Energy and metric gauging in the covariant theory of gravitation

Transcription

1 Aksaray Unversty Journal of Sene and Engneerng, Vol., Issue,. - (18). htt://dx.do.org/1.9/asujse.4947 Energy and metr gaugng n the ovarant theory of gravtaton Sergey G. Fedosn PO box 61488, Svazeva str. -79, Perm, Perm Kra, Russa, E-mal: fedosn@hotmal.om Relatons for the relatvst energy and metr are analyzed nsde and outsde the body n the framework of the ovarant theory of gravtaton. The methods of otmal energy gaugng and equatons for the metr are hosen. It s shown that for the matter nsde the body a roedure s requred to average the hysal quanttes, nludng the osmologal onstant and the salar urvature. For the ase of the relatvst unform system, the osmologal onstant and the salar urvature are exltly alulated, whh turn out to be onstant values nsde the body and are assumed to be equal to zero outsde the body. Comarson of the osmologal onstants nsde a roton, a neutron star and n the observable Unverse allows us to exlan the osmologal onstant roblem arsng n the Lambda-CDM model. MSC: 8A5; 8D5; Q1. PACS:..+; 4.4.-b; 95..Sf. Keywords: Cosmologal onstant, Salar urvature, Relatvst unform system, Gravtatonal feld, Aeleraton feld, Pressure feld. 1. Introduton The relatvst energy of the hysal system s art of the tme omonent of the fourmomentum of the system and s one of the most mortant haratersts, along wth the momentum. In ths ase, the energy s determned wth an auray u to a onstant, seleted arbtrarly based on the onvenene of alulaton. Thus, the roblem of energy gaugng arses n eah theory. In the ovarant theory of gravtaton, the energy s gauged based on the fat that the value of the osmologal onstant s roortonal wth an auray to a onstant multler to the energy densty of the matter artles n the roer felds of the system under onsderaton [1]. The use of the osmologal onstant for energy gaugng results n ertan hanges n the equaton for the metr, n whh the osmologal onstant s resent alongsde wth the salar 1

2 urvature. Therefore, we wll further analyze both the exressons for the metr and for the energy. The urose of ths artle s to larfy the queston how the osmologal onstant and the salar urvature n the matter nsde bodes should be understood. The ont s that, as a rule, reresentatve volumes oued by tyal artles should be seleted n the matter, and alulatons should be arred out for suh artles, nludng soluton of the equaton of moton. Alyng the method of tyal artles mles we need to use the arorate averagng of the hysal quanttes atng on suh artles. Suh quanttes as the osmologal onstant and the salar urvature are not exetons. Thus, they should also be onsdered as some averaged quanttes. As a result of our analyss for the ase of the relatvst unform system, we wll alulate the osmologal onstant and the salar urvature nsde the body, and wll show that they are onstant quanttes. In addton, we wll try to larfy the osmologal onstant roblem n onneton wth ts nonssteny wth the zero energy of the vauum.. Equatons for the metr and the energy The use of the rnle of least aton n the framework of the ovarant theory of gravtaton leads to the followng relaton for the metr [1]: k R k R g kg D J g U A j g W U J g B J g P, (1) where s the seed of lght; k s the onstant, whh s art of the Lagrangan n the terms wth the salar urvature R and the osmologal onstant ; R s the R tensor; g s the metr tensor; J s the mass four-urrent; j s the harge four-urrent; D, A, U and are the four-otentals of the gravtatonal and eletromagnet felds, the aeleraton feld and the ressure feld, resetvely; U, W, B and P are the stress-energy tensors of these felds, resetvely. Equaton (1) an be ontrated by means of multlyng by the metr tensor, takng nto aount that g U, g W, g B, g P, gr R, and n the four-dmensonal saetme g g 4 :

3 kr 4k D J A j U J J. () Substtuton of () nto (1) gves an equaton for the metr: 1 1 R R g U W B P. () 4 k Let us take the ovarant dervatve of both sdes of equaton (): 1 1 R g R U W B P. (4) 4 k For the tensors on the rght-hand sde, the relaton U W B P s vald as an exresson of the equaton of moton [1]. Consequently, the rght-hand sde of (4) vanshes. The R tensor and the salar urvature are art of the Ensten tensor, the ovarant dervatve of whh s equal to zero due to the roertes of the urvature tensor and the dfferental Banh dentty: 1 R g R. (5) From omarson of (5) wth the left-hand sde of (4), whh must also be equal to zero, t follows that R. It means that the ovarant dervatve of the salar urvature must be equal to zero at any ont n sae, both nsde and outsde the system. In addton to relaton (), whh ontans the salar urvature R and the osmologal onstant, there s another relaton n [1], whh ontans these quanttes. In artular, for the Hamltonan and the relatvst energy of the hysal system wth ontnuous dstrbuton of matter we found the followng:

4 1 1 E q u g dx dx dx 1 k R k Φ Φ F F 16G 4 1 g dx dx dx. u u f f (6) In (6) and q denote the nvarant denstes of mass and harge, resetvely;,, and are the salar otentals of the gravtatonal and eletromagnet felds, the aeleraton feld and the ressure feld, resetvely; Φ, F, u and f are the tensors of these felds, resetvely; u s the tme omonent of the four-veloty of the matter unt; g s the metr tensor determnant; 1 dx dx dx s the rodut of the dfferentals of sae oordnates; G s the gravtatonal onstant; s the magnet onstant; s the aeleraton feld onstant; s the ressure feld onstant.. Gaugng outsde the body We wll use (6) to alulate the ontrbuton nto the system s energy outsde the body, where there s no matter and there are only the gravtatonal and eletromagnet felds, so t s suffent to take nto aount only the seond ntegral. In ths ase, the mass and harge foururrents are equal to zero and the ondton R 4 remans n (), where the symbol o refers o to the quanttes outsde the body. Under ths ondton, the ontrbuton nto the energy (6) outsde the body wll be: o 1 Eo k Φ Φ F F g dx dx dx 1 o 16G 4. (7) It s onvenent to assume that n (7) the osmologal onstant o, that s, the ontrbuton nto the relatvst energy n the volume outsde the system deends nether on the salar urvature nor on the osmologal onstant. Then the ondton R 4 mles the equalty Ro. As a result, we obtan the equalty R o, whh follows from (4) and (5). o o 4

5 The fat that both the salar urvature R o and the osmologal onstant o are assumed to be zero outsde the body was used n [] to alulate the metr tensor omonents and to smlfy the equaton for the metr () to the followng form: 1 R U W. (8) k 4. Gaugng nsde the body We wll now ass on to the stuaton nsde the body, where all the stress-energy tensors on the rght-hand sde of the equaton for the metr () are non-zero. Sne aordng to (4) and (5) the ondton R must hold, where the symbol refers to the quanttes n the matter nsde the body, then after alyng the ovarant dervatve to all the terms n () the followng remans: ( k D J A j U J J ). (9) We wll now substtute the salar urvature from () nto the exresson for the energy (6): 1 1 E q u g dx dx dx k D J A j U J J Φ Φ F F u u f f G 1 g dx dx dx. (1) We fnd the osmologal onstant n two relatons n (9) and n (1). In (9), the ovarant dervatve k must behave wth an auray u to a sgn lke the ovarant dervatve ( D J A j U J J ). And n (1) the term k s some addtonal energy densty. The hoe of the value of for gaugng uroses s not ntally lmted by anythng, exet that t must be nvarant wth reset to the ovarant transformatons of oordnates and tme. For onvenene we wll use the smlest varant, whh sgnfantly smlfes the exresson for the energy. Just as n [1], we wll suose that the osmologal onstant body s suh that the followng relaton would hold: 5 n the matter nsde the

6 k D J A j U J J. (11) Then, aordng to (1) and (11), the energy n the sae nsde the body oued by the matter and felds wll no longer deend on the osmologal onstant: 1 1 E q u g dx dx dx 1 1 Φ Φ F F u u f f g dx dx dx. 16 G (1) Assumng that the osmologal onstant outsde the body s equal to zero beause of the absene of matter there, o, we wll omare the relatons for the energy (7) and (1). From these relatons we an see that the general exresson for the energy s (1), n whh the energy E must be relaed wth E. The system s energy E wth the rght-hand sde n the form of (1) was derved by us earler n [1]. The denser the osm objet s, the hgher s the energy densty of the artles n the feld otentals on the rght-hand sde of (11), and the greater s the osmologal onstant nsde the body. Sne (11) and () mly the relaton R, then the salar urvature R nsde the bodes s not equal to zero and vares roortonally to the osmologal onstant. Consequently, n denser bodes the salar urvature has greater value. We wll exress the four-otentals of the felds n (11) n terms of the resetve salar and vetor otentals of these felds: D, D for the gravtatonal feld, A, A for the eletromagnet feld, U, U for the aeleraton feld,, Π for the ressure feld. In the lmt of the seal theory of relatvty, the four-urrents have the followng form: J u (, ) v, j (, ) u v, where s the Lorentz fator of the q q artle of the movng and ontnuously dstrbuted matter of the system, the artle s moton. Ths gves the followng: v s the veloty of k D v A v U v Πv. (1) q q 6

7 Relatons (11) and (1) must hold true not only for the matter nsde the body, but also for the matter n suh a state, when ths matter has not yet aggregated nto a losely onneted system and was n the form of artles dstant from eah other. In the latter ase, the osmologal onstant and the salar urvature nsde ndvdual artles have ther roer values. In the lmt of low velotes we an neglet the terms ontanng the veloty artles and the vetor otentals D, A, U and Π. Then n (1) the Lorentz fator s v of the 1 and only the terms wth salar feld otentals are left. For the artles sattered at nfnty n osm sae we an assume that these otentals arse only from the artles roer felds and are the otentals averaged over the volume of artles. In ths ase, aordng to [],, where s the Lorentz fator for the matter at the enter of the artles. Denotng the osmologal onstant for ndvdual artles n osm sae by we an wrte: k. (14) q From (11) and (14) t follows that n the absene of matter the osmologal onstant vanshes. Ths s onsstent wth the fat that n Seton we assumed that the osmologal onstant outsde the body s equal to zero. Aordng to (14), s defned by the rest energy densty of the artles wth a ertan addton from the energy densty of the artles n the gravtatonal and eletromagnet felds and n the ressure feld n the matter. Now we an average over the entre sae, as well as the mass densty and the harge densty q, wthout hangng the values of the feld otentals. To do ths, we wll take nto aount that n the frst aroxmaton the rodut s the rato of the artle s mass to the artle s volume as some average densty. Averagng over the entre sae wll take lae f we dstrbute the artle s mass over the entre volume, whh, on the average, an be attrbuted to one artle n osm sae. In ths ase s hanged to and to. Leavng n (14) only the rest energy as the bas term due to ts value, we an aroxmately wrte: k. 7

8 Usng the defnton n the followng form: k, where 16G s a onstant of the order of unty, we fnd the averaged value 16G. Substtutng nstead of the estmate of the osmologal onstant we fnd the orresondng densty: m aordng to the Lambda-CDM model [4], 7 kg/m, whh s lose enough to the observed average mass densty of the matter. If we onsder the roton, whh s stable n all resets, as the bas artle n osm sae, then we an estmate the osmologal onstant for t n (14). To do ths, nstead of we should use the average roton densty of the order of kg/m wth ts radus m, aordng to [5]. Ths gves the value whh s 44 orders of magntude greater than the osmologal onstant over the entre osm sae. 16G 8. 1 m, 1 5 m averaged The next ste an be made by takng nto aount the strong gravtaton desrbed n [6, 7], and assumed as the bass for desrbng the strong nteraton at the hadron level. If we substtute the gravtatonal onstant G wth the strong gravtatonal onstant G m /(kgˑs ) aordng to [8], then the orresondng osmologal onstant for the roton wll equal 16G a m. The relaton for the salar urvature matter nsde the roton wll be wrtten as a R 9 for the R 1 m. Assumng n the frst aroxmaton that the saetme nsde the roton has onstant urvature, we wll estmate the radus of urvature, based on the exresson n [9], whh relates the salar urvature and the radus of urvature: r 1 R m. The value r, alulated n the feld of strong gravtaton, s of the order of the roton radus. 5. Averagng of hysal quanttes nsde the body Whle estmatng the matter arameters the usual roedure s to sngle out artles or volume elements of suh szes, that they ould haraterze on the average the bas roertes of the matter. For examle, n a rystallne sold body a tyal element s a rystal ell, so that the whole body an be dvded nto a number of suh ells. If we onsder the ntervals between the tyal artles of matter to be small, then to suh matter n the form of lqud we an aly 8

9 the aroxmaton of ontnuous medum. In ths ase, the artles reman ndeendent to some extent and an move at dfferent velotes. However, due to the lose nteraton of artles, n eah statonary system ertan deendenes of hysal quanttes on the oordnates and tme are establshed, whh haraterze the system on the average. We wll assume that the tyal artles of the system have exatly suh arameters, whh defne the average hysal quanttes n the matter. Atually ths means that n all equatons used to desrbe the matter all quanttes refer to tyal artles. We wll next onsder a non-rotatng body of a sheral shae, whh reresents a hysal system of losely nteratng artles and felds, held n equlbrum by gravtaton, and wll use a relatvst unform model to desrbe all hysal quanttes. Wthn the framework of the seal theory of relatvty, the ovarant dervatves are relaed wth the four-gradent, and ths relaton follows from (11) and (9): k ( D J A j U J J ). (15) On the rght-hand sde of (15) we wll exress the roduts of the four-otentals of the felds by the four-urrents n the same way as t was done n (1). For the hysal system under onsderaton, the vetor feld otentals D, A, U and Π averaged over a suffent number of tyal artles vansh due to the haot moton of these artles, and the Lorentz fator of the artles s the quantty as a funton of the urrent radus. Consequently, n (1), n a frst aroxmaton we an neglet the terms wth the vetor otentals, and the followng remans n (15): k ( q ). (16) [1]: The salar otentals of the felds nsde the shere wth the radus a were found n [], G a G r G ( r a ) os 4 sn 4, r 4 ( r a ), 4 6 q a q r q os 4 sn 4 4 r 4 9

10 r r sn 4, r 4 r r sn 4, (17) r 4 where r s the urrent radus, s the eletr onstant, s the ressure feld otental at the enter of the shere, and s the Lorentz fator at the enter of the shere. The salar otentals (17) deend on the urrent radus and gve the averaged values as a onsequene of nteraton of the entre set of tyal artles. Before substtutng these otentals nto (16), equaton (16) should be averaged over the volume of a tyal artle. Ths also means that when usng the osmologal onstant and the salar urvature nsde the body, these quanttes should be onsdered as some averaged quanttes. If we denote by V the roer volume of a tyal artle and by V the aarent volume of a movng artle from the vewont of an observer, who s statonary relatve to the body, then averagng of the left-hand sde of (16) over the volume of the movng artle yelds: 1 k dv k V, (18) where artle. s the averaged salar urvature nsde the body at the loaton of the gven tyal For the rght-hand sde of (16), averagng leads to the followng: 1 V ( q ) dv. (19) The volume element dv n the ntegrals (18) and (19) s the volume element of a movng artle from the vewont of an observer, who s statonary relatve to the body, so that 1

11 V dv. The value 1 V n the aroxmaton of the seal theory of relatvty s a onstant value for the artle under onsderaton, and therefore n (19) t was taken outsde the dervatve sgn. Sne the whole set of artles densely fll the shere, for the gven observer the sum of the volumes of all movng artles should gve the volume of the shere: Vs V. Hene t follows that the volume element dv an also be onsdered as the volume element dv of a fxed shere, so that by summng all these volume elements ths observer an determne the volume of the shere. On the other hand, the tyal artle hosen by us moves at a ertan averaged veloty v and wth the orresondng Lorentz fator. As a result, f a artle at rest has the volume V, then a movng artle, from the ont of vew of the theory of relatvty, has the redued volume V, whle V V, as well as dv dv dv, takng nto aount the equaton dv dv. Ths fat was used n [11] when onsderng the vral theorem. In (19), the quanttes dv dv dm and dv dv dq reresent q q the elements of mass and harge of the artle. Wth ths n mnd, equaton (19) an be rewrtten usng the averaged salar otentals (17) of the felds nsde the shere: 1 V m ( ) q ( ) q. () Exresson () s a ertan four-vetor, eah omonent of whh must be zero. The tme omonent of ths four-vetor vanshes, sne the otentals n the statonary shere do not deend on tme, just as the Lorentz fator of the artles. It remans to onsder the sae omonents n (), for whh we wll use the relaton for the feld oeffents, derved n [1] from the equaton of moton of matter: q G 4 4. (1) q G m 11

12 If we substtute the otentals (17) nto () and take nto aount (18) and (1), then we see that ndeed for the tme and sae omonents of the four-gradent of the averaged osmologal onstant the followng relaton holds true: k ( ) q. () 6. Cosmologal onstant and salar urvature nsde the body Sne (11) and () mly the relaton R, a smlar equaton must also exst for the averaged quanttes. Ths means that the salar urvature nsde the body must also be averaged and transformed nto R, whle the relatons R and R must be satsfed. Aordngly, equaton for the metr () nsde the body must be wrtten for the averaged quanttes: 1 1 R R g U W B P. () 4 k We an assume that relaton () was obtaned by averagng (11) and subsequent takng of the four-gradent. Removng the sgn of the four-gradent (17) and (1), we fnd the followng nsde the body: from (), takng nto aount k ( ) q G os 4 os 4. a q a 4 (4) Let us now use the value of the system s total harge q b, as well as the value of the system s gravtatonal mass m g, whh, aordng to [1], s equal to the total mass of the system s artles m b : q a a qb dq q dv sn 4 aos

13 a a mg mb dm dv sn 4 aos 4. 4 Exressng n (4) the orresondng osnes n terms of the mass m g and the harge q b, and then exandng the sne aordng to the rule x sn x x, n vew of (1) we fnd: 6 G a k sn 4 mg a 4 q q a sn 4 qb 4 a 4 Gm Gm q q. a a a a g b q q 4 8 (5) 4a 4q In (5) the auxlary mass m and the auxlary harge q a are used. Gmg Introdung then the salar otental of the gravtatonal feld a and the salar a qb otental of the eletr feld a on the surfae of the body at r a, we obtan: 4 a Gm q k. (6) q a q a a 8a Thus, the averaged osmologal onstant fxed body. The same s true for the averaged salar urvature R ase the requred ondton R s met automatally. 1 s non-zero and s a onstant value nsde the n () sne R. In ths Atually relatons (5-6) for the body reeat relaton (14) for an ndvdual artle. However, (5-6) are muh more nformatve. In artular, from (6) t follows that the osmologal onstant deends on the salar otentals of the felds. In ths ase, the otental of the ressure feld and the otental of the aeleraton feld are taken at the enter of the body, but the otentals of the gravtatonal feld a and the eletr feld a are taken not at the enter, but on the surfae of the body. The latter s assoated wth a seal

14 way of gaugng the energy and otentals of the gravtatonal and eletr felds they are gauged so that as the dstane to nfnty nreases, they vansh. We an also sefy the values of the quanttes and so that n (6) all the quanttes were determned more resely. In [11], the exresson was found for the square of the artles velotes v at the enter of a sheral body, wth the hel of whh we an estmate the value of the orresondng Lorentz fator: v v m m v 8 1a a In ths ase, the salar otental of the ressure feld at the enter of the body s aroxmately equal to: m 9 1 1a 14, and the onstant of the aeleraton feld and the onstant of the ressure feld are exressed by the formulas: q G, 5 4 q G Conluson Aordng to the onlusons n Seton, outsde the body both the osmologal onstant o and the salar urvature R o are assumed to be zero. Ths leads to exresson (7) for the ontrbuton of the feld energy outsde the body nto the total relatvst energy of the system and to equaton for the external metr (8). As for the stuaton nsde the body, t s neessary to erform an oeraton of averagng the hysal quanttes n suh a way that they would orresond to the tyal artles, whh most fully haraterze the hysal system. The order of averagng of the hysal quanttes s desrbed n Seton 5. After averagng, the salar urvature and the osmologal onstant nsde the body are onneted by the relaton R, where they are onstant quanttes. In (5-6) the osmologal onstant s exressed n terms of the otentals of all the felds 14

15 exstng n the system. The same s true for the salar urvature R, n ths ase the otentals of the aeleraton feld and the ressure feld are taken at the enter of the body, and the otentals of the gravtatonal and eletr felds are taken on the surfae of the body. For gravtatonally bound bodes the seond most mortant value after the rest energy s the gravtatonal energy. It follows from (6) that f the oeffent mass densty k s negatve, then as the nsde the shere wth the onstant radus a nreases, then due to the nrease n the rest energy densty both the osmologal onstant and the salar urvature R nrease as well. But sne the absolute value of the gravtatonal energy densty nsde the body nreases roortonally to the square of the mass densty, ths somewhat slows down the nrease of the and R. Takng nto aount k 16G we wll aly (6) to estmate the osmologal onstant n the matter nsde a neutron star, leavng on the rght-hand sde only the rest energy densty: 16G s 8 s m. Here we used the average mass densty of the star of the order of 17 s.7 1 kg/m wth ts radus of 1 km and the mass of a tyal star of 1.5 solar masses. Passng on to the salar urvature wth the hel of the equaton R, we an estmate the radus of stat saetme urvature nsde the star as n a sheral Remannan s s sae: r s 1 R 4 m. s.1 1 As a rule, artles are loated nsde the bodes n suh a way that some gas reman between the artles. Ths leads to the fat that n massve objets the average denstes of mass and energy do not exeed the orresondng values of the denstes nsde the artles. As a onsequene, n the roess of transton to suh objets the values of the averaged osmologal onstant and the salar urvature derease. In addton, the osmologal onstant n eah system turns out to be lmted to a ertan value, whh s, aordng to (11), roortonal to the rest energy densty of ths matter wth regard to the roer felds, and whh s a ertan referene ont n gaugng of the relatvst energy (1). We an omare our alulaton for the neutron star and alulaton for the roton made n Seton 4. Aordng to the theory of nfnte nestng of matter [1], these objets are analogous to eah other n many resets, whle for the roton we used both the ordnary gravtatonal onstant and the strong gravtatonal onstant. Both for the neutron star, and usng the strong gravtatonal onstant for the roton, we obtaned that the radus of the saetme urvature 15

16 does not dffer muh n the order of magntude from the radus of the orresondng objet. The rato of the sefed rad s the followng: rs r R (7) R s We wll now use the oeffents of smlarty between the neutron star and the roton. The rato of the star s mass of to the roton s mass gves the oeffent of smlarty n mass , the rato of the rad of these objets gves the oeffent of smlarty n szes 19 P 1.4 1, and the oeffent of smlarty n seeds of same-tye roesses equals. It seems that the rato rs r S. should equal P, but ths s not so, beause the radus of urvature s derved through the energy densty and s not smly a lnear dmenson. In order to show ths, we wll take nto aount that n (7) the salar urvature nsde eah objet s roortonal to the orresondng gravtatonal onstant and mass densty: r G P r G S s a. s Meanwhle, aordng to the dmenson theory, the rato of the gravtatonal onstants s G G a RS s and the rato of the mass denstes s P. From the results of Seton 4 t follows that nsde the roton the osmologal onstant, takng nto aount the strong gravtatonal onstant, should be of the order of a m. Ths s 8 orders of magntude greater than the osmologal onstant 16G 1 5 m, whh follows from the general theory of relatvty as aled to the observable Unverse. In ths onneton we reall that n osmology there s stll unexlaned roblem of the osmologal onstant. The essene of ths roblem s that the osmologal onstant, alulated wth the hel of the general theory of relatvty for the osm sae, s almost 1 orders of magntude less than the osmologal onstant for the zero vauum energy, aordng to quantum hyss. The osmologal onstant s a requred element n the Lambda- CDM model, and n ths ase t beomes unlear why the exeted large magntude of the zero 16

17 vauum energy s transformed nto suh a small osmologal onstant of the osm sae of the Unverse [14]. Our exlanaton of the roblem of the osmologal onstant s as follows. Dsreany between the onlusons of the quantum hyss and the general theory of relatvty n reset of the osmologal onstant s assoated wth the geometral aroah of the general theory of relatvty, whh relaes the gravtaton, as a fore aton, wth the saetme urvature. Ths leads to the absene n ths theory of the stress-energy tensor of the gravtatonal feld and to the mossblty of alulatng the energy of the nternal arts of the hysal system under study, whh also sgnfantly omlates quantzaton of the general theory of relatvty. In the ovarant theory of gravtaton there are no suh roblems. We fnd searate omonents of the energy nsde the body, and wth ther hel we determne the orresondng osmologal onstant for eah body. The observed art of the Unverse an be onsdered as the nternal art of some global body, and n ths ase the osmologal onstant s of the order of 1 5 m, besdes t s a onstant value, whh haraterzes the entre sae flled wth stars and galaxes. For the neutron star, the osmologal onstant reahes the value of 8 s m 8, and for the roton t equals. 1 m - for the ordnary gravtaton and m n the strong gravtatonal feld. Aordng to the aroah to energy and metr gaugng n the ovarant theory of gravtaton, outsde the body n the sae wthout matter the osmologal onstant and salar urvature of saetme vansh. Thus, the alleged relaton between the osmologal onstant and the zero vauum energy of the quantum hyss outsde the body s broken. As for the dfferene between the osmologal onstants nsde the roton and the neutron star on the one hand, and the osm sae on the other hand, t s exlaned by the fat that the osmologal onstant of the observable Unverse s the osmologal onstant of all the artles and bodes of the Unverse averaged over the sae. We should note that n the modernzed Le Sage s model gravtaton an arse as a onsequene of the aton of the fluxes of relatvst artles of the vauum feld on the bodes [15, 16]. In ths ase, the standard roblems of the Le Sage s model are elmnated by exlanng the way n whh the vauum artles nterat wth the matter [17]. In ths ase, nstead of searhng for the quantum zero energy of the vauum t s ossble to determne the energy densty of the vauum feld artles, to derve the gravtatonal onstant and the eletr onstant through the vauum feld arameters and to exlan the effet of the Newton s and Coulomb s laws for gravtatonal and eletr fores. 17

18 The equatons of the ovarant theory of gravtaton desrbe only the onsequenes of nteraton of the vauum feld s artles wth the matter, whh are exressed n hangng of the atng fores, hangng of the matter energy, as well as n reatng nerta of the bodes and martng mass to them. Ths means that the energy densty of the vauum feld s artles, deste ts largeness, s not taken nto aount as an essental omonent of the osmologal onstant. However, the energy of the vauum feld s artles nfluenes ndretly the magntude of the osmologal onstant nsde a artular system through the averaged densty of the rest energy of the system s artles. Referenes 1. Fedosn S.G. About the osmologal onstant, aeleraton feld, ressure feld and energy. Jordan Journal of Physs, Vol. 9, No. 1,. 1- (16). htt://dx.do.org/1.581/zenodo Fedosn S.G. The Metr Outsde a Fxed Charged Body n the Covarant Theory of Gravtaton. Internatonal Fronter Sene Letters, ISSN: , Vol. 1, No. 1, (14). htt://dx.do.org/1.185/ Fedosn S.G. The Integral Energy-Momentum 4-Vetor and Analyss of 4/ Problem Based on the Pressure Feld and Aeleraton Feld. Ameran Journal of Modern Physs, Vol., No. 4, (14). htts://dx.do.org/ /j.ajm Tegmark, Max; et al. Cosmologal arameters from SDSS and WMAP. Physal Revew D. Vol. 69, Issue 1, 151 (4). htts://dx.do.org/1.11%fphysrevd Fedosn S.G. The radus of the roton n the self-onsstent model. Hadron Journal, Vol. 5, No. 4, (1). htt://dx.do.org/1.581/zenodo Salam, A., and Strathdee, J. Confnement Through Tensor Gauge Felds. Physal Revew D, Vol.18, Issue 1, (1978). htts://do.org/1.11/physrevd Oldershaw R.L. Dsrete Sale Relatvty. Astrohyss and Sae Sene, Vol. 11, N. 4, (7). htt://dx.do.org/1.17/s x. 8. Fedosn S.G. Fzka flosofa odoba ot reonov do metagalaktk. Perm, ages 544 (1999). ISBN Fok V. A. The Theory of Sae, Tme and Gravtaton. Mamllan. (1964). 1. Fedosn S.G. Relatvst Energy and Mass n the Weak Feld Lmt. Jordan Journal of Physs. Vol. 8, No. 1, (15). htt://dx.do.org/1.581/zenodo

19 11. Fedosn S.G. The vral theorem and the knet energy of artles of a maroso system n the general feld onet. Contnuum Mehans and Thermodynams, Vol. 9, Issue, (16). htts://dx.do.org/1.17/s Fedosn S.G. Estmaton of the hysal arameters of lanets and stars n the gravtatonal equlbrum model. Canadan Journal of Physs, Vol. 94, No. 4, (16). htt://dx.do.org/1.119/j Fedosn S.G. The hysal theores and nfnte herarhal nestng of matter, Volume 1, LAP LAMBERT Aadem Publshng, ages 58 (14). ISBN Martn J. Everythng you always wanted to know about the osmologal onstant roblem (but were afrad to ask). Comtes Rendus Physque, Vol. 1, Issue 6, (1). htt://dx.do.org/1.116/j.rhy Fedosn S.G. The gravton feld as the soure of mass and gravtatonal fore n the modernzed Le Sage s model. Physal Sene Internatonal Journal, ISSN: 48-1, Vol. 8, Issue 4, (15). htt://dx.do.org/1.974/psij/15/ Fedosn S.G. The fore vauum feld as an alternatve to the ether and quantum vauum. WSEAS Transatons on Aled and Theoretal Mehans, Vol. 1, Art. #,. 1-8 (15). htt://dx.do.org/1.581/zenodo Fedosn S.G. The harged omonent of the vauum feld as the soure of eletr fore n the modernzed Le Sage s model. Journal of Fundamental and Aled Senes, Vol. 8, No., (16). htt://dx.do.org/1.414/jfas.v