Introduction to General Relativity Lectures by Pietro Fré

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Introduction 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

  • 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 space—time

  • Point-like particles move in space—time following special world-lines that are “straight”

  • The laws of physics are the same for all observers

Hence the mathematical model of space time is a pair:

Manifolds are:

Open Charts:

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

Differentiable structure continued....


Tangent spaces and vector fields

Parallel Transport

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


Still continuing



Light like

Let us now review the general case

the Christoffel symbols are:

Connection 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 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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