PDF sources of the lectures 2nd, 3rd, 4th, 5th, 6th, 6bth=7th, 8th, 9th, 10th, 11th, 12th lecture
One of the most striking attribute of general relativity is that spacetime geometry is dynamical. Since the fundamental discovery by Yvone Choquet-Bruhat there have been considerable developments in studying the evolution of the geometry of the spacetime. In formulating these dynamical features one inevitably ends up with the hyperbolic formulations of Einstein's equations. These are solved in an initial value problem, also known as the Cauchy problem, of general relativity. The primary aim of the Cauchy problem is to show that there is a one-to-one correspondence between solutions and initial data specifications. The first part of my lecture series is to introduce the basics of the evolutionary aspects of general relativity.
These lectures are to be followed by the introduction of a set of geometrically distinguished variables which provide a remarkable flexibility in dealing with the constraint equations and the Cauchy problem of general relativity. In particular, this new approach allows of solving the constraints as evolutionary systems, and also identification of the true degrees of freedom in metric theories of gravity. The series of lectures will be completed by demonstrating the applicability of the proposed new method in solving various problems of physical interest. In proving the positive mass theorem and also the Penrose inequality in the generic setup. In demonstrating its use in numerical relativity, and, in particular, in constructing physically adequate initial data for binary black hole systems. Last but not least the use of the proposed new method in solving the initial-boundary value problem in a fully constrained and geometrically unique way will also be demonstrated.
More details at the homepage of FUW1. Wald R M: General relativity, University of Chicago Press, Chicago (1984)
2. Choquet-Bruhat Y: General relativity and Einstein's equations, Oxford University Press (2009)
3. Ringstrom H: The Cauchy Problem in General Relativity (ESI Lectures in Mathematics and Physics)-European Mathematical Society (2009)
Some additional references:4. Rácz, I.: Is the Bianchi identity always hyperbolic?, CQG 31 155004 (2014)
5. I. Rácz: Cauchy problem as a two-surface based `geometrodynamics', Class. Quantum Grav. 32 015006 (2015)
6. I. Rácz: Dynamical determination of the gravitational degrees of freedom, arXiv:1412.0667 (2015)
7. I. Rácz: Constraints as evolutionary systems, Class. Quantum Grav. 33 015014 (2016)
8. I. Rácz and J. Winicour: Black hole initial data without elliptic equations, Phys. Rev. D 91 , 124013 (2015)
9. I. Rácz and J. Winicour: On solving the constraints by integrating a strongly hyperbolic system, arXiv:1601.05386
10. I. Rácz: A simple method of constructing binary black hole initial data, Astronomy Reports 62 , 953-958 (2018)
11. I. Rácz: On the ADM charges of multiple black holes, arXiv:1608.02283