The Symmetries of Kibble s Gauge Theory of Gravitational Field, Conservation Laws of Energy-Momentum Tensor Density and the

Transcription

1 The Symmetres of Kbble s Gauge Theory of Gravtatonal Feld, Conservaton aws of Energy-Momentum Tensor Densty and the Problems about Orgn of Matter Feld Fangpe Chen School of Physcs and Opto-electronc Technology,Dalan Unversty of Technology,Dalan Emal:chenfap@dlut.edu.cn Abstract Based on the analyss of the Kbble gravtatonal gauge feld theory, we studed the Noether theorem for the transformaton of the Poncare local group n the physcs systems, derved the energymomentum tensor densty conservaton laws, and proved the equvalence of ths conservaton laws to the orentz and ev-cvta energy-momentum tensor densty conservaton laws. Moreover, we dscussed the problems about the orgn of matter feld. 1

2 Keywords agrangan; matter feld; gravtatonal feld; energy-momentum tensor densty; conservaton law; orgn of matter feld 1. Introducton Kbble gravtatonal gauge feld theory [1] s a standard gravtatonal feld theory wth Poncare group as ts gauge group. The symmetry of ths gauge theory s prmarly the symmetry related to the Poncare group transformatons, whch s mportant n at least the followng two aspects: 1 To eplan the emergence of the gravtatonal feld from the global to local change of the Poncare group transformaton. Its eplanaton of the emergence of gravtatonal feld [1, 2] s as follows: when a 2

3 gravtatonal feld does not est, and the matter feld possesses the symmetry of the Poncare group global transformaton, then by further requrng the symmetry of the global transformaton of the Poncare group to be the symmetry of the local transformaton of Poncare group.e. the parameters of the group are the functon of space-tme coordnates, one arrves at the concluson that a gravtatonal feld must est. Ths can be also descrbed as follows, accordng to the Kbble gravtatonal feld gauge theory, the estence of the symmetry of the local transformaton of the Poncare group mples the estence of the gravtatonal feld, and the estence of the gravtatonal feld s represented by the estence of the symmetry of the local transformaton of Poncare group. 2 To eamne the relatons of conservaton current. 3

4 Accordng to the Noether theorem, whch s generally applcable n theoretcal physcs, f the acton varable s unchanged under the transformaton of some group, then ths must lead to the correspondng conservaton current. By applyng ths theorem, the local transformaton of Poncare group n a physcs system has also a conserved current. But ths current contans some parameters of the local transformaton of the Poncare group. These parameters are the functon of space-tme coordnates, whch vary ndependently from each other, and can be mutually separated. Therefore, multple dentty relatons can be derved from the conserved current of the Noether theorem, and these dentty relatons reflect the mportant dynamc propertes of the physcal system. Wth regard to the above two aspects of the symmetry, there have been more numerous and comple 4

5 studes of the frst aspect, but fewer and less comprehensve nvestgaton of the second aspect. Ths paper descrbes my recent study of the second aspect wth partcular focus on the dervaton of the conservaton law of the energy-momentum tensor densty from the Noether theorem appled to the local transformaton of the Poncare group. 2. The local transformaton of the Poncare group and the Noether theorem For a physcs system, f t only contans a matter feld, then ts agrangan can be epressed as: M M [ ; ], 1 5

6 Here for smplcty, we assume the matter feld has only one component, although n realty the matter feld can be a spnor, tensor, vector or scalar. Wth the ecepton of scalar, all the other quanttes are of multple components. Under the local transformaton of the Poncare group, the space-tme coordnates are changed to ' 2 the matter feld s transformed to ' ' 1 2 S 3 where the group parameters or are all varables. So are dffcult to 6

7 dstngush. One can defne,then Equaton 2 can be wrtten as: ' 2 from now on we wll use to represent. The agrangan, acton ntegrals, feld equatons and conservaton laws are all determned for a certan physcs system. Therefore, one needs to select a relevant physcs system for studyng these problems. When there s no or a neglgble gravtatonal force, the agrangan only contans the matter feld. But t s noted that, apart from the gravtatonal force, all other elementary nteractons can be ncluded n the agrangan of matter feld. When there s a nonneglgble gravtatonal feld, the agrangan of a physcs system s usually wrtten n two parts [1, 2]: 7

8 M G 4 Here, not only descrbes the pure matter feld, M but also descrbes the gravtatonal force acted on the matter. Therefore can be regarded as a M generalzed matter agrangan. Whereas only descrbes pure gravtatonal force, and t can thus be called as the agrangan of a pure gravtatonal feld. G In the Kbble gravtatonal gauge feld theory, and M G can be epressed by the followng generalzed functonal [1]: M [ ; ; ; ] M, h 5 [ ; ; ] G G, h 6 8

9 Where h s a Verben feld that can be used to determne the metrc of space-tme, s a tetrad connecton feld that can be used for determnng the connecton of space-tme. For a flat space-tme, there s always h and. For a curved space-tme, there always be h and, then curvature and torson emerge n the space-tme. Accordng to the gravtatonal gauge feld theory, curvature and torson are all manfestatons of the gravtatonal force, hence the Verben feld h and the tetrad connecton feld n the epresson of 5 and 6 mply the estence of gravtatonal force. When dscussng a physcs system wth gravtatonal force phenomenon, n the acton ntegral: 9

10 I g 4 d 7 the agrangan must be the total agrangan M G. Therefore,,,,, h h 8 where represents the varaton of a functon at fed value of, and represents the varaton of a functon under changng value of. g s n equaton 7 because of the need to consder the Jacob matr of coordnate transformaton n the estence of a gravtatonal force. It can be proved [3] that: g' ' ' ' g 1

11 11 Under the local transformaton of the Poncare group, the varaton of the acton varable becomes[1] d d d d I g g g g 4 ] [ 4 ] [ 4 4 ' ' ' ' ' } { ' 9 where g g g g g g g h h h h,, ] [ ] [,, ] [ ] [,, ] [ ] [ ] [ 1 Notce that there s an addtonal term h g, ] [ n equaton 1, whch s dentcal to because s not a functon of h,. Therefore t does not affect the result but makes t convenent for dervaton studes. After calculatons, one can rewrte the terms

12 nsde the brackets {} of the ntegral n Equaton 9 as g g g {[ h, h g g [ h h,,,, ] g, g ] } d 4 11 where g g g g h h g g h g, g, g, 12 are the Euler-agrange equatons for matter feld [4],Verben feld and tetrad connecton feld respectvely. If these 12

13 equatons are all satsfed, and under the local transformaton of Poncare group the followng condton holds I ', I' I then the followng conserved current may be derved from Equatons 9-12: g g g [ g ] h, h,, 13 whch s the Noether theorem under the local transformaton of the Poncare group. 3. The conservaton law of the energymomentum tensor densty under the local transformaton of Poncare group 13

14 Based on the mathematcs relatonshps, one can obtan [5]: 1 mn, 2 s mn, mn h h h m nj, h, j mn, mn m m j n nk kj, mn j m nk k, By substtutng, h,, nto Eq. 13, after complcated calculatons, one can dvde Eq. 13 nto several dentty equatons because the parameters,,, mn, mn, mn,,,, are mutually ndependent.these dentty equatons are ether conservaton laws or other mportant relatons. In ether case, they are mportant dynamc propertes of the physcal systems. Ths paper cannot cover all of 14

15 these propertes, so we focus on how to derve the conservaton law of the energy-momentum tensor densty from the Noether theorem of the local transformaton of the Poncare group., Due to the set of parameters,,, wth the set of parameters mn, mn, mn,, are ndependent of each other, the part contanng the parameters,, and the part contanng the,, parameters mn, mn, mn,, n Eq.13 should be conserved respectvely. Hence from Eq. 13, one can obtan: g { [,, g ] [ h, h,, h ] g [,,, ] 15 g } 14

16 Eq. 14 can be dvded further nto three dentty equatons: g [,, g h h,, g,, g ] 15, g [,,, g h h g [ h h,,, g g,, \ ], g ] 16 g g [ ], h h,, 17 16

17 , Snce the parameters,, are also mutually, ndependent, hence Eqs 15,16 and 17 can reduce to, other three equatons wthout,,., One can defne: g T M g, M, g h M, h, g M,, g M as the energy-momentum tensor densty of the generalzed matter feld, and defne: g T G g, G, g h G, h, g G,, g G as the energy-momentum tensor densty of the pure, gravtatonal feld. Because the parameter, 17

18 and represents the translaton of the space-tme,, these defntons are approprate. From Eq. 15, one can obtan: gt gt M G 18 from Equaton 16,17, one obtans: gt M gt G 19 Equaton 18,19 are orentz and ev-cvta energymomentum tensor densty conservaton laws. In , ev-cvta and other scentsts had a dspute wth Ensten on the energy-momentum conservaton laws [6][7]. The readers are referred to Reference [7] for the detals of the dspute. The focus of ths paper s on the physcal concept nstead of the dspute. 18

19 4. The problems about orgn of matter feld In the present studes of physcs, there have been nsuffcent nvestgatons about the orgn of matter feld. Just lke n the nvestgaton of the orgn of lfe, wheren one has to address the queston of how lvng thngs arse from non-lvng matter, n the nvestgaton of the orgn of matter feld, one also needs to address the queston of how the Unverse evolves from a matter-less state to a state wth matter feld. But there has been a lack of study of how matter feld arse from matter-less state, and t s unclear how to nvestgate ths problem. The basc property of matter feld s the possesson of a postve energy,whch s determned by the energy-momentum tensor densty. To study the orgn of 19

20 matter feld, there s a need to eplan how the energymomentum densty evolved from ts non-estence state wth zero energy to ts estence state wth postve energy. In theoretcal physcs, there are dfferent defntons of the energy-momentum tensor densty, whch can be used to derve dfferent energymomentum tensor densty conservaton laws. A generally applcable theory for orgn of matter feld must be based on a generally applcable energy-momentum tensor densty conservaton law. Under the local transformaton of the Poncare group, the energy-momentum tensor densty conservaton law derved from the Noether theorem s equvalent to the orentz and ev-cvta energy-momentum tensor densty conservaton laws. Because Noether theorem s generally applcable to any physcs system, so do the orentz and ev-cvta energymomentum tensor densty conservaton laws. 2

21 The orentz and ev-cvta energy-momentum tensor densty conservaton laws dctate that when the energymomentum tensor densty of the matter feld of a physcs system ncreases, the energy-momentum tensor densty of the gravtatonal feld wll decrease, whereas when the energy-momentum tensor densty of the matter feld of a physcal system decreases, the energy-momentum tensor densty of the gravtatonal feld wll ncreases, but the sum remans constant. Ths mples that the energymomentum tensor densty of the gravtatonal feld can be converted nto the energy-momentum tensor densty of the matter feld. Under specal condtons, the energy-momentum tensor densty may be zero, If the above mentoned transformaton of the energy-momentum tensor densty stll ests, then correspondngly, the system n ths partcular space-tme locaton changes from the state of 21

22 non-estence of energy-momentum tensor densty of matter feld to that of the estence of energy-momentum tensor densty of matter feld wth smultaneous emergence of a negatve energy-momentum tensor densty of gravtatonal feld. Because the matter feld s always lnked wth ts energy-momentum tensor densty, the estence of the energy-momentum tensor densty of the matter feld means the estence of matter feld. The non-estence of the energy-momentum tensor densty of the matter feld means the non-estence of matter feld. The above analyss ndcates that the Unverse can evolve from a state wthout matter feld to a state wth matter feld. The above opnon has been dscussed n a recent publcaton [8]. As ths problem s closely related to the problems dscussed n ths paper, t wll be further descrbed below. 22

23 There's a queston of whether the energy-momentum tensor densty s dentcal to matter feld? From the followng consderatons, the two seem to be nondentcal but closely related. The energy and ts related energy-momentum tensor densty must have a bearer, whch s lkely the vacuum, namely Mnkowsk space-tme. When there s no matter feld completely,the space-tme s a vacuum. A matter feld emerges only when the vacuum carres postve energy and the correspondng energy-momentum tensor densty. It s recognzed that there are dfferent opnons about ths queston, and further nvestgatons are needed to fully resolve ths queston. Physcs s an eperment-based scence. Physcs theores must be based on the laboratory studes and observatons. Although there s no eperment to defntvely prove t, the concept of the creaton of 23

24 matter feld from the state wthout matter feld s not nconsstent wth estng eperments and observatons. For nstance, the great energy phenomena of the quasar and at the center of galaes had been eplaned by the estence of black holes, but there have been opnons that black holes may not est at all. An alternatve eplanaton s that these great energy phenomena are due to the creaton of matte feld, whch s defntely a very promsng eplanaton. Whle ths s stll a hypothess to be further verfed by epermental evdences, the current evdences seem to suggest that t s not a totally mpossble event for matter feld to emerge from the non-estence state, and ths problem s worth future nvestgatons. References 24

25 [1] Kbble T.W.B.1961, orentz nvarance and the gravtatonal feld, J.Math.Phys.2:212. [2] Chen F P 214, Space-tme, the basc concept and laws of physcs Scentfc Publsher Beng [3] Carmel M.1982, Classcal felds:general Relatvty and Gauge Theory,John Wley & Sons.New York. [4] Held F.W.,von der Heyde P.,Kerlck G.D.1976, General relatvty wth spn and torson:foundatons and prospects Rev.Mod.Phys.,48,393. [5] Chen F P.199, Internatonal Journal of Theoretcal Physcs, 29: 16 [6] Chen F P 2, J Herbe Normal Unversty, 24: 326 [7] Cattan C, De Mara M. 1993, Conservaton aws and Gravtatonal Waves n General Relatvty. // Earman J, Janssen M, Norton J D. The Attracton of Gravtaton, Boston: Brkhauser. [8] Chen F.P.215, The Conservaton aw of Energy- Momentum Tensor Densty and the Orgn of Matter,Astronomy and Astrophyscs, 3: