A New Approach to General Relativity

Transcription

1 Apeion, Vol. 14, No. 3, July 7 7 A New Appoach to Geneal Relativity Ali Rıza Şahin Gaziosmanpaşa, Istanbul Tukey aizasahin@gmail.com Hee we pesent a new point of view fo geneal elativity and/o space-time metics that is emakably diffeent fom the well-known viewpoint of geneal elativity. Fom this unique standpoint, we attempt to deive a new metic as an altenative to the chwazschild metic fo any planet in the sola system. Afte detemining the metic by means of some simple mathematical and physical manipulations, we used this altenative metic to ecalculate the peihelion pecession of any planet in the sola system and deflection of light that passes nea the sun, as examples of this new viewpoint. While we obtained the esult of classical geneal elativity fo the peihelion pocession, we found a slightly diffeent esult, elative to classical geneal elativity, fo the deflection of light. Keywods: Geneal elativity, chwazschild metic, basis vectos, peihelion pecession, deflection of light 1) Intoduction: Oigins of a New Metic As is shown in [1], the main equations of geneal elativity, known as the Einstein field equations, can be expessed using only fou basis 7 C. Roy Keys Inc.

2 Apeion, Vol. 14, No. 3, July 7 71 vectos, o tetads. Additionally, it is seen that the Einstein tenso in tems of tetads has the same fom as the stess-enegy tenso of electomagnetism/dynamics and the tetads satisfy α αeν = j ν. (1) As stated in [1], (1) is not only a diffeential equation but also a mathematical ule which basis vectos of any metic must satisfy. ince evey metic can be expessed in tems of some basis vectos, (1) also bings some cetain limits fo all metics. As a esult of this fact it is expected that all basis vectos of all metics must satisfy (1). ubsequently, if thee is such a ule that basis vectos must obey, it is expected to check this condition fo the known cases. Obviously, the best-known case is the chwazschild solution, which will be checked in ou study. The chwazschild solution is the most famous exact solution of the Einstein equations, thus, it has an exclusive place and a cucial ole in geneal elativity. It can coectly pedict the peihelion pocession of a planet in the sola system, and can give a close pediction to obsevations fo the deflection of light that passes nea the sun. In addition, some mysteious concepts such as black holes o hoizon poblems can be investigated by means of the chwazschild metic. Despite this, howeve, thee is a significant discepancy between the chwazschild metic and [1]. The chwazschild metic was oiginally deived fo a planet obit aound the sun, so it is expected that basis vectos of the chwazschild metic must satisfy the vacuum case of (1), that is α αeν= (because thee is no mass density and mass flux outside of the sun fo the sola system) o α equivalently j =. ince the chwazschild metic is α 7 C. Roy Keys Inc.

3 Apeion, Vol. 14, No. 3, July GM GM = dσ c dt d c c + dθ + inθdϕ, the basis vectos will be ( GM c ) 1/ ( GM c ) 1/ e = 1 / n, * e = 1 / n, (x33q) * whee n and n ae unit vectos. α * Fom the last two equations it can easily be seen that αe α * and αe. This suggests that the chwazschild metic is not a pope metic accoding to the point of view of [1] and a new metic fo the sola system whose basis vectos satisfy (1) needs to be found. Hence, this was the fist basis fo seaching a new metic as an altenative to the chwazschild metic. Besides the above mathematical necessity, we can also give a physical condition fo finding a new metic. The foundation of this condition poposes a new and significant limit condition between the space-time metic and classical mechanics. Fo this limit condition, the metic in tems of tetads fo the most geneal case should be witten fist. As seen fom () in [1], all basis vectos ae pependicula to each othe, the metic in tems of tetads fo the most x ict geneal case is ( ) ds = c e e dt + e e dx + e e dy + e e dz. () If () is divided by witten as cdτ, whee τ is the pope time, it can be 7 C. Roy Keys Inc.

4 Apeion, Vol. 14, No. 3, July 7 73 ds dt v1 v v = c dτ e e dτ e e c e e c e e c, (3) dx dy dz whee v1 =, v = and v3 =. Allowing dτ dτ dτ d 1 ds = dτ c dτ (3) to become dt v1 v v3 = d e e e e e e e e. (4) dτ dτ c c c Hee, (4) is a elativistic expession and fo classical cases it is v known that i 1 ( 1,, 3 c i = ), consequently v i can be taken. c Although thee ae inne poducts of basis vectos, these do not distub this limit condition, especially fo planets in the sola system. Thus (4) can be witten as d dt e e dτ dτ o d e e dt. (5) On the othe hand, it is known that fom classical mechanics d = L( v, q ; t) dt (6) i whee is the action function, L is the Lagangian, v i ae velocities and q i ae coodinates ( i = 1,,3 ). i 7 C. Roy Keys Inc.

5 Apeion, Vol. 14, No. 3, July 7 74 Now we will make the limit assumption: If (5) and (6) ae elated with classical cases, we assume that actions functions in (5) and (6) ae the same action functions, o at least, ae popotional with each othe. Then we can wite L ( v, q; t) = L e e i i whee L is a constant popotion coefficient and it must be equal to mc in ode to obtain flat space-time metic fo a stationay object whose est mass is m, and L is the elativistic Lagangian fo the most geneal case. But fo small speeds L becomes equal to the classical mechanics Lagangian. o in the light of this limit condition, () can be witten as L = mc ds c dt e e dx e e dy e e dz, (7) whee now m denotes the est mass of the any planet aound sun. It can be easily seen fom (7) that the time component of the L chwazschild metic is not equal to fo a planet in the sola mc U U system, but appoximately equal to , whee mc mc U is the potential enegy of the planet. Thus, thee is also a need of the finding a new metic whose time component of it is equivalent to L. mc o fa we have pesented ou two motivations fo finding a new metic. One motivation is based on a mathematical calculation and the 7 C. Roy Keys Inc.

6 Apeion, Vol. 14, No. 3, July 7 75 othe is based on a physical eason. In the emainde of the study we will attempt to find this new metic fo any planet in the sola system, and calculate the mathematical esults of it. Afte we find the new metic fo the sola system, we will pesent two well-known examples fo it: peihelion pecession of a planet and deflection of light. As is known, these two examples ae well-known examples used to test the chwazschild metic o classical geneal elativity. Theefoe, these examples ae chosen to compae ou new metics esults with esults fom classical geneal elativity. It is expected that the metic, which is deived as a esult of the above-mentioned motivations, will give moe satisfying esults, o at least the same esults as those fom the chwazschild metic. Intepetation of these esults will be ou last task in this study, although some intepetations will be left fo futue studies. ) Detemination of the New Metic and Applications of It Due to spheical symmety in the sola system due to we can wite the metic as ds = c e e dt + e e d + e e dθ + e e in θdϕ. θ θ ϕ ϕ ince the sun is fixed elative to planets and thee is no mass change o cuent, fom α αeν=, we undestand that the spatial pat of the metic must be flat. This enables us to wite the metic as ds = c e e dt + d + dθ + in θdϕ and detemine the esults. ince e e is popotional with Lagangian, we need to find Lagangian fo a planet aound the sun. In ode to find the elativistic Langangian of the planet aound the sun, we assume that the planet 7 C. Roy Keys Inc.

7 Apeion, Vol. 14, No. 3, July 7 76 has no any kinetic enegy at (the potential enegy is also zeo, but ou planet possesses mc as est enegy), and that it has acquied some kinetic enegy when it is placed to its own obit at = (fo GMm this case the potential enegy is, whee G is the univesal gavitational constant and M is the mass of the sun). Then, the kinetic enegy of the planet must be equal to the change in the potential enegy (we give anothe way fo finding elativistic Lagangian of a planet aound sun in the Appendix I section). Thus, GMm L= mc +, whee GM c e e = 1+, = /. Then the metic can be witten as ds c 1 dt d d in d = θ + θ ϕ. (8) We have obtained the metic (8), which satisfy ou two conditions that wee stated in the pevious section. Now we can ecalculate the peihelion pecession and the deflection of light fo (8). Afte some tedious calculations, whose details ae given in the Appendix II section, the metic in (8) leads to the following pecession angle: 6π GM δ =, (9) ac (1 e ) 7 C. Roy Keys Inc.

8 Apeion, Vol. 14, No. 3, July 7 77 whee δ is the pecession angle, a is the obit majo semi axis, and e is the eccenticity. Notice that δ in (9) is equivalent to the pecession pedicted by classical geneal elativity (see [], p. 197). Howeve, we cannot find the same ageement between (8) and the chwazschild metic, fo the deflection of light because (8) gives the deflection of light as (see Appendix III section fo detailed calculations): 3π GM Δ=, (1) cr whee Δ is the total deflection and R is the smallest distance of light fom the cente of the sun. Accoding to classical geneal elativity (see [], p. 19), the deflection is GM Δ = 4. cr 3) Conclusions Although, it seems that the main ideas and esults of this study conflict with the majoity of the ideas of geneal elativity, the pimay disageement is esulted fom the Einstein s limit condition between geneal elativity and Newtonian gavity. Einstein poposed as a limit condition that (see [], p. 15) g Gρ, whee ρ is the mass density. In classical geneal elativity, Einstein connected metic tensos with only gavitation by this condition. Howeve in this study it is poposed that g L (11) 7 C. Roy Keys Inc.

9 Apeion, Vol. 14, No. 3, July 7 78 is the limit condition. L can be the elativistic Lagangian of any system o inteaction, so thee is no any constaint fo L to belong to any system o any inteaction. As a esult of the above facts, in classical geneal elativity all deived metics o solutions ae based on the Einstein s limit condition. Consequently, all deived metics, solutions and pedictions ae some outcomes of this condition. ince the chwazschild metic is one of these outcomes, it also contains featues of the Einstein s limit condition. ubsequently, black holes, hoizons and all othe pedictions of the chwazschild metic ae esults of the Einstein s limit condition. Howeve, in this study the metic in (8) is a esult of (1) and (11). It contains outcomes of these two equiements and does not lead to the well-known concepts of classical geneal elativity o the chwazschild metic, and actually changes some of them. At this stage, all these points ae left as some extaodinay points to be investigated in futue studies. But thee is a pleasantly clea point: although (8) gives the same peihelion pocession amount with classical geneal elativity, it gives a diffeent deflection amount fo light. We know that obsevations do not exactly confim the chwazschild metic s pediction on the deflection of light. Theefoe, the discepancy between the chwazschild metic s pediction and ou metics pediction fo the deflection of light is a key facto to test the validity of (8) and consequently to test the validity of (1) and (11). Appendix I In this section we deive elativistic Lagangian fo some objects whose est mass is m and speed is adequately small, and we give some othe useful calculations. 7 C. Roy Keys Inc.

10 Apeion, Vol. 14, No. 3, July 7 79 Conside a feely moving object whose est mass is m. As a esult of easons that ae mentioned in pevious sections, the metic fo this object will be L ds = c dt + dx + dy + dz. (AI.1) L Dividing (AI.1) by dτ, whee τ is the pope time, yields ds L c t v dτ =, (AI.) L whee the dot denotes d/ dτ and We can choose becomes dx dy dz v = dτ dτ dτ. ds dτ as any constant (say L 7 C. Roy Keys Inc. b ). Thus (AI.) b = c t + v (AI.3) L (AI.3) gives the following geodesic equation fo t olution of (AI.4) is d L t =. (AI.4) dτ L L t = ε, (AI.5) L

11 Apeion, Vol. 14, No. 3, July 7 8 whee ε is the integation constant and can be detemined easily. ince (AI.5) is valid fo all values of L and v, fo an object at est, in ode to obtain a flat space-time metic, L= L (see (AI.)) and dt t = = 1 can be witten, so ε = 1. By substituting (AI.5) into (AI.3) dτ we have L b = c + v. (AI.6) L (AI.6) is also valid fo all values of L and v. Conside again an object at est, thus b = c (AI.7) can be found. o (AI.6) can be witten as L c = c + v L o L L = 1 v / c which is the elativistic Lagangian and the elativistic kinetic enegy of a feely moving object. We can find easily that L = mc, then mc L =. (AI.8) 1 v / c 7 C. Roy Keys Inc.

12 Apeion, Vol. 14, No. 3, July 7 81 ince (AI.8) is the elativistic Lagangian of a feely moving object, fo an object that is moving in any potential the Lagangian should be mc 1 L= + m v U 1 v / c whee (do not confuse d/ dτ with d/ dt) 7 C. Roy Keys Inc. (AI.9) dx dy dz v = + + dt dt dt If we do not wish to see elativistic effects fo slowly moving objects, we can assume that v / c. Thus 1 L mc + mv U (AI.1) can be obtained. Notice that (AI.1) is equivalent to the classical mechanics Lagangian, since the additional constant tems in the Lagangian can always be ignoed in classical mechanics. In this case the action function satisfies (6). If we want to see elativistic effects we assume that v / c but v / c 1. This enables us to wite v v and to expand (AI.9) as (fo this case the action function satisfies (7)) 1 1 L mc + mv + mv U, L m c + m v U. Fo a planet aound the sun, since mv GMm =,

13 can be found. Apeion, Vol. 14, No. 3, July 7 8 GMm L mc + Appendix II In this section, we calculate peihelion pecession fo the metic in (8). Geodesic equations of (8) ae: dt t dτ =, (AII.1) + d ( + ) = c t + θ + in θϕ, (AII.) 3 dτ d θ θ Cosθ inθϕ dτ = +, (AII.3) d ϕ ( + θcotθ) = ϕ, (AII.4) dτ whee dots denote d/ dτ, and τ denotes the pope time. We can choose axes such that θ = π /, so θ =. Now we can solve (AII.1) and (AII.4) immediately. (AII.1) gives t = ε 1+, (AII.5) whee ε is the integation constant. (AII.4) gives: j ϕ =, (AII.6) whee j is the integation constant. 7 C. Roy Keys Inc.

14 Apeion, Vol. 14, No. 3, July 7 83 Instead of solving (AII.) we can deive a simple equation fo. Fo this we divide (8) by dτ, and we choose ( ds / dτ ) = 1. o we have 1= c 1+ t + + θ + in θϕ. (AII.7) By substituting (AII.5) and (AII.6) into (AII.7) we get j ε 1= c (AII.8) In ode to get a simple equation, we change u = 1/ and wite deivates with espect to ϕ. (AII.8) becomes ( ) 1 c ε 1 u j u j u = + + +, (AII.9) whee pime denotes d/ dϕ. The integation constant ε can be detemined by using (AII.9). ince (AII.9) is valid fo all when, u = 1/, and u = /. Thus, ε = 1/c can be obtained. Now the metic (AII.9) becomes ( ) u ju ju 1= (AII.1) We diffeentiate (AII.1) with espect to ϕ, and we have u + u = ( 1+ ) 3 u. (AII.11) j In ode to detemine u we need to solve (AII.11). Unfotunately, howeve, it is a non-linea diffeential equation and cannot be solved 7 C. Roy Keys Inc.

15 Apeion, Vol. 14, No. 3, July 7 84 exactly. Fotunately, since u 1 fo the sola system, ( u) u, can be witten. Then (AII.11) can be witten as u + u = ( 1 3 u), j 3 u + 1+ u =. (AII.1) j j ince the path of a planet is investigated, and the path of a planet is an ellipse, j must be such that 1 =. j a(1 e ) Consequently, =. j a(1 e ) Now taking into consideation that / j 1, (AII.1) can be solved and u can be found as 1 ecosϕ u, a(1 e ) whee 1/ ϕ 3 = ϕ 1 + a(1 e ) and ϕ is the essential facto to detemine peihelion pecession. ince 3 / a(1 e ) 1 7 C. Roy Keys Inc.

16 Apeion, Vol. 14, No. 3, July / a(1 e ) a(1 e ). Thus ϕ becomes ϕ 3GM ϕ 1+. (AII.13) ac (1 e ) Consequently, total peihelion pecession is (using that fo complete obit ϕ = π ) 6π GM δ. (AII.14) ac (1 e ) Appendix III In this section, we calculate the deflection of light fo the metic in (8). ince light, which tavels in a staight line, is consideed in this case, the ight hand side of (AII.1) can be dopped and u can be taken as u = (1/ R ) Cosϕ (AIII.1) whee R is the smallest distance of light fom the cente of the sun, 1/ and ϕ ϕ( 1 3 / ) ( 1 3 / j ϕ j ) = + + again. Natually, this time the constant j has a diffeent value and in ode to detemine the deflection of light, j must be found in tems of the known quantities. (AII.1) can be used, noting that ϕ uns fom π / to + π / and that when = R, ϕ =. 1= ( 1+ u ) + ju + ju = R 7 C. Roy Keys Inc.

17 Apeion, Vol. 14, No. 3, July 7 86 j 1= 1+ + j + R R By assuming that R, the last equation can be witten as Thus, j R R j 7 C. Roy Keys Inc. R. (AIII.) With the coection, ϕ uns fom ( π / +Δ /) to ( / /) π +Δ, whee Δ is the full angle of the deflection. etting u = ( ) fo ϕ = π / +Δ /, and teating Δ and / R as small, fom (AIII.1) and (AIII.) π Δ 3 Cos 1 + =, 4R π Δ 3π 3Δ Cos + =. (AIII.3) 8R 8R In (AIII.3) 3 Δ /8R is negligible when compaed with othe tems, so it can be dopped. By expanding (AIII.3), Δ 3π in 8R can be obtained. Fom the last expession Δ 3π, 8R

18 Apeion, Vol. 14, No. 3, July π 3π GM Δ =. (AIII.4) 4R c R Refeences [1] A.R. Şahin, Einstein Equations fo Tetad Fields, Apeion, Vol. 13, No. 4, (Octobe 6), []. Weinbeg, Gavitation and Cosmology: Pinciples and Applications of the Geneal Relativity, John Wiley & ons (197). 7 C. Roy Keys Inc.