1. Lorentz Symmetry Breaking: Monopole Lensing
Department of Physics Astronomy, and Geophysics, Connecticut College
Jianbin Guan ‘19 (Jguan@conncoll.edu) Instructor: Michael Seifert
Background
 Special Relativity:
i. The laws of physics are identical in all inertial
reference frames
ii. The speed of light in a vacuum is the same for all
inertial reference frames (Lorentz Symmetry)
 Tensor Field: Generalization of a vector field or
scalar field at each point in space. Example: Electric
field, Magnetic field, etc
 Monopole: Hypothetical non-constant tensor field
with spherical symmetry
 Fermat’s Principle (The Principle of Least Time): The
“shortest” path between two points are the path
which will minimize the travel time
 The phenomenon which speed of light varies at each
point in space is an example of Lorenz Symmetry
Breaking
 Coupling Parameter: 𝜉 ---“Knob”, it determines how
much influence the tensor field has on the light
bending effect
 As 𝜉 increase, light bending effect gets stronger,
and vice versa
 Attractive Case (𝜉 > 0): light will bend toward to
the monopole
 Repulsive Case (𝜉 <0): light will bend away from
the monopole
Method
Steps to set pp the mathematical model (Mathematica):
1. Polar coordinate----dS
2. Interpolation of monopole profile to get g(r) function
3. Euler-Lagrange Equation
4. Set up the numerical integration
Analysis
Result
 Three Hypothesis:
 Accuracy and precision goal are too low
 Limit of integration is at infinity
 Extension of the interpolation function fails
 Solutions
 Increase accuracy and precision goal
 U-Substitution
 Extension of g(r)
 We recorded where the bump occurs and what the bump
ratios are
 Bump Ratio =
𝛼 𝑚𝑎𝑥
𝛼𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
.
Could it be numerical error?
The bump seems to persist!
Repulsive Case
𝜉 Bump
Ratio
(𝛽 𝑚𝑎𝑥, 𝛼max)
-0.001 1.138 (2.4, 3.57 × 10−3
)
-0.002 1.138 (2.4, 7.15 × 10−3
)
-0.005 1.137 (2.4, 1.79 × 10−2
)
-0.01 1.136 (2.4, 3.57 × 10−2
)
-0.02 1.136 (2.5, 7.14 × 10−2
)
-0.05 1.137 (2.7, 1.79 × 10−1
)
Attractive Case
𝜉 Bump
Ratio
(𝛽 𝑚𝑎𝑥, 𝛼max)
0.001 1.144 (2.3 , 3.59 × 10−3
)
0.002 1.133 (2.1, 7.12 × 10−3
)
0.005 1.131 (2.3, 1.78 × 10−2
)
0.01 1.130 (2.3, 3.56 × 10−2
)
0.02 1.129 (2.2, 7.11 × 10−2
)
0.05 1.130 (2.0, 1.78 × 10−1
)
Conclusion
Thus far, we have discovered the relationship between the
deflection angle and the impact parameter. More specifically,
we found out that for each value of 𝜉, there is a maximum
impact parameter which will maximize the light bending
effect and the bump ratios are relatively consistent.
The following diagrams illustrate the light bending trajectories:
The following graphs illustrate the relationship between the
deflection angle and the impact parameter:
Acknowledgement
I would like to thank Professor Michael Seifert for giving me
the opportunity to research with him, as well as the
Connecticut College Summer Science Research Program.