▁horizon ") ▁in ▁a ▁region ▁where ▁the ▁weak ▁energy ▁condition ▁is ▁satisfied , ▁meaning ▁that ▁the ▁region ▁contains ▁no ▁matter ▁with ▁negative ▁energy ▁density ▁( ex otic ▁matter ). ▁Sol utions ▁such ▁as ▁Ti pler ' s ▁assume ▁cyl ind ers ▁of ▁infinite ▁length , ▁which ▁are ▁easier ▁to ▁analyze ▁mathemat ically , ▁and ▁although ▁Ti pler ▁suggested ▁that ▁a ▁finite ▁cyl inder ▁might ▁produce ▁closed ▁tim eli ke ▁curves ▁if ▁the ▁rotation ▁rate ▁were ▁fast ▁enough , ▁he ▁did ▁not ▁prove ▁this . ▁But ▁Haw king ▁points ▁out ▁that ▁because ▁of ▁his ▁theorem , ▁" it ▁can ' t ▁be ▁done ▁with ▁positive ▁energy ▁density ▁everywhere ! ▁I ▁can ▁prove ▁that ▁to ▁build ▁a ▁finite ▁time ▁machine , ▁you ▁need ▁negative ▁energy ." ▁This ▁result ▁comes ▁from ▁Haw king ' s ▁ 1 9 9 2 ▁paper ▁on ▁the ▁chron ology ▁protection ▁conject ure , ▁where ▁he ▁exam ines ▁" the ▁case ▁that ▁the ▁caus ality ▁viol ations ▁appear ▁in ▁a ▁finite ▁region ▁of ▁spac etime ▁without ▁curv ature ▁singular ities " ▁and ▁proves ▁that ▁" there ▁will ▁be ▁a ▁C auch y ▁horizon ▁that ▁is ▁compact ly ▁generated ▁and ▁that ▁in ▁general ▁contains ▁one ▁or ▁more ▁closed ▁null ▁ge odes ics ▁which ▁will ▁be ▁incomplete . ▁One ▁can ▁define ▁geomet rical ▁quantities ▁that ▁measure ▁the ▁Lor ent z ▁boost ▁and ▁area ▁increase ▁on ▁going ▁round ▁these ▁closed ▁null ▁ge odes ics . ▁If ▁the ▁caus ality ▁viol ation ▁developed ▁from ▁a ▁non comp act ▁initial ▁surface , ▁the ▁aver aged ▁weak