Relativistic Jets and Apparent Superluminal Motion

How can a jet knot appear to move faster than light? A knot moving toward us emits its next photon from closer by. The second photon has less distance to travel, so we see a large sideways displacement in a short arrival interval. The knot itself stays slower than light.

Drag the viewing direction of two moving jet knots. Compare their apparent motion, unequal brightness and observed polarization.

Approaching apparent speed $v_{\rm app,+}/c=3.03$

Observed ordered-field intensity ratio $6.06\times10^{4}$

Uniform-field linear polarization $\Pi=72.4\,\%$

Observer arrival time $T/T_0=2$

Look along the jet

Knot and plasma speed

0.95

\beta=v/c

Angle from the approaching jet

20^\circ

\theta

Ordered comoving field pitch

60^\circ

\psi

Drag the gold sightline handle to change the viewing angle.

Both knots contribute to one arrival-time image.

Times, brightness and display

The two positions were emitted at different times. The main light markers propagate from those events to one common host-frame time. Up to three smaller rings per side sample steady radiation at equal host-frame emission intervals; they generally arrive at other observer times. Their three-dimensional gap is not an arrival interval. These marks leave a fixed displayed flight-length window; they have not reached a detector. Host-frame light speed is one, while playback advances the observer’s arrival clock.

\begin{gathered}T=t_{\rm em}-\boldsymbol n\cdot\boldsymbol r/c,\\c=L_0=T_0=1\end{gathered}

Intensity is dimensionless, normalized to an intrinsic emissivity coefficient of one. Glow uses one fixed logarithmic scale. Hollow rings mark positions even when a knot is too faint to see. Knot sizes, electric-vector marks and light-front rings are schematic; the position ratios and common length scale are preserved.

\ell(I)=\frac{\ln(1+I)}{\ln(1+10^4)}

The field is one uniform comoving patch per knot. Its linear polarization reaches the uniform-field limit. Pattern speed equals plasma speed in this example.

Native three-dimensional scene.

Model assumptions

Two compact, optically thin synchrotron knots move at one constant speed in flat spacetime. Pattern and radiating-plasma speeds are equal. The displayed positions are emission events whose photons reach the distant observer at the same arrival time, after subtracting the common origin-to-observer travel time. Lengths share one view scale during a run; knot sizes and electric-vector glyphs are enlarged. Glow uses a fixed logarithmic mapping of the physical ordered-field intensity; hollow rings and thin paths mark geometry, not brightness. The uniform comoving magnetic patches have a fixed optically thin electron spectrum. Pitch changes directional emission and the transformed electric-vector angle; it does not represent turbulent depolarization. The separate isotropic beaming ratio is a reference. There is no acceleration, cooling, absorption, Faraday effect, cosmological fitting or global magnetic-field reconstruction.

Approaching emissionReceding emissionGold handle: toward the observer

Small rings sample steady emission; leaving the displayed ray window does not mean reaching a detector.

Where the knots appear

ApproachingReceding

T/T_0

x_\perp/L_0

Electric vectors in the observer’s sky

ApproachingReceding

Line length gives the linear fraction; direction is defined modulo a half turn.

Read emission times and Stokes parameters

\begin{gathered}t_{\rm em,+}/T_0=18.6,\\t_{\rm em,-}/T_0=1.06\end{gathered}

\begin{gathered}\delta_+=2.91,\\\delta_-=0.165\end{gathered}

\begin{gathered}I_+=54.8,\\Q_+=20,\\U_+=34.3\end{gathered}

\begin{gathered}I_-=9.04\times10^{-4},\\Q_-=6.54\times10^{-4},\\U_-=-4.26\times10^{-5}\end{gathered}

\begin{gathered}\chi_+=29.9^\circ,\\\chi_-=178^\circ,\\V=0\end{gathered}

All three Stokes intensities share the same normalized unit. The positive first linear component uses the sky direction along the approaching jet. These uniform knots do not change their intensity or polarization with time at fixed controls.

R_{\rm iso}=4.73\times10^{4}

The isotropic beaming expression is a separate reference. The primary intensity ratio also includes the ordered field’s directional emission.

Focus question: Which interval belongs in a speed measured from images?
One-sentence intuition: Use the arrival interval. The apparent transverse speed can exceed light speed while the emitting knot remains slower than light.

Core mathematical model

The same emission event

T=t_{\rm em}-\boldsymbol n\cdot\boldsymbol r/c

Arrival time subtracts the common origin-to-observer light travel time. The sightline points from the source toward the distant observer.

Motion measured on the sky

\beta_{\rm app,+}=\frac{\beta\sin\theta}{1-\beta\cos\theta}

The numerator is transverse motion. The denominator compresses the arrival interval on the approaching side. The receding side has the opposite sign in that denominator.

Brightness with an ordered field

\begin{gathered}I=\delta^{3+\alpha}\lvert\boldsymbol n'\times\boldsymbol B'\rvert^{1+\alpha},\\\alpha=0.75\end{gathered}

This discrete-knot model combines Doppler beaming with directional synchrotron emission in the comoving frame. Dropping the angular factor gives a separate isotropic reference, not the displayed intensity.

The observed electric vector

\begin{gathered}Q=\Pi I\cos(2\chi),\\U=\Pi I\sin(2\chi),\\\Pi=\frac{21}{29}\end{gathered}

The electric-vector angle is measured in the observer’s sky. The field is perfectly ordered here, so changing its pitch changes angle and intensity while the linear fraction stays at its uniform-field limit.

Common difficulties

One picture, different emission times

Typical misconceptionThe two displayed knots are simultaneous positions in the host frame.

Better mental modelThey are the two retarded events at one arrival time. Their light markers lie on one equal-arrival plane when propagated to a common host-frame time.

A visible ring is a position mark

Typical misconceptionThe faint counterjet must be as bright as its hollow ring.

Better mental modelThe ring and thin path mark geometry. Only the glow encodes physical intensity, through the same fixed logarithmic scale on both sides.

Run the experiment

01
Watch the default pair
Let the knots move, then pause and compare their sky tracks.
What to observe: The approaching image covers more sky per arrival-time interval. Its apparent speed exceeds light speed without a superluminal emitter.
02
Move the sightline
Drag the gold handle, or use the viewing-angle slider. Compare Looking into the beam with Nearly side by side.
What to observe: The apparent speed, retarded positions and brightness ratio all change together from the same trajectory.
03
Turn the ordered field
Move the comoving field-pitch control and watch the electric-vector compass.
What to observe: The observed electric vector rotates and intensity changes. The line’s fractional length stays fixed because this model has no field disorder.
04
Keep an exact record
Pause and save this view. Inspect the two emission times, sky positions and Stokes values.
What to observe: The saved record keeps the unrounded model values and the separate isotropic beaming reference.

Relativistic Jets and Apparent Superluminal Motion

A fast image does not need a faster-than-light knot

Core mathematical model

The same emission event

Motion measured on the sky

Brightness with an ordered field

The observed electric vector

Common difficulties

One picture, different emission times

A visible ring is a position mark

Run the experiment

Watch the default pair

Move the sightline

Turn the ordered field

Keep an exact record

Related experiments and articles

Articles

How a relativistic jet reaches our telescope