The Effect of Modified Gravity on Solar System Scales
Dane Pittock
Physics Department
Case Western Reserve University
Cleveland, Ohio 44106 USA
Abstract
During my senior project, I have explored the effects of Modified Gravity on Solar System scales. Infrared modifications of gravity are typically expected to modify Newton’s law on very large distance scales, but such theories usually come hand in hand with additional degrees of freedom which could already provide smoking gun signatures on much smaller distance scales. I have paid special attention to the effects of the acceleration of the Universe on small distance scales.
During this senior project I have familiarized myself with standard techniques of General Relativity and applied them to known systems and observations. As a second stage I have worked out how to parameterize modifications of gravity on solar system scales and derived some relevant observable signatures.
Contents
1 Background and Literature Review 1
2 General Relativity Solution 3
2.1 Derivation of General Relativity Solutions . . . 3
2.1.1 Spherically Symmetric Vacuum Solution . . . 3
2.1.2 Schwarzschild de-Sitter Solution . . . 7
2.2 Geodesics . . . 9
3 Massive Gravity Solution 10 3.1 Dynamic Metric and Einstein Tensor . . . 10
3.2 Reference Metric and Stress Energy Tensor . . . 14
4 Discussion and Review 18
5 Acknowledgements 19
6 Engineering Physics Appendix 20
1
Background and Literature Review
There are four know fundamental forces in physics: the strong force, the weak force, electromagnetism, and gravity. These forces are mediated by force-carrying particles. The strong force is mediated by gluons, the weak force by the W and Z bosons, and electromagnetism by the photon. A force carrying particle for gravity has not yet been discovered. However, we can theoretically determine this particle’s characteristics based upon our knowledge of gravitation.
The theoretical particle believed to be the force-carrier for gravity is the graviton. This particle is believed to be massless, have no charge, and is expected to be a spin-2 boson. General relativity is the theory which governs the graviton, and it recovers Newtonian gravitation in the weak-field limit.
modern particle physics. Most prominent, from observations of supernovae, data has surfaced which implies a late-time acceleration in the expansion of our universe [1]. This implies a constant dark energy density, believed to be due to a cosmological constant providing a repulsive gravitation. It would be reasonable to believe that this cosmological constant was the vacuum energy density predicted by particle physics, however a cosmological constant that would provide the repulsive gravity necessary to fuel this accelerated expansion is extremely small compared to the vacuum energy density expected from particle physics [2].
We can attempt to solve this problem by modifying gravity itself, where new degrees of freedom are introduced into the gravitational field that lead to the expected accelerated expansion of the universe on large distance scales. By introducing a non-zero value for the mass of the graviton, these additional degrees of freedom are given to the graviton.
Fierz and Pauli (FP) first studied massive gravity at the linear level [5]. The added mass term breaks the gauge invariance of general relativity, leading to five degrees of freedom in the graviton, as opposed to the two from general relativity. An effective theory approach restores the gauge invariance by introducing additional Stuckelberg fields that represent the helicity-0 mode of the graviton [6]. Once departing from the linearized theory, however, massive gravity has run into issues with ghosts at the non-linear level such as the one discovered by Boulware and Deser [7] and continuity issues. Fierz and Pauli’s theory did not recover general relativity in the limit where the mass of the graviton reduces to zero. This became known as the van Dam, Veltman, Zakharov discontinuity [8], [9].
A recent theory by de Rham, Gabadadze, and Tolley (dRGT) has been formulated that is ghost-free at the fully non-linear level [10], [11]. Here general relativity is recovered in the massless limit via a Vainshtein mechanism [12]. The approach to massive gravity that will be utilized in this project stems from dRGT.
While these modifications are meant to modify gravity on large distances, we may be able to observe these additional degrees of freedom on smaller scales. By comparing precisely known values in our solar system to general relativity and this theory of massive gravity, we can determine the accuracy of both theories on small scales. The goal of this project was to study the accuracy of geodesic paths of particles in theoretical gravitational fields of the solar system as defined by general relativity and massive gravity.
applied to real solar system tests, such as the precession of the perihelion of Mercury, the bending of light around the sun, etc. It is expected to see massive gravity recover general relativity in the limit that the mass of the graviton is zero.
2
General Relativity Solution
The first section of this project was to create a model of the solar system in general relativity, with which the geodesics (equations of motion) of massive and massless particles could be derived and then compared to geodesics of particles in the massive gravity solution. The model of the solar system in general relativity would also go on to become the reference metric for the massive gravity solution, which will be explained more in Section 3.
2.1
Derivation of General Relativity Solutions
2.1.1 Spherically Symmetric Vacuum Solution
We began with the simple assumption that the solution would be spherically symmetric and due to a point source. The mass of the sun, much larger than the mass of the rest of the solar system, dominates the gravitational field, resulting in a spherically symmetric field originating from the sun. The sun’s radius is negligible compared to the radius of the solar system, and we could approximate it as a point source.
A spherically symmetric solution of Einstein’s field equations is defined as a spacetime with rotational invariance. This means rotations around any radial direction will not change the metric. We can use Killing vectors to build a definition for the metric. In a four dimensional spacetime, this gives rise to three Killing vectors:
R = −y∂x+x∂y (1)
S = z∂x+x∂z (2)
T = −z∂y+y∂z (3)
coordinates, these vectors are:
R = r2sin2θ∂φ (4)
S = r2cosφ∂θ−r2sin2θcotθsinφ∂φ (5)
T = −r2sinφ∂θ−r2sin2θcotθcosφ∂φ (6)
These three Killing vectors must solve the Killing equation:
ξa∂agµν+ (∂µξa)gaν+ (∂νξa)gaµ= 0 (7)
Whereξais a Killing vector. This gives us the components for our metric. The non-zero metric components
are: gaa = gaa(a, b) (8) gab = gab(a, b) (9) gba = gab(a, b) (10) gbb = gbb(a, b) (11) gθθ = r2(a, b) (12) gφφ = r2sin2θ(a, b) (13)
The general metric is:
ds2=gaa(a, b)da2+ 2gab(a, b)dadb+gbb(a, b)db2
+r2(a, b)dθ2+r2(a, b)sin2θdφ2 (14)
We can use some tricks to simplify the metric to something easier to work with. By changing the variable b to r, the radial component, the metric becomes:
ds2=gaa(a, r)da2+ 2gar(a, r)dadr+grr(a, r)dr2
+r2(a, r)dθ2+r2(a, r)sin2θdφ2 (15)
By changing the variable a to t, or time:
dt = ∂t ∂ada+ ∂t ∂rdr (16) dt2 = ∂t ∂a 2 da2+ 2∂t ∂a ∂t ∂r dadr+∂t ∂r 2 dr2 (17)
The simplified metric is:
ds2=gtt(t, r)dt2+grr(t, r)dr2+r2dθ2+r2sin2θdφ2 (18)
Now, we can apply standard general relativity practices to find the values ofgttandgtr. We begin by finding
the Christoffel symbols from the metric using:
Γλµν = 1 2g
λσ(∂
µgνσ+∂νgσν−∂σgµν) (19)
The non-vanishing Christoffel symbols for the spherically symmetric vacuum solution are:
Γttt = 1 2g tt∂ tgtt (20) Γtrt = 1 2g tt∂ rgtt (21) Γtrr = −1 2g tt∂ tgrr (22) Γrtt = − 1 2g rr ∂rgtt (23) Γrrt = 1 2g rr∂ tgrr (24) Γrrr = 1 2g rr∂ rgrr (25) Γrθθ = −rgrr (26) Γrφφ = −rsin2θgrr (27) Γθrθ = 1 r (28) Γθφφ = −cosθsinθ (29) Γφrφ = 1 r (30) Γφθφ = cotθ (31)
The next step is to find the Ricci tensor using:
Rµν =∂ρΓρνµ−∂νΓρρµ+ Γ ρ ρλΓ λ νµ−Γ ρ νλΓ λ ρµ (32)
But first, we know that in a vacuum, the Ricci tensor vanishes:
By solvingRtr, it is discovered:
∂tgrr = 0 (34)
The metric component grr does not depend on time. This also proves the metric component gtt does not
depend on time either, after correcting the Christoffel symbols the Ricci tensor no longer has any ∂tgtt
components. With no more time dependence, we can see the metric is invariant under time translation. The corrected Ricci tensor components are (g0µν represents partial derivative with respect to r):
Rtt = rgttgrr0 gtt0 +grr r(gtt0 )2−2gtt(2gtt0 +rgtt00) 4rg2 rrgtt (35) Rrr = grr0 4r+gtt0 gtt +grr (g 0 tt)2−2gttgtt00 g2 tt 4grr (36) Rθθ = 1 2 2 + rgrr0 g2 rr −2 + rg0tt gtt grr ! (37) Rφφ = Rθθsin2θ (38)
All other components of the Ricci tensor are trivial. Solving the Ricci tensor by applying (33), the final metric components are found:
grr = 1 + 1 C2r −1 (39) gtt = C1 1 + 1 C2r (40)
In the weak gravitational field limit, they are expected to resemble Newton’s theory of gravity. We can use this classical gravitation theory to determine values forC1 andC2:
grr = 1−2GM r −1 (41) gtt = − 1−2GM r (42)
Where M is the mass of the point source (the sun) and G is the gravitational constant. Thus, the spherically symmetric solution is the Schwarzschild solution, with the metric:
ds2=−1−2GM r dt2+1−2GM r −1 dr2+r2dθ2+r2sin2θdφ2 (43)
2.1.2 Schwarzschild de-Sitter Solution
The Schwarzschild metric derived in 2.1.1 took the following form:
ds2=−f(r)dt2+f(r)−1dr2+r2dθ2+r2sin2θdφ2 (44)
In the vacuum solution,f(r) was seen to be:
f(r) = 1−2GM
r (45)
Now, the solution will be expanded to incorporate a cosmological constant. We can begin with the same standard general relativity practices as before The non-vanishing Christoffel symbols are:
Γtrt = f 0 2f (46) Γrtt = f f 0 2 (47) Γrrr = f 0 2f (48) Γrθθ = −rf (49) Γrφφ = −rfsin2θ (50) Γθrθ = 1 r (51) Γθφφ = −cosθsinθ (52) Γφrφ = 1 r (53) Γφθφ = cotθ (54)
The next step is to apply (32), and find the non-zero components of the Ricci tensor:
Rtt = f f0 r + f f00 2 (55) Rrr = − f0 rf − f00 2f (56) Rθθ = 1−f−rf0 (57) Rφφ = Rθθsin2θ (58)
The cosmological constant now comes into play with the stress-energy tensor,Tµν. The connection between
the curvature of spacetime and the stress-energy tensor we will use in this situation is:
Rµν = 8πG(Tµν−
1
Here the cosmological constant, Λ, is described as a perfect fluid. This lead to a stress-energy tensor like this: Tνµ= −Λ 0 0 0 0 −Λ 0 0 0 0 −Λ 0 0 0 0 −Λ (60)
Contracting the indices with the metric gives us this:
Tµν = Λf 0 0 0 0 −Λ f 0 0 0 0 −r2Λ 0 0 0 0 −r2sin2θΛ (61)
By taking the trace of (60), we find:
T =Tµµ=−4Λ (62)
Now we have everything we need to solve (59) forf(r). It turns out to be:
f = c1
r +c2−
8πGr2Λ
3 (63)
In the weak field limit, this will resemble Newton’s theory of gravity (just like in the vacuum solution) with a new cosmological constant term:
f = 1−2GM
r −
8πGr2Λ
3 (64)
Defining the Hubble parameter as:
H2≡ 8πGΛ
3 (65)
Gives us this final version off(r):
f = 1−2GM
r −H
2r2 (66)
This metric, known as the Schwarzschild de-Sitter metric, is now solved for:
ds2=−(1−2GM r −H 2 r2)dt2+ (1−2GM r −H 2 r2)−1dr2 +r2dθ2+r2sin2θdφ2 (67)
2.2
Geodesics
Now that we have the Schwarzschild de-Sitter metric ready to be tested, the next step is to find the equations of motion, or geodesics for the general relativity solution. We can find the governing equations of motion for both massive, such as a planet, and massless, such as a photon, particles in the gravitational field of the sun. The geodesic equation is:
d2xµ dλ2 + Γ µ ρσ dxρ dλ dxσ dλ = 0 (68)
For null, or massless geodesics, the norm of the tangent vectors will be zero with respect to an affine parameter. Note how the norm of the tangent vectors is the metric:
gµν dxµ dλ dxν dλ = 0 =ds 2 (69)
For massive geodesics, the norm of the tangent vectors will be negative:
gµν dxµ dλ dxν dλ < 0 (70) ds2 < 0 (71)
And with respect to proper time:
gµν
dxµ
dτ dxν
dτ =−1 (72)
The Lagrangian for the null geodesics will equal 0. The Lagrangian is the metric divided by an affine parameter:
L= ds
2
dλ2 = 0 (73)
The Lagrangian for massive geodesics will be−1:
L= ds
2
dτ2 =−1 (74)
Utilizing the Euler-Lagrange formula, the equations of motion can be obtained. The conserved quantites from the equations of motion are below. We chose to keep θ at π
2, for the metric is rotationally invariant and keeping a consistent value ofθ simplified the work. L is the angular momentum,E is the energy, and
Lis the Lagrangian: θ = π 2 (75) L = r2φ˙ (76) E = (1−2GM r −H 2r2) ˙t (77) L = − E 2 1−2GM r −H2r2 + r˙ 2 1−2GM r −H2r2 +L 2 r2 (78)
This leads to the geodesic equation for massless particles in the general relativity solution:
dr dφ 2 = E 2r4 L2 −r 2(1−2GM r −H 2r2) (79)
And the geodesic equation for massive particles in the general relativity solution:
dr dφ 2 =E 2r4 L2 −(1− 2GM r −H 2r2)r2− r 4 L2 (80)
3
Massive Gravity Solution
Now, we will carefully examine the possibility for the graviton mass to be non-zero. We will take the Schwarzschild de-Sitter solution from 2.1.2 and create with it a reference metric to base the new, dynamic metric on. This reference metric will represent the matter in the system (the sun and the cosmological constant), and is necessary to create the potential due to the graviton mass later on. This potential will be a scalar function of both the reference metric and the dynamic metric.
3.1
Dynamic Metric and Einstein Tensor
Let’s start by creating the dynamic metric:
ds2=−C(r)dt2+A(r)dr2+ 2D(r)dtdr+B(r)dθ2+B(r)sin2θdφ2 (81)
This is the most generic spherically symmetric metric. We cannot use the same tricks as we did with the general relativity solution to simplify it, as we have broken the gauge invariance. We will need to find the
Einstein tensor to solve forA,B,C, and D. As a matrix, the dynamic metric looks like: gµν = −C(r) D(r) 0 0 D(r) A(r) 0 0 0 0 B(r) 0 0 0 0 B(r)sin2θ (82)
Now we can see that this metric is not diagonal like the Schwarzschild metric was. Therefore, the inverted metric must be explicitly calculated for the Christoffel Symbols:
gµν = −A(r) ∆(r) D(r) ∆(r) 0 0 D(r) ∆(r) C(r) ∆(r) 0 0 0 0 B1(r) 0 0 0 0 B(r)sin1 2θ (83) Where ∆(r) =A(r)C(r) + D(r)2
. The calculated Christoffel Symbols for the dynamic metric, using (19), are: Γttt = D(r)C 0(r) 2∆ (84) Γtrt= Γttr = A(r)C 0(r) 2∆ (85) Γtrr = D(r)A 0(r)−2A(r)D0(r) 2∆ (86) Γtθθ = −D(r)B 0(r) 2∆ (87) Γtφφ = −D(r)B 0(r)sin2θ 2∆ (88) Γrtt = C(r)C 0(r) 2∆ (89) Γtrt= Γttr = −D(r)C 0(r) 2∆ (90) Γrrr = C(r)A 0(r)−2D(r)D0(r) 2∆ (91) Γrθθ = −C(r)B0(r) 2∆ (92) Γrφφ = −C(r)B 0(r)sin2θ 2∆ (93) Γθrθ= Γθθr = B 0(r) 2B(r) (94) Γθφφ = −cosθsinθ (95)
Γφrφ= Γφφr = B
0(r)
2B(r) (96)
Γφθφ = cotθ (97)
Applying (32), the components of the Ricci tensor are:
Rtt= C(r) 2∆(r)B0(r)C0(r)−B(r)A(r) C0(r)2+ 2D(r)C0(r)D0(r) 4B(r) ∆(r)2 −2 D(r)2C00(r) +C(r) A0(r)C0(r)−2A(r)C00(r) 4B(r) ∆(r)2 (98) Rrt= −D(r) 2∆(r)B0(r)C0(r)−B(r)A(r) C0(r)2+ 2D(r)C0(r)D0(r) 4B(r) ∆(r)2 −2 D(r)2C00(r) +C(r) A0(r)C0(r)−2A(r)C00(r) 4B(r) ∆(r)2 (99) Rrr = 1 4 2 B0(r)2+B(r)B 0(r) C(r)A0(r)+2D(r)D0(r) ∆(r) −2B(r)B 00(r) B(r)2 + A(r)A(r) C0(r)2+C(r) A0(r)C0(r)−2A(r)C00(r)+ 2D(r) C0(r)D0(r)−D(r)C00(r) ∆(r)2 ! (100) Rθθ= 4 A(r)2 C(r)2 + 4 D(r)4 + C(r)2 A0(r)B0(r) + 2C(r)D(r)B0(r)D0(r) 4 ∆(r)2 − 2 D(r)2B0(r)C0(r) +C(r)B00(r)−A(r)C(r)−8 D(r)2+B0(r)C0(r) + 2C(r)B00(r) 4 ∆(r)2 (101) Rφφ = sinθ2Rθθ (102)
We will approach the Einstein tensor,Gµν, through:
Gµν =Rµν−
1
2Rgµν (103)
The Ricci scalar,R, is calculated as a contraction of the inverse dynamic metric and the Ricci tensor:
R=Rµµ=gµνRµν (104) R= 4 A(r)2 B(r) C(r)2 +C(r) D(r)2 B0(r)2 + 2B(r) 2 D(r)4 + C(r)2 A0(r)B0(r) 2 B(r)2 ∆(r)2 +2C(r)D(r)B0(r)D0(r)−2 D(r)2B0(r)C0(r) +C(r)B00(r) +A(r) C(r)2 B0(r)2 2 B(r)2 ∆(r)2
−2B(r)C(r)−4 D(r)2 +B0(r)C0(r) + 2C(r)B00(r)+ B0(r)2 C0(r)2 −2C(r)C00(r) 2 B(r)2 ∆(r)2 + B(r)2 C(r)A0(r)C0(r) + 2D(r)C0(r)D0(r)−D(r)C00(r) ! 2 B(r)2 ∆(r)2 (105)
Finally, theGtt andGtr Einstein tensor components are:
Gtt= C(r) 4 A(r)2 B(r) C(r)2 +C(r) D(r)2 B0(r)2 +A(r)C(r) C(r) B0(r)2 4 B(r)2 ∆(r)2 +B(r)8 D(r)2−4C(r)B00(r) + 2B(r) 2 D(r)4+ C(r)2A0(r)B0(r) + 2C(r)D(r)B0(r)D0(r) 4 B(r)2 ∆(r)2 − D(r)2 B0(r)C0(r) + 2C(r)B00(r) ! 4 B(r)2 ∆(r)2 (106) Gtr = −D(r) 4 A(r)2 B(r) C(r)2 +C(r) D(r)2 B0(r)2 +A(r)C(r) C(r) B0(r)2 4 B(r)2 ∆(r)2 +B(r)8 D(r)2−4C(r)B00(r) + 2B(r) 2 D(r)4+ C(r)2A0(r)B0(r) + 2C(r)D(r)B0(r)D0(r) 4 B(r)2 ∆(r)2 − D(r)2 B0(r)C0(r) + 2C(r)B00(r) ! 4 B(r)2 ∆(r)2 (107) It is important to notice: D(r)Gtt+C(r)Gtr= 0 (108) This means: D(r)TttU+C(r)TtrU = 0 (109) Through: −TµνU =Gµν (110)
This simplifies the equation to a relationship between two components of the stress-energy tensor that is created due to the potential of the massive graviton. DenotedTµνU so as not to be confused with the
3.2
Reference Metric and Stress Energy Tensor
In order to find TU
µν and solve for the metric components A, B, C, and D, we must study the reference
metric. More specifically, we must look at the relationship between the reference metric and the dynamic metric. The reference metric as derived in 2.1.2, now in matrix form:
fµν = −(1−2GM r −H 2r2) 0 0 0 0 (1−2GM r −H 2r2)−1 0 0 0 0 r2 0 0 0 0 r2sin2θ (111)
For convenience, we will define:
(1−2GM
r −H
2r2) =F(r) (112)
Now we will delve into the relationship between the reference metric and dynamic metric. The contraction of the inverse dynamic metric and the reference metric will be crucial in the construction of the potential.
gµαfαν= AF ∆ D ∆F 0 0 −DF ∆ C ∆F 0 0 0 0 rB2 0 0 0 0 rB2 (113)
More specifically, the square root of this contraction is the vital piece of information that we need:
p gµαf αν= α β 0 0 γ δ 0 0 0 0 √r B 0 0 0 0 √r B (114)
The roots α, β, γ, and δ were solved for, and their values are listed below. We now define M(r), in an
attempt to keep the equations easier to read:
M(r) =pC2−2ACF2−4D2F2+A2F4 (115) α= −(−C+AF2−M)qC+AF2−M ∆F + (−C+AF 2+M)qC+AF2+M ∆F 2√2M (116)
β= −(−C+AF2−M)(−C+AF2+M) q C+AF2−M ∆F 4√2DF2M −(C−AF 2+M)(−C+AF2+M)qC+AF2+M ∆F 4√2DF2M (117) γ= DF2qC+AF2−M ∆F −DF 2qC+AF2+M ∆F √ 2M (118) δ= (−C+AF2+M)qC+AF2−M ∆F + (C−AF 2+M)qC+AF2+M ∆F 2√2M (119)
We can now construct the potential due to the massive gravity from here. TheKterms are connections from
the reference metric and dynamic metric that the potentialU is built out of. α3andα4are free parameters.
Kνµ(g, f) =δνµ−pgµαf αν (120) U2= 1 2! [K] 2 −[K2] (121) U3= 1 3! [K] 3−3[K][K2] + 2[K3] (122) U4= 1 4! [K] 4−6[K2][K]2+ 8[K3][K] + 3[K2]2−6[K4] (123) U(g, f) =U2+α3U3+α4U4 (124)
The Lagrangian for massive gravity is the Lagrangian for general relativity plus a new term due to that potential: L= M 2 P l 2 √ −g R+ 2m2U(g, f) +LM (125)
This leads to the stress-energy tensor due to the potential:
TµνU =m2 Kµν−[K]gµν− 1 +α3 Kµν2 −[K]Kµν+ 1 2gµν [K] 2−[K2] + α3+α4 Kµν3 −[K]Kµν2 +1 2Kµν [K] 2−[K2] −1 6gµν [K] 3−3[K][K2] + 2[K3] (126)
Now, we find all of the terms. TheKνµ terms are found to be:
Ktt= 1−α (127)
Krt=−β (128)
Ktr=−γ (129)
Kθθ= 1−√r
B (131)
Kψψ= 1−√r
B (132)
The tensorKµν comes fromgµαKνα. Its terms are found to be:
Ktt=−C+Cα−Dγ (133) Ktr=Cβ+D−Dδ (134) Krt=D−Dα−Aγ (135) Krr =−Dβ+A−Aδ (136) Kθθ=B−r √ B (137) Kψψ=Kθθsin2θ (138)
The tensorKµν2 comes from the contractionKµαKνα. Its terms are:
Ktt2 = (−C+Cα−Dγ)(1−α) + (Cβ+D−Dδ)(−γ) (139)
Ktr2 = (−C+Cα−Dγ)(−β) + (Cβ+D−Dδ)(1−δ) (140)
The tensorKµν3 comes from the contractionKµαKβαK β
ν. Its terms are:
Ktt3 = (−C+Cα−Dγ)(1−α)2+ (−C+Cα−Dγ)βγ
+(Cβ+D−Dδ)(−γ)(1−δ) + (Cβ+D−Dδ)(−γ)(1−α) (141)
Ktr3 = (−C+Cα−Dγ)(1−α)(−β) + (−C+Cα−Dγ)(1−δ)(−β)
+(Cβ+D−Dδ)(1−δ)2+ (Cβ+D−Dδ)γβ (142)
The trace terms ofK, denoted by [...], are:
[K] = 4−α−δ−√2r B (143) [K2] = 4−2α−2δ−√4r B + AF ∆ + C ∆F + 2r2 B (144) [K3] = 4−3(α+δ+√4r B) + 3( AF ∆ + C ∆F + 2r2 B )− AF α ∆ − Cδ ∆F − 2r3 B3/2 (145)
Now that we have the tools, the TttU and TtrU components of the stress energy tensor are the ones we are
interested in. Using theKterms above, we can find TU tt: TttU =m2 (−C+Cα−Dγ) +C(4−α−δ−√2r B)− 1 +α3 ((−C+Cα−Dγ)(1−α) +(Cβ+D−Dδ)(−γ))−(4−α−δ−√2r B)(−C+Cα−Dγ)− C 2 (4−α−δ− 2r √ B) 2 −(4−2α−2δ−√4r B + AF ∆ + C ∆F + 2r2 B ) + α3+α4 ((−C+Cα−Dγ)(1−α)2 +(−C+Cα−Dγ)βγ+ (Cβ+D−Dδ)(−γ)(1−δ) + (Cβ+D−Dδ)(−γ)(1−α)) −(4−α−δ−√2r B)((−C+Cα−Dγ)(1−α) + (Cβ+D−Dδ)(−γ)) +1 2(−C+Cα−Dγ) (4−α−δ− 2r √ B) 2−(4−2α−2δ−√4r B + AF ∆ + C ∆F + 2r2 B ) +C 6 (4−α−δ− 2r √ B) 3−3(4−α−δ−√2r B)(4−2α−2δ− 4r √ B + AF ∆ + C ∆F +2r 2 B ) + 2(4−3(α+δ+ 4r √ B) + 3( AF ∆ + C ∆F + 2r2 B )− AF α ∆ − Cδ ∆F − 2r3 B3/2) (146)
We follow a similar plan to solve forTU tr: TtrU =m2 (Cβ+D−Dδ)−D(4−α−δ−√2r B)− 1 +α3 ((−C+Cα−Dγ)(−β) +(Cβ+D−Dδ)(1−δ))−(4−α−δ−√2r B)(Cβ+D−Dδ) + D 2 (4−α−δ− 2r √ B) 2 −(4−2α−2δ−√4r B + AF ∆ + C ∆F + 2r2 B ) + α3+α4 ((−C+Cα−Dγ)(1−α)(−β) +(−C+Cα−Dγ)(1−δ)(−β) + (Cβ+D−Dδ)(1−δ)2+ (Cβ+D−Dδ)γβ) −(4−α−δ−√2r B)((−C+Cα−Dγ)(−β) + (Cβ+D−Dδ)(1−δ)) +1 2(Cβ+D−Dδ) (4−α−δ− 2r √ B) 2−(4−2α−2δ−√4r B + AF ∆ + C ∆F + 2r2 B ) −D 6 (4−α−δ− 2r √ B) 3−3(4−α−δ−√2r B)(4−2α−2δ− 4r √ B + AF ∆ + C ∆F + 2r2 B ) +2(4−3(α+δ+√4r B) + 3( AF ∆ + C ∆F + 2r2 B )− AF α ∆ − Cδ ∆F − 2r3 B3/2) (147)
Now, we will plug these values for TU
tt and TtrU into (109), which I have repeated below, and simplify the
expression:
The relationship is seen to be: CDα+C2β−D2γ−CDδ3−√2r B +α3(2− 2r √ B) + (α3+α4) 1 + r2 B − 2r √ B −AF 2∆ − C 2∆F +βγ+ α2 2 + δ2 2 = 0 (149)
By looking at equation (149), we can see that there are several ways to approach it. We can set the free parameters to zero, and find the solution whereB=4r92. We can setα4to−α3, leaving a solution in which:
B=1 +α3 3 + 2α3
2
4r2 (150)
Which will give the same solution in the limitα3 approaches zero. The left side of the equation is quadratic
in bothCandD. It may be possible to find a solution there, as well. More work can be done by simplifying
the equations forα,β,γ, andδ in an attempt to find the whole solution.
4
Discussion and Review
During the beginning of this project, a solution was derived which modeled our solar system in general relativity. Equations of motion were found for this solution. Once this was completed, the theory of massive gravity was presented to me, and time was taken to gain a familiarity with it so that I could make reasonable progress on this project. I studied papers written on the subject and practiced working out the solutions which they described. Once I had achieved a familiarity with the subject, work began on the massive gravity solution presented in this paper. The parameters were defined and the techniques which were used during the project were presented. The Einstein tensor was found, leading to a relationship between two components of the stress energy tensor which finally led to equation (149). From there, a solution can be found.
The original final goal of this project was to study the massive gravity geodesics in comparison to those in general relativity. Unfortunately, this was not achieved. There are several steps left in the process which will need to be continued for this goal to be reached. First, solutions would need to be found and validated as real. Once a solution is found, the equations of motion could be found in a manner similar to how the geodesics were in 2.2. Once found, the equations of motion were intended to be tested with real solar system phenomena, such as the precession of the perihelion of Mercury or the bending of light around the sun.
5
Acknowledgements
Despite not reaching my final goal, I quite enjoyed this project. I continually encountered new concepts and techniques which have greatly improved my knowledge on the subject. I would sincerely like to thank Professors Claudia de Rham and Andrew Tolley for support and assistance throughout this project.
References
[1] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201]; S. Perlmutteret al.[Supernova Cosmology Project Collaboration], Astrophys. J.517, 565 (1999) [arXiv:astro-ph/9812133].
[2] S. Weinberg, ”The Cosmological Contant Problem,” Rev. Mod. Phys.61, 1 (1989).
[3] M. Fasiello and A. J. Tolley, ”Cosmological perturbations in Massive Gravity and the Higuchi bound”, arXiv:1206.3852v3 [hep-th].
[4] K. Koyama, G. Niz, and G. Tasinato, ”Analytic solutions in non-linear massive gravity”, arXiv:1103.4708v2 [hep-th].
[5] M. Fierz and W. Pauli, ”On relativistic wave equations for particles of arbitrary spin in an electromag-netic field,” Proc. Roy. Soc. Lond. A173, 211 (1939).
[6] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, ”Effective field theory for massive gravitons and gravity in theory space,” Annals Phys.305, 96 (2003) [hep-th/0210184].
[7] D. G. Boulware and S. Deser, ”Can gravitation have a finite range?,” Phys. Rev. D6, 3368 (1972).
[8] H. van Dam and M. J. G. Veltman, ”Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys. B22, 397 (1970).
[9] V. I. Zakharov, ”Linearized gravitation theory and the graviton mass,” JEETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz.12, (1970)].
[10] C. de Rham and G. Gabadadze, ”Generalization of the Fierz-Pauli Action,” Phys. Rev. D82, (2010) 044020 [arXiv:1007.0443 [hep-th]].
[11] C. de Rham, G. Gabadadze, A. J. Tolley, ”Resummation of Massive Gravity,” Phys. Rev. Lett. 106,
231101(2011). [arXiv:1011.1232 [hep-th]].
[12] A. I. Vainshtein, ”To the problem of nonvanishing gravitation mass,” Phys. Lett. B39, 393 (1972).
6
Engineering Physics Appendix
During this project, it was my goal to create two gravitational models of our solar system in which the theory of massive gravity could be studied. They were constrained to be spherically symmetric around the sun, and filled with a cosmological constant. The first, the Schwarzschild de-Sitter solution, was created through standard techniques of general relativity. The massive gravity solution was looked at from several different ways while I familiarized myself with the theory and saw slightly different approaches towards finding solutions. I eventually decided upon an approach based on [3] and [4]. I based my approach on these papers because they explained their methods quite well, and for a beginner, that was a very important factor. I also was grateful of the accessibility of the Professors de Rham and Tolley if I ran into any issues understanding their theory. Quite a bit of my project was spent learning the theory, and as such I practiced following solutions in papers, which were similar to what I planned on doing. I tried a couple of different solutions on different background metrics, which others had used in their papers, and learned a great deal from struggling through them.
There were no instruments used during my senior project. Nearly all of the work was done by hand, with one or two difficult exceptions for which I consulted Mathematica.
