Journal of Theoretics

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1 Junal f Theetics Junal Hme Page The Classical Pblem f a Bdy Falling in a Tube Thugh the Cente f the Eath in the Dynamic They f Gavity Iannis Iaklis Haanas Yk Univesity Depatment f Physics and Astnmy A Petie Building Nth Yk, Ontai MJ-P CANADA iannis@yku.ca Abstact Thee is a new they gavity called the dynamic they, which is deived fm themdymical pinciples in a five dimenal space. In this they we will examine the classical pblem f a bdy falling in a tube thugh the eath s cente. F simplicity and t an idealized scenai the eath is assumed t be a sphee f cnstant density equals t the mean density f the Eath. The deived equatin f mtin will be slved f a vaiety f initial cnditins, and the esults will be cmpaed t thse f Newtnian gavity. ey wds: Dynamic they f gavity, geneal elativity, enegy-mmentum tens. Intductin Thee is a new they called the Dynamic They f Gavity (DTG). It is deived fm classical themdynamics and equies that Einstein s pstulate f the cnstancy f the speed f light hlds. []. Given the validity f the pstulate, Einstein s they f special elativity fllws ight away []. The dynamic they f gavity (DTG) thugh Weyl s quantum pinciple als leads t a nn-gula electstatic ptential f the fm: () = e. () whee is a cnstant and is a cnstant defined by the they. The DTG descibes physical phenmena in tems f five dimens: space, time and mass. [] By cnsevatin f the fifth dimen we btain equatins which ae

2 identical t Einstein s field equatins and descibe the gavitatinal field. These field equatins ae simila t thse f geneal elativity and ae given belw: αβ αβ αβ αβ g T = G =. () Nw T αβ is the suface enegy-mmentum tens which may be fund within the space tens and is given by: αβ αβ αβ α β h ν T = Tsp F F F F ν () c and Tsp µν is the space enegy-mmentum tens f matte unde the influence f the gauge fields and is given by:[] ij i j i kj ij k Tsp = γu u F k F a F F k () c which futhe can be witten in tems f the suface metic as fllws:[] αβ α β α kβ α β αβ αβ µν ν Tsp = γu u F F F F ( g h )( F Fµν F F ν ) k (5) c ce: dy y u = y u = 0 (6) dt t is the statement equied by the cnsevatin f the fifth dimen, and the suface indices ν, α, β. = 0,,, and space index i, j, k, l = 0,,,,, and i j g = a y y = a h = a a y a y y whee the suface field αβ ij α α αβ αβ αβ α β α β tens is given by: (7) F ij E = E E F αβ E B B = F y ij E B i α B y j β and y E B B i α y = x i α = δ i α f i =,,, and y α y = x 0.. (8) It was shwn by Weyl that the gauge fields may be deived fm the gauge ptentials and the cmpnents f the 5-dimenal field tens Fij given by the 5 5 matix given in (8). [] Nw the deteminatin f the fifth dimen may be seen, f the nly physically eal ppety that culd give Einstein s equatins is the gavitating mass it s equivalent, mass [5]. Finally the dynamic they f gavity futhe agues that the gavitatinal field is a gauge field linked t the electmagnetic field in a 5-dimenal manifld f space-time and mass, but, when cnsevatin f mass is impsed, it may be descibed by the gemety f the - dimenal hypesyface f space-time, embedded int the 5-dimenal α

3 manifld by the cnsevatin f mass. The 5 dimenal field tens can nly have ne nnze cmpnent 0 which must be elated t the gavitatinal field and the fifth gauge ptential must be elated t the gavitatinal ptential. The they makes its pedictins f ed shifts by wking in the five dimenal gemety f space, time, and mass, and detemines the unit f actin in the atmic states in a way that can be calculated with the help f quantum Pissn backets when cvaiant diffeentiatin is used: [6] µ ν νq µ s [,p ] Φ = i g { δ Γ x }Φ x η. (9) µ q s,q In (9) the vect cuvatue is cntained in the Chistffel symbls f the secnd kind and the gauge functin Φ is a multiplicative fact in the metic tens g ν q, whee the indices take the values ν,q = 0,,,,. In the cmmutat, x µ and p ν ae the space and mmentum vaiables espectively, and finally δ µ q is the Cnecke delta. In DTG the mmentum ascibed, as a vaiable cannically cnjugated t the mass is the ate at which mass may be cnveted int enegy. The cannical mmentum is defined as fllws: (0) p = mv whee the velcity in the fifth dimen is given by: v γ = α () and gamma dt is a time deivative and gamma has units f mass density ( kg/m ) and α is a density gadient with units f kg/m. In the absence f cuvatue (8) becmes: µ ν νq x,p Φ = iη δ. () [ ] Φ The equatin f mtin in the dynamic they f gavity T pceed let us assume that a test bdy f mass m is falling thugh a tube that passes thugh the cente f the eath. The test bdy is at a distance away fm the cente f the eath. The fce that acts n the mass m is assciated nly with the mass M f the eath that lies within a sphee f adius. Thus the shell f the eath that lies utside this sphee exets n fce n the bdy. Theefe we can wite: ' ' πρ M () = ρ () = () whee ρ is the density functin assumed t be cnstant and equal t the mean density f the eath mateial, and is the vlume f the sphee f mass M. The gavitatinal ptential in the they f dynamic gavity can be descibed as sme sht f mdified Newtnian ptential and is given by the elatin belw: [ ] GM = e () () a slutin f the fllwing diffeential equatin, an equatin that is deived fm Weyl s quantizatin pinciple and has the fm:

4 () d ( ) = O. (5) d Next the fce acting n the bdy f mass m nw takes the fm: GM g() = which can be futhe witten as fllws: () = e πgρ g() = e finally the diffeential equatin f mtin in the tube becmes: d πgρ e O dt = (8) which is sme kind f a nn-linea hamnic scillat equatin. The paamete f the they depends n the ttal mass f the bdy M() and is equal t = G M /c =. 0 - m. Theefe duing the mtin acss the tube thugh the cente f the eath >. Expanding the expnential tem t secnd de and keeping nly fist de tems in / we btain the fllwing diffeential equatin f mtin: d πgρ 8πGρ O dt =, (9) which has the fllwing slutin: πgρ πgρ () t = c t c cs t (0) and c and c ae tw cnstants t be detemined by the initial cnditins. Applying diffeent initial cnditins Applying the initial cnditin indicated belw that we btain the cespnding slutins, if = = (πgρ /) / : (6) (7) i) Initial cnditins: (0)= (0)=0 Newtnian gavity slutin: (t) = 0 () Dynamic gavity: (t) = cs( t) () ii) Initial cnditins: (0)=0, (0)= Newtnian gavity slutin: (t) ( t) = ()

5 5 Dynamic gavity slutin: (t) = cs( t) ( t) () iii) Initial cnditins: (0)=, (0)= Newtnian gavity slutin (t) = cs( t) ( t) (5) Dynamic gavity slutin (t) = ( ) cs( t) ( t) (6) iv) Initial cnditins: (t) =, (t)= Newtnian gavity slutin cs( t ) (t) = ( t) ( t ) cs( ( t ) t) cs( t ) Dynamic gavity slutin ( t) (t) = cs( t) cs( t) ( ( t )) ( cs( t) t ) (7) (8) v) Initial cnditins (t)=, (t)=0 Newtnian gavity slutin (t) = ( cs( t) cs( t) ( t )( t) ) (9) Dynamic gavity slutin (t) = ( )( cs( t )cs( t) ( t )( t) ) (0) vi) Initial cnditins (t)=0, (t)= Newtnian gavity slutin t t cs t t (t) = ()

6 6 Dynamic gavity slutin cs t (t) = t t () t cs t cs t vii) Initial cnditins (t)=0, (t)=0 Newtnian gavity slutin (t) = 0 () Dynamic gavity slutin (t) = cs t cs t t () [ ] t viii) Initial cnditins (t)=, (t)= 0 Newtnian gavity slutin (t) = ( cs t cs t t t ) (5) Dynamic gavity slutin (t) = ( )( cs t cs t t t ) (6) elcity and acceleatin functins In paticula the expess f the velcity and acceleatin f the bdy mving unde Newtnian and dynamic gavity as elate t equatins (5), (6) and als (7) and (8). Fm equatins (5) and (6) we btain the velcity and acceleatin functins unde Newtnian gavity: (t) ( t) ( t = (t) = cs ) (7) and next the acceleatin functin t be: ( t) ( t) (t) = cs, (8) a(t) = next in the case f mtin unde dynamic gavity we btain: (t) ( t) ( ) ( t = (t) = cs ). (9) Nw making use f equatins (7) and (8) we btain f Newtnian gavity: (t) = (t) = cs cs( t ) t ( t ) ( t ) t cs t (0)

7 7 and finally a(t) = cs (t) = cs( t ) t ( t ) ( t ) t cs t () 5 Pltting the slutins f the diffeential equatins T btain an idea between mtin unde Newtnian gavity and mtin unde dynamic gavity sme numeical paametes shuld be calculated. Fist cnstant has the value: πgρ = = =. 0 sec (7) whee the mean density f the eath ρ has been taken equal t ρ = 5.5 g/cm [7]. Next fu equatins f all eight cases will be chsen, namely (5),(6), (7) and (8) and thei gaphs will pltted and cmpaed f Newtnian and dynamic gavity. Taking = km = 0 m, and = 0 m/sec we btain the gaphs belw f a numbe f a thusand pints pltted We actually bseve that thee is a diffeence between dynamic gavity and Newtnian gavity displacement amplitude The Newtnian amplitude appeas t be slightly lage than the dynamic n e in bth cases whee elatins have been deived f the cespnding velcities and acceleatins. Displacement Fig Displacement vesus time gaph f the Newtnian and dynamic gavity slutins with initial cnditins (0)=(0)=0.

8 Displacement Fig Displacement vesus time gaph f the Newtnian and dynamic gavity with initial cnditins (0)=000 m, (0)=00 m/sec. Theefe we have the fllwing amplitude elatins: Case Newtnian gavity scillatin amplitude: A N = (8) Dynamic gavity scillatin amplitude: AN A D = ( ) = (9) Case Newtnian gavity scillatin amplitude: A N ( t ) ( t ) ( cs( t ) ( t )) = cs (0) Dynamic gavity scillatin amplitude A D ( ) cs( t ) ( t ) ( cs( t ) ( ) ( t )) = ()

9 9 6. Applying an appximate methd f slving the same equatin Obseve that equatin (8) can be witten as fllws, if secnd de tems ae kept in the expan and / ae mitted: d dt =. () This equatin can be classified as ne having the geneal fm belw: d ε = 0 F, dt s if we assume a slutin f the fm (t) = A (tφ) whee bth A and φ ae assumed functins f t t be detemined s that (t) = A (tφ) =A ψ becmes a slutin f (). This is knwn as the methd f equivalent lineaizatin. Fllwing the analysis in [8] we have that: da dt ε ε = π π d ψ ε = π π d F dt A The abve equatins give that: ( A) = F( A φ, Acs φ) cs φ φ ( A φ, Acs φ) φdφ () (). (5) da = 0 A = cnst = dt (6) ψ = t θ A (7) which makes the fist appximatin t the slutin t be: (t) = t θ, (8) this is a hamnic scillatin with cnstant amplitude and angula fequency given by the expes (- / 0) which depends n the cnstant amplitude as well as the dynamic they paametes and is itself a cnstant.

10 Displacement Fig Displacement vesus time gaph f the lineaized slutin which has been deived as fist appximatin t the slutin f the nn linea hamnic scillat. The nn linea equatin is deived fm the dynamic gavity ptential. 7 Tying anthe density functin We next ae ging t ty the same pblem given that the density at a distance fm the cente f the eath vaies accding t the functin: ρ () = ρc (9) whee ρc is the cental density and is the adius f the eath. Taking int accunt the dynamic gavity acceleatin f gavity which nw becmes: G π ρ () c g = e (50) we can wite dwn the diffeential equatin f the mtin f the mass m inside the tube: d dt e = 0. (5) Afte expanding the expnential tems as befe the fist appximate equatin descibing the mtin can be: d dt = (5)

11 which has the fllwing slutin: = t cs C t C (t) (5) If we apply the initial cnditin (0)=0, (0)=0 (5) becmes: = t (t). (5) Diffeent initial cnditins namely (0)=0 and (0)=0 we btain: = t t cs (t) (55 If nw (0)=0 and (0) = 0 the slutin is: = t t (t). (56) Next cnside pssible initial cnditins t be (0)=0 and (0)=0, the slutin becmes: = cs (t) (57)

12 8 Pltting the slutins Ug equatin (5) deived f the given density functin we btain the fllwing gaph: Dynamic gavity Case (0)=0 and v(0)=0 Displacement Fig Displacement vesus time gaph. Slutin t the nn linea hamnic scillat equatin deived fm the dynamic gavity ptential and a vaiable density functin. Dynamic gavity Case (0)= 0 and v(0)=0 Displacement Fig 5 Displacement vesus time gaph. Slutin t the nn linea hamnic scillat equatin deived fm the dynamic gavity ptential and f a vaiable density functin.

13 Dynamic gavity Case (0)=0 and v(0)=00 m/sec Displacement Fig 6 Displacement vesus time gaph. Slutin t the nn linea hamnic scillat equatin deived fm the dynamic gavity ptential and f a vaiable density functin, and f the initial cnditins given abve. Dynamic gavity Case (0)=000m and v(0)=0 Displacement Fig 7 Displacement vesus time gaph. Slutin t the nn linea hamnic scillat equatin deived fm the dynamic gavity ptential and f a vaiable density functin, and f the initial cnditins given abve.

14 Next applying the same methd as in (7) we can als btain a fist appximate slutin t the fllwing nn linea scillat equatin belw: d dt = e (58) the slutin can be witten as fllws: (t) = t θ (59) Dynamic gavity Plt f the appximate slutin Displacement Fig 8 Displacement vesus time gaph f the lineaized slutin which has been deived as fist appximatin t the slutin f the nn linea hamnic scillat. The nn linea equatin is deived fm the dynamic gavity ptential. Cnclus The gavitatinal ptential f a new they f gavity namely the dynamic they f gavity was used t study the classical pblem f a mass m falling thugh a tube at the eath s cente. As a fist idealizatin the eath was cnsideed t be a sphee f cnstant density. The diffeential equatin f the mtin deived can be thught as sme kind f nn linea hamnic scillat. Next a vaiety f slutins wee btained f a vaiety f diffeent initial cnditins and sme f the slutins wee pltted. F the slutins chsen t be pltted we can see that the mtin is peidic with an amplitude f

15 5 scillatin slightly smalle in the case f dynamic gavity when cmpaed t that f the Newtnian gavity. Afte that and f the slutins which wee pltted subjected t the apppiate initial cnditins expess f the amplitudes f the mtin wee als given. Taking anthe appach the methd f equivalent libealizatin was used and a fist de appximatin f the slutin f the nn linea equatin was btained and pltted. This plt als demnstated peidic mtin simila t that f figues ne and tw. Finally a density functin was assumed f the intei f the eath and slutins f the new diffeential equatin f mtin wee btained subject t fu diffeent initial cnditins. These slutins wee pltted demnstating again the peidic natue f the mtin, except figue fu which demnstates a mtin that is peidic but des nt css the cente f the eath. Again the lineaized slutin f the new equatin was btained and pltted demnstating again the peidic natue f the mtin. In clg we cnclude that the mtin f a bdy in a tube tugh the cente f the eath in the case f dynamic gavity esembles that f the peidic mtin unde Newtnian gavity. efeences [] P., E., Williams, Themdynamic Basis f the Cnstancy f the Speed f Light, Mden Physics Lettes A, N 5, 997, [] P., E., Williams, Apein, l. 8, n., Apil, 00, p [] P., E., Williams, Mechanical Entpy and its Implicatins, Entpy, 00,, p [] P., E., Williams, p. cit., p. 06. [5] G. Hunte, S. Jeffes, J., P., igie, Causality and Lcality in Mden Physics, luwe Academic Publishes, p [6] P., E., Williams, p. cit., p. 06. [7] C., W., Misne,., S., Thne, J., A., Wheele, Gavitatin, W. H., Feeman and Cmpany 97, p. 9. [8] N., ylff, and N., Bgliubff, Intductin t Nn Linea Mechanics, Pincetn Univesity Pess 97, p. 9-5 Junal Hme Page

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