Introduction to General Relativity

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Introduction to General Relativity

• Lectures by Pietro Fré
• Virgo Site May 26th 2003

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The issue of reference frames and observers
Since oldest antiquity the humans have looked at
the sky and at the motion of the Sun, the Moon
and the Planets. Obviously they always did it
from their reference frame, namely from the
EARTH, which is not at rest, neither in
rectilinear motion with constant velocity!
Who is at motion? The Sun or the Earth? A famous
question with a lot of history behind it
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The Copernican Revolution....
According to Copernican and Keplerian theory ,
the orbits of Planets are Ellipses with the Sun
in a focal point. Such elliptical orbits are
explained by NEWTONs THEORY of GRAVITY
But Newtons Theory works if we choose the
Reference frame of the SUN. If we used the
reference frame of the EARTH, as the ancient
always did, then Newtons law could not be
applied in its simple form
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Seen from the EARTH
The orbit of a Planet is much more complicated
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Actually things are worse than that..

• The true orbits of planets, even if seen from
  the SUN are not ellipses. They are rather curves
  of this type

For the planet Mercury it is
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Were Ptolemy and the ancients so much wrong?

• Who is right Ptolemy or Copernicus?
• We all learned that Copernicus was right
• But is that so obvious?
• The right reference frame is defined as that
  where Newtons law applies, namely where

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Classical Physics is founded.......

• on circular reasoning
• We have fundamental laws of Nature that apply
  only in special reference frames, the inertial
  ones
• How are the inertial frames defined?
• As those where the fundamental laws of Nature
  apply

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The idea of General Covariance

• It would be better if Natural Laws were
  formulated the same in whatever reference frame
• Whether we rotate with respect to distant
  galaxies or they rotate should not matter for the
  form of the Laws of Nature
• To agree with this idea we have to cast Laws of
  Nature into the language of geometry....

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Equivalence Principle a first approach
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This is the Elevator Gedanken Experiment of
Einstein
There is no way to decide whether we are in an
accelerated frame or immersed in a locally
constant gravitational field
The word local is crucial in this context!!
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G.R. model of the physical world
Physics
Geometry

• The when and the where of any physical physical
  phenomenon constitute an event.
• The set of all events is a continuous space,
  named space-time
• Gravitational phenomena are manifestations of the
  geometry of spacetime
• Point-like particles move in spacetime following
  special world-lines that are straight
• The laws of physics are the same for all
  observers

• An event is a point in a topological space
• Space-time is a differentiable manifold M
• The gravitational field is a metric g on M
• Straight lines are geodesics
• Field equations are generally covariant under
  diffeomorphisms

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Hence the mathematical model of space time is a
pair
Differentiable Manifold
Metric
We need to review these two fundamental concepts
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Manifolds are
Topological spaces whose points can be labeled by
coordinates. Sometimes they can be globally
defined by some property. For instance as
algebraic loci
In general, however, they can be built, only by
patching together an Atlas of open charts
The concept of an Open Chart is the Mathematical
formulation of a local Reference Frame. Let us
review it
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Open Charts
The same point ( event) is contained in more
than one open chart. Its description in one chart
is related to its description in another chart by
a transition function
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Gluing together a Manifold the example of the
sphere
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We can now address the proper Mathematical
definitions

• First one defines a Differentiable structure
  through an Atlas of open Charts
• Next one defines a Manifold as a topological
  space endowed with a Differentiable structure

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Differentiable structure
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Differentiable structure continued....
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Manifolds
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Tangent spaces and vector fields
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Parallel Transport
A vector field is parallel transported along a
curve, when it mantains a constant angle with the
tangent vector to the curve
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The difference between flat and curved manifolds
In a flat manifold, while transported, the vector
is not rotated.
In a curved manifold it is rotated
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To see the real effect of curvature we must
consider.....
Parallel transport along LOOPS
After transport along a loop, the vector does not
come back to the original position but it is
rotated of some angle.
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On a sphere
The sum of the internal angles of a triangle is
larger than 1800 This means that the curvature is
positive
How are the sides of the this traingle
drawn? They are arcs of maximal circles, namely
geodesics for this manifold
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The hyperboloid a space with negative curvature
and lorentzian signature
This surface is the locus of points satisfying
the equation
We can solve the equation parametrically by
setting
Then we obtain the induced metric
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The metric a rule to calculate the lenght of
curves!!
A curve on the surface is described by giving the
coordinates as functions of a single parameter t
This integral is a rule ! Any such rule is a
Gravitational Field!!!!
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Underlying our rule for lengths is the induced
metric
Where a and q are the coordinates of our space.
This is a Lorentzian metric and it is just
induced by the flat Lorentzian metric in three
dimensions
using the parametric solution for X0 , X1 ,
X2
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What do particles do in a gravitational field?
Answer They just go straight as in empty
space!!!!
It is the concept of straight line that is
modified by the presence of gravity!!!!
The metaphor of Eddingtons sheet summarizes
General Relativity. In curved space straight
lines are different from straight lines in flat
space!! The red line followed by the ball
falling in the throat is a straight line
(geodesics). On the other hand space-time is
bended under the weight of matter moving inside
it!
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What are the straight lines
They are the geodesics, curves that do not change
length under small deformations. These are the
curves along which we have parallel transported
our vectors
On a sphere geodesics are maximal circles
In the parallel transport the angle with the
tangent vector remains fixed. On geodesics the
tangent vector is transported parallel to itself.
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Let us see what are the straight lines
(geodesics) on the Hyperboloid

• ds2 lt 0 space-like geodesics cannot be
  followed by any particle (it would travel faster
  than light)
• ds2 gt 0 time-like geodesics. It is a possible
  worldline for a massive particle!
• ds2 0 light-like geodesics. It is a possible
  world-line for a massless particle like a photon

Three different types of geodesics
Is the rule to calculate lengths
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Deriving the geodesics from a variational
principle
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The Euler Lagrange equations are
The conserved quantity p is, in the time-like or
null-like cases, the energy of the particle
travelling on the geodesic
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Continuing...
This procedure to obtain the differential
equation of orbits extends from our toy model in
two dimensions to more realistic cases in four
dimensions it is quite general
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Still continuing
Let us now study the shapes and properties of
these curves
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Space-like
These curves lie on the hyperboloid and are
space-like. They stretch from megative to
positive infinity. They turn a little bit around
the throat but they never make a complete loop
around it . They are characterized by their
inclination p. This latter is a constant of
motion, a first integral
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Time-like
These curves lie on the hyperboloid and they can
wind around the throat. They never extend up to
infinity. They are also labeld by a first
integral of the motion, E, that we can identify
with the energy
Here we see a possible danger for
causality Closed time-like curves!
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Light like
These curves lie on the hyperboloid , are
straight lines and are characterized by a first
integral of the motion which is the angle shift
a
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Let us now review the general case
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the Christoffel symbols are
wherefrom do they emerge and what is their
meaning?
They are the coefficients of an affine
connection, namely the proper mathematical
concept underlying the concept of parallel
transport.
ANSWER
Let us review the concept of connection
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Connection and covariant derivative
A connection is a map
From the product of the tangent bundle with
itself to the tangent bundle
with defining properties
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In a basis...
This defines the covariant derivative of a
(controvariant) vector field
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Torsion and Curvature
Torsion Tensor
Curvature Tensor
The Riemann curvature tensor
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If we have a metric........
An affine connection, namely a rule for the
parallel transport can be arbitrarily given, but
if we have a metric, then this induces a
canonical special connection THE LEVI CIVITA
CONNECTION
This connection is the one which emerges from the
variational principle of geodesics!!!!!
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Now we can state the.......
Appropriate formulation of the Equivalence
Principle
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Harmonic Coordinates and the exponential map
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A view of the locally inertial frame
The geodesic equation, by definition, reduces in
this frame to
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The structure of Einstein Equations

• We need first to set down the items entering the
  equations
• We use the Vielbein formalism which is simpler,
  allows G.R. to include fermions and is closer in
  spirit to the Equivalence Principle
• I will stress the relevance of Bianchi identities
  in order to single out the field equations that
  are physically correct.

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The vielbein or Repère Mobile
We can construct the family of locally inertial
frames attached to each point of the manifold
q
p
M
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The vielbein encodes the metric
Indeed we can write
Mathematically the vielbein is part of a
connection on a Poincarè bundle, namely it is
like part of a YangMills gauge field for a gauge
theory with the Poincaré group as gauge group
This 1- form substitutes the affine connection
Poincaré connection
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Using the standard formulae for the curvature
2-form
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The Bianchi Identities
The Bianchis play a fundamental role in building
the physically correct field equations. It is
relying on them that we can construct a tensor
containing the 2nd derivatives of the metric,
with the same number of components as the metric
and fulfilling a conservation equation
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Bianchis and the Einstein tensor
Allows for the conservation of the stress energy
tensor
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It suffices that the field equations be of the
form

• Source of gravity in Newtons theory is the mass
• In Relativity mass and energy are
  interchangeable. Hence Energy must be the source
  of gravity.
• Energy is not a scalar, it is the 0th component
  of 4-momentum. Hence 4momentum must be the
  source of gravity
• The current of 4momentum is the stress energy
  tensor. It has just so many components as the
  metric!!
• Einstein tensor is the unique tensor, quadratic
  in derivatives of the metric that couples to
  stress-energy tensor consistently

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TORSION EQUATION
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Action Principle
plus the action of matter
 
where
Lagrangian density of matter being a 4-form
EINSTEIN EQUATION
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We have shown that.......

• The vanishing of the torsion and the choice of
  the Levi Civita connection is the yield of
  variational field equation
• The Einstein equation for the metric is also a
  yield of the same variational equation
• In the presence of matter both equations are
  modified by source terms.
• In particular Torsion is modified by the presence
  of spinor matter, if any, namely matter that
  couples to the spin connection!!!

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A fundamental example the Schwarzschild solution
Using standard polar coordinates plus the time
coordinate t
Is the most general static and spherical
symmetric metric
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Finding the solution
And from this, in few straightforward steps we
obtain the EINSTEIN TENSOR
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The solution
This yields the final form of the Schwarzschild
solution
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The Schwarzschild metric and its orbits
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Energy Angular Momentum
Newtonian Potential. Is present for time-like but
not for null-like
Centrifugal barrier
G.R. ATTRACTIVE TERM RESPONSIBLE FOR NEW EFFECTS
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The effects Periastron Advance
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Bending of Light rays
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More to come in next lectures....Thank you for
your attention