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Covariant Cherenkov Radiation and its Friction Force

Will Price, Martin S. Formanek, Johann Rafelski

Abstract

We derive the covariant generalization of the Frank-Tamm formula describing the Cherenkov radiation by a charged particle moving uniformly with a speed faster than the local speed of light within a homogeneous dielectric medium. We use our result to derive the covariant Cherenkov radiation reaction force and obtain a four-force explicitly orthogonal to particle four-velocity consistent with a relativistic friction force. We present the photon emission spectrum that is dependent primarily on the dielectric properties of the medium. We hint at a possible use of this work to interpret an excess of soft photons seen in relativistic hadron collisions.

Covariant Cherenkov Radiation and its Friction Force

Abstract

We derive the covariant generalization of the Frank-Tamm formula describing the Cherenkov radiation by a charged particle moving uniformly with a speed faster than the local speed of light within a homogeneous dielectric medium. We use our result to derive the covariant Cherenkov radiation reaction force and obtain a four-force explicitly orthogonal to particle four-velocity consistent with a relativistic friction force. We present the photon emission spectrum that is dependent primarily on the dielectric properties of the medium. We hint at a possible use of this work to interpret an excess of soft photons seen in relativistic hadron collisions.
Paper Structure (16 sections, 172 equations, 1 figure)

This paper contains 16 sections, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Covariant form of the constitutive equations
  4. Field of a uniformly moving particle in an isotropic medium
  5. Covariant form of the Cherenkov radiation friction force
  6. Conclusions and discussion
  7. Coordinate system
  8. Calculation of momentum space propagator and choice of gauge
  9. Contour integral for the position space propagator
  10. Proper time integral for $A^\mu$
  11. Case $n\mathcal{V} < 1$
  12. Case $n\mathcal{V}>1$
  13. Overall solution for arbitrary $n\mathcal{V}$
  14. Electromagnetic field tensor
  15. Displacement field tensor
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Schematic 2D representation of the 4D problem. Within a material medium with a constant four-velocity $\eta^\mu$ a charged particle is moving along a trajectory $z^\mu(\tau)$ (dash-dotted line). The four-velocity of the particle is a constant $u^\mu$. If the particle is superluminal within this medium it emits radiation represented by a spectral component with a four-wavevector $k^\mu$ which can be decomposed to an invariant wavenumber $\widetilde{k}$ (dashed line) and perpendicular four-vector $\widetilde{k}_\perp^\mu$ (dashed arrow). A general position four-vector $x^\mu$ can be also decomposed into components (dotted lines): (a) parallel to medium four-velocity $\widetilde{x}$ [Eq. (\ref{['eq:x_tilde']})], and in a subspace orthogonal to medium four-velocity into components (b) parallel to particle motion $x_\parallel$ [Eq. (\ref{['eq:x_parallel']})] and (c) transverse to four-velocity $x_\perp$ [Eq. (\ref{['eq:x_perp']})].