Asymptotic structure of space-time with a positive cosmological constant

Open Access
Kesavan, Aruna
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
June 02, 2016
Committee Members:
  • Abhay Vasant Ashtekar, Dissertation Advisor
  • Abhay Vasant Ashtekar, Committee Chair
  • Sarah Elizabeth Shandera, Committee Member
  • Eugenio Bianchi, Committee Member
  • Steinn Sigurdsson, Outside Member
  • Asymptotics
  • Dark Energy
  • Gravitational Radiation
  • Cosmological Constant
In general relativity a satisfactory framework for describing isolated systems exists when the cosmological constant $\Lambda$ is zero. The detailed analysis of the asymptotic structure of the gravitational field, which constitutes the framework of asymptotic flatness, lays the foundation for research in diverse areas in gravitational science. However, the framework is incomplete in two respects. First, asymptotic flatness provides well-defined expressions for physical observables such as energy and momentum as `charges' of asymptotic symmetries at null infinity, $\scri^+$. But the asymptotic symmetry group, called the Bondi-Metzner-Sachs group is infinite-dimensional and a tensorial expression for the `charge' integral of an arbitrary BMS element is missing. We address this issue by providing a charge formula which is a 2-sphere integral over fields local to the 2-sphere and refers to no extraneous structure. The second, and more significant shortcoming is that observations have established that $\Lambda$ is not zero but positive in our universe. Can the framework describing isolated systems and their gravitational radiation be extended to incorporate this fact? In this dissertation we show that, unfortunately, the standard framework does not extend from the $\Lambda =0$ case to the $\Lambda >0$ case in a physically useful manner. In particular, we do not have an invariant notion of gravitational waves in the non-linear regime, nor an analog of the Bondi `news tensor', nor positive energy theorems. In addition, we argue that the stronger boundary condition of conformal flatness of intrinsic metric on $\scri^+$, which reduces the asymptotic symmetry group from $\Diff$ to the de Sitter group, is insufficient to characterize gravitational fluxes and is physically unreasonable. To obtain guidance for the full non-linear theory with $\Lambda > 0$, linearized gravitational waves in de Sitter space-time are analyzed in detail. i) We show explicitly that conformal flatness of the boundary removes half the degrees of freedom of the gravitational field by hand and is not justified by physical considerations; ii) We obtain gauge invariant expressions of energy-momentum and angular momentum fluxes carried by gravitational waves in terms of fields defined at $\scrip$; iii) We demonstrate that the flux formulas reduce to the familiar ones in Minkowski space-time in spite of the fact that the limit $\Lambda \to 0$ is discontinuous (since, in particular, $\scrip$ changes its space-like character to null in the limit); iv) We obtain a generalization of Einstein's 1918 quadrupole formula for power emission by a linearized source to include a positive $\Lambda$; and, finally v) We show that, although energy of linearized gravitational waves can be arbitrarily negative in general, gravitational waves emitted by physically reasonable sources carry positive energy.