THE GRAVITATIONAL POTENTIAL OF A MULTIDIMENSIONAL SHELL BY MUGUR B. RĂUŢ Abstact. This pape is a attept to geealize the well-kow expessio of the gavitatioal potetial fo oe tha thee diesios. We used the Seddo-Thohill appoach of the Newto s theoe ad the the esults ae passed though the filte of Poisso s equatio. The copaiso with othe theoies iplies soe estictios, but the oveall esults ae valid util the expeiet will dispove the. Key wods: gavitatioal potetial; exta-diesios. 1. Itoductio I efeece (Seddo & Thohill, 1948) the authos wee tyig to accedit a ew deostatio of Newto s theoe ad have foud soe ew esults coceig the classical gavitatioal potetial. So, the elatio: has the solutios: 2 2 (1) ( ) A 1 2 B e C 3 e (2) whee Φ ae the potetials pe uit ass ad A=B=C=G, the gavity costat. The easoig used to ceate the elatio (1) is siple. The gavitatioal potetial ceated by a poit ass, i which all the ass of a spheical ateial shell is cocetated, is equivalet to the gavitatioal potetial of the spheical shell itself, i a whateve exteio efeece poit P. I (Baes & Keoghi, 1984) oe ca fid aothe solutio fo the gavitatioal potetial: 2 4 D (3) e-ail: _b_aut@yahoo.co
esult developed i (Răuţ, 2010) fo the expasio of the uivese case. Fiely, i (Răuţ, 2011) these esults ae geealized fo all the easuable cases. Accodig to (Seddo & Thohill, 1948) the well-kow Newtoia potetial 1 ca be obtaied fo the coditio that the solutio (2) should be capable to ceate sufaces of equal potetial i spheical shell iteio, so that:, fo <α. (4) ( ) ( ) I the followig we will state that coditio (4) is o loge take ito cosideatio. Sice the shell seas to ot exist because it has the sae potetial as the poit which is its cete, thei potetials ae equivalet. Theeby the gavitatioal potetial geeated by the cetal poit ass it ceates equipotet sufaces aoud its poit exteio whateve the cosideate distace is. The iteio of the shell is the exteio of the poit ass. Thus the gavitatioal potetial soewhee ito the shell is the sae as the gavitatioal potetial of the poit ass coespodet, because they ae calculated i the sae poit, idiffeetly whee it is. I additio, at diffeet scales it is a ufogettable istake to use 1, so to pesue coditio (4) to be ot cosideed seas to be quite easoable. O the othe had we do t kow what the expessio of gavitatioal potetial i shell iteio is. We ca oly ituit it. If we iagie that we ca iiize the iteio of the shell i viciity of the cetal poit, the solutio (2) is valid. As a cosequece, i the followig cosideatios we will show a ew appoach of this poble. Additioally, we will geealize the solutio (2) fo oe tha thee diesios. 2. The N-Diesioal Case I equatio (1) we ca eglect the secod left te. If the coditio (4) is o oe valid the the costat γ, coespodig to the additioal potetial i equatio (1), te which ca be added to the gavitatioal potetial without the esulted law foce to be alteed, has ow eithe sigificatio. I cosequece the elatio (1) becoes: 2 (5) ( ) ( ) ( )
This elatio allows the solutios (2) fo the gavitatioal potetial pe uit ass, as i pevious case. The diffeece ow is that with (5) we ae ot able to geeate solutios which to adit othe costats tha those fo (2). I pevious case we wee foced to ake these costats ull. With elatio (5) we ca ow thik about the ultidiesioal case. Suppose: V ( ) 2 (6) is the elatio betwee the potetial of the cetal poit ass ad the potetial of its coespodig -diesioal shell. I this elatio σ is the 2-diesioal ass desity of the -diesioal shell, thus (6) has o eaig fo less tha thee diesios. The coespodet equivalet asses will be defied depedig o the gavitatioal potetial. I equatio (6) we ade the hypothesis that the gavitatioal potetial of the -diesioal shell is due to oe diesio oly, defied as a thickess. A (-1)-diesioal suface, with evey diesio defied by the adius is, (Weeks, 1985): Fo eve we have: ad fo odd: Equatio (6) has the solutios: with the equivalet ass: S 1 V /2 (1/ ( / 2)!) V ( 1)/2 V (2 (( 1) / 2)! G () (7) 1 V ad the coespodig Yukawa-like potetials: 1Y 2Y (8) with the equivalet ass:
The geeal solutio: with the equivalet ass: V 1Y sh( ) (9) i i (10) i is the sae as the oe obtaied i (Radall & Sudu, 1999). Nevetheless, 2 (Ehefest, 1918) was stated that i a -diesioal space 1/. To be i accodace with this stateet we ust odify the equatio (6) as follows: This equatio has the solutios: ad: V ( ) ( ) ( ) (11) 2 ' G (12) 2 '1Y '2Y (13) with the sae equivalet asses as (7) ad (8). The solutio (13) is a ovel oe but i soe espect it is equivalet with (8). The uivesal gavity costat is iflueced by scale ad this depedece ca be expessed by the distace at which the iteactio takes place. The lowe is the distace the lage is the 3 gavity costat, G G. The sae coets ae valid fo (7) ad (12). The oveall solutio i this case ust have the equivalet ass (10). Regadig the solutios (8), they ust veify the Poisso s equatio: 4 ( ) G vac Although they ae geeated by the atte we ust do this copoise to be i ageeet with Poisso s equatio, as solutio (7) is. The solutios (13) ust veify the Poisso s equatio:
as solutio (12) does (Ehefest, 1918). 4 G( ) vac 3. Coclusios I this pape the esults give i (Răuţ, 2011) ae geealized fo the ultidiesioal case. The esults ae ot foud i (El-Nabulsi, 2012) but they ae i ageeet with (Radall & Sudu, 1999). Nevetheless, it ca occu the situatio whe (12) ad (13) ae valid. I ay case, the attept to uify the two solutios ca lead to soe logical coclusios. O oe had, if the uivesal gavity costat depeds o scale, this depedece ca be expessed by the distace at which the iteactio takes place. O the othe had, the size of iteactig asses ca ifluece the stegth of iteactio ad theefoe the physical value of the uivesal gavity costat. Futue expeietal esults will deteie which of (7)-(8) ad (12)-(13) solutios ae valid. REFERENCES Baes A., Keoghi C.K., Math. Gaz., 68, 138, 1984, cited by Baow D. ad Tipple F.J. I The Athopic Cosological Piciple, 307 308, Oxfod Uiv. Pess 1986. Ehefest P., I what way does it Becoe Maifest i the Fudaetal Laws of Physics that Space has Thee Diesios? Poc. Koi. Akad., 20, 200 209, 1918. El-Nabulsi A.R., New Astophysical Aspects fo Yukawa Factioal Potetial Coectio to the Gavitatioal Potetial i D Diesios. INJP, 86, 763 768, 2012. Radall L., Sudu R., A Alteative to Copactificatio. Phys. Rev. Lett., 83, 4690 4693, 1999. Răuţ M.B., A Newtoia Equivalet fo the Cosological Costat. Bul. Ist. Polit. Iaşi, LVI (LX), 2, 33 38, 2010. Răuţ M.B., The ealized Gavitatiol Potetial. Bul. Ist. Polit. Iaşi, LVII (LXI), 3, 107 114, 2011. Seddo J.N., Thohill C.K., A Popety of Yukawa Potetial. MPCPS, 46, 318 320, 1948. Weeks J.R., The Shape of Space: How to Visualize Sufaces ad Thee-Diesioal Maifolds. Macel Dekke, Chap. 14, 1985.