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5.2 Izolirovannye sistemy na prostrannstvennoi beskonechnosti

5.2.1 Asimptoticheski ploskoe prostranstvo-vremya

My nazyvaem prostranstvo-vremya asimptoticheski ploskim, esli ono sootvetstvuet izolirovannoi (ostrovnoi) sisteme v ploskom prostranstve-vremeni (sm., naprimer, raboty [4] $^{\!, }$ [5]), i v kachestve opredeleniya udovletvoryaet sleduyushim ishodnym ,,naivnym'' trebovaniyam:

(1) Na prostranstvennoi beskonechnosti mirovye tochki parametrizuyutsya koordinatami $\{ x^\alpha\}$ takimi, chto $-\infty < x^\alpha < \infty$ .
(2) V odnoi iz etih sistem metrika imeet povedenie:

$\begin{displaymath} g_{\mu\nu} = \eta_{\mu\nu} + O(r^{-1}), g_{\mu\nu,\alpha} = O(r^{-2}) \end{displaymath}$ (5.13)

pri $r \rightarrow \infty$ , gde $\eta_{\mu\nu} \equiv {diag (-1, 1, 1, 1)}$ i $r^2 \equiv {{x^1}^2 + {x^2}^2 + {x^3}^2}$ . Oboznachenie $O(r^\omega)$ sootvetstvuet asimptoticheskomu povedeniyu pri $r \rightarrow \infty$ .
(3) Ne sushestvuet koordinatnyh sistem, dlya kotoryh padenie potencialov gravitacionnogo polya bylo by bystree, chem v (5.13).
(4) Tenzor energii-impul'sa istochnikov v sisteme (5.13) imeet povedenie:

$\begin{displaymath} T_{\mu\nu} = O(r^{-3 - \alpha}), \alpha > 0. \end{displaymath}$ (5.14)

Pereformuliruem eto opredelenie v ramkah polevogo podhoda. Dlya etogo nuzhno vybrat' fon. My vybiraem prostranstvo Minkovskogo v lorencevyh koordinatah (5.13). Posle vybora fona vybor koordinat (v silu kovariantnosti vseh vyrazhenii polevoi formulirovki) ne vazhen, no lorencevy koordinaty zdes' udobny, poskol'ku v nih ochevidno asimptoticheskoe povedenie. Po etoi zhe prichine my ne ispol'zuem kryshki v etoi chasti lekcii. Itak, punkt (1) sohranitsya.

(2) V odnoi iz kalibrovok gravitacionnye potencialy imeyut povedenie:

$\begin{displaymath} l^{\mu\nu} = O(r^{-1}), l^{\mu\nu}_{ ,\alpha} = O(r^{-2}). \end{displaymath}$ (5.15)
(3) Ne sushestvuet kalibrovok, gde ubyvanie $l^{\mu\nu}$ bylo by bystree, chem v (5.15).
(4) Uslovie dlya material'nyh istochnikov takzhe sohranyaetsya:

$\begin{displaymath} t^m_{\mu\nu} = O(r^{-3 - \alpha}), \alpha > 0. \end{displaymath}$ (5.16)

5.2.2 Global'nye sohranyayushiesya velichiny

Dlya togo, chtoby opredelit' integraly dvizheniya ostrovnoi sistemy my ispol'zuem vektory Killinga fonovogo prostranstva Minkovskogo:

$\begin{displaymath} \lambda^\mu_{(\alpha)} =\delta^\mu_\alpha, \lambda^\mu_{[... ...)} = \{ \lambda^\mu_{(\alpha)}, \lambda^\mu_{[\alpha\beta]}\}. \end{displaymath}$

(5.17)

Kak v obychnoi polevoi teorii, global'naya velichina sootvetstvuyushaya $\lambda^\mu_{(K)}$ opredelyaetsya kak

$\begin{displaymath} P^{(K)} = {1 \over c} \lim_{r \rightarrow \infty} \int_\Sigma t^{0\mu}_{(tot)} \lambda^{(K)}_\mu dx^3, \end{displaymath}$

(5.18)

gde secheniya $\Sigma$ opredelyayutsya kak t = const. Kak vsegda velichiny P^(K) sohranyayutsya, esli granichnye usloviya ustroeny tak, chto

$\begin{displaymath} {\lim_{r \rightarrow \infty}}\oint{ d S_k t^{k\mu}_{tot} \lambda^{(K)}_\mu} = 0, \end{displaymath}$

(5.19)

gde d S_k element koordinatnogo ob'ema na ,,stenkah'' cilindra (sm. Ris. 1), okruzhayushego izolirovannuyu sistemu.

Uravneniya Einshteina v polevoi formulirovke na ploskom fone mogut byt' perepisany v vide (sm. (12) v lekcii 4):

$\begin{displaymath} t^{\mu\nu}_{(tot)} = {1 \over{\kappa}} G^{\mu\nu}_L(l) = {1 \over {\kappa}} U^{\mu\nu\beta}_{ ,\beta}, \end{displaymath}$

(5.20)

gde my opredelili

$\begin{displaymath} U^{\mu\nu\beta} \equiv {{1\over2} (l^{\mu\nu,\beta} + \eta^{... ...{\mu\beta,\nu} - \eta^{\mu\beta} l^{\nu\alpha}_{ ,\alpha})}. \end{displaymath}$

(5.21)

Podstanovka (5.20) i (5.21) v (5.18) i ispol'zovanie antisimmetrii $U^{\mu\nu\beta} = -U^{\mu\beta\nu}$ privodit eti integraly k vidu:

$\displaystyle P^{(\alpha)}$	=	$\displaystyle {1\over{c \kappa}} {\lim_{r \rightarrow \infty}} \int_{\Sigma} \left[ U^{\alpha 0 i}_{ ,i}\right] dx^3$
	=	$\displaystyle {1\over{2 c \kappa}} {\lim_{r \rightarrow \infty}} \oint_{\parti... ...eta}_{ ,\beta} - l^{\alpha i, 0} - \eta^{\alpha i} l^{0 \beta}_{ ,\beta})},$	(5.22)

$\displaystyle P^{(\left [mn\right ])}$	=	$\displaystyle {1\over {2 c \kappa}} {\lim_{r \rightarrow \infty}} \int_{\Sigma}{ \left [(U^{n0i} x^m - U^{m0i} x^n)_{,i} + U^{m0n} - U^{n0m} \right]dx^3}$
	=	$\displaystyle {1\over {4 c \kappa}} {\lim_{r \rightarrow \infty}} \oint_{\parti... ...n0,i} - l^{ni,0} - \delta^{ni}l^{0\alpha}_{ ,\alpha}) x^m + \delta^{ni}l^{m0}$
	-	$\displaystyle (l^{m0,i} - l^{mi,0} - \delta^{mi} l^{0\alpha}_{ ,\alpha})x^n - \delta^{mi}l^{n0}),$	(5.23)

$\displaystyle P^{(\left[m0\right])}$	=	$\displaystyle {1\over{2 c \kappa}} {\lim_{r \rightarrow \infty}} \int_{\Sigma}{ \left[(U^{00i} x^m - U^{m0i} x^0)_{,i} - U^{00m}\right]dx^3}$
	=	$\displaystyle {1 \over {4 c \kappa}} {\lim_{r \rightarrow \infty}} \oint_{\partial\Sigma} {d s_i} ((l^{00,i} - l^{ik}_{ ,k}) x^m$
	-	$\displaystyle (l^{m0,i} - l^{mi,0} - \delta^{mi} l^{0\alpha}_{ ,\alpha})x^0 - \delta^{mi} l^{00} + l^{mi}),$	(5.24)

gde ds_i -- koordinatnyi element integrirovaniya na 2-sfere, okruzhayushei izolirovannuyu sistemu.

5.2.3 Naislabeishee ubyvanie gravitacionnyh potencialov

Podstanovka potencialov s povedeniem (5.15) v integraly dvizheniya privodit k tomu, chto (5.22) okazyvayutsya horosho opredelennymi, v to vremya kak (5.23) i (5.24) rashodyatsya. Chtoby izbezhat' etogo, dostatochno usilit' usloviya padeniya do

$\displaystyle l^{\mu\nu}$	=	$\displaystyle O^{+}(r^{-1}) + O^{-}(r^{-\beta}), \beta \ge 2,$
$\displaystyle l^{\mu\nu}_{ ,\pi}$	=	$\displaystyle O^{-}(r^{-2}) + O^{+}(r^{-1 - \beta}),$	(5.25)

gde (+) i (-) oznachayut chetnuyu i nechetnuyu funkcii po otnosheniyu k izmeneniyu znaka 3-vektora ${\vec n} = \lbrace {x^k}/r \rbrace$ . Takaya asimptotika vpervye byla vvedena Redzhe i Teitel'boimom [6].

Usloviya (5.25) dolzhny byt' Puankare invariantny, poetomu neobhodimo potrebovat' [5]

$\displaystyle l^{\mu\nu}_{ ,\pi\rho}$	=	$\displaystyle O^{+}(r^{-3}) + O^{-}(r^{-2 - \beta}),$
$\displaystyle l^{\mu\nu}_ { ,\pi\rho\sigma}$	=	$\displaystyle O^{-}(r^{-4}) + O^{+}(r^{-3 -\beta}),$
.........	=	................................. ,	(5.26)

kak eto chasto trebuyut pri posleduyushih integrirovaniyah [7].

Dlya real'noi ostrovoi sistemy padenie bystree, chem v (5.15) (ili v (5.25)) nevozmozhno v silu neobhodimosti sootvetstvovat' zakonu N'yutona v slabopolevom priblizhenii. A vot mozhet li ono byt' slabee bez izmeneniya osnovnyh integral'nyh harakteristik? Okazyvaetsya, chto mozhet [7] $^{\!- }$ [9] $^{\!, }$ [5]. Chtoby opredelit' usloviya padeniya bolee slabye, chem (5.25) my ispol'zuem kalibrovochno invariantnye svoistva polevoi formulirovki [5]. Kak my ustanovili, uravneniya dvizheniya kalibrovochno invariantny na samih sebe. Eto oznachaet, chto dlya ploskogo fona polnyi tenzor energii-impul'sa pri preobrazovaniyah (5.9) preobrazuetsya kak:

$\begin{displaymath} {t^\prime}^{(tot)}_{\mu\nu} = t^{(tot)}_{\mu\nu} + {1 \over... ... \right\vert _{\overline {\hat g^{\mu\nu}} = {\eta^{\mu\nu}}}, \end{displaymath}$

(5.27)

to est', v silu opredeleniya operatora $G^L_{\mu\nu}$ v (5.20), on invarianten s tochnost'yu do kovariantnoi divergencii. Togda, kalibrovochnaya neinvariantnost' dlya integralov dvizheniya (5.22) - (5.24) takzhe skazhetsya v poverhnostnyh integralah, a ih znacheniya mogut regulirovat'sya asimpototicheskim povedeniem $\xi^\alpha$ i ih proizvodnyh. My stavim zadachu naiti samoe slaboe povedenie $\xi^\alpha$ i ih proizvodnyh, kotoroe garantiruet kalibrovochnuyu invariantnost' (5.22) - (5.24), to est' invariantonost' integralov (5.18) otnositel'no podstanovki (5.27).

Vazhno otmetit', chto vyrazhenie $G^L_{\mu\nu}(l)$ invariantno (v absolyutnom smysle, bezotnositel'no k usloviyam padeniya) otnositel'no


		$\displaystyle {l^\prime}^{\mu\nu} = l^{\mu\nu} + \left.\hbox{$\pounds$}_\xi \le... ...^{\mu\nu}}\right) \right\vert _{\overline {\hat g^{\mu\nu}} = {\eta^{\mu\nu}}},$

chto yavlyaetsya samym pervym chlenom summy v (5.27), i, takim obrazom, bol'she ne rassmatrivaetsya.

Teper' budem iskat' ogranicheniya na povedenie $\xi^\alpha$ . Ispol'zuem sleduyushie ochevidnye trebovaniya:

(i) Dinamicheskie polya $l^{\mu\nu}$ i $\phi^A$ udovletvoryayut uravneniyam Einshteina i usloviyam padeniya (5.16) i (5.25), (5.26).
(ii) Kazhdaya iz ishodnyh komponent $\xi^\alpha, \xi^\alpha_{ ,\beta}, \xi^\alpha_{ ,\beta\gamma},...$ yavlyaetsya proizvol'noi nezavisimoi velichinoi v kazhdoi tochke prostranstva Minkovskogo. Estestvenna simmetriya po nizhnim indeksam.
(iii) Velichiny $\xi^\alpha, \xi^\alpha_{ ,\beta}, \xi^\alpha_{ ,\beta\gamma},...$ izmenyayutsya kak tenzory pri Puankare preobrazovaniyah.
(iv) Funkcii $\xi^\alpha$ yavlyayutsya klassa $C^\infty$ i kazhdaya sleduyushaya proizvodnaya ot $\xi^\alpha$ padaet bystree predydukshei na odin poryadok po stepeni r. Takim obrazom, vse svoistva $l^{\mu\nu}$ sohranyayutsya dlya $l'^{\mu\nu}$ .

Predpolozhim usloviya padeniya dlya $\xi^\alpha$ v forme:

$\begin{displaymath} \xi^\alpha = O^{-}(r^{1 - \varepsilon}) + O^{+}(r^{1 - \delta}). \end{displaymath}$

(5.28)

Chtoby 4-impul's $P^{(\alpha)}$ byl invarianten otnositel'no kalibrovochnyh preobrazovanii (5.9) neobhodimo predpolozhit', chto nechetnaya chast' ot kalibrovochnoi dobavki ubyvaet bystree chem r^-2, to est'

$\begin{displaymath} O^{-}\left( \partial_\alpha \left[\sum^{\infty}_{k = 1}{1\ov... ...at l^{\mu\nu}\right)\right]\right) < O^{-}\left(r^{-2}\right). \end{displaymath}$

(5.29)

V silu punkta (ii) rassmatrivaem vse chleny tipa $\xi\xi_{,\alpha\beta\gamma}$ v (5.29) kak nezavisimye. Togda, v silu punkta (iv) neravenstvo (5.29) daet ogranichenie na povedenie $\xi^\alpha$ :

$\begin{displaymath} \varepsilon > {1\over 2}, \delta > {1\over 2}. \end{displaymath}$

(5.30)

Etogo usloviya dostatochno, chtoby vse ostavshiesya chleny v kalibrovochnoi summe ne davali vklada v $P^{(\alpha)}$ . Analogichno, trebovanie sohraneniya pri kalibrovochnyh preobrazovaniyah $P^{(\left[\alpha\beta \right])}$ privodit k

$\begin{displaymath} \varepsilon + \delta > 2, \delta > 1, \varepsilon \ge ... ...i} \beta > 2, \varepsilon > 0 {\rm esli} \beta = 2. \end{displaymath}$

(5.31)

Kombinirovanie (5.30) i (5.31) privodit k ogranicheniyam na povedenie (5.28)

$\begin{displaymath} \varepsilon + \delta > 2, 1 \ge \varepsilon > {1\over 2}, \delta > 1, \end{displaymath}$

(5.32)

kotoroe odnovremenno garantiruet invariantnost' otnositel'no (5.9) $P^{(\alpha)}$ i $P^{(\left[\alpha\beta \right])}$ . Uslovie $\varepsilon \le 1$ vyrazhaet tot fakt, chto v real'noi ostrovnoi sisteme gravitacionnye peremennye ne mogut padat' bystree n'yutonova potenciala.

Takim obrazom, podstanovka (5.28) s usloviyami (5.32) v (5.9) daet povedenie dlya gravitacionnyh peremennyh v vide:

$\begin{displaymath} {l^\prime}^{\mu\nu} = O^{+}(r^{-\varepsilon}) + O^{-}(r^{-\delta}). \end{displaymath}$

(5.33)

Asimptotika (5.33) s usloviyami (5.32) zametno slabee, chem ishodnaya (5.25). Tem ne menee vse integraly (5.18) (ili v yavnom vyrazhenii -- (5.22) - (5.24)) sohranyayut svoi znacheniya. Malo togo, pri kazhdom posleduyushem kalibrovochnom preobrazovanii (5.9) s (5.28) i (5.32):


		$\displaystyle {l^{\prime\prime}}^{\mu\nu} = {l^\prime}^{\mu\nu} + \sum^{\infty}... ...ox{$\pounds$}_\xi^k \left(\overline {\hat g^{\mu\nu}} + \hat l'^{\mu\nu}\right)$

my uzhe ne narushim povedenie (5.33), znacheniya P^(K) takzhe ostayutsya prezhnimi.

Naislabeishee asimptoticheskoe povedenie (5.33) s usloviyami (5.32) yavlyayutsya novymi rezul'tatami, poskol'ku oni utochnyayut i ispravlyayut izvestnye rezul'taty [7] $^{\!- }$ [9].

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