# Lorentz Violation at the Level of Undergraduate Classical Mechanics

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## Abstract

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## 1. Introduction

## 2. General Theory

#### 2.1. Transformations and Symmetries

Key Idea 1:In the Standard-Model Extension, the vacuum is not symmetrical under rotations. It has a directionality to it.

- In the flat-spacetime limit of the SME, momentum and energy are conserved. This implies that the background vector field cannot be a
**force**field; a background force field would generate momentum (and kinetic energy) from nothing. Instead, the background fields that appear in the SME are more abstract.It is reasonable to instead imagine a background**torque**field, i.e., a field that encourages everything in the universe to rotate around a particular axis. Such effects do occur in the SME, but they are not directly relevant to this work; - The background field we’ll be studying in this work is a matrix (a 2nd-rank tensor) rather than a vector. Details of its behavior appear later in this section;
- The full SME has many fields: A torque-producing field that acts on electrons, a torque-producing field that acts on protons, and so on. The fields that act on different particles do not necessarily point in the same direction or have the same magnitude.Moreover, there are other effects that cannot be cleanly represented as torques: a vector field that behaves in many ways, like the electromagnetic vector potential, a matrix field that causes a particle’s kinetic energy to depend on the direction of its velocity, and so on. The fundamental goal of the SME is to include
**all**effects that violate Lorentz symmetry while respecting most other important aspects of the Standard Model; this leads to a broad smorgasbord of effects.

Key Idea 2:An important technique for searching for Lorentz violation is to create a system, such as an atomic clock, that should keep ticking at a constant rate according to conventional physics. We then watch it very carefully as it rotates with Earth through the background field.

Key Idea 3:Since Lorentz-violating effects are correlated to the background vacuum, they are correlated to sidereal days.

#### 2.2. Lagrangian, Momentum, and Newton’s 2nd Law

## 3. Example Systems

#### 3.1. Simple Pendulum

#### 3.1.1. Theory: Period

#### 3.1.2. Simulated Data

#### 3.1.3. Pendulum Experiment

#### 3.2. Block on Inclined Plane

**The net force and the acceleration are not parallel to each other.**

#### 3.3. Binary Star System

#### 3.3.1. Theory: Equations of Motion

#### 3.3.2. Theory: Energy

#### 3.3.3. Theory: Angular Momentum

#### 3.3.4. Binary Star System Computational Simulation

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SME | Standard-Model Extenshion |

EJS | Easy Java Simulation |

## References

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**Figure 1.**Background vector field that violates rotational symmetry. (Image constructed by Charles D. Lane with POV-Ray software. Earth map from http://visibleearth.nasa.gov/).

**Figure 3.**Transformation of coordinates [8] between a rotating Earth-based frame $x,y,z$ and a nonrotating Sun-based frame $X,Y,Z$. The origins of the frames are shown as coinciding to ease understanding of relative directions.

**Figure 9.**Diagram of a binary star system where the only force on Star 1 (red) is the gravitational force from Star 2 (blue) and vice versa. The center of mass is denoted with a “⊗” symbol and all calculations are done in a frame where it is at rest.

**Figure 10.**EJS screenshot of a binary star system with SME coefficient ${c}_{xy}=0.1$ to illustrate that momentum (black vector) and velocity (gray vector) are no longer parallel.

**Figure 11.**Plot of the z-component of angular momentum of Star 2 vs. time when the SME coefficient ${c}_{xy}=0.05$ to illustrate the rate that angular momentum changes.

**Figure 12.**Example where ${c}_{xy}=0.05$ to illustrate that angular momentum is no longer conserved under Lorentz-symmetry-violating effects.

**Figure 15.**Long-term trajectories for a variety of values of ${c}_{xz}$. From left to right, the values are ${c}_{xz}=0.01$, $0.05$, $0.1$, and $0.2$.

Constants | Value | Uncertainty |
---|---|---|

Mass m | 0.1 kg | ±0.01 kg |

Length ℓ | 0.47 m | ±0.01 m |

Initial Angle ${\theta}_{0}$ | 25${}^{\circ}$ | $\pm {1}^{\circ}$ |

Co-latitude $\chi $ | 0.9724 rad | ±0.0005 rad |

**Table 2.**SME coefficients relative to the sun’s non-rotating frame and their determined values and uncertainties using multiple linear regression.

SME Coefficient | Value | Uncertainty |
---|---|---|

${c}_{XX}$ | −0.00012 | ±0.00079 |

${c}_{XY}$ | −0.00058 | ±0.00066 |

${c}_{YY}$ | 0.00014 | ±0.00081 |

**Table 3.**Initial Conditions based off of calculated variables for Sirius A (Star 1) and Sirius B (Star 2) for the Easy Java Simulation (EJS) simulation.

Parameter | Value |
---|---|

${m}_{1}$ | 2.063 M${}_{\odot}$ |

${m}_{2}$ | 1.018 M${}_{\odot}$ |

${\tau}_{op}$ | 50.124 years |

$({x}_{1},{y}_{1},{z}_{1})$ | (−6.536, 0, 0) AU |

$({x}_{2},{y}_{2},{z}_{2})$ | (13.246, 0, 0) AU |

$({v}_{{x}_{1}},{v}_{{y}_{1}},{v}_{{z}_{1}})$ | (0, 0.819,0) AU/years |

$({v}_{{x}_{2}},{v}_{{y}_{2}},{v}_{{z}_{2}})$ | (0, −1.66,0) AU/years |

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**MDPI and ACS Style**

Clyburn, M.; Lane, C.D.
Lorentz Violation at the Level of Undergraduate Classical Mechanics. *Symmetry* **2020**, *12*, 1734.
https://doi.org/10.3390/sym12101734

**AMA Style**

Clyburn M, Lane CD.
Lorentz Violation at the Level of Undergraduate Classical Mechanics. *Symmetry*. 2020; 12(10):1734.
https://doi.org/10.3390/sym12101734

**Chicago/Turabian Style**

Clyburn, Madeline, and Charles D. Lane.
2020. "Lorentz Violation at the Level of Undergraduate Classical Mechanics" *Symmetry* 12, no. 10: 1734.
https://doi.org/10.3390/sym12101734