184 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Space-time as Dark Energy and Dark Matter Nalin de Silvaa*, Kumudumalee Jayakodyb, Hemantha Don Maddumagec, Wasantha Katugampalad a109/1, Railway Avenue, Maharagama, Sri Lanka bNo: 54, Mihindu Mawatha, Gampaha,. Sri Lanka cResearch Computing Center, Florida State University, 151, Dirac Science Library, Tallahassee, FL32306, USA dDepartment of Mathematics, University of Kelaniya, Kelaniya, Sri Lanka aEmail: nalink2003@gmail.com bEmail: jan_kumudu@yahoo.com cEmail: mdphemantha@yahoo.com dEmail: wasantha@kln.ac.lk Abstract In this paper we modify Einstein’s field equations, and write down the cosmological term and the metric tensor as an energy momentum tensor of space-time,and interpret space-time as a form of energy, not vacuum energy as such.The energy momentum tensor of space-time has equivalents of both mass and pressure components.The acceleration of the universe, Dark Energy and Dark Matter are explained in terms of the energy momentum tensor of space-time. Keywords: Space-time energy momentum tensor; Space-time as energy; Einstein’s field equations; Dark Energy; Dark Matter; Universe with acceleration. 1. Introduction We write,following Hemantha and de Silva [1, 2],Einstein’s Field Equations in General Relativity in the form, π‘…π‘…πœ‡πœ‡πœ‡πœ‡ βˆ’ 1 2 π‘…π‘…οΏ½π‘”π‘”πœ‡πœ‡πœ‡πœ‡ = πœ…πœ…πœ…πœ…πœ‡πœ‡πœ‡πœ‡ βˆ’ π›¬π›¬π‘”π‘”πœ‡πœ‡πœ‡πœ‡ (1.1) ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 185 where RΞΌΟ… ,gΞΌΟ…, TΞΌΟ… are the Ricci tensor, metric tensor and the energy momentum tensor of ordinary matter and radiation respectively,𝑅𝑅� is the Ricci scalar,𝛬𝛬 is the cosmological term and πœ…πœ… = βˆ’8πœ‹πœ‹πœ‹πœ‹ 𝑐𝑐2 . The 𝛬𝛬 term with the metric tensor, is written on the right hand side of the equations as we take it to represent the energy momentum tensor due to space- time. This is a new interpretation and the first term on the right hand side of the equation represents the usual energy momentum tensor due to matter and radiation, and the second term represents the energy momentum tensor due to space-time. The space-time itself is considered as a source of energy- momentum. The space time is determined by not only ordinary matter and radiation but by space time itself as well. The space time (space and time) in Newtonian formulation was given and matter and radiation existed in a given space time. In the Einsteinian formulation space time was determined by matter and radiation. In the present formulation space time is determined by space time itself in addition to matter and radiation. In the original version of equations of Einstein the 𝛬𝛬 term is written on the left hand side of the equations as a geometrical tensor, and the right hand side of the equations consists only of the energy- momentum tensor due to matter and radiation. The 𝛬𝛬 term, as a positive constant, and written on the left hand side of the field equations in the given sign convention, gives rise to a field that repels particles and objects, rather than to one that attracts them. Einstein, as well known, introduced the term in order to obtain a static solution instead of the expanding universe that was obtained by solving the original equations without the 𝛬𝛬 term. In our formulation 𝛬𝛬 term could be a variable either positive or negative. 2. 𝜦𝜦gΞΌΟ… as a tensor representing energy momentum When the 𝛬𝛬 term with the metric tensor is written on the right hand side of the equations as a term that represents the energy momentum tensor of the space-time itself,it presents a different picture altogether. The 𝛬𝛬 term with the metric tensor now represents a space-time that is a source for the space- time itself, while having an energy momentum of its own. This energy of the space time is not the same as the vacuum energy in general, however, we do not rule out the possibility of space time energy giving the vacuum energy at some epoch in the evolution of the universe. Now the left hand side of the equation (1.1) is divergenceless, and hence the right hand side also should be divergenceless. If 𝛬𝛬is considered as a constant, then since the metric tensor is divergenceless it implies that the energy momentum tensor of ordinary matter and radiation is also divergenceless as in the original equations of Einstein. In this case the energy of the space time is β€œconserved” as the energy momentum tensor of the space- time is also divergenceless. However, if 𝛬𝛬 is not a constant, then 𝛬𝛬gΞΌΟ… is not divergenceless implying that the energy momentum tensor is also not divergenceless, for the sum of the two terms on the right hand side of the equation (1.1) has to be divergenceless. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 186 This implies that it is the total of the energy momentum of matter and radiation, and that of the space-time that remains a constant. This is a new phenomenon arising out of consideration of space-time as a form of energy. The space-time being a form of energy can be converted to other forms of energy and vice versa. This is different from the C-field introduced by Hoyle, (see for example [3]), in his formulation of the steady state theory. In the formulation of Hoyle the C –field contributed to continuous creation of matter, increasing the energy of the universe. There was no β€œexchange” of energy between matter and energy, and space - time. The above formulation does not imply that the space - time would interact with electromagnetic radiation, it only means that the energy of space - time could be converted to other forms of energy and vice versa. 3. Robertson Walker Space Times For different values of πœ‡πœ‡ and 𝜈𝜈 in the equation (1.1), we obtain the following two independent equations with four unknown variables𝑅𝑅, 𝜌𝜌, 𝛬𝛬 and 𝑝𝑝, in the case of the Robertson-Walker metrics. βˆ’ 3�̇�𝑅2 𝑅𝑅2𝑐𝑐2 βˆ’ 3π‘˜π‘˜ 𝑅𝑅2 = πœ…πœ…πœŒπœŒ βˆ’ 𝛬𝛬 (3.1) π‘˜π‘˜ 𝑅𝑅2 + �̇�𝑅2+2π‘…π‘…οΏ½ΜˆοΏ½π‘… 𝑅𝑅2𝑐𝑐2 = πœ…πœ…πœ…πœ… 𝑐𝑐2 + 𝛬𝛬 (3.2) where π‘˜π‘˜ = βˆ’1,0,1 and a dot denotes differentiation with respect to cosmic time. From the above equations we find that the space time energy tensor gives rise to a component equivalent to that of ρ the density of ordinary matter and radiation as in the equation (3.1), and to a component equivalent that of p the pressure due to ordinary matter and radiation as in the equation (3.2). Since πœ…πœ… = βˆ’8πœ‹πœ‹πœ‹πœ‹ 𝑐𝑐2 is negative, it is clear that when 𝛬𝛬is positive, it represents a positive density of space time energy as in equation (3.1), and a negative pressure due to space time energy as in equation (3.2). When 𝛬𝛬is negative, it represents a negative density and a positive pressure. We define 𝛬𝛬′ equal to 𝛬𝛬𝑐𝑐 2 8πœ‹πœ‹πœ‹πœ‹ so that 𝛬𝛬′has same dimensions as of density𝜌𝜌[π‘€π‘€πΏπΏβˆ’3] and could be compared with 𝜌𝜌. Thus 𝛬𝛬′[π‘”π‘”π‘π‘π‘π‘βˆ’3] gives the density of energy due to the space-time, and 𝜌𝜌[π‘”π‘”π‘π‘π‘π‘βˆ’3] gives the energy density of ordinary matter and radiation.𝛬𝛬′ resembles the vacuum energy but has a different identification as the energy density of space - time. We may define the vacuum energy as the energy of space- time in the absence of ordinary matter and radiation. From equations (3.1) and (3.2) we obtain 𝑑𝑑 𝑑𝑑𝑑𝑑 [(𝜌𝜌 + Ξ›β€²)𝑅𝑅3] + 3 οΏ½ πœ…πœ… 𝑐𝑐2 βˆ’ Ξ›β€²οΏ½ 𝑅𝑅2�̇�𝑅 = 0 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 187 (3.3) When 𝛬𝛬′ = 0 ,the above equation reduces to the usual equation of β€œconservation of energy” (see for example Mc.Vittie [4]), 𝑑𝑑 𝑑𝑑𝑑𝑑 (πœŒπœŒπ‘…π‘…3) + 3 οΏ½ πœ…πœ… 𝑐𝑐2 �𝑅𝑅2�̇�𝑅 = 0 (3.4) The equation (3.3) can also be written as 𝑑𝑑 𝑑𝑑𝑑𝑑 [πœŒπœŒπ‘…π‘…3] + 3 οΏ½ πœ…πœ… 𝑐𝑐2 �𝑅𝑅2�̇�𝑅 + Λ̇′𝑅𝑅3 = 0 (3.5) The equations (3.3) and (3.5) demonstrate that the energy of space - time can be converted to energy of ordinary matter and radiation, and vice versa. When 𝛬𝛬′ is a constant, the equation (3.5) reduces to the usual equation (3.4), since the space - time energy is β€œconserved β€œseparately in that case. It should be noted that 𝛬𝛬′appears with both the density and pressure of ordinary matter and energy in equation (3.3) as it should be. 𝛬𝛬′acts as a density and as well as pressure. It implies that when off diagonal terms are zero, the trace of π›¬π›¬π‘”π‘”πœ‡πœ‡πœ‡πœ‡gives rise toβˆ’2𝛬𝛬, due to negative pressure arising out of 𝛬𝛬 and this makes the space –time expands, when 𝛬𝛬is positive. 4. de - Sitter space We illustrate some of these ideas in the case when 𝛬𝛬(Ξ›β€²) is a constant with the well known de-Sitter Universe, where the energy momentum tensor of matter and radiation is null. Eliminating k in equations (3.1) and (3.2), we have, in the case of a matter-radiation free universe 3R̈ βˆ’ Ξ›c2R = 0 (4.1) where 𝛬𝛬 is a constant. If 𝛬𝛬 is a positive constant,we obtain the de-Sitter universe that expands forever. If 𝛬𝛬 is a negative constant, the equation is similar to that of simple harmonic motion. We have solutions where R takes the form𝑅𝑅 = 𝐴𝐴 cos(πœ”πœ”πœ”πœ” + 𝛼𝛼)and we take the modulus of𝑅𝑅as it has to be positive. Thus 𝑅𝑅would change from zero to zero for suitable values of cosmic timeπœ”πœ”,and then would start a different cycle. This corresponds to a cyclic universe. If 𝛬𝛬 is zero, 𝑅𝑅increases linearly with time. In all the three cases the energy density of space-time remains constant and is β€œconserved” on its own as there is American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 188 no ordinary matter and radiation. In this case the energy of the space-time could be identified as the vacuum energy. Now consider the de Sitter space-time written in the form 𝑑𝑑𝑠𝑠2 = (1 βˆ’ 1 3 π›¬π›¬π‘Ÿπ‘Ÿ2 )𝑐𝑐2π‘‘π‘‘πœ”πœ”2 βˆ’ π‘‘π‘‘π‘Ÿπ‘Ÿ2 (1 βˆ’ 1 3π›¬π›¬π‘Ÿπ‘Ÿ 2 ) βˆ’ π‘Ÿπ‘Ÿ2 (𝑑𝑑ϴ2 + 𝑠𝑠𝑠𝑠𝑠𝑠2ϴ𝑑𝑑φ2 ) Jayakody[5] has shown that the angular momentum β„Žper unit mass of a particle that describes a circle of coordinate radius a can be written as β„Ž2 = βˆ’π›¬π›¬π‘Žπ‘Ž 4 3 𝑐𝑐, implying that no circular motion is possible if 𝛬𝛬is non negative, as could be expected since positive 𝛬𝛬gives rise to a repulsion. 5. 𝜦𝜦 considered as a variable In this case the energy of space - time does not remain a constant and can be converted to other forms of energy represented by the energy momentum tensor for ordinary matter and radiation. In order to illustrate this and also to illustrate that the space - time energy could be interpreted as dark energy that accelerates the universe as observed by Perlmutter and his colleagues [6,7] and Reiss and his colleagues [8] we proceed as follows. This is only an illustration. We now begin a discussion on solving the two equations (3.1) and (3.2), for 𝑅𝑅 under the boundary conditions stated below. However in the process, 𝜌𝜌 and 𝛬𝛬 also have to be found as they are not prescribed as functions of time. Only 𝑝𝑝 is prescribed as we consider zero pressure models. As there are three variables in effect ( namely𝑅𝑅, 𝜌𝜌 and 𝛬𝛬), in addition to k, we could assume a solution of the form 𝑅𝑅 = 𝑅𝑅(πœ”πœ”) and substitute it in the two equations (3.1) and (3.2), to obtain 𝜌𝜌 and 𝛬𝛬 as functions of πœ”πœ”, the cosmic time, and see whether the solution would agree with the values of 𝜌𝜌 and𝛬𝛬 at the present epoch.. Further since we are interested in solutions that give an expanding universe with an acceleration in the present epoch, and since the Universe had been expanding with a deceleration in the previous epoch commencing with the big bang, it is seen that οΏ½ΜˆοΏ½π‘…(πœ”πœ”) has changed from negative to positive at the onset of acceleration. This implies that οΏ½ΜˆοΏ½π‘…(πœ”πœ”) is equal to zero at the onset of acceleration. In solving the above equations we assume the following boundary conditions. 𝑅𝑅(πœ”πœ”) = 0 at πœ”πœ” = 0, this would correspond to the usual big bang model at πœ”πœ” = 0. The present observations tell us that the ratio of dark energy to that of ordinary matter is 7 3 , (see for example[9]). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 189 Since we assume that 𝛬𝛬′ correspond to dark energy or space-time energy,we should take 𝛬𝛬 β€² ρ = 7 3 at the present epoch. Since according to observations the onset of acceleration has taken place when the redshift was in the range 1.2 - 1.6 (see for example[8]), we could take the range of 𝑧𝑧to be 𝑧𝑧 = 0.2 βˆ’ 0.6 at the onset of acceleration. This implies that οΏ½ΜˆοΏ½π‘…(πœ”πœ”)should be equal to zero in the range 𝑧𝑧 = 0.2 βˆ’ 0.6 . We demand that the density 𝜌𝜌 should be positive for all values ofthe cosmic timeπœ”πœ”, and should lie between 4.5 Γ— 10βˆ’30 π‘”π‘”π‘π‘π‘π‘βˆ’3and 1.8 Γ— 10βˆ’29 π‘”π‘”π‘π‘π‘π‘βˆ’3 at the present epoch. We take the density to lie in the above range as different authors have quoted these values. We make no demand on 𝛬𝛬′in general, as 𝛬𝛬′ could be either positive or negative, in different epochs. However we would prefer to have 𝛬𝛬′ in the range 1.9 Γ— 10βˆ’30 π‘”π‘”π‘π‘π‘π‘βˆ’3<𝛬𝛬′<7.7 Γ— 10βˆ’29 π‘”π‘”π‘π‘π‘π‘βˆ’3 , at the present epoch, as a value in this interval is quoted by different authors as the density of dark energy. Using equations (3.1) and (3.2) with 𝑝𝑝 = 0 we obtain the following expressions for density 𝜌𝜌 of the homogeneous universe and the density of energy due to space-time which is represented by 𝛬𝛬′. 𝜌𝜌 = 1 4πœ‹πœ‹πœ‹πœ‹ οΏ½π‘˜π‘˜π‘π‘ 2+�̇�𝑅2βˆ’π‘…π‘…οΏ½ΜˆοΏ½π‘… 𝑅𝑅2𝑐𝑐2 οΏ½ (5.1) 𝛬𝛬′ = 1 8πœ‹πœ‹πœ‹πœ‹ οΏ½π‘˜π‘˜π‘π‘ 2+�̇�𝑅2+2π‘…π‘…οΏ½ΜˆοΏ½π‘… 𝑅𝑅2𝑐𝑐2 οΏ½ (5.2) Using the boundary condition𝛬𝛬 β€² ρ = 𝑠𝑠, where n is a constant at the present epoch, and substituting the above expressions for ρ and Ξ›' , we have 1 (8πœ‹πœ‹πœ‹πœ‹) οΏ½π‘˜π‘˜π‘π‘ 2+�̇�𝑅2+2π‘…π‘…οΏ½ΜˆοΏ½π‘… 𝑅𝑅2𝑐𝑐2 οΏ½ οΏ½ (4πœ‹πœ‹πœ‹πœ‹) 𝑅𝑅2𝑐𝑐2 π‘˜π‘˜π‘π‘2+�̇�𝑅2βˆ’π‘…π‘…οΏ½ΜˆοΏ½π‘… οΏ½ = 𝑠𝑠, (5.3) which reduces to (2𝑠𝑠 βˆ’ 1)(π‘˜π‘˜π‘π‘2 + �̇�𝑅2 ) βˆ’ 2(𝑠𝑠 + 1)π‘…π‘…οΏ½ΜˆοΏ½π‘… = 0 i.e. οΏ½ΜˆοΏ½π‘… = (2π‘›π‘›βˆ’1)(π‘˜π‘˜π‘π‘2+�̇�𝑅2) 2(𝑛𝑛+1) 𝑅𝑅 , at the present epoch. (5.4) At the present epoch this implies that οΏ½ΜˆοΏ½π‘… > 0, if π‘˜π‘˜ = 0 π‘œπ‘œπ‘Ÿπ‘Ÿ 1 andn >1 2 . If n =1 2 , οΏ½ΜˆοΏ½π‘… =0 for all values of π‘˜π‘˜. It implies that for values of π‘˜π‘˜ = 0 or 1 the universe changes from deceleration to acceleration when n =1 2 . Further if π‘˜π‘˜ = βˆ’1, οΏ½ΜˆοΏ½π‘… < 0, when n >1 2 at the present epoch assuming that �̇�𝑅 is less than c, indicating a deceleration. Under these conditions the universe changes from acceleration to deceleration when n =1 2 . It follows that, assuming π‘˜π‘˜ = 0 or 1, the Universe expands with acceleration at the present epoch since n = 7 3 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 190 (boundary condition (ii)). Further even if 𝛬𝛬′ is a constant the universe changes from deceleration to acceleration when ρ decreases to the value 2𝛬𝛬′ , if π‘˜π‘˜ = 0 or 1, and the other way around whenπ‘˜π‘˜ = βˆ’1. Reference [10] has found many solutions but unfortunately none of them satisfies all the boundary conditions, though he has solutions that give inflation as well. Katugampala and de Silva [11] also have given solutions involving both acceleration and deceleration of the universe though the solutions do not satisfy some of the boundary conditions. 6. A solution with deceleration and acceleration Though it is difficult to find a model for the universe satisfying all the boundary conditions stated, Katugamapala [10] has shown that the model 𝑅𝑅 = 𝑏𝑏�(1 βˆ’ cos3πœ”πœ”πœ”πœ”) (6.1) where b and πœ”πœ” are constants, satisfies inflation, an acceleration at the present epoch, taken as πœ”πœ”0 = 4.26 Γ— 1017𝑠𝑠 (13.69 Γ— 109 π‘¦π‘¦π‘¦π‘¦π‘¦π‘¦π‘Ÿπ‘Ÿπ‘ π‘ ), time from the big bang. From (5.4) we have �̇�𝑅 =3𝑏𝑏𝑏𝑏 cos2𝑏𝑏𝑑𝑑 sin𝑏𝑏𝑑𝑑 2οΏ½1βˆ’cos3Ο‰t (6.2) οΏ½ΜˆοΏ½π‘… =οΏ½3𝑏𝑏𝑏𝑏 2 2 οΏ½ οΏ½βˆ’3 cos6𝑏𝑏𝑑𝑑+cos4𝑏𝑏𝑑𝑑+6cos3π‘π‘π‘‘π‘‘βˆ’4 cos𝑏𝑏𝑑𝑑 2(1βˆ’cos3 𝑏𝑏𝑑𝑑)3/2 οΏ½ (6.3) cosπœ”πœ”πœ”πœ” = 0 , is a solution of οΏ½ΜˆοΏ½π‘… = 0 , and it is clear that when cosπœ”πœ”πœ”πœ” = 0 , �̇�𝑅 = 0 , making values of t corresponding to cosπœ”πœ”πœ”πœ” = 0points of inflection for 𝑅𝑅as a function of πœ”πœ”. cosπœ”πœ”πœ”πœ” = 0 Implies πœ”πœ”πœ”πœ” = πœ‹πœ‹ 2 (0 < πœ”πœ”πœ”πœ” < πœ‹πœ‹). Let π‘…π‘…π‘Žπ‘Ž be the value of 𝑅𝑅 at the point of inflection, which is the point at which acceleration of the Universe begins after a phase of deceleration. Thenπ‘…π‘…π‘Žπ‘Ž = 𝑏𝑏�(1 βˆ’ cos3 Ο€ 2 ) =𝑏𝑏 Writing πœ…πœ… = cosπœ”πœ”πœ”πœ”0, and substituting (6.1), (6.2) and (6.3) in the equation (5.3) with n = 7 3 for the present epoch πœ”πœ”0and rearranging the terms we have the following expression for the unknown 𝑏𝑏, in terms of πœ”πœ”andπœ”πœ”0. 𝑏𝑏2 = �𝑐𝑐 2 𝑏𝑏2οΏ½ 44οΏ½1βˆ’π‘‡π‘‡3οΏ½ (βˆ’81𝑇𝑇6βˆ’39𝑇𝑇4+360𝑇𝑇3βˆ’240𝑇𝑇) (6.4) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 191 Taking the redshift at the onset of acceleration as 1.4 we have π‘…π‘…π‘œπ‘œ π‘…π‘…π‘Žπ‘Ž = 1.4 , which givescosπœ”πœ”πœ”πœ”0 = βˆ’ 0.9864, and hence πœ”πœ”πœ”πœ”0 = 2.96 π‘Ÿπ‘Ÿπ‘¦π‘¦π‘‘π‘‘ . As πœ”πœ”0 = 4.26 Γ— 1017𝑠𝑠, we have πœ”πœ” = 6.94 Γ— 10βˆ’18π‘Ÿπ‘Ÿπ‘¦π‘¦π‘‘π‘‘π‘ π‘ βˆ’1. However, this leads to negative values of 𝜌𝜌for certain values of t, and we have to discard this value. In order to obtain positive values for 𝜌𝜌 for all time t we take the value πœ”πœ” = 5.32 Γ— 10βˆ’18π‘Ÿπ‘Ÿπ‘¦π‘¦π‘‘π‘‘π‘ π‘ βˆ’1 which gives b = 6.14 Γ— 1027cm, corresponding to a redshift of 1.26, at the onset of acceleration. The density of the universe at the present epoch is 1.22 Γ— 10βˆ’29 𝑔𝑔. π‘π‘π‘π‘βˆ’3 agreeing with the observations as stated under boundary condition (iv). From 𝑅𝑅 = 𝑏𝑏�(1 βˆ’ cos3πœ”πœ”πœ”πœ”), we find that 𝑑𝑑𝑅𝑅 𝑑𝑑𝑑𝑑 asπœ”πœ” tends to zero is οΏ½3 2 π‘π‘πœ”πœ”, which is of the order of the velocity of light for the stated values. An interesting feature is that 𝛬𝛬′decreases from very high values at t = 0 to zero at πœ”πœ” = 1.5 Γ— 1017 s, and is negative till πœ”πœ” = 1.02 Γ— 1018 s becoming negative again at πœ”πœ” = 1.35 Γ— 1018 s. The significance of negative values of 𝛬𝛬′ will be discussed under dark matter in the next section. 6. Schwarzschild – de Sitter Metric with 𝜦𝜦 It is not only dark energy that could be represented by 𝛬𝛬. We may identify 𝛬𝛬as representing dark matter (see for example Trimble [12]) as well. We write down the field equations in the absence of ordinary matter and radiation, in the form π‘…π‘…πœ‡πœ‡πœ‡πœ‡ βˆ’ 1 2 π‘…π‘…οΏ½π‘”π‘”πœ‡πœ‡πœ‡πœ‡ = βˆ’π›¬π›¬π‘”π‘”πœ‡πœ‡πœ‡πœ‡ (6.1) and write the Schwarzschild metric in the form 𝑑𝑑𝑠𝑠2 = (1 βˆ’ 2π‘šπ‘š π‘Ÿπ‘Ÿ βˆ’ 1 3 π›¬π›¬π‘Ÿπ‘Ÿ2 )𝑐𝑐2π‘‘π‘‘πœ”πœ”2 βˆ’ π‘‘π‘‘π‘Ÿπ‘Ÿ2 (1βˆ’2π‘šπ‘šπ‘Ÿπ‘Ÿ βˆ’13π›¬π›¬π‘Ÿπ‘Ÿ 2 ) βˆ’ π‘Ÿπ‘Ÿ2 (𝑑𝑑ϴ2 + 𝑠𝑠𝑠𝑠𝑠𝑠2ϴ𝑑𝑑φ2 ) (6.2) for a spherically symmetric object. The geodesic equations for constant r = a, and ΞΈ = Ο€ 2 can be written as (Jayakody [5]) 𝑦𝑦2 𝑑𝑑φ 𝑑𝑑𝑑𝑑 = β„Ž (6.3) and American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 192 β„Ž2 = [3π‘šπ‘šβˆ’π›¬π›¬π‘Žπ‘Ž3 3(π‘Žπ‘Žβˆ’3π‘šπ‘š) ] 𝑦𝑦2𝑐𝑐2 (6.4) eliminating 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 , whereh is a constant, m = πœ‹πœ‹πΊπΊ 𝑐𝑐2 , M being the mass of the spherically symmetric object. Since a can be assumed to be greater than 3m,(6.4) implies that 𝛬𝛬<3π‘šπ‘š π‘Žπ‘Ž3 . The equation (6.3)gives the angular momentum h per unit mass of a particle describing a circular path of coordinate radius a around a spherically symmetric distribution of mass M, and the equation (6.4) expresses h at a given a in terms of m and 𝛬𝛬. If 𝛬𝛬 =0 and m is negligible the equation (6.4) reduces to β„Ž2 = 𝑐𝑐𝑦𝑦𝑐𝑐2 (6.5) corresponding to the Newtonian equation. When m =0, the equation (6.4) reduces to β„Ž2 = βˆ’π›¬π›¬π‘Žπ‘Ž 4 3 𝑐𝑐 discussed under section 4. Dark matter had to be formulated [12] as it was noted that the right hand side of the equation (6.5) with the mass of the object could not account for the large h that was observed. However, from the equation (6.4) it is clear that a larger h could be accounted for at given a, for suitable values of 𝛬𝛬 . Assuming that the mass required to generate the velocity observed is five times the mass of the object, we find that 𝛬𝛬 that satisfies the equation [3π‘šπ‘šβˆ’π›¬π›¬π‘Žπ‘Ž3 3(π‘Žπ‘Žβˆ’3π‘šπ‘š) ] 𝑦𝑦2𝑐𝑐2 = 5𝑐𝑐𝑦𝑦𝑐𝑐2 could account for the velocities around the central body. This 𝛬𝛬 = βˆ’3π‘šπ‘š (4π‘Žπ‘Žβˆ’15π‘šπ‘š) π‘Žπ‘Ž4 (6.6) which could be satisfied if 4a> 15 m, and 𝛬𝛬 < 0. In our formulation 𝛬𝛬is a variable, and for some models of the universe it should be possible to find suitable values satisfying (6.6). The model discussed under the previous section gives negative 𝛬𝛬 for certain values of cosmic time t. If the mass required to account for high values of h is n times the mass of the object, then we have [3π‘šπ‘šβˆ’π›¬π›¬π‘Žπ‘Ž3 3(π‘Žπ‘Žβˆ’3π‘šπ‘š) ] 𝑦𝑦2𝑐𝑐2 = 𝑠𝑠𝑐𝑐𝑦𝑦𝑐𝑐2, giving 𝛬𝛬 = βˆ’3π‘šπ‘š[(π‘›π‘›βˆ’1)π‘Žπ‘Žβˆ’3π‘›π‘›π‘šπ‘š] π‘Žπ‘Ž4 . 𝛬𝛬 can be positive if𝑐𝑐 > (π‘›π‘›βˆ’1)π‘Žπ‘Ž 3𝑛𝑛 . Thus even if the pressure due to space –time is negative, it is possible to have circular motion of high values of h, for suitable values of m, n and a, as m would overcome the effect of 𝛬𝛬, provided that 𝛬𝛬<3π‘šπ‘š π‘Žπ‘Ž3 as seen from (6.4). Jayakody [5] has also shown that for a ray of light passing at a distance r0 from the centre of the spherically symmetric body the angle of deflection is given by 2Ξ΄ where American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 193 𝛿𝛿 = οΏ½ 2πœ€πœ€ 3π‘Ÿπ‘Ÿ0 + 5πœ‹πœ‹πœ€πœ€2 24π‘Ÿπ‘Ÿ02 οΏ½ οΏ½1 + π‘Ÿπ‘Ÿ02Ξ› 6 βˆ’ 2πœ€πœ€2 9π‘Ÿπ‘Ÿ02 οΏ½ βˆ’1 where Ξ΅ = 3m. Neglecting terms of order higher than 1 of Ξ› and terms of order higher than 2 of πœ€πœ€, Jayakody [5] has shown that 2𝛿𝛿 β‰ˆ 4𝑐𝑐 π‘Ÿπ‘Ÿ0 + 15πœ‹πœ‹π‘π‘2 4π‘Ÿπ‘Ÿ02 βˆ’ οΏ½ 2π‘Ÿπ‘Ÿ0𝑐𝑐 3 + 5πœ‹πœ‹π‘π‘2 8 οΏ½Ξ› This agrees with the Schwarzschild value 4π‘šπ‘š π‘Ÿπ‘Ÿ0 as a first approximation, and also could be seen to be greater than that value when 𝛬𝛬<0. Thus negative 𝛬𝛬 increases the angle of deflection of a light ray passing near a spherically symmetric object. It is possible to obtain values of 2Ξ΄>4π‘šπ‘š π‘Ÿπ‘Ÿ0 even when 𝛬𝛬>0 for suitable values of r0 and m. 7. Conclusion Field equations in General Relativity are written with the cosmological term considered as a variable, on the right hand side, interpreting the corresponding tensor not as a geometric tensor but as the energy momentum tensor of the space – time. With this formulation the space – time itself contributes to the energy momentum that determines the metric tensor. The energy of the space- time and that of the ordinary matter and radiation is conserved as a whole, thus the energy of the space – time is β€œconverted” to other forms of energies, and vice versa. This implies that matter could be created not as in the C – field theory but at the expense of the energy of space – time. This energy is different from the vacuum energy, eliminating the so called coincidence problem and the cosmological constant problem, as 𝛬𝛬is a variable. 𝛬𝛬also gives rise to acceleration of the universe at the present epoch with possible deceleration in the future as οΏ½ΜˆοΏ½π‘… could be negative for certain values of 𝛬𝛬. The energy of the space - time is identified as dark energy. It should be emphasized that it is the energy momentum tensor of the space - time that is important as the acceleration of the universe is caused by the negative pressure due to 𝛬𝛬 in the relevant energy momentum tensor. 𝛬𝛬 is also identified with dark matter, especially when it has negative values. As Cho [13] has mentioned there is little hope for finding WIMPS to explain dark matter and we are of the view that our hypothesis is an answer to the problems discussed in literature. 8. Limitations and recommendations We are limited and constrained by not coming out with a detailed model to explain the expansion of the universe and we recommend the working out of a detailed model and also calculations in regard to weak lensing in connection with dark matter. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No 1, pp 184-194 194 Acknowledgements We thank the Faculty of Science, University of Kelaniya, Sri Lanka, where almost all the work was done when we were there at one time or the other, for providing us all the facilities. It is a pleasure to thank Dr. V. K. Senanayake of the University of Kelaniya for helpful discussions, and Ms. B. B. U. P. Perera and Ms. S. Devadithya of University of Kelaniya and University of Washington, USA, respectively for helping us with computations. References [1] M . D. P Hemantha ,Nalin de Silva,Proceedings of the Annual Research Symposium, University of Kelaniya, 61,2003. [2] M D P Hemantha , Nalin de Silva, Proceedings of the Annual Research Symposium, University of Kelaniya, 55,2004. [3] F, FHoyle., J.V.Narlikar, Proc. R. Soc. (Lond.) A,273, 1,1963. [4] Mc. Vittie, General Relativity and Cosmology, London;Chapman& Hall Ltd,1956. [5] J.A.N.K Jayakody, β€œA study on the effects of the cosmological constant with respect to the Schwarzschild - de Sitter Metric in comparison with the Schwarzschild Metric” MPhil thesis,University of Kelaniya,2013. [6] S. Perlmutter et. Al., Apj, 483, 565,1997. [7] S. Perlmutter et. Al., Perlmutter, S., et al. Nature, 391, 51,1998. [8] A G.Reiss et.al., The Astronomical Journal, 116, 1009,1998. [9] L N K de Silva, Journal of Physical Science and Application, 1 , 184,, 2011. [10] K D W J Katugampala, , β€œSome cosmological models with variable Lamda”M. Phil. Thesis, University of Kelaniya, Sri Lanka,2013. [11] W.Katugampala, and de Silva Nalin,,Proceedings of the Annual Research Symposium, University of Kelaniya, 135,2007. [12] V .Trimble, Annual Rev. of Astron. And Astrop., 25,472,1987. [13] A.Cho, Science 351, 1376, 2016.