▁spatial ▁slice ▁and ▁the ▁metric ' s ▁conjug ate ▁momentum ▁, ▁which ▁is ▁related ▁to ▁the ▁extr insic ▁curv ature ▁and ▁is ▁a ▁measure ▁of ▁how ▁the ▁induced ▁metric ▁evol ves ▁in ▁time . ▁These ▁are ▁the ▁metric ▁canonical ▁coordinates . ▁ ▁In ▁ 1 9 8 6 ▁Ab h ay ▁Ash te kar ▁introduced ▁a ▁new ▁set ▁of ▁canonical ▁variables , ▁Ash te kar ▁( new ) ▁variables ▁to ▁represent ▁an ▁unusual ▁way ▁of ▁re writing ▁the ▁metric ▁canonical ▁variables ▁on ▁the ▁three - dimensional ▁spatial ▁s lices ▁in ▁terms ▁of ▁an ▁SU ( 2 ) ▁gauge ▁field ▁and ▁its ▁complement ary ▁variable . ▁ ▁Over view ▁Ash te kar ▁variables ▁provide ▁what ▁is ▁called ▁the ▁connection ▁representation ▁of ▁canonical ▁general ▁relativ ity , ▁which ▁led ▁to ▁the ▁loop ▁representation ▁of ▁quantum ▁general ▁relativ ity ▁and ▁in ▁turn ▁loop ▁quantum ▁gravity ▁and ▁quantum ▁hol onomy ▁theory . ▁ ▁Let ▁us ▁introduce ▁a ▁set ▁of ▁three ▁vector ▁fields ▁, ▁ ▁that ▁are ▁orthogonal , ▁that ▁is , ▁ ▁. ▁ ▁The ▁ ▁are ▁called ▁a ▁tri ad ▁or ▁drei - be in ▁( G erman ▁literal ▁translation , ▁" three - leg "). ▁There ▁are ▁now ▁two ▁different ▁types ▁of ▁indices , ▁" space " ▁indices ▁ ▁that ▁behave ▁like ▁regular ▁indices ▁in ▁a ▁cur ved ▁space , ▁and ▁" internal " ▁indices ▁ ▁which ▁behave ▁like ▁indices ▁of ▁flat - space ▁( the ▁corresponding ▁" metric " ▁which ▁ra ises ▁and ▁low ers ▁internal ▁indices ▁is ▁simply ▁). ▁Define ▁the ▁dual ▁drei