Another Collimation Mechanism of Astrophysical Jet

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1 Journal o Earth Sene and Engneerng 7 ( do:.765/59-58x/7.. D DAVID PUBLISHING Another Collmaton Mehansm o Astrophysal Jet Yoshnar Mnam Advaned Sene-Tehnology esearh Organzaton (Formerly NEC Spae Development Dvson, 35-3, Hgashkubo-Cho, Nsh-Ku, Yokohama -6, Japan Abstrat: The aeleraton mehansm o astrophysal et and the ollmaton mehansm narrowng down to a long dstane have been examned so ar. It s a ollmaton problem o how to narrow the astrophysal et narrowly. Further, the model o the astrophysal et aeleraton mehansm s requred to solve ths ollmaton problem at the same tme as well as ts aeleraton. At the present tme, the magnet ore model (magnet entrugal ore and magnet pressure s regarded as the most domnant theory whh solves the two problems o astrophysal et aeleraton and ollmaton at the same tme. In addton to the present astrophysal et narrow ollmaton mehansm by magnet tenson (pnh ore, n ths artle, another ollmaton mehansm whh narrows down an astrophysal et s newly ntrodued. That s, sne a urvature s generated n the spae around the astrophysal et by magnet eld, a knd o pressure equvalent to the gravtatonal eet s generated n the dreton o the nteror o astrophysal et as well as the pnh ore rom the outer rumerental surae o the astrophysal et. Key words: Astrophysal et, areton dsk, blak hole, ollmaton mehansm, aeleraton mehansm, magnet eld, ontnuum, urvature, spae-tme, osmology.. Introduton The astrophysal et s a narrow etted plasma et at hgh speed ( km/s to near the speed o lght that emts n both dretons vertally rom areton dsk around the ompat entral obet suh as a neutron star or blak hole. Its length s an enormous, long and narrow et reahng rom lght year lght years mllon lght years. A et propagatng at a speed lose to the speed o lght s alled a relatvst et. The aeleraton mehansm o the astrophysal et and the ollmaton mehansm narrowng down to a long dstane have been examned so ar. They are due to thermal gas pressure, lght radaton pressure, and magnet eld pressure. Currently, adatve Aeleraton model aelerated by the radaton eld o the areton dsk and Magnet Aeleraton model aelerated by magnet eld Correspondng author: Yoshnar Mnam, admnstratve dretor, researh elds: satellte desgn and engneerng, propulson theory and propulson physs, laser propulson, nterstellar navgaton theory. E-mal: y-mnam@mt.bglobe.ne.p. penetratng the areton dsk are representatve models. The hgh veloty, hghly ollmated gas streams ets rase two maor problems, namely how the et materal s aelerated, and how t s ollmated (Fg. a. It s a ollmaton problem o how to narrow the et narrowly, and the model o the et aeleraton mehansm s requred to solve ths ollmaton problem at the same tme as well as aeleraton. At the present tme, the magnet ore model (magnet entrugal ore and magnet pressure s regarded as the most domnant theory whh solves the two problems o et aeleraton and ollmaton at the same tme. That s, the areton dsk generates a helal magnet eld by twstng the magnet eld lnes, aelerates by magnet ore, and narrows the et by magnet tenson (pnh. The sel-pnhng ore o magnet eld twsted by the rotaton ours naturally as a ore to ollmate the et thnly (Fg. b [-8]. However, there are also ontroverses among researhers as ollows: the global magnet eld annot be MHD (magnetohydrodynam et due to

2 Another Collmaton Mehansm o Astrophysal Jet 75 (a (b Fg. (a Astrophysal Jet rom an Areton Dsk o Blak Hole; (b Formaton o Astrophysal Jet wound by twstng magnet eld lnes. ( h?rk=&e=utf-8&gdr=&p=astrophysal+jet weak magnet eld; even the global magnet eld s weak, t an do MHD et, and the loal magnet eld an do MHD et, urthermore the astrophysal et an ollmate by magnet eld; 3 the magnet eld hypothess s dult beause the magnet eld n the et s weak. Even so, the magnet eld lnes are n the et, there s a possblty that a strong magnet eld regon s loally generated due to loal turbulene and shok waves n the plasma. In addton to the onventonal et narrow ollmaton mehansm by magnet tenson (pnh ore, n ths artle, we ntrodue another ollmaton mehansm whh narrows down a et. That s, sne a magnet eld s present n the et, a urvature s generated n the surroundng spae by magnet eld; a spatal pressure equvalent to the gravtatonal eet n the surroundng spae s generated n the dreton o the nteror o the et [9-]. Ths spatal pressure ats as well as the pnh ore rom the outer rumerental surae o the astrophysal et,.e., ollmaton mehansm.. Astrophysal Jet Aeleraton and Collmaton An astrophysal et s a phenomenon oten seen n astronomy, where streams o matter are emtted along the axs o rotaton o a ompat entral obet (suh as a blak hole or neutron star. Many stellar obets wth areton dsks have ets. Whle t s stll the subet o ongong researh to understand how ets are ormed and powered, the two most oten proposed orgns are dynam nteratons wthn the areton dsk, or a proess assoated wth the ompat entral obet. When matter s emtted at speeds approahng the speed o lght, these ets are alled relatvst ets. Whle t s not known exatly how areton dsks would aelerate ets or produe postron-eletron plasma, they are generally thought to generate tangled magnet elds that ause the ets to aelerate and ollmate. One o the astonshng propertes o astrophysal ets s that they reman ollmated over qute large dstanes. Agan, MHD proesses seem to be most

3 76 Another Collmaton Mehansm o Astrophysal Jet lkely responsble or ths behavor: the same pnh mehansm, whh ored the plasma gas nto a beam dreted along the polar axs o the drvng soure, s also ollmatng the astrophysal et urther out. The dea o magnet ollmaton o ets n the asymptot regme (.e., ar rom the drvng soures has been proposed rst or galat rado ets showed that any axsymmetr (nonrelatvst magnetzed wnd wll approah a ylndrally ollmated struture, the eletr urrent arred by the low s non-zero. The ollmaton mehansm s straghtorward to understand or a urrent arryng low: the urrent reates a magnet eld wrappng around the urrent va Ampère s law. The aton o ths (torodal eld then pnhes the urrent bak to the low axs va the Lorentz ore. In the ase o a vanshng urrent, the low would stll be parabolodally ollmated. The mportane o magnet elds or et ollmaton s vald. Sne the one-lud approxmaton holds, the equaton o moton o the magnet lud s gven by F J B ( B B / ( B B / ( B / ( where F s the Lorentz ore, J s the urrent densty, 7 B s the magnet eld, ( H / m. We use the ollowng equaton (Ampère s rule B J. The rst term n Eq. ( denotes magnet tenson and seond term denotes magnet pressure. Fg. a shows ylndral plasma. A urrent I lows n the plasma and a magnet eld B θ ours n the θ dreton. Although the ore o ontratng the plasma by B θ s Lorentz ore, magnet pressure P B s externally appled and the ylndral plasma s ontrated (Fg. b. Usng Eqs. ( and (3, rotbds Bds JdS I ( S C S I Bds rb (3 C magnet eld B θ s obtaned: Then, magnet pressure s obtaned: B I ( r B P B (5 Fg. 3a shows General elatvst MHD smulaton on the nteraton between ergosphere and magnet eld lne o rotatng blak hole. A blak hole dynams has ts plasma algned to the nterstellar magnet eld lnes that thread through the equatoral plane o the areton dsk ust outsde the event horzon, so that the plasma evolves nto areton dsk, Fg. (a ylndral plasma; (b Contrated ylndral plasma by magnet pressure.

4 Another Collmaton Mehansm o Astrophysal Jet 77 (a (b Fg. 3 (a MHD ets rom Kerr hole magnetosphere (Kode et al. Sene; (b KatoY, Mneshge, Shbata (; 3D sm. (ApJ. Ths torodal eld domnated et s launhed by magnet pressure (smlar to Shbata and Uhda 985, Turner et al. 999, Kudoh et al., and s also Smlar to magnet tower o Lynden-Bell (996. whh ould be desrbed as a ondensate o eletron-postron pars. Sne the plasma beng hghly ondutve, t wll be expeted that the nterstellar magnet eld lnes wll beome rozen nto that plasma whh rotates wthn areton dsk, and as the areton dsk rotates t wll drag and twst those magnet eld lnes, pullng them together. Fg. 3b shows a torodal eld domnated et s launhed by magnet pressure. As the magnet eld penetratng the areton dsk s twsted, the energy o the magnet eld s aumulated, and at the same tme t propagates along the magnet eld lnes, the et eets rom the areton dsk, and not only the magnet entrugal ore but also magnet pressure also ontrbutes to aeleraton o et. When the magnet eld s twsted n the dreton o rotaton by the atuaton rotaton o the plasma materal, the twsted magnet eld ats lke a sprng to aelerate the plasma materal urther upward. In other words, t s aeleraton by magnet pressure. I the et s magnetally aelerated, the et s expeted to have a twsted helal magnet eld (Fg. b. Furthermore, the twsted magnet eld ats lke a rubber strng, and the ore o the rubber band shrnks (magnet pnh so that the low o the plasma substane s dreted n the rotaton axs dreton. Ths s the ollmaton o the et by magnet eld. Collmaton also ours voluntarly n addton to aeleraton n a model where a et s drven rom an areton dsk by magnet eld and rotaton. Even the magnet eld penetratng the areton dsk s very weak, the rotaton o the areton dsk auses the magnet eld to twst and nrease more and more, and the energy s stored n the magnet

5 78 Another Collmaton Mehansm o Astrophysal Jet eld to the same extent as the rotatonal energy o the areton dsk. Even n the ase o a loal magnet eld n the dsk nstead o the global magnet eld, the magnet eld s twsted n the dsk, so that magnet pressure s generated and t s possble to aelerate the et [-7]. The energy o the magnet eld s nreased by ompressng the gas or strethng the magnet eld lnes due to the plasma gas. The phenomenon n whh energy stored n the orm o a magnet eld s released loally and n large quanttes n a short tme s well known or solar lares. The aeleraton mehansm or these ets may be smlar to the magnet reonneton proesses observed n the Earth s magnetosphere and the solar wnd. The energy o magnet eld s gven by: B 3 E mag 3 (6 where s the radus o sunspot. As magnet rotaton nstablty grows, magnet eld lnes are strethed n the azmuth dreton, the magnet eld s strengthened, and the magnet energy exponentally nreases. As long as there s a weak magnet eld at the begnnng, the magnet eld s ampled by magnet rotaton nstablty. 3. Collmaton Mehansm o Astrophysal Jet Indued by Spatal Curvature Generated n Spae around Magnet Feld 3. Introduton The prnple o ths dea s derved rom General elatvty and the theory o ontnuum mehans. We assume that the so-alled vauum o spae ats as an nnte elast body lke rubber. The urvature o spae plays a sgnant role. A hypothess or mehanal property o spae-tme s ntrodued by Mnam n 988 []. Conernng the bas onept, please reer to Appendx A: Contnuum Mehans o Spae-Tme. Furthermore, the maor omponent o urvature o spae an be produed by not only mass densty but also magnet eld. In the subsequent setons, Generaton o Surae Fore Indued by Spatal Curvature n the rst, Curvature Control by Magnet Feld n the seond, and nally, Collmaton Mehansm Indued by Spatal Curvature around Magnet Feld are ntrodued. 3. Generaton o Surae Fore Indued by Spatal Curvature On the supposton that spae s an nnte ontnuum, ontnuum mehans an be appled to the so-alled vauum o spae. Ths means that spae an be onsdered as a knd o transparent eld wth elast propertes. Fg. shows the urvature o spae. I spae urves, then an nward normal stress P s generated (Fg. a. Ths normal stress,.e. surae ore serves as a sort o pressure eld. P N ( / N (/ / (7 where N s the lne stress,, are the radus o prnpal urvature o urved surae, and s the maor omponent o spatal urvature. A large number o urved thn layers orm the undretonal surae ore,.e. aeleraton eld. Aordngly, the spatal urvature produes the aeleraton eld (Fg. b. The undamental three-dmensonal spae struture s determned by quadrat surae struture. Thereore, a Gaussan urvature K n two-dmensonal emann spae s sgnant. The relatonshp between K and the maor omponent o spatal urvature s gven by: K (8 ( g g g where s non-zero omponent o emann urvature tensor. It s now understood that the membrane ore on the urved surae and eah prnpal urvature generates the normal stress P wth ts dreton normal to the

6 Another Collmaton Mehansm o Astrophysal Jet 79 urved surae as a surae ore. The normal stress P ats towards the nsde o the surae as shown n Fg. a. A thn-layer o urved surae wll take nto onsderaton wthn a spheral spae havng a radus o and the prnpal rad o urvature that are equal to the radus ( = =. Sne the membrane ore N (servng as the lne stress an be assumed to have a onstant value, Eq. (7 ndates that the urvature generates the nward normal stress P o the urved surae. The nwardly dreted normal stress serves as a pressure eld. When the urved suraes are nluded n a great number, some type o undretonal pressure eld s ormed. A regon o urved spae s made o a large number o urved suraes and they orm the eld as a undretonal surae ore (.e. normal stress. Sne the eld o the surae ore s the eld o a knd o ore, the ore aelerates matter n the eld,.e. we an regard the eld o the surae ore as the aeleraton eld. A large number o urved thn layers orm the undretonal aeleraton eld (Fg. b. Aordngly, the spatal urvature produes the aeleraton eld. Thereore, the urvature o spae plays a sgnant role to generate pressure eld. Applyng membrane theory, the ollowng equlbrum ondtons are obtaned n quadrat surae, N b P (9 where N s a membrane ore,.e. lne stress o urved spae, b s seond undamental metr o urved surae, and P s the normal stress on urved surae [9-, 3-6]. The seond undamental metr o urved spae b and prnpal urvature K ( has the ollowng relatonshp usng the metr tensor g, Thereore we get: b K ( g ( N b N K g ( g N K N K N K ( ( ( From Eqs. (9 and (, we get: ( N K P ( ( As or the quadrat surae, the ndes and take two derent values,.e. and, thereore Eq. ( beomes: N K ( N K ( P (3 where K ( and K ( are prnpal urvature o urved surae and are nverse number o radus o prnpal urvature (.e. / and /. The Gaussan urvature K s represented as: K K K / (/ ( Aordngly, suppose ( ( ( N N N, we get: N / / P ( (5 It s now understood that the membrane ore on the urved surae and eah prnpal urvature generate the normal stress P wth ts dreton normal to the (a (b Fg. Curvature o Spae: (a urvature o spae plays a sgnant role. I spae urves, then nward stress (surae ore P s generated A sort o pressure eld; (b a large number o urved thn layers orm the undretonal surae ore,.e. aeleraton eld.

7 8 Another Collmaton Mehansm o Astrophysal Jet urved surae as a surae ore. The normal stress P s towards the nsde o surae as shown n Fg.. A thn-layer o urved surae wll be taken nto onsderaton wthn a spheral spae havng a radus o and the prnpal rad o urvature whh are equal to the radus ( = =. From Eqs. (8 and (, we then get: K (6 Consderng N ( / P o Eq. (5, and substtutng Eq. (6 nto Eq. (5, the ollowng equaton s obtaned: P N (7 Sne the membrane ore N (servng as the lne stress an be assumed to have a onstant value, Eq. (7 ndates that the urvature generates the nward normal stress P o the urved surae. The nwardly dreted normal stress serves as a knd o pressure eld. Here, we gve an aount o urvature n advane. The soluton o metr tensor g s ound by gravtatonal eld equaton as the ollowng: 8G g T (8 where s the tensor, s the salar urvature, G s the gravtatonal onstant, s the veloty o lght, T s the energy momentum tensor. Furthermore, we have the ollowng relaton or salar urvature : g, g g, g (9 where k s emannan onneton oeent. I the urvature o spae s very small, the term o hgher order than the seond an be negleted, and tensor beomes: (,, The maor urvature o tensor ( s alulated as ollows: g g ( As prevously mentoned, emannan geometry s a geometry that deals wth a urved emann spae, thereore emann urvature tensor s the prnpal quantty. All omponents o emann urvature tensor are zero or lat spae and non-zero or urved spae. I an only non-zero omponent o emann urvature tensor exsts, the spae s not lat spae but urved spae. Although tensor has ndependent omponents, the maor omponent s the ase o,.e.,. Thereore, the maor urvature o tensor plays a sgnant role. 3.3 Curvature Control by Magnet Feld Let us onsder the eletromagnet energy tensor M. In ths ase, the soluton o metr tensor g s ound by 8G g M (3 Eq. (3 determnes the struture o spae due to the eletromagnet energy. Here, we multply both sdes o Eq. (3 by g, we obtan g g g g g ( tensor s represented by: (,, ( 8G g M 8G 8G 8G gm M M (5

8 Another Collmaton Mehansm o Astrophysal Jet 8 The ollowng equaton s derved rom Eqs. ( and (5 8G M Substtutng Eq. (6 nto Eq.(3, we obtan 8G M 8G M g g M (6 (7 On the other hand, 6 omponents o antsymmetr tensor are gven by eletr eld E and magnet eld B rom the relaton to Maxwell s eld equatons Ex, E 3 3 Ez Bz, 3 3 B x, y, Usng antsymmetr tensor whh denotes the 3 3 B y magntude o eletromagnet eld, the eletromagnet energy tensor M s represented as ollows; M g, g (8 (9 Aordngly, substtutng M nto Eq. (7, we get (3 Although tensor has ndependent omponents, the maor omponent s the ase o,.e.,. Thereore, Eq. (3 beomes g Thereore, or M, we have M M g M g g g 8G 8 G M M (3 33 Substtutng Eq. (3 nto Eq. (8, we have M E Fnally, rom Eqs. (3 and (33, we have G (3. (33 (3 7 where we let ( H /, 8, 3 ( m / s, ( Nm / kg, s a magnet eld n Tesla and s a maor omponent o spatal urvature (/ m. The relatonshp between urvature and magnet eld was derved by Mnam and ntrodued t n 6th Internatonal Symposum on Spae Tehnology and Sene (988 []. Eq. (3 s derved rom general method usng gravtatonal eld equaton. On the other hand, Lev-Cvta also nvestgated the gravtatonal eld produed by a homogeneous eletr or magnet eld, whh was expressed by 3 Paul []. I x s taken n the dreton o a B 38 B 8. B B ( B n Tesla m 9 /(36 ( F / m G 6.67 B,

9 8 Another Collmaton Mehansm o Astrophysal Jet magnet eld o ntensty F (Gauss unt, the square o the lne element s o the orm; where r= a ds (35, and are onstants,, k s Newtonan gravtatonal onstant (G, and x x are Cartesan oordnates (x x 3 =spae, x =t wth orthograph proeton. The spae s ylndrally symmetr about the dreton o the eld, and on eah plane perpendular to the eld dreton the same geometry holds as n Euldean spae on a sphere o radus a, that s, the radus o urvature a s gven by (36 Sne the relaton o between magnet eld B n SI unts and magnet eld F n CGS Gauss unts are desrbed as ollows: 3 ( dx ( dx ( dx ( x dx x dx a r 3 3 exp( x / a exp( x / a ( dx k F ( x ( x B F, then the radus o urvature a n Eq. (36 s expressed n SI unts as the ollowng (hangng symbol, k G, F B : a 8 (3.8 meters B Whle, salar urvature s represented by a GF GB k F G B a G B (37 (38 whh ondes wth Eq. (3. 3. Collmaton Mehansm Indued by Spatal Curvature around Magnet Feld As mentoned above, the ollmaton mehansm s summarzed as ollows. ( On the supposton that spae s an nnte ontnuum, ontnuum mehans an be appled to the so-alled vauum o spae. Ths means that spae an be onsdered as a knd o transparent elast eld. That s, spae as a vauum perorms the moton o deormaton suh as expanson, ontraton, elongaton, torson and bendng. We an regard spae as an nnte elast body lke rubber. ( From General elatvty, the maor omponent o urvature o spae an be produed by not only mass densty but also the magnet eld B as ollows (See 3.3 Curvature Control by Magnet Feld: G 38 B 8. B (39 Eq. (39 ndates that the maor omponent o spatal urvature an be ontrolled by magnet eld B. (3 I spae urves, then an nward normal stress P s generated (See 3. Generaton o Surae Fore Indued by Spatal Curvature. Ths normal stress,.e. surae ore serves as a sort o a pressure eld. P N / N (/ / ( ( where N s the lne stress o membrane o urved surae,, are the rad o prnpal urvature o urved surae. A large number o urved thn layers orm the undretonal surae ore,.e. aeleraton eld. Aordngly, the spatal urvature produes the aeleraton eld. ( From the ollowng lnear approxmaton sheme or the gravtatonal eld equaton (See

10 Another Collmaton Mehansm o Astrophysal Jet 83 Appendx B: Aeleraton Indued by Spatal Curvature: ( weak gravtatonal eld,.e. small urvature lmt, ( quas-stat, ( slow-moton approxmaton (.e. v/<<, we get the ollowng relaton between aeleraton o urved spae and urvature o spae: b g x dx a ( ( Eq. ( ndates that the aeleraton eld s produed n urved spae. (5 In the urved spae regon, the massve body m (kg exstng n the aeleraton eld s subeted to the ollowng ore F (N: Settng = 3 (.e. dreton o radus o urvature: r, we get: 3 F F m m g ( r dr ( From Eqs. (39, ( and (, we obtan the ollowng equatons: P N 8 G 9 N B. NB b b a (3 g ( r dr ( b a ( a Eq. (3 ndates the spatal pressure ndued by magnet eld. Eq. ( ndates the aeleraton atng on the surae o astrophysal et. It may be easy to understand by usng the spatal pressure as ompared wth the aeleraton. Next, we desrbe the pressure o the ylndral spae rom the shape o the astrophysal et. Fg. 5 shows the spatal pressure or aeleraton ndued by magnet eld atng on the surae o astrophysal et. Fg. 6 shows the spatal pressure as an nward normal stress P about spheral urved spae and ylndral urved spae. As shown n Fg. 6b, the prnpal rad o urvature and a yeld the ollowng: / P N( N(/ / N/ N/ a From Eq. (37, a G B Then we get P N a G B G N B (5 (6 (7 Aordngly, spatal pressure (normal stress rom the onentr spatal surae surroundng the et ollmates the et. However, sne the shape and ntensty o the magnet eld hange dependng on the turbulene o the plasma and the ondton o the shok wave nsde the astrophysal et, t may be possble to apply a (a (b Fg. 5 (a Spatal pressure rom the onentrally urved multlayer spae to the vnty surae o the ylnder et; (b Extended gure o (a.

11 8 Another Collmaton Mehansm o Astrophysal Jet (a (b Fg. 6 (a Spheral urved spae. I spae urves, then nward stress (surae ore P s generated A sort o pressure eld; (b Cylndral urved spae or Astrophysal Jet. spheral urved spae (Fg. 6a rather than ylndral urved spae (Fg. 6b. Sne the present spae s rgd, the lne stress o spae N seems to be expeted as large value.. Conluson Magnet ore model (magnet entrugal ore and magnet pressure s regarded as the most domnant theory whh solves the two problems o astrophysal et aeleraton and ollmaton at the same tme. That s, the areton dsk generates a helal magnet eld by twstng the magnet eld lnes, aelerates by magnet ore, and narrows the et by magnet tenson (pnh. On the other hand, sne the magnet eld s present n the astrophysal et, a spatal urvature s generated ndued by magnet eld n the surroundng spae; a spatal pressure n spae equvalent to the gravtatonal eet s generated n the dreton o the nteror o the astrophysal et as well as the pnh ore rom the outer rumerental surae o the astrophysal et. Aordngly, although ts eet may be small than magnet pressure, another ollmaton mehansm o astrophysal et an be possble to exst. eerenes [] Contopoulos, I., Gabuzda, D., and Kylas, N., eds. 5. The Formaton and Dsrupton o Blak Hole Jets. Sprnger. [] Dermer, C. D., and Menon, G. 9. Hgh Energy adaton rom Blak Holes. Prneton Unversty Press. [3] Kato, S., Fukue, J., and Mneshge, S. 8. Blak-Hole Areton Dsks Towards a New Paradgm. Kyoto Unversty Press. [] Shbata, K., Fukue, J., Matsumoto,., and Mneshge, S., eds Atve Unverse Physs o Atvty n Astrophysal Obets. Tokyo: Shokabo. [5] Fukue, J. 7. Shnng Blak-Hole Areton Dsks. Pleades Publshng Co. Ltd. [6] Mneshge, S. 6. Blak Hole Astrophyss. Nppon Hyoron sha Co. Ltd. [7] Koyama, K., and Mneshge, S. 7. Blak Hole and Hgh-Energy Phenomena, Nppon Hyoron sha Co., Ltd. [8] Mnam, Y. 6. A Journey to the Stars: Spae Propulson Brought about by Astrophysal Phenomena Suh as Areton Dsk and Astrophysal Jet. Global Journal o Tehnology & Optmzaton 7:. do:

12 Another Collmaton Mehansm o Astrophysal Jet 85.7/ [9] Mnam, Y. 5. Contnuum Mehans o Spae Seen rom the Aspet o General elatvty An Interpretaton o the Gravty Mehansm. Journal o Earth Sene and Engneerng 5: 88-. [] Wllams, C., (ed.; Mnam, Y., (Chap. 3, et al. 5. Advanes n General elatvty esearh. Nova Sene Publshers. [] Mnam, Y Spae Stran Propulson System. 6th Internatonal Symposum on Spae Tehnology and Sene (6th ISTS, Vol., [] Paul, W. 98. Theory o elatvty. New York: Dover Publatons, In. [3] Flügge, W. 97. Tensor Analyss and Contnuum Mehans. Berln, Hedelberg, New York: Sprnger-Verlag. [] Fung, Y. C.. Classal and Computatonal Sold Mehans. World Sent Publshng Co. Pre. Ltd. [5] Hans, Z An Introduton to Thermomehans. North-Holland Publshng Company. [6] Borg, S. F Matrx-Tensor Methods n Contnuum Mehans. D. Van Nostrand Company.

13 86 Another Collmaton Mehansm o Astrophysal Jet Appendx A: Contnuum Mehans o Spae-Tme Gven a pror assumpton that spae as a vauum has a physal ne struture lke ontnuum, t enables us to apply a ontnuum mehans to the so-alled vauum o spae. Assumng that spae as vauum s an nnte ontnuum, spae an be onsdered as a knd o transparent elast eld, and ts struture s determned by emannan geometry. That s, spae as a vauum perorms the motons o deormaton suh as expanson, ontraton, elongaton, torson and bendng. The latest expandng unverse theores (Fredmann, de Stter, nlatonary osmologal model support ths assumpton. Sne the subet o our study s a our-dmensonal emann spae as a urved spae, we asrbe a great deal o mportane to the urvature o spae. We a pror aept that the nature o atual physal spae s a our-dmensonal emann spae, that s, three dmensonal spae (x = x, y = x, z = x 3 and one dmensonal tme (w = t = x, where s the veloty o lght. These our oordnate axes are denoted as x ( =,,, 3. The square o the nntesmal dstane ds between two nntely proxmate ponts x and x + dx s gven by equaton o the orm: ds g dx dx (A where g s a metr tensor. The metr tensor g determnes all the geometral propertes o spae and t s a unton o ths spae oordnate. In emann spae, the metr tensor g determnes a emannan onneton oeent k, and urthermore determnes the emann urvature tensor or pk, thus the geometry o spae s determned by a metr tensor. p k emannan geometry s a geometry whh provdes a tool to desrbe urved emann spae, thereore a emann urvature tensor s the prnpal quantty. All the omponents o emann urvature tensor are zero or lat spae and non-zero or urved spae. I a non-zero omponent o emann urvature tensor exsts, the spae s not lat spae, but urved spae. In urved spae, t s well known that the result o the parallel dsplaement o vetor depends on the hoe o the path. Further, the omponents o a vetor der rom the ntal value, ater we dsplae a vetor parallel along a losed urve untl t returns to the startng pont. An external physal aton suh as the exstene o mass energy or eletromagnet energy yelds the strutural deormaton o spae. In the deormed spae regon, the nntesmal dstane s gven by: ds g dx dx (A g the metr tensor o deormed spae regon, and we use the onveted oordnates ( where x x. As shown n Fg. A, the lne element between the arbtrary two near ponts (A and B n spae regon S (beore strutural deormaton s dened as ds g dx, the nntesmal dstane between the two near ponts s gven by Eq. (A: ds g dx dx. Let us assume that a spae regon S s struturally deormed by an external physal aton and transormed to spae regon T. In the deormed spae regon T, the lne element between the dental two near pont (A and B o the dental spae regon newly hanges, ders rom the length and dreton, and beomes. ds g dx

14 Another Collmaton Mehansm o Astrophysal Jet 87 Fg. A Fundamental struture o Spae. Thereore, the nntesmal dstane between the two near ponts usng the onveted oordnate ( x ds g dx dx x s gven by: (A3 The g s the transormed base vetor rom the orgnal base vetor orgnal metr tensor get: g and the g s the transormed metr tensor rom the g. Sne the degree o deormaton an be expressed as the hange o dstane between the two ponts, we ds ds g dx dx g dx dx ( g g dx dx r dx dx (A Hene the degree o geometral and strutural deormaton an be expressed by the quantty denoted hange o metr tensor,.e. r g g (A5 On the other hand, the state o deormaton an be also expressed by the dsplaement vetor u (see Fg. A. From the ontnuum mehans [3-6], usng the ollowng equatons: du : g u dx (A6 d s ds du ds g u: dx (A7 Here we use the usual notaton : or ovarant derentaton. As s well known, the partal dervatve tensor equaton. The ovarant dervatve systems. u u, u k k u u x, s not : s tensor equaton and an be arred over nto all oordnate From usual ontnuum mehans, the nntesmal dstane ater deormaton beomes as ollow [3]: The terms o hgher order than seond k ds rdx dx ( u: u : u : u k: dx dx (A8 d s u k u k: : an be negleted the dsplaement s o small enough value. As the atual physal spae an be dealt wth the mnute dsplaement rom the tral alulaton o stran, we get: r u : u : (A9 Whereas, aordng to the ontnuum mehans [3], the stran tensor e s gven by:

15 88 Another Collmaton Mehansm o Astrophysal Jet e r ( u : u : (A So, usng Eqs. (A5 and (A we get: ds ( g g dx dx edx dx (A ds where g, g s a metr tensor, e s a stran tensor, and s the square o the nntesmal dstane between ds ds two nntely proxmate ponts x and x + dx. From Eq. (A, the stran o spae s desrbed as ollows: e / ( g ' g (A Eq. (A ndates that a ertan geometral strutural deormaton o spae s shown by the onept o stran. In essene, the hange o metr tensor ( g g due to the exstene o mass energy or eletromagnet energy tensor produes the stran eld e. Namely, a ertan strutural deormaton o spae-tme s desrbed by stran tensor e ; the physal stran s generated by the derene o a geometral metr o spae-tme. Appendx B: Aeleraton Indued by Spatal Curvature A massve body auses the urvature o spae-tme around t, and a ree partle responds by movng along a geodes n that spae-tme. The path o ree partle s a geodes lne n spae-tme and s gven by the ollowng geodes equaton: d x d s emannan onneton oeent, k k dx dx d d where k s proper tme, x s our-dmensonal emann spae, that s, three dmensonal spae (x = x, y = x, z = x 3 and one dmensonal tme (w = t = x, where s the veloty o lght. These our oordnate axes are denoted as x ( =,,, 3. Proper tme s the tme to be measured n a lok restng or a oordnate system. We have the ollowng relaton desrbed rom an nvarant lne element ds between Speal elatvty (lat spae and General elatvty (urved spae: From Eq. (B, the aeleraton o ree partle s obtaned by d (B g dx g dt (B d x d k k dx dx d d (B3 As s well known n General elatvty, n the urved spae regon, the massve body m (kg exstng n the aeleraton eld s subeted to the ollowng ore F (N : F k dx dx k mk m g k u u m d d (B where u, u k are the our veloty, Г k s the emannan onneton oeent, and τ s the proper tme. From Eqs. (B3 and (B, we obtan:

16 Another Collmaton Mehansm o Astrophysal Jet 89 d x d dx d dx d k k k g ku u (B5 Eq. (B5 yelds a more smple equaton rom the ondton o lnear approxmaton, that s, weak-eld, quas-stat, and slow moton (speed v << speed o lght : u : g (B6 On the other hand, the maor omponent o spatal urvature n the weak eld s gven by (B7 In the nearly Cartesan oordnate system, the value o are small, so we an neglet the last two terms n Eq. (B7, and usng the quas-stat ondton we get (B8 From Eq. (B8, we get ormally ( x dx (B9 Substtutng Eq. (B9 nto Eq. (B6, we obtan b g x dx ( (B a where : aeleraton (m/s, g : tme omponent o metr tensor, a-b: range o urved spae regon(m, x : omponents o oordnate ( =,,,3, : veloty o lght, : maor omponent o spatal urvature (/m. Eq. (B ndates that the aeleraton eld s produed n urved spae. The ntensty o aeleraton produed n urved spae s proportonal to the produt o spatal urvature and the length o urved regon. Eq. (B yelds more smple equaton rom above-stated lnear approxmaton ( u, F m g b u u m g m m g ( x dx (B a Settng = 3 (.e., dreton o radus o urvature: r, we get Newton s seond law: b 3 3 F F m m g ( r dr m g (B The aeleraton ( o urved spae and ts emannan onneton oeent ( a 3 are gven by: g g 3, g 3, 3 33 (B3 where : veloty o lght, g and g 33 : omponent o metr tensor, g,3 : g / x 3 = g / r. We hoose the spheral oordnates t = x, r = x 3, θ = x, = x n spae-tme. The aeleraton s represented by the equaton both n the derental orm and n the ntegral orm. Pratally, sne the metr s usually gven by the soluton o gravtatonal eld equaton, the derental orm has been ound to be advantageous.