Relativistic Discs

Selfgravitating, rotating, relativistic discs may have played an important role in the early universe where these objects are possible predecessors of the centres of galaxies.

I have studied the structure of rotating, self-gravitating discs by means of numerical methods where an infinitesimally thin disc in otherwise empty space is considered. The analysis covered stationary solutions for the Newtonian and fully relativistic case and also a time-dependent analysis within the first post-Newtonian approximation.

  • Stationary Solutions

      Infinitesimally thin configurations of discs with internal (two-dimensional) pressure are studied. These discs can be considered as the limit of three-dimensional bodies, for example after a vertical integration. For a Newtonian object with a constant (3D) mass density one obtains after vertical integration the classical MacLaurin discs.

      Using a polytropic relation of the surface pressure and surface density, equilibrium sequences of rotating discs were studied. Since the disc is infinitesimally thin its gravitational potential is obtained by solving the vacuum field equations where the surface density of the disc acts only as a boundary condition. The complete solution of the problem requires the iterative solution of the non-linear hydrostatic equation simultaneously with the field equations. For a given central redshift z sequences of disc solutions for varying rotation rate can be constructed. Only rigidly rotating discs were considered here.

      Already the Newtonian case (MNRAS 282, 234 (1996)) revealed that in addition to the standard MacLaurin disc solutions a second solution branch exists, which consists of a self-gravitating ring without central object. The bifurcation proceeds through a sequence of dumbbell shaped density distributions. It is completely analogous to the three-dimensional case of rigidly rotating homogeneous (MacLaurin) ellipsoids.

      In the fully relativistic calculation (MNRAS 287, 26 (1997)) the similar feature of ring formation was found for discs with small central redshifts. For larger z the disc sequences end at the so called mass-shed limit where the outer radius of the disc is just marginally bound to the system.

  • Oscillating Solutions

      Time dependent calculations of oscillating discs in full general relativity are highly complex. To simplify the problem we studied radial oscillations of a disc of self gravitating non-interacting (dust) particles in the first Post-Newtonian approximation. The corresponding Newtonian problem was solved by Hunter (MNRAS, 129, 321, 1965) by assuming a self-similar form of the solution. Assuming rigid rotation throughout, the equation of motion for the outer radius of the disc is identical to the equation for the radial distance in the standard Keplerian two-body problem.

      In the post-Newtonian case all equations are augmented by terms of the order (1/c**2), and three additional gravitational "potentials" have to be calculated. The ansatz of a modified similarity solution yields also in this case to a second order differential equation for the radius of the disc. This is now equivalent to the first post-Newtonian equation of motion for a binary system. Thus, the solution of the disc can be written in parameterized form using two eccentricities.

      This work was performed together with Gerhard Schäfer and is published in (PRD, 50, 6217, 1994). The form of the gravitational waves emitted from such oscillating discs are calculated in (Schäfer & Kley, PRD, 50, 6227, 1994)

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