Pessue Calculation of a Constant Density Sta in the Dynamic Theoy of Gavity Ioannis Iaklis Haanas Depatment of Physics and Astonomy Yok Univesity A Petie Science Building Yok Univesity Toonto Ontaio CANADA e-mail: ioannis@yoku.ca Abstact In a new theoy called Dynamic Theoy of Gavity, the gavitational potential is is deived fom gauge elations and has a diffeent fom than the classical Newtonian potential. In the same theoy an analytical expession fo the pessue is deived fom the equation of the hydonamic equilibium which is solved fo a sta of constant density and the esults ae compaed with those of Newtonian gavity. Changes then in the cental pessue and adius ae also calculated and finally a edshift calculation is pefomed so that the dynamic gavity effects if any might be shown to be of some detectabe magnitude. Key Wods Dynamic theoy of gavity, gauge fields, Weyl s quantum pinciple, field equtions, Chistoffel symbols, enegy-momentum tenso.. Intoduction Thee is a new theoy called the Dynamic Theoy of Gavity [DTG]. It is deived fom classical themodynamics and equies that Einstein s postulate of the constancy of the speed of light holds. []. Given the validity of the postulate, Einstein s theoy of special elativity follows ight away []. The dynamic theoy of gavity (DTG) though Weyl s quantum pinciple also leads to a non-singula electostatic potential of the fom: K o V ( ) = e. () whee K o is a constant and is a constant defined by the theoy. The DTG descibes physical phenomena in tems of five dimensions: space, time and mass. [] By consevation of the fifth dimension we obtain equations which ae identical to Einstein s field equations and descibe the gavitational field. These field equations ae simila to those of geneal elativity and ae given below: K T o g = G =. ()
Now T is the suface enegy-momentum tenso which may be found within the space tenso and is given by: α β h ν = Tsp F F F F ν c T () and T sp µν is the space enegy-momentum tenso fo matte unde the influence of the gauge fields and is given by:[] T ij sp i kj ij kl Fk F a F Fk c () i j = γu u l which futhe can be witten in tems of the suface metic as follows:[] α β α kβ α β = γu u Fk F F F µν ν c µν ν ( g h )( F F F F ) Tsp (5) since: dy y u = y u = (6) dt t is the statement equied by the consevation of the fifth dimension, and the suface indices ν, α, β. =,,, and space index i, j, k, l =,,,,, and i j g = a y y = a h = a a y a y y whee the suface field tenso is given by: ij α α α β α β F = F y ij i α y j β i i y i y and yα = = δ α α fo i =,,, and yα =. (7) α x x o E E E V o E o B B V F = ij E B o B V. (8) E B B o V V V V V o o It was shown by Weyl that the gauge fields may be deived fom the gauge potentials and the components of the 5-dimensional field tenso F ij given by the 5 5 matix given in (8). [] Now the detemination of the fifth dimension may be seen, fo the only physically eal popety that could give Einstein s equations is the gavitating mass o it s equivalent, mass [5]. Finally the dynamic theoy of gavity futhe agues that the gavitational field is a gauge field linked to the electomagnetic field in a 5-dimensional
manifold of space-time and mass, but, when consevation of mass is imposed, it may be descibed by the geomety of the -dimensional hypesyface of space-time, embedded into the 5-dimensional manifold by the consevation of mass. The 5 dimensional field tenso can only have one nonzeo component V which must be elated to the gavitational field and the fifth gauge potential must be elated to the gavitational potential. The theoy makes its pedictions fo ed shifts by woking in the five dimensional geomety of space, time, and mass, and detemines the unit of action in the atomic states in a way that can be calculated with the help of quantum Poisson backets when covaiant diffeentiation is used: [6] µ ν νq µ s [ x, p ] Φ i g { δ Γ x }Φ = h. (9) µq s, q In (9) the vecto cuvatue is contained in the Chistoffel symbols of the second kind and the gauge function Φ is a multiplicative facto in the metic tenso g νq, whee the indices take the values ν, q =,,,,. In the commutato, x µ and p ν ae the space and momentum vaiables espectively, and finally δ µq is the Conecke delta. In DTG the momentum ascibed, as a vaiable canonically conjugated to the mass is the ate at which mass may be conveted into enegy. The canonical momentum is defined as follows: p = mv () whee the velocity in the fifth dimension is given by: γ v = () α o and gamma dot is a time deivative and gamma has units of mass density (kg/m ) and α o is a density gadient with units of kg/m. In the absence of cuvatue (8) becomes: µ ν νq [ x p ] Φ = ihδ Φ,. (). The Gavitational Potential of Dymamic Gavity As it tuns out in the DTG the gavitational potential takes the fom: GMm V ( ) = exp () Whee M is the mass of the main body and m is the mass of the test paticle body and is the distance fom the cente of main body to the cente of the test paticle body, and finally is a distance facto, which may be diffeent fo each paticle, and can be defined as: [6]
GM = () c We can easily note that equation () is a non-singula equation, almost of the same fom as a Yukawa equation. Yukawa equation goes as athe that / in the exponent. Fo distances much geate has the familia / fom. When = the potential has its maximum value. At = the potential becomes zeo due to the oveiding effect of the exponential.. Equations of Hydostatic Equilibium fo a Sta The stuctue of any stable sta whee non-elativistic effects ae not included in hydostatic equilibium, obeys the following diffeential equations: dp( ) GM ( ) = g( ) ρ () = ρ( ) d dm ( ) = π ρ () d (5) If elativistic effects ae taken into account then fist of the equations in (5) becomes the so called Oppenheime-Volkoff diffeential equation of stella stuctue below: p() π p( ) Gm( ) ρ () - dp( ) = ρ ()c m( ) c d Gm( ) c (6) Whee: in equations, () and (), the L.H.S of the fist and second equations indicate, the pessue and the mass gadients as a function of adius. And in the.h.s, of the fist equation ρ() is the density of the sta as a function of distance, and M() is the mass of the sta whithin the distance. In paticula in equation (6) the exta appeaing tems ae the elativistic coection tems.. Stella Stuctue in the Dynamic Theoy of Gavity To study any possible effects in Dynamic Gavity of gavity lets fist find using equation () the foce due to such potential as given by the DTG: V ( ) GMm F( ) = = exp (7) Then equation () finally gives the following foce function:
5 GMm F( ) = ( ) exp (8) Fom which we find that the acceleation of gavity pe unit mass takes the fom: GM ( ) g( ) = ( ) exp (9) If we now substitute equation (9) into the fist equation of hydostatic equilibium we obtain: dp( ) GM ( ) = g( ) ρ( ) = ( ) ρ( )exp d () As the next step we will have to expess the mass M() as function of distance. Fist we will assume constant density function ρ() = ρ Then the second equation fo hydonamic equilibioum gives that: π M ( ) = ρo fo () Which finally makes equation () equal to: dp( ) πg = ρ ( ) exp d () Integating now fom (, ) and changing vaiable back to we obtain: πg = P ( ) P() ρ ( ) exp exp Γ, () 6 Whee Γ(,/) is the Incomplete Gamma function of agument (, /). This expession can be futhe simplified to: πg = GM P( ) P() ρ exp Γ, c sta () If we now apply the bounday condition that P(=) =, we obtain the cental pessue P( ) to be equal to: πg = P () ρ Γ exp,, 5)
6 which can be futhe simplified as follows: ( ). ρ πg P (5a) whee the value of the incomplete gamma function Γ(,/) = Γ(,.7-6 )=.88 fo sta of adius = Sun 7]. The cental pessue effect due to dynamic gavity is zeo when =. =.986 Km. That would pobably mean that, dynamic gavity effects on the cental pessue die out within a sphee of adius =.986 Km fom the cente of the sta having a mass equal to M sun. Finally the total pessue at any point can be witten as follows: =, exp, exp ) ( c GM Γ c GM Γ ρ πg P sta sta (6) In the case whee the adius of the sta is much geate than lamda then equation (7) can be simplified as follows, afte expanding the exponentials to fist ode: ( ) = Γ Γ ρ πg P,, ) ( ) ( (7) Taking into account that = G M/ c equation () can be futhe witten as follows: ( ) ( ) = Γ c GM Γ c GM c GM πgρ P,, ) ( (8) If we look at equation (8) we can see that the fist tem in the ight hand sight goes as ( - ) and is simila to the density pofile of a Newtonian gavity sta of constant density [8]. Thee is now a diffeence in the pessue pofile due to the DTG and its new gavitational potential. This diffeence can be shown as two exta coection tems in equation (8). 5. Change in Cental Pessue The change in cental pessue P() between dynamic and Newtonian gavity sta becomes:
7 P P cn P P cd cn. = =. (9) P c cn Theefoe we now futhe have: [9] ' GM. P = P P = cd cn c 8π. () Table adius of the Sta In Sola adii ( sun ) Mass of the Sta In Sola Masses (M S ) Value of. (km)...8.9999 n.8.668.9999 8.8.7.67.78 Apat fom the sola type constant density sta two moe stas wee used so that the dynamic gavity effects could be calculated on the cental pessue. These wee neuton stas [9] and we must say these stas can not be eally thought as constant density stas to eally fit a simple constant density calculation. A diffeent elativistic teatment has to be pefomed in ode to study these effects and that is the title of ou ongoing publication. We can now see that the value fo the dynamic coection of the cental pessue takes a value of.9999 fo the two fist stas, and.78 times fo the neuton sta of adius equal to 8.8 Sun. Next an expession fo the pessue change at any is also deived below: P P () = () ( ) Γ, Γ, () which can be futhe witten if the numeical factos taken into account: P P( ) ( ) 6 = Γ, 8.68. () Plots of P()/P() vs ae shown fo the adial anges of zeo to thity kilometes fom the cente of a sun like sta, and whee hunded thousand points have been plotted fo this and all the gaphs below.
DPHLPHL.7 8.6.5.... 5 5 5 HKmL Fig. P()/P() vesus adial distance fom the cente of a sola mass constant density sta fo Km Km. DPHLPHL.5.... 5 6 7 HKmL Fig. P()/P() vesus adial distance fom the cente of a sola mass constant density sta fo Km 7 Km.
DPHLPHL 9..5. 5-6 6 8 HkmL -5-6 Fig. P()/P() vesus adial distance fom the cente of a sola mass constant density sta and fo Km Km. DPHLPHL -8. -6-8.5-6 -8. -6-8.5-6 6 8 HKmL -8.5-6 Fig P()/P() vesus adial distance fom the cente of a sola mass constant density sta and fo 7 Km Km.
DPHLPHL -8.6-6 -8.65-6 -8.66-6 5 6 7 HKmL Fig 5 P()/P() vesus adial distance fom the cente of a sola mass constant density sta and fo Km 7 Km. 6. Change in the Sta adius A change in the adius of the sta if any due to the change in the cental pessue becomes: D N 6 = =.6 () N N which makes the adius of the sta to incease by: D = N 6 (.6 ) =.6 N. () 7. edshift in the Dynamic Gavity The edshift in the dynamic theoy of gavity is given by the fomula below []: e G exp M e = = e M e e z (5) c e whee M is the mass of the body whee the edshifted photons ae eceived, M e the mass of the body whee the edshifted photons ae eceived, and similaly and e the adii of the coesponding bodies, and and e the values of dynamic theoy paamete at the eceiving and emitting bodies. Using the eally small incease in the adius of the sta e = D =.6 N we find that:
z = = exp (6) 6 6 (.7 ) =.7 whee the mass and adius of the eceiving body is taken to be the mass and the adius of the eath. The dynamic gavity edshift contibution due to the minute change in the sta s adius is calculated above and it is exactly identical to that of a sta whee the adius emains unchanged unde Newtonian gavity and equal to = sola. It is also of the same numeical value as the edshift ceated by Newtonian gavity. Taking o not the eally small change of the adius into account in dynamic gavity we see that the coesponding edshift is vey small and of the ode of appoximately two pats pe million. Fo such a sta it will be a eally difficult task fo an obseve to diffeentiate with cetainty if he is eally obseving dynamic o Newtonian gavity effects, not to mention that a edshift of two pats pe million is a eally small change to be obseved fo Newtonian o dynamic gavity Fo such a sta dynamic gavity effects do not poduce a dastic change in the edshift which can be eassily identified fom those of which ae caused by Newtonian gavity. 8. Conclusions In the dynamic theoy of gavity a vey simple calculation has been pefomed using the non-elativistic equation of hydodynamic equilibium fo a sta of constant density and thus an analytical expession fo the pessue at any point has been obtained. The pessue P() is found to be given in tems of the incomplete gamma function of agument (, /) whee is the dynamic gavity paamete. Next by applying bounday conditions, an expession of the cental pessue was also obtained. Both expessions esemble those of a constant density sta in Newtonian gavity but, they involve exta coection tems. The cental pessue of such a sta appeas to be slightly smalle than a constant density Newtonian sta. If that s the case we anticipated some incease in the adius of the sta which was estimated to be of the ode of two pats pe million small enough, to poduce any noticeable edshiftt. Fo ou simple model calculations it appeas to be no diffeence in magnitude between Newtonian and dynamic gavity edshift, and especially when the photons ae emitted fom such M sola sta and obseved on the eath. The small incease in adius does not effect the edshift which is identically the same as the Newtonian o that of dynamic gavity with sta = sola. An expession fo the changes in pessue P()/P() at any has also been deived and plotted fo fou diffeent egions of, namely km 9 Km fom the cente of the sta, next fo the anges 9 km 7 km, 7 Km km, and finally, Km 7 km. All gaphs have been plotted fo one hunded thousand points. Fom the behaviou of the gaphing function we can see that close to the cente thee is a positive change in pessue due to dynamic gavity which has a maximum value of appoximately.7 and at appoximately Km fom the cente. That would agee with the maximum value af the dynamic gavity potential at = =.8 km fo a sola type of sta. Next at appoximately Km the change in pessue dops to positive -5 and finally at becoming zeo at appoximately km. Finally in the ange of km and 8 km fom the cente it takes the values of 8.5-6 and 8.66-6 espectively. Ou expession deived fo the cental pessue pedicts a
coection of.999, fo M = M sola and = sola sta. It seems that thoughout the sta and at diffeent adial distances the coections to the pessue due to dynamic gavity changes sign fom positive to negative. To conclude we admit inspie the fact that this is not a vey ealistic kind of sta, it could give us an idea of how the dynamic gavity might effect stella stuctue. efeences [] P. E. Williams, On a Possible Fomulation of Paticle Dynamics in Tems of Themodynamic Conceptualizations and the ole of Entopy in it. Thesis U.S. Naval Postgaduate School, 976. [] P.E. Williams, The Pinciples of the Dynamic Theoy eseach epot EW-77-, U.S. Naval Academy, 977. [] P. E. Williams, The Dynamic Theoy: A New View of Space, Time, and Matte, Los Alamos Scientific Laboatoy epot LA-87-MS, Feb 98. [] P. E. Williams, Quantum Measuement, Gavitation, and Locality in the Dynamic Theoy, Symposium on Causality and Locality in Moden Physics and Astonomy: Open Questions and Possible Solutions, Yok Univesity, Noth Yok, Canada, August 5-9, 997, Kluwe Academic Publishes, p.6-68. [5] P. E.Williams, Mechanical Entopy and Its Implications, Entopy,,, 76-5 www.mdpi.og/entopy/ [6] P. E. Williams, The Dynamic Theoy: A New View of Space-Time Matte, 99, p: 7 http:// www.nmt.edu/~phais/ [7] Mathematica. fo windows. [7] Dina Pialnik, An intoduction to the Theoy of Stella Evolution, Cambidge Univesity Pess,, p: 9. [8] Phillips A. C., The Physics of Stas, John Wiley and Sons, 999, p: 99. [9] Stuad L. Shapio, Saul A., Teukolsky, Black Holes, White Dwafs, and Neuton Stas, John Wiley&Sons, 98, p. 8, [] P. E. Williams, Using the Hubble Telescope to Detemine the Split of a Cosmological Object s edshift into its Gavitational and Distance Pats, Apeion, Vol 8, No, Apil,, p.9.