Introduction to General Relativity Lectures by Pietro Fré
to General Relativity
Lectures by Pietro Fré
Virgo Site May 26th 2003
The issue of reference frames and observers
The Copernican Revolution....
Seen from the EARTH
Actually things are worse than that..
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 Newton’s law applies, namely where
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
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....
Equivalence Principle: a first approach
This is the Elevator Gedanken
Experiment of Einstein
G.R. model of the physical world
of any physical physical phenomenon constitute an
The set of all events is a continuous space, named
are manifestations of the
move in space—time following special world-lines that are
laws of physics
are the same for
Hence the mathematical model of space time is a pair:
Gluing together a Manifold: the example of the sphere
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
Differentiable structure continued....
Tangent spaces and vector fields
The difference between flat and curved manifolds
To see the real effect of curvature we must consider.....
On a sphere
The hyperboloid: a space with negative curvature
and lorentzian signature
The metric: a rule to calculate the lenght of curves!!
Underlying our rule for lengths is the induced metric:
What do particles do in a gravitational field?
What are the straight lines
Let us see what are the straight lines (=geodesics) on the Hyperboloid
Deriving the geodesics from a variational principle
The Euler Lagrange equations are
Let us now review the general case
the Christoffel symbols are:
and covariant derivative
In a basis...
Torsion and Curvature
If we have a metric........
Now we can state the.......
Harmonic Coordinates and the exponential map
A view of the locally inertial frame
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.
The vielbein or Repère Mobile
The vielbein encodes the metric
Using the standard formulae for the curvature 2-form:
The Bianchi Identities
Bianchi’s and the Einstein tensor
It suffices that the field equations be of the form:
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!!!
A fundamental example:
the Schwarzschild solution
Finding the solution
The Schwarzschild metric and its orbits
Energy & Angular Momentum
The effects: Periastron Advance
Bending of Light rays
More to come in next lectures....
Thank you for your attention
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