aa r X i v : . [ phy s i c s . g e n - ph ] J u l Noname manuscript No. (will be inserted by the editor)
Model of a Solar System in the ConservativeGeometry
Edward Lee Green
Received: date / Accepted: date
Abstract
Pandres has shown that an enlargement of the covariance groupto the group of conservative transformations leads to a richer geometry thanthat of general relativity. Using orthonormal tetrads as field variables, thefundamental geometric object is the curvature vector denoted by C µ . From anappropriate scalar Lagrangian field equations for both free-field and the fieldwith sources have been developed. We first review models which use a free-fieldsolution to model the Solar System and why these results are unacceptable.We also show that the standard Schwarzschild metric is also unacceptablein our theory. Finally we show that there are solutions which involve sourceswhich agree with general relativity PPN parameters and thus approximate theSchwarzschild solution. The main difference is that the Einstein tensor is notidentically zero but includes small values for the density, radial pressure andtangential pressure. Higher precision experiments should be able to determinethe validity of these models. These results add further confirmation that thetheory developed by Pandres is the fundamental theory of physics. PACS · · · We assume a 4-dimensional space with the orthonormal tetrad h iµ and internaltetrad L iI as field variables. (Please refer to earlier papers by Pandres andGreen for a thorough discussion of the physical motivation and for additionalexamples of how to interpret the theory [1,2,3,4,5,6].) In the present paper,which is concerned with gravity and cosmology, the internal tetrad may be E. GreenUniversity of North Georgia, Dahlonega, GA 30597Tel.: +706-864-1809Fax: +706-864-1678E-mail: [email protected] Edward Lee Green chosen to be constants. The metric is defined by g µν = η ij h iµ h jν where η ij = diag (cid:8) − , , , (cid:9) . (Note: our sign conventions are the same as Misner, Thorneand Wheeler [7].) The following condition (1) defines the conservation group, agroup of coordinate transformations that include the group of diffeomorphismsas a proper subgroup [1]: x ν,α (cid:0) x α,ν,µ − x α,µ,ν (cid:1) = 0 . (1)This group preserves the wave equation and other conservation laws [2], whichare of the form V α ; α = 0 .The geometry determined by (1) is more general than a manifold [1,2,3,4,5]. We regard a conservative, non-diffeomorphic transformation as a mappingbetween two manifolds [6]. The geometrical content of space-time is determinedby the curvature vector: C α ≡ h νi (cid:0) h iα,ν − h iν,α (cid:1) = γ µαµ (2)where the Ricci rotation coefficient is given by γ i µν = h iµ ; ν [1,2,3]. The curva-ture vector, C α , is covariant under transformations from x µ to x µ if and onlyif the transformation is conservative, i.e., it satisfies (1).When sources are present, the Lagrangian is of the form L = L f + L s = Z (cid:18) π C α C α + L s (cid:19) h d x (3)where L s is the appropriate source Lagrangian density function. Using theRicci rotation coefficients, one finds that C α C α = R + γ αβν γ ανβ − C α ; α where R is the usual Ricci scalar curvature. Thus, the free-field part containsadditional terms which may correspond to other forces or dark matter [3,4,6].Variation of (3) with respect to the tetrad results in (see [8]) Z " π (cid:18) C ( µ ; ν ) − C α γ α ( µν ) − g µν C α C α − g µν C α ; α (cid:19) − ( T s ) µν h h iν δh µi d x = 0Thus the source stress-energy tensor is8 π ( T s ) µν = C ( µ ; ν ) − C α γ α ( µν ) − g µν C α C α − g µν C α ; α (4)and we also find G µν = (cid:18) γ α ( µ ν ); α + γ ασ ( ν γ σµ ) α + 12 g µν γ αβσ γ ασβ (cid:19) + 8 π ( T s ) µν (5)The first term in the parentheses represents the free-field part of the totalstress energy tensor which is the dark matter/dark energy part. The secondterm given by (4) transforms covariantly under the conservation group (1). odel of a Solar System in the Conservative Geometry 3 In this section we develop the formulae for the curvature vector, metric andtotal stress-energy tensor. In spherical coordinates, an arbitrary sphericallysymmetric tetrad may be expressed by h i µ = e Φ ( r ) e Λ ( r ) sin θ cos φ r cos θ cos φ − r sin θ sin φ e Λ ( r ) sin θ sin φ r cos θ sin φ r sin θ cos φ e Λ ( r ) cos θ − r sin θ (6)where the upper index refers to the row. The curvature vector for this tetradfield is given by C µ = e Λ r (cid:20) , − e − Λ (cid:0) rΦ ′ + 2 (cid:1) , , (cid:21) (7)where components are in the order [ t, r, θ, φ ] and the prime denotes the deriva-tive with respect to r . The tetrad (6) leads to the metric ds = − e Φ ( r ) dt + e Λ ( r ) dr + r dθ + r sin θdφ . (8)When ( rΦ ′ + 2) = 2 e Λ , then C µ in equation (7) is identically zero and hence( T s ) µν is identically zero leading to a free-field solution.The metric (8) leads to a diagonal Einstein tensor with nonzero elements: G tt ≡ − πρ = 1 r (cid:0) − re − Λ Λ ′ + e − Λ − (cid:1) = − r ddr (cid:20) r (cid:16) − e − Λ (cid:17)(cid:21) (9) G rr ≡ πp r = 1 r (cid:16) re − Λ Φ ′ + e − Λ − (cid:17) (10) G θθ = G φφ ≡ πp T = e − Λ r (cid:18) rΦ ′′ + r ( Φ ′ ) − rΦ ′ Λ ′ + Φ ′ − Λ ′ (cid:19) (11)We note that G tt = − πρ depends only on Λ ( r ). Depending on whether C µ is zero via (7) we will have a free-field or a field with sources. The source termof the Lagrangian (3) that we will use is L s = ρ s ( r ), where ρ s ( r ) is the densityof the source as a function of r . Previous results suggest the interpretationthat the density of the source is given by [6]:8 πρ s = 12 C µ C µ (12)We assume that for realistic densities, ρ s < ρ and we define ρ dm ≡ ρ − ρ s torepresent the density of dark matter. Edward Lee Green δ ǫ , and following [9] we add a term L p = ρ p ( x ) = µ Z δ ǫ ( x − γ ( s ))( − u µ u µ ) ds (13)to the Lagrangian of (3). Note: the space-like volume is represented by ǫ whichis assumed to be small. The mass of the particle will be denoted by µ . Thepath of the particle is represented by γ ( s ) and its velocity is u α = dx α dτ . Wewill use the ”dot” notation for the components of u α , i.e. u α = h ˙ t, ˙ r, ˙ θ, ˙ φ i . Thecondition T βα ; β = 0 leads to (see [9] and [6]) µ √− u ν u ν u β u α ; β = δ α F p (14)with F p ≡ ǫ e − Λ (cid:0) − p ′ R + r (cid:0) p T − p R (cid:1) (cid:1) . When α = 1, (14) implies u β u α ; β = 0which is the usual geodesic equation.We will only be concerned with orbits and motions that can be confined to asurface which we henceforth assume to be the θ = π surface. This is consistentwith θ component of (14) which implies that ¨ θ + r ˙ r ˙ θ − (sin θ cos θ ) ˙ φ = 0 .The φ component of (14) using metric (8) implies that¨ φ + 2 r ˙ r ˙ φ = 0 ⇒ ˙ φ = Lr (15)where L is a constant interpreted as the conserved angular momentum. From(14) we also find that the t component is¨ t + 2 ˙ Φ ˙ t = 0 ⇒ ˙ t = E e − Φ (16)where Φ ( r ) is the function appearing in h (see 6) and E is a constant (energy)of the motion. For the metric (8) with θ ≡ π , the r component of the particlemotion is determined by¨ r + e Φ − Λ Φ ′ ( r ) ˙ t + Λ ′ ( r ) ˙ r − r e − Λ ˙ φ = 1 µ F p (17)Using (15), (16), (10) and (11) along with the normalization ˙ t = e − Φ (i.e., wechoose E = 1 in (16) - see [8], page 186), we find that¨ r + Λ ′ ˙ r + e − Φ − Λ Φ ′ − e − Λ r L = 2 ǫ e − Λ Φ ′ ( Λ ′ + Φ ′ ) µ r (18) odel of a Solar System in the Conservative Geometry 5 We first consider the free-field case where C µ = 0. This implies that C µ C µ = 0and hence the density of the source, ρ s is identically zero. Let w ( r ) = r (cid:16) − e − Λ (cid:17) and assume that lim r →∞ w ( r ) = W , a constant which is related to theasymptotic value of the mass of the Sun. In this case, using (7), Φ ′ ( r ) = 2 r (cid:18) e Λ − (cid:19) = 2 r (cid:18) − w ( r ) r (cid:19) − − r (19)Asymptotically, we have Φ ′ ( r ) ≈ Wr ⇒ e Φ ( r ) ≈ − Wr (20)To match with the weak field solution of general relativity, we thus concludethat W = M , where M is the total mass-energy of the Sun. Thus we seethat in the free-field models, G tt represents of the density of mass-energy(nevertheless, we shall continue to denote G tt by − πρ in the free-field case).In [6], we investigated this case in-depth and found several positive results.The tetrad solutions that were investigated were determined by w ( r ) = M (Case 1) and w ( r ) = M − M r (Case 2). In both cases we found that theEinstein tensor was nonzero (see (9) - (11)). In Case 1, asymptotically, wehad 8 πρ = 0, 8 πp r ≈ Mr and 8 πp T ≈ − M r . In Case 2, we had the sameasymptotically for p r and p T , but we had 8 πρ = M r .We found asymptotic agreement with the gravitational redshift and withKepler’s Law for both Case 1 and Case 2. Additionally, we were able to explainthat the Pioneer anomaly was produced by the nonzero and nonequal radialand tangential pressures. About the same time, another explanation of thePioneer anomaly was given in terms thermal corrections [10].However, discrepancies were found with the precession of perihelia. Specifi-cally, the precession of perihelia calculation for Case 1 produced a value whichwas of the standard value and the calculation for Case 2 produced a valuewhich was of the standard value. Arguments were made to justify thediscrepancy in Case 2, since the nonzero density could contribute to the pre-cession.We also found discrepancies in the light deflection and time delay pre-dictions. For both the light deflection and time delay calculations our theoryyielded values which were 75% of the standard value (for both Case 1 and Case2). It was argued that the nonzero density of the inertial mass (given by ρ + p r )along with reasonable refraction index values could explain the discrepancy. Itis difficult to imagine why the numbers would agree with the standard resultas observed in solar system experiments, so we reject the free-field model dueto a lack of naturalness. Edward Lee Green
When setting up a model with our theory, one should keep in mind that thethere is a great deal of freedom due to the larger covariance group. This meansthat additional conditions may be needed to obtain a model that classicallyrepresents a given physical situation. One may view solutions that do notmeet these conditions as being non-physical or as non-classical. The additionalconditions given below will enable us to obtain a reasonable model of the SolarSystem.Recall that the scalar formed from the curvature vector (2) by C µ C µ exactly determines the density of mass-energy of the sources according to8 πρ s = C µ C µ [6]. We note that C µ is a vector under the group of conserva-tive transformations and thus C µ C µ is an invariant scalar. From (7) and (8)we find that C µ C µ = e − Λ (cid:18) e Λ r − Φ ′ − r (cid:19) .For convenience in our model development, we define β ( r ) = p C µ C µ .Hence Φ ′ = 2 e Λ r − r − β ( r ) e Λ (21)As a strategy for model development, we first will propose a function for Λ ( r ),or equivalently e Λ . We will require that, in contrast to the free-field case, G tt corresponds to − πρ , the usual density of mass-energy (9). (See condition C1)below.) This implies that m ( r ) = 12 r (1 − e − Λ ) (22)The next factor (see condition C2) below) to consider is whether the corre-sponding metric asymptotically agrees with the weak-field metric, i.e., g tt ≈− (1 − Mr ). The third factor to consider (see condition C3) below) is whetherthe density of dark matter is nonnegative, i.e., ρ − ρ s ≥ e Λ = − Mr and e Φ = 1 − Mr . From (21) wethen find β = (1 − √ − Mr )(1 − √ − Mr )2 r √ − Mr and hence C µ C µ = (1 − q − Mr ) (1 − q − Mr ) r (1 − Mr ) ≈ M (1 − Mr ) r (23)where the approximation is for r >> M .This solution is not acceptable however, because G tt = 0 and thus ρ ≡ ρ s ≈ M (1 − Mr )16 πr . Since ρ s + ρ dm = ρ , this implies that ρ dm <
0. Ifwe require that the density of mass-energy be positive in the case of the solarsystem (no extreme energies, such as extreme gravity near a black hole) thenwe must abandon this solution as being physically unreasonable. odel of a Solar System in the Conservative Geometry 7
The Parameterized-Post Newtonian (PPN) formulism utilizes a parameter-ized metric which is in the isotropic form. First we exhibit the general sphericalisotropic tetrad (where Φ represents Φ ( r ) and Λ represents Λ ( r )): h i µ = e Φ e Λ sin θ cos φ re Λ cos θ cos φ − re Λ sin θ sin φ e Λ sin θ sin φ re Λ cos θ sin φ re Λ sin θ cos φ e Λ cos θ − re Λ sin θ (24)where the upper index refers to the row. The curvature vector for this tetradfield is given by C µ = (cid:20) , − dΦdr − dΛdr , , (cid:21) (25)where components are in the order [ t, r, θ, φ ]. Hence we have C µ C µ = e − Λ (cid:18) dΦdr + 2 dΛdr (cid:19) (26)This tetrad (24) yields a spherically symmetric metric in isotropic coordinatesgiven by ds = − e Φ dt + e Λ (cid:20) dr + r dθ + r sin θdφ (cid:21) (27)The metric (27) leads to a diagonal Einstein tensor with nonzero elements: G tt ≡ e − Λ r (cid:20) Λ ′ + 2 rΛ ′′ + r ( Λ ′ ) (cid:21) (28) G rr ≡ e − Λ r (cid:20) Φ ′ + 2 Λ ′ + 2 rΦ ′ Λ ′ + r ( Λ ′ ) (cid:21) (29) G θθ = G φφ ≡ e − Λ r (cid:20) Φ ′ + Λ ′ + rΦ ′′ + r ( Φ ′ ) + r ( Λ ′ ) (cid:21) (30)where primes indicate derivatives with respect to r .In the PPN system, the value of e Φ is parameterized up to second orderin Mr , but the value of e Λ is specified only up to first order in Mr . The unac-ceptable solution (23) for exact matching with the Schwarzschild metric leadsus to take a different approach. We start with the isotropic form (27) with e Λ ≈ Mr + bM r (31)where b is a free parameter. We also require that our resulting metric (inisotropic coordinates) agrees with the standard external Schwarzschild solutionso that there will be agreement with Solar System experiments. Thus e Φ ≈ − Mr + 2 M r (32) Edward Lee Green
We now proceed to find values for ρ s and ρ up to the terms of order M r (note: Mr is of order 10 − in natural units). We first calculate ρ s using (12) and (26)and inserting (32) and (31). The result is8 πρ s ≈ M r + (4 b − M r (33)We will use a bar over the radial coordinate (¯ r ) to represent the fact thatwe are using the standard radial coordinate and use the usual symbol, r , forisotropic coordinates. The conversion formula is simply,¯ r = r e Λ ( r ) ⇒ r ≈ ¯ r (cid:18) − M ¯ r + (1 − b ) M r (cid:19) (34)Because of (34) the expression for g tt = − e Φ depends on b : e Φ ≈ − M ¯ r + (4 − b ) M ¯ r (35)(notice that the term of order M ¯ r is zero). From (33) we find that in thestandard radial coordinates,8 πρ s ≈ M r + (2 b − M ¯ r (36)We now find the value of ρ (total density) for (27) with conditions (31) and(32). The result is 8 πρ ≈ − (2 b − M r + 4(2 b − M r (37)After converting to standard coordinates we find8 πρ ≈ − (2 b − M ¯ r (38)where the M ¯ r term is zero.4.1 Acceptability Conditions.For the model to be acceptable, we require that its:C1) Einstein tensor component G tt match with − πρ ;C2) metric match with the asymptotic (weak field) approximation;and C3) (since 8 πρ s = C µ C µ ) value of C µ C µ must be such that ρ s < ρ .Conditions C1) and C2) imply that C µ is nonzero. Thus in order thatour Solar System models be acceptable, C µ C µ must be nonzero with 8 πρ s = C µ C µ < − G tt . This rules out the value of b = that matches with theSchwarzschild metric in isotropic coordinates. It is seen then that b < . FromTable 1 we see that for b ≤ we have acceptable models for the Solar System. odel of a Solar System in the Conservative Geometry 9 Table 1
Asymptotic approximations for various values of parameter bb πρ (¯ r ) 8 πρ s (¯ r ) e Φ (¯ r ) m (¯ r ) Acceptable? M r − M ¯ r + M ¯ r M no M r M r (cid:16) − M ¯ r (cid:17) − M ¯ r + M r M (cid:16) − M r (cid:17) yes1 M ¯ r M r (cid:16) − M ¯ r (cid:17) − M ¯ r + M ¯ r M (cid:16) − M r (cid:17) yes
12 2 M ¯ r M r (cid:16) − M ¯ r (cid:17) − M ¯ r + M ¯ r M (cid:16) − M ¯ r (cid:17) yes0 M ¯ r M r (cid:16) − M ¯ r (cid:17) − M ¯ r + M ¯ r M (cid:16) − M r (cid:17) yes b = 1If we choose b = 1, we may easily develop a model with exact (not approxi-mate) solutions. For this model, we equate e Λ to the value 1 + Mr + M r sothat e Λ = (cid:18) Mr (cid:19) (39)In order to satisfy (32) we choose specifically e Φ = 1 + Mr (cid:16) Mr (cid:17) ≈ − Mr + 2 M r (40)From (21) we solve for β ( r ) and thus determine C µ C µ and hence 8 πρ s . Theresult is 8 πρ s = M r (1 + Mr ) (1 + Mr ) (41)The resulting metric in line-element form is given by ds = − (1 + Mr )(1 + Mr ) dt + (cid:16) Mr (cid:17) (cid:18) dr + r dθ + r sin θdφ (cid:19) (42)From (34) we see that the regular Schwarzschild-type radial coordinate isobtained by the replacement, ¯ r = r + M . The resulting metric is found to be(in line element form): ds = − (1 − M ¯ r ) (1 + M ¯ r ) dt + 1(1 − M ¯ r ) d ¯ r + ¯ r dθ + ¯ r sin θ dφ (43)and is defined on the interval ¯ r > R , where R is the Schwarzschild radiusof the Sun. We note that g tt ≈ − ( 1 − Mr ). For this metric and tetrad, theresulting densities and pressures are C µ C µ = M ¯ r (cid:16) M ¯ r (cid:17) πρ s = M r (cid:16) M ¯ r (cid:17) (44) πρ = M ¯ r πρ dm = M (1 + M ¯ r + M ¯ r )2¯ r (1 + M ¯ r ) (45)8 πp r = M (1 − M ¯ r )¯ r (1 + M ¯ r ) 8 πp T = − M (1 − M ¯ r − M ¯ r )¯ r (1 + M ¯ r ) (46)This matches the standard external metric of Schwarzschild up to the level ofaccuracy. The total mass-energy as a function of ¯ r is m (¯ r ) = M − M r (47)All solar system results match since the only difference would be due to F p given by (14) and for this model, we find that 8 πF p ≈ ǫM ¯ r , which is negligible.Thus, the solar system experiments, as reported by Will [11] , would agree withthis metric just as it does for standard general relativity. The only differenceof note is due to the small difference in acceleration due to (47). The impliesthat there is an extra outward acceleration at Mercury which is approximately5 . × − meters per second per second. As ¯ r increases this extra outwardacceleration decreases and could be interpreted as an extra inward accelerationin the standard theory. We note that this is slightly less than the Pioneeranomaly value of 8 . × − meters per second per second. We claim thatthe Pioneer anomaly is mostly explained by (47) and partially explained bythermal forces [10]. The results of this paper show that our theory can reasonably agree with SolarSystem tests of general relativity. The value of the parameter b which we useto represent the M r term of g rr is seen to be strictly less than , which is thevalue of b for general relativity. There is a need for higher precision tests todetermine whether b is indeed less than . Acknowledgements
The author is deeply thankful for the long-time collaboration withProfessor Dave Pandres, Jr., who began this work and who passed away in August 2017.The author also thanks Peter Musgrave, Denis Pollney and Kayll Lake for the GRTensorIIsoftware package which was very helpful. The author thanks the University of North Georgiafor travel support.