General Relativity
Physics Honours 2007
A/Prof. Geraint F. Lewis
Rm 557, A29
gfl@physics.usyd.edu.au
Lecture Notes 1
Why are we here?
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Why are we here?
GPS
Black Holes
Extreme Stars
The Universe
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Chapter 1
Prior Knowledge
There is no escaping the fact that General Relativity is a
mathematically challenging physical theory. However, this
course is structured differently to typical courses on relativity;
the goal will be to develop a physical understanding of
underlying theory and learn how to apply the mathematical
framework in various physical situations.
To tackle this course effectively, prior knowledge includes
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Differential Equations
Special Relativity
Lagrangian Mechanics (desirable)
Maxwell’s Equations (desirable)
Tensors (desirable)
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Textbook
Gravity: An Introduction to Einstein’s GR
J. B. Hartle
à Course Book
You should obtain a copy of the textbook as it contains required
reading and the assignment questions. Note that we will not
step through the book linearly! Other useful texts are;
Introducing Einstein’s Relativity
General Relativity
A First Course in GR
by Hobson, Efstathiou & Lazenby
by B. F. Schutz
Spacetime and Geometry
Lecture Notes 1
by R. D’Inverno
by S. M. Carroll
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Additional Resources
There are many additional general relativity resources that I
encourage you to explore. These include;
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http://www.physics.usyd.edu.au/~gfl/Lecture/
http://spacetimeandgeometry.net/
“A no-nonsense introduction to general relativity” by Sean
Carroll (http://pancake.uchicago.edu/~carroll/notes)
“Living Reviews in Relativity”
(http://www.livingreviews.org)
“The meaning of Einstein’s equations” by John Baez and
Emory Bunn (http://arxiv.org/gr-qc/0103044)
Preprint archive (http://www.arxiv.org)
Type “general relativity” etc into google & wikipedia
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Assessment
Assessment for this course will consist of;
Three assignments
Final exam
30%
70%
Assignments are to be handed into the Student Support Office
on the due date. Late assignments will be penalized 20% for
each day they are late. Assignments more than one week late
will not be accepted without a formal special consideration.
You can bring one hand written A4 page into the exam.
Postgraduate students must achieve >70% on all assignments
and will sit the exam in an open book environment.
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Geometry and Physics
We will hear a lot about geometry,
especially the geometry of curved
surfaces. This can be quite different to
Euclidean geometry (flat surface) and
requires a way to characterize the
curvature (the metric).
We will rely on differential geometry, whereby geometry
can be understood in terms of a line element.
Cartesian coordinates
Polar coordinates
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Chapter 2
Geometry and Physics
If we consider the surface of a sphere,
the line element is more complicated
(remember the surface of a sphere is 2dimensional). If we use spherical polar
coordinates;
In relativity we will be using the concept of spacetime, in
which we consider 4-dimensional surfaces;
[This is the line element for the Schwarzschild black hole]
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Coordinates and Invariance
When discussing distances, motions etc through any general
curved surface (a manifold), we rely on coordinates. Such
coordinates are not fundamental to the surface and different
coordinate systems can be used of the surface.
For example, we can cover a flat plane with a Cartesian or
Polar coordinate system.
The physical predictions we make should be invariant and
not depend on the choice of coordinates. We should be able to
transform from one coordinate system to another.
What we will see is that we can write all physical laws in a
tensor form that provides a method to transform between the
differing coordinate systems.
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Newtonian Relativity
Newtonian mechanics assumes flat space and absolute time.
To describe the motion of particles we can use a flat-space
coordinate system (e.g. Cartesian), and events have a unique
set of coordinates (t,x,y,z).
Newtonian mechanics gives us a special set of observers,
those in inertial reference frames, who will see Newton’s
laws hold. These are stationary, or move with constant
velocity, with respect to one another.
All inertial observers are equivalent as far
as dynamical experiments are concerned.
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Chapter 3
Galilean Transformations
Observers can transform between the coordinates of an event
in different inertial frames with Galilean Transformations. If
the frames S and S’ have collinear x-axes, and S’ moves along
the x-axis with constant velocity v, then;
How do you use the rules of Newtonian mechanics to make
predictions in a particular physical situation?
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Review 3.3-3.5
Relativity in a Nutshell
Einstein Field Equation
Geodesic Equation
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Special Relativity
The foundation of Special Relativity comes from Einstein’s
understanding of Maxwell’s equations. He realised that while
they predicted the velocity of light, they do not say with
respect to what this velocity should be measured.
In 1905, Einstein proposed the postulate of the constancy
of the speed of light.
The velocity of light in free space is
the same for all inertial observers.
Note that while the famous null result of the MichelsonMorley experiment can be explained in terms of this
postulate, the experiment was not Einstein’s motivation!
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Chapter 4
Special Relativity
If the speed of light c is the same in all inertial frames, then
something else has to give. Einstein proposed that the
Newtonian ideas of space and absolute time had to be
abandoned.
As well as carrying their own spatial coordinates, all inertial
observers also carry their own clock. Observers now disagree
on where an event happens, and also when it happens.
If we consider two nearby events for a
particular observer, we can define an
invariant interval;
[Review section 4.3]
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Light Cones
The negative sign in ds2 has
some interesting effects.
Intervals in flat space are
always positive, but this is not
the case with spacetime.
This breaks spacetime into
distinct regions. These are
very important to the notion
of causality.
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World Lines
Massless particles travel along null world lines.
Massive particles travel along timelike world lines. Note that
their path always lies within their past and future light cones.
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The Interval Again!
Consider a person moving
along a particular world line.
To them, they are at rest and
so dx=0.
So, the only thing that
changes along a path is the
time they experience (i.e.
what is seen on their watch).
This is known as the proper
time d.
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The Interval Again!
Remember that the interval is invariant,
so everyone agrees on its value.
This allows us to connect how spacetime
is experienced between observers.
Rewriting;
Therefore there is time dilation between the observers.
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The Twin Paradox
An awful lot has been written
about the twin paradox, much
of it complete rubbish.
The important point to note is
that the straight-line path on
a spacetime diagram
represents the longest
4-distance (i.e. through
spacetime).
www.phys.vt.edu
All other paths are shorter.
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Wikipedia!
Lorentz Boosts
To transform between inertial
frames, we use Lorentz
Boosts.
where
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Simultaneity & Contraction
Consider the green dots as
two separate events. In the
primed frame, these occur on
a line of constant ct’ and are
simultaneous.
In the unprimed frame, these
two events occur on two
different lines of ct, and so
are not simultaneous.
Furthermore, if we take the
events to denote the ends of
a rod, clearly the spatial
length as measured in the
two frames will be different
and we get length
contraction.
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Addition of Velocities
The Lorentz boosts also give relativistic velocity addition.
and similarly
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4-Vectors
When dealing with 4-dimensional spacetime, it is natural to
consider ‘things’ in terms of 4-vectors. In an inertial frame,
we can define basis 4-vectors of unit length, pointing along
(t,x,y,z). Then a 4-vector a can we written as;
We will be dealing with the components of the vector
a=(at,ax,ay,az) when considering relativity. Note that the
components are often written as a=(a0,a1,a2,a3). We can
write a 4-vector as;
The last term here is the Einstein
Summation Convention, his
greatest contribution to maths!
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Chapter 5
Lorentz Boosts
The components of a 4-vector can be transformed between
inertial frames using Lorentz Boosts.
Here we have set c=1.
It should be clear that we can write Lorentz Boosts as matrix
multiplications.
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Scalar Products
The concept of a scalar product is very important for 4-vectors
Here we have an
implicit summation
over  and .
Here,  is the metric of
flat spacetime.
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Scalar Products
As with 3-vectors, two 4-vectors are orthogonal if
The length (or magnitude) of a 4-vector is
Note, the length of a vector can be positive (spacelike), zero
(null) or negative (timelike).
Importantly, the invariant interval can be written as
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Orthogonality
Consider two unit 4-vectors in
the primed frame, one along
t’ and the other along x’.
This is orthogonal as a¢b= 0
a & b in the unprimed frame;
But;
Orthogonality is preserved!
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Kinematics
Any particle is constantly moving
through spacetime (even if it is
stationary in space).
We can describe location in spacetime
in terms of parameterized coordinates;
For massive objects, the natural choice
of parameter is the proper time .
We can now define the velocity of a
particle through spacetime;
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Kinematics
The components of the 4-velocity are;
So, 4-velocity is
Note;
All massive particle have the same speed through spacetime.
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Newton’s laws
Newton’s first law holds in relativistic mechanics;
We also have Newton’s second law;
Here, m is the rest mass of the particle. But we are now
dealing with the 4-force and 4-acceleration.
You should convince yourself that f and u are orthogonal
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Other 4-Vectors
Another important 4-vector which we will see is the 4momentum;
The components of the 4-momentum are
The magnitude of p is
And we can write Newton’s second law as;
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Review 5.4
Light Rays
Light rays do not have a proper time (as ds=0) and so we
cannot use it to parameterize path through spacetime.
We can use an affine parameter which we use to describe
the motion, but it has no physical meaning.
As a light rays has a straight-line path in a spacetime
diagram, we can parameterize the path as;
Where u = dx/d. This is a null vector and u¢u=0.
Again, the 4-momentum is important and
The 4-momentum and wave 4-vector k are null!
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Doppler Shift
Consider a source emitting photons of frequency  in all directions (in
its rest frame). The wave 4-vector is;
Suppose to an observer, this source is moving along the x’axis with velocity V. What frequency will this observer see a
photon detected at angle ’? In the primed frame
We can connect the two using a Lorentz Boost
But kx’ = ’ cos ’ so
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Observers
The orthonormal basis is an important concept in relativity.
This consists of one time and three spatial unit vectors, with
the time vector tangent to the world line (i.e. it points along
the 4-velocity u). The components of the orthonormal basis
must be orthogonal to each other. What an observer
measures is relative to the orthonormal basis.
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Observers
Suppose a particle with 4-momentum p passes through a lab
with a defined orthonormal basis. We can express the 4momentum in terms of the orthonormal basis;
The first component in this basis is the energy of the particle
as seen by the observer. We can extract the components of the
4-momentum via
So the energy of the particle can be written as
Work through the examples in Chapter 5.
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Steps to General Relativity
While special relativity overturned established ideas in physics,
Einstein realised that it was incompatible with gravity
The problem is positions according to who, and at what time?
Einstein’s revelation began with the realization of equivalence
of gravitational and inertial mass. Simply put, all masses fall
at the same rate. What happens if I drop a laboratory?
Clearly everything falls together.
What Einstein realized is that
effectively gravity has vanished.
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Chapter 6
Equivalence Principle
Einstein’s “happiest
thought” came from
the realization he could
take the equivalence
principle further.
Simply put, Einstein
reasoned that;
There is no experiment that can distinguish
between uniform acceleration and a uniform
gravitational field.
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Implications
There were several implications of Einstein’s view of the
equivalence principle, including the fact that light should be
deflected in a gravitational field.
http://physics.syr.edu/courses/modules/LIGHTCONE/equivalence.html
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Time & Acceleration
Suppose the rocket is accelerating
at g. The location of the two
astronauts is
Suppose a pulse is fired from A to
B at t=0. Then
Now a second pulse is emitted
from A at A and is received at
t1+B
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Time & Acceleration
If we assume  is small, and so we can expand these expressions in
only the linear terms, we find
But the equivalence principle tells us that this should also
occur in a uniform gravitational field, and so we would expect
Where  is the Newtonian gravitational potential. This
immediately suggests that the rocket and gravitational field
should see photons red/blue-shifted.
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GPS 6.4
Spacetime is Curved
We could suppose that spacetime is flat and the gravitational
potential somehow influence the running of the clock. This is a
little like measuring distances on a flat map, but having to
change the length of the ruler.
While some aspects of relativity do adopt this view, it is
“simpler, more economical and ultimately more powerful” to
keep our measuring devices (rulers and clocks) fixed and
assume the geometry in which they sit is curved.
Newtonian gravity can be expressed in the Static Weak-Field
Metric in which time and space are curved
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Time Dilation
Consider two observers, xA and
xB, in a gravitational field. A
fires two photons to B; these do
not travel as straight lines in
this picture, but they will have
the same shape. Hence, if
photons from A are separated
with a coordinate time t,
then B will receive them with
the same separation.
But how much time will each measure on their watches (their
proper time). Neither move spatially, so x=y=z=0 and so
the interval is;
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Time Dilation
Taking the square root then;
As t is common for both observers, we can equate their
proper times and find that;
Again, we recover the time dilation, but find it is due to the
geometry of spacetime.
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Spacetime Diagrams
Understanding the paths of
light rays over a spacetime
diagram is an important
aspect of relativity.
In general, they do not
look like those seen in
special relativity, as they
can be squeezed & rotated.
Here we have a spacetime
diagram for a white hole
in Eddington-Finkelstein
Coordinates.
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