Universal 2+1-Dimensional Plane Equations in General Relativity and
  Evolutions of Disk Nebula
Microsoft Word - EvoluGRG.doc
Universal 2+1-Dimensional Plane Equations in General 
Relativity and Evolutions of Disk Nebula 
Yi-Fang Chang 
Department of Physics, Yunnan University, Kunming, 650091, China 
(e-mail: yifangchang1030@hotmail.com)  
ABSTRACT 
The general relativity is the base for any exact evolutionary theory of large scale 
structures. We calculate the universal 2+1-dimensional plane equations of 
gravitational field in general relativity. Based on the equations, the evolutions of disk 
nebula are discussed. A system of nebula can form binary stars or single star for 
different conditions. While any simplified linear theory forms only a single star 
system. It is proved that the nonlinear interactions are very general, so the binary stars 
are also common. 
Key words: general relativity, evolution, disk nebula, binary stars. 
PACS: 04.20. q; 97.10. Cv; 98.38. Ly; 97.80. d.  
At the present the general relativity with a precise formulation of the theory is the best 
astronomic theory for various large scale structures of space-time [1,2]. Since the general solutions 
of Einstein ¯s fi el d equati ons are ver y co mpl ex, so far,some exact solutions [1,2,3], including from 
the Schwarzschild solution, Eddington-Robertson solution, and various Kerr solutions to the 
gravitational lensing in metric theories of gravity [4], spherically symmetric space-times in 
massive gravity [5], and all static circularly symmetric perfect fluid solutions of 2+1 gravity [6]. 
They are mostly some static solutions. 
On the other hand, the evolutions of various celestial bodies exist widely in the universe. In 
particular, astronomers observed that binary star systems are very common from 1989 [7,8]. The 
generality of binary star is proposed. Fact, Spyrou presented the results of a systematic study of 
relativistic celestial mechanics of binary stars in the post-Newtonian approximation (PNA) of 
general relativity [9], and proposed star models determined the inertial and rest masses of binary 
stars [10]. Itoh, et al., discussed the equation of motion for relativistic compact binaries with the 
strong field [11]. Alvi and Liu studied the dynamics of a cluster of collisionless particles orbiting a 
non-rotating black hole, which is part of a widely separated circular binary, and found that the 
most stable orbits are close to the companion's orbital plane and retrograde with respect to the 
companion's orbit [12]. Gu¨¦ro and Letelier studied binary systems around a black hole [13]. 
Hansen discussed the motion of the binary system composed of an oscillating and rotating 
coplanar dusty disk and a point-like object [14]. 
The most exact celestial evolutionary theory should be based on the general relativity. Wilson 
and Mathews [15,16] reported preliminary results obtained with a relativistic numerical evolution 
code. Their dynamical calculations suggest that the neutron stars may collapse to black holes prior 
to merger. Baumgarte, et al., studied the quasiequilibrium model on binary neutron stars in general 
relativity [17]. Then they presented a new numerical method for the construction of 
quasiequilibrium formulation of black hole-neutron star binaries, and solved the constraint 
equations of general relativity, and solved these coupled equations in the background metric of a 
Kerr-Schild black hole, which accounts for the neutron star's black hole companion [18]. 
Taniguchi, et al., presented new sequences of general relativistic, quasiequilibrium black 
hole-neutron star binaries. They solved the gravitational field equations coupled to the equations of 
relativistic hydrostatic equilibrium for a perfect fluid [19]. Shibata, et al., presented a new 
implementation for magnetohydrodynamics (MHD) simulations in full general relativity 
(involving dynamical space-times), and performed numerical simulations for standard test 
problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic 
magnetized Bondi ow in stationary space-time, and a long-term evolution for self-gravitating 
system composed of a neutron star and a magnetized disk in full general relativity [20]. Faber, et 
al., calculated the dynamical evolutions of merging black hole-neutron star binaries that construct 
the combined black hole-neutron star space-time in a general relativistic framework. They treated 
the metric in the conformal flatness approximation, and assumed that the black hole mass is 
sufficiently large compared to that of the neutron star so that the black hole remains fixed in space 
[21].  
One of more successful theories of the formation of binary stars is the fragmentation 
proposed by Boss, et al.[22-24], which supposes that binary stars are born during the protostellar 
collapse phase under their own gravity. Using computer simulation they obtained that an initial 
spherical cloud in rapidly rotation collapses and flattens to a disk, which later fragments into a 
binary system. Based on the basic equations of a rotating disk on the nebula, we use the qualitative 
analysis theory of nonlinear equation and obtain a nonlinear dynamical model of formation of 
binary stars [25]. Under certain conditions a pair of singular points results in the course of 
evolution, which corresponds to the binary stars. Under other conditions these equations give a 
single central point, which corresponds to a single star. This new method and model may be 
extended and developed. Steinitz and Farbiash established the correlation between the spins 
(rotational velocities) in binaries, and show that the degree of spin correlation is independent of 
the components separation. Such a result might be related for example to the nonlinear model for 
the formation of binary stars from a nebula [26]. 
So far, the general relativity is mainly applied to neutron star-neutron star binaries, or black 
hole-neutron star binaries. Wilson, et al., chose the 3-metric to be conformally flat [15,16]. In 
Cartesian coordinates the line element can be written [17]         
))((4222 dtdxdtdxdtds jjiiij .                        (1) 
In more general cases, for a flatten disk nebula we take a universal 2+1-dimensional plane 
metric into diagonal form with 2+1-paramers         
2 ),,( drtfdredxedxdxgds vkiik .                       (2) 
where ),,( rtvv
and ),,( rt . Denoting by 210 ,, xxx , respectively, the 
(2+1)-dimensional plane coordinates ct, r, , we have for the nonzero components of the metric 
tensor the expressions [27]:         
),,(,, 221100 rtfgegeg
,                                  (3) 
and     
1221100 ,, fgegeg v .                                     (4) 
With these values the calculation leads to the following Christoffel symbols: 
00 vvev
                                   
(5a) 
v                               
(5b) 
11 fef
                               
(5c) 
f                                 
(5d) 
11 fef
.                        (5e) 
Here the prime means differentiation with respect to r, while a dot on a symbol means 
differentiation with respect to ct. All other components are zero. Therefore, we may derive the 
components of the field equations: 
''''2[
                     
(6a) 
                          
(6b) 
])'(''''2[
vvveveG v
                    
(6c) 
                        
(7a) 
                   
(7b) 
                  
(7c) 
These equations may possess various solutions. 
The energy-momentum tensor of perfect fluid is [3] 
abbaab pguupT )( .                                            (8) 
Assume that b
0 and d
0 , so    
1 vdf
,                     (9a) 
)'''''2(''
''''2[
2 vvvdvfb
,                (9b) 
)(2[]
f                     
(9c) 
Here bbbb 321 and dddd 321 . For the evolutions of disk nebula, we consider 
a system only with time. Let yffdtfdx //)(ln, , the equations (9a) become 
xxvxy
,                                           (10a) 
yyvxy
,                                           (10b) 
From dx/dt=0 and dy/dt=0 we derive four solutions: 
(),0,(),,0(),0,0(),(
vvyx .                    (11) 
Using a qualitative analysis theory of the nonlinear equations, the characteristic matrix of the 
equations (10) is       
2/2/ 2/ 
2/2/2/
yvbxby
dxxvdy
.                                (12) 
Its characteristic equation is          
02 DT .                                                   (13) 
While DT 42 . For a singular point (0,0), 
0,04/, 1
11 vDvT .                                      (14) 
It is a critical nodal point. For a singular point (0, v ), 
1(,0)
T .        (15) 
It is a saddle point. For a singular point ( v ,0), 
1(,0)
T .         (16) 
It is also a saddle point. For a singular point )
.])()1(4[
)1)(1(
11112
vdbdb
                    
. (17) 
In the usual case it is a nodal point. When 11db >1, i.e., T<0, the point is a stable sink, this system 
may form the binary stars. When 11db <1, i.e., T>0 and D<0, the point is a saddle point, this 
system forms a single star for (0,0) singularity. They seem to correspond to thin nebula and dense 
nebula, respectively. 
When the singularity is a nodal point, ffyx /,
may be represented by 
)0(, 11 khbeyaex
. So 0ln,0 11
ktht ebydtfeaxdt
is also 
a nodal point. 
The equations (6) are simplified to           
                                       
(18a) 
2 aTf
                                 
(18b) 
2 ]2[
aTve v .                                      (18c) 
Then 
04 aTexy
, and the equations (18b) and (18c) become 
aTexxv
,                                       (19a) 
aTeyyv
,                                       (19b) 
From dx/dt=0 and dy/dt=0 we derive solutions:    
2,1 XvaTevvx
,                         (20a)    
2,1 YvaTevvy
.                          (20b) 
Using a qualitative analysis theory, the characteristic matrix of the equations (19) is 
6       
2/ 0 
.                                             (21) 
Its characteristic equation is (13). In this case, for a singular point ),( 11 yx , 
111 YXXYDYXT .     (22a) 
It is a nodal point, and is a stable sink. For a singular point ),( 21 yx , 
222 YXXYDYXT .      (22b) 
It is a saddle point. For a singular point ),( 12 yx , 
333 YXXYDYXT .        (22c) 
It is also a saddle point. For a singular point ),( 22 yx , 
444 YXXYDYXT .     (22d) 
It is a nodal point, and is an unstable source. For the linear equations of simplified (10) or 
(19), only singularity (0,0) is a nodal point, and forms only a single star system. 
Based on the universal 2+1-dimensional plane equations of gravitational field in general 
relativity, the evolutions of disk nebula are discussed. A system of nebula can form binary stars or 
single star for different conditions. Here the nonlinear interactions play an important role. They 
are necessary conditions of the formation of binary stars, but are not sufficient conditions. 
Generally, any stable star should be a stable fixed point in the evolutionary process in 
astrophysics. Assume that an evolutionary equation is y=f(x), which corresponds to an equation of 
the invariant point x*=f(x*). Therefore, if f(x) is a nonlinear n-order function, there will be 
possibly multiple (stable or unstable) invariant points. While f(x)=ax+b is a linear function, there 
will be only an invariant point x*=b/(1-a), namely, it can form only a single star. Since the 
nonlinear interactions are very general, the binary and multiple stars are also common.  
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