1. Mitch Begelman JILA, University of Colorado SPECIAL RELATIVITY FOR JET MODELERS

2. 2 DISTINCT EFFECTS: Lorentz transformation –Connects observers in different frames “Rest” (jet) frame → ”Lab” (observer) frame –Depends on relative speed but not location of sources –For radiation use Doppler factor Light travel-time effects –Connects different observers in same frame –Depends on location of sources (nearer, further)

3. VARIABILITY TIMESCALE COMPRESSION: A LIGHT TRAVEL-TIME (LTT) EFFECT Suppose source emits flashes 1 day apart, while moving toward you at 0.8c Flash 1 1 lt-day 0.8 lt-day Flash 2 Flash 1 Flash 2 Flash 1 0.2 lt-day Day 0 Day 1 Flashes emitted 1 day apart, received 0.2 days apart.

4. SUPERLUMINAL MOTION: THE MOST FAMOUS LIGHT TRAVEL-TIME EFFECT Consider continuously glowing blob, moving almost directly toward you at 0.8c 0.8 lt-day Day 0 Actual dist. traveled: 0.8 lt-day Apparent travel time: (1- 0.8 cos  ) day Sideways dist: 0.8 sin  Apparent sideways speed: 0.8 sin  / (1- 0.8 cos  ) c Day 1  0.8 sin  0.8 cos 

5. ANOTHER LTT EFFECT: ASYMMETRIC EXPANSION OF A SYMMETRIC SOURCE Receding Approaching... as seen in Sco X-1! (Fomalont et al. 2001)

6. ULTRARELATIVISTIC LIMIT γ>>1 Doppler factor Light travel-time factor LTT effect is more intense – why? Because Doppler factor has extra γ -1 factor, due to time dilation: “transverse Doppler shift” (not present in Newtonian Doppler shift)

7. SS 433: Mixture of LTT + Doppler precessing jets – 0.26 c “skywriting” with LTT asymmetry (VLA: Blundell & Bowler 2004) jets in plane of sky – offset from rest wavelength due to transverse Doppler effect

8. ABERRATION OF LIGHT Aberration of rain (Galilean effect) Aberration of light (Newtonian idea, corrected by Einstein)

9. Synchrotron emission from a single electron: another combination of Doppler+LTT

10. DOPPLER BEAMING 0.5 c 7x brighter 0.75 c 30x brighter 0.94 c 440x brighter 0.98 c 3100x brighter Amazing fact: power radiated (over all ν and all directions) is Lorentz- invariant! Doppler boost of each photon’s energy exactly compensates decrease in rate of photon emission due to time-dilation. Doppler Beaming effect primarily due to transformation of solid angles!!

11. ... but it’s more complicated if one looks at spectral flux or surface brightness

12. RADIATIVE TRANSFER Observer’s view Same ray as viewed by jet Complication: conditions in jet frame can change in time it takes ray to cross emitting region (simultaneity different in different frames).

13. JET-COUNTERJET RATIOS Steady jets: path length through jet and counterjet the same. Surface brightness and flux ratios both proportional to Expanding hotspots: add LTT effect – emission from near-side is received sooner, faster (and decay is sooner) obs.

14. BRIGHTNESS TEMPERATURES Direct observation of surface brightness (resolved source): Deducing brightness temp. from variability (unresolved source): Applications: synchrotron self-absorption, induced Compton scattering, intraday variability (scintillation vs. intrinsic) (radio, sub-mm); “compactness” to pair production (X-ray, gamma, TeV); synchrotron “efficiency” (cooling times, energy requirements) (X-ray, gamma)

15. COMPTON SCATTERING Compton power  radiation energy density in jet frame. 2 generic sources of seed photons: –Synchrotron self-Compton: –External radiation Compton: ~isotropic ambient radiation density boosted by factor γ 2 (γ for photon density  γ for photon energy) Applications: gamma-ray blazars, large-scale X-ray jets, Compton emission from compact radio lobes