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....



Manifolds



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



Continuing...



Still continuing



Space-like



Time-like



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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