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Äíè ãåîìåòðèè-2014,ïîñâÿùåííûå 85-ëåòèþ àêàäåìèêàÞðèÿ Ãðèãîðüåâè÷à ÐåøåòíÿêàÈíñòèòóò ìàòåìàòèêè ÑÎ ÐÀÍÍîâîñèáèðñê, 24-27 ñåíòÿáðÿ 2014 ã.
Ñîçäàíèå óñëîâèé äëÿ îáðàçîâàíèÿ
âðåìåííûõ ïåòåëü â 5-ìåðíîì
ëîðåíöåâîì ìíîãîîáðàçèè
À.Ê. Ãóö
Îìñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Ô.Ì. Äîñòîåâñêîãî
2014
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Âðåìåííûå ïåòëè
Âðåìåííàÿ ïåòëÿ â n-ìåðíîãî ïñåâäîðèìàíîâîì ïðîñòðàíñòâåMn ëî-ðåíöåâîé ñèãíàòóðû � ýòî ãëàäêàÿ çàìêíóòàÿ âðåìåíèïîäîáíàÿ êðèâàÿ.
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Êðîòîâûå íîðû
a) Ðîæäåíèå 4-ìåðíîé êðîòîâîé íîðû. Ïðîñòðàíñòâî ñ òîïîëîãèåé3-ñôåðû òåðÿåò ñâÿçíîñòü. Îáðàçóþòñÿ äâà ïðîñòðàíñòâà, êàæäîå èç
êîòîðûõ ãîìåîìîðôíî 3-ñôåðå.b) Ðîæäåíèå 3-ìåðíîé êðîòîâîé íîðû â ïðîñòðàíñòâå ñ òîïîëîãèåé
3-ñôåðû. Ïðîñòðàíñòâî òåðÿåò îäíîñâÿçíîñòü.
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a) Ðîæäåíèå 4-ìåðíîé êðîòîâîé íîðû. Ïðîñòðàíñòâî ñ òîïîëîãèåé3-ñôåðû òåðÿåò ñâÿçíîñòü. Îáðàçóþòñÿ äâà ïðîñòðàíñòâà, êàæäîå èç
êîòîðûõ ãîìåîìîðôíî 3-ñôåðå.b) Ðîæäåíèå 3-ìåðíîé êðîòîâîé íîðû â ïðîñòðàíñòâå ñ òîïîëîãèåé
3-ñôåðû. Ïðîñòðàíñòâî òåðÿåò îäíîñâÿçíîñòü.
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Êîñìè÷åñêèå ïîëåòû
Ñâåðõáûñòðûé ïåðåëåò ê äàëåêîé çâåçäå ïî 4-ìåðíîé êðîòîâîé íîðå.
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Ïîÿâëåíèå âðåìåííîé ïåòëè
â 5-ïðîñòðàíñòâå
Âîçìîæíûé ïåðåõîä ïî êðèâîé L â ïðîøëîå b ñîáûòèÿ a â ñëîåíèè.
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Ïðóæèííîå ïðîñòðàíñòâî-âðåìÿ
Ïðîñòðàíñòâî-âðåìÿ, ñâåðíóòîå â ïðóæèíó â 5-ìåðíîìïðîñòðàíñòâå-âðåìåíè.
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Ïðóæèííîå ïðîñòðàñòðî-âðåìÿ â 5-Ãèïåðïðîñòðàíñòâå ñîçäàåò óñëîâèåäëÿ ïîñòðîåíèÿ ñîåäèíÿþùåé íàñòîÿùåå è áóäóùåå 4-ìåðíîé êðîòîâîéíîðû
4-êðîòîâàÿ íîðà ìåæäó íàñòîÿùèì è ïðîøëûì.
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Òåîðèÿ ñëîåíèé
Îïðåäåëåíèå Ñëîåíèå F ðàçìåðíîñòè p è êîðàçìåðíîñòè q íà ãëàä-êîì ìíîãîîáðàçèèMn (n = p+q) � ýòî àòëàñ ëîêàëüíûõ êàðò íàMn
òàêîé, ÷òî êàðòû ϕi : Ui ⊂ Mn → Vi ⊂ IRq × IRp óäîâëåòâîðÿþòóñëîâèþ ñîõðàíåíèÿ ôóíêöèÿìè ïåðåõîäà ϕi ◦ ϕ−1j : ϕj(Ui ∩ Uj)→ϕi(Ui ∩ Uj) ðàçëîæåíèÿ IRq × IRp íà ìíîæåñòâà âèäà {x} × IRp,x ∈ IRq.
Àòëàñ ñëîåíèÿ: ëèñòû ïåðåõîäÿò â ëèñòû
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Êëàññ Ãîäáèéîíà-Âåÿ
Ñëîåíèå F êîðàçìåðíîñòè 1 çàäàåòñÿ äèôôåðåíöèàëüíîé 1-ôîðìîéγ, êîòîðàÿ â ëîêàëüíûõ êîîðäèíàòàõ xA(A = 0, 1, 2, 3, 5) èìååò âèäγ = γAdx
A. Ôîðìà γ äîëæíà óäîâëåòâîðÿòü óñëîâèþ èíòåãðèðóåìî-ñòè Ôðîáåíèóñà γ ∧ dγ = 0.
Ýòî îçíà÷àåò, ÷òî ñóùåñòâóåò 1-ôîðìà α (îïðåäåëåííàÿ ñ òî÷íîñòüþäî êðàòíîãî γ) òàêàÿ, ÷òî dγ = α ∧ γ.
Êëàññ êîãîìîëîãèé 3-ôîðìû α ∧ dα íàçûâàåòñÿ êëàññîìÃîäáèéîíà-Âåÿ ñëîåíèÿ F è îáîçíà÷àåòñÿ GV (F).
Òåîðåìà (Äóìèíè). Åñëè GV (F) 6= 0, òî ñëîåíèå F èìååò ïðó-æèííûå ñëîè
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Ïëîòíûå ñëîè
Ñëîé L ⊂ M 5 íàçûâàåòñÿ ïëîòíûì (â M 5), åñëè åãî çàìûêàíèåñîâïàäàåò ñM 5, ò.å. L = M 5.Åñëè L ïëîòíûé ñëîé, òî äëÿ ëþáîé òî÷êè x0 ∈ M 5 ñóùåñòâóåò ïî-ñëåäîâàòåëüíîñòü òî÷åê {xk} ⊂ L, ñòðåìÿùèõñÿ ê x0. Èíà÷å ãîâîðÿ,âñåãäà íàéäóòñÿ ñêîëü óãîäíî áëèçêèå ê äðóã äðóãó â òîïîëîãèè Ãè-ïåðïðîñòðàíñòâà M 5 òî÷êè xk1, xk2, îäíà èç êîòîðûõ, ïóñòü ýòî xk1,îáîçíà÷èì å�å ÷åðåç b, ëåæèò â ïðîøëîì äðóãîé òî÷êè xk2, îáîçíà÷èìå�å ÷åðåç a. a
Êàê âèäèì, â ïëîòíîì ñëîå åñòü âîçìîæíîñòü ñîâåðøèòü ïåðåõîä â ïðî-øëîå, âûéäÿ â Ãèïåðïðîñòðàíñòâî è ïðîéäÿ ñðàâíèòåëüíî íåáîëüøîåðàññòîÿíèå. Âîïðîñ: â ëþáîé ëè ìîìåíò, ò.å. èç ëþáîé ëè òî÷êè ïëîò-íîãî ñëîÿ òàêîé ïåðåõîä âîçìîæåí? Íî ïðèíöèïèàëüíàÿ âîçìîæíîñòüòàêîãî ïóòåøåñòâèÿ ñóùåñòâóåò.
aÑ òî÷íîñòüþ äî çàìåíû, åñëè ýòî íå òàê, îäíîé èç ýòèõ äâóõ òî÷åê íà áëèçêóþ ê íåé â
ñëîå, íî óæå óäîâëåòâîðÿþùóþ èñêîìîìó óñëîâèþ.
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Ðàñøèðÿþùèåñÿ ñëîåíèÿ è òåìíàÿ ýíåðãèÿ
Äëÿ òîãî ÷òîáû îáðàçîâàëèñü ïðóæèííûå ñëîè íåîáõîäèìî, íàïðèìåð,äåôîðìèðîâàòü ñëîåíèå òàê, ÷òîáû îíî ïðåâðàòèëîñü â ðàñøèðÿþ-ùååñÿ (expansive) ñëîåíèå, â êîòîðîì êàæäûé ñëîé óáåãàåò ïðî÷ü îòáëèæàéøåãî ê íåìó äðóãîãî ñëîÿ.
ßñíî, ÷òî äëÿ ýòîãî íóæíî âêëþ÷èòü èñòî÷íèê ýíåðãèè, ñïîñîáñòâóþ-ùèé îòòàëêèâàíèþ îäíîãî ñëîÿ, ò. å. îäíîé áðàíû îò îò äðóãîãî ñëîÿ,ò. å. äðóãîé áðàíû. (Îòòàëêèâàíèå çàòðàãèâàåò è òðàåêòîðèè â ñëîÿõ).Åñòåñòâåííûì íåîáõîäèìûì èñòî÷íèêîì ýíåðãèè äëÿ äàííîé äåôîðìà-öèè ÿâëÿåòñÿ òåìíàÿ ýíåðãèÿ â áàëêå.
Îòìåòèì, ÷òî ñîâðåìåííîå ðàçáåãàíèå ãàëàêòèê â íàøåé Âñåëåííîéîáúÿñíÿåòñÿ äåéñòâèåì òåìíîé ýíåðãèè.
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Îïðåäåëåíèå ðàñøèðÿþùåãîñÿ ñëîåíèÿ
Óáåãàíèå ñëîÿ Fx, ïðîõîäÿùåãî ÷åðåç òî÷êó x, îò ñëîÿ Fy, ïðîõîäÿùåãî÷åðåç òî÷êó y, èçìåðÿåòñÿ ñëåäóþùèì îáðàçîì. Âîçüìåì R > 0 èðàññìîòðèì ïóòü γx â Fx ñ íà÷àëîì x è ñ äëèíîé íå áîëüøåé, ÷åì R,è ñïðîåêòèðóåì åãî ëîêàëüíî íà Fy, íà÷èíàÿ ñ òî÷êè x. Ïóñòü ploc(γx)ðåçóëüòèðóþùèé ïóòü â Fy. Ïðîäåëàåì ýòî æå ñ àíàëîãè÷íûì ïóòåì γyâ Fy ñ íà÷àëîì y è ñïðîåöèðóåì åãî íà Fx. Ïóñòü
d1 = supγx, l(γx)≤R
suptd(γx(t), plocγx(t)),
d2 = supγy, l(γy)≤R
suptd(γy(t), plocγy(t)),
dR(x, y) = max(d1, d2).
Ñëîåíèå F ðèìàíîâà ìíîãîîáðàçèÿ (M 5, h(5)) íàçûâàåòñÿ ðàñøèðÿ-þùèìñÿ, åñëè ñóùåñòâóåò ε > 0 òàêîå, ÷òî äëÿ êàæäîé ïàðû òî÷åêx è y â M 5, äîñòàòî÷íî áëèçêèõ, ÷òîáû äîïóñêàëàñü âûøåîïèñàííàÿêîíñòðóêöèÿ, íàéäåòñÿ R > 0, äëÿ êîòîðîãî dR(x, y) > ε.
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Ïðóæèííûå ñëîè â ðàñøèðÿþùèõñÿ ñëîå-
íèÿõ
Inaba è Tsuchiya äîêàçàëè, ÷òî ðàñøèðÿþùååñÿ ñëîåíèå êîðàçìåðíîñòè1 çàìêíóòîãî ìíîãîîáðàçèÿ îáëàäàåò ïðóæèííûì ñëîåì, êîòðûå ÿâëÿ-þòñÿ ïëîòíûìè.
Ñëåäîâàòåëüíî, â òàêîé ãåîìåòðè÷åñêîé 5-ìåðíîé Âñåëåííîé óñëîâèéäëÿ ñîçäàíèÿ âðåìåííûõ ïåòåëü ïðåäîñòàòî÷íî è, ñëåäîâàòåëüíî, ìà-øèíà âðåìåíè ðàñïðîñòðàíåííîå êîñìè÷åñêîå ÿâëåíèå.
Åñëè äîïîëíèòåëüíî ó÷åñòü, ÷òî êâàíòîâûå ôëóêòóàöèè 5-ìåòðèêè g(5)
è òîïîëîãèè (îáðàçîâàíèå 4-ðó÷åê) â 5-ìåðíîì ïðîñòðàíñòâå-âðåìåíè
∆g(5) ∼L∗
L
√T
L0
,
ãäå L∗ ∼ 10−33 ñì � ïîñòîÿííàÿ Ïëàíêà, à L4 × L0 � õàðàêòåðíûéðàçìåð 5-ìåðíîé îáëàñòè, T � êîíñòàíòà ñ ðàçìåðíîñòüþ [ñì], ñâÿ-çàííàÿ ñ 5-ì èçìåðåíèåì, ìîãóò èìåòü ìàêðîñêîïè÷åñêèé õàðàêòåð [?],òî âåðîÿòíîñòü îáíàðóæèâàòü ñïîíòàííûå ïðèðîäíûå âðåìåííûå ïåòëèêðàéíå âûñîêà.
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Áëàãîäàðíîñòè
Àâòîð áëàãîäàðèò òðåõ ìàòåìàòèêîâ, êîòîðûì îí îáÿçàí ñâîèì ìàòå-ìàòè÷åñêèì îáðàçîâàíèåì.
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Ëèòåðàòóðà
[1]. K. G�odel, �An Example of a New Type of Cosmological Solutions ofEinstein's Field Equtions of Gravitation�, Rev. Mod. Phys., Vol. 21, No. 3.447�450, (1949).[2]. A. Ãóö, Ôèçèêà ðåàëüíîñòè, Îìñê: Èçä-âî ÊÀÍ, (2012).[3]. R. Langevin, �A List of Questions about Foliations�, Workshop onTopology "Di�erential topology, foliations, and group actions". January 6-17,1992. Rio de Janeiro, Brazil. AMS Publ., (1994).[4]. N. Inaba, N. Tsuchiya, �Expansive foliations�, Hokkaido Math. J., Vol 21,39�49 (1992).[5]. A. Ãóö, Ýëåìåíòû òåîðèè âðåìåíè, Ì.: Èçäàòåëüñòâî ËÊÈ, (2011).[6]. A. Bejancu, H.R. Farran, Foliations and Geometric Structures, SpringerPubl., (2006).
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Ñïàñèáî çà âíèìàíèå!