*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** |