[Paper Review] Radiation and Boundary Conditions in the Theory of Gravitation

Andrzej Trautman|arXiv (Cornell University)|Apr 11, 2016

Cosmology and Gravitation Theories2 references48 citations

TL;DR

This paper proposes a generalized boundary condition framework for gravitational radiation in general relativity, extending Einstein's energy-momentum pseudotensor formalism to allow for radiative fields. By introducing null vector fields and asymptotic conditions on metric derivatives, it ensures finite, coordinate-invariant energy-momentum integrals and defines radiated energy via the difference in total energy between spacelike hypersurfaces, with a key result that radiated energy is non-negative and well-defined in the wave zone.

ABSTRACT

The Sommerfeld boundary conditions, applied to an asymptotically weak gravitational field, are shown to imply that the 1/r part of the curvature tensor of a space-time, satisfying the Einstein equations, is of type null in the Petrov classification and that there is then a flux of energy carried away by the outgoing gravitational wave.

Motivation & Objective

• To resolve the ambiguity in defining total energy and momentum in general relativity for radiating systems by extending boundary conditions beyond the standard Lichnerowicz conditions.
• To formulate a consistent prescription for computing the total energy radiated by isolated gravitational systems, particularly in the presence of outgoing radiation.
• To ensure that the energy-momentum integrals remain finite and invariant under coordinate transformations that preserve the asymptotic structure at spatial infinity.
• To connect the proposed radiation conditions with existing definitions of pure radiation fields by Pirani and Lichnerowicz, showing consistency in the wave zone.

Proposed method

• Introduce a null vector field $ k^ u = n^ u + t^ u $, where $ n^ u $ is a unit spacelike normal to spatial hypersurfaces and $ t^ u $ is a unit timelike normal, to define the asymptotic direction of radiation.
• Propose generalized boundary conditions: $ g_{ u au} = ar{g}_{ u au} + h_{ u au} $, with $ h_{ u au} = O(r^{-1}) $, and $ h_{ u au, ho} o b_{ u, au}k_ ho + O(r^{-2}) $, ensuring asymptotic behavior compatible with radiation.
• Use the superpotential formalism $ ar{rak{A}}_{ u}^{ ho au} $ derived from the Einstein tensor to define the energy-momentum pseudotensor $ rak{t}_{ u}^{ ho} $, ensuring divergencelessness via Einstein's equations.
• Define the total energy-momentum $ P_ u[ au] $ via surface integrals over spacelike hypersurfaces $ au $, and the radiated energy as the difference $ p_ u = P_ u[ au] - P_ u[ au'] $, computed via the flux integral over a timelike hypersurface $ ar{ au} $.
• Establish invariance of $ P_ u $ under coordinate transformations that preserve the boundary conditions, using transformation rules for $ g_{ u au} $ and $ h_{ u au} $, showing $ rak{A}'_{ u}{}^{ ho au}k'_{ ho}n'_{ au} = rak{A}_{ u}{}^{ ho au}k_{ ho}n_{ au} + O(r^{-3}) $.
• Derive the leading-order behavior of the pseudotensor: $ rak{t}_{ u}^{ ho} = au k_ u k^ ho + O(r^{-3}) $, with $ 4 auar{ ho} = h^{ ho au}(h_{ ho au} - rac{1}{2}ar{ ho}^{ ho au}h_{ ho au}) $, ensuring non-negative radiated energy.

Experimental results

Research questions

• RQ1 How can boundary conditions in general relativity be generalized to consistently describe gravitational radiation fields, beyond the standard $ O(r^{-1}) $ fall-off?
• RQ2 What conditions ensure the finiteness and coordinate invariance of the total energy-momentum integral $ P_ u[ au] $ in the presence of outgoing radiation?
• RQ3 How does the proposed formalism relate to the definitions of pure radiation fields by Pirani and Lichnerowicz, particularly in the asymptotic wave zone?
• RQ4 Can the total radiated energy be rigorously defined as the difference in total energy between two spacelike hypersurfaces, and is it non-negative?
• RQ5 What is the role of the null vector field $ k^ u $ in characterizing the direction and structure of gravitational radiation at infinity?

Key findings

• The proposed boundary conditions, involving $ h_{ u au} = O(r^{-1}) $ and $ h_{ u au, ho} o b_{ u, au}k_ ho + O(r^{-2}) $, allow for asymptotic radiation fields while preserving the finiteness and invariance of $ P_ u[ au] $ under admissible coordinate transformations.
• The total energy-momentum $ P_ u[ au] $ is well-defined and independent of the choice of hypersurface $ au $, provided the generalized boundary conditions are satisfied, and remains invariant under coordinate changes that reduce to identity at infinity.
• The radiated energy $ p_ u = P_ u[ au] - P_ u[ au'] $ is given by the flux integral $ ar{p}_ u = ar{ au} k_ u k^ u $, with $ ar{ au} o O(r^{-2}) $, and is non-negative due to the positivity of $ au $, ensuring physical consistency.
• In the wave zone, the curvature tensor behaves as $ R_{ u au ho heta} o rac{1}{2}k_{[ u}i_{ au][ ho}k_{ heta]} $, with $ i_{ u au} = O(r^{-1}) $, and the Weyl tensor is of Petrov-Pirani type II, consistent with pure radiation fields.
• The conditions $ R_{ u au} o ho k_ u k_ au + O(r^{-3}) $ and $ k^ u R_{ u au ho heta} o 0 $, $ k^{[ u}R^{ ho] au} ho heta o 0 $, match Lichnerowicz’s definition of pure radiation fields in the limit $ r o ty $, showing consistency with established formalisms.
• The formalism remains valid in the presence of electromagnetic fields, where $ ar{ au} = O(r^{-2}) $, and the radiated energy remains non-negative, confirming robustness of the approach.