A bright knot in a jet can appear to cross the sky faster than light because it moves toward us between the flashes we receive. The later flash has less distance to travel. Its arrival follows unusually close behind the earlier one, so a large sideways displacement appears to happen in a short time.
Imagine a camera taking two pictures of the same moving light. You divide the separation on the pictures by the time between the exposures. That is a sensible way to measure an ordinary cyclist. For a distant jet aimed partly toward you, the pictures contain a second clock: the changing travel time of the light itself.
Follow the second flash
Let a knot move at ninety-five percent of light speed. Watch it from twenty degrees away from its direction of motion. In one year measured beside the galaxy, it travels 0.95 light-years. Most of that journey brings it closer to you; about 0.325 light-years lies sideways across the sky.
A flash emitted at the end of that year starts about 0.893 light-years nearer to you than a flash emitted at its beginning. Light takes about 0.893 years less to make the second trip. Your telescope therefore receives the two flashes only about 0.107 years apart. Divide the sideways distance by that short interval and the apparent speed is about three times light speed.
The knot has travelled less than one light-year in a year beside the galaxy. Every flash has travelled at light speed. The surprisingly large number belongs to a distance across the sky divided by an interval between arrivals.
An original geometric example: during one source-frame year the knot advances 0.95 light-years. The later flash saves about 0.893 years of travel, leaving an arrival interval of about 0.107 years. The sideways displacement is about 0.325 light-years. Distances are schematic; the numbers follow the stated geometry.
The back jet tells the same story in reverse
Now launch an identical knot in the opposite direction. Its second flash starts farther away. The same one-year journey produces an arrival interval of about 1.893 years, and its apparent sideways speed is only about 0.172 times light speed. Two knots with equal speeds can therefore look very unequal in a sequence of images.
A photograph taken at one telescope time also samples different ages. Light from the approaching knot can have been emitted much later in the galaxy than light from the receding knot. The two sides of the image are a view of equal arrival time, not a simultaneous photograph made beside the galaxy.
For readers who want the compact rule, let β be the actual speed divided by light speed, and θ the angle between the approaching motion and our line of sight. The apparent transverse speed, also divided by light speed, is
βapp=1−βcosθβsinθ.
The numerator is the sideways part of the motion. The denominator tells us how much the interval between arrivals shrinks. The next chapter derives both from two emission events.
Looking straight down the jet is a different limit
Tilting a jet toward us initially increases the apparent speed. A jet pointed exactly at us, however, has no sideways displacement at all. Its apparent transverse speed is zero. The largest apparent speed occurs at a small but nonzero viewing angle; brightness and sideways motion need not peak at the same angle.
A useful test is to picture the jet in the plane of the sky. The knot then neither approaches nor recedes, so there is no changing light-travel distance. The arrival interval equals the galaxy-frame interval, and the apparent sideways speed equals the actual speed. These two limits explain why “more face-on means faster” eventually fails.
What is the moving thing?
The knot is a bright feature that we can follow from image to image. It might mark a concentration of emitting plasma, or a pattern such as a shock moving through that plasma. Tracking the feature measures the pattern's apparent motion. Inferring the speed of the radiating matter requires a physical relation between that matter and the pattern.
Our two-knot example assumes that the feature and emitting plasma move together, each at a constant speed. It isolates the travel-time effect. Astronomers combine repeated images with spectra and polarization when they ask how a real jet carries matter and energy.
Try this thought experiment before calculating anything: leave the knot's actual speed unchanged but move the observer from twenty degrees to a sideways view. The apparent speed falls from roughly three times light speed to ninety-five percent of light speed. The material has not slowed down; the second flash has lost its shorter route.
Sources
C. M. Urry and P. Padovani, Unified Schemes for Radio-Loud Active Galactic Nuclei (1995), author review hosted by NED, Appendix A, especially equation A4 and its limiting angles. Kinematic derivation and limits.
The same authors' Appendix B explains the connection between catching up to earlier photons and the arrival intervals of a moving source. Arrival intervals and Doppler enhancement.
