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Physics & Astronomy International Journal

Letter to Editor Volume 2 Issue 4

Theorem on spaces and long–ranged interaction forces forming these spaces

Veitsman EV

Research and Production Enterprise ?Tekhnolazer?, Moscow, Russia

Correspondence: Emil Viktorovich Weizmann, Research and Production Enterprise ?Tekhnolazer?, Veitsman?s Science Project, 28 department, 5 Klimashkin Street, 123557, Moscow, Russia

Received: August 02, 2018 | Published: August 28, 2018

Citation: Veitsman EV. Theorem on spaces and long–ranged interaction forces forming these spaces. Phys Astron Int J. 2018;2(1):399-401. DOI: 10.15406/paij.2018.02.00116

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Abstract

The aim of this paper is to find connection between the spaces of different dimensions i(from zero up to ‘n’) and the long–range attractive forces that create these spaces and have (forces) its dimension j (from zero up to ‘m’). A theorem is formulated and strictly proved showing in which cases the long–ranged attractive forces can form real spaces of different dimensions (from zero up to i=1,2,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaigdacaGGSaGaaGOm aiaacYcacqGHMacVcaWHUbaaaa@4305@ ). The existence of the attraction between masses is defined by divergence the vector of interaction between masses.

Keywords: attractive forces, spaces of different dimensions, real spaces, attraction between masses,divergence

Pacs

04.50.+h ;02.40.–k

Letter to editor

As is well–known from affine geometry,1 there are spaces with the allowable systems of orthogonal coordinates having the common origin, an identical unit volume and the same orientation. Such is our real 3d–space. Why? Because our real space could be created only owing to long ranged attractive forces, e.g., by the forces of gravitation. An empty space, i.e., the space without any matter, can have any dimension–from zero up to ‘n’. The mathematical space is the empty.

The main goal of this article is to find connection between the spaces of different dimensions i (from zero up to ‘n) and the long–range attractive forces that create these spaces and have (forces) its dimension j (from zero up to ‘m). By the dimension of long–range attraction forces Fis meant the value of the exponent j in the denominator of the formula F=k m 1 m 2 / r j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadAeacqGH9aqpcaWGRbGaamyBaKqbaoaaBaaajeaibaqc LbmacaaIXaaajeaibeaajugibiaad2gajuaGdaWgaaqcbasaaKqzad GaaGOmaaqcbasabaqcLbsacaGGVaGaamOCaOWaaWbaaSqabKqaGeaa jugWaiaadQgaaaaaaa@4B55@ ,

where m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaigdaaKqaG8aabeaaaaa@3FBD@ and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaikdaaKqaG8aabeaaaaa@3FBE@ are interacting masses (kg); k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4Aaaaa@3CA1@ is a coefficient;the distance between these masses (m); j( j )=1,2,3,,m( m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOAaOWdamaabmaabaqcLbsapeGaamOA aaGcpaGaayjkaiaawMcaaKqzGeWdbiabg2da9iaaigdacaGGSaGaaG OmaiaacYcacaaIZaGaaiilaiabgAci8kaacYcacaWHTbGcpaWaaeWa aeaajugib8qacaWGTbaak8aacaGLOaGaayzkaaaaaa@4C56@ .

Our problem should not be confused with problem that P. Ehrenfest was solving 100 years ago.2–5 He made attempt to link the dimension of space with fundamental laws of physics but he did not concern the problems connected with the creature of spaces under influence of long–ranged interaction forces.

Call any space containing matter real space. Any real space has to contain the sources of long–range interactions, in our case of the attraction between masses. The existence of these sources is defined by divergence the vector of interaction a between masses. For example, for our 3–D real space the divergence of awill have the following form if ais only the function of the coordinate ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqyWdihaaa@3D71@ (spherical coordinate system):

di v 3 a V0 = Lim 3 a ρ n ρ dS V , V= 4 3 π r 3 ,S=4π r 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aadaWfqaqaaKqzGeaeaaaaaaaaa8qacaWGKbGaamyAaiaadAhak8aa daWgaaqcbasaaKqzadWdbiaaiodaaSWdaeqaaKqzGeWdbiaahggaaK qaG8aabaqcLbmapeGaamOva8aacqGHsgIRcaaIWaaaleqaaKqzGeGa eyypa0JcdaWgaaWcbaqcLbsapeGaamitaiaadMgacaWGTbaal8aabe aakmaalaaabaWaa8qfaeaajugibiaadggakmaaBaaajeaibaqcLbma cqaHbpGCaSqabaqcLbsacaWGUbGcdaWgaaqcbasaaKqzadGaeqyWdi haleqaaKqzGeGaamizaiaadofaaKqaGeaajugWaiaaiodaaSqabKqz GeGaeSyeUhTaey4kIipaaOqaaKqzGeGaamOvaaaacaGGSaGcdaWgaa WcbaqcLbsapeGaamOvaiabg2da9aWcpaqabaGcdaWcaaqaaKqzGeGa aGinaaGcbaqcLbsacaaIZaaaaiabec8aWjaadkhajuaGdaahaaqcba sabeaajugWaiaaiodaaaqcLbsacaGGSaGaam4uaiabg2da9iaaisda cqaHapaCcaWGYbqcfa4aaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGe Gaaiilaaaa@77BA@    (1)

F=k m 1 m 2 r/ r 3 ; aE( F/ m 2 )=r k m 1 r 3 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOCaiaac+cacaWGYbqcfa4damaaCaaajeaibeqaaKqzadWdbi aaiodaaaqcLbsacaGG7aGaaiiOaiaadggacaWGfbGcpaWaaeWaaeaa jugib8qacaWGgbGaai4laiaad2gajuaGpaWaaSbaaKqaGeaajugWa8 qacaaIYaaajeaipaqabaaakiaawIcacaGLPaaajugibiabg2da98qa caWGYbGcpaWaaSaaaeaajugibiaadUgacaWGTbGcdaWgaaqcbasaaK qzadGaaGymaaWcbeaaaOqaaKqzGeGaamOCaOWaaWbaaSqabKqaGeaa jugWaiaaiodaaaaaaKqzGeGaai4oaaaa@65F3@  (2)

where index “3” in (1) indicates that the above formulae refer to 3D– dS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadofaaaa@3D72@ space; n ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaad6gakmaaBaaajeaibaqcLbmacqaHbpGCaSqabaaaaa@3FD2@ is a surface element (in our case of the spherical surface);the unit vector perpendicular to dS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadofaaaa@3D72@ ;Vvolume F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aaiqWajugibabaaaaaaaaapeGaa8Nraaaa@3C87@ the interaction force between masses m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaigdaaKqaG8aabeaaaaa@3FBD@ and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaikdaaKqaG8aabeaaaaa@3FBE@ ; k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadUgaaaa@3C81@ the constant of gravitation ( k g 1 · m 3 · s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4AaiaadEgajuaGpaWaaWbaaKqaGeqa baqcLbmapeGaeyOeI0IaaGymaaaajugibiabl+y6Njaad2gak8aada ahaaWcbeqcbasaaKqzadWdbiaaiodaaaqcLbsacqWIpM+zcaWGZbqc fa4damaaCaaajeaibeqaaKqzadWdbiabgkHiTiaaikdaaaaaaa@4F7F@ );Ethe vector of gravitation field intensity ( m· s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaiabl+y6NjaadohajuaGpaWaaWba aKqaGeqabaqcLbmapeGaeyOeI0IaaGOmaaaaaaa@43DB@ ), as m 2 =1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaikdaaKqaG8aabeaajugibiabg2da9iaaigdacaGGUaaaaa@42C0@

If V0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadAfacqGHsgIRcaaIWaaaaa@3F13@ , then we can write down (1) as

di v 3 a= 3 n ρ k r 2 δ m 1 δV dS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIZaaal8aabeaajugib8qacaWHHbWdaiabg2da9O Waa8qfaeaajugibiaad6gakmaaBaaajeaibaqcLbmacqaHbpGCaSqa baGcdaWcaaqaaKqzGeGaam4AaaGcbaqcLbsacaWGYbqcfa4aaWbaaK qaGeqabaqcLbmacaaIYaaaaaaakmaalaaabaqcLbsacqaH0oazcaWG TbGcdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaOqaaKqzGeGaeqiTdq MaamOvaaaaaKqaGeaajugWaiaaiodaaSqabKqzGeGaeSyeUhTaey4k IipacaWGKbGaam4uaaaa@5FCA@  . (3)

Since ds= ρ 2 sinθdθdϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacaWGZbGaeyypa0JaeqyWdixcfa4aaWbaaKqaGeqa baqcLbmacaaIYaaaaKqzGeGaci4CaiaacMgacaGGUbGaeqiUdeNaam izaiabeI7aXjaadsgacqaHvpGzaaa@4D69@ in spherical system of coordinate (ρ,θ,ϕ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiabew9aMjaa cMcacaGGSaaaaa@4438@ we have after integration (3):

di v 3 a=4πk n ρ δ m 1 δV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaG4maaqcbaYdaeqaaKqzGeWdbiaahggapaGaey ypa0JaaGinaiabec8aWjaadUgacaWGUbGcdaWgaaqcbasaaKqzadGa eqyWdihaleqaaOWaaSaaaeaajugibiabes7aKjaad2gakmaaBaaaje aibaqcLbmacaaIXaaaleqaaaGcbaqcLbsacqaH0oazcaWGwbaaaaaa @548B@   (4)

If we would use another formula instead, F=k m 1 m 2 r/ r 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gak8aadaWgaaqcbasaaKqzadWdbiaaikdaaSWdaeqaaKqzGeWdbiaa hkhacaGGVaGaamOCaKqba+aadaahaaqcbasabeaajugWa8qacaaIZa aaaaaa@4CAF@ e.g.,

F=k m 1 m 2 r/ r 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbGcpaWa aSbaaKqaGeaajugWa8qacaaIXaaal8aabeaajugib8qacaWGTbqcfa 4damaaBaaajeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaa hkhacaGGVaGaamOCaKqba+aadaahaaqcbasabeaajugWa8qacaaIYa aaaKqzGeGaaiilaaaa@4DED@    (5)

then we have

di v 3 a=4πkr n ρ δ m 1 δV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIZaaal8aabeaajugib8qacaWHHbWdaiabg2da9i aaisdacqaHapaCcaWGRbGaamOCaiaad6gakmaaBaaajeaibaqcLbma cqaHbpGCaSqabaGcdaWcaaqaaKqzGeGaeqiTdqMaamyBaOWaaSbaaK qaGeaajugWaiaaigdaaSqabaaakeaajugibiabes7aKjaadAfaaaaa aa@54DF@     (6)

It means that di v 3 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIZaaal8aabeaajugib8qacaWHHbaaaa@4285@ depends on “ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOCaaaa@3CA8@ ’, it means, in turn, that the law of energy conservation is broken. Indeed, if V0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadAfacqGHsgIRcaaIWaaaaa@3F13@ (see (1)), then r0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadkhacqGHsgIRcaaIWaaaaa@3F2F@ and di v 3 a=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaG4maaqcbaYdaeqaaKqzGeWdbiaahggacqGH9a qpcaaIWaaaaa@44E8@ . It means that the gravitation source at this point of space is not observed. The analogous picture takes place at other points of our space. In fact, it means as well that there is no any real space, since the interaction (5) cannot maintain its existence.

Now take such a law instead (5):

F=k m 1 m 2 r/ r 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbGcpaWa aSbaaKqaGeaajugWa8qacaaIXaaal8aabeaajugib8qacaWGTbqcfa 4damaaBaaajeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaa hkhacaGGVaGaamOCaKqba+aadaahaaqcbasabeaajugWa8qacaaI0a aaaKqzGeGaaiilaaaa@4DEF@      (7)

then instead (6) we have

di v 3 a=4πk r 1 n ρ δ m 1 δV , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIZaaal8aabeaajugib8qacaWHHbWdaiabg2da9i aaisdacqaHapaCcaWGRbGaamOCaKqbaoaaCaaajeaibeqaaKqzadGa eyOeI0IaaGymaaaajugibiaad6gakmaaBaaajeaibaqcLbmacqaHbp GCaSqabaGcdaWcaaqaaKqzGeGaeqiTdqMaamyBaKqbaoaaBaaajeai baqcLbmacaaIXaaajeaibeaaaOqaaKqzGeGaeqiTdqMaamOvaaaaki aacYcaaaa@5A7B@    (8)

and di v 3 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIZaaal8aabeaajugib8qacaWHHbWdaiabgkziUk abg6HiLcaa@45F2@ if V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOvaaaa@3C8C@ and, consequently, r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOCaaaa@3CA8@ tends to zero. It means that our space collapses into a point, i.e., we obtain a black hole.

Here we should make an important remark. As seen, studying the above case, we have used the sphere of dimension 3. It means that we have been studying an isotropic space. If the space investigated had a fractional dimension, e.g., 2.9, then we had to take for our investigations not a sphere but an ellipsoid. Consequently, we cannot take the relation for the element of the ellipsoid surface as ds= ρ 2 sinθdθdϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacaWGZbGaeyypa0JaeqyWdixcfa4aaWbaaKqaGeqa baqcLbmacaaIYaaaaKqzGeGaci4CaiaacMgacaGGUbGaeqiUdeNaam izaiabeI7aXjaadsgacqaHvpGzaaa@4D69@ , since in this case ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqyWdihaaa@3D71@ should be a function of the angles φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqOXdOgaaa@3D6E@ and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqiUdehaaa@3D67@ . In this work we have been studying only isotropic spaces.

 

Now we can put a question: how will things be going for spaces of other dimensions–from zero to n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOBaaaa@3CA8@ ? To answer this question, first of all write down expressions for volumes and surfaces of different ranks. We begin to study spaces whose dimensions , i.e. i3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAa8aacqGHKjYOpeGaaG4maaaa@3F34@ , 0,1,2.

If i=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaikdaaaa@3E65@ , then we consider a circumference and a space inside it (we we call this space a flat sphere). Therefore we have:

di v 2 a=Lim 2 a ρ n ρ dL S ,S= π r 2 ,L=2πr, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jqwaG9FaaKqzadWdbiaaikdaaKqaG8aabeaajugib8qacaWHHbGaey ypa0JaamitaiaadMgacaWGTbGcpaWaaSaaaeaadaWdvaqaaKqzGeGa amyyaKqbaoaaBaaajeaibaqcLbmacqaHbpGCaKqaGeqaaKqzGeGaam OBaOWaaSbaaKqaGeaajugWaiabeg8aYbWcbeaajugibiaadsgacaWG mbaajeaibaqcLbmacaaIYaaaleqajugibiablgH7rlabgUIiYdaake aajugibiaadofaaaWdbiaacYcacaWGtbGaeyypa0JaaiiOa8aacqaH apaCcaWGYbqcfa4aaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaai ilaiaadYeacqGH9aqpcaaIYaGaeqiWdaNaamOCaiaacYcaaaa@6E1E@    (9)

S0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadofacqGHsgIRcaaIWaaaaa@3F10@  

F=k m 1 m 2 r/ r 2 ; aE( F/ m 2 )r k m 1 r 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOCaiaac+cacaWGYbqcfa4damaaCaaajeaibeqaaKqzadWdbi aaikdaaaqcLbsacaGG7aGaaiiOaiaadggacaWGfbGcpaWaaeWaaeaa jugib8qacaWGgbGaai4laiaad2gak8aadaWgaaqcbasaaKqzadWdbi aaikdaaSWdaeqaaaGccaGLOaGaayzkaaqcLbsapeGaamOCaOWdamaa laaabaqcLbsacaWGRbGaamyBaOWaaSbaaKqaGeaajugWaiaaigdaaS qabaaakeaajugibiaadkhajuaGdaahaaqcbasabeaajugWaiaaikda aaaaaKqzGeGaai4oaaaa@64C1@   (10)

here the coefficient kin k g 1 · m 2 · s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4AaiaadEgajuaGpaWaaWbaaKqaGeqa baqcLbmapeGaeyOeI0IaaGymaaaajugibiabl+y6Njaad2gajuaGpa WaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaajugibiabl+y6Njaadoha juaGpaWaaWbaaKqaGeqabaqcLbmapeGaeyOeI0IaaGOmaaaaaaa@4FF7@ units.

If S0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadofacqGHsgIRcaaIWaaaaa@3F10@ , then we can write down (9) as

di v 2 a= 2 n ρ k r δ m 1 δS dL. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaahggapaGaey ypa0JcdaWdvaqaaKqzGeGaamOBaOWaaSbaaKqaGeaajugWaiabeg8a YbWcbeaakmaalaaabaqcLbsacaWGRbaakeaajugibiaadkhaaaGcda WcaaqaaKqzGeGaeqiTdqMaamyBaKqbaoaaBaaajeaibaqcLbmacaaI XaaajeaibeaaaOqaaKqzGeGaeqiTdqMaam4uaaaaaKqaGeaajugWai aaikdaaSqabKqzGeGaeSyeUhTaey4kIipacaWGKbGaamitaiaac6ca aaa@5EF2@   (11)

Since dL=ρdϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacaWGmbGaeyypa0JaeqyWdiNaamizaiabew9aMbaa @42C2@ in polar system of coordinate (ρ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaacIcacqaHbpGCcaGGSaGaeqy1dyMaaiykaaaa@4122@ , we have after integration in (11):

di v 2 a=2πk n ρ δ m 1 δS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaahggapaGaey ypa0JaaGOmaiabec8aWjaadUgacaWGUbGcdaWgaaqcbasaaKqzadGa eqyWdihaleqaaOWaaSaaaeaajugibiabes7aKjaad2gakmaaBaaaje aibaqcLbmacaaIXaaaleqaaaGcbaqcLbsacqaH0oazcaWGtbaaaaaa @5485@  . (12)

If we would have another formula instead F=k m 1 m 2 r/ r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOCaiaac+cacaWGYbqcfa4damaaCaaajeaibeqaaKqzadWdbi aaikdaaaaaaa@4D51@ , e.g.,

F=k m 1 m 2 n ρ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOBaOWdamaaBaaajeaibaqcLbmapeGaeqyWdihal8aabeaaju gib8qacaGGSaaaaa@4D8B@    (13)

Then we would have

di v 2 a=2πkr n ρ δ m 1 δS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIYaaal8aabeaajugib8qacaWHHbWdaiabg2da9i aaikdacqaHapaCcaWGRbGaamOCaiaad6gajuaGdaWgaaqcbasaaKqz adGaeqyWdihajeaibeaakmaalaaabaqcLbsacqaH0oazcaWGTbqcfa 4aaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaaGcbaqcLbsacqaH0oaz caWGtbaaaaaa@561F@    (14)

It means that di v 2 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIYaaal8aabeaajugib8qacaWHHbaaaa@4284@ depends on “r’, it means, in turn, that the law of energy conservation is broken. Indeed, if S0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadofacqGHsgIRcaaIWaaaaa@3F10@ (see (9)), then r0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadkhacqGHsgIRcaaIWaaaaa@3F2F@ and di v 2 a=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIYaaal8aabeaajugib8qacaWHHbGaeyypa0JaaG imaaaa@4444@ . It means that the gravitation source at this point of space is not observed. The analogous picture takes place at other points of this space. In fact, it means as well that there is no any real 2D–space. Since the interaction (13) cannot maintain its existence.

Now take such a law instead (10):

F=k m 1 m 2 r/ r 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOCaiaac+cacaWGYbqcfa4damaaCaaajeaibeqaaKqzadWdbi aaiodaaaqcLbsacaGGSaaaaa@4E91@    (15)

Then instead (14) we have

di v 2 a=2πk r 1 n ρ δ m 1 δS , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaIYaaal8aabeaajugib8qacaWHHbWdaiabg2da9i aaikdacqaHapaCcaWGRbGaamOCaKqbaoaaCaaajeaibeqaaKqzadGa eyOeI0IaaGymaaaajugibiaad6gakmaaBaaajeaibaqcLbmacqaHbp GCaSqabaGcdaWcaaqaaKqzGeGaeqiTdqMaamyBaOWaaSbaaKqaGeaa jugWaiaaigdaaSqabaaakeaajugibiabes7aKjaadofaaaGaaiilaa aa@59C8@  , (16)

and di v 3 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaG4maaqcbaYdaeqaaKqzGeWdbiaahggapaGaey OKH4QaeyOhIukaaa@4695@ , if r0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadkhacqGHsgIRcaaIWaaaaa@3F2F@ , i.e., the flat sphere collapses into point and we have black hole but in 2D–space. Below the Greek letters φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqOXdOgaaa@3D6E@ and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaeqiUdehaaa@3D67@ will be replaced by Greek letter ζ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabeA7a6PWaaSbaaKqaGeaajugWaiaad2gaaSqabaaaaa@3FCE@ with index m=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaiabg2da9iaaigdacaGGSaGaaGOm aaaa@3FD0@ , since we shall study spaces with i4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAa8aacqGHLjYScaaI0aaaaa@3F36@ .

 

If i=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaigdaaaa@3E64@ , a straight–line segment will be as if an analogue of the above flat sphere and a pair of points will be as if analogue of the above circumference bounding the above flat one. At this case we have:

di v 1 a=2a, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaahggacqGH9a qpcaaIYaGaamyyaiaacYcaaaa@467E@    (17)

F=k m 1 m 2 ,a=E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbGcpaWa aSbaaKqaGeaajugWa8qacaaIXaaal8aabeaajugib8qacaWGTbqcfa 4damaaBaaajeaibaqcLbmapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaa cYcapaGaamyyaiabg2da9iaadweaaaa@4A9B@   (18)

Here the coefficient kis in k g 1 ·m· s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4AaiaadEgajuaGpaWaaWbaaKqaGeqa baqcLbmapeGaeyOeI0IaaGymaaaajugibiabl+y6Njaad2gacqWIpM +zcaWGZbqcfa4damaaCaaajeaibeqaaKqzadWdbiabgkHiTiaaikda aaaaaa@4C85@ units. At last, if, then the space is a point here and its, di v 0 a= 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGimaaqcbaYdaeqaaKqzGeWdbiaadggapaGaey ypa0JcdaWcaaqaaKqzGeGaaGimaaGcbaqcLbsacaaIWaaaaaaa@46EC@ i.e. we have uncertainty. Now we shall study the spaces having the dimensions from i=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaisdaaaa@3E67@ up to i=n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaah6gaaaa@3EA0@

If i=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaisdaaaa@3E67@ , then we have:6

di v 4 A( 4 )Lim 4 a ρ n ρ d Λ (4) Ω (4) ,   Ω (4) = 1 2 π 2 R (4) 4 ;   Λ (4) =2 π 2 R (4) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jqwaG9FaaKqzadWdbiaaisdaaKqaG8aabeaajugib8qacaWHbbGcpa WaaeWaaeaajugib8qacaaI0aaak8aacaGLOaGaayzkaaqcLbsapeGa amitaiaadMgacaWGTbGcpaWaaSaaaeaadaWdvaqaaKqzGeGaamyyaO WaaSbaaKqaGeaajugWaiabeg8aYbWcbeaajugibiaad6gakmaaBaaa jeaibaqcLbmacqaHbpGCaSqabaqcLbsacaWGKbGaeu4MdWKcdaWgaa qcbasaaKqzadGaaiikaiaaisdacaGGPaaaleqaaaqcbasaaKqzadGa aGinaaWcbeqcLbsacqWIr4E0cqGHRiI8aaGcbaqcLbsacqqHPoWvju aGdaWgaaqcbasaaKqzadGaaiikaiaaisdacaGGPaaajeaibeaaaaqc LbsacaGGSaGcdaWgaaWcbaqcLbsapeGaaiiOaaWcpaqabaqcLbsacq qHPoWvjuaGdaWgaaqcbasaaKqzadGaaiikaiaaisdacaGGPaaajeai beaajugibiabg2da9OWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaaG OmaaaacqaHapaCkmaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa amOuaKqbaoaaDaaajeaibaqcLbmacaGGOaGaaGinaiaacMcaaKqaGe aajugWaiaaisdaaaqcLbsacaGG7aGcdaWgaaWcbaqcLbsapeGaaiiO aaWcpaqabaqcLbsacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikai aaisdacaGGPaaajeaibeaajugibiabg2da9iaaikdacqaHapaCjuaG daahaaqcbasabeaajugWaiaaikdaaaqcLbsacaWGsbqcfa4aa0baaK qaGeaajugWaiaacIcacaaI0aGaaiykaaqcbasaaKqzadGaaG4maaaa aaa@9A99@    (19)

Ω (4) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaaGinaiaa cMcaaKqaGeqaaKqzGeGaeyOKH4QaaGimaaaa@449D@  

F ( 4 ) =k m 1 m 2 R (4) R (4) 4 ;  A ( 4 ) E ( 4 ) ( F (4) / m 2 ) R ( 4 ) k m 1 R (4) 4 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaaGinaaqcbaYdaiaawIcacaGLPaaaaeqaaK qzGeWdbiabg2da9iaadUgacaWGTbGcpaWaaSbaaKqaGeaajugWa8qa caaIXaaal8aabeaajugib8qacaWGTbGcpaWaaSbaaKqaGeaajugWa8 qacaaIYaaal8aabeaajugib8qacaWHsbqcfa4damaaBaaajeaibaqc LbmacaGGOaWdbiaaisdapaGaaiykaaqcbasabaqcLbsacaWGsbqcfa 4aa0baaKqaGeaajugWaiaacIcacaaI0aGaaiykaaqcbasaaKqzadGa aGinaaaajugibiaacUdapeGaaiiOaiaadgeak8aadaWgaaqcbasaaK qbaoaabmaajeaibaqcLbmapeGaaGinaaqcbaYdaiaawIcacaGLPaaa aSqabaqcLbsapeGaamyraKqba+aadaWgaaqcbasaaKqbaoaabmaaje aibaqcLbmapeGaaGinaaqcbaYdaiaawIcacaGLPaaaaeqaaOWaaeWa aeaajugib8qacaWGgbqcfa4damaaBaaajeaibaqcLbmacaGGOaWdbi aaisdapaGaaiykaaqcbasabaqcLbsapeGaai4laiaad2gak8aadaWg aaqcbasaaKqzadWdbiaaikdaaSWdaeqaaaGccaGLOaGaayzkaaqcLb sapeGaamOuaOWdamaaBaaajeaibaqcfa4aaeWaaKqaGeaajugWa8qa caaI0aaajeaipaGaayjkaiaawMcaaaWcbeaakmaalaaabaqcLbsaca WGRbGaamyBaKqbaoaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaOqa aKqzGeGaamOuaKqbaoaaDaaajeaibaqcLbmacaGGOaGaaGinaiaacM caaKqaGeaajugWaiaaisdaaaaaaKqzGeGaai4oaaaa@895E@   (20)

Where index “4” in (19–20) indicates that these formulae refer to 4D–space; d Λ (4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikaiaa isdacaGGPaaajeaibeaaaaa@4237@ is the element of 3D–surface; n ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaad6gajuaGdaWgaaqcbasaaKqzadGaeqyWdihajeaibeaa aaa@4075@ component of unit vector perpendicular to each point of this 3D–surface; Ω 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaaI0aaajeaibeaa aaa@400E@ 4–D volume; F ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaaGinaaqcbaYdaiaawIcacaGLPaaaaeqaaa aa@41DA@ interaction force between masses m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqbaoaaBaaajeaibaqcLbmacaaI Xaaajeaibeaaaaa@3F8F@ and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqbaoaaBaaajeaibaqcLbmacaaI Yaaajeaibeaaaaa@3F90@ in 4D–space; k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4Aaaaa@3CA1@ constant of gravitation in 4D–space ( k g 1 · m 4 · s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4AaiaadEgajuaGpaWaaWbaaKqaGeqa baqcLbmapeGaeyOeI0IaaGymaaaajugibiabl+y6Njaad2gak8aada ahaaWcbeqcbasaaKqzadWdbiaaisdaaaqcLbsacqWIpM+zcaWGZbqc fa4damaaCaaajeaibeqaaKqzadWdbiabgkHiTiaaikdaaaaaaa@4F80@ ); E vector of gravitation field intensity m· s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaiabl+y6NjaadohajuaGpaWaaWba aKqaGeqabaqcLbmapeGaeyOeI0IaaGOmaaaaaaa@43DB@ ( m 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaOWdamaaBaaajeaibaqcLbmapeGa aGOmaaWcpaqabaqcLbsacqGH9aqpcaaIXaaaaa@416B@ ), as.

If Ω (4) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaaGinaiaa cMcaaKqaGeqaaKqzGeGaeyOKH4QaaGimaaaa@449D@ , then we can write down (19) as

di v 4 A ( 4 ) = 4 n ρ k R 3 δ m 1 δ Ω (4) d Λ (4) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGinaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaaisdaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabg2da9OWaa8qfaeaajugibiaad6 gakmaaBaaajeaibaqcLbmacqaHbpGCaSqabaGcdaWcaaqaaKqzGeGa am4AaaGcbaqcLbsacaWGsbGcdaahaaWcbeqcbasaaKqzadGaaG4maa aaaaGcdaWcaaqaaKqzGeGaeqiTdqMaamyBaOWaaSbaaKqaGeaajugW aiaaigdaaSqabaaakeaajugibiabes7aKjabfM6axLqbaoaaBaaaje aibaqcLbmacaGGOaGaaGinaiaacMcaaKqaGeqaaaaaaeaajugWaiaa isdaaSqabKqzGeGaeSyeUhTaey4kIipacaWGKbGaeu4MdWucfa4aaS baaKqaGeaajugWaiaacIcacaaI0aGaaiykaaqcbasabaqcLbsacaGG Uaaaaa@708C@    (21)

Since d Λ (4) = ρ 3 F( ζ 1 , ζ 2 , ζ 3 )d ζ 1 d ζ 2 d ζ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikaiaa isdacaGGPaaajeaibeaajugibiabg2da9iabeg8aYPWaaWbaaSqabK qaGeaajugWaiaaiodaaaqcLbsacaWGgbGaaiikaiabeA7a6Lqbaoaa BaaajeaibaqcLbmacaaIXaaajeaibeaajugibiaacYcacqaH2oGEju aGdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaqcLbsacaGGSaGaeqOT dONcdaWgaaqcbasaaKqzadGaaG4maaWcbeaajugibiaacMcacaWGKb GaeqOTdOxcfa4aaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaKqzGeGa amizaiabeA7a6LqbaoaaBaaajeaibaqcLbmacaaIYaaajeaibeaaju gibiaadsgacqaH2oGEjuaGdaWgaaqcbasaaKqzadGaaG4maaqcbasa baaaaa@6CCA@ in a spherical system of coordinate (ρ, ζ 1 , ζ 2 , ζ 3 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaacIcacqaHbpGCcaGGSaGaeqOTdOxcfa4aaSbaaKqaGeaa jugWaiaaigdaaKqaGeqaaKqzGeGaaiilaiabeA7a6PWaaSbaaKqaGe aajugWaiaaikdaaSqabaqcLbsacaGGSaGaeqOTdOxcfa4aaSbaaKqa GeaajugWaiaaiodaaKqaGeqaaKqzGeGaaiykaiaacYcaaaa@5072@ then we have after integration (21):

di v 4 A ( 4 ) =2 π 2 k n ρ δ m 1 δ Ω (4) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaI0aaal8aabeaajugib8qacaWHbbqcfa4damaaBa aajeaibaqcfa4aaeWaaKqaGeaajugWa8qacaaI0aaajeaipaGaayjk aiaawMcaaaqabaqcLbsacqGH9aqpcaaIYaGaeqiWdaNcdaahaaWcbe qcbasaaKqzadGaaGOmaaaajugibiaadUgacaWGUbGcdaWgaaqcbasa aKqzadGaeqyWdihaleqaaOWaaSaaaeaajugibiabes7aKjaad2gaju aGdaWgaaqcbasaaKqzadGaaGymaaqcbasabaaakeaajugibiabes7a KjabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaaGinaiaacMcaaK qaGeqaaaaakiaac6caaaa@62D9@   (22)

If we would have another formula instead F=km1m2R(4)/, e.g. R (4) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaDa aaleaacaGGOaGaaGinaiaacMcaaeaacaaI0aaaaaaa@39D0@ ,

F=k m 1 m 2 R (4) / R (4) 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbGcpaWa aSbaaKqaGeaajugWa8qacaaIXaaal8aabeaajugib8qacaWGTbGcpa WaaSbaaKqaGeaajugWa8qacaaIYaaal8aabeaajugib8qacaWHsbqc fa4damaaBaaajeaibaqcLbmacaGGOaWdbiaaisdapaGaaiykaaqcba sabaqcLbsacaGGVaGaamOuaKqbaoaaDaaajeaibaqcLbmacaGGOaGa aGinaiaacMcaaKqaGeaajugWaiaaiodaaaqcfaOaaiilaaaa@555F@   (23)

Then we would have

di v 4 A ( 4 ) =2 π 2 k R (4) n ρ δ m 1 δ Ω (4) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGinaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaaisdaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabg2da9iaaikdacqaHapaCjuaGda ahaaqcbasabeaajugWaiaaikdaaaqcLbsacaWGRbGaamOuaKqbaoaa BaaajeaibaqcLbmacaGGOaGaaGinaiaacMcaaKqaGeqaaKqzGeGaam OBaOWaaSbaaKqaGeaajugWaiabeg8aYbWcbeaakmaalaaabaqcLbsa cqaH0oazcaWGTbGcdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaOqaaK qzGeGaeqiTdqMaeuyQdCvcfa4aaSbaaKqaGeaajugWaiaacIcacaaI 0aGaaiykaaqcbasabaaaaKqzGeGaaiOlaaaa@6985@    ( 24)

It means that di v 4 A ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGinaaqcbaYdaeqaaKqzGeWdbiaahgeak8aada WgaaqcbasaaKqbaoaabmaajeaibaqcLbmapeGaaGinaaqcbaYdaiaa wIcacaGLPaaaaSqabaaaaa@47EE@ depends on “ R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOuaaaa@3C88@ dq&ruo;, it means, in turn, that the law of energy conservation is broken. Indeed, if Ω (4) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaaGinaiaa cMcaaKqaGeqaaKqzGeGaeyOKH4QaaGimaaaa@449D@ (see (19), then R0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadkfacqGHsgIRcaaIWaaaaa@3F0F@ and di v 4 A ( 4 ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaaGinaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaaisdaaKqaG8aa caGLOaGaayzkaaaabeaajugib8qacqGH9aqpcaaIWaaaaa@4AC6@ . The analogous picture takes place at other points of our space. In fact, it means as well that there is no any real space, since the interaction (23) cannot maintain its existence.

Now take such a law instead (23):

F=k m 1 m 2 R (4) / R (4) 5 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraiabg2da9iaadUgacaWGTbqcfa4d amaaBaaajeaibaqcLbmapeGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2 gajuaGpaWaaSbaaKqaGeaajugWa8qacaaIYaaajeaipaqabaqcLbsa peGaaCOuaKqba+aadaWgaaqcbasaaKqzadGaaiika8qacaaI0aWdai aacMcaaKqaGeqaaKqzGeGaai4laiaadkfajuaGdaqhaaqcbasaaKqz adGaaiikaiaaisdacaGGPaaajeaibaqcLbmacaaI1aaaaKqzGeGaai ilaaaa@56A8@    (25)

Then instead (24) we have

di v 4 A ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaI0aaal8aabeaajugib8qacaWHbbqcfa4damaaBa aajeaibaqcfa4aaeWaaKqaGeaajugWa8qacaaI0aaajeaipaGaayjk aiaawMcaaaqabaqcLbsacqGHsgIRcqGHEisPaaa@4BB1@    (26)

And di v 4 A ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaaI0aaal8aabeaajugib8qacaWHbbqcfa4damaaBa aajeaibaqcfa4aaeWaaKqaGeaajugWa8qacaaI0aaajeaipaGaayjk aiaawMcaaaqabaqcLbsacqGHsgIRcqGHEisPaaa@4BB1@ if Ω (4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaaGinaiaa cMcaaKqaGeqaaaaa@4167@ and, consequently R ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOuaWWdamaaBaaajeaibaqcfa4aaeWa aKqaGeaajugWa8qacaaI0aaajeaipaGaayjkaiaawMcaaaqcKfaG=h qaaaaa@4351@ ,tend to zero. It means that our space collapses into a point, i.e., we obtain a black hole.

 

In principle, we get a similar picture for the cases i=5n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaiwdacaGGtaIaaCOB aaaa@4016@ . Show it for the case i=n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaah6gaaaa@3EA0@ .

di v n A ( n ) = lim Ω (n) 0 n a ρ n ρ d Λ (n1) Ω (n) ;  Ω (n) = C (n) ρ n . Λ (n1) =n C (n) ρ n1 ,   C (n) = π n/2 Γ( n 2 +1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaamOBaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaah6gaaKqaG8aa caGLOaGaayzkaaaabeaajugib8qacqGH9aqpkmaaxababaqcLbsaci GGSbGaaiyAaiaac2gaaKqaGeaajugWa8aacqqHPoWvjuaGdaWgaaqc cauaaKqzadGaaiikaiaad6gacaGGPaaajiaqbeaajugWaiabgkziUk aaicdaaSWdbeqaaOWdamaalaaabaWaa8qfaeaajugibiaadggakmaa BaaajeaibaqcLbmacqaHbpGCaSqabaqcLbsacaWGUbGcdaWgaaqcba saaKqzadGaeqyWdihaleqaaKqzGeGaamizaiabfU5amLqbaoaaBaaa jeaibaqcLbmacaGGOaGaamOBaiabgkHiTiaaigdacaGGPaaajeaibe aaaeaajugWaiaad6gaaSqabKqzGeGaeSyeUhTaey4kIipaaOqaaKqz GeGaeuyQdCvcfa4aaSbaaKqaGeaajugWaiaacIcacaWGUbGaaiykaa qcbasabaaaaKqzGeGaai4oa8qacaGGGcWdaiabfM6axLqbaoaaBaaa jeaibaqcLbmacaGGOaGaamOBaiaacMcaaKqaGeqaaKqzGeGaeyypa0 Jaam4qaKqbaoaaBaaajeaibaqcLbmacaGGOaGaamOBaiaacMcaaKqa GeqaaKqzGeGaeqyWdixcfa4aaWbaaKqaGeqabaqcLbmacaWGUbaaaK qzGeGaaiOlaiabfU5amLqbaoaaBaaajeaibaqcLbmacaGGOaGaamOB aiabgkHiTiaaigdacaGGPaaajeaibeaajugibiabg2da9iaad6gaca WGdbqcfa4aaSbaaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqcbasa baqcLbsacqaHbpGCjuaGdaahaaqcbasabeaajugWaiaad6gacqGHsi slcaaIXaaaaKqzGeGaaiila8qacaGGGcGaaiiOa8aacaWGdbqcfa4a aSbaaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqcbasabaqcLbsacq GH9aqpkmaalaaabaqcLbsacqaHapaCjuaGdaahaaqcbasabeaajugW aiaad6gacaGGVaGaaGOmaaaaaOqaaKqzGeGaeu4KdCKcdaqadaqaam aalaaabaqcLbsacaWGUbaakeaajugibiaaikdaaaGaey4kaSIaaGym aaGccaGLOaGaayzkaaaaaKqzGeGaaiilaaaa@C144@  (27)

F ( n ) =k m 1 m 2 R (n) / R (n) n ;  A ( n ) E ( n ) ( F (n) / m 2 ) R ( n ) k m 1 R (n) n ;a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaK qzGeWdbiabg2da9iaadUgacaWGTbqcfa4damaaBaaajeaibaqcLbma peGaaGymaaqcbaYdaeqaaKqzGeWdbiaad2gajuaGpaWaaSbaaKqaGe aajugWa8qacaaIYaaajeaipaqabaqcLbsapeGaaCOuaKqba+aadaWg aaqcbasaaKqzadGaaiika8qacaWGUbWdaiaacMcaaKqaGeqaaKqzGe Gaai4laiaadkfajuaGdaqhaaqcbasaaKqzadGaaiikaiaad6gacaGG PaaajeaibaqcLbmacaWGUbaaaKqzGeGaai4oa8qacaGGGcGaamyqaK qba+aadaWgaaqcbasaaKqbaoaabmaajeaibaqcLbmapeGaamOBaaqc baYdaiaawIcacaGLPaaaaeqaaKqzGeWdbiaadweajuaGpaWaaSbaaK qaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaKqaG8aacaGLOaGa ayzkaaaabeaakmaabmaabaqcLbsapeGaamOraKqba+aadaWgaaqcba saaKqzadGaaiika8qacaWGUbWdaiaacMcaaKqaGeqaaKqzGeWdbiaa c+cacaWGTbqcfa4damaaBaaajeaibaqcLbmapeGaaGOmaaqcbaYdae qaaaGccaGLOaGaayzkaaqcLbsapeGaamOuaKqba+aadaWgaaqcbasa aKqbaoaabmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPa aaaeqaaOWaaSaaaeaajugibiaadUgacaWGTbGcdaWgaaqcbasaaKqz adGaaGymaaWcbeaaaOqaaKqzGeGaamOuaKqbaoaaDaaajeaibaqcLb macaGGOaGaamOBaiaacMcaaKqaGeaajugWaiaad6gaaaaaaKqzGeGa ai4oaiaadggaaaa@8F41@     (28)

where index “n” in (27–28) indicates that these formulae refer to nD–space; d Λ (n1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikaiaa d6gacqGHsislcaaIXaGaaiykaaqcbasabaaaaa@4414@ is the element of nD–surface; n ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaad6gajuaGdaWgaaqcbasaaKqzadGaeqyWdihajeaibeaa aaa@4075@ a component of unit vector perpendicular to each point of this (n–1)D–surface; Ω (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaamOBaiaa cMcaaKqaGeqaaaaa@419C@ nD–volume; F ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaa aa@420F@ the interaction force between of masses m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaOWaaSbaaKqaGeaajugWaiaaigda aSqabaaaaa@3EEC@ and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaOWaaSbaaKqaGeaajugWaiaaikda aSqabaaaaa@3EED@ in nD–space;the constant of gravitation in nD–space ( k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaam4Aaaaa@3CA1@ ); E the vector of gravitation field intensity ( m· s 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaiabl+y6NjaadohajuaGpaWaaWba aKqaGeqabaqcLbmapeGaeyOeI0IaaGOmaaaaaaa@43DB@ ), m 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamyBaKqba+aadaWgaaqcbasaaKqzadWd biaaikdaaKqaG8aabeaajugibiabg2da9iaaigdaaaa@420E@ as; Γ( n 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfo5ahPWaaeWaaeaadaWcaaqaaKqzGeGaamOBaaGcbaqc LbsacaaIYaaaaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaaa@431A@ gamma function.

If Ω (n) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaamOBaiaa cMcaaKqaGeqaaKqzGeGaeyOKH4QaaGimaaaa@44D2@ , then we can write down (27) as

di v n A ( n ) = n n ρ k R n1 δ m 1 δ Ω (n) d Λ (n) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaamOBaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabg2da9OWaa8qfaeaajugibiaad6 gakmaaBaaajeaibaqcLbmacqaHbpGCaSqabaGcdaWcaaqaaKqzGeGa am4AaaGcbaqcLbsacaWGsbqcfa4aaWbaaKqaGeqabaqcLbmacaWGUb GaeyOeI0IaaGymaaaaaaGcdaWcaaqaaKqzGeGaeqiTdqMaamyBaKqb aoaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaOqaaKqzGeGaeqiTdq MaeuyQdCvcfa4aaSbaaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqc basabaaaaaqaaKqzadGaamOBaaWcbeqcLbsacqWIr4E0cqGHRiI8ai aadsgacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikaiaad6gacaGG Paaajeaibeaajugibiaac6caaaa@748F@    (29)

Since d Λ (n) = ρ n1 F( ζ 1 , ζ 2 , ζ 3 ,..., ζ n1 )d ζ 1 d ζ 2 d ζ 3 ,...,d ζ n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadsgacqqHBoatjuaGdaWgaaqcbasaaKqzadGaaiikaiaa d6gacaGGPaaajeaibeaajugibiabg2da9iabeg8aYLqbaoaaCaaaje aibeqaaKqzadGaamOBaiabgkHiTiaaigdaaaqcLbsacaWGgbGaaiik aiabeA7a6PWaaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGSa GaeqOTdOxcfa4aaSbaaKqaGeaajugWaiaaikdaaKqaGeqaaKqzGeGa aiilaiabeA7a6LqbaoaaBaaajeaibaqcLbmacaaIZaaajeaibeaaju gibiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaeqOTdOxcfa4aaSba aKqaGeaajugWaiaad6gacqGHsislcaaIXaaajeaibeaajugibiaacM cacaWGKbGaeqOTdOxcfa4aaSbaaKqaGeaajugWaiaaigdaaKqaGeqa aKqzGeGaamizaiabeA7a6LqbaoaaBaaajeaibaqcLbmacaaIYaaaje aibeaajugibiaadsgacqaH2oGEjuaGdaWgaaqcbasaaKqzadGaaG4m aaqcbasabaqcLbsacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaads gacqaH2oGEjuaGdaWgaaqcbasaaKqzadGaamOBaiabgkHiTiaaigda aKqaGeqaaaaa@855B@ in a spherical system of coordinate (ρ, ζ 1 , ζ 2 , ζ 3 ,..., ζ n1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaacIcacqaHbpGCcaGGSaGaeqOTdONcdaWgaaqcbasaaKqz adGaaGymaaWcbeaajugibiaacYcacqaH2oGEjuaGdaWgaaqcbasaaK qzadGaaGOmaaqcbasabaqcLbsacaGGSaGaeqOTdOxcfa4aaSbaaKqa GeaajugWaiaaiodaaKqaGeqaaKqzGeGaaiilaiaac6cacaGGUaGaai OlaiaacYcacqaH2oGEkmaaBaaajeaibaqcLbmacaWGUbGaeyOeI0Ia aGymaaWcbeaajugibiaacMcacaGGSaaaaa@5A5D@ we have after integration (29):

di v n A ( n ) =k n ρ n π n/2 Γ( n 2 +1 ) δ m 1 δ Ω (n) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaWGUbaal8aabeaajugib8qacaWHbbqcfa4damaaBa aajeaibaqcfa4aaeWaaKqaGeaajugWa8qacaWGUbaajeaipaGaayjk aiaawMcaaaqabaqcLbsacqGH9aqpcaWGRbGaamOBaOWaaSbaaKqaGe aajugWaiabeg8aYbWcbeaakmaalaaabaqcLbsacaWGUbGaeqiWdaNc daahaaWcbeqcbasaaKqzadGaamOBaiaac+cacaaIYaaaaaGcbaqcLb sacqqHtoWrkmaabmaabaWaaSaaaeaajugibiaad6gaaOqaaKqzGeGa aGOmaaaacqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaWaaSaaaeaaju gibiabes7aKjaad2gakmaaBaaajeaibaqcLbmacaaIXaaaleqaaaGc baqcLbsacqaH0oazcqqHPoWvjuaGdaWgaaqcbasaaKqzadGaaiikai aad6gacaGGPaaajeaibeaaaaGccaGGUaaaaa@6CE4@    (30)

If we would have another formula instead, F ( n ) =k m 1 m 2 / R n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaK qzGeWdbiabg2da9iaadUgacaWGTbGcpaWaaSbaaKqaGeaajugWa8qa caaIXaaal8aabeaajugib8qacaWGTbqcfa4damaaBaaajeaibaqcLb mapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaac+cacaWGsbGcpaWaaWba aSqabKqaGeaajugWa8qacaWGUbGaeyOeI0IaaGymaaaaaaa@532B@ e.g.,

F ( n ) =k m 1 m 2 R (n) / R (n) n1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaK qzGeWdbiabg2da9iaadUgacaWGTbGcpaWaaSbaaKqaGeaajugWa8qa caaIXaaal8aabeaajugib8qacaWGTbGcpaWaaSbaaKqaGeaajugWa8 qacaaIYaaal8aabeaajugib8qacaWHsbqcfa4damaaBaaajeaibaqc LbmacaGGOaWdbiaad6gapaGaaiykaaqcbasabaqcLbsapeGaai4la8 aacaWGsbqcfa4aa0baaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqc basaaKqzadGaamOBaiabgkHiTiaaigdaaaqcLbsapeGaaiilaaaa@5E09@    (31)

Then we would have

di v n A ( n ) =k n ρ n R (n) π n/2 Γ( n 2 +1 ) δ m 1 δ Ω (n) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaamOBaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabg2da9iaadUgacaWGUbGcdaWgaa qcbasaaKqzadGaeqyWdihaleqaaOWaaSaaaeaajugibiaad6gacaWG sbqcfa4aaSbaaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqcbasaba qcLbsacqaHapaCkmaaCaaaleqajeaibaqcLbmacaWGUbGaai4laiaa ikdaaaaakeaajugibiabfo5ahPWaaeWaaeaadaWcaaqaaKqzGeGaam OBaaGcbaqcLbsacaaIYaaaaiabgUcaRiaaigdaaOGaayjkaiaawMca aaaadaWcaaqaaKqzGeGaeqiTdqMaamyBaKqbaoaaBaaajeaibaqcLb macaaIXaaajeaibeaaaOqaaKqzGeGaeqiTdqMaeuyQdCvcfa4aaSba aKqaGeaajugWaiaacIcacaWGUbGaaiykaaqcbasabaaaaKqzGeGaai Olaaaa@7492@    (32)

It means that di v n A ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGcpaWaaSbaaKqa GeaajugWa8qacaWGUbaal8aabeaajugib8qacaWHbbqcfa4damaaBa aajeaibaqcfa4aaeWaaKqaGeaajugWa8qacaWGUbaajeaipaGaayjk aiaawMcaaaqabaaaaa@482E@ depends on " R (n) ", MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaackcacaWGsbqcfa4aaSbaaKqaGeaajugWaiaacIcacaWG UbGaaiykaaqcbasabaqcLbsacaGGIaGaaiilaaaa@4370@ it means, in turn, that the law of energy conservation is broken. Indeed, if " R (n) ", MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaackcacaWGsbqcfa4aaSbaaKqaGeaajugWaiaacIcacaWG UbGaaiykaaqcbasabaqcLbsacaGGIaGaaiilaaaa@4370@ (see (27), then R (n) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiaadkfajuaGdaWgaaqcbasaaKqzadGaaiikaiaad6gacaGG PaaajeaibeaajugibiabgkziUkaaicdaaaa@441B@ and divn A ( n ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bGaamOBaiaahgea juaGpaWaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaK qaG8aacaGLOaGaayzkaaaabeaajugib8qacqGH9aqpcaaIWaaaaa@4832@ . It means that the gravitation source at this point of space is not observed. The analogous picture takes place at other points of this space. In fact, it means as well that there is no any real space since the interaction (31) cannot maintain its existence.

Now take such a law instead (31):

F ( n ) =k m 1 m 2 R (n) / R (n) n+1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaK qzGeWdbiabg2da9iaadUgacaWGTbGcpaWaaSbaaKqaGeaajugWa8qa caaIXaaal8aabeaajugib8qacaWGTbqcfa4damaaBaaajeaibaqcLb mapeGaaGOmaaqcbaYdaeqaaKqzGeWdbiaahkfajuaGpaWaaSbaaKqa GeaajugWaiaacIcapeGaamOBa8aacaGGPaaajeaibeaajugibiaac+ cacaWGsbqcfa4aa0baaKqaGeaajugWaiaacIcacaWGUbGaaiykaaqc basaaKqzadGaamOBaiabgUcaRiaaigdaaaqcLbsacaGGSaaaaa@5E72@    (33)

Then instead (24) we have

di v n A ( n ) =k n ρ n R (4) 1 π n/2 Γ( n 2 +1 ) δ m 1 δ Ω (n) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaamOBaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabg2da9iaadUgacaWGUbqcfa4aaS baaKqaGeaajugWaiabeg8aYbqcbasabaGcdaWcaaqaaKqzGeGaamOB aiaadkfajuaGdaqhaaqcbasaaKqzadGaaiikaiaaisdacaGGPaaaje aibaqcLbmacqGHsislcaaIXaaaaKqzGeGaeqiWdaxcfa4aaWbaaKqa GeqabaqcLbmacaWGUbGaai4laiaaikdaaaaakeaajugibiabfo5ahP WaaeWaaeaadaWcaaqaaKqzGeGaamOBaaGcbaqcLbsacaaIYaaaaiab gUcaRiaaigdaaOGaayjkaiaawMcaaaaadaWcaaqaaKqzGeGaeqiTdq MaamyBaKqbaoaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaOqaaKqz GeGaeqiTdqwcLbmacqqHPoWvjuaGdaWgaaqcbasaaKqzadGaaiikai aad6gacaGGPaaajeaibeaaaaqcLbsacaGGSaaaaa@797C@    (34)

And di v n A ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamizaiaadMgacaWG2bqcfa4damaaBaaa jeaibaqcLbmapeGaamOBaaqcbaYdaeqaaKqzGeWdbiaahgeajuaGpa WaaSbaaKqaGeaajuaGdaqadaqcbasaaKqzadWdbiaad6gaaKqaG8aa caGLOaGaayzkaaaabeaajugibiabgkziUkabg6HiLcaa@4CBE@ if Ω (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibiabfM6axLqbaoaaBaaajeaibaqcLbmacaGGOaGaamOBaiaa cMcaaKqaGeqaaaaa@419C@ and, consequently, R ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaamOuaKqba+aadaWgaaqcbasaaKqbaoaa bmaajeaibaqcLbmapeGaamOBaaqcbaYdaiaawIcacaGLPaaaaeqaaa aa@421B@ tend to zero. It means that our space collapses to point, i.e., we obtain a black hole.

Now we can assume that vacuum is a nD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOBaiaahseaaaa@3D75@ space where the interaction law between masses has the rank n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOBaiabgUcaRiaaigdaaaa@3E45@ . There are fluctuations of the number n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOBaiabgUcaRiaaigdaaaa@3E45@ in the interaction one and the rank of the interaction may become less than the dimension i=0,1, 2,,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaaicdacaGGSaGaaGym aiaacYcacaqGGaGaaGOmaiaacYcacqGHMacVcaGGSaGaaCOBaaaa@45C2@ of the space. As a result, there occurs the Big Bang. Thus we have shown that long–raged interaction forces of the dimensionscan form real isotropic Euclidean spaces if and only if, when the dimensionsof these spaces equals j=i+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOAaiabg2da9iaahMgacqGHRaWkcaaI Xaaaaa@4039@ . Then we can affirm, using the method of mathematical induction, that long–raged interaction forces of the dimensions i=n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCyAaiabg2da9iaah6gacqGHRaWkcaaI Xaaaaa@403D@ can form a real isotropic Euclidean space of the rank j=i+1=n+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPqFH0xe9v8qqaqFD0xXdHaVhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aajugibabaaaaaaaaapeGaaCOAaiabg2da9iaahMgacqGHRaWkcaaI XaGaeyypa0JaaCOBaiabgUcaRiaaikdaaaa@43D4@ .

This is a theorem which we name “Theorem on spaces and long–ranged interaction forces forming these spaces” or, more shortly, “Theorem on spaces and forces forming them”.

Corrigendum

We have omitted the sign “–“ (minus) in the right sides formulae of the type (2), (3), (5) and so on after the sign “=” (equality), since we are only interested in an absolute volume of this divergence.

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest.

References

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  5. Vladimirov YS. The Nature of Space and Time. Anthology of Ideas. Russia: URSS; 2015.
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