Relativistic jets compress arrival times and shift radiation

Let the emitting region move with speed βc and bulk Lorentz factor Γ relative to its host. The viewing angle θ is measured in that host frame between the velocity and the observer’s direction. Define δ as the reciprocal of Γ times one minus βcosθ. During a comoving emission interval, the source moves closer to or farther from the observer, changing photon path lengths. This produces an arrival interval proportional to one over δ, rather than merely the familiar time dilation Γ.

Cosmological redshift reduces every arriving photon frequency by one plus z and stretches arrival intervals by that same factor. For a causally coordinated approximately homogeneous emitting region, its comoving light-crossing time cannot exceed the inferred comoving variability time. This gives an upper bound on that region’s size. It does not measure the entire jet radius or the distance from the central engine. Small patches, relativistic substructure, angular patterns or external modulation can require a different causal interpretation.

Specific intensity divided by frequency cubed is invariant under local Lorentz transformations. A single unresolved isotropically emitting moving blob therefore has a bolometric isotropic-equivalent host-frame luminosity δ⁴ times its comoving luminosity, under the usual compact-blob assumptions. A fixed observer band additionally samples a shifted spectrum. Continuous jets, anisotropic external seed radiation and velocity gradients require different angular and temporal accounting. Neither an isotropic-equivalent luminosity nor that simple fourth-power boost is a universal angle-integrated jet power.

Inside the comoving plasma, use γ for an electron’s random Lorentz factor. Classical synchrotron critical frequency depends on γ², magnetic field and pitch angle; the broad single-electron spectrum peaks below that critical frequency. For an isotropic pitch-angle population the average synchrotron loss rate involves the magnetic energy density. In SI units this is B′²/(2μ₀), not the Gaussian expression B′²/(8π). Electron energy and radiation density must be evaluated in the same frame before forming a cooling time.

A useful Thomson condition for the seed photons being considered is 4γε′ much smaller than the electron rest energy, all in the comoving plasma frame. When this fails, electron-frame recoil and the Klein–Nishina cross section alter both the scattered spectrum and cooling. Extrapolating the mean Thomson γ² law into that regime can predict impossible photon energies. A finite source also needs acceleration and escape times compared in that same frame. A compact bright photon field can further absorb gamma rays by pair production, so radiation generated is not necessarily radiation escaping.

The spacetime lab checks a prerequisite Lorentz transformation only. Reset and set both controls to β=0,ct=1: with fixed x=0.6 predict transformed coordinates (0.6,1). Reset and set β=0.6,ct=1: predict (0,0.8). These are coordinates of one event, not two photon arrivals, and its sliders do not specify a viewing angle or Doppler factor. The separate path-length argument is essential for converting a source clock interval into an observed light curve.