# (Gravitational) Vacuum Cherenkov Radiation

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of the Process

## 3. Properties of Vacuum Cherenkov Radiation in Minkowski Spacetime

#### 3.1. Classical Description

#### 3.2. Quantum Effects

#### 3.3. Radiation of Particles Other Than Photons

## 4. Cherenkov-Type Radiation in Modified Gravity

#### 4.1. Lorentz Violation in Gravity

#### 4.2. Gravitational Vacuum Cherenkov Radiation

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Standard nullcone (black, dashed), standard mass shell (blue), and modified photon dispersion relation (red) in momentum space with the momentum $\mathbf{p}=({p}^{1},{p}^{2},{p}^{3})$. Photon emissions from a standard Dirac fermion with a certain energy are indicated by green, wiggly curves. (

**a**) CPT-even isotropic modification of the photon sector with ${\tilde{\kappa}}_{\mathrm{tr}}=3/5$. The wiggly lines run parallel to parts of the modified nullcone. In this particular example, two subsequent photon emissions are possible until the process ceases; (

**b**) CPT-odd Maxwell–Chern–Simons (MCS) theory with ${k}_{AF}^{3}=5{m}_{\psi}$ where only one of the two modified photon dispersion laws is illustrated. The wiggly lines run along properly shifted parts of the deformed nullcone such that the minimum is at the lower one of the two points that it connects. Two subsequent photon emissions are shown.

**Figure 2.**Standard nullcone (black), standard mass shell (blue, dashed), and modified mass shell (red) in momentum space. (

**a**) isotropic spin-degenerate modification of Dirac theory with ${c}_{00}=-1/2$. From the initial energy of the modified fermion, two photon emissions are possible until the fermion stops radiating; (

**b**) isotropic spin-nondegenerate modification of Dirac theory with ${b}_{0}={m}_{\psi}$. There are two distinct modified dispersion laws. Photon emission can occur from one to the other branch.

**Figure 3.**(

**a**) two-sphere (${S}^{2}$) parameterized in terms of the polar angle $\vartheta $ and the azimuthal angle $\phi $. The coordinate lines of constant $\vartheta $ are circles parallel to the equator where the coordinate lines of constant $\phi $ are great circles linking the north and south poles with each other; (

**b**) illustration of a general coordinate transformation for ${S}^{2}$ that deformes the original coordinate lines; (

**c**) illustration of a diffeomorphism for ${S}^{2}$. To distinguish it from a general coordinate transformation, the unchanged coordinate lines are not shown. The graphics demonstrates how an original geodesic connecting the north and the south pole gets deformed by the diffeomorphism; (

**d**) ${S}^{2}$ endowed with an additional vector field that could correspond to a matter field in physics; (

**e**) general coordinate transformation applied to ${S}^{2}$. The coordinate lines are deformed, but the vector field itself stays untouched. As the coordinates change, however, the explicit representation of the vector field changes as well; (

**f**) diffeomorphism applied to ${S}^{2}$ with an intrinsic vector field. In contrast to before, the coordinates remain unchanged, but the vector field transforms in a nontrivial manner.

**Figure 4.**(

**a**) tangent plane of ${S}^{2}$ with coordinate lines shown; (

**b**) observer Lorentz transformation rotating the coordinate lines; (

**c**) diffeomorphism (translation) that simply maps the points of a geodesic (straight line) to the points of a parallel geodesic. The unmodified coordinate lines are omitted for clarity; (

**d**) tangent plane endowed with a vector field; (

**e**) observer Lorentz transformation that changes the coordinate lines, but leaves the vector field untouched; (

**f**) particle Lorentz transformation that induces a nontrivial transformation of the vector field. The coordinate lines are not modified.

**Table 1.**Compilation of constraints on SME coefficients obtained from the absence of Cherenkov-type radiation in the vacuum. Most of the constraints can be found in the data tables [32]. As the number of constraints in the gravity sector is extensive, approximate constraints on groups of coefficients are given where $0<j\le d-2$ (with the mass dimension d) and $0\le m\le j$.

Sector | Constraint | Reference |
---|---|---|

Proton | $-5\times {10}^{-23}<{\stackrel{\u02da}{c}}^{\mathrm{UR}\left(4\right)}$ | [12] |

Proton, Photon | ${\tilde{\kappa}}_{\mathrm{tr}}-(4/3){c}_{00}^{\left(4\right)\mathrm{p}}<6\times {10}^{-20}$ | [19] |

Electron, Photon | ${\tilde{\kappa}}_{\mathrm{tr}}-(4/3){c}_{00}^{\left(4\right)\mathrm{e}}<1.2\times {10}^{-11}$ | [20] |

Gravity | ${\overline{s}}_{00}^{\left(4\right)}>-3\times {10}^{-14}$ | [25] |

Gravity | $|{\overline{s}}_{jm}^{\left(4\right)}|\lesssim {10}^{-13}$ | [25] |

Gravity | ${\overline{s}}_{00}^{\left(6\right)}<2\times {10}^{-31}\phantom{\rule{0.166667em}{0ex}}Ge{V}^{-2}$ | [25] |

Gravity | $|{\overline{s}}_{jm}^{\left(6\right)}|\lesssim {10}^{-29}\phantom{\rule{0.166667em}{0ex}}Ge{V}^{-2}$ | [25] |

Gravity | ${\overline{s}}_{00}^{\left(8\right)}>-7\times {10}^{-49}\phantom{\rule{0.166667em}{0ex}}Ge{V}^{-4}$ | [25] |

Gravity | $|{\overline{s}}_{jm}^{\left(8\right)}|\lesssim {10}^{-45}\phantom{\rule{0.166667em}{0ex}}Ge{V}^{-4}$ | [25] |

Pion | ${\delta}^{\pi}>-7\times {10}^{-13}$ | [26] |

Quark, Photon | $-3\times {10}^{-23}\le {c}_{00}^{\left(4\right)\mathrm{u}}-(3/4){\tilde{\kappa}}_{\mathrm{tr}}-(3/8){\left({f}_{0}^{\left(4\right)\mathrm{u}}\right)}^{2}$ | [28] |

Quark | $|{d}_{00}^{\left(4\right)\mathrm{u}}|<3\times {10}^{-23}$ | [28] |

Quark | $|{e}_{0}^{\left(4\right)\mathrm{u}}|<9\times {10}^{-12}$ | [28] |

Quark | $|{\stackrel{\u02da}{g}}_{1}^{\left(4\right)\mathrm{u}}|<9\times {10}^{-12}$ | [28] |

W boson | $|{\left({k}_{1}\right)}^{\mu}|<1.7\times {10}^{-8}\phantom{\rule{0.166667em}{0ex}}GeV$ | [29] |

W boson | $|{\left({k}_{2}\right)}^{\mu}|<1.1\times {10}^{-7}\phantom{\rule{0.166667em}{0ex}}GeV$ | [29] |

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**MDPI and ACS Style**

Schreck, M.
(Gravitational) Vacuum Cherenkov Radiation. *Symmetry* **2018**, *10*, 424.
https://doi.org/10.3390/sym10100424

**AMA Style**

Schreck M.
(Gravitational) Vacuum Cherenkov Radiation. *Symmetry*. 2018; 10(10):424.
https://doi.org/10.3390/sym10100424

**Chicago/Turabian Style**

Schreck, Marco.
2018. "(Gravitational) Vacuum Cherenkov Radiation" *Symmetry* 10, no. 10: 424.
https://doi.org/10.3390/sym10100424