Hamilton s principle and the Generalized Field Theory : A comparative study

Transcription

1 Hamlton s prncple and the Generalzed Feld Theory : A comparatve study arxv:gr-qc/ v1 27 Dec 1998 F. I. Mkhal Department of mathematcs, faculty of scence, An shames unversty, Egypt M. I. Wanas Astronomy Department, Faculty of Scence, Caro unversty Egypt December 7, 2013 Abstract The feld equatons of the generalzed feld theory (GFT) are derved from an acton prncple. A comparson between (GFT), Møller s tetrad theory of gravtaton (MTT), and general relatvty s carred out regardng the Lagrangan of each theory. The results of solutons of the feld equatons, of each theory, are compared n case of sphercal symmetry. The dfferences between the results are dscussed and nterpreted. 1

2 1 Introducton: In the last ffteen years several feld theores appeared n lterature, each of whch s assumed to be a natural generalzaton of the theory of general relatvty. It s hgh tme to carry out a thorough comparson between these theores wth regard to three fundamental pvots of any generalzed feld theory,.e: () the geometrc structure used n the formalsm of the theory, () the procedure used n the dervaton of ts feld equatons, () the models to whch these feld equatons are appled and the results, of physcal nterest, obtaned. We beleve that such comparson wll be more feasble f we are able to unfy these three aspects n the theores under consderaton. The present work s a tral to carry out such a comparson along the lnes specfed above. Strctly speakng, we are gong to examne the two generalzed feld theores : (a) the generalzed feld theory (GFT) constructed by Mkhal & Wanas (1977), (b) Møller s tetrad theory of gravtaton (MTT) (1978), versus two dfferent versons of general relatvty (GR): the standard orthodox theory n whch the lorentz sgnature s beng mposed on the metrc used from the begnnng, and a modfed verson gven by Wanas (1990) n whch the Lorentz sgnature s ntroduced at a later stage. Fortunately, the two theores (a), (b) under consderaton depend on a Absolute Parallelsm (Ap-Space) n ther formalsm. Thus the pvot() mentoned above s already unfed. To satsfy pvot() we have to derve the feld equatons of (GFT) usng a varatonal prncple,.e. Hamlton s acton prncple. Ths s gven n secton 4 below. Besdes, for comparson wth (GR), we are gong to use a Lagrangan functon gven by Møller (1978) as shown n secton 6. Wth regards to pvot (), we are gong to apply the two theores (a),(b) and both versons of (GR) to the same absolute parallelsm (AP) space, namely the most general AP-Space havng sphercal symmetry derved by Robertson (1932). The results obtaned by applyng the four dfferent theores to ths same AP-space are compared and tabulated n secton 7. 2 Basc Geometrc Structure (AP-space) We are manly nterested n feld theores usng for ther structure a geometry admttng absolute parallelsm (AP-geometry). The AP-space s a 4-dmensonal vector space (T 4 ), each pont of whch s labelled by 4-ndependent varables x ν (ν = 0, 1, 2, 3). At each pont, 4-lnearly ndependent contravarant vectors λ µ, ( = 0, 1, 2, 3), are ned. Assumng that the λ µ, s the normalzed cofactor of λ µ n the determnant λ µ. Hence they satsfy the relatons, 2

3 λ µ λ j µ = δ j (2.1) λ µ λ ν = δ µ ν (2.2) In what follows we use Greek letters (µ, ν, α, β,...) to ndcate coordnate components (world ndces), and Latn letters (, k,...) to ndcate vector numbers (mesh ndces). Summaton conventon wll be appled to both types of ndces. Usng these vectors we can ne the followng 2nd order tensor: ĝ νµ = λ ν λ µ, (2.3) It can be shown that where ĝ = ĝ µν, and ˆλ = λµ. ĝ νµ = λ ν λ µ. (2.4) ĝ µα g να = δ µ ν, (2.5) ĝ = ˆλ 2, (2.6) we can use ĝ µν and ĝ αβ to rase and lower world ndces. It can also be used as a metrc tensor nng a Remannan space assocated wth the AP-space when needed. A non symmetrc connecton Γ α µν s then ned such that, λ µ + ν = λ µ,ν Γ α µν λ α = 0. (2.7) Ths s the condton of absolute parallelsm. Equaton (2.7) can drectly be solved to gve, It can be shown that (cf.mkhal (1962)): Γ α µν = λ α λ µ,ν. (2.8) Γ α µν = { α µν } + γ α µν, (2.9) where { α µν} s Chrstoffel symbols ned as usual usng the tensors (2.3), (2.4), and γ α µν s a 3rd order tensor ned by, = λ α λµ;ν. (2.10) γ α µν So,n the AP-space, we have three dfferent connexons: { } α µν, Γ α µν and Γ α (µν) (= 1 2 (Γα µν + Γ α νµ )).Consequently, we can ne the followng types of absolute dervatves: A µ ;ν { } µ = A µ,ν + A α, (2.11) να 3

4 µ + A ν µ A ν A µ ν = A µ,ν + Γµ α ν A α, (2.12) = A µ,ν + Γ µ ναa α, (2.13) = A µ,ν + Γµ (αν) Aα, (2.14) where A µ s an arbtrary contravarant vector and the comma (,) denotes ordnary partal dfferentaton. The torson tensor s ned by Λ α µν = Γ α µν Γ α νµ = Λ α νµ. (2.15) Contractng ths tensor by settng ν = α,we get the basc vector For more detals see Mkhal (1952). C µ = Λ α µα = γα µα. (2.16) 3 Feld Equatons of (GFT): Usng the AP-space, as descrbed n the above secton, Mkhal & Wanas (1977) were able to construct a generalzed feld theory (GFT). Usng a certan Lagrangan functon, they were able to derve an dentty of the form, µ + E ν = 0. (3.1) + µ Followng some analogy wth (GR), they were led to chose ther feld equatons nto the form where E µ ν s a non-symmetrc tensor gven by: E µ ν = 0, (3.2) E µ ν = δ µ ν L 2Lµ ν 2δµ ν C α α 2C µ C ν 2δ µ αc β Λ α βν + 2ĝ αµ C ν 2ĝ αβ δ µ + + α ǫλ ǫ ν, (3.3) + + α β and L µν s gven by L µν L = Λ α βµ Λβ αν C µc ν, = ĝ µν L µν. (3.4) Several promsng solutons of ths theory have been obtaned,e.g. Wanas (1985, 1987, 1989). 4

5 However, t s of great nterest to fnd out the partcular lagrangan functon whch wll gve rse to the feld equatons (3.2) by usng a varatonal prncple. Ths s what we ntend to do n the followng secton. In fact we were able to fnd out the Lagrangan functon L gvng rse to the scalar densty, L = ˆλL, and then usng the varatonal prncple, δ L d 4 x = 0, we wll try to derve the feld equatons (3.2). 4 Dervaton of the feld equaton of GFT usng Hamlton s acton prncple Generally speakng, startng wth a Lagrangan functon L whch depends only on the tetrad vectors and ther frst coordnate dervatves.e. L = L(λ µ, λ µ,ν), (4.1) we can ne the scalar densty, and thus : Then the acton prncple wll gve δ L = ˆλL, (4.2) L = L(λ µ, λ µ,ν). (4.3) L(λ µ, λ µ,ν) d 4 x = 0. (4.4) where d 4 x = dx 0 dx 1 dx 2 dx 3. Then consderng an arbtrary varaton (δ λµ) n the tetrad vectors, (4.4) can be put n the form, δl δ λµ d 4 x = 0, (4.5) δ λµ where δl δ λµ = L L λµ x γ λµ,γ, (4.6) s the Hamltonan dervatve of L w.r.t. the arbtrary varaton (δ λ µ). On the other hand we can wrte the Hamltonan dervatve (4.6) n the form, δl δ λµ = ˆλS µ. (4.7) 5

6 It can be easly shown that S µ s a vector for each value of (Wanas 1975). Usng S µ, we can ne the followng 2nd order tensor, S β σ = λ σs β. (4.8) We assume that (δ λ β) are lnearly ndependent, and arbtrary as stated before, then for (4.5) to be satsfed, we wrte the Euler-Lagrange equaton for the present problem : δl δ λµ = L λ β L x α = 0. (4.9) λβ,α Then usng the same Lagrangan functon ned by (3.4) n (GFT), the 1st term of (4.9) wll take, after some manpulaton, the form : L λβ = ˆλ λ β ĝ µν L µν ˆλ ( ) ĝ µβ λ ν + ĝ νβ µ λ L µν ) +ˆλĝ (λ µν α Λ β µǫ Λǫ αν + λ ǫ Λ α ǫµ Λβ να λ α Λ β αµ C ν λ α Λ β αν C µ. (4.10) Smlarly, we get for the 2nd term of (4.9) the expresson : x γ L λ β,γ = 2ˆλΓ µ µγ (λ ǫĝ γα Λ βǫα λ ǫĝ αβ Λ γǫα λ γĝ αβ C α + λ βĝ γα C α ) ( ǫ +2ˆλ λ,γ ĝ γα Λ β ǫα + λ ǫĝ γα,γ Λβ ǫα + λ ǫĝ γα Λ β ǫα,γ ǫ λ,γ ĝ αβ Λ γ ǫα λ ǫĝ αβ,γ Λγ ǫα λ ǫĝ αβ Λ γ ǫα,γ ) γ λ,γ C β λ γ C β β,γ λ,γ C γ + λ β C γ,γ. (4.11) Substtutng from (4.10), (4.11) nto Euler-Lagrange equaton (4.9) we get, after comparng results wth (4.7) & (4.8) the followng set of feld equatons, (snce ˆλ 0 and λ σ are lnearly ndependent) where S β σ = 0, (4.12) S β σ = δ β σ L 2Lβ σ + 2ĝµα Λ β µν (δǫ σ Λν ǫα + δν σ C α) 2 [ ( ) Γ µ µγ δ ǫ σ ĝ γα Λ β ǫα δ ǫ σĝ αβ Λ γ ǫα δ γ σĝ αβ C α + δ β σĝ γα C α Γ ǫ σγĝγα Λ β ǫα + δǫ σĝγα,γ Λβ ǫα + δǫ σĝγα Λ β ǫα,γ Γǫ σγĝαβ Λ γ ǫα δ ǫ σĝαβ,γ Λγ ǫα δǫ σĝαβ Λ γ ǫα,γ + Γγ σγ Cβ δ γ σ Cβ,γ Γβ σγ Cα + δ β σ Cγ,γ]. (4.13) 6

7 Usng the followng results (Wanas ((1975)) C ν σ C ν + σ = Cǫ Λ ν ǫσ, and the dentty (Ensten (1929)) then (4.13) can be wrtten n the form, then (4.13) can be wrtten n the form γ + ĝ α γ = 0, ĝ + α + ν σ = 0, γ Λ = C σ + + γ α + α σ, Cα + σ S β σ = δ β σ L 2Lβ σ 2δβ σ C γ γ 2C β C σ 2δ β ν Cǫ Λ ν ǫσ + 2ĝαβ C σ + α 2ĝ γα δ β ν Λ ν + σ + α + γ, (4.14) whch s dentcal wth the tensor E µ ν gven by (3.3) n the Mkhal- Wanas dervaton. Ths shows that the feld equatons of (GFT) can be derved from an acton prncple wth the Lagrangan functon gven by (3.4), (4.2). 5 Role of the sgnature In metrc theores of gravty, the Lorentz sgnature plays a very mportant role. The sgnature s ned as the dfference between the number of +ve and -ve egenvalues of the metrc tensor. Lorentz sgnature s mposed on the metrc, for purely physcal reasons, namely to account for the specal relatvty n the lmtng case. Ths reflects the fact that we dstngush between spatal and temporal sectons of space-tme. It was realzed, snce the appearence of specal relatvty, that our unverse s a 4-dmensonal one. So, there s no fundamental need to dstngush between space and tme from the begnnng. We are unable nether, to construct equpment or experment, nor to measure n 4-dmensons, but rather n (3+1) dmensons. Thus, n order to compare the results of feld theores wth observaton and experment, Lorentz sgnature should be ntroduced at a later stage n the theory to reflect our dstncton between space & tme. In fact, the ntroducton of Lorentz sgnature transfers the theory from 4-dmensons to (3+1) dmensons. Wanas (1990) suggested that the nserton of Lorentz sgnature s to be done just before matchng the results of the theory wth observatons or experments,.e. after solvng the feld equatons. He speculated that ths may gve rse to new physcs, and gve an example confrmng hs speculaton. There are dfferent ways for nsertng the Lorentz sgnature n a theory constructed n the AP-space. One way s just to change the sgn of some constants of ntegraton after solvng the feld equatons. Another way s to replace (2.3) by g µν = e λ µ λ ν 7

8 where e = (+1, 1, 1, 1) s the Lev-Cvta ndcator. A 3rd way s to take the vector λ 0 µ to be magnary. For AP-space whose assocated metrc s dagonal (or t could be dagonalzed), t doesn t matter whether we nsert the Lorentz sgnature before or after solvng the feld equatons. 6 Drect comparson between the four theores : So far we have satsfed the two factors () & () mentoned n the ntroducton. It may be of nterest to compare the four theores, under consderaton, wth regard to the Lagrangan functon used and the feld equatons derved n each. For standard (GR) (wrtten n the AP-space ) the Lagrangan s usually taken to be (cf. Møller 1978), L GR = g (γ αµν γ νµα C ν C ν ). (6.1) Consequently, the modfed verson of (GR) as speculated by Wanas (1990) wll be, The correspondng Lagrangan of (GFT), as gven above, L GFT = L ˆ GR = ĝ (γ αµν γ νµα C ν C ν ). (6.2) ĝ (3γ αµν γ νµα γ αµν γ αµν C ν C ν ). (6.3) Møller (1978) has chosen for hs theory the Lagrangan functon, L MTT = g ((1 2ψ)γ αµν γ νµα + ψγ αµν γ αµν C ν C ν ), (6.4) where ψ s a free parameter of order unty. It s clear that L GR,L GFT can be obtaned as specal cases of Møller s Lagrangan (5.3) by takng, ψ = 0, ψ = 1, respectvely. However, although the lagrangan used by Møller n (MTT) appears to be more general than those for dervng the feld equatons of (GR), and (GFT), yet Møller s theory (MTT) as a whole s no so general. Ths wll be shown clearly n the next secton. The reason, n our opnon, s due to the followng two factors : () In Møller s theory, the Lorentz sgnature s mposed on the theory from the begnnng. () Møller used a phenomenologcal nton for the materal-energy tensor T µν, whle t s ned geometrcally n (GFT). Formally, (6.1) & (6.2) are smlar, but the results of applcaton are dfferent. We are gong to consder the theory dependng on (6.2) as dfferent compared to (GR). The followng table summarzes the man features of each theory. 8

9 Table 1: Comparson between the four feld theores Feld Feld Feld Feld Gravtatonal T µν Theory Lagrangan Equatons Varables Potental GR (1916) L GR G µν = κt µν g µν g µν Phenom. GFT (1977) L GFT Ĝ µν = B µν λ µ g µν Geomet. E [µν] = 0 MTT (1978) L MTT G µν + H µν = κt µν λ µ g µν Phenom. V [µν] = 0 ˆ GR (1990) ˆLGR Ĝ µν = κt µν ĝ µν g µν Phenom. where G µν s Ensten tensor. 7 Comparson n the case of sphercal symmetry Snce all theores tabulated n table 1 are wrtten n the AP-space, t s convenent to compare them by usng the most general AP-space havng sphercal symmetry. The structure of ths space s gven by Robertson (1932). The tetrad gvng the structure of ths space can be wrtten n sphercal polar coordnates as, A Dr B snθ cos φ B λ µ r cos θ cos φ B snφ r sn θ =, (7.5) B B cos φ 0 B snθ sn φ cos θ snφ r r snθ 0 B cos θ B r sn θ 0 where A,B,D are unknown functons of r only. Calculatng the necessary tensors for ths space we fnd that (see table 1) : (1) For Møller s tetrad theory H µν = 0, V [µν] = 0 s satsfed dentcally. (2) For GFT B µν = 0, E [µν] = 0 s satsfed dentcally. Furthermore, f we drect our attenton to free space solutons (T µν = 0) we found the results summarzed n table 2. 9

10 Table 2: Comparson n the case of sphercally symmetrc solutons Feld Feld Schwarz. Other Reference Theory equatons soluton? Solutons? GR G µν = 0 Yes No Brkhoff Theorem GFT Ĝ µν = 0 Yes Yes Wanas (1985) MTT G µν = 0 Yes No Mkhal et. al. (1991) ˆ GR Ĝ µν = 0 Yes Yes Wanas (1990) It s clear from the second column n table(2) above that although the feld equatons of the four theores reduce formally to those of GR n free space, the results obtaned are not dentcal. It s clear that ths dfference s a drect consequence to the stage at whch Lorentz sgnature s ntroduced n the theory. 10

11 References: A. Ensten, Stz. der Preuss. Akad. der Wss. 1, 2 (1929). F. I. Mkhal, Ph.D. Thess, Unversty of London (1952). F. I. Mkhal, An Shams Sc. Bul. 6, 87 (1962). F. I. Mkhal and M. I. Wanas, Proc. Roy. Soc. Lond. A 356, 471 (1977). F. I. Mkhal and M. I. Wanas, Int. J. Theoret. Phys. 20, 671 (1981). F. I. Mkhal, M. I. Wanas, A. A. Hndaw, E.I.Lashn Int. J. Theoret. Phys. 32, 1627 (1993). C. Møller, Mat. Fys. Skr. Dan. Vd. Selsk. 39, No. 13, 1 (1978). H. P. Robertson, Annals of Mathematcs Prnceton 33, 496 (1932). M. I. Wanas, Ph.D. Thess, Caro Unversty (1975). M. I. Wanas, Int. J. Theor. Phys. 24, 639 (1985). M. I. Wanas, ICTP Preprnt IC/87/39. M. I. Wanas, Astrophys. Space Sc. 154, 165 (1989). M. I. Wanas, Astron. Nachr. 311, 253 (1990). 11