The Effect of Modified Gravity on Solar System Scales

Transcription

1 The Effect of Modified Gavity on Sola System Scales Dane Pittock Physics Depatment Case Westen Reseve Univesity Cleveland, Ohio USA May 3, 013

2 Abstact Duing my senio poject, I have exploed the effects of Modified Gavity on Sola System scales. Infaed modifications of gavity ae typically expected to modify Newton s law on vey lage distance scales, but such theoies usually come hand in hand with additional degees of feedom which could aleady povide smoking gun signatues on much smalle distance scales. I have paid special attention to the effects of the acceleation of the Univese on small distance scales. Duing this senio poject I have familiaized myself with standad techniques of Geneal Relativity and applied them to known systems and obsevations. As a second stage I have woked out how to paameteize modifications of gavity on sola system scales and deived some elevant obsevable signatues.

3 Contents 1 ackgound and Liteatue Review 1 Geneal Relativity Solution 3.1 Deivation of Geneal Relativity Solutions Spheically Symmetic Vacuum Solution Schwazschild de-sitte Solution Geodesics Massive Gavity Solution Dynamic Metic and Einstein Tenso Refeence Metic and Stess Enegy Tenso Discussion and Review 18 5 Acknowledgements 19 6 Engineeing Physics Appendix 0 1 ackgound and Liteatue Review Thee ae fou know fundamental foces in physics: the stong foce, the weak foce, electomagnetism, and gavity. These foces ae mediated by foce-caying paticles. The stong foce is mediated by gluons, the weak foce by the W and Z bosons, and electomagnetism by the photon. A foce caying paticle fo gavity has not yet been discoveed. Howeve, we can theoetically detemine this paticle s chaacteistics based upon ou knowledge of gavitation. The theoetical paticle believed to be the foce-caie fo gavity is the gaviton. This paticle is believed to be massless, have no chage, and is expected to be a spin- boson. Geneal elativity is the theoy which govens the gaviton, and it ecoves Newtonian gavitation in the weak-field limit. Howeve, key issues have been found concening the integation of the theoy of geneal elativity into 1

4 moden paticle physics. Most pominent, fom obsevations of supenovae, data has sufaced which implies a late-time acceleation in the expansion of ou univese [1]. This implies a constant dak enegy density, believed to be due to a cosmological constant poviding a epulsive gavitation. It would be easonable to believe that this cosmological constant was the vacuum enegy density pedicted by paticle physics, howeve a cosmological constant that would povide the epulsive gavity necessay to fuel this acceleated expansion is extemely small compaed to the vacuum enegy density expected fom paticle physics []. We can attempt to solve this poblem by modifying gavity itself, whee new degees of feedom ae intoduced into the gavitational field that lead to the expected acceleated expansion of the univese on lage distance scales. y intoducing a non-zeo value fo the mass of the gaviton, these additional degees of feedom ae given to the gaviton. Fiez and Pauli FP) fist studied massive gavity at the linea level [5]. The added mass tem beaks the gauge invaiance of geneal elativity, leading to five degees of feedom in the gaviton, as opposed to the two fom geneal elativity. An effective theoy appoach estoes the gauge invaiance by intoducing additional Stuckelbeg fields that epesent the helicity-0 mode of the gaviton [6]. Once depating fom the lineaized theoy, howeve, massive gavity has un into issues with ghosts at the non-linea level such as the one discoveed by oulwae and Dese [7] and continuity issues. Fiez and Pauli s theoy did not ecove geneal elativity in the limit whee the mass of the gaviton educes to zeo. This became known as the van Dam, Veltman, Zakhaov discontinuity [8], [9]. A ecent theoy by de Rham, Gabadadze, and Tolley drgt) has been fomulated that is ghost-fee at the fully non-linea level [10], [11]. Hee geneal elativity is ecoveed in the massless limit via a Vainshtein mechanism [1]. The appoach to massive gavity that will be utilized in this poject stems fom drgt. While these modifications ae meant to modify gavity on lage distances, we may be able to obseve these additional degees of feedom on smalle scales. y compaing pecisely known values in ou sola system to geneal elativity and this theoy of massive gavity, we can detemine the accuacy of both theoies on small scales. The goal of this poject was to study the accuacy of geodesic paths of paticles in theoetical gavitational fields of the sola system as defined by geneal elativity and massive gavity. Once the geodesic equations fo massive gavity and geneal elativity have been deived, they can be

5 applied to eal sola system tests, such as the pecession of the peihelion of Mecuy, the bending of light aound the sun, etc. It is expected to see massive gavity ecove geneal elativity in the limit that the mass of the gaviton is zeo. Geneal Relativity Solution The fist section of this poject was to ceate a model of the sola system in geneal elativity, with which the geodesics equations of motion) of massive and massless paticles could be deived and then compaed to geodesics of paticles in the massive gavity solution. The model of the sola system in geneal elativity would also go on to become the efeence metic fo the massive gavity solution, which will be explained moe in Section 3..1 Deivation of Geneal Relativity Solutions.1.1 Spheically Symmetic Vacuum Solution We began with the simple assumption that the solution would be spheically symmetic and due to a point souce. The mass of the sun, much lage than the mass of the est of the sola system, dominates the gavitational field, esulting in a spheically symmetic field oiginating fom the sun. The sun s adius is negligible compaed to the adius of the sola system, and we could appoximate it as a point souce. A spheically symmetic solution of Einstein s field equations is defined as a spacetime with otational invaiance. This means otations aound any adial diection will not change the metic. We can use Killing vectos to build a definition fo the metic. In a fou dimensional spacetime, this gives ise to thee Killing vectos: R = y x + x y 1) S = z x + x z ) T = z y + y z 3) Whee R is a otation about the z-axis, S is about the y-axis, and T is about the x-axis. In spheical 3

6 coodinates, these vectos ae: R = sin θ φ 4) S = cos φ θ sin θcot θsin φ φ 5) T = sin φ θ sin θcot θcos φ φ 6) These thee Killing vectos must solve the Killing equation: ξ a a g µν + µ ξ a )g aν + ν ξ a )g aµ = 0 7) Whee ξ a is a Killing vecto. This gives us the components fo ou metic. The non-zeo metic components ae: g aa = g aa a, b) 8) g ab = g ab a, b) 9) g ba = g ab a, b) 10) g bb = g bb a, b) 11) g θθ = a, b) 1) g φφ = sin θa, b) 13) The geneal metic is: ds = g aa a, b)da + g ab a, b)dadb + g bb a, b)db + a, b)dθ + a, b)sin θdφ 14) We can use some ticks to simplify the metic to something easie to wok with. y changing the vaiable b to, the adial component, the metic becomes: ds = g aa a, )da + g a a, )dad + g a, )d + a, )dθ + a, )sin θdφ 15) y changing the vaiable a to t, o time: dt = t t da + d 16) a t ) da t ) t ) t ) d dt = + dad + 17) a a 4

7 The simplified metic is: ds = g tt t, )dt + g t, )d + dθ + sin θdφ 18) Now, we can apply standad geneal elativity pactices to find the values of g tt and g t. We begin by finding the Chistoffel symbols fom the metic using: Γ λ µν = 1 gλσ µ g νσ + ν g σν σ g µν ) 19) The non-vanishing Chistoffel symbols fo the spheically symmetic vacuum solution ae: Γ t tt = 1 gtt t g tt 0) Γ t t = 1 gtt g tt 1) Γ t = 1 gtt t g ) Γ tt = 1 g g tt 3) Γ t = 1 g t g 4) Γ = 1 g g 5) Γ θθ = g 6) Γ φφ = sin θg 7) Γ θ θ = 1 8) Γ θ φφ = cos θsin θ 9) Γ φ φ = 1 30) Γ φ θφ = cot θ 31) The next step is to find the Ricci tenso using: R µν = ρ Γ ρ νµ ν Γ ρ ρµ + Γ ρ ρλ Γλ νµ Γ ρ νλ Γλ ρµ 3) ut fist, we know that in a vacuum, the Ricci tenso vanishes: R µν = 0 33) 5

8 y solving R t, it is discoveed: t g = 0 34) The metic component g does not depend on time. This also poves the metic component g tt does not depend on time eithe, afte coecting the Chistoffel symbols the Ricci tenso no longe has any t g tt components. With no moe time dependence, we can see the metic is invaiant unde time tanslation. The coected Ricci tenso components ae g µν epesents patial deivative with espect to ): R tt = g ttg g tt + g g tt ) g tt g tt + g tt) ) R = R θθ = 1 g ) 4 + g tt g tt + g + g g 4gg 35) tt ) g tt ) g ttg tt gtt 36) 4g + g ) tt g tt 37) g R φφ = R θθ sin θ 38) All othe components of the Ricci tenso ae tivial. Solving the Ricci tenso by applying 33), the final metic components ae found: g = ) 1 39) C g tt = C ) 40) C In the weak gavitational field limit, they ae expected to esemble Newton s theoy of gavity. We can use this classical gavitation theoy to detemine values fo C 1 and C : g = 1 GM ) 1 41) g tt = 1 GM ) 4) Whee M is the mass of the point souce the sun) and G is the gavitational constant. Thus, the spheically symmetic solution is the Schwazschild solution, with the metic: ds = 1 GM ) dt + 1 GM ) 1d + dθ + sin θdφ 43) 6

9 .1. Schwazschild de-sitte Solution The Schwazschild metic deived in.1.1 took the following fom: ds = f)dt + f) 1 d + dθ + sin θdφ 44) In the vacuum solution, f) was seen to be: f) = 1 GM 45) Now, the solution will be expanded to incopoate a cosmological constant. We can begin with the same standad geneal elativity pactices as befoe The non-vanishing Chistoffel symbols ae: Γ t t = f f Γ tt = ff Γ = f f 46) 47) 48) Γ θθ = f 49) Γ φφ = fsin θ 50) Γ θ θ = 1 51) Γ θ φφ = cos θsin θ 5) Γ φ φ = 1 53) Γ φ θφ = cot θ 54) The next step is to apply 3), and find the non-zeo components of the Ricci tenso: R tt = ff + ff R = f f f f 55) 56) R θθ = 1 f f 57) R φφ = R θθ sin θ 58) The cosmological constant now comes into play with the stess-enegy tenso, T µν. The connection between the cuvatue of spacetime and the stess-enegy tenso we will use in this situation is: R µν = 8πGT µν 1 g µνt ) 59) 7

10 Hee the cosmological constant, Λ, is descibed as a pefect fluid. This lead to a stess-enegy tenso like this: T ν µ = Λ Λ Λ Λ 60) Contacting the indices with the metic gives us this: Λf Λ f 0 0 T µν = 0 0 Λ sin θλ y taking the tace of 60), we find: 61) T = T µ µ = 4Λ 6) Now we have eveything we need to solve 59) fo f). It tuns out to be: f = c 1 + c 8πG Λ 3 63) In the weak field limit, this will esemble Newton s theoy of gavity just like in the vacuum solution) with a new cosmological constant tem: f = 1 GM 8πG Λ 3 64) Defining the Hubble paamete as: H 8πGΛ 3 65) Gives us this final vesion of f): f = 1 GM H 66) This metic, known as the Schwazschild de-sitte metic, is now solved fo: ds = 1 GM H )dt + 1 GM H ) 1 d + dθ + sin θdφ 67) 8

11 . Geodesics Now that we have the Schwazschild de-sitte metic eady to be tested, the next step is to find the equations of motion, o geodesics fo the geneal elativity solution. We can find the govening equations of motion fo both massive, such as a planet, and massless, such as a photon, paticles in the gavitational field of the sun. The geodesic equation is: d x µ dλ + dx ρ dx σ Γµ ρσ dλ dλ = 0 68) Fo null, o massless geodesics, the nom of the tangent vectos will be zeo with espect to an affine paamete. Note how the nom of the tangent vectos is the metic: g µν dx µ dλ Fo massive geodesics, the nom of the tangent vectos will be negative: dx ν dλ = 0 = ds 69) g µν dx µ dλ dx ν dλ < 0 70) ds < 0 71) And with espect to pope time: g µν dx µ dτ dx ν dτ The Lagangian fo the null geodesics will equal 0. = 1 7) The Lagangian is the metic divided by an affine paamete: The Lagangian fo massive geodesics will be 1: L = ds dλ = 0 73) L = ds = 1 74) dτ Utilizing the Eule-Lagange fomula, the equations of motion can be obtained. The conseved quantites fom the equations of motion ae below. We chose to keep θ at π, fo the metic is otationally invaiant and keeping a consistent value of θ simplified the wok. L is the angula momentum, E is the enegy, and 9

12 L is the Lagangian: θ = π 75) L = φ 76) E = 1 GM H )ṫ 77) E L = 1 GM H + ṙ 1 GM H + L 78) This leads to the geodesic equation fo massless paticles in the geneal elativity solution: d ) E 4 = dφ L 1 GM H ) 79) And the geodesic equation fo massive paticles in the geneal elativity solution: d ) E 4 = dφ L 1 GM ) H ) 4 L 80) 3 Massive Gavity Solution Now, we will caefully examine the possibility fo the gaviton mass to be non-zeo. We will take the Schwazschild de-sitte solution fom.1. and ceate with it a efeence metic to base the new, dynamic metic on. This efeence metic will epesent the matte in the system the sun and the cosmological constant), and is necessay to ceate the potential due to the gaviton mass late on. This potential will be a scala function of both the efeence metic and the dynamic metic. 3.1 Dynamic Metic and Einstein Tenso Let s stat by ceating the dynamic metic: ds = C)dt + A)d + D)dtd + )dθ + )sin θdφ 81) This is the most geneic spheically symmetic metic. We cannot use the same ticks as we did with the geneal elativity solution to simplify it, as we have boken the gauge invaiance. We will need to find the 10

13 Einstein tenso to solve fo A,, C, and D. As a matix, the dynamic metic looks like: C) D) 0 0 D) A) 0 0 g µν = 0 0 ) )sin θ 8) Now we can see that this metic is not diagonal like the Schwazschild metic was. Theefoe, the inveted metic must be explicitly calculated fo the Chistoffel Symbols: g µν = A) ) D) ) D) ) 0 0 C) ) ) )sin θ 83) Whee ) = A)C) + D) ). The calculated Chistoffel Symbols fo the dynamic metic, using 19), ae: Γ t tt = D)C ) Γ t t = Γ t t = A)C ) Γ t = D)A ) A)D ) Γ t θθ = D) ) Γ t φφ = D) )sin θ Γ tt = C)C ) Γ t t = Γ t t = D)C ) Γ = C)A ) D)D ) Γ θθ = C) ) Γ φφ = C) )sin θ Γ θ θ = Γ θ θ = ) ) 84) 85) 86) 87) 88) 89) 90) 91) 9) 93) 94) Γ θ φφ = cos θsin θ 95) 11

14 Γ φ φ = Γφ φ = ) ) 96) Γ φ θφ = cot θ 97) Applying 3), the components of the Ricci tenso ae: C) ) )C ) ) A) C ) ) + D)C )D ) R tt = 4) ) ) D) ) C ) + C) A )C ) A)C ) ))) 4) ) ) 98) D) ) )C ) ) A) C ) ) + D)C )D ) R t = 4) ) ) D) ) C ) + C) A )C ) A)C ) ))) 4) ) ) 99) R = 1 ) ) ) ) ) + ) C)A )+D)D ) ) ) ) ) 4 ) A) A) C ) ) + C) A )C ) A)C ) ) + D) C )D ) D)C ) )) ) + ) 100) ) R θθ = 4 A) ) ) ) 4 ) A C) + 4 D) + C) ) ) + C)D) )D ) 4 ) ) D) ) ) )C ) + C) ) A)C) 8 D) ) ) + )C ) + C) ) 4 ) ) 101) R φφ = sin θ R θθ 10) We will appoach the Einstein tenso, G µν, though: G µν = R µν 1 Rg µν 103) The Ricci scala, R, is calculated as a contaction of the invese dynamic metic and the Ricci tenso: R = R µ µ = g µν R µν 104) 4 A) ) ) C) ) + C) D) ) ) ) + ) D) ) 4 + C) ) A ) ) R = ) ) ) ) +C)D) )D ) D) ) )C ) + C) )) ) C) ) + A) ) ) ) ) ) ) 1

15 )C) 4 D) ) ) + )C ) + C) ) + ) ) C ) ) ) ) C)C ) ) ) ) ) + ) ) ) )) C)A )C ) + D) C )D ) D)C ) ) ) ) ) 105) Finally, the G tt and G t Einstein tenso components ae: C) 4 A) ) ) ) ) C) + C) D) ) ) + A)C) C) ) ) G tt = 4 ) ) ) ) +) 8 D) ) 4C) )) ) + ) D) ) 4 ) A + C) ) ) + C)D) )D ) 4 ) ) ) ) D) ) ) )) )C ) + C) ) 4 ) ) ) 106) ) D) 4 A) ) ) ) ) C) + C) D) ) ) + A)C) C) ) ) G t = 4 ) ) ) ) +) 8 D) ) 4C) )) ) + ) D) ) 4 ) A + C) ) ) + C)D) )D ) It is impotant to notice: 4 ) ) ) ) D) ) ) )) )C ) + C) ) 4 ) ) ) ) 107) D)G tt + C)G t = 0 108) This means: D)T U tt + C)T U t = 0 109) Though: T U µν = G µν 110) This simplifies the equation to a elationship between two components of the stess-enegy tenso that is ceated due to the potential of the massive gaviton. Denoted T U µν so as not to be confused with the stessenegy tenso seen in the deivation of the Schwazschild de-sitte solution. 13

16 3. Refeence Metic and Stess Enegy Tenso In ode to find T U µν and solve fo the metic components A,, C, and D, we must study the efeence metic. Moe specifically, we must look at the elationship between the efeence metic and the dynamic metic. The efeence metic as deived in.1., now in matix fom: f µν = Fo convenience, we will define: 1 GM H ) GM H ) sin θ 111) 1 GM H ) = F ) 11) Now we will delve into the elationship between the efeence metic and dynamic metic. The contaction of the invese dynamic metic and the efeence metic will be cucial in the constuction of the potential. g µα f αν = AF DF D F 0 0 C F ) Moe specifically, the squae oot of this contaction is the vital piece of infomation that we need: α β 0 0 γ δ 0 0 g µα f αν = ) The oots α, β, γ, and δ wee solved fo, and thei values ae listed below. We now define M), in an attempt to keep the equations easie to ead: M) = C ACF 4D F + A F 4 115) α = C + AF C+AF M) M F + C + AF + M) M 14 C+AF +M F 116)

17 β = C + AF M) C + AF C+AF + M) M F 4 DF M C AF + M) C + AF + M) 4 DF M γ = DF C+AF M F C+AF +M F 117) DF C+AF +M F 118) M δ = C + AF C+AF + M) M F + C AF C+AF + M) +M F M We can now constuct the potential due to the massive gavity fom hee. The K tems ae connections fom the efeence metic and dynamic metic that the potential U is built out of. α 3 and α 4 ae fee paametes. 119) K ν µ g, f) = δ ν µ g µα f αν 10) U =! 1 [K] [K ] ) 11) U 3 = 3! 1 [K] 3 3[K][K ] + [K 3 ] ) 1) U 4 = 4! 1 [K] 4 6[K ][K] + 8[K 3 ][K] + 3[K ] 6[K 4 ] ) 13) Ug, f) = U + α 3 U 3 + α 4 U 4 14) The Lagangian fo massive gavity is the Lagangian fo geneal elativity plus a new tem due to that potential: This leads to the stess-enegy tenso due to the potential: L = M P l g R + m Ug, f) ) + L M 15) Tµν U = m K µν [K]g µν ) 1 + α 3 Kµν [K]K µν + 1 g µν [K] [K ] )) + α 3 + α 4 ) K 3 µν [K]K µν + 1 K µν [K] [K ] ) 1 6 g µν [K] 3 3[K][K ] + [K 3 ] ))) 16) Now, we find all of the tems. The K µ ν tems ae found to be: Kt t = 1 α 17) K t = β 18) Kt = γ 19) K = 1 δ 130) 15

18 Kθ θ = 1 131) K ψ ψ = 1 13) The tenso K µν comes fom g µα K α ν. Its tems ae found to be: K tt = C + Cα Dγ 133) K t = Cβ + D Dδ 134) K t = D Dα Aγ 135) K = Dβ + A Aδ 136) K θθ = 137) K ψψ = K θθ sin θ 138) The tenso K µν comes fom the contaction K µα K α ν. Its tems ae: K tt = C + Cα Dγ)1 α) + Cβ + D Dδ) γ) 139) K t = C + Cα Dγ) β) + Cβ + D Dδ)1 δ) 140) The tenso K 3 µν comes fom the contaction K µα K α β Kβ ν. Its tems ae: K 3 tt = C + Cα Dγ)1 α) + C + Cα Dγ)βγ +Cβ + D Dδ) γ)1 δ) + Cβ + D Dδ) γ)1 α) 141) K 3 t = C + Cα Dγ)1 α) β) + C + Cα Dγ)1 δ) β) +Cβ + D Dδ)1 δ) + Cβ + D Dδ)γβ 14) The tace tems of K, denoted by [...], ae: [K 3 ] = 4 3α + δ + 4 ) + 3 AF + [K ] = 4 α δ 4 + AF + [K] = 4 α δ 143) C F + 144) C F + ) AF α Cδ F 3 145) 3/ 16

19 Now that we have the tools, the T U tt and T U t components of the stess enegy tenso ae the ones we ae inteested in. Using the K tems above, we can find T U tt : T U tt = m C + Cα Dγ) + C4 α δ ) 1 + α 3 ) C + Cα Dγ)1 α) +Cβ + D Dδ) γ)) 4 α δ ) C + Cα Dγ) C 4 α δ ) 4 α δ 4 + AF + C F + ))) + α 3 + α 4 ) C + Cα Dγ)1 α) + C + Cα Dγ)βγ + Cβ + D Dδ) γ)1 δ) + Cβ + D Dδ) γ)1 α)) 4 α δ ) C + Cα Dγ)1 α) + Cβ + D Dδ) γ)) + 1 C + Cα Dγ) 4 α δ ) 4 α δ 4 + AF + + C 6 C F + )) 4 α δ ) 3 34 α δ )4 α δ 4 + AF + C F + 4 ) + 4 3α + δ + ) + 3 AF + C F + ) AF α Cδ F 3 )))) 3/ 146) We follow a simila plan to solve fo T U t : T U t = m Cβ + D Dδ) D4 α δ ) 1 + α 3 ) C + Cα Dγ) β) +Cβ + D Dδ)1 δ)) 4 α δ )Cβ + D Dδ) + D 4 α δ ) 4 α δ 4 + AF + C F + ))) + α 3 + α 4 ) C + Cα Dγ)1 α) β) + C + Cα Dγ)1 δ) β) + Cβ + D Dδ)1 δ) + Cβ + D Dδ)γβ) 4 α δ ) C + Cα Dγ) β) + Cβ + D Dδ)1 δ)) D Cβ + D Dδ) 4 α δ ) 4 α δ 4 + AF + 4 α δ ) 3 34 α δ )4 α δ 4 + AF α + δ + 4 ) + 3 AF + C F + )) C F + ) C F + ) AF α Cδ F 3 )))) 3/ 147) Now, we will plug these values fo T U tt and T U t into 109), which I have epeated below, and simplify the expession: D)T U tt + C)T U t = 0 148) 17

20 The elationship is seen to be: ) CDα + C β D γ CDδ 3 + α 3 ) + α 3 + α 4 ) 1 + AF C F α + βγ + + δ ) ) = 0 149) y looking at equation 149), we can see that thee ae seveal ways to appoach it. We can set the fee paametes to zeo, and find the solution whee = 4 9. We can set α 4 to α 3, leaving a solution in which: = 1 + α3 3 + α 3 ) 4 150) Which will give the same solution in the limit α 3 appoaches zeo. The left side of the equation is quadatic in both C and D. It may be possible to find a solution thee, as well. Moe wok can be done by simplifying the equations fo α, β, γ, and δ in an attempt to find the whole solution. 4 Discussion and Review Duing the beginning of this poject, a solution was deived which modeled ou sola system in geneal elativity. Equations of motion wee found fo this solution. Once this was completed, the theoy of massive gavity was pesented to me, and time was taken to gain a familiaity with it so that I could make easonable pogess on this poject. I studied papes witten on the subject and pacticed woking out the solutions which they descibed. Once I had achieved a familiaity with the subject, wok began on the massive gavity solution pesented in this pape. The paametes wee defined and the techniques which wee used duing the poject wee pesented. The Einstein tenso was found, leading to a elationship between two components of the stess enegy tenso which finally led to equation 149). Fom thee, a solution can be found. The oiginal final goal of this poject was to study the massive gavity geodesics in compaison to those in geneal elativity. Unfotunately, this was not achieved. Thee ae seveal steps left in the pocess which will need to be continued fo this goal to be eached. Fist, solutions would need to be found and validated as eal. Once a solution is found, the equations of motion could be found in a manne simila to how the geodesics wee in.. Once found, the equations of motion wee intended to be tested with eal sola system phenomena, such as the pecession of the peihelion of Mecuy o the bending of light aound the sun. 18

21 5 Acknowledgements Despite not eaching my final goal, I quite enjoyed this poject. I continually encounteed new concepts and techniques which have geatly impoved my knowledge on the subject. I would sinceely like to thank Pofessos Claudia de Rham and Andew Tolley fo suppot and assistance thoughout this poject. Refeences [1] A. G. Riess et al. [Supenova Seach Team Collaboation], Aston. J. 116, ) [axiv:astoph/980501]; S. Pelmutte et al. [Supenova Cosmology Poject Collaboation], Astophys. J. 517, ) [axiv:asto-ph/981133]. [] S. Weinbeg, The Cosmological Contant Poblem, Rev. Mod. Phys. 61, ). [3] M. Fasiello and A. J. Tolley, Cosmological petubations in Massive Gavity and the Higuchi bound, axiv: v3 [hep-th]. [4] K. Koyama, G. Niz, and G. Tasinato, Analytic solutions in non-linea massive gavity, axiv: v [hep-th]. [5] M. Fiez and W. Pauli, On elativistic wave equations fo paticles of abitay spin in an electomagnetic field, Poc. Roy. Soc. Lond. A 173, ). [6] N. Akani-Hamed, H. Geogi and M. D. Schwatz, Effective field theoy fo massive gavitons and gavity in theoy space, Annals Phys. 305, ) [hep-th/010184]. [7] D. G. oulwae and S. Dese, Can gavitation have a finite ange?, Phys. Rev. D 6, ). [8] H. van Dam and M. J. G. Veltman, Massive and massless Yang-Mills and gavitational fields, Nucl. Phys., ). [9] V. I. Zakhaov, Lineaized gavitation theoy and the gaviton mass, JEETP Lett. 1, ) [Pisma Zh. Eksp. Teo. Fiz. 1, 1970)]. 19

22 [10] C. de Rham and G. Gabadadze, Genealization of the Fiez-Pauli Action, Phys. Rev. D 8, 010) [axiv: [hep-th]]. [11] C. de Rham, G. Gabadadze, A. J. Tolley, Resummation of Massive Gavity, Phys. Rev. Lett. 106, ). [axiv: [hep-th]]. [1] A. I. Vainshtein, To the poblem of nonvanishing gavitation mass, Phys. Lett. 39, ). 6 Engineeing Physics Appendix Duing this poject, it was my goal to ceate two gavitational models of ou sola system in which the theoy of massive gavity could be studied. They wee constained to be spheically symmetic aound the sun, and filled with a cosmological constant. The fist, the Schwazschild de-sitte solution, was ceated though standad techniques of geneal elativity. The massive gavity solution was looked at fom seveal diffeent ways while I familiaized myself with the theoy and saw slightly diffeent appoaches towads finding solutions. I eventually decided upon an appoach based on [3] and [4]. I based my appoach on these papes because they explained thei methods quite well, and fo a beginne, that was a vey impotant facto. I also was gateful of the accessibility of the Pofessos de Rham and Tolley if I an into any issues undestanding thei theoy. Quite a bit of my poject was spent leaning the theoy, and as such I pacticed following solutions in papes, which wee simila to what I planned on doing. I tied a couple of diffeent solutions on diffeent backgound metics, which othes had used in thei papes, and leaned a geat deal fom stuggling though them. Thee wee no instuments used duing my senio poject. Nealy all of the wok was done by hand, with one o two difficult exceptions fo which I consulted Mathematica. 0