Manifolds, Charts, Atlases, Coordinates
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Take Rn to be the set of all n-tuples of real numbers, that is the set of all (x1 , x2 , . . . , xn )
where xi R for i = 1, . . . , n.
Let
Geometric Objects, Tensors, and Tensor Algebra
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Scalars
Scalars are simply real numbers, ie. they are elements of . While a scalar may be a
function of position in the manifold, as
Time and Simultaneity
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In Special Relativity it is not possible to determine a unique denition of the simultaneity of two spacetime events when they are not at the same location in
The Covariant Derivative
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1
Dening the Covariant Derivative
Consider the vector eld with covariant components Ba (xc ) and compute the partial derivative:
Ba
Ba|b =
,
(1)
xb
where
The Einstein Field Equations
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Basic Ingredients
We list here some basic requirements to be met in developing a set of eld equations to
generalize the classical gravitational equati
Vector as a Derivative
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Consider a dierentiable function, f (x1 , ., xn ), on an manifold M of dimension n, so
that f maps M to the real line R1 .
In the same manifold, c
The Fermi-Walker Derivative
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Consider a time-like curve , with parameter , and tangent vector u. We will dene
the Fermi-Walker derivative of the vector t along , which we
The Energy Momentum Tensor
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Since the aim is to consider a theory of spacetime where the geometry and the physical
content of the spacetime are closely related, in some s
Frequency and the Redshift Factor
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Waves and Rays
We consider the propagation of electromagnetic radiation in a curved spacetime. We expect
to have to consider both the wave aspect
Robertson-Walker Metrics, Friedmann-LeMa
tre
Equations
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In the study of the large-scale structure of the universe, now and in the past, that is, in
cosmology, it is dicult to proce
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Christoel Symbols and Special Coordinate Systems
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Some Relations for a
bc
We can write the Christoel symbols (of the second kind) as:
m = 1 g me gae|b + gbe|a gab|e
ab
2
(1)
Since
Equation of Geodesic Deviation
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ua
Q
va
ua
P
Figure 1: Deviation between two geodesics.
Consider two geodesic curves, , and , which are diverging from each other. Suppose
that alon
Equations for a Geodesic
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A Simple Lagrangian Choice
Consider the space with metric tensor gmn , depending on the coordinates xa . The line
element is then:
ds2 = gmn dxm dxn
(1)
A
Just How Singular is r = 2m
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The line element for the Schwarzschild spherically symmetric solution can be written
as
2m
2m 1 2
2
ds = 1
dt 1
dr r2 d2
(1)
r
r
At rst glance, this
Spherically Symmetric Spacetimes
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We wish to develop the line-element, and the corresponding metric tensor, for a spherically symmetric spacetime. We choose coordinates (
Spherically Symmetric Spacetimes
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We wish to develop the line-element, and the corresponding metric tensor, for a spherically symmetric spacetime. We choose coordinates (
The Lie Derivative
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1
The Lie Bracket
Suppose that we have two vector elds, U and V , dened as directional derivatives along
two dierent families of curves dened on a die
Newtonian Limit in General Relativity
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Consider the Einstein eld equations in the context where we impose the requirements:
1. The coordinate system is cartesian with coo
Manifolds and Maps
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Topological Spaces
Consider a set M , and T , a family of subsets of M . We will normally think of the
elements of M as points. The pair (M, T ) is a topologica
The Kerr Metric
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While perfect spherical symmetry is a powerful assumption in considering many astrophysical bodies, virtually all astrophysical systems will involve some rotation.
Parallel Propagation
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1
Propagation Along Geodesics
va
ua
Figure 1:
Consider a geodesic curve, say , with ane parameter , given as the four functions
x (). Taking ua = dx
Orbits and Paths in Schwarzschild Spacetime
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1
Geodesic Equations for Null and Time-Like paths
The paths of photons and time-like particles in the Schwarzschild spacetime dier from
Problem Set 1 - GR-I - 2012
Hand in at class on Friday, October 18 2013. Write on one side only, in ink
no pencil! See the note on the course web-page. Do NOT use any computer
algebra system to solve
Problem Set 2 - GR-I - 2013
Hand in at class on Tuesday, December 3, 2013. Write on one side only, in ink
no pencil! See the note on the course web-page. Do NOT use any computer
algebra system to sol
Riemannian Curvature and Parallelism
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1
Traversing the Parallelogram
One simple characterization of Euclidean geometry regarding parallelism is that if one
transports a v
Properties of the Riemann Curvature Tensor
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Symmetries
We have the expression for the Riemann curvature tensor:
Ramsq = gar r |s r |q + r n r n
nq ms
mq
ms
ns mq
(1)
Since the symm
The Schwarzschild Solutions
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We have already shown that with a suitable choice of coordinates, we can write any
spherically symmetric metric in the form:
ds2 = e (t,r) dt2 e(t,r) d
Simple Dust Friedmann Equation
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Consider the line element:
ds2 = c2 dt2 a2
dr 2
+ r2 d2
1 kr2
(1)
where a = a(t) and k = 0, 1. Dening the Ricci tensor as Rab = Rmamb , computing th