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.




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.




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

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


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