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Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26

RESEARCH ARTICLE OPEN ACCESS

Negative theory().

Garmsiri member of the Astrophysical Society of Bushehr Province in Persian golf, dehdashti koi hosseinieh, bushehr state.

ABSTRACT: According to the theory of effective negative mass, published in the journal of physical Review letters, If force is applied to a negative mass That mass is accelerating That it accelerates unlike the direction Force applied. Therefore, based on this theory, the material can have both positive and negative . here we show that negative mass has negative gravity and create curvature in spacetime which basis the shape of the cosmos is after the . The combination of two curves created by positive mass and negative mass basis of the orbits of the planets is around the stars….. ------Date Of Submission: 30-08-2019 Date Of Acceptance: 16-09-2019 ------

I. INTRODUCTION According to the theory of effective The Fields Equation Of Negative Mass negative mass, published in the journal of physical In order to prove that the negative mass in Review letters, If force is applied to a negative spacetime produces a curvature, we know mass That mass is accelerating That it accelerates impact tensor must be related to the spacetime unlike the direction Force applied [1]. Therefore, curvature. We show the energy density flux with based on this theory, the material can have both the symbol ρ and the hydrostatic density positive and negative masses. But the important with p symbol. Energy impact tensor thing is that the acceleration of the positive mass is is as follows: opposed to the acceleration of the negative mass, Tμv = p + ρ uμ uv + pgμv , Thus, according to the theory of general relativity, In the energy impact tensor, the energy density flux , Given the two Hypothesis, he says: and the hydrostatic pressure are the result of a 1) Accelerated mass and gravitational mass are positive mass, and we call it a stress-energy tensor, both one and the same. or a material-energy, which is a positive mass. But 2) The function of the accelerator and the gravity in order to obtain a stress- tensor, device are the same. we must have μv μ v μv Therefore, we can state the theory of gravity of T′ = pnm + ρnm u u + pnm g , negative mass It is proved with these two where μv hypotheses but for a negative mass, we must add T′ : negative energy impact Tensor, two another hypothesis, in general, with the ρnm : Energy density flux of negative mass, following three hypotheses so that: pnm : hydrostatic pressure, 1) we assume that the negative mass has a constant uμ : Four velocity vector, mass (that is, it has size, volume and radius). gμv : Metric, 2) The acceleration that the positive mass takes due Now this negative energy impact tensor to the force applied be equal to the acceleration that should have a linear relationship with the Ricci negative mass due to the same force applied. tensor. If the Covariant derivative is zero for the 3) Accelerated negative mass and gravitational negative energy impact tensor and the Ricci tensor negative mass are both one and the same. is zero, then the problem is solved. But we know 4) The function of the accelerator and the gravity that Albert Einstein in general relativity proved that device are the same. the Covariant derivative material-energy tensor is Therefore, expresses gravitational zero, but the Ricci tensor, with the help of bianchi negative mass theory that the movement of a identity, turned into the Ricci tensor and the Ricci smaller negative mass leads to a larger negative scalar In this case, the covariant derivative for this mass, It is due to the curvature that generates a tensor is zero. So, since Einstein proved that the larger negative mass in spacetime And gravity is new tensor has a zero covariant derivative, here is due to this curvature. (In this paper, for a smaller an negative energy impact tensor with the Einstein negative mass, the negative mass of the substance equalizer, and since Einstein's tensor has a is smaller and for a larger negative mass, the covariant derivative of zero, here we should only negative mass of the substrnce is larger). find the covariant derivative for the negative www.ijera.com DOI: 10.9790/9622- 0909022026 20 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26

μv 1 energy impact tensor to gain. If T′ ,is an negative Rμv − gμv R = −kT′μv , energy impact tensor symbol, we have 2 (2) μv μ v μv T′ = pnm + ρnm u u + pnm g , where where ′μv μ v T :negative impact energy tensor. u u gμv = 1, In the Einstein equation, the constant k is to be μ u = (c, 0,0,0), determined. Earlier, when we obtained the −2 −c 0 0 0 Newoton’s equation in the weak and static field gμv = 0 1 0 0 , approximation, we had found that [5], 0 0 1 0 2ф 2GM g = 1 + = 1 − , 0 0 0 1 44 c2 c2r We calculate the covariant derivative for this tensor (3) is the metric and the Covariant derivative is zero The Newotonian theory in the presence of for the [2] (appendix A). so, gij,k = gives poisson equation [5], 0, 2 ф = 4πGρ , i ∇ We can also the covariant derivative vector u is (4) zero [3] (appendix B). so: ui = 0, and We know that in Newton’s law of gravity so: T′μv = 0, the force of gravity between two positive masses is We want to say that, as if the mass too with the product of two masses direct proportion hight, it creates a curvature in the spacetime, the and with square of distance two masses reverse negative mass if too hight also creates a curvature ratio [6]. This law also applies for a negative mass, in spacetime. Therefore, we proved that the but to make proportions equal to that we must derivative of negative Energ impact tensor is equal multiply the negative mass gravitational constant zero, and derivative of Ricci tensor that Einstein it that’s it showing with symbol Gnm ,(appendix D) to turned to Ricci tensor and scalar Ricci is zero . so, we have so, There should be a linear relationship between Fn1 = Fn2 = Gnm (m*1) (m*2) /( r²). the tow. Therefore should a side be geometrical So, based on the first hypothesis, we can define involving the spacetime metric, Ricci tensor and Newoton’s equation for negative mass, so that we Ricci scalar, and the another side involves the had found that negative Energy- tensor of the matter 2ф′ 2G (M∗) g = 1 + = 1 − nm , with negative mass fields multiplier k , Therefore 44 c2 c2r we use Einstein-Hilbert action, the difference is (5) that we here use Eistein-Hilbert action including where matter with negative mass. Gnm :gravity constant of negative mass. We know Einstein-Hilbert action including matter So, the Newotonian theory in the presence of is [4], matter with negative mass gives poisson equation 2 ′ ф′ = 4πGnm (ρnm ) , I = Ig + Im , ∇ (1) (6) Now if a matter be with negative mass equation (1) where changes to Gnm :gravity constant of negative mass. I = Ig + I(m∗) , Now we define equation (6) gravitational potential of negative mass and show it with ′2ф′ Therefor in continuum mechanics, the action is the ∇ (flat) spacetime integral of a lagrangian negative ,We want to express the gravity theory with the density . so, negative mass, so that the negative mass produces a I = Ig + I(m∗) curvature in spacetime. Multiplying the equation 1 4 4 μv = −gd xR + −gd xℒ , (2) by g μ v and denoting T′ g μ v ,by T', we find R = 2k (m∗) Where k T, Then, this equation can be as 1 R = −k(T′ − g T′ ) , ℒ(m∗) , is the lagrangian negative density for μv μv 2 μv matter with negative mass, (7) We have proved that the negative impact Compare this equation with (6). We know that 4- dxμ energy tensor derivative is zero and We know that velocity uμ = .uμ (x), and also ds the Einstein tensor derivative that obtained from μ v the Ricci and Ricci scalar tensors is zero so, We gμv u u =1 ,Therefore, using the result that the use Einstein’s experience and postulate the metric gμv ,is covariantly constant, we have μ equation for matter with negative mass field g uμ uv = g uμuv + u uv = 2g uμ uv = μv ;λ μv ;λ ;λ μv ;λ (appendix C) 0. . Leading to As www.ijera.com DOI: 10.9790/9622- 0909022026 21 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26

v ρ ρ ρ ς ρ ς uv u;λ = 0. R μ v = ( ∂ ρ Γ μ v − ∂ v Γ μρ + Γ ςρ Γ μ v − Γ ς v Γ ρμ ) , (8) The terms involving the derivatives of Γs are μ Introduce a scalarρnm (x), so that ρnm u , gives that retained. They are, using (13): flow of matter with negative mass and negative ρ 1 ρλ ∂ρ Γμv ≃ η (∂ρ ∂μhvλ + ∂ρ ∂v hμλ − ∂ρ ∂λ hμv ) , i 4 2 density, as ρnm u −g, and ρnm u −g , (16) respectively. Then the conservation of matter with And negative mass is gowerned by ρ 1 ρλ ∂v Γμρ ≃ ∂v ∂μ h , h = η hρλ , ∂ ρ 풳 uμ −g = 0. 2 μ nm (17) (9) So, Expanding 1 R ≃ ηρλ ∂ ∂ h + ∂ ∂ h − ∂ ∂ h − ∂ ρ 풳 uμ −g = ∂ ρ 풳 uμ −g + μv 2 ρ μ vλ ρ v μλ ρ λ μv μ nm μ nm 1 μ ∂ ∂ h , (ρnm 풳 )u ∂μ −g = 0 , . 2 v μ Using (18) ς 1 We have considered weak field approximation. For Γμς = (∂μ −g), . −g static approximation, (16) gives The above equation becomes 1 ρλ 1 2 R44 ≃ η ∂ρ ∂λ h44 = ′ h44. −g ∂ ρ 풳 uμ + Γς ρ 풳 uμ = 2 2 ∇ μ nm μς nm (19) μ −g Dμ ρnm 풳 u = 0 , . Now we use consider (13) in the above That is approximation. In this case uμ = (c, 0,0,0), (ρ 풳 uμ ) = 0. 2 nm :μ and g44 ≃ c ,Then, the (44) component of the (10) right side of (13) is The matter with negative mass has an negative 1 1 −k(ρnm 풳 ) c² − g44 = − k(ρnm 풳 )c², energy density ρ (x)u4u4 −g ,and momentum 2 2 nm Then using (19), we have flux ρ (x)uiui −g ,and if we take 2 nm ′ h44 = −k(ρ x )c2 , μv μ v ∇ nm T′ = ρnm 풳 u u , , (11) 2Φ′ From (5), h = Then −gT′μv , gives the negative energy density 44 c² 4 μv 2 kc and the momentum flux. Thus this T′ , is the And so ′ Φ′ = − (ρnm x ) , ∇ 2 negative energy-momentum tensor for the matter When compared with (6), k is determined as μv with negative mass. We use this T′ 8πG κ = − nm , , ,In equation (2). We have to C4 verify T′μv = 0 And equation (2) becomes 1 8πG T′μv = (ρ 풳 uμ uv ) = uμ (ρ 풳 uv ) + Rμv − gμv R = nm T′μv . ;v nm ;v nm ;v 2 C4 v μ (ρnm 풳 )u u;v , (12) (20) In above, the first term on the right side vanishes The left side (20) is geometrical involving by (10) and the second term vanishes if the the spacetime metric, Ricci tensor and Ricci scalar. elements of the matter with negative mass move Once the spacetime metric is given, the left side μ p along the .(flow u u;μ = 0 , and can be calculated and in this sense it is geometrical, dx μ property of spacetime. The right side involves the uμ = . ) now we use equation (11) we have ds negative energy – momentum tensor of the matter μ v μ v T′=gμv {ρnm 풳 u u } = ρnm 풳 sincegμv u u = with negative mass fields (nongravitational). 1 , Then (7) becomes 1 CNCLUDE Rμv = −k(ρnm 풳 ) uμ uv − gμv , 2 The substance has two positive and (13) negative masses. Given Einstein's relationship, As we want to compare (7) with (6) with later in (Assuming space time if flat) If a positive mass the Newtonian case, we consider weak increases, it will curvature in spacetime (figure 1). gravitational field by expanding gμv , as We also proved that the negative mass also μv μv μv gμv ≃ ημv + hμv ; g ≃ η − h , increases, It causes curvature in spacetime (fig 2). (14) Assuming space time if flat, The curvature of Then, spacetime in Einstein's relationship is due positive 1 Γμ = gμλ Γ = gμλ ∂ g + ∂ g − ∂ g ≃ mass, Different from the curvature of spacetime by vρ vρ,λ 2 v ρλ ρ vλ λ vρ 1 negative mass according to equation (20) Because ημλ ∂ h + ∂ h − ∂ h . (15) 2 v ρλ ρ vλ λ vρ the acceleration that takes two positive and In the expression for negative masses of a substance is in the opposite direction. Consequently, the curvature of spacetime www.ijera.com DOI: 10.9790/9622- 0909022026 22 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26 by matter, consequence is curvature of spacetime a positive mass and a negative mass. This means that the combination of these two curves together, Causes cut off the space-time lines of positive mass and negative mass. The result of this intersection is a groove that Surrounded by a material that has curvature in spacetime. Here is an example of Earth and the sun. We know that the mass of the sun is greater than the mass of the earth Therefore, the curvature of the space-time created by the sun is more than Earth When the Earth is on the slope of the curvature of the Sun's spacetime The motion of the earth toward the sun is the same as the positive mass motion in the gravitational field of the sun At the same time, the negative mass of the Earth moves to the gravitational field by the negative mass of the sun, Two positive and negative masses of the earth go up to the ridge. if the mass be large, the groove getting closer.( so, the mass that going up on the slope, it causes the groove to move toward itself that this displacement depends on the FIG 2. space-time curvature is created with size of the mass.) But in the groove, Positive mass negative mass and this slop same gravity of center encounters an obstacle Which is the curvature of negative mass.(garmsiri). space-time generated by a negative mass And the negative mass also hits the barrier That is, the curvature of spacetime is a positive mass Here, two positive and negative masses can only have rotational motion Which is the transition of the earth around the sun But the movement of the positive and negative masses is in the opposite direction. Finally, the groove is the planetary orbit that orbiting around the stare. (fig 3).

FIG 3. spacetime that is black (color online), created by positive mass and located a larger mass at the center, and a smaller mass at the slope of curvature. spacetime that is blue (color online), created by negative mass and located a larger negative mass at the center, and a smaller negative mass at the slope of curvature. A combination of two spacetimes, causes two grooves around negative and positive mass becomes central and small masses fall into these grooves, so that the small negative mass in the lower groove rotates around the central negative mass and the small positive mass in the upper groove rotates around the central pocitive mass. The direction of rotation of the two smaller masses is opposite to each other.(garmsiri).

ACKNOWLEDGMENTS Here are not financial support.

APENDICES FIG 1. space-time curvature is created with positive mass and this slop same gravity of center mass.(garmsiri).

www.ijera.com DOI: 10.9790/9622- 0909022026 23 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26

훍퐯 Appendix A: We have 훅퐠 = 퐠퐠 훅퐠훍퐯 ,and so We calculate the covariant derivative for this ퟏ 훍퐯 훅 −퐠 = −퐠퐠 훅퐠 , tensor is the metric and the Covariant derivative ퟐ 훑퐯 (C6) is zero for the metric tensor. so: 훍퐯 훍 훍퐯 퐩 퐩 And know that form 퐠 퐠퐯훒 = 훅훒, 훅퐠 퐠퐯훒 = 퐠퐢퐣,퐤 = 퐠퐢퐣,퐤 − 퐠퐩퐣훕 − 퐠퐢퐩훕 , 퐢퐤 퐣퐤 훍퐯 (A1) − 퐠 훅 퐠 퐯 훒 , from which we have 퐩 퐩 훍퐯 훍훒 훔퐯 훅퐠 = −퐠 퐠 훅퐠훒훔 , 퐠퐩퐣훕퐢퐤 = 퐢퐤, 퐣 , 퐠퐢퐩훕퐣퐤 = 퐣퐤, 퐢 , (A2) (C7) 퐠 = 퐢퐤, 퐣 + 퐣퐤, 퐢 , Substituting (C6) and (C7) i n (C5), we find 퐢퐣,퐤 훅퓛 = (A3) 퐠 ퟏ ퟏ −퐠퐑퐠훍퐯 훅퐠 − −퐠퐠훍훒퐠퐯훔퐑 훅퐠 + 퐢퐤, 퐣 + 퐣퐤, 퐢 − 퐢퐤, 퐣 − 퐣퐤, 퐢 = ퟎ , ퟐ 퐤 ퟐ 훍 퐯 훍 퐯 훒훔 (A4) −퐠퐠훍퐯훅퐑훍퐯, (C8)

Appendix B: Now We need to examine훅퐑훍퐯 , We know that 퐝퐱훍 훒 훒 훒 훔 훒 훔 We know that 4-velocity 퐮훍 = .퐮훍 (퐱), and 퐑훍퐯 = 훛훒횪훍퐯 − 훛퐯횪훍훒 + 횪훔훒횪훍퐯 − 횪훔퐯횪훒훍 , 퐝퐬 also (C9) 퐠 퐮훍 퐮퐯 =1 , Therefore, using the result that 훍퐯 And so the metric 퐠훍퐯 ,is covariantly constant, we have 훒 훒 훒 훔 훍 퐯 훍 퐯 훍 퐯 훅퐑훍퐯 = 훛훒 훅횪훍퐯 − 훛퐯 훅횪훍훒 + 훅횪훔훒 횪훍퐯 + 퐠훍퐯퐮 퐮 = 퐠훍퐯 퐮 퐮 + 퐮 퐮 = ;훌 ;훌 ;훌 횪훒 훅횪훔 − 훅횪훒 횪훔 − 횪훒 훅횪훔 . (C10) ퟐ퐠 퐮훍퐮퐯 = ퟎ, 훔훒 훍퐯 훔퐯 훒훍 훔퐯 훒훍 훍퐯 ;훌 We help from (C7), we have (B1) 훅횪훒 = 훅 퐠훒훌횪 , Leading to 훍퐯 훍퐯,훌 퐯 = 훅퐠훒훌 횪 + 퐠훒훌 훅횪 , 퐮퐯퐮;훌 = ퟎ. 훍 퐯 , 훌 훍 퐯 , 훌 훒훌 훔 훒훌 (B2) = 훅퐠 퐠훌훔횪훍퐯 + 퐠 훅횪훍퐯,훌 , 훒훂 훃훌 훔 훒훌 Appendix C: = −퐠 퐠 훅퐠훂훃 퐠훌훔횪훍퐯 + 퐠 훅횪훍퐯,훌 , ퟏ We know Einstein-Hilbert action including = −퐠훒훂 훅퐠 횪훔 + 퐠훒훌 훛 훅퐠 + matter is 훂훔 훍퐯 ퟐ 훍 퐯훌 훛퐯훅퐠훍훌−훛훌훅퐠훍퐯. (C11) 퐈 = 퐈퐠 + 퐈퐦 , (C1) Now if a matter be with negative mass equation Consider ퟏ (C1) changes to 퐠훒훌 훅퐠 + 훅퐠 − 훅퐠 , ퟐ 퐯훌 ;훍 훍훌 ;퐯 훍퐯 ;훌 퐈 = 퐈퐠 + 퐈(퐦∗) (C12) Therefor in continuum mechanics, the action is the (flat) spacetime integral of a lagrangian As퐠훍퐯 is a covariant tensor of (rank 2),훅퐠훍퐯 is negative density . so, also a covariant tensor of rank 2. Then, using 퐈 = 퐈퐠 + 퐈(퐦∗) , the covariant derivative of rank two covariant ퟏ = −퐠퐝ퟒ퐱퐑 + −퐠퐝ퟒ퐱퓛 , tensors. So the above expression is ퟐ퐤 (퐦∗) ퟏ (C2) 퐠훒훌 훛 훅퐠 − 횪퐩 훅퐠 − 횪퐩 훅퐠 + ퟐ 훍 퐯훌 퐯훍 퐩훌 훌훍 퐩퐯 Where 훛퐯훅퐠훍훌−횪훍퐯퐩훅퐠퐩훌−횪훌퐯퐩훅퐠퐩훍−훛훌훅퐠훍퐯+횪훍훌 퓛(퐦∗) ,is the lagrangian negative density for 퐩훅퐠퐩퐯+횪퐯훌퐩훅퐠퐩훍=−퐠훒훌횪훍퐯퐩훅퐠퐩훌+ퟏퟐ퐠훒훌훛훍 matter with negative mass, 훅퐠퐯훌+훛퐯훅퐠훍훌−훛훌훅퐠훍퐯. We vary the above action with respect to the metric퐠훍퐯(퐱) ,and write ퟏ 훅퐈 = 퐝ퟒ퐱훅 −퐠퐑 + 훅 −퐠퐝ퟒ퐱퓛 , Substituting the expression in the curly bracket ퟐ퐤 (퐦∗) ≡ 퐝ퟒ퐱훅퓛 + 훅 −퐠퐝ퟒ퐱퓛 , in the right side of the above in (C11), We see 퐠 (퐦∗) ퟏ 훅횪훒 = 퐠훒훌 훅퐠 + 훅퐠 − 훅퐠 , (C3) 훍퐯 ퟐ 퐯훌 ;훍 훍훌 ;퐯 훍퐯 ;훌 Whit (C13) Expression 훅횪훒 , in terms of the covariant ퟏ 훍퐯 훒 훅퓛퐠 = 훅 −퐠 퐑 + −퐠훅퐑 , ퟐ퐤 derivatives. Thus 훅횪훍퐯 , is a mixed tensor of (C4) rank 3. It is to be noted that thought횪훒 , is not a 훍퐯 훍퐯 Since 퐑 = 퐠 퐑훍퐯, we have 훒 ퟏ mixed tensor of rank 3,훅횪훍퐯 , is. This result 훅퓛 = 훅 −퐠 퐑 + −퐠 훅퐠훍퐯 퐑 + 퐠 ퟐ퐤 훍퐯 allows us to counsider the covariant derivative −퐠퐠훍퐯훅퐑훍퐯, (C5) 훒 of 훅횪훍퐯 , as the covariant derivative of a mixed rank 3. Then it easy to see that www.ijera.com DOI: 10.9790/9622- 0909022026 24 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26

훔 훔 (훅횪훍퐯);훔 − (훅횪훍훔);퐯 = 훅퐑훍퐯. Diameter of circle = AB, ab , (C14) We have from term of 3 in the right side Circle center = S, s , of equation (C8) Two particles located along the diameter = P, p , 훍퐯 훍퐯 훔 The radius of negative gravity force emitted to the −퐠퐠 훅퐑훍퐯 = −퐠퐠 (훅횪훍퐯);훔 − particle = (PK, PL), (pk, pl), (훅횪훍훔훔);퐯=−퐠(퐠훍퐯훅횪훍퐯훔);훔−(퐠훍퐯훅횪훍훔훔);퐯, (C15) As can be seen in the figure, when the negative gravity force radius are emitted to the particle, 훍퐯 훍퐯 훔 −퐠퐠 훅퐑훍퐯 = 훛훔 −퐠퐠 훅횪훍퐯 − some arcs are obtained on the circle: The arcs obtained from circles = HK, hk, IL, il , 훛 −퐠퐠훍퐯 훅횪훔 , 퐯 훍훔 Hk = hk, . IL = il, Which isa total divergence (the difference of (SE) ┴ (PL), (se) ┴ (pl), total divergences). In the variation of the action (SD) ┴ (PK), (sd) ┴ (pk), integral, such terms vanish when we integrate (IQ) ┴ (AB), (iq) ┴ (ab), over all spacetime. Or, such total divergences (IR) ┴ (PK), (and) ┴ (pk), will not contribute to the classical equation of Location of intersection = F, f , motion. Then the remaining terms in (C5) are Now if the following lines which collide the 훅퓛퐠 = particle at an angle, simultaneously approach zero, ퟏ ퟏ 훍퐯 훍훒 퐯훔 we will have −퐠퐑퐠 훅퐠훍퐯 − −퐠퐠 퐠 퐑훍퐯 훅퐠훒훔 . ퟐ퐤 ퟐ (PL, pl) → 0, (PK, pk) → 0, −퐠 ퟏ = 퐠훍퐯퐑 − 퐑훍퐯 훅퐠 . Because, ퟐ퐤 ퟐ 훍퐯 With this, the variation of action becomes (DS, ds) = (ES, es), ퟏ ퟏ The following lines can be considered equal, 훅퐈 = −퐠 퐠훍퐯퐑 − 퐑훍퐯 훅퐠 퐝ퟒ퐱 + ퟐ퐤 ퟐ 훍퐯 (PE, PF) = (pe, pf), ퟒ 훅 −퐠퓛(퐦∗)퐝 퐱. DF = df, We introduce the’negative energy – momentum Because, as said above, when the following angles tensor’퐓′훍퐯 , for the matter with negative mass simultaneously approach zero, their proportions fields by eventually equate to each other. ퟒ ퟏ 훍퐯 ퟒ Angle DPE → 0, 훅 −퐠퓛(퐦∗)퐝 퐱 ≡ − 퐓′ (훅퐠훍퐯) −퐠퐝 퐱. ퟐ Angle dpe → 0, So that for arbitrary variation the classical So, considering all the conditions stated above, we equations are given by ퟏ can write: 퐑훍퐯 − 퐠훍퐯퐑 = −퐤퐓′훍퐯 ퟐ (PI) /( PF) =( RI) /( DF), (Pf) /( pi) =( DF) /( ri), . By multiplying the two above terms and Appendix D: simplifying the product, we have we can by supposing tow negative (PI.pf) / (PF.pi) =( RI) /( ri) = arc (IH) / arc (ih), masses and it is proved that force of gravity (D1) between tow negative mass, by the This means that the negative gravity force emitted multiplication of tow negative masses, the ratio from the surface of the sphere affects the particles is direct and with distance square of tow masses outside the surface of the sphere. It is directly the ratio is indirect. proportional to the surface of the sphere. Again, we can have (PI) /( PS) =( IQ) /( SE), (Ps) /( pi) = (SE) /( iq), By multiplying these two terms and simplifying its product, we have (PI.ps) / (PS.pi) =( IQ) /( iq) , (D2) By multiplying equations (D1), and (D2), we have (PI².pf.ps) / (pi².PF.PS) = (HI.IQ) / (ih.iq), From the above result, it can be said that the arc of the spherical surface, where the negative gravity force emitted from the surface of the sphere to the FIG 4. (garmsiri). particle, is inversely proportional to the squared

distance between the sphere surface from which the there are two spheres, then, according to above (fig negative gravity force emitted and the particle. 4), the following assumptions are available: Since the negative gravity force is directly Two spherical surfaces = AHKB, ahkb ,

www.ijera.com DOI: 10.9790/9622- 0909022026 25 | P a g e

Garmsiri Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622 Vol. 9,Issue 9 (Series -II) September 2019, pp 20-26 proportional to the arc, as proved before (Eq.(D1)), the negative gravity force is inversely proportional to the distance between the sphere surface and the particle. Then the negative gravity force between the two masses can be equation as follows: 2 Fn1=Fn2= Gnm (m*1) (m*2) /( r ), where Gnm : Negative gravity constant, Fn : negative gravity force, M*1: Negative mass 1, M*2: Negative mass 2, r : The distance between the center of negative mass 1 and the center of negative mass 2.

REFERENCES [1]. sci_news.com [2]. Joshi A. W, 1995, Matrices and Tensors in physics, Publisher: (New york: wiley). [3]. Parthasarthy, R. introduction to general relativity, alpha science international Ltd. Oxford, U.K. ISBN 978-1-84265-949-6. [4]. Parthasarthy, R. introduction to general relativity, alpha science international Ltd. Oxford, U.K. ISBN 978-1-84265-949-6. [5]. Parthasarthy, R. introduction to general relativity, alpha science international Ltd. Oxford, U.K. ISBN 978-1-84265-949-6. [6]. Newton, I, 1803, Ed., the mathematica principhes of natural philosophy (the principia), proposition LXXI, theorem XXXI, pag 812 – 817, this edition of sir isac newton’s principia, translated into English by andrew motte, in three volumes.

Garmsiri" Negative mass gravity theory (general relativity)." International Journal of Engineering Research and Applications (IJERA), Vol. 09, No.09, 2019, pp. 20-26

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