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9.2 Dvizhenie chastic v pole Shvarcshil'da

Teper' rassmotrim dvizhenie chastic v etom pole. Po n'yutonovskoi teorii dlya chastic, pokoyashihsya na beskonechnosti, ispol'zuya zakon sohraneniya energii, imeem

$\displaystyle -{GMm\over r}+{mv^2\over 2}=0,$

t. e.

$\displaystyle v^2=GM/r.$

Opredelim naivno

kak radius, gde

. Togda (sr. razdel 1.7)

$\displaystyle r_g={2GM\over{c^2}}$

Etot rezul'tat poluchil eshe Laplas. Podcherknem, odnako, chto eto ne bol'she chem mnemonicheskii priem. Ochevidno, chto n'yutonovskaya mehanika neprimenima pri $v\sim c$ .

Teper' napishem strogo uravnenie dvizheniya chastic v pole Shvarcshil'da. V metrike, ne zavisyashei ot vremeni, sohranyaetsya energiya, t. e. komponenta (imenno s nizhnim indeksom) 4-vektora impul'sa chasticy:

$\displaystyle p_0=-mc^2\,g_{00}{dx^0\over{ds}}.$

V staticheskom pole

$\displaystyle ds^2=g_{00}dt^2-dl^2$

ili

$\displaystyle ds=dt\,\sqrt{g_{00}}\sqrt{1-\frac{dl^2}{g_{00}\,dt^2}}.$

Velichina

$\displaystyle {dl\over{\sqrt{g_{00}}dt}}={v\over c}$

eto skorost', izmerennaya lokal'nym nablyudatelem. Togda imeem

$\displaystyle p_0={mc^2\sqrt{g_{00}}\over{\sqrt{1-v^2/c^2}}}=$ const $\displaystyle$

(eto sootnoshenie ne zavisit ot napravleniya

Pust' pri $r\longrightarrow\infty\;(g_{00}\longrightarrow 1)$ $v\longrightarrow 0$ , t.e. . V etom sluchae vsegda

$\displaystyle 1-v^2/c^2=g_{00}=1-r_g/r\,,\quad v^2=c^2r_g/r\,,$

t.e. v OTO imeet mesto svyaz' mezhdu

v tochnosti takaya, kak v n'yutonovskoi mehanike i teorii tyagoteniya. V chastnosti,

-- eto kak raz tot radius, gde

, esli pri $r=\infty,\,v=0$ v takoi teorii. Esli $v\ne 0$ na beskonechnosti, to

$\displaystyle 1-{v^2\over{c^2}}=\left(1-{r_g\over r}\right)\left(1-{v^2_{\infty}\over{c^2}}\right).$

Takim obrazom, v tochnoi teorii v OTO

tol'ko na

nezavisimo^9.1 ot znacheniya $v=v_{\infty}$ pri $r=\infty$ .

Eshe raz podcherknem, chto -- eto skorost', izmerennaya lokal'nym nablyudatelem.

V dal'neishem polozhim . Pri dvizhenii po radiusu v sluchae pri $r\longrightarrow\infty$

$\displaystyle v={dl\over{d\tau}}={\sqrt{g_{11}}\over{\sqrt{g_{00}}}}\,{dr\over{dt}}=\sqrt{{r_g\over r}}.$

Togda, ispol'zuya vyrazhenie dlya $g_{00}$ i $g_{11}$ ,

$\displaystyle {dr\over{dt}}=\sqrt{{r_g\over r}}\left(1-{r_g\over r}\right),$

$\displaystyle t=\int\limits_R^{r_g}\sqrt{{r\over{r_g}}}\left(1-{r_g\over r}\right)^{-1}\,dr.$

Pri padenii s konechnogo

etot integral logarifmicheski rashoditsya pri $r\longrightarrow r_g$ . Otmetim, chto

prohodit cherez maksimum i stremitsya k nulyu pri $r\longrightarrow r_g$ . Konechno, v shvarcshil'dovskom pole net ottalkivaniya, hotya $dr/dt\longrightarrow0$ pri $r\longrightarrow r_g$ , tak kak

-- eto ne skorost'.

Sushestvuet horoshee sravnenie. Esli my budem brosat' chasticy na chernuyu dyru, to my uvidim, chto oni zamedlyayut dvizhenie i skuchivayutsya vblizi . Eta ostanovka ne sootvetstvuet polozheniyu ravnovesiya. Predstavim sebe, chto snimaetsya na kinoplenku prygun, a potom pri prosmotre zamedlyaetsya skorost' dvizheniya plenki, tak chto v nekotoryi moment on ostanovitsya. To zhe samoe pri dvizhenii chastic. Esli ustanovit' chasy na chasticah, kotorye deistvitel'no ostanavlivayutsya v polozhenii ravnovesiya, to chasy vse ravno idut. V sluchae dyry chasticy ostanavlivayutsya, no ostanavlivayutsya i chasy na nih. Po sobstvennomu vremeni nikakih ostanovok ne proishodit. Prosto s tochki zreniya udalennogo nablyudatelya chasy zamedlyayutsya iz-za dopler-effekta i iz-za vliyaniya gravitacionnogo potenciala.

Predstavim sebe padenie gaza, sostoyashego iz otdel'nyh chastic s tochki zreniya dalekogo nablyudatelya. Okazyvaetsya, chto gaz skaplivaetsya vblizi , no ne otdaet svoyu kineticheskuyu energiyu. Nesmotrya na to, chto kolichestvo skopivshegosya gaza neogranichenno vozrastaet s techeniem vremeni, ego plotnost', izmerennaya v sobstvennoi sisteme (nablyudatelya, dvizhushegosya vmeste s gazom), ostaetsya konechnoi v kazhdoi tochke prostranstva, v chastnosti vblizi .

Zadacha. Rasschitat' radial'noe dvizhenie dlya sveta (fotonov) v metrike Shvarcshil'da.

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