Cherenkov emission by a fast-moving uncharged Schwarzschild black hole

We demonstrate that, in the presence of an external magnetic field, an uncharged classical Schwarzschild black hole moving superluminally in a dielectric with permittivity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ε</a:mi><a:mo>&gt;</a:mo><a:mn>1</a:mn></a:math> produces Cherenkov emission. This is a new physical effect: classical (nonquantum) emission of electromagnetic waves by a completely charge-neutral “particle.” The governing equations (involving general relativity, electromagnetism, and the physics of continuous media) have no external electromagnetic source—it is the distortion of the initial electromagnetic fields by the gravity of the black hole that plays the role of a superluminally moving source. The effect relies on nonzero values of both the magnetic field and the gravitational radius, as well as on the usual Cherenkov condition on the velocity, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>v</c:mi><c:mo>/</c:mo><c:mi>c</c:mi><c:mo>&gt;</c:mo><c:mn>1</c:mn><c:mo>/</c:mo><c:msqrt><c:mi>ε</c:mi></c:msqrt></c:math>. Unlike Cherenkov emission by a point charge, the effective source in this case is spatially distributed, with emission generated along the single Cherenkov emission cone. The emitted spectrum is red dominated, with power <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>∝</e:mo><e:mi>d</e:mi><e:msub><e:mi>k</e:mi><e:mi>z</e:mi></e:msub><e:mo>/</e:mo><e:mo stretchy="false">|</e:mo><e:msub><e:mi>k</e:mi><e:mi>z</e:mi></e:msub><e:mo stretchy="false">|</e:mo></e:math> for wave numbers <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">|</i:mo><i:msub><i:mi>k</i:mi><i:mi>z</i:mi></i:msub><i:mo stretchy="false">|</i:mo><i:mo>≤</i:mo><i:mn>1</i:mn><i:mo>/</i:mo><i:msub><i:mi>R</i:mi><i:mi>G</i:mi></i:msub></i:math>, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>R</m:mi><m:mi>G</m:mi></m:msub></m:math> is the Schwarzschild radius. We comment on possible observability of this process during black hole–neutron star mergers.