General Relativity as a fully singular Lagrange system
1 General Relativity as a fully singular Lagrange system
T. Mei (Department of Journal, Central China Normal University, Wuhan, Hubei PRO, People’s Republic of China E-Mail: [email protected] [email protected] ) Abstract:
We present some gauge conditions to eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations of general relativity; at the same time, we present the corresponding Lagrangian in which there is not any quadratic term of first time derivative that can leads to those vierbein forms of the Einstein equations without second time derivative term by the corresponding Euler-Lagrange equations. General relativity thus becomes a fully singular Lagrange system.
Keywords: general relativity; the simplest constraint conditions; a fully singular Lagrange system Introduction
As well known, if x is the time coordinate of a frame of reference, then the ten Einstein equations mnmnmn p Tc GRgR =- can be divided into two categories: One is the four equations lll p Tc GRgR =- in which there is not any second time derivative term, another is the six equations ijijij Tc GRgR p=- including the six second time derivative terms ,, ij g . On the other hand, we have presented a group of gauge conditions to eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations in Ref. [1], however, for which so far I cannot find out the corresponding Lagrangian density. In this paper, we not only present some gauge conditions that can eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations, but also, at the same time, present the corresponding Lagrangian in which there is not any quadratic term of first time derivative that can leads to those vierbein forms of the Einstein equations without second time derivative term by the corresponding Euler-Lagrange equations. General relativity thus becomes a fully singular Lagrange system. In Sect. 1 of this paper, we present a Lagrangian density of general relativity, which is separated to kinetic energy and potential energy terms naturally; In Sect. 2, we investigate in detail a group of gauge conditions such that general relativity becomes a fully singular Lagrange system; In Sect. 3, we generate the group of gauge conditions discussed in Sect. 2 to more general cases. All symbols and conventions follow Ref. [1]. A Lagrangian density of general relativity
As well known, in general relativity, the Einstein-Hilbert action reads ∫∫∫ --¶ ¶+-=-= , )(dπ16 dπ16dπ16 srsmrmrsrsm GG gggxxGcLxgGcxRgGcS (1-1) )( G(1) srsrabsbrrasab
GGGG -= gL . (1-2) We introduce ~ g gggg jiijij -= , which satisfies ijkjik gg d= ~ , ij g ~ is thus the inverse matrix of ij g , and, further, the 3-dimentional Christoffel symbol can be written to the form ( ) jlmljmmjlijilm gggg , , , ~21~ -+=G . Based on ij g ~ and ilm G ~ , we define ( ) , ~~~~ 1 lmijlmijjmil gggggL GG--= (1-3) ( ) ( )( ) ( )( ) ( ) ; ~~~~~~~~~ 21 ~~~~~~~~~121 ~~~~~~~ mlmlijmjllimijjijikkijijk mlmlijmjllimijjijikkijijk mlmlijmjllimkikjkijkij ggggg ggggggg gggL
GGGGGG GGGGGG GGGGGGGG rssr lll ----= ---= ---= (1-4) It can be proved that the remainder of
G(1) Lg - and ( ) GUGK
LLg -- is a total derivative: ( ) ¶¶-¶¶-¶ ¶+--=- nnmnmnm xggxggg gxLLgLg , (1-5) we therefore can use GUGKG(2)
LLL -= (1-6) as a Lagrangian density of general relativity. As well known, the coordinate variables ( ) , , , xxxx in general relativity are four parameters. In the following discussion, we shall assume that the decomposition of time and space has been finished by the ADM decomposition [2] and by x we denote time coordinate. If x is the time coordinate of a frame of reference, from (1-4) we see that there is not any time derivative term in GU L , and all time derivative terms appear in GK L ; hence, GK L and GU L can be regarded as kinetic energy and potential energy terms, respectively. Especially, for the metric of an accelerated, rotating frame of reference [3, 4, 5] ijijiii gctggctctg dvv = ·== ·- (cid:215)+-= , )( , )()(1
002 2 200 rrrW , (1-7) we have GUGK == LL since = ij G and = ilm G , this is a due result, because there is not real gravitational field for an accelerated, rotating frame of reference.
2 General relativity as a fully singular Lagrange system under a group of gauge conditions
In principle, for removing all second time derivative terms in the ten Einstein equations, we must use tetrad am ˆ e rather than metric tensor mn g as basic variables. Because if we use metric tensor mn g as basic variables, then for choosing special form of metric tensor, what we can use are only four gauge conditions provided by coordinate transformation )~( nmm xxx = , since metric tensor cannot be changed by local Lorentz transformation )(~)()( ˆˆˆˆ xexxe bmabam L= ; however, there are the six dynamical variables ij g , i.e., the six second time derivative terms ,, ij g , in the ten Einstein equations. Generally speaking, four conditions cannot eliminate six terms. On the other hand, if we use tetrad am ˆ e as basic variables, then there are only six dynamical equations including second time derivative terms in the ten Einstein equations yet, but for choosing special form of tetrad, ten gauge conditions can now be used, the four are provided by coordinate transformation )~( nmm xxx = and the six are provided by local Lorentz transformation )(~)()( ˆˆˆˆ xexxe bmabam L= . We therefore regard tetrad field am ˆ e as basic variables of gravitational field, in the following discussion, mn g and mn g are as the abbreviation for naam ˆˆ ee and nama ˆˆ ee , respectively. Substituting naammn ˆˆ eeg = and namamn ˆˆ eeg = to (1-3) and (1-4), we obtain the vierbein forms of GK L and GU L immediately. However, we write GK L given by (1-3) to another form (2-1), which is convenient for the following discussion. Defining
00ˆ 0ˆ0ˆˆˆ eeeee iaiaia -= , which satisfies babiiaijajia eeee ˆˆˆˆˆˆ , dd == , ia e ˆ is thus the inverse matrix of ai e ˆ ; and notice that
00 0ˆ0ˆˆˆˆˆ ~ g eeeeg bababjaiij -= h . Using these formulas, we obtain ( ) ( ) , 1 ~~~~ 1 ˆˆˆˆ00 0ˆ0ˆˆˆ00 0ˆ0ˆˆˆ00 0ˆ0ˆˆˆ00 0ˆ0ˆˆˆ00 0ˆˆ0ˆˆˆˆˆˆ00GK dcbadcdcbabadbdbcaca lmmdlcijjbiadvcubqapuvpqqvpu YYgeeg eeg eeg eeg eeeeeeeegggggL - -- - --= --= hhhh GG (2-1) where ( ) ( ) [ ] ( ) ( ) . 12121 21
0ˆ , 0ˆ , ˆ0ˆˆ0ˆ000ˆˆ , ˆ , 0ˆˆˆ , ˆ , ˆˆ , ˆ , ˆˆ0 0ˆˆˆˆ -+-++ -+--= = iiiabibaijjijbia iaiaibibibiaijjbiaba eeeeeegeeeeee eeeeeeg eeY lllggg lllll G (2-2) The vierbein forms of the Einstein equations have been given in Ref. [1]. For removing all quadratic terms of first time derivative of the Lagrangian (2-1), we first prove a theorem. As well known, when we investigate a system with constraint conditions, an algebraic constraint in which there is not derivative term can be putted directly in the Lagrangian density of the system, but, generally speaking, we cannot do like so for differential constraint including derivative term. However, if a differential constraint appears in quadratic form in the Lagrangian density of a system, then we can prove the following theorem. Theorem.
If the Lagrangian density ) ,( , l jj aa L of a system whose basic variables are ) , 2, 1,( Na a L = j has the form [ ] ) ,() ,( ) ,() ,( ,2 , , , llll jjjjcjjjj aaaaaaaa LWL ¢+= , then if we add the constraint condition
0) ,( , = l jjc aa , then the term [ ]
2 , , ) ,( ) ,( ll jjcjj aaaa W in ) ,( , l jj aa L can be removed directly, namely, we can use ) ,( , l jj aa L ¢ as the Lagrangian density of the field a j . The proof of the theorem is quite simple. The Euler-Lagrange equation corresponding to ) ,( , l jj aa L is [ ] { } [ ] { } [ ][ ] , ) ,() ,( ) ,() ,() ,(2) ,() ,( ) ,() ,() ,(2) ,() ,( ) ,() ,( ) ,( ) ,( ) ,( ) ,( ) ,() ,( , , , , , , ,2 , , , , , ,2 , , , , , , 2 , ,2 , , , , , l tlt l ttttl tl ttttt l tlt l ttltt l tlt j jjj jj j jjcjjcjjjjcj jj j jjcjjcjjjjcj jj j jjj jj j jjcjjj jjcjj j jjj jj a bba bb a bbbbbbbba bb a bbbbbbbba bb a bba bb a bbbba bbbb a bba bb LL WW WW LL WW LL ¶¢¶¶-¶¢¶+ ¶¶+¶¶¶- ¶¶+¶¶= ¶¢¶¶-¶¢¶+ ¶¶¶-¶¶= ¶¶¶-¶¶ if we add the constraint condition
0) ,( , = l jjc aa , then the above Euler-Lagrange equation becomes l tltl tlt j jjj jjj jjj jj , , , , , , ) ,() ,() ,() ,( a bba bba bba bb LLLL ¶¢¶¶-¶¢¶=¶¶¶-¶¶ . This means that we can obtain the equation of motion of the field a j by the Euler-Lagrange equation corresponding to ) ,( , l jj aa L ¢ . For example, the action of electromagnetic field in flat spacetime is ( ) ( ) , d212121 d ))((41 d , , 42 , , ,4 , , , ,4EM ∫∫∫ ¶ -¶+ -+-= ----= s r rss rrmml lsrsr mmmnnmmnnm x AAAAxAejAAAx AejAAAAxS (2-3) where the last term is an integral of a total derivative, which does not impact on the derivation of the equations of motion. If we add the Lorenz gauge condition , = l l A , then we can remove the quadratic term ( )
2 , l l A in (2-3) and, thus, (2-3) becomes ∫ --= mmsrsr AejAAxS , ,4EM
21 d~ , (2-4) from which we obtain the equation of motion mnm n ejA = , , . (2-5) Under local Lorentz transformation, the manner of transformation of am ˆ e reads )(~)()( , )(~)()( ˆˆˆˆˆˆˆˆ xexxexexxe mbbamabmabam LL == , (2-6) where , ˆˆˆˆˆ ˆ ˆˆˆˆˆˆ abgbagabaggb dLLdLL == . We first choose three gauge-fixing terms, so called the Schwinger time gauge condition: , 3 2, 1, 0, == ae a (2-7) which can be implemented by choosing appropriate )( ˆ0ˆ x a L of )( ˆˆ x ab L indicated by (2-6). Under the condition (2-7) we have ( ) . , , , ;~ , , , ; , , , 3; 2, 1, 0, eeegeegeegeeeg eegeeeegeegeg eeeeeeeeeeie jaaiijiaaiaa jaiaijjaiajiijii jiaijababiiaiaaii ==-==+ -= =+-=-=-= ==-= === - dd (2-8) where [ ] am ˆ4 det ee = and [ ] ai ee ˆ3 det = are the determinants of the 4 × [ ] am ˆ e and the 3 × [ ] , ˆ ai e respectively. Under the time gauge condition (2-7), (2-1) and (2-2) become ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 2 2 22126132 2 +++-+--+- = +++--- = - = YYYYYYYYYe YYYYYYYYYe YYeL aa dcbadcbadbca hhhh (2-9) ( ) ( ) [ ] , 21 , ˆ , ˆˆ , ˆ , ˆˆ0ˆ00ˆ0ˆˆˆˆ iaiaibibibiaijjbiaba eeeeeeeeeeY lllll
G -+-== (2-10) respectively. For ba Y ˆˆ we have ( ) , ~ , , , ,0000000ˆˆˆˆ ˆ , ˆ , ˆ0ˆ00ˆ3ˆ3ˆ2ˆ2ˆ1ˆ1ˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ll lll GGh h hhh --=== -=++== === ggggggee eeeeeYYYYY YYYYYY lmijijijjbiaba a iaiiababaaa dcdbcababccaababba (2-11) notice that aa Y ˆˆ can be expressed by metric tensor. According to (1-3), (2-9) and (2-11), we see that under the time gauge condition (2-7), we have ( ) ( ) ( ) ( ) ( ) ( ) . 0 2 2 221261 ~31~~ ‡+++-+--= - YYYYYYYY ggg ijillmijjmil
GGG (2-12) It is obvious that there is a negative kinetic energy term in (2-9), of which the concrete form is ( ) ( ) . 3 2 ,32~132132
2 ,000GNK 2 ,00020002 ˆˆ00GNK --=- ==--= ll ll G ggggLg ggggggYgL lm lmijijaa (2-13) The above forms of the negative kinetic energy term have been discussed in detail in Ref. [1]. Using the formulas in (2-7) and (2-8), it can be proved that GU L given by (1-4) becomes ¶¶-¶¶¶ ¶+--= jiajajjaiaikk xeexeeeexeeLUeeL ˆˆˆˆ30ˆ030ˆ0GV0ˆ , 000ˆGU , (2-14) ( ) ( ) . 2141 , ˆ , ˆ , ˆ , ˆ , ˆˆˆˆˆˆˆˆˆˆGV ˆ , ˆ , ˆˆˆ -- +--= -= b jmb mja ila lijbiaibjajdidbamclc a ija jikbjbiak eeeeeeeeeeeeL eeeeeU h (2-15) Hence, up to a total derivative, we can employ MGV0ˆ , 000ˆGK3Total(1)Total(1)430ˆ0)1(
LLUeeLGcLxxLeeS kk + ++== ∫ (2-16) as the action of the system whose thirteen basic variables are { } ai ee ˆˆ0 , g , where GK L , i U and GV L are given by (2-9) and (2-15), respectively; )( M xL is the Lagrangian density of matter. The Euler-Lagrange equations corresponding to the action (2-16) are: =-=¶ ¶¶-¶ ¶ gg nng Q eGece LeeeLee , (2-17) ˆ0ˆ033ˆ , Total(1)30ˆ0ˆTotal(1)30ˆ0 =-=¶ ¶¶-¶ ¶ iaaiai eGece LeeeLee Q nn ; (2-18) where ( ) ( )
821 1
Tec GLLUexee kk pQ --+¶ ¶= , (2-19) ( ) aijbjibbjiabkkbaa Tec GeeSeSexee phQ --+ ¶ ¶= , (2-20) ( ) ( ) ( ) ( ) ; 821~ ~ 1 21 1 1 ˆ0ˆ04GVˆ0ˆ0 , ˆ , ˆˆˆ0ˆ0 , ˆ , ˆˆˆˆˆˆˆ0ˆ , 0ˆˆ0ˆ , 0 ˆ0ˆ0ˆˆˆˆˆˆˆ0ˆ , 033 , ˆ , ˆ0ˆˆ0ˆˆ0ˆ0 , ˆ , ˆˆ0ˆˆ0ˆ0ˆ0GKˆ0ˆ0 ˆ0ˆˆˆ0ˆ0ˆˆˆ033ˆˆˆ0ˆ303ˆˆˆ0ˆ0 iaiajkbkjbbikja lmcmlcmcibmbiclajbjjiaijaj bijbaibjajbiakbkj jbjbbijajkbkjbbjkaiia bjiccbijccjbabibaia
Tec GLeeeesee eeeeeeeeeUeUee seeeeeeeexe eeeSeeeeSeeeLee SeeSeeexeSexee phhhQ lll -- -+ - -+-+ + -¶ ¶+ ----- -¶ ¶+ ¶ ¶-= (2-21) In (2-20) and (2-21), -= -= baibbbiaabibbbiaai YeYeeYeYeeS ˆˆˆˆˆˆ0ˆ0ˆˆˆˆˆˆ0ˆ0ˆ0ˆ , (2-22) ( ) ( ) ( )( ) ; 21 21~ ˆ , ˆ , ˆˆˆˆˆˆ ˆ , ˆ , ˆˆˆˆˆˆˆ , ˆ , ˆˆˆˆˆ c lmc mlmclbjaibjbia c lmc mlicjbjcibmblaa lma mlmcjclbibaij eeeeeeee eeeeeeeeeeeeees - -- --+--= (2-23) . , ˆ , M30ˆ0ˆ M30ˆ030ˆ0ˆ ˆ , 0 M30ˆ0ˆ0 M30ˆ030ˆ00ˆ ¶ ¶¶-¶ ¶= ¶ ¶¶-¶ ¶= aiaiia e Leee Leeee cT e Leee Leeee cT nn g nngg (2-24) All the action )1( S , the Lagrangian Total(1) L and the vierbein forms of the Einstein equations under the time gauge condition given by (2-16) and (2-17) ~ (2-24) respectively have been presented and analyzed in Ref. [1] (some formulas rewritten here are in new forms); for example, in Ref. [1] it has been pointed out that we can only obtain ten independent equations from (2-17) ~ (2-24) since in which there are three identities ) 3; 2, 1, ,( ˆ0ˆ0ˆ0ˆ0ˆˆ0ˆ0ˆ0ˆ0ˆ jijieeeeee iaajjajaaiia „= -= - QQQQ . From (2-19) and (2-20) we see that there is not any second time derivative term in the four equations (2-17), and from (2-21) we see that in the nine equations (2-18), all second time derivative terms only appear in the term ¶ ¶- biba Sexe ˆ0ˆ303ˆˆ h . Hence, if we can eliminate all first time derivative terms in ai S ˆ0ˆ given by (2-22) via appropriate gauge conditions, then there is not any second time derivative term in the all equations (2-17) and (2-18). If both two tetrads { } am ˆ ~ e and { } am ˆ e satisfy the time gauge condition, namely, (2-7) holds for { } am ˆ e and == ae a (2-25) for { } am ˆ ~ e , then a local Lorentz transformation ab L ˆˆ between such { } ai ee ˆˆ0 , g and { } ai ee ˆˆ0 ~ ,~ g has the characteristics: babdacdcaa xxx ˆˆˆ ˆ ˆ ˆ ˆˆ0ˆ ˆ ˆ 0ˆ 0ˆ 0ˆ , 0 )( , 0 )( , 1 )( hLLhLLL ==== . (2-26) Under the above special local Lorentz transformation, the relation between { } ai ee ˆˆ0 , g and { } ai ee ˆˆ0 ~ ,~ g reads mmmmmmmmmm LLd bbaababa exeexegeeggee ˆˆˆˆˆˆ ˆ ˆ0000ˆ0ˆ0000ˆ0ˆ ~)( , ~)( ; ~ , ~ ==-==--== . (2-27) All formulas (2-8) ~ (2-24) hold for the tetrad { } ai ee ˆˆ0 ~ ,~ g , e.g., corresponding to (2-22) and (2-9) we have baibbbiaai YeeYeeS ˆˆˆ0ˆ0ˆˆˆ0ˆ0ˆ0ˆ ~~~~~~ -= , (2-28) ( ) ( ) ( ) ( ) ( ) ( ) , ~ 2~ 2~ 2~~21~~~261~32~ +++-+--+- = YYYYYYYYYeL aa (2-29) respectively; And by the symbol “~” we denote those quantities corresponding to { } ai ee ˆˆ0 ~ ,~ g , for example, the expression (2-10) of ba Y ˆˆ corresponds to the tetrad { } ai ee ˆˆ0 , g , but the expression ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ] { } ~~~~~~ ~~~~~~~~ ~21 ~~~~~~ ~~21~~~ , ˆ , ˆˆ , ˆ , ˆˆˆˆ0 , 0ˆ0 , ˆˆ , 0ˆ0 , ˆˆ200ˆ , ˆ , ˆˆ , ˆ , ˆˆ0ˆ00ˆ0ˆˆˆˆ ijajiaibijbjibiajcciaiaibibibia iaiaibibibiaijjbiaba eeeeeeeeeeeeeee eeeeeeeeeeY -+---+-= -+-== lllll G (2-30) of ba Y ˆˆ ~ corresponds to the tetrad { } ai ee ˆˆ0 ~ ,~ g . Because local Lorentz transformation (2-6) cannot change metric tensor ij g , according (2-27), the manner of transformation of ba Y ˆˆ given by (2-10) under the transformation matrix (2-26) is dcdbcaijjddbiccaijjbiaba YeeeeY ˆˆˆ ˆ ˆ ˆ 0ˆˆ ˆ ˆˆ ˆ 0ˆˆˆˆ ~~~
LLGLLG === . (2-31) According to the corresponding mathematical theorem, if the non diagonal elements of the 3 ×
3 symmetric matrix ba Y ˆˆ are not zero, then we can make an orthogonal transformation by an orthogonal matrix ba ˆˆ L such that the new 3 ×
3 symmetric matrix ba Y ˆˆ ~ is diagonal; Namely, for any tetrad { } ai ee ˆˆ0 , g satisfying the time gauge condition (2-7), if ) ;3 ,2 ,1 ,( 0 ˆˆ babaY ba „=„ , then we can choose ba ˆˆ L such that = Y , = Y , = Y , (2-32) for a new group of tetrad { } ai ee ˆˆ0 ~ ,~ g satisfying the time gauge condition (2-25) yet. Eq. (2-32) is in fact just so called the simplest constraint conditions in Ref. [1], it originates from the simplest case in Ref. [6]. For the new group of tetrad { } ai ee ˆˆ0 ~ ,~ g that satisfy (2-32), we use (2-32) to eliminate the terms )( ~ ˆˆ abY ba „ in ai S ˆ0ˆ given by (2-28) and, thus, ai S ˆ0ˆ becomes )3 ,2 ,1( ~~~~~~ ˆˆˆ0ˆ0ˆˆˆ0ˆ0ˆ0ˆ =-= aYeeYeeS aaiabbiaai . (2-33) On the other hand, according to the Theorem proved in Sect. 2.1 and (2-32), the term ( ) ( ) ( ) ++ ~ 2~ 2~ 2~ YYYe in (2-29) can be removed directly and, thus, GK L becomes ( ) ( ) ( ) . ~~~~~231~21~~32 -+-- + -= YYYYYeYeL aa (2-34) So far, all six gauge conditions provided by local Lorentz transformation bmabam L ˆˆˆˆ ~)( exe = have run out. If we want to continue to eliminate all first time derivative terms in ai S ˆ0ˆ given by (2-33), then we have to employ some coordinate conditions to eliminate all first time derivative terms in the three terms )3 ,2 ,1( ~ ˆˆ = aY aa in ai S ˆ0ˆ . For this purpose, we first find out the expressions in which )3 ,2 ,1( ~ ˆˆ = aY aa are expressed by metric tensor. Since for determining the tetrad { } ai ee ˆˆ0 ~ ,~ g , all six gauge conditions provided by local Lorentz transformation bmabam L ˆˆˆˆ ~)( exe = have run out, all elements of the transformation matrix )( ˆˆ x ab L have been determined fully corresponding to a given tetrad, and the tetrad { } ai ee ˆˆ0 ~ ,~ g has been determined fully corresponding to metric tensor mn g . Concretely, according to jaiaij eeg ˆˆ ~~~ = given in (2-8) and (2-32) we can determine nine quantities ) , ; (~~ , ,0ˆˆ tmn ijjiiaia gggee = (2-35) by solving the nine algebraic equations ==== ijjiijjiijjiijjaia eeeeeegee GGG . (2-36) Notice that according to (2-27) we have had ~ m e and m ~ e expressed by mn g , hence, after we have the expression (2-35), it is easy to obtain the expressions ) , ; (~~ , ,0ˆˆ tmn ijjiaiai gggee = and ) , ; (~~ , ,0ˆ0ˆ0 tmn ijjiaa gggee = , since ai e ˆ ~ is the inverse matrix of ia e ˆ ~ and according to (2-8) we have aiia eeee ˆ0ˆ0ˆ0ˆ0 ~~~~ -= . In principle, we can obtain the expression )3 ,2 ,1( ) , ; (~~ , ,0ˆˆˆˆ == agggYY ijjiaaaa tmn by substituting (2-35) to )3 ,2 ,1( ~~~ == aeeY ijjaiaaa G . However, the quite complex process obtaining (2-35) by solving the nine algebraic equations in (2-36) forces us to use a different method to determine )3 ,2 ,1( ~ ˆˆ = aY aa . According to the corresponding mathematical theorem, for determining the orthogonal matrix ba ˆˆ L , we first solve the equation =+-+-=- bbbIY ba llll (2-37) of eigenvalues of 3 × ba Y ˆˆ . The coefficients of the above cubic equation are ( ) ( ) ( ) ( ) ijijijlmijiaijjbiaba lmijlmijjmilijijaa ggeeeYb ggggLgYYYYYYYYYb gYYYYb GGGG GGG ===== --==---++= ==++= where ij G means the determinant of the 3 ×
3 matrix [ ] ij G . In the expression of b , we have used (2-9). Although the formulas for the roots of cubic equation are well known, we are not going to write out the concrete forms of the three roots )3 ,2 ,1( )( = a a l of the cubic equation (2-37) here and only point out the following three characteristics about )3 ,2 ,1( )( = a a l . ① According to the corresponding mathematical theorem, all the three roots )3 ,2 ,1( )( = a a l of (2-37) are real; ② From (2-38) we see that all the coefficients of the cubic equation (2-37) have been expressed as functions of metric tensor mn g , ji g ,0 and t , ij g , all the three roots )3 ,2 ,1( )( = a a l of (2-37) thus as well: )3 ,2 ,1( ) , ; ( , ,0)()( =” aggg ijjiaa tmn ll . (2-39) ③ According to Viète's formulas, we have )3()2()1(3)3()2()3()1()2()1(2)3()2()1(1 , , llllllllllll =++=++= bbb . (2-40) After obtaining the three real roots )3 ,2 ,1( )( = a a l of (2-37), we can obtain the corresponding three eigenvectors )3 ,2 ,1( )( = aU a determined by the equation [ ] )3 ,2 ,1( )()()(ˆˆ == cUUY cccba l , where all )3 ,2 ,1( )( = aU a are 3 ×
1 matrices and satisfy )3 ,2 ,1 ,( ))(()(T )( == baUU baba d . Taking advantage of )3 ,2 ,1( )( = aU a , we can construct the orthogonal matrix [ ] baba UUU )3()2()1(ˆ ˆ =L , the matrix form of (2-31) is [ ] [ ] [ ] [ ][ ] . 00 00 00 ~~~ ~~~ ~~~~ )3()2()1()3()3()2()2()1()1(T)3(T )2(T)1( )3()2()1(ˆˆT)3(T )2(T)1(ˆˆT3ˆ3ˆ2ˆ3ˆ1ˆ3ˆ 3ˆ2ˆ2ˆ2ˆ1ˆ2ˆ 3ˆ1ˆ2ˆ1ˆ1ˆ1ˆˆˆ = = == = llllll LL UUUUUU UUUYUUUYYYY YYY YYYY dcdcba
The above result not only verifies (2-32) again, but also presents the following conclusion: )3 ,2 ,1( ) , ; (~~~ , ,0)()(0ˆˆˆˆ =”== agggeeY ijjiaaijjaiaaa tmn llG , (2-41) namely, )3 ,2 ,1( ~ ˆˆ = aY aa are just the three roots of the cubic equation (2-37). GK L Generally speaking, coordinate transformation can provide four coordinate conditions, and from (2-41) we see that all )3 ,2 ,1( ~ ˆˆ = aY aa are functions of mn g , ji g , 0 and l , ij g and independent with second derivative srmn , , g , we therefore can choose special combinations of )3 ,2 ,1( ~ ˆˆ = aY aa as coordinate conditions. We first choose ,0000003ˆ3ˆ2ˆ2ˆ1ˆ1ˆˆˆ = --==++= ll G ggggggYYYY lmijijaa (2-42) as one of four coordinate conditions. After adding the coordinate condition (2-42), (2-33) becomes )3 ,2 ,1( ~~~ ˆˆˆ0ˆ0ˆ0ˆ =-= aYeeS aaiaai ; (2-43) The special meaning of the coordinate condition (2-42) is that it can eliminate the negative kinetic energy term expressed by (2-13) in GK L . In fact, according to the Theorem proved in Sect. 2.1 and considering the condition (2-42), the negative kinetic energy term ( ) ~32 aa Ye - in (2-29) or (2-34) can be removed and, thus, GK(1) L given by (2-34) becomes ( ) ( ) . ~~~~~231~21 -+-- = YYYYYeL (2-44) So far, we have used seven gauge conditions, the six are provided by local Lorentz transformation bmabam L ˆˆˆˆ ~)( exe = , the one is a coordinate condition provided by coordinate transformation )~( nmm xxx = . All these seven gauge conditions have been given in Ref. [1]. We further choose the following two coordinate conditions ~~~2 F gYYY -=-- , (2-45) ~~ F gYY -=- , (2-46) where both F and F are functions of mn g and first space derivative k g , mn , whose concrete forms will be determined in the following discussion. Considering (2-41), the three coordinate conditions (2-42), (2-45) and (2-46) can be written to the following forms: ; 3121~ , 3121~ , 31~ +== -==-== FFlFFlFl gYgYgY (2-47) And, further, according to (2-38) and (2-40), the above three coordinate conditions are equivalent to ( ) ( ) . 91 121 , 3121~~ , 0~ -=- +=-= FFFGFFGGG ijijlmijjmilijij gggggg (2-48) Notice that the second of the above three coordinate conditions asks ‡ lmijjmil gg GG , but this inequality is only a corollary of (2-12). Under the three coordinate conditions (2-47) and considering ( ) ~ eg -= given in (2-8), ai S ˆ0ˆ given by (2-43) becomes += -=-= FFFFF iiiiii eeSeeSeeS ; (2-49)
GK(2) L given by (2-44) can be written to the form ( ) ( ) ( ) ( ) ( ) ( ) , 31~21 ~~ ~~~231~ ~~~21~~~2~61 + - -+-- - +- ++-- = FFFF FF ggegYYgYYYe gYYegYYYeL according to the Theorem proved in Sect. 2.1 and considering (2-45) and (2-46), both the terms ( ) ~~~2~61 F gYYYe +-- and ( ) ~~~21 F gYYe +- in the above expression can be removed directly and, thus, GK(2) L becomes ( ) ( ) ( )( ) ( ) ( ) ( ) [ ] { ( ) ( ) ( ) [ ] } ( ) ( ) ( ) [ ] ( ) ( ) [ ] { } ( ) . 31~21 ~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~2 ~~~~~~~~~2 31~ 31~21 ~~ ~~~231 +- -------+ ------ -----= +--+--= FF FFFFFF e eeeeeeeeeeeeeeeee eeeeeeeeeeeeeee eeeeeeeeee eYYYYYL ijjiijccijjiijcciiiiii ijjiijccijjiijccijjiijcc iiiiiiiii (2-50) GK(3) L can be written to the form ( ) ( ) ( ) , 31~21 ~~ ~~~231 +++-++--= FFFFFF egYYgYYYL if GK(3) L appears in Lagrangian density, then although according to (2-45) and (2-46) we have =+-- F gYYY and , 0~~ =+- F gYY neither of ( ) ~~~231 FF gYYY +-- and ( ) ~~31 FF gYY +- can be removed from GK(3) L , since both ~~~2 F gYYY +-- and ~~ F gYY +- appear in linear but not quadratic forms in GK(3) L . On the other hand, if GK(3) L appears in equation of motion, for example, the term GK(3) L in the equations =Q e and ˆ0ˆ0 = ia e Q given by (2-17) ~ (2-24), then the both terms ( ) ~~~231 FF gYYY +-- and ( ) ~~31 FF gYY +- in GK(3) L can be eliminated immediately and, thus, the term GK(3) L in (2-19) and (2-21) becomes ( ) . 31~41 + FF e From (2-49) and (2-50) we see that if both F and F are functions of mn g and k g , mn , in other words which are independent with time derivative mn g , then after which are expressed by tetrad { } ai ee ˆˆ0 ~ ,~ g via iaai eegeeg ˆˆ000ˆˆ000 ~~ ,~~ == gg and jaaiij eeg ˆˆ ~~ = , there is not any time derivative g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ in ai S ˆ0ˆ given by (2-49) and there is not any quadratic term of first time derivative in GK(3) L given by (2-50). And, further, since the term GK L in the equation =Q e and ˆ0ˆ0 = ia e Q is replaced with ( ) , 31~41 + FF e we see that there is not any time derivative term in the equations of motion given by (2-17), (2-19) and (2-20), and there is not any second time derivative term in the equations of motion given by (2-18) and (2-21). We emphasize again that there is not any time derivative term in ai S ˆ0ˆ only when ai S ˆ0ˆ given by (2-49) is expressed by tetrad, if it is expressed by metric tensor, then even if both F and F are independent with time derivative terms mn g , from (2-35) we see that ia e ˆ ~ in the expression (2-49) of ai S ˆ0ˆ is dependent with time derivative terms ij g , ai S ˆ0ˆ thus as well. Furthermore, notice + ++-= kijkkjikkijkijij ggggggggggg ,000 ,000 ,0000 , 000 G , (2-51) if we choose both F and F are functions of jilmi ggggg ,000000 , , and nlm g , : , , ; , , , ; , , ,00000022 , ,00000011 ” ” nlmjilminlmjilmi gggggggggggg FFFF (2-52) then in the three coordinate conditions (2-48), g only appear in the combined-item gg i but not alone; And, further, the three coordinate conditions in (2-48) are consistent and independent, since we can regard them as three equations determining three functions gg i . Substituting ( ) ~ eg -= and ( ) iaai eeeg ˆˆ0200ˆ0 ~~~ = obtained by (2-8) and jaaiij eeg ˆˆ ~~ = to (2-52), we have ( ) ( ) ( ) ( ) ( ) ( ) , ~~ , ~~ ; ~~ , ~~ , ~~ , ~~ ; ~~ , ~~ ,ˆˆ ,ˆˆ0ˆˆˆˆ022 ,ˆˆ ,ˆˆ0ˆˆˆˆ011 nmaaljiaamaaliaanmaaljiaamaaliaa eeeeeeeeeeeeeeee FFFF ”” (2-53) we see that both F and F are independent with ~ e . Not both F and F vanish, or from (2-49) we have ˆ0ˆ = ai S and, according to (2-17) and (2-20), we obtain a contradictory equation =-= aa Tc Ge Q . A simple choose of F and F is ( )
0 , ~~ == -= FF iiaaii eegg , (2-54) a different choose is ( ) ( ) iiaaiikjccikbjbiaakijjki eebggbeeeeeeagggga ,ˆˆ0 ,0002 ,ˆˆˆˆˆˆ0 ,0001 ~~ , ~~~~~~~ = -=-== FF , (2-55) where both a and b are constants, etc. When F and F are given by (2-54) or (2-55), the three coordinate conditions in (2-48) hold for the metric tensor given by (1-7). Of course, one can try to choose other forms of F and F . We first summarize some characteristics of the group of tetrad { } am ˆ ~ e given in the above discussion. At first, we rewrite the nine gauge conditions given by (2-25), (2-32) and (2-47) as follows. ; 3) 2, 1,( 0 ~ == ae a (2-56) ; 0~~ , 0~~ , 0~~ = = = YeYeYe (2-57) ; 03121~~ , 03121~~ , 031~~ = ++ = -+ =- FFFFF
YeYeYe (2-58) According to (2-30) we see that if both F and F are in the form (2-53), then all six conditions in (2-57) and (2-58) are independent with ~ e . In the above nine gauge conditions, the six given by (2-56), (2-57) and the three given by (2-58) are provided by local Lorentz transformation bmabam L ˆˆˆˆ ~)( exe = and coordinate transformation )~( nmm xxx = , respectively. Although coordinate transformation can provide four coordinate conditions, what we have used is only three of four coordinate conditions. As an example, we can add the fourth coordinate condition ( ) -= -=-= - eeg , since ~ e is independent with the nine gauge conditions (2-56) ~ (2-58). In fact, we can choose any relation about mn g and trs , g being independent with the three coordinate conditions (2-58), in other words (2-48), as the fourth coordinate condition. However, we would rather to remain one undetermined coordinate condition for other significant purpose, the two functions F and F as well. Although two chooses have been given in (2-54) and (2-55), we can choose different forms of F and F for significant purpose. We now can investigate a system whose basic variables are { } ai ee ˆˆ0 ~ ,~ g and the action with matter is ∫ = )(d )(~)(~ Total(2)430ˆ0)2( xxLxexeS , MGV0ˆ , 000ˆGK(3)3Total(2) ~~2π16
LLUeeLGcL kk + ++= , (2-59) where GK(3) L is given by (2-50) but in which F and F are in the form (2-53), k U and GV L are still given by (2-15) but in which { } ai e ˆ is replaced with { } ai e ˆ ~ . Using the six gauge conditions (2-57) and (2-58) and according to the proof process of the Theorem in Sect. 2.1, it is easy to prove that the equations of motion of the system obtained by the Euler-Lagrange equations corresponding to the action (2-59) are equivalent to the equations (2-17) ~ (2-24) but in which ai S ˆ0ˆ is expressed by (2-49) and in (2-19) and (2-21), the term GK L is replaced with ( ) . 31~41 +FF e For the system described by the action (2-59), the momenta conjugate to g ˆ0 ~ e and ai e ˆ ~ are , ~ ~~~ ~~π16~ ~~~ , ~ ~~~ ~~~ ˆ0 , M30ˆ0ˆ0 , GK(3)30ˆ03ˆ0 , Total(2)30ˆ0ˆ ˆ 0 , 0M30ˆ0ˆ 0 , 0 Total(2)30ˆ00ˆ aiaiaiia eLeeeLeeGce Lee eLeee Lee ¶ ¶+¶¶=¶ ¶= ¶ ¶=¶ ¶=pp ggg (2-60) by the expression (2-50) of GK(3) L , we obtain the concrete forms of ia ˆ ~ p : . ~ ~~ 31~ ~~π16~ , ~ ~~ 31~ ~~π16~ , ~ ~~~ ~~4π2~
3ˆ 0 , M30ˆ0213ˆ300ˆ33ˆ 2ˆ 0 , M30ˆ0212ˆ300ˆ32ˆ1ˆ 0 , M30ˆ011ˆ300ˆ31ˆ iii iiiiii eLeeeeeGc eLeeeeeGceLeeeeeGc ¶ ¶+ +-= ¶ ¶+ --=¶ ¶+= FFp FFpFp (2-61) If we want to obtain the corresponding Hamiltonian representation of (2-59), then we first should express g ˆ 0 , 0 ~ e and ai e ˆ0 , ~ via ~ g p and ia ˆ ~ p , namely, obtain the expressions ( ) ia ee ˆ0ˆˆ 0 , 0ˆ 0 , 0 ~ , ~~~ pp ggg = and ( ) iaaiai ee ˆ0ˆˆ0 , ˆ0 , ~ , ~~~ pp g = from (2-60) and (2-61). However, generally speaking, there is not any quadratic term of first time derivative g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ in M L of matter, namely, there is not any time derivative term g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ in g ˆ 0 , 0M ~ eL ¶ ¶ and ai eL ˆ0 , M ~ ¶ ¶ , all the Dirac field, the Klein-Gordon field, the Maxwell field and the Yang-Mills field in curve spacetime have such characteristic. Hence, if both F and F are in the form (2-54) in which there is not any time derivative term g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ , then none of the expressions ( ) ia ee ˆ0ˆˆ 0 , 0ˆ 0 , 0 ~ , ~~~ pp ggg = and ( ) iaaiai ee ˆ0ˆˆ0 , ˆ0 , ~ , ~~~ pp g = can be obtained by (2-60) and (2-61). As well-known, if the determinant of the matrix ¶¶¶ ba cc L ff ff t && ) ,( ,2 of a Lagrangian density ) ,( , l ff aa L vanishes, then the system described by the Lagrangian ) ,( , l ff aa L is so called singular Lagrange system. However, for the system whose basic variables and the action are { } ai ee ˆˆ0 ~ ,~ g and (2-59), respectively, denoting { } aia ee ˆˆ0 ~ ,~ g f ” , Total(2)30ˆ0 , ~~) ,(
LeeL aa ” l ff , where Total(2) L is given by (2-59), we see that not only the determinant of the matrix ¶¶¶ ba cc L ff ff t && ) ,( ,2 vanishes, but also the rank of ¶¶¶ ba cc L ff ff t && ) ,( ,2 is zero, since from (2-60) and (2-61) we see that every matrix element
0) ,( ,2 =¶¶¶ ba cc L ff ff t && . We call such system fully singular Lagrange system . Formally, an example of fully singular Lagrange system is the Dirac field in flat spacetime, since from the Lagrange ygy mm i D -¶ ¶= mcxL h of the Dirac field we have + =¶ ¶= aaa yyp cL h& i D and D2 =¶¶ ¶ ba yy && L . As well-known, the system described by (2-4) and (2-5) is not equivalent to the theory of electromagnetic field described by (2-3), the equivalence asks to add the Lorenz condition , = l l A , or , = l l A for state vector in the corresponding quantum theory. Similarly, the fully singular Lagrange system described by (2-59) is not equivalent to general relativity, the equivalence asks to add the additional conditions (2-57) and (2-58), or add the six conditions = ++ = -+ = - = = = FFFF F
YeYe YeYeYeYe (2-62) for state vector in the corresponding quantum theory, where the expression of ba Y ˆˆ ~ is given by (2-30). We can try to use the Dirac-Bargmann method for a singular Lagrangian system or the method of path integral to realize quantization of the fully singular Lagrange system described by (2-59). This will be studied further. On the other hand, all second time derivative terms in the vierbein forms of the ten Einstein equations can be eliminated, this characteristic shows that general relativity is great different from other fields, e.g., the Yang-Mills field. For non-Abelian gauge field, we can choose appropriate gauge conditions, for example, the space-axial gauge = a A [7] , to eliminate some dynamic variables. However, no matter how to choose gauge-fixing terms, we cannot remove all second time derivative terms in the equations of motion of the non-Abelian gauge field a A m . Hence, it is impossible to ascribe general relativity to non-Abelian gauge field.
3 The generation of the simplest constraint conditions
In this section, we still discuss the Lagrangian
G(2) L given by (1-6) and the vierbein forms of the Einstein equations under time gauge condition, hence, all formulas in Sect. 2.2 hold. For the purpose that eliminates all first time derivative terms in ai S ˆ0ˆ given by (2-22) and all the quadratic terms of first time derivative in GK L given by (2-9), we are not limited to the simplest constraint conditions discussed in Sect. 2.3 but consider a more general quantity ij W : ( ) jiijkijijijijij gggg WWWWWW mnmn ==+-= , , , 21 ,0 , 00 , (3-1) namely, ij W is a function of metric tensor mn g and first space derivative k g , mn , and symmetric in the pair of indices. For example, kijkijij gg ,0
21 , 0 == WW , etc. And then, we introduce ijjbiabaijjbiaba eeYeeZ wW ˆˆˆˆˆˆˆˆ -== , (3-2) where ba Y ˆˆ is given by (2-10); from (2-10) and (2-51) we have ( ) . , 21 , ,000 ,000 ,000000 kijijkijkkjikkijkijijij gggggggggggg mnmn wWWGw ”- + +-=-= (3-3) If we take + +-= kijkkjikkijkij gggggggggg ,000 ,000 ,00000 W , then ijij GW = , baba YZ ˆˆˆˆ = and = ij w , we therefore return to the case discussed in Sect. 2.3. It is obvious that ba Z ˆˆ is symmetric in the pair of indices and the manner of transformation under the transformation matrix (2-26) is dcdbcaijjddbiccaijjbiaba ZeeeeZ ˆˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆˆˆˆ ~~~
LLWLLW === , (3-4) where ( ) ( ) [ ] , ~~ ~~~~~~ ~~21~~~~ ˆˆ , ˆ , ˆˆ , ˆ , ˆˆ0ˆ00ˆˆˆˆˆˆˆ ijjbiaiaiaibibibiaijjbiababa eeeeeeeeeeeeYZ ww lllll --+-=-= (3-5) in (3-5), we have used the expression (2-30) of ba Y ˆˆ ~ . Hence, similar to ba Y ˆˆ , for a tetrad { } am ˆ e satisfying the time gauge condition, if ) ;3 ,2 ,1 ,( 0 ˆˆ babaZ ba „=„ , then we can choose ba ˆˆ L such that = Z , = Z , = Z (3-6) for a new group of tetrad { } am ˆ ~ e satisfying the time gauge condition yet. Substituting ijjbiababa eeZY w ˆˆˆˆˆˆ ~~~~ += obtained by (3-5) to (2-28) and using (3-6) to eliminate the terms )( ~ ˆˆ abZ ba „ in ai S ˆ0ˆ , we obtain ; )3 ,2 ,1( ~~~~~~~~~~ ˆˆˆ0ˆ0ˆˆˆ0ˆ0ˆˆˆ0ˆ0ˆ0ˆ =--= aeeeeZeeYeeS lmmblaibaaiabbiaai w (3-7) On the other hand, substituting ijjbiababa eeZY w ˆˆˆˆˆˆ += obtained by (3-5) to (2-9) and using some formulas given in Sect. 2, it is easy to prove that GK L can be written to the form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ20ˆ0 2 1ˆ3ˆ2 3ˆ2ˆ2 2ˆ1ˆ2 3ˆ3ˆ2ˆ2ˆ2 3ˆ3ˆ2ˆ2ˆ1ˆ1ˆ20ˆ02 ˆˆ20ˆ0GK lmmdlcijjbiabadcbadbcaaa eeeeYe ZZZZZZZZeYeL wwhhhh - - + +++-+-- + -= If in the above expression, { } ai ee ˆˆ0 , g , ba Y ˆˆ and ba Z ˆˆ are replaced with { } ai ee ˆˆ0 ~ ,~ g , ba Y ˆˆ ~ and ba Z ˆˆ ~ , respectively, then according to the Theorem proved in Sect. 2.1 and (3-6), we can remove the term ( ) ( ) ( ) ++ ~ 2~ 2~ 2 ZZZe and, thus, GK L becomes ( ) ( ) ( ) ( ) . ~~ ~~~2 31~ ~~21~~~261~~~32 ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ20ˆ0 2 3ˆ3ˆ2ˆ2ˆ2 3ˆ3ˆ2ˆ2ˆ1ˆ1ˆ20ˆ02 ˆˆ20ˆ0GK(4) lmmdlcijjbiabadcbadbca aa eeeeYe ZZZZZeYeL wwhhhh - - + -+-- + -= (3-9) For the equation =+-+-=- dddIZ ba tttt (3-10) of eigenvalues of 3 × ba Z ˆˆ , the coefficients are ( ) ( ) ( ) ( ) , ~ , ~~~~ 21 , ~ lmijijlmijiaijjbiaba lmijlmijjmilijijijjaiaaababa ggeeeZd ggggZZZZZZZZZd geeZZZZZd WWWW WWWWh ===== --=---++= ====++= (3-11) where ij W means the determinant of the 3 × [ ] ij W . Similar to the three characteristics about )3 ,2 ,1( )( = a a l of the cubic equation (2-37) discussed in Sect. 2.3, all the three roots )3 ,2 ,1( )( = a a t of (3-10) are real and functions of mn g , ji g ,0 and t , ij g : )3 ,2 ,1( ) , ; ( , ,0)()( =” aggg ijjiaa tmn tt ; (3-12) and according to Viète's formulas, we have )3()2()1(3)3()2()3()1()2()1(2)3()2()1(1 , , tttttttttttt =++=++= ddd . (3-13) Similar to )3 ,2 ,1( ~ ˆˆ = aY aa , for )3 ,2 ,1( ~ ˆˆ = aZ aa we can prove )3 ,2 ,1( ) , ; (~~~ , ,0)()(ˆˆˆˆ =”== agggeeZ ijjiaaijjaiaaa tmn ttW . (3-14) namely, )3 ,2 ,1( ~ ˆˆ = aZ aa are just the three roots of the cubic equation (3-10). So far, all six gauge conditions provided by local Lorentz transformation bmabam L ˆˆˆˆ ~)( exe = have run out, from (3-14) we see that all )3 ,2 ,1( ~ ˆˆ = aZ aa are functions of mn g , ji g , 0 and l , ij g and independent with second derivative srmn , , g , we therefore can further choose special combinations of )3 ,2 ,1( ~ ˆˆ = aZ aa as coordinate conditions to eliminate all first time derivative terms in ai S ˆ0ˆ expressed by (3-7). We can employ the coordinate condition (2-42) to eliminate aa Y ˆˆ ~ in (3-7) and the negative kinetic energy term ( ) ~32 aa Ye - in (3-9), respectively, however, formally, we consider a more general coordinate condition ~~~~~ XG ==++= ijijaa gYYYY , (3-15) where X is appropriate function of mn g and first space derivative k g , mn , ( ) . ; , 00 k gg mnmn XX ” (3-16) Using the coordinate condition (3-15), ai S ˆ0ˆ expressed by (3-7) becomes lmmblaibaaiaiaai eeeeZeeeeS wX ˆˆˆ0ˆ0ˆˆˆ0ˆ00ˆ0ˆ0ˆ0ˆ ~~~~~~~~~ --= ; (3-17) On the other hand, the negative kinetic energy term ( ) ~32 aa Ye - can be written to the form ( ) ( ) ~32~~34~~32~~32 XXX + -- -= - eYeYeYe aaaaaa , this term appears in the expression (3-9) of GK(4) L yet, hence, according to the Theorem proved in Sect. 2.1 and (3-15), we can remove the term ( ) ~~32 X - - aa Ye in the expression (3-9) and, thus, GK(4) L becomes ( ) ( ) ( ) . ~~ ~~~2 31~ ~~21~~~261~~32~~34 ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ20ˆ0 2 3ˆ3ˆ2ˆ2ˆ2 3ˆ3ˆ2ˆ2ˆ1ˆ1ˆ20ˆ02020ˆ0ˆˆ020ˆ0GK(5) lmmdlcijjbiabadcbadbca aa eeeeYe ZZZZZeeYeL wwhhhh XX - - + -+-- + + -= (3-18) According to (3-5) and considering jaiaij eeg ˆˆ ~~~ = , the coordinate condition (3-15) can be written to the form ijij gZZZ wX ~~~~ -=++ . (3-19) We further choose the following two coordinate conditions ~~~2 X=--
ZZZ , (3-20) ~~ X=- ZZ , (3-21) where X and X are appropriate functions of mn g and first space derivative k g , mn , ( ) ( ) . ; , ; , 22 , 11 kk gggg mnmnmnmn XXXX ”” (3-22) Considering (3-14), the three coordinate conditions (3-19) ~ (3-21) can be written to the following forms: ( ) ; 313221~31~ , 313221~31~ , 31~31~ --+-== +-+-==++-== XXXwt XXXwtXXwt ijij ijijijij gZ gZgZ (3-23) According to (3-11) and (3-13), the above three coordinate conditions are equivalent to ( ) ( )( ) ; 4121~91 ~ 31 , 3121~32 ~~~~ , ~~
222 1010 22212 00 - +-++-= ++--=- +-= XXXwXXwW XXXwWWXwW ijijijijlmij ijijlmijlmijjmil ijijijij ggg ggggg gg (3-24) Notice that the first condition in (3-24) is just ~ XG = ijij g , and substituting ijijij wGW -= obtained by (3-3) to the above three coordinate conditions, (3-24) thus be written to the form: ( )( ) ( )( ) . 4121~91 ~ 31 , 3121~31 ~~ , ~
222 10100 22212 00000 - +-++-==- +=----= XXXwXXwwG XXXwwGwGXG ijijijijlm ijij ijijlmlmijijjmilijij ggg gggg (3-25) Under the three coordinate conditions (3-23) and considering malalm eeg ˆˆ ~~~ = in (2-8), ai S ˆ0ˆ given by (3-17) becomes ( ) . ~ ~~~~31~216132~~ ~~~~313221~31~~~~ , ~ ~~~~31~216132~~ ~~~~313221~31~~~~ , ~ ~~~~31~2~~31 ~~~~3131~31~~~~ ˆ3ˆˆˆ3ˆ0ˆ02103ˆ0ˆ0 ˆ3ˆˆ0ˆ02103ˆ0ˆ003ˆ0ˆ03ˆ0ˆ ˆ2ˆˆˆ2ˆ0ˆ02102ˆ0ˆ0 ˆ2ˆˆ0ˆ02102ˆ0ˆ002ˆ0ˆ02ˆ0ˆ ˆ1ˆˆˆ1ˆ0ˆ0101ˆ0ˆ0 ˆ1ˆˆ0ˆ0101ˆ0ˆ001ˆ0ˆ01ˆ0ˆ lmmalialaii lmmblibijijiii lmmalialaii lmmblibijijiii lmmalialaii lmmbliblmlmiii eeeeeeee eeeegeeeeS eeeeeeee eeeegeeeeS eeeeeeee eeeegeeeeS wXXX wXXXwX wXXX wXXXwX wXX wXXwX -+ ++= - --+--= -+ -+= - +-+--= -+-= - ++--= (3-26) In Sect. 2.5 we point out that not both F and F introduced by (2-45) and (2-46) vanish, however, even if we now take === XXX , it is not incompatible with the equation = a e Q since from (3-26) we see that ˆ0ˆ „ ai S when === XXX but „ ij w . On the other hand, we rewrite GK(5) L given by (3-15) to the form ( ) ( )( ) ( ) ( ) , ~~ ~~~2 31~ 31~21 ~~ ~~~231~ ~~~21~~~2~61~32~~34 ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ20ˆ0 222120ˆ0 23ˆ3ˆ2ˆ2ˆ13ˆ3ˆ2ˆ2ˆ1ˆ1ˆ20ˆ0 2 23ˆ3ˆ2ˆ2ˆ20ˆ0 2 13ˆ3ˆ2ˆ2ˆ1ˆ1ˆ20ˆ0 2020ˆ0ˆˆ020ˆ0GK(5) lmmdlcijjbiabadcbadbca aa eeeeYe eZZZZZe ZZeZZZeeYeL wwhhhh XXXX XXXX - - + + - -+-- + -- +--- + + -= according to the Theorem proved in Sect. 2.1 and (3-20) and (3-21), in the above expression both the terms ( ) ~~~2~61 X --- ZZZe and ( ) ~~~21 X -- ZZe can be removed directly, and, further, using (3-5),
GK(5) L becomes ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ++ --+ - --+ - + ++ +-+- + --+-- + -= ijjiijjiijjiijjiijji ijjiijjiijji ijjiijjiijjiijjiijji ijjiijjiijjiaa eeeeeeeeeeee eeeeeeee eeYeeYeeYeeeeeYYe eeeeeeYYYeYeL If GK(6) L appears in equation of motion, for example, the term GK(6) L in the equations = Q e and ˆ0ˆ0 = ia e Q given by (2-17) ~ (2-24), then we can substitute (3-5), (3-6) and (3-23) to (3-27) immediately and, thus, the term GK(6) L in (2-19) and (2-21) becomes ( )( ) ( ) ( ) ( ) . ~~ ~~ ~~~~~~~~41 ~~~~~~2~121~31 ++ +-+ + --+ + - ijjiijjiijjiijjiijji ijjiijjiijji eeeeeeeeeeee eeeeeeee (3-28) Considering iaai eegeeg ˆˆ000ˆˆ000 ~~ ,~~ == gg and jaaiij eeg ˆˆ ~~ = , ij w , X , X and X introduced by (3-3), (3-16) and (3-22) become ( ) ( )( ) ( ) , ~ , ~ ; ~ ,~ , ~ , ~ ; ~ ,~ , ~ , ~ ; ~ ,~ , ~ , ~ ; ~ ,~ ˆ , ˆ , 0ˆˆ022ˆ , ˆ , 0ˆˆ011 ˆ , ˆ , 0ˆˆ000ˆ , ˆ , 0ˆˆ0 amlnakamlnak amlnakamlnakijij eeeeeeee eeeeeeee gggg gggg XXXX XXww ”” ”” (3-29) Substituting (3-29) to (3-26) and (3-27), we see that there is not any time derivative term g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ in ai S ˆ0ˆ and there is not any quadratic term of first time derivative in GK(6) L . We now can investigate a system whose basic variables are { } ai ee ˆˆ0 ~ ,~ g and the action with matter is ∫ = )(d )(~)(~ Total(3)430ˆ0)3( xxLxexeS , MGV0ˆ , 000ˆGK(6)3Total(3) ~~2π16
LLUeeLGcL kk + ++= , (3-30) where GK(6) L is given by (3-27), k U and GV L are still given by (2-15) but in which { } ai e ˆ is replaced with { } . ~ ˆ ai e From (3-30) we obtain the momenta conjugate to g ˆ0 ~ e and ai e ˆ ~ are ( ) . ~ ~~~23212~31π8 ~~~ ~~~ , ~ ~~~23212~31π8 ~~~ ~~~ , ~ ~~ ~2~31π8 ~~~ ~~~ , ~ ~~~ ~~~
3ˆ 0 , M3ˆ2103ˆ330ˆ03ˆ 0 , Total(3)30ˆ03ˆ 2ˆ 0 , M2ˆ2102ˆ330ˆ02ˆ 0 , Total(3)30ˆ02ˆ 1ˆ 0 , M1ˆ101ˆ330ˆ01ˆ 0 , Total(3)30ˆ01ˆ ˆ 0 , 0M30ˆ0ˆ 0 , 0 Total(3)30ˆ00ˆ ¶ ¶+ + ----=¶ ¶= ¶ ¶+ + -+--=¶ ¶= ¶ ¶+ +-+-=¶ ¶= ¶ ¶=¶ ¶= ijkkijlmlmiii ijkkijlmlmiii ijkkijlmlmiii eLeggeGceee Lee eLeggeGceee Lee eLeggeGceee Lee eLeee Lee wwXXXp wwXXXp wwXXpp ggg (3-31) Hence, if ij w , X , X and X are expressed by (3-29) in which there is not any time derivative term g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ and there is not any quadratic term of first time derivative g ˆ 0 , 0 ~ e or ai e ˆ0 , ~ in M L of matter, then none of the expressions ( ) ia ee ˆ0ˆˆ 0 , 0ˆ 0 , 0 ~ , ~~~ pp ggg = and ( ) iaaiai ee ˆ0ˆˆ0 , ˆ0 , ~ , ~~~ pp g = can be obtained by (3-31). And, further, similar to the system described by the action (2-59), the system described by the action (3-30) is also a fully singular Lagrange system and equivalent to general relativity under the six conditions (3-6) and (3-23) but in which all ij w , X , X and X are expressed by (3-29), or six conditions ( ) = ---+= +--+ = +-+=== XXXwXXXw XXw ijijijij ijij gZgZ gZZZZ for state vector in the corresponding quantum theory, where the expression of ba Z ˆˆ ~ is given by (3-5). For example, under the six gauge conditions (3-6) and (3-23) and according to the proof process of the Theorem in Sect. 2.1, it is easy to prove that the equations of motion of the system obtained by the Euler-Lagrange equations corresponding to the action (3-30) are equivalent to the equations (2-17) ~ (2-24) but in which ai S ˆ0ˆ is expressed by (3-26) and in (2-19) and (2-21), the term GK L is replaced with (3-28). We can try to choose appropriate functions ij w , X , X and X to simplify the system described by the action (3-30), this will be studied further. References [1]
T. Mei . Gen. Rel. Grav. (9), 1913-1945 (2008). arXiv: 0707.2639. [2] R. Arnowitt, S. Deser and C. W. Misner. Gen. Rel. Grav. (9): 1997-2027 (2008). arXiv: gr-qc/0405109. [3] R. A. Nelson. J. Math. Phys. (10), 2379-2383 (1987). [4] R. A. Nelson. J. Math. Phys. , 6224-6225 (1994). [5] V. V. Voytik. Corrections: Generalized Lorentz transformation for an accelerated, rotating frame of reference
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