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Asymptotics with a positive cosmological constant: II. Linear fields on de Sitter space-time

Abhay Ashtekar, Béatrice Bonga, Aruna Kesavan

TL;DR

Problem: develop a framework for gravitational radiation with a positive cosmological constant. Approach: analyze linearized gravitational waves on the de Sitter Poincaré patch, construct a covariant phase space, and derive Hamiltonians for seven preserved symmetries to define fluxes at $\mathcal{I}^+$. Key findings: conformal flatness of $\mathcal{I}^+$ is physically untenable; fluxes can be expressed in terms of data at $\mathcal{I}^+$; energy can be negative in general but is positive for physically realistic sources; the $\Lambda\to 0$ limit reproduces Minkowski results despite discontinuities. Significance: lays groundwork for a nonlinear theory of gravity with $\Lambda>0$ and informs the reliability of $\Lambda=0$ approximations in gravitational-wave astrophysics.

Abstract

Linearized gravitational waves in de Sitter space-time are analyzed in detail to obtain guidance for constructing the theory of gravitational radiation in presence of a positive cosmological constant in full, nonlinear general relativity. Specifically: i) In the exact theory, the intrinsic geometry of $\scri$ is often assumed to be conformally flat in order to reduce the asymptotic symmetry group from $\Diff$ to the de Sitter group. Our {results show explicitly} that this condition is physically unreasonable; ii) We obtain expressions of energy-momentum and angular momentum fluxes carried by gravitational waves in terms of fields defined at $\scrip$; iii) We argue that, although energy of linearized gravitational waves can be arbitrarily negative in general, gravitational waves emitted by physically reasonable sources carry positive energy; and, finally iv) We demonstrate that the flux formulas reduce to the familiar ones in Minkowski space-time in spite of the fact that the limit $Λ\to 0$ is discontinuous (since, in particular, $\scri$ changes its space-like character to null in the limit).

Asymptotics with a positive cosmological constant: II. Linear fields on de Sitter space-time

TL;DR

Problem: develop a framework for gravitational radiation with a positive cosmological constant. Approach: analyze linearized gravitational waves on the de Sitter Poincaré patch, construct a covariant phase space, and derive Hamiltonians for seven preserved symmetries to define fluxes at . Key findings: conformal flatness of is physically untenable; fluxes can be expressed in terms of data at ; energy can be negative in general but is positive for physically realistic sources; the limit reproduces Minkowski results despite discontinuities. Significance: lays groundwork for a nonlinear theory of gravity with and informs the reliability of approximations in gravitational-wave astrophysics.

Abstract

Linearized gravitational waves in de Sitter space-time are analyzed in detail to obtain guidance for constructing the theory of gravitational radiation in presence of a positive cosmological constant in full, nonlinear general relativity. Specifically: i) In the exact theory, the intrinsic geometry of is often assumed to be conformally flat in order to reduce the asymptotic symmetry group from to the de Sitter group. Our {results show explicitly} that this condition is physically unreasonable; ii) We obtain expressions of energy-momentum and angular momentum fluxes carried by gravitational waves in terms of fields defined at ; iii) We argue that, although energy of linearized gravitational waves can be arbitrarily negative in general, gravitational waves emitted by physically reasonable sources carry positive energy; and, finally iv) We demonstrate that the flux formulas reduce to the familiar ones in Minkowski space-time in spite of the fact that the limit is discontinuous (since, in particular, changes its space-like character to null in the limit).

Paper Structure

This paper contains 11 sections, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Poincaré patch
  4. Maxwell fields in de Sitter space-time
  5. Linearized gravitational fields
  6. The Hamiltonian framework
  7. The covariant phase space
  8. 3-momentum, angular momentum and energy carried by gravitational waves
  9. 3-momentum and angular momentum
  10. Energy
  11. Discussion

Figures (2)

  • Figure 1: Left Panel: The Penrose diagram of a spherical isolated star in general relativity with $\Lambda>0$. The solid diagonal line denotes $E^{+}(i^{-})$, the future event horizon of $i^{-}$. The star and the radiation it emits are invisible to all observers whose world-lines are confined to the lower portion of the de Sitter space-time below $E^{+}(i^{-})$. Therefore in the discussion of this isolated system, it is natural to restrict oneself to the upper half. The dashed diagonal line is $E^{-}(i^{+})$, the past event horizon of $i^{+}$. Right Panel: The Poincaré patch of de Sitter space-time of interest is the upper triangle, to the future of the event horizon $E^{+}(i^{-})$ where $\eta=-\infty$. The $\eta= {\rm const}$ lines denote the cosmological slices, i.e., flat Cauchy surfaces.
  • Figure 2: Left Panel: Integral curves of the time translation Killing field $T^{a}$ in the Poincaré patch. $T^{a}$ is future directed and time like in region I and space-like in region II. It is future directed and null on portion of the event horizon $E^{-}(i^{+})$ to the future of the cross-over 2-sphere (bifurcate horizon) $C$ and on the portion of the null event horizon $E^{+}(i^{-})$ to the past of $C$. It is past directed and null on the portion of $E^{+}(i^{-})$ to the future of $C$. Right Panel: A time changing quadrupole emitting gravitational waves. Radiation crossing the cosmological horizon $E^{-}(i^{+})$ and reaching $\mathcal{I}^{+}$ originates in region I; there is no incoming radiation on $E^{+}(i^{-})$.