Superluminal Motion in Jets

Point a very-long-baseline radio array at the heart of a quasar and, over a few years, you can watch it move. The parsec-scale jet resolves into a stationary "core" and bright "knots" that drift outward, and converting their angular proper motion into a transverse velocity using the source's distance gives a number routinely several times the speed of light — close to forty times $c$ in the most extreme sources. This is superluminal motion, and after its discovery in 1971 it briefly looked like a crisis for relativity. It is not. The resolution needs no new physics: it is a pure light-travel-time projection effect for material streaming toward us at nearly $c$ . Because an approaching blob almost keeps pace with the photons it emits, successive flashes arrive bunched together, and the sky-projected displacement divided by that compressed interval can exceed $c$ though nothing physical does. Remarkably, Martin Rees wrote the prediction down in 1966, five years before anyone saw it. What the observations added was a measurement: a single apparent speed places a hard lower bound on the bulk Lorentz factor, $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ , turning "are these jets relativistic?" into a settled yes. The same geometry that fakes faster-than-light motion also Doppler-boosts the approaching jet, dims its receding twin below detection, and swells the variability — so superluminal sources are the pole-on blazars of the AGN unified model. Apparent kinematics and observed luminosity are two readings of one orientation, and this article derives the effect from scratch, extracts the Lorentz factor and viewing angle from data, and traces the line from Rees to the MOJAVE survey.

Key takeaways

Compact jet knots in quasars and radio galaxies cross the sky at apparent transverse speeds $\beta_{\rm app}$ well above 1, sometimes reaching $\beta_{\rm app}\sim 10$ – $40$ in units of $c$ .
It is a geometric illusion: plasma at bulk speed $\beta\to 1$ at a small angle $\theta$ to the line of sight nearly overtakes its own light, compressing photon arrival times by $(1-\beta\cos\theta)$ .
The apparent speed is $\beta_{\rm app}=\dfrac{\beta\sin\theta}{1-\beta\cos\theta}$ , peaking at $\cos\theta=\beta$ at $\beta_{\rm app,max}=\beta\gamma$ , with $\gamma=(1-\beta^2)^{-1/2}$ the bulk Lorentz factor.
A single measured $\beta_{\rm app}$ forces a hard bound $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ — kinematic proof that AGN jets carry highly relativistic plasma.
The identical geometry drives relativistic Doppler beaming, tying superluminal sources to blazars and the orientation-based AGN unified model.
Key idea: an emitter moving at $\beta\to 1$ at a small angle $\theta$ to the line of sight nearly catches its own emitted light, so the observed interval between two positions is compressed by $(1-\beta\cos\theta)$ ; the sky-projected speed $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ can then exceed unity though the true speed never does.

The core insight

An emitter moving at $\beta\to 1$ at a small angle $\theta$ to the line of sight nearly overtakes its own emitted light, so the observed time between two positions is compressed by the factor $(1-\beta\cos\theta)$ ; the sky-projected speed $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ can then exceed unity even though the true speed cannot.

Where this lives today

The modern engine of the field is systematic, multi-epoch VLBI monitoring. The MOJAVE program has tracked hundreds of AGN jets at 15 GHz for over two decades, measuring proper motions, accelerations, and knot bending. Its apparent-speed distribution — most sources modestly superluminal, a tail reaching $\beta_{\rm app}\sim40$ — is just what an orientation-selected population with intrinsic $\gamma\sim5$ – $40$ should produce, and fitting it with Doppler factors yields the speed and orientation statistics that anchor the AGN unified model. Superluminal motion is now a routine diagnostic: with a Doppler factor it delivers $\gamma$ and $\theta$ (worked example 2), and its mere presence certifies a jet as relativistic. The Event Horizon Telescope connects the jet base in M87 near the black hole to the superluminal knots further out, while Fermi links the fastest blazar jets to the flaring gamma-ray sky. Open questions remain sharp — how and where jets accelerate to these Lorentz factors, and whether a structured "spine–sheath" flow shows up in the kinematics — but the backbone, Rees's 1966 projection effect, is settled physics.

The physical picture

Picture a blob that emits a flash at position A, then moves toward you and emits a second at B. Because it has travelled closer to you in between, the second flash has less distance to cover, so the two reach you separated by less time than the true flight time from A to B. Divide the real transverse displacement by that compressed interval and the quotient — the apparent sky speed — can exceed $c$ . Nothing outran light; the messenger simply ran alongside the source. The first figure fixes the geometry and reads the formula off directly.

Figure

A blob moves at speed v at angle θ to the line of sight. Between two flashes it advances Δz = vΔt cosθ toward the observer, so the second flash arrives only Δt_obs = Δt(1 − βcosθ) later, while it sweeps Δx⊥ = vΔt sinθ across the sky — hence β_app = βsinθ ⁄ (1 − βcosθ). Angle exaggerated for clarity.

Two competing factors are the whole story. The numerator's $\sin\theta$ vanishes head-on — a blob aimed straight at you shows no transverse motion. The denominator's $1/(1-\beta\cos\theta)$ blows up head-on, where compression is strongest. Their product peaks at the "sweet spot" $\cos\theta=\beta$ , where $\beta_{\rm app}=\beta\gamma\approx\gamma$ . The apparent speed is thus a Lorentz-factor detector: to fake a large $\beta_{\rm app}$ the flow must genuinely move at $\gamma\gtrsim\beta_{\rm app}$ . The second figure plots this and marks the peaks.

Figure

Apparent transverse speed β_app versus viewing angle θ for bulk Lorentz factors γ = 5 and γ = 10. Each curve climbs past the β_app = 1 threshold, peaks at cosθ = β where β_app = βγ (9.95 at θ = 5.7° for γ = 10, and 4.90 at θ = 11.5° for γ = 5), then falls back to β near θ = 90°. The peak apparent speed is essentially the Lorentz factor itself.

The amber curve ( $\gamma=5$ ) tops out near 4.9, the blue curve ( $\gamma=10$ ) near 9.95: doubling the Lorentz factor doubles the peak apparent speed while shrinking the angle at which it occurs, since $\sin\theta_{\rm max}=1/\gamma$ . Fast jets beam their fastest motion into an ever-narrower cone — which is why the superluminal sky is dominated by the rare few pointed almost straight at us.

The core equations

The master relation follows from the first figure: transverse displacement over light-compressed observed time. Everything downstream — maximum, Lorentz bound, Doppler links — is algebra on this one expression.

\beta_{\rm app}=\frac{\beta\sin\theta}{1-\beta\cos\theta} \qquad (1)

where: $\beta_{\rm app}$ — apparent transverse speed in units of $c$ ; $\beta=v/c$ — true bulk speed; $\theta$ — angle between the velocity and the line of sight. Reading it: the denominator is the compression factor, always below 1 for an approaching source, and it pushes the ratio above unity even though the numerator $\beta\sin\theta$ is itself below 1. Maximising equation (1) over $\theta$ at fixed $\beta$ gives a clean optimum: the largest apparent speed a jet of Lorentz factor $\gamma$ can display is $\beta\gamma$ , when it points a little off the line of sight.

\beta_{\rm app,max}=\beta\gamma,\qquad \text{at}\quad\cos\theta=\beta,\ \ \sin\theta=\frac{1}{\gamma} \qquad (2)

where: $\gamma=(1-\beta^2)^{-1/2}$ — bulk Lorentz factor. Reading it: for a relativistic jet $\beta\to1$ , so $\beta_{\rm app,max}\approx\gamma$ — the peak apparent speed reads off the Lorentz factor, at an optimal angle of roughly $1/\gamma$ radians. Because the observed value can never exceed that maximum, a single $\beta_{\rm app}$ — viewing angle unknown — already forces a floor on $\gamma$ , the most model-independent statement in the subject.

\gamma\ \ge\ \sqrt{1+\beta_{\rm app}^2} \qquad (3)

where: $\beta_{\rm app}$ — measured apparent speed; $\gamma$ — minimum Lorentz factor consistent with it. Reading it: $\beta_{\rm app}=10$ demands $\gamma\ge\sqrt{101}\approx10.05$ ; no angle lets a slow jet fake a fast one. Turning a raw observation into $\beta_{\rm app}$ needs cosmology: an interferometer measures an angular proper motion $\mu$ , not a velocity, and distance plus a redshift factor convert it.

\mu=\frac{\beta_{\rm app}\,c}{D_A\,(1+z)} \qquad (4)

where: $\mu$ — angular proper motion (rad s $^{-1}$ ); $D_A$ — angular-diameter distance; $z$ — redshift, whose $(1+z)$ converts observed to emitted time. Reading it: two sources at equal true speed but different $z$ show different $\mu$ , so a cosmological model is unavoidable when quoting $\beta_{\rm app}$ . The same geometry also boosts the flux, through the relativistic Doppler factor — the same denominator as equation (1), now controlling brightness.

\delta=\frac{1}{\gamma\,(1-\beta\cos\theta)} \qquad (5)

where: $\delta$ — relativistic Doppler factor. Reading it: near the optimal angle $\delta\approx\gamma$ , so a $\gamma=15$ jet is boosted by $\delta\sim15$ in frequency and time, and by a large power in flux. Both $\beta_{\rm app}$ and $\delta$ are functions of the same $(\beta,\theta)$ , so they satisfy exact identities; measuring both pins down $\gamma$ and $\theta$ .

\beta_{\rm app}=\gamma\,\delta\,\beta\sin\theta,\qquad \beta_{\rm app}^2+\delta^2=2\gamma\delta-1 \qquad (6)

where: $\delta$ — Doppler factor; $\beta_{\rm app}$ — apparent speed. Reading it: the second identity inverts to $\gamma=(\beta_{\rm app}^2+\delta^2+1)/(2\delta)$ , from which $\beta$ and $\theta$ follow — the standard route to full jet kinematics. (This corrects a common misprint: the left identity is $\beta_{\rm app}=\gamma\delta\beta\sin\theta$ , since $\gamma\delta=1/(1-\beta\cos\theta)$ .) The flux boost scales as a power of the Doppler factor, the exponent set by geometry and the synchrotron spectrum.

S_{\rm obs}=\delta^{\,p}\,S_{\rm em},\qquad p=3-\alpha\ \ (\text{discrete blob}),\quad p=2-\alpha\ \ (\text{continuous jet}) \qquad (7)

where: $S_{\rm obs},S_{\rm em}$ — observed and rest-frame flux densities; $\alpha$ — spectral index, $S_\nu\propto\nu^{\alpha}$ . Reading it: a synchrotron jet with $\alpha\approx-0.7$ has $p\approx2.7$ – $3.7$ , so even modest $\delta$ boosts enormously. Applied to a symmetric jet's two sides, equation (7) predicts a colossal asymmetry — why parsec-scale jets look one-sided.

\frac{S_{\rm jet}}{S_{\rm counterjet}}=\left(\frac{1+\beta\cos\theta}{1-\beta\cos\theta}\right)^{p} \qquad (8)

where: $S_{\rm jet},S_{\rm counterjet}$ — flux of approaching versus receding jet. Reading it: for $\beta\approx1$ , small $\theta$ this reaches millions, so the counterjet drops below detection — the geometry that fakes superluminal motion also hides the far side.

Derivation AThe superluminal formula from light-travel time

The goal is to derive the master relation $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ from first principles, using nothing but the geometry of two emission events and the finite speed of light — no velocity addition, no Lorentz transformation, just bookkeeping of when photons leave the blob and when they reach us. We compare the true transverse motion to the compressed interval over which we receive the light.

Step 1. The blob emits a flash at coordinate time $t_1=0$ and a second at $t_2=\Delta t$ , moving a coordinate distance $v\,\Delta t$ in between. Why: two events define a motion, and coordinate (source-frame) time keeps the light-travel effect separate from any intrinsic clock effect.

Step 2. Resolve onto the line of sight: the blob advances $\Delta z=v\,\Delta t\cos\theta$ toward the observer between flashes. Why: this approach shortens the second photon's path and so shrinks the interval over which the flashes are received.

Step 3. The second photon starts $\Delta z$ nearer, so it needs $\Delta z/c$ less travel time; the received interval is $\Delta t_{\rm obs}=\Delta t-\Delta z/c=\Delta t\,(1-\beta\cos\theta)$ . Why: this single factor $(1-\beta\cos\theta)$ is the whole physics of the illusion — as $\beta\to1$ , $\theta\to0$ it drives the observed interval toward zero.

Step 4. Resolve perpendicular to the line of sight: the transverse motion an interferometer measures is $\Delta x_\perp=v\,\Delta t\sin\theta$ . Why: only motion across the sky gives the changing angular position we call proper motion; line-of-sight motion shows up only through the Step 3 timing effect.

Step 5. Divide the transverse displacement by the received interval and normalise by $c$ : $\beta_{\rm app}=\Delta x_\perp/(c\,\Delta t_{\rm obs})=\beta\sin\theta/(1-\beta\cos\theta)$ . Why: this is the ratio an observer forms, and it exceeds 1 whenever $\beta\sin\theta>1-\beta\cos\theta$ — easily met by fast jets at small angles. *Note: $\beta_{\rm app}>1$ requires only $\beta>1/(\sin\theta+\cos\theta)$ — for example, $\beta>0.71$ at $\theta=45°$ — so even modestly relativistic jets can appear superluminal at favourable angles.*

Go deeper · Derivation B · 1 minThe optimal viewing angle and the peak apparent speed

Derivation BThe optimal viewing angle and the peak apparent speed

The goal is to find the viewing angle that maximises the apparent speed at fixed bulk speed, and to evaluate the maximum there. This matters because the largest apparent speeds across a jet population trace an envelope set by the true Lorentz factors, so knowing where and how high the curve peaks lets us read $\gamma$ almost directly off the fastest sources. We differentiate $\beta_{\rm app}(\theta)$ at fixed $\beta$ , solve for the stationary point, and substitute back.

Step 1. Differentiate $\beta_{\rm app}(\theta)=\beta\sin\theta/(1-\beta\cos\theta)$ by the quotient rule, giving the numerator $\beta\cos\theta\,(1-\beta\cos\theta)-\beta^2\sin^2\theta$ over $(1-\beta\cos\theta)^2$ . Why: the maximum is a stationary point, so setting the numerator to zero locates it.

Step 2. Expand: $\beta\cos\theta-\beta^2\cos^2\theta-\beta^2\sin^2\theta=\beta(\cos\theta-\beta)$ , which vanishes at $\cos\theta_{\rm max}=\beta$ . Why: the Pythagorean identity collapses the trigonometric terms into a constant $\beta^2$ , so the optimum is startlingly simple — the cosine of the best angle equals the speed itself.

Step 3. At the optimum, $\sin\theta_{\rm max}=\sqrt{1-\beta^2}=1/\gamma$ and $1-\beta\cos\theta_{\rm max}=1-\beta^2=1/\gamma^2$ . Why: expressing both through $\gamma$ makes the substitution a one-line cancellation and shows the optimal angle shrinks as $1/\gamma$ .

Step 4. Substitute back: $\beta_{\rm app,max}=\beta(1/\gamma)/(1/\gamma^2)=\beta\gamma$ . Why: for $\gamma\gg1$ this is $\approx\gamma$ , so the peak apparent speed is essentially the Lorentz factor — the key to interpreting the fastest sources. *The calculus confirms the textbook maximum: $\gamma=10$ gives $\beta_{\rm app,max}=9.95$ at $\theta=5.7°$ , and $\gamma=5$ gives $4.90$ at $\theta=11.5°$ (Figure 2).*

Go deeper · Derivation C · 1 minThe kinematic lower bound on the Lorentz factor

Derivation CThe kinematic lower bound on the Lorentz factor

The goal is to extract the most robust statement from a single apparent-speed measurement when the viewing angle is unknown: the smallest bulk Lorentz factor that could reproduce it. This is what makes superluminal motion evidence rather than curiosity — it converts one measured number into a hard floor on the jet's true speed, and historically it settled whether AGN jets are relativistic. We need only the peak relation of the previous derivation and one identity.

Step 1. The observed speed cannot exceed the maximum for the true Lorentz factor: $\beta_{\rm app}\le\beta_{\rm app,max}=\beta\gamma$ . Why: whatever the unknown angle, equation (2) caps the apparent speed at $\beta\gamma$ , so the measurement constrains $\beta\gamma$ from below.

Step 2. Square and use $\beta^2\gamma^2=(1-\gamma^{-2})\gamma^2=\gamma^2-1$ , giving $\beta_{\rm app}^2\le\gamma^2-1$ . Why: the identity eliminates $\beta$ , leaving a clean inequality in $\gamma$ alone — no need to know speed and angle separately.

Step 3. Solve: $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ . Why: this bound means $\beta_{\rm app}\gg1$ forces $\gamma\gtrsim\beta_{\rm app}$ , proving the emitting matter carries relativistic bulk kinetic energy and is not a slow-moving pattern. *For example, $\beta_{\rm app}=8\Rightarrow\gamma\ge\sqrt{65}\approx8.06$ and $\beta_{\rm app}=10\Rightarrow\gamma\ge10.05$ , consistent with the $\gamma\sim5$ – $40$ range seen across VLBI surveys.*

Limiting cases

Reading the master relation $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ in its extreme regimes shows both when the effect switches on and why it must vanish in the everyday limit.

Case	Result	Meaning
Non-relativistic ( $\beta\ll1$ )	$\beta_{\rm app}\approx\beta\sin\theta\ll1$	Ordinary projection; the apparent speed is just the true transverse speed, always sub-luminal.
In the plane of the sky ( $\theta=90^\circ$ )	$\beta_{\rm app}=\beta<1$	No line-of-sight motion, so no arrival-time compression and no enhancement.
Head-on ( $\theta\to0$ )	$\beta_{\rm app}\to0$	Maximum time compression but zero transverse displacement; the knot barely shifts on the sky.
Optimal angle ( $\cos\theta=\beta$ )	$\beta_{\rm app,max}=\beta\gamma\approx\gamma$	Largest apparent speed for a given $\gamma$ , at $\sin\theta=1/\gamma$ .
Ultra-relativistic near optimum	$\beta_{\rm app}\approx\gamma\approx\delta$	Apparent speed, Doppler factor, and Lorentz factor coincide — the blazar regime.

Common trapSuperluminal motion breaks special relativity

It does not, and the point is worth stating flatly: no matter, energy, or signal in these jets ever travels faster than light. The physical speed obeys $\beta<1$ always; only the apparent transverse speed $\beta_{\rm app}$ exceeds 1, purely because the emitting blob nearly keeps pace with its own photons, compressing their arrival times. $\beta_{\rm app}$ is not a velocity in any frame — it is a sky-projected distance over a light-distorted observed time, and it may exceed $c$ for the same reason a searchlight spot can sweep a distant wall faster than light. A subtler trap: $\beta_{\rm app}$ is not the true speed, and you cannot read $\beta$ off it without the viewing angle. A lone measurement yields only $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ ; recovering the actual $\gamma$ and $\theta$ needs a second observable, such as the Doppler factor.

Worked examples

1 · From proper motion to Lorentz factor (fully worked)

A quasar at $z=0.5$ has a jet knot drifting at $\mu=0.2$ mas yr $^{-1}$ . In a standard flat cosmology ( $H_0=70$ , $\Omega_m=0.3$ ) the angular-diameter distance is $D_A\approx1267$ Mpc. Find the apparent speed and minimum Lorentz factor.

Step 1. Convert $\mu$ to SI. One mas is $10^{-3}/206265$ rad, one year $3.156\times10^{7}$ s, so $\mu=0.2\times10^{-3}/206265/(3.156\times10^{7})=3.07\times10^{-17}$ rad s $^{-1}$ .

Step 2. Distance: $D_A=1267\times3.086\times10^{22}=3.91\times10^{25}$ m.

Step 3. Equation (4): $\beta_{\rm app}=\mu D_A(1+z)/c=(3.07\times10^{-17})(3.91\times10^{25})(1.5)/(2.998\times10^{8})=\mathbf{6.0}$ — six times $c$ .

Step 4. Apply bound (3): $\gamma\ge\sqrt{1+36}=\sqrt{37}=\mathbf{6.08}$ . If the jet sits at the optimal angle the bound is tight, with $\sin\theta=1/\gamma=0.164$ , $\theta\approx\mathbf{9.4^\circ}$ . Result: $\beta_{\rm app}=6.0$ , so the plasma moves at $\gamma\ge6.1$ — genuinely relativistic.

2 · Solving for the jet's speed and angle (fully worked)

VLBI gives a blazar knot $\beta_{\rm app}=15$ , and variability plus the synchrotron self-Compton limit give a Doppler factor $\delta=12$ . Find the bulk Lorentz factor, true speed, and viewing angle.

Step 1. Solve equation (6)'s second identity for $\gamma$ : $\gamma=(\beta_{\rm app}^2+\delta^2+1)/(2\delta)=(225+144+1)/24=\mathbf{15.4}$ . One inversion delivers the Lorentz factor from the two observables.

Step 2. Recover the speed: $\beta=\sqrt{1-1/\gamma^2}=\mathbf{0.9979}$ , i.e. $99.79\%$ of $c$ .

Step 3. Recover the angle from $\delta=1/[\gamma(1-\beta\cos\theta)]$ : $1-\beta\cos\theta=1/(\gamma\delta)=1/185.0$ , so $\cos\theta=(1-1/185.0)/\beta=0.9967$ , $\theta\approx\mathbf{4.7^\circ}$ . Just inside the optimal $\theta_{\rm max}=\arcsin(1/\gamma)\approx3.7^\circ$ , so both observables are large at once.

Step 4. Check: $1-\beta\cos\theta=0.00541$ gives $\beta_{\rm app}=15.0$ ✓ and $\delta=1/(15.4\times0.00541)=12.0$ ✓. Result: $\gamma\approx15.4$ , $\beta\approx0.998$ , $\theta\approx4.7^\circ$ — a fast jet pointed almost at us.

3 · Why the counterjet disappears

Take the same jet, $\beta=0.998$ at $\theta=5^\circ$ , with a synchrotron spectrum $\alpha=-0.7$ so the continuous-jet exponent is $p=2-\alpha=2.7$ . Estimate the approaching-to-receding brightness ratio.

Step 1. With $\beta\cos\theta=0.998\times0.99619=0.9942$ , the bracket is $(1+\beta\cos\theta)/(1-\beta\cos\theta)=1.9942/0.00582=\mathbf{343}$ .

Step 2. Raise to $p=2.7$ : $S_{\rm jet}/S_{\rm counterjet}=343^{2.7}\approx\mathbf{7\times10^{6}}$ . The approaching jet outshines its receding twin seven million to one — the counterjet is undetectable, which is why superluminal jets look one-sided. (For a discrete blob, $p=3.7$ gives $\sim2\times10^{9}$ .)

Physics of the portrait

Superluminal motion is one of the rare cases where theory arrived before data. Five years before any interferometer could catch a jet moving, a graduate student wrote down the projection effect, the formula, and the prediction that expanding radio sources should appear to fly apart faster than light.

Origins · 1 minMartin Rees and how this was discovered

Martin Rees

1942– · relativistic astrophysics, black holes, and cosmology

Martin Rees was a 24-year-old research student at Cambridge when, in 1966, he published a short Nature paper titled "Appearance of Relativistically Expanding Radio Sources." The radio astronomy of the day had found quasars varying so fast that, taken naïvely, the emitting region seemed to be expanding at many times the speed of light. Rees showed no such thing was needed: a source moving relativistically toward the observer would look superluminal purely through light-travel-time compression, and he wrote down the geometry that gives $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ — a prediction with almost no supporting evidence, since the resolving power to test it did not yet exist. When very-long-baseline interferometry matured around 1971, the archetypal quasars 3C 273 and 3C 279 were resolved into components separating superluminally, precisely as his picture required. Rees went on to co-author, with Blandford and McKee, the relativistic-beaming framework that ties superluminal motion to Doppler boosting and jet one-sidedness, and became one of the architects of modern high-energy astrophysics — arguing early that quasars are powered by accretion onto supermassive black holes, and later ranging across gamma-ray bursts and cosmology as Astronomer Royal and President of the Royal Society. The superluminal prediction remains a textbook case of theory-led astrophysics: a back-of-envelope relativity argument that dissolved an apparent paradox before the instruments could even pose it.

History

1966

— Rees predicts that relativistically expanding radio sources will appear to move superluminally

1971

— early VLBI resolves 3C 279 and 3C 273 into components separating faster than light

1977

— Blandford, McKee and Rees set out the relativistic-beaming interpretation of the phenomenon

1979

— Blandford and Königl publish the relativistic jet model unifying superluminal motion with one-sidedness

1994

— the Very Long Baseline Array begins routine multi-epoch monitoring of AGN jet kinematics

2018

— the MOJAVE survey catalogues apparent speeds and accelerations for hundreds of AGN jets

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §28 (Active Galaxies) — the standard treatment of superluminal motion and beaming · Rybicki & Lightman, Radiative Processes in Astrophysics, §4.8 (relativistic beaming) · Dodelson, Modern Cosmology (angular-diameter distance, equation 4) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 14 (AGN and unification) · Binney & Tremaine, Galactic Dynamics · Ryden, Introduction to Cosmology, 2nd ed.

Primary sources. Rees 1966, Nature 211, 468 (the prediction) · Blandford, McKee & Rees 1977, Nature 267, 211 (beaming) · Blandford & Königl 1979, ApJ 232, 34 (jet model) · Lister et al. (MOJAVE) 2019, ApJ — VLBI apparent-speed statistics. The kinematics rest on pure special-relativistic light-travel-time geometry. The key relations are: (1) $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ ; (2) the maximisation giving $\cos\theta=\beta$ and $\beta_{\rm app,max}=\beta\gamma$ ; (3) the bound $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ , derived via $\beta^2\gamma^2=\gamma^2-1$ ; (4) the identity $\beta_{\rm app}^2+\delta^2=2\gamma\delta-1$ and its inversion $\gamma=(\beta_{\rm app}^2+\delta^2+1)/(2\delta)$ ; (5) the corrected left identity $\beta_{\rm app}=\gamma\delta\beta\sin\theta$ , since $\gamma\delta=1/(1-\beta\cos\theta)$ . Numerical anchors: $\beta_{\rm app}=6.0$ from $\mu=0.2$ mas yr $^{-1}$ , $z=0.5$ , $D_A=1267$ Mpc; the joint solve $\beta_{\rm app}=15,\ \delta=12\Rightarrow\gamma=15.4,\ \theta=4.7^\circ$ ; and the counterjet brightness ratio $\sim7\times10^{6}$ .

Prerequisites. Special Relativity and the Lorentz Factor · Relativistic Jets · AGN: the Unified Model · Cosmological Distances and Redshift. Leads to. Blazars and Relativistic Beaming · Black Hole Spin · Quasars.

Ask the tutor: Give me a measured apparent speed (and a Doppler factor if you have one) and I will walk you through extracting the jet's Lorentz factor, true speed, and viewing angle step by step.

Does superluminal motion mean something actually moves faster than light?

No. The physical bulk speed of the jet plasma is always below $c$ — in the worked examples it is $99.8\%$ of $c$ , never more. What exceeds $c$ is the apparent transverse speed $\beta_{\rm app}$ : the sky-projected distance a knot seems to cover divided by the light-distorted time over which we receive its light. Because an approaching blob nearly keeps pace with its own photons, they arrive bunched together, and dividing a real transverse motion by an artificially short arrival interval gives a ratio that can top $c$ . No matter, energy, or signal travels superluminally, so relativity is untouched — the effect is a projection, like a searchlight spot sweeping a distant wall.

Why does the apparent speed peak at an intermediate angle instead of head-on?

Two effects fight each other. The transverse motion you can see scales as $\sin\theta$ , which vanishes head-on — a jet aimed straight at you shows no sideways drift. The arrival-time compression scales as $1/(1-\beta\cos\theta)$ , strongest head-on and weakening as the angle grows. Their product $\beta_{\rm app}=\beta\sin\theta/(1-\beta\cos\theta)$ must therefore peak in between, and calculus pins the sweet spot at $\cos\theta=\beta$ , i.e. $\sin\theta=1/\gamma$ . For a fast jet that angle is only a few degrees, and the peak value there is $\beta\gamma\approx\gamma$ . The largest apparent speeds come not from jets aimed exactly at us, but from those aimed just slightly off.

How does one measured β_app become a limit on the Lorentz factor?

The apparent speed can never exceed the maximum available to the true Lorentz factor, $\beta_{\rm app,max}=\beta\gamma$ , so a measurement forces $\beta_{\rm app}\le\beta\gamma$ . Squaring and using $\beta^2\gamma^2=\gamma^2-1$ eliminates the speed, leaving $\beta_{\rm app}^2\le\gamma^2-1$ , i.e. $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ . No viewing angle is needed: whatever the geometry, the Lorentz factor must be at least this large. A knot at $\beta_{\rm app}=10$ requires $\gamma\ge10.05$ , and no slow flow can counterfeit a fast apparent speed — which is why one clean proper-motion measurement certifies a jet as relativistic.

If a single β_app only gives a lower bound, how do we get the real γ and θ?

By measuring a second, independent quantity — usually the Doppler factor $\delta$ , from the source's variability timescale and brightness temperature or from synchrotron self-Compton arguments. Since both $\beta_{\rm app}$ and $\delta$ depend on the same $(\beta,\theta)$ , they satisfy exact identities; the cleanest, $\beta_{\rm app}^2+\delta^2=2\gamma\delta-1$ , inverts to $\gamma=(\beta_{\rm app}^2+\delta^2+1)/(2\delta)$ , and then $\beta$ and $\theta$ follow. Worked example 2 runs this through for $\beta_{\rm app}=15$ , $\delta=12$ , giving $\gamma\approx15.4$ , $\theta\approx4.7^\circ$ . Two observables break the degeneracy and fix the full kinematics.

Why do these jets so often show only one side?

The same geometry that fakes superluminal motion boosts the approaching jet and suppresses the receding one. Flux scales as $S\propto\delta^{p}$ with $p\approx2$ – $4$ , and the two sides have Doppler factors set by $1\mp\beta\cos\theta$ , so their brightness ratio is $[(1+\beta\cos\theta)/(1-\beta\cos\theta)]^{p}$ — reaching millions to billions for $\beta\approx1$ and small $\theta$ (worked example 3). The counterjet is not absent; it is de-boosted below any realistic detection threshold. This one-sidedness is itself evidence for relativistic beaming.

Why is a cosmological model needed to quote an apparent speed?

Because an interferometer measures an angular rate — a proper motion $\mu$ in mas yr $^{-1}$ — not a velocity. Converting it to a transverse velocity needs the physical distance, which comes from the redshift through a cosmological model: $\beta_{\rm app}=\mu D_A(1+z)/c$ . The angular-diameter distance $D_A$ depends on $H_0$ , $\Omega_m$ , $\Omega_\Lambda$ , and the $(1+z)$ accounts for time dilation between emitted and observed frames. Two knots at the same true speed but different redshifts show different proper motions, and the wrong cosmology gives the wrong $\beta_{\rm app}$ . The bound $\gamma\ge\sqrt{1+\beta_{\rm app}^2}$ inherits this dependence only through $\beta_{\rm app}$ .