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Fundamental Cosmology 4.General Relativistic
Cosmology
Matter tells space how to curve.
Space tells
matter how to move. John Archibald Wheeler
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4.1 Introduction to General Relativity

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• Introduce the Tools of General Relativity
• Become familiar with the Tensor environment
• Look at how we can represent curved space times
• How is the geometry represented ?
• How is the matter represented ?
• Derive Einsteins famous equation
• Show how we arrive at the Friedmann Equations
  via General Relativity
• Examine the Geometry of the Universe

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• Suggested Reading on General Relativity
• Introduction to Cosmology - Narlikar 1993
• Cosmology - The origin and Evolution of Cosmic
  StructureColes Lucchin 1995
• Principles of Modern Cosmology - Peebles 1993

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4.1 Introduction to General Relativity

• Newton v Einstein

• Newton
• Mass tells Gravity how to make a Force
• Force tells mass how to accelerate

• Newton
• Flat Euclidean Space
• Universal Frame of reference

• Einstein
• Mass tells space how to curve
• Space tells mass how to move

• Einstein
• Space can be curved
• Its all relative anyway!!

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4.1 Introduction to General Relativity

• The Principle of Equivalence

A more general interpretation led Einstein to his
theory of General Relativity
Implications of Principle of Equivalence

• Imagine 2 light beams in 2 boxes
• the first box being accelerated upwards
• the second in freefall in gravitational field
• For a box under acceleration
• Beam is bent downwards as box moves up
• For box in freefall ? photon path is bent down!!

• Fermats Principle
• Light travels the shortest distance between 2
  points
• Euclid straight line
• Under gravity - not straight line
• ? Space is not Euclidean !

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4.2 Geometry, Metrics Tensors

• The Interval (line element) in Euclidean Space

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4.2 Geometry, Metrics Tensors

• The Interval in Special Relativity (The Minkowski
  Metric)

• The Laws of Physics are the same for all inertial
  Observers (frames of constant velocity)
• The speed of light, c, is a constant for all
  inertial Observers
• Events are characterized by 4 co-ordinates
  (t,x,y,z)
• Length Contraction, Time Dilation, Mass increase
• Space and Time are linked
• The notion of SPACE-TIME

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4.2 Geometry, Metrics Tensors

• The Interval in Special Relativity (The Minkowski
  Metric)

Causally Connected Events in Minkowski Spacetime

• Area within light cone dS2gt0
• Events that can affect observer in
  past,present,future.
• This is a timelike interval.
• Observer can be present at 2 events by selecting
  an appropriate speed.

• Area outside light cone dS2lt0
• Events that are causally disconnected from
  observer.
• This is a spacelike interval.
• These events have no effect on observer.

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4.2 Geometry, Metrics Tensors

• The Interval in General Relativity

• In General

The interval is given by

• gij is the metric tensor (Riemannian Tensor)
• Tells us how to calculate the distance between 2
  points in any given spacetime
• Components of gij
• Multiplicative factors of differential
  displacements (dxi)
• Generalized Pythagorean Theorem

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4.2 Geometry, Metrics Tensors

• The Interval in General Relativity

Euclidean Metric
Minkowski Metric
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4.2 Geometry, Metrics Tensors

• A little bit about Tensors - 1

• Consider a matrix M(mij)
• Tensor arbitray number of indicies (rank)
• Scalar zeroth rank Tensor
• Vector 1st rank Tensor
• Matrix 2nd rank Tensor matrix is a tensor
  of type (1,1), i.e. mij
• Tensors can be categorized depending on certain
  transformation rules
• Covariant Tensor indices are low
• Contravariant Tensor indices are high
• Tensor can be mixed rank (made from covariant
  and contravariant indices)
• Euclidean 3D space Covariant Contravariant
  Cartesian Tensors
• ?? mijk? mijk

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4.2 Geometry, Metrics Tensors

• A little bit about Tensors - 2
• Covariant Tensor Transformation

Generalize for 2nd rank Tensors, Covariant 2nd
rank tensors are animals that transform as -
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4.2 Geometry, Metrics Tensors

• A little bit about Tensors - 3
• Contravariant Tensor Transformation

Generalize for 2nd rank Tensors, Contravariant
2nd rank tensors are animals that transform as -
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4.2 Geometry, Metrics Tensors

• More about Tensors - 4
• Index Gymnastics

• Raising and Lowering indices

• Einstein Summation

Repeated indices (in sub and superscript) are
implicitly summed over
Einstein c.1916 "I have made a great discovery
in mathematics I have suppressed the summation
sign every time that the summation must be made
over an index which occurs twice..."
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4.2 Geometry, Metrics Tensors

• More about Tensors - 5
• Tensor Manipulation

• Tensor Contraction

Set unlike indices equal and sum according to the
Einstein summation convention. Contraction
reduces the tensor rank by 2. For a second-rank
tensor this is equivalent to the Scalar Product
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4.2 Geometry, Metrics Tensors

• More about Tensors - 6
• Tensor Calculus

• The Covariant Derivative of a Tensor

Derivatives of a Scalar transform as a vector

• Christoffel Symbols
• Defines parallel vectors at neighbouring points
• Parallelism Property affine connection of S-T
• Christoffel Symbols fn(spacetime)

, normal derivative covariant derivative
Redefine the covariant derivative of a tensor as
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4.2 Geometry, Metrics Tensors

• Riemann Geometry

General Relativity formulated in the
non-Euclidean Riemann Geometry
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4.2 Geometry, Metrics Tensors

• The Riemann Tensor

Interchange differentiation w.r.t xn xk and use
Ai,nkAi,kn
Rimkn is the Riemann Tensor

• Defines geometry of spacetime (0 for flat
  spacetime).
• Has 256 components but reduces to 20 due to
  symmetries.

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4.2 Geometry, Metrics Tensors

• The Einstein Tensor

• Lowering the second index of the Riemann Tensor,
  define

• Contracting the Riemann Tensor gives the Ricci
  Tensor describing the curvature

• Contracting the Ricci Tensor gives the Scalar
  Curvature (Ricci Scalar)

Rimkn is the Riemann Tensor

• Define the Einstein Tensor as

The Einstein Tensor has ZERO divergence
or
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4.3 Einsteins Theory of Gravity

• The Energy Momentum Tensor

• Non relativistic particles (Dust)

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4.3 Einsteins Theory of Gravity

• The Energy Momentum Tensor

• Relativistic particles (inc. photons, neutrinos)

• Perfect Fluid

For any reference frame with fluid 4 velocity u
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4.3 Einsteins Theory of Gravity

• The Einstein Equation

Basically Relativistic Poisson Equation
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4.3 Einsteins Theory of Gravity

• The Einstein Equation

Matter tells space how to curve. Space tells
matter how to move.
Fabric of the Spacetime continuum and the energy
of the matter within it are interwoven
Einstein inserted the Cosmological Constant term L
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4.3 Einsteins Theory of Gravity

• Solutions of the metric

just spatial coords

• Cosmological Principle
• Isotropy ? ga,b da,b, only take ab terms
• Homogeneity ? g0,b g0,a 0

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4.3 Einsteins Theory of Gravity

• Solutions of the metric

Introduce spherical polar identities in 3 Space

1. Multiply spatial part by arbitrary function of
   time R(t) wont affect isotropy and
   homogeneity because only a f(t)
2. Absorb elemental length into R(t) ?r becomes
   dimensionless
3. Re-write a-2 k

The Robertson-Walker Metric

• r, q, f are co-moving coordinates and dont
  changed with time - They are SCALED by R(t)
• t is the cosmological proper time or cosmic time
  - measured by observer who sees universe
  expanding around him
• The co-ordinate, r, can be set such that k -1,
  0, 1

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4.3 Einsteins Theory of Gravity

• The Robertson-Walker Metric and the Geometry of
  the Universe

The Robertson-Walker Metric
k defines the curvature of space time
k 0 Flat Space
k -1 Hyperbolic Space
k 1 Spherical Space
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4.3 Einsteins Theory of Gravity

• The Robertson-Walker Metric and the Geometry of
  the Universe

k 0 Flat Space
k -1 Hyperbolic Space
k 1 Spherical Space
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4.3 Einsteins Theory of Gravity

• The Friedmann Equations in General Relativistic
  Cosmology

• Need
• metric gik
• energy tensor Tik

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4.3 Einsteins Theory of Gravity

• The Friedmann Equations in General Relativistic
  Cosmology

Sub ? Einstein
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4.3 Einsteins Theory of Gravity

• The Friedmann Equations in General Relativistic
  Cosmology

Non-zero components of Energy Tensor Tik
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4.3 Einsteins Theory of Gravity

• The Friedmann Equations in General Relativistic
  Cosmology

Our old friends the Friedmann Equations
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4.4 Summary

• The Friedmann Equations in General Relativistic
  Cosmology

• We have come along way today!!!

THESE WILL BE OUR TOOLBOX FOR OUR COSMOLOGICAL
STUDIES
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4.4 Summary
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Fundamental Cosmology 4. General Relativistic
Cosmology
Fundamental Cosmology 5. The Equation of state
the Evolution of the Universe
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