General Relativity Physics Honours 2005

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General RelativityPhysics Honours 2005

• Dr Geraint F. Lewis
• Rm 557, A29
• gfl_at_physics.usyd.edu.au

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The Field Equations
It is time to look at the source of the metric
and this involves understanding the field
equations of General Relativity
where T is the stress-energy tensor and the R
terms are the Ricci tensor and scalar. The left
terms contain the metric (and its derivatives)
whereas the right describes the distribution of
energy. The field equations are ten 2nd order
PDEs which must be solved to determine the metric
for a particular energy distribution.
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Manifolds (Ch 5)
General relativity is based upon non-Euclidean
geometry. To understand this, we must consider
the concept of a manifold. Crudely speaking a
manifold is an object which is locally isomorphic
to a piece of n-dimensional euclidian space Rn .
Hence, in a small region about any point it can
be parameterized by n coordinates. If each point
has a unique set of coordinates, then the
coordinate system is non-degenerate.
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Manifolds
A non-degenerate coordinate system cannot always
be extended over the manifold and generally
overlapping coordinate patches are used. Hence,
we need to understand coordinate transforms to
take us between the patches on a
manifold. Example A plane can be covered with a
single non-degenerate coordinate system (i.e.
cartesian), but can also be covered with polar
coordinates which are degenerate at the origin
(i.e. ? is undefined). Example A sphere cannot
be covered with a single non-degenerate
coordinate system.
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Coordinate Transforms
Suppose a point has coordinates xa (a1,,n) in
one set of coordinates. In a passive coordinate
transform we can find xa in a new coordinate
system with the new coordinates being functions
of the old coordinates (i.e. xa xa
(x)). Example cartesian coordinates can be
expressed as a function of spherical polar
coordinates.
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Coordinate Transforms
Differentiating the mapping between coordinate
systems give the transformation matrix
The determinant J of the matrix is the Jacobian
and if it is non-zero we can define the inverse
transform xaxa(x). The total differential is
then
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Coordinate Transforms
The cartesian and spherical polar example
The Jacobian is Jr2 sin? and
Etc.
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Contravariant Tensors
Definition tensors are quantities that satisfy
certain transformation laws. Tensors contain
vectors and scalars as special cases. We just saw
the transformation for the infinitesimal total
differential vector. This is a contravariant
vector (or tensor of rank 1) i.e. the quantity
Xa at a point which transforms as
The infinitesimal vector dxa and the tangent
vector dxa/du to the curve x(u) are contravariant
vectors.
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Contravariant Tensors
We can generalize this idea to higher tensors. A
contravariant tensor of rank 2 is defined to be
the object at point P that transforms as
Higher order tensors as defined analogously. A
scalar is a contravariant tensor of rank 0 and is
invariant under a coordinate transformation.
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Covariant Tensors
Consider the function ?(xb). If we regard the xb
as functions of the xa then
Thus, the gradient operator ?/? xb transforms in
a different way to contravariant tensors. Hence a
covariant tensor of rank 1 can be defined as
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Mixed Tensors
A mixed tensor can be defined such as
which is contravariant rank 1 and covariant rank
2 and is called a type (1,2). An example of a
mixed rank tensor is the conductivity tensor (s)
that relates the ?th component of the 3-current j
to the ?th component of the electric field E.
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Tensor Fields
A tensor is defined at a particular point. A
tensor field associates a tensor at each point on
a manifold. Rank 0 potential energy Rank 1
electric field Rank 2 conductivity Remember,
transformations between coordinate systems depend
upon the location on the manifold.
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Elementary Operations
Tensors can only be added to, or subtracted from,
tensors of the same type. Scalar multiplication
of a tensor results in a tensor of the same type.
Note, indicies must match up in every term. A
tensor can be symmetrized or antisymmetrized to
give
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Elementary Operations
The alternating sum has a positive sign for even
permutations of ai and negative signs for odd
permutations.
We can define a tensor multiplication such that
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Contraction
We will see contraction throughout the course.
The idea is to sum over a pair of contra and
covariant indicies. Hence, Xabcd can be
contracted on a and b through
Hence a tensor of type (1,3) contracts to a
tensor of type (0,2). For example
the left is type (0,1), while the right is (1,2)
contracted to (0,1).
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Index-Free Notation
Section 5.9 discusses an index free
interpretation of tensors. We will not consider
them further here, but you should be familiar
with the concept (although most discussions of
general relativity appear to prefer the indexed
version of tensors).
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Tensor Calculus
We need to consider the differentiation of
tensors. This would involve an entire formal
course, but here we will focus on issues we need
in General Relativity. We denote the partial
derivative of a contravariant tensor as
How does this quantity transform?
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Tensor Calculus
The partial derivative of a tensor is
This is not a tensor (why?)! This is not
surprising as the limiting process to obtain a
partial derivative involves evaluating the tensor
at two different places.
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Lie Derivative
Consider a congruence of curves such that a
unique curve goes through each point on the
manifold. We can parameterize one of these curves
as xaxa(u) and can define a field of tangent
vectors
Conversely, a non-zero vector field yields a
unique congruence of curves over at least a
subset of the manifold (Fig 6.1 6.2).
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Lie Derivative
Suppose the field Xa is given, and we can
construct the corresponding congruence of curves.
To differentiate the tensor T at the point P we
drag T along the curve through P to a
neighbouring point Q. The derivative therefore
compares the dragged tensor with the tensor
evaluated at Q in the limits that Q! P. 1) If the
coordinates at P are xa, those at Q are
for a small displacement ?u along the curve
though P.
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Lie Derivative
2) We can treat the shift from xa to xa as an
active coordinate transformation with
3) Consider the tensor Tab. The components of the
dragged along tensor at Q are
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Lie Derivative
4) The value of the tensor at Q can be obtained
by Taylor-expanding T about P to the same order
5) Define the Lie Derivative of Tab with respect
to Xa as
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Lie Derivative
We can always introduce a coordinate system such
that
Then the Lie derivative reduces to
In this special coordinate system, the Lie
derivative reduces to an ordinary directional
derivative.
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Lie Derivative

• Linear
• Satisfies the product rule
• Preserves tensor type
• Is the directional derivative for a scalar field
• The Lie derivative for a general tensor field is

with a negative term for each contravariant
index, and a positive term for each covariant
index.
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Covariant Derivative
Now we have the Lie derivative, we need to
another (more useful) form of tensor derivative.
Again, we will consider a contravariant vector
field Xa(x) evaluated at Q(x? x) near P(x).
Taylors theorem gives
Again the difference is not tensorial as we are
subtracting tensors at two different points. We
will define the covariant derivative by
comparing X(x? x) with a vector which is in some
sense parallel to the one at P.
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Covariant Derivative
Writing the parallel vector as
To lowest order the barred term must be linear in
Xa and dxb such that
where the properties of ? have yet to be defined.
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Covariant Derivative
We can define the covariant derivative to be
Clearly the G term (the affine connection) does
not transform like a tensor. This is not
surprising considering that its role is to
compensate for non-tensorial aspects of the
derivative. The covariant derivative of a scalar
satisfies
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Covariant Derivative
Furthermore, for a covariant vector
and in general
Throughout this course we will restrict ourselves
to symmetric connections such that
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Covariant Derivative
When considering the derivatives of the
components of a tensor, the affine connection can
be seen to include contributions from changes in
the tensor, plus contributions from the twisting
and turning of the coordinates in general
curvilinear coordinates. When we differentiate
aj we are really differentiating the vector ajej,
where ej is the jth unit vector of the coordinate
system.
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Affine Geodesics
Introducing the notation
which represents the covariant derivative
contracted with X cf. the usual directional
derivative
We can define the absolute derivative to be
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Affine Geodesics
The tensor is said to be parallel-transported
along the curve if its absolute derivative
vanishes i.e we hold it locally as constant as
possible along the congruence curve defined X. As
an example, we could compare a vector to a
gyroscope also taken along the curve. An affine
geodesic is a curve along which its own tangent
vector is propagated parallel to itself, or the
straightest possible curve
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Affine Geodesic
This is equivalent to (exercise)
If we parameterize the curve such that ?(u)
vanishes, then the tangent vector is parallely
transported (or propagated) along the curve onto
itself.
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Affine Geodesic
The affine parameter s is only defined up to an
affine transformation s! ? s ?. Here, ? and ?
are constants. Hence, we can calculate the affine
length along a geodesic as
Note that we cannot compare lengths along
different geodesics without a metric.
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Riemann Tensor
Generally, covariant derivatives do not commute
(assuming that the ? terms are symmetric). The
Riemann Tensor is related to the curvature of a
manifold and vanishes when the manifold is flat
(i.e. where the covariant derivatives commute).
We can use this as the definition of flat.
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Affine Flatness

• Properties of a manifold
• There exists a special coordinate system in
  which the connection coefficients vanish
  everywhere (affine flat).
• Manifold is integrable if vectors transported
  along various paths between two points are
  transported to the same vector.
• Manifold is integrable iff its Riemann tensor
  vanishes everywhere.
• Manifold is affine flat iff its connection is
  symmetric integrable.
• Manifold is flat iff Riemann tensor vanishes
  everywhere.

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Geodesic Coordinates
While affine flatness is a global concept, we
can also describe a local flatness. The Geodesic
Coordinates are those in which the connection
coefficients vanish, but not their derivatives
(or the curvature tensor).
This yields ?abcP 0. This coordinate system
is the flattest looking one at the points P and
for a 2-D surface this would represent the
tangent plane at P. More generally it represents
a tangent space.
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Metric Reminder
The square of the infinitesimal distance between
two points xa and xa dxa is
where the metric tensor gab is a symmetric tensor
that defines the metric. Furthermore, we define
the norm of a vector X
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Metric Reminder
The metric is non-singular if the determinant is
non-zero and an inverse exists so
The metric tensor can be used to raise and lower
indicies on tensors by defining
In general
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Metric Flatness
A metric is flat if there exists a coordinate
system in which the metric tensor reduces to a
diagonal form
The metric tensor is flat iff it Riemann tensor
vanishes. (Note the signature of the metric in
this form is the sum of the diagonal components).
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Metric Geodesic
The curve with maximum or minimum (i.e.
stationary) between two points is the metric
geodesic (Ch 7)
with
(Note that this is a particular case of a more
general argument. Affine and metric geodesics
coincide only if the connection is symmetric i.e.
torsion free. Read the details in Ch 6).
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Curvature Tensor
The curvature tensor Rabcd embodies the curvature
of the metric. As we have seen, this quantity
depends upon the metric, as well as its first and
second derivatives. Hence, Rabcd is really a
series of coupled differential equations (but
what is it equal to?) On an n-dimensional
manifold, this tensor has n4 components, but it
possesses a number of symmetries which reduces
the number of independent components.
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Curvature Tensor
These symmetries are (Pg 86)
This reduces the number of independent components
to
Hence, for the case we are interested in (n4)
this reduces the number of independent components
from 256 to 20!
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Curvature Tensor
Rabcd also satisfies the Bianchi identities
Two important quantities are contractions of the
curvature tensor
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Einstein Tensor
The Ricci tensor and scalar are combined to give
another symmetric tensor, the Einstein Tensor
In terms of relativity, this is the most useful
representation of the curvature tensor
satisfies the contracted Bianchi identities
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Spheres Cylinders
Remember, what we have seen is related to
non-Euclidean geometry, and is not specific to
relativity! We can consider two simple examples,
the sphere and the cylinder, to explore these
concepts. In the following, the shapes may be
three dimensional, but the manifold refers to
only the two dimensional surfaces. The third
dimension only aids visualization. The higher
dimensions required to visualize 4-d manifolds in
relativity just make your head hurt!
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Spheres Cylinders
Coordinates on a sphere can be labeled with ? and
?. The line element on a sphere is given by
Remember, a 2-d being on the surface of the
sphere has no physical concept of r. The metric
is simply given by
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Spheres Cylinders
Given that det(gab) r4 sin2? ? 0 the metric is
non-singular except at ? 0 ?. Hence
We can now calculate the connections
i.e. two independent coefficients.
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Spheres Cylinders
The 2416 element Riemann tensor is (all other
cmpts are zero)
Lowering the first index gives
and the symmetries noted earlier are verified.
The tensor does not vanish, and so the metric is
not flat. Neither is it affine flat or integrable.
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Spheres Cylinders
The components of the Ricci tensor and scalar are
The resulting Einstein tensor is Gab 0 (note
that in ngt2 this can only happen if Rab0).
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Spheres Cylinders
The geodesic equations are
Rotating to ?p/2 and taking d?/ds0 initially
These are great circle paths over the sphere.
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Spheres Cylinders
The covariant derivatives of a vector Xa are
As noted previously, these latter terms describe
the change of the unit vectors from one place to
another on the sphere.
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Spheres Cylinders
On the surface of a cylinder, use cylindrical
coordinates ? and z
The connections involve derivatives of ? z but
the metric does not depend on these. Hence the
Riemann, Ricci Einstein tensors and curvature
scalar are all zero and the cylinder is flat!