**
Theoretical Physics**

Prof. Remo RUFFINI

**
First Part**

**
Newtonian equilibrium systems:
**Politropic
Equations of state. Equations of Lane-Emden.Numerical and analitic solutions of
the Lane Emden equations. Scaling laws in the solutions: homoteties. Isothermal
configurations. Self gravitating systems in uniform rotation. Self gravitating
systems with uniform vorticity. The solutions of Mac Laurin, Jacobi, Dedekind,
Riemann. Inhomogeneous self gravitating systems with rotation and vorticity. The
virial theorem of order *
n*.
A functional approach to the configurations of equilibrium.

**
Equilibrium configurations of
fermions and bosons: **
The Thomas-Fermi atom. Thomas-Fermi
atoms at finite temperature. Self gravitating systems of fermions at zero and at
finite temperature. The concept of cut-off in the phase space. Self gravitating
systems of bosons at zero temperature and at finite temperature. White dwarfs
and neutron stars. Scaling laws. The concept of critical mass.

**
Newton gravitational theory:
**Tests of inverse
square law. The gravitational potential. Gravitational multipoles. The
equivalence of inertial and gravitational mass and its experimental verification,
on the ground and in space. Tidal forces.

**
Special relativity: **
The principle of special
relativity. Lorentz transformations. The Minkowski space. Relativistic invariant
field equations. The relativistically invariant formulation of the motion of a
free particle. Geodesics. The relativistically invariant formulation of the
field equations for a free field: the case of the electric field. The
electromagnetic interactions: the Maxwell equations and the Proca equations.
Doppler shift. The equations of motion of a charged particle. Derivation of the
Maxwell equations from a Lagrangian formulation.

**
Relativistically invariant
theory of Gravitation: **
The linear field equations of
gravitation. The interaction of gravitation and matter. The variational
principle and the equation of motion. The non relativistic limit and newton
theory. The geometric interpretation. Curved space-time.

**
Applications of the linear
theory: **The field
of a spherical mass. The gravitational time dilatation. The deflection of light.
The retardation of light. Gravitational lenses. The field of a rotating mass.
The Lense-Thirring effect.

**
Gravitational waves:
**Plane waves. The emission
of gravitational waves. Emission by a vibrating quadrupole. Emission by a
rotating quadrupole. Emission of bursts of gravitational radiation. The
quadrupole detector and its cross section. Experiments with Detectors of
gravitational radiation.

**
References:**

1. |
H. Ohanian, R. Ruffini, “Gravitation and Space-Time”, W. W. Norton, also Zanichelli |

2. |
J. Jackson, “Classical Electrodynamics”, Wiley |

3. |
H. Gursky, R. Ruffini, “Neutron Stars, Black Holes and Binary X-Ray Sources”, Reidel |

4. |
L. Landau, Y. Lifshitz, “Teoria dei campi”, Editori riuniti |

5. |
S. Chandrasekhar, “An introduction to the study of stellar structures”, Dover |

6. |
S. Chandrasekhar, “On ellipsoidal figures of Equilibrium”, Dover |

7. |
S. Filippi, R. Ruffini, A. Sepulveda, “Self Gravitating systems with rotation and vorticity” (in preparation) |

**
Second Part**

**
Riemannian geometry:
**General coordinates and
tensors. Affine Spaces. Parallel transport: the covariant derivative. The affine
geodesic equations. The Riemann tensor. The metric spaces. The metric geodesic
equations. Geodesic deviation and tidal fields. Isometries of

space time: killing vectors. Conserved quantities.

**
The principle of general
relativity: **
Einstein field equations. Variational principles. The Palatini approach.
Stationary and static space-times. Solutions with spherical symmetry. The
Birkhoff theorem. The Schwarschild solution. The motion of planets: the
perihelion

precession. The Hamilton Jacobi Equation. Positive and negative energy states. The propagation of light: the gravitational red shift. Geodetic precession. The energy momentum tensor. The relativistic equations of equilibrium of a star. The Tolmann-Oppenheimer-Volkoff equation of equilibrium. Stabilty of the equilibrium configurations. Eigenfrequencies of pulsation of a star. The maximum mass of a Neutron Star.

**
Singularities and pseudo
singularities: **The
black hole and the horizon. Kruskall diagrams. The maximal extension of a
Schwarschild geometry. The Reisnner-Nordström solution. The Kerr solution. The
Kerr- Newmann solution. Capture of particles from black holes. Effective
potential. Circular orbits. The positive and negative energy states. CPT.
Horizons and singularities in the Reissner-Nordström black hole. Horizons and
singularities in the Kerr and Kerr-Newmann geometries. Electric and magnetic
fields of black holes. Black holes in our galaxy. The paradigm of identification
of Black Holes.

**
The dynamical formation of a
Black Hole: **The
Christodoulou-Ruffini mass-energy formula of Black Holes. Collapseng shells .The
formula for the irreducible mass of a black hole. Upper limit on the energy
extractable from a Black Hole. Process of vacuum polarization in a Black Hole.
The relativistic hydrodynamics equations and the Taub Equations. Observational
consequences and Gamma ray Bursts.

**
Gravitational waves emission in
the field of Black Holes: **
Radial infall and circular
orbits. Emission of electromagnetic radiation in the field of black holes.
Gravitational induced electromagnetic radiation. Electromagnetic induced
gravitational radiation.

**
Cosmology: **
The Einstein universes. The
Godel universe. The large scale structure of the Universe. Cosmic distances. The
homogeneous universes. The Bianchi models. The Friedmann universe. The
cosmological redshift. Hubble law. Gamow cosmology. The Fermi-Turkevich
nucleosynthesis computations. Cosmological nucleosynthesis. The cosmic
background radiation. The mass density. Dark matter. Galaxies rotation curves.
Propagation of light. The particles horizon.

**
Early universe and structures
formation: **The
temperature of the early universe. Particles in thermal equilibrium. Neutrinos
decoupling. Densities perturbations. The role of massive neutrinos in cosmology.
The fractal structure of the universe. The upper cut-off to the fractal
structure. The cellular structure of the universe.

**
References:**

1. |
H. Ohanian, R. Ruffini, “Gravitation and Space-Time”, W. W. Norton, also Zanichelli |

2. |
R. Giacconi, R. Ruffini, “Physics and astrophysics of neutron stars and black holes”, North Holland |

3. |
L. Landau, Y. Lifshitz, “Teoria dei campi”, Editori riuniti |

4. |
C. Sigismondi and R. Ruffini, “Nonlinear gravitodynamics”, World Scientific |

5. |
V. Gurzadyan and R. Ruffini, “Fermi and Astrophysics”, World Scientific |

6. |
S. Weinberg, “Gravitation and Cosmology”, Wiley |

7. |
C. Misner, K. Thorne, J.A. Wheeler, “Gravitation”, Freeman |