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
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
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