Jump to ContentJump to Main Navigation
Advanced General RelativityGravity Waves, Spinning Particles, and Black Holes$

Claude Barrabès and Peter A. Hogan

Print publication date: 2013

Print ISBN-13: 9780199680696

Published to Oxford Scholarship Online: September 2013

DOI: 10.1093/acprof:oso/9780199680696.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 23 March 2017

(p.135) Appendix C Some useful scalar products

(p.135) Appendix C Some useful scalar products

Source:
Advanced General Relativity
Publisher:
Oxford University Press

For Chapter 3 the following list of scalar products involving the vector field U i defined in (3.114) are useful in calculating the line-element (3.119):

(C.1)
k i ξ U i ξ = 0 ,
(C.2)
k i η U i ξ = P 0 2 F ,
(C.3)
v i U i ξ = 0 ,
(C.4)
k i U i ξ = F η ,
(C.5)
k i η U i η = 0 ,
(C.6)
k i ξ U i η = P 0 2 F ,
(C.7)
v i U i η = 0 ,
(C.8)
k i U i η = F ξ ,
(C.9)
U i u U i u = F 2 P 0 2 ( ( h 0 ξ ) 2 + ( h 0 η ) 2 ) ,
(C.10)
v i U i u = P 0 2 ( h 0 ξ F η h 0 η F ξ ) ,
(C.11)
U i u U i ξ = P 0 2 ( h 0 ξ F η h 0 η F ξ ) F η h 0 ξ F 2 ,
(C.12)
U i u U i η = P 0 2 ( h 0 ξ F η h 0 η F ξ ) F ξ h 0 η F 2 ,
(C.13)
k i U i u = 0 ,
(C.14)
k i ξ U i u = h 0 η F ,
(p.136)
(C.15)
k i η U i u = h 0 ξ F ,
(C.16)
U i ξ U i ξ = P 0 2 F 2 F η 2 ,
(C.17)
U i η U i η = P 0 2 F 2 F ξ 2 ,
(C.18)
U i ξ U i η = F ξ F η .