensional Euclidean space,
and fortunately by that fact, very easy to work with.
The differential of distance (ds) in cartesian 3D space is defined as:
In the geometry of special relativity, a fourth dimension, time, is added, with units of
c, so that the equation for the differential of distance becomes:
.
If we wished to make the time coordinate look like the space coordinates, we could
treat time as imaginary: x4 = ict . In this case the above equation becomes symmetric:
.
This suggests what is in fact a profound theoretical insight as it shows that special
relativity is simply a rotational symmetry of our space-time, very similar to rotational
symmetry of Euclidean space.
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D
space
We see that the null geodesics lie along a dual-cone:
defined by the equation
or
Which is the equation of a circle with r=c*dt. If we extend this to three spatial
dimensions, the null geodesics are the 4-dimensional cone:
This null dual-cone represents the "line of sight" of a point in space. That is, when we
look at the stars and say "The light from that star which I am receiving is X years
old", we are looking down this line of sight: a null geodesic.
Having recognised the four-dimensional nature of spacetime, we are driven to employ
the Minkowski metric, η, given in components (valid in any inertial reference frame)
as:
Its reciprocal is:
Lorentz transformation.. Then we recognise that co-ordinate transformations
between inertial reference frames are given by the Lorentz transformation tensor Λ.
For the special case of motion along the x-axis, we have:
which is simply the matrix of a boost (like a rotation) between the x and t coordinates
where  indicates the row and  indicates the column. Also, remember,
More generally, a transformation from one inertial frame (ignoring translations for
simplicity) to another must satisfy:
where there is an implied summation of
and
from 0 to 3 on the right-hand side
in accordance with the Einstein summation convention.
All proper physical quantities are given by tensors. So to transform from one frame to
another, we use the well known tensor transformation law
where
is the reciprocal matrix of
.
To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate
which is the Lorentz transformation given above. All tensors transform by the same
rule.
The squared length of the differential of the position four-vector
using
constructed
is an invariant. Being invariant means that it takes the same value in all inertial
frames.
Recognising other physical quantities as tensors also simplifies their transformation
laws.
Interval: the metric for flat spacetime
ds 2  c 2 dt 2  dx 2  dy 2  dz 2
Positive: timelike, define the proper time ds/c
Negative: spacelike, define a proper distance (-ds2)1/2
4. The Schwarzschild metric
Matter tells spacetime how to curve.
Curved spacetime tells matter how to move.
How does mass act on spacetime?
With a spheroidal mass M, the Schwarzschild metric can be
derived:

ds 2  cdt 1 - 2GM/rc 2

2

dr
 
2
 1 - 2GM/rc
2

  r 2 d 2  sin 2  d 2




where the coordinates are defined by a distant observer at rest. Note what
occurs if M is zero: back to Minkowski space of SR.
This means that the spatial distance between two points on the
same radial line is greater than the coordinate distance.
The metric also includes information concerning time dilation and
gravitational redshift. The proper time dto is
dt 1 - 2GM/rc 2
…..less time has passed in the proper frame time passes slower closer to the mass.
For a rising photon:
dr


cdt  
2 ,
 1 - 2GM/rc 
a longer time is required.
For a falling photon:
dr


 cdt  
2 .
 1 - 2GM/rc 
How does spacetime act on mass? An object moves along
the shortest path between two points in spacetime.
Inertial motion: Take d  dt , dr  0 and d  90 (equator)
intergrating over one orbit and maximising the interval yields the
coordinate orbital speed

v  r 
GM
r
.
EXAMPLE:
How fast does time pass on the Earth’s equator?
Proper time is given by teq = ds/c
Take dr = 0, and d = 0
At the equator: 
 = dt
where = 2 / 1day = 7.27 10-5 radians s-1 .
Substitute into the Schwarzschild metric:

c dt eq  cdt 1 - 2GM/rc
2
2
2

 
2

 r 2 2 dt 2

 
c 2 dt eq  c 2 dt 2 1  2GM / rc 2  r 2 2 dt 2
2

dteq  dt 2 1  2GM / rc 2  r 2 2 / c 2
2

= 1 - 2 x 6.67x10-11 x 6x1024 / (6.4x106 x 9x1016) –
(6.4x106 x 7.27x10-5 / 3x108)2
= 1 – 1.4x10-9 - 2.4x10-12
Take square root:
dteq = dt (1 – 7.0x10-10 - 1.2x10-12)
So, in one year, less time has passed when under the influence of
the earth’s gravity….dt = 3.16x107 seconds in a year
5. Black Holes
The behaviour of space and time at
RS  2GM / c 2
is remarkable for the Schwarzschild metric.
1. Time slows to a complete stop at RS
2. RS acts as a barrier: the event horizon. Anything inside the
event horizon is gone from sight forever (nothing can
escape).
3. At the centre is the singularity, infinite density.
4. Hawking radiation: particle-antiparticel pair creation implies
that black holes can evaporate
5. Absence of hair: three numbers describe black holes
completely: mass, angular momentum and electric charge.
Black holes are completely specified by their mass M,
angularmomentum J, and charge Q (likely ~zero).
6. Q=0, J=0: Schwarzschild black hole
Spherically symmetric. Solution has two important radii for
our purposes:
1. An event horizon at Rs = 2GM/c2 (Schwarzschild radius)
No matter, radiation, or information can propagate
outwards through this radius
• Rs = 3 km for one Solar mass black hole
• Rs = 2 AU for a 108 Solar mass black hole
Object
Mass (Msun)
Rs
Star
10
30 km
Star
3
9 km
Sun
1
3 km
Earth 3 x 10-6
9 mm
2. A last stable circular orbit at Rms = 6GM/c2
Outside Rms test particles can orbit indefinitely in stable circular
orbits
Inside Rms orbits are unstable, particles spiral rapidly
past the event horizon and into the black hole
Defines the inner edge of the gas disk in AGN
Sets a minimum orbital period. Roughly,
[t = 2 Rms/ vms
= 2  Rms/ (c2 /6)1/2
Newtonian - so this is not quite correct… ]
For 108 Solar masses, this is about 12 hours
For 10 Solar masses, ~ 5 ms
Kerr Black Holes
Q=0, J and M arbitrary: Kerr black hole
Axisymmetric solution - hole has a preferred rotation axis
Define the amount of angular momentum via a dimensionless
spin parameter:
a = c J/ (GM2)
Maximum angular momentum of a Kerr black hole
corresponds to a spin parameter a = 1.
Cannot spin a Kerr hole up beyond this limit.
THE END