Astronet > 3.2 Konformnye vektory Killinga dlya fridmanovskih modelei

po tekstam po klyuchevym slovam v glossarii po saitam perevod po katalogu

<< 3.1 Vvedenie | Oglavlenie | 3.3 Zakony sohraneniya i ... >>

Razdely

3.2 Konformnye vektory Killinga dlya fridmanovskih modelei

3.2.1 Metrika Fridmana. Konformnye uravneniya Killinga

Napishem metriku $d\bar s^2$ fridmanovskogo fona v bezrazmernyh koordinatah $x^\mu= (x^0=\eta, x^k)$ s k,l,m=1,2,3 dlya kotoryh simmetrichnaya rol' x^k ochevidna:

$\begin{displaymath} d\bar s^2= \bar g_{\mu\nu}dx^\mu dx^\nu= a^2(d\eta^2-f_{kl} dx^kdx^l)= a^2 e_{\mu\nu}dx^\mu dx^\nu, \end{displaymath}$

(3.1)

gde $a(\eta)$ -- masshtabnyi faktor, a f_kl, f^kl i f=det(f_kl) zadany kak

$\begin{displaymath} f_{kl} = \delta_{kl} + k {{\delta_{km}x^m \delta_{ln}x^n}\ov... ...kr^2 }, f^{kl}=\delta^{kl}-kx^k x^l, f={1\over 1-kr^2} \end{displaymath}$

(3.2)

dlya vseh $k=0, \pm 1$ , i $r^2=\delta_{kl}x^k x^l$ . Nenulevye simvoly Kristoffelya imeyut vid:

$\begin{displaymath} \bar \Gamma^0_{00}=\dot a, \bar \Gamma^0_{kl}=\dot a f_{kl... ...a^m_{0l}=\dot a \delta^m_l, \bar \Gamma^m_{kl}=kx^m f_{kl}, \end{displaymath}$

(3.3)

gde $\dot a$ -- ,,bezrazmernaya'' postoyannaya Habbla:

$\begin{displaymath} \dot a = {1\over a}{da\over d\eta}. \end{displaymath}$

(3.4)

V etih oboznacheniyah nenulevye komponenty fonovyh uravnenii Einshteina priobretayut vid:

$\begin{displaymath} \overline{G}^0_0={3\over a^2}(k+\dot a^2)=\kappa\overline{T}... ...dot a^2+2\partial_0 \dot a)\delta^m_l=\kappa\overline {T}^m_l. \end{displaymath}$

(3.5)

Dlya sravneniya vspomnim opredelenie obychnyh killingovyh vektorov. Oni udovletvoryayut uravneniyam ${\pounds_\xi} \bar g_{\mu\nu} = 0$ , to est' smesheniya vdol' etih vektorov ne induciruyut izmenenii v metrike $\bar g_{\mu\nu}(x) \rightarrow \bar g_{\mu\nu}(x)$ . Konformnye killingovy vektory udovletvoryayut uravneniyam

$\begin{displaymath} {\pounds_\xi} \bar g_{\mu\nu} = {1\over 4} \bar g_{\mu\nu}\bar g^{\rho\sigma} {\pounds_\xi} \bar g_{\rho\sigma}, \end{displaymath}$

(3.6)

a smesheniya vdol' etih vektorov induciruyut konformnye preobrazovaniya metriki

$\begin{displaymath} \bar g_{\mu\nu}(x) \rightarrow \Omega(x)\bar g_{\mu\nu}(x). \end{displaymath}$

(3.7)

I, naoborot, preobrazovaniya (3.7) ne izmenyayut uravnenii (3.6). Takim obrazom, nam budut interesny resheniya uravnenii (3.6) s fonovoi metrikoi (3.1). Sushestvuet 15 lineino nezavisimyh reshenii. V silu togo, chto sami uravneniya (3.6) ne zavisyat ot masshtabnogo faktora, oni mogut byt' zapisany v prostoi 3-kovariantnoi forme:

$\begin{displaymath} \partial_0{\xi^0}={1\over 3}\nabla_k\xi^k,\qquad \partial_0 ... ... \xi^0,\qquad \nabla^{(k} \xi^{l)} = f^{kl}\partial_0{\xi^0}, \end{displaymath}$

(3.8)

gde $\nabla_k$ -- 3-kovariantnaya proizvodnaya postroennaya s pomosh'yu f_kl, i $\nabla^k=f^{kl}\nabla_l$ . Otmetim takzhe, chto pervoe iz etih uravnenii est' sled poslednego. Nesmotrya na vidimuyu prostotu (3.8), reshat' ih ,,napryamuyu'' neprosto. K schast'yu, mozhno vospol'zovat'sya ,,konformnymi'' svoistvami. V rabote [10] konstruiruyutsya i izuchayutsya konformnye killingovy vektory ploskogo mira v koordinatah Minkovskogo $X^\mu$ . Metrika $e_{\mu\nu}$ yavlyaetsya konformnoi po otnosheniyu k metrike Minkovskogo $\eta_{\mu\nu}$ , to est' v sootvetstvuyushih koordinatah $x^\alpha$ : $e_{\mu\nu}=\Omega^2 \eta_{\mu\nu}$ global'no. V to zhe samoe vremya konformnye killingovy vektory ostayutsya temi zhe samymi, kak dlya prostranstva Minkovskogo, tak i dlya konformnogo prostranstva Minkovskogo, -- oni ne zavisyat ot $\Omega$ . Dalee neobhodimo tol'ko komponenty $\xi^\mu$ v koordinatah $X^\mu$ preobrazovat' v koordinaty $x^\mu$ v metrike (3.1). Preobrazovaniya koordinat ot $X^\mu$ k $x^\mu$ v (3.1) uzhe postroeny v knige Penrouza i Rindlera [11]. Ostalos' eti preobrazovaniya ispol'zovat', a dlya proverki podstavit' v (3.8).

3.2.2 Konformnye vektory Killinga v prostranstve Minkovskogo

Forma konformnyh vektorov Killinga v prostranstve Minkovskogo v lorencevyh koordinatah naibolee ochevidna i stanovitsya naibolee yasnoi ih interpretaciya. Poetomu my predstavlyaem ih yavnyi vid kak eto sdelano v rabote [10] i daem ih kratkoe opisanie.

Rassmatrivaetsya metrika Minkovskogo

$\begin{displaymath} d\bar s^2= \eta_{\mu\nu}dX^\mu dX^\nu = dT^2 - \delta_{kl} dX^k dX^l. \end{displaymath}$

(3.9)

Sushestvuet 15-parametricheskaya gruppa konformnyh preobrazovanii $X^\alpha \rightarrow {\tilde X}^\alpha$ , takih, chto (3.9) perehodit v


		$\displaystyle d\bar s^2= \Phi({\tilde X})\eta_{\mu\nu}d{\tilde X}^\mu d{\tilde X}^\nu.$

Eti preobrazovaaniya imeyut vid

$\begin{displaymath} {\tilde X}^\mu = a^\mu + A^\mu_\rho X^\rho + b X^\mu + {{X^\mu - B^\mu X^2}\over {1 - 2B_\mu X^\mu + B^2 X^2}}, \end{displaymath}$

(3.10)

gde $\eta_{\mu\nu}A^\mu_\rho A^\nu_\sigma = \eta_{\rho\sigma}$ , $B^2 = \eta_{\mu\nu}B^\mu B^\nu$ , $X^2 = \eta_{\mu\nu}X^\mu X^\nu$ . Variaciya uravneniya (3.10) privodit k vyrazheniyu, koefficienty kotorogo i est' komponenty konformnyh vektorov Killinga:

$\begin{displaymath} \delta {\tilde X}^\mu = \xi^\mu_{(\alpha)} \delta a^\alpha +... ...\xi^\mu_{[(0)]} \delta b + \xi^\mu_{[\alpha]} \delta B^\alpha. \end{displaymath}$

(3.11)

Pervye dva slagaemyh predstavlyayut obychnuyu gruppu dvizhenii, kotoraya opredelyayutsya ,,obychnymi'' vektorami Killinga, vektorami 4-translyacii:

$\begin{displaymath} \xi^\mu_{(\alpha)} = \delta^\mu_\alpha \end{displaymath}$

(3.12)

i 4-vrashenii:

$\begin{displaymath} \xi^\mu_{([\alpha\beta])} =\left(\delta^\mu_\alpha\eta_{\beta\gamma} - \delta^\mu_\beta\eta_{\alpha\gamma}\right)X^\gamma. \end{displaymath}$

(3.13)

Tretii chlen v (3.11) otvechaet tak nazyvaemym dilatonnym (dilatation) ili masshtabnym preobrazovaniyam:

$\begin{displaymath} \xi^\mu_{[(0)]} = X^\mu. \end{displaymath}$

(3.14)

I, nakonec, poslednii chlen v (3.11) sootvetsvuet ,,4-uskoreniyam'':

$\begin{displaymath} \xi^\mu_{[\alpha]} =\left(\delta^\mu_\gamma\eta_{\alpha\beta... ... - \delta^\mu_\alpha\eta_{\beta\gamma}\right)X^\beta X^\gamma. \end{displaymath}$

(3.15)

3.2.3 Konformnye vektory Killinga v geometrii Fridmana

Poskol'ku komponenty (3.12) - (3.15) te zhe samye kak i v konformno-ploskoi kosmologicheskoi metrike, neobhodimo lish' s pomosh'yu preobrazovanii [11] (kotorye svyazyvayut metriku (3.1) s konformno-ploskoi metrikoi Fridmana) perepisat' ih v koordinatah ( $\eta, x^k$ ) metriki (3.1). Estestvenno, chto vse 15 lineino nezavisimyh vektorov ostanutsya konformnymi killingovymi dlya resheniya (3.1), odnako ne vse 10 vektorov (3.12) - (3.13) ostanutsya ,,obychnymi'' killingovymi: killingov vektor vremennyh translyacii iz (3.12) i 3 vektora lorencevyh vrashenii iz (3.13) perestayut byt' killingovymi dlya (3.1). Posle neslozhnyh, no gromozdkih vychislenii poluchaem komponenty vseh 15 vektorov. Zdes' net neobhodimosti davat' yavnyi vid etih vektorov, my privodim tol'ko ih spisok. Ih nazvaniya sootvetstvuyut konformnym killingovym vektoram prostranstva Minkovskogo. Eto vektor vremennyh translyacii t, 3 vektora prostranstvennyh translyacii ${\bf s}_a, a=1,2,3$ , 3 vektora prostranstvennyh vrashenii ( ${\bf r}_a$ ) i 3 vektora lorencevyh vrashenii ${\bf l}_a$ . Krome etih vektorov s ,,obychnymi'' nazvaniyami sushestvuyut eshe dilatonnyi vektor ${\bf d}$ , vektor vremennnyh ,,uskorenii'' ${\bf a}$ i 3 vektora prostranstvennyh ,,uskorenii'' ${\bf b}_a$ . Bezotnositel'no k nashim prilozheniyam, eti vektory interesny sami po sebe. V nedavnei rabote [12] byli podrobno izucheny ih gruppovye svoistva.

<< 3.1 Vvedenie | Oglavlenie | 3.3 Zakony sohraneniya i ... >>

Publikacii s klyuchevymi slovami: zakony sohraneniya - Obshaya teoriya otnositel'nosti - gravitaciya Publikacii so slovami: zakony sohraneniya - Obshaya teoriya otnositel'nosti - gravitaciya
Sm. takzhe: Padenie shara v Solnechnoi sisteme Gravitacionnye anomalii na Merkurii Galaktika Arp 94 Sputniki GRAIL sostavlyayut kartu lunnoi gravitacii 70 let Stivenu Hokingu Molotok protiv peryshka na Lune Sputnik Gravity Probe B podtverdil nalichie gravimagnetizma Vse publikacii na tu zhe temu >>

Obsudit' etu publikaciyu

Versiya dlya pechati

Astrometriya - Astronomicheskie instrumenty - Astronomicheskoe obrazovanie - Astrofizika - Istoriya astronomii - Kosmonavtika, issledovanie kosmosa - Lyubitel'skaya astronomiya - Planety i Solnechnaya sistema - Solnce