Editing Orientability (section)

===Lorentzian geometry===

In [[Lorentzian geometry]], there are two kinds of orientability: [[space orientability]] and [[time orientability]].  These play a role in the [[causal structure]] of spacetime.<ref>{{cite book | author1-link=Stephen Hawking |first1=S.W. |last1=Hawking |author2-link=George Francis Rayner Ellis |first2=G.F.R. |last2=Ellis  | title=The Large Scale Structure of Space-Time | publisher=Cambridge University Press | year=1973 | isbn=0-521-20016-4| title-link=The Large Scale Structure of Space-Time }}</ref>  In the context of [[general relativity]], a [[spacetime]] manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at the same spacetime point, and then meet again at another point, they remain right-handed with respect to one another.  If a spacetime is time-orientable then the two observers will always agree on the direction of time at both points of their meeting.  In fact, a spacetime is time-orientable if and only if any two observers can agree which of the two meetings preceded the other.<ref>{{cite journal |first=Mark J. |last=Hadley |year=2002 |url=http://www.iop.org/EJ/article/0264-9381/19/17/308/q21708.pdf?request-id=49d1e985-bf89-4203-b020-48367545e3c0 |title=The Orientability of Spacetime |journal=[[Classical and Quantum Gravity]] |volume=19 |issue=17 |pages=4565–71 |doi=10.1088/0264-9381/19/17/308 |citeseerx=10.1.1.340.8125 |arxiv=gr-qc/0202031v4|bibcode=2002CQGra..19.4565H }}</ref>

Formally, the [[pseudo-orthogonal group]] O(''p'',''q'') has a pair of [[character theory|characters]]: the space orientation character &sigma;<sub>+</sub> and the time orientation character &sigma;<sub>&minus;</sub>,
:<math>\sigma_{\pm} : \operatorname{O}(p, q)\to \{-1, +1\}.</math>

Their product &sigma;&nbsp;=&nbsp;&sigma;<sub>+</sub>&sigma;<sub>&minus;</sub> is the determinant, which gives the orientation character.  A space-orientation of a pseudo-Riemannian manifold is identified with a [[section (fiber bundle)|section]] of the [[associated bundle]]
:<math>\operatorname{O}(M) \times_{\sigma_+} \{-1,+1\}</math>

where O(''M'') is the bundle of pseudo-orthogonal frames.  Similarly, a time orientation is a section of the associated bundle
:<math>\operatorname{O}(M) \times_{\sigma_-} \{-1,+1\}.</math>