Unified treatment of null and spatial infinity IV: angular momentum at null and spatial infinity

Abstract

A bstract In a companion paper [1] we introduced the notion of asymptotically Minkowski spacetimes . These space-times are asymptotically flat at both null and spatial infinity, and furthermore there is a harmonious matching of limits of certain fields as one approaches i ° in null and space-like directions. These matching conditions are quite weak but suffice to reduce the asymptotic symmetry group to a Poincaré group $$ {\mathfrak{p}}_{i{}^{\circ}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> </mml:msub> </mml:math> . Restriction of $$ {\mathfrak{p}}_{i{}^{\circ}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> </mml:msub> </mml:math> to future null infinity $$ {\mathcal{I}}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> yields the canonical Poincaré subgroup $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> <mml:mi>bms</mml:mi> </mml:msubsup> </mml:math> of the BMS group $$ \mathfrak{B} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> selected in [2, 3] and that its restriction to spatial infinity i °, the canonical subgroup $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> <mml:mi>spi</mml:mi> </mml:msubsup> </mml:math> of the Spi group $$ \mathfrak{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> selected in [4, 5]. As a result, one can meaningfully compare angular momentum that has been defined at i ° using $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> <mml:mi>spi</mml:mi> </mml:msubsup> </mml:math> with that defined on $$ {\mathcal{I}}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> using $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> <mml:mi>bms</mml:mi> </mml:msubsup> </mml:math> . We show that the angular momentum charge at i ° equals the sum of the angular momentum charge at any 2-sphere cross-section S of $$ {\mathcal{I}}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> and the total flux of angular momentum radiated across the portion of $$ {\mathcal{I}}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> to the past of S . In general the balance law holds only when angular momentum refers to SO(3) subgroups of the Poincaré group $$ {\mathfrak{p}}_{i{}^{\circ}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>°</mml:mo> </mml:mrow> </mml:msub> </mml:math> .

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