The sky that crowds into a cone

Ask what a traveller at 0.99c sees and the usual answer is about colour: things ahead blueshifted, things behind redshifted. That is true and it is the smaller half of the effect. The larger half is that the stars themselves have moved. Half of the entire sky — everything that was at or behind right angles when the ship was at rest — has been crowded into a patch eight degrees across, dead ahead.

The map #

Take a source moving at $\beta c$ along the $z$ axis. A photon that leaves at angle $\theta'$ to that axis in the source’s own frame is seen in the laboratory at $\theta$, with

$$\cos\theta = \frac{\cos\theta' + \beta}{1 + \beta\cos\theta'}.$$

This is not a new law. It is the velocity-addition rule applied to something moving at $c$: the photon’s velocity components transform like any other velocity’s, and the ratio of the transformed components is the tangent of the new angle. That the speed comes out $c$ again in both frames is the input, not the output.

The aberration formula is a special case of the composition rule, and the case that matters is the one at the boundary: anything moving at $c$ in one frame moves at $c$ in every other. A photon’s speed is untouched by a boost, so the only thing a boost can do to it is change its direction. Aberration is precisely what happens to the direction when the speed is not allowed to change.

It is worth writing the inverse too, because the two directions of the map are constantly muddled. Swapping $\beta \to -\beta$ gives

$$\cos\theta' = \frac{\cos\theta - \beta}{1 - \beta\cos\theta},$$

which is the same function with the sign of the velocity reversed — as it must be, since the source sees the observer receding at $-\beta$. Whether a given curve is a crowding or a spreading therefore depends entirely on which frame is being called the laboratory, and the physical content is only the relation between the two.

The map has three features worth naming. It is monotone, so directions do not cross — the sky is rearranged and not scrambled. It is not linear, so equal angles do not go to equal angles. And it takes $\theta' = 90°$ to $\theta = \arccos\beta$, which is the half-sky angle: everything emitted into the forward hemisphere arrives inside that cone.

For $\beta \to 1$ that angle tends to $1/\gamma$, which is the number always quoted. The figure draws both because the approximation is worse than its reputation: at 0.5c it is 17% out.

The cone’s angle is quoted in terms of $\gamma$, which is 1.15 at 0.5c, 2.29 at 0.9c and 7.09 at 0.99c. That is why the same fractional change in speed produces such different amounts of crowding, and why almost nothing interesting happens below about half the speed of light: the factor has to get large before the geometry does anything a reader would notice, and it gets large only in the last few per cent.

The two aberrations, and why one of them is not this one #

Bradley discovered stellar aberration in 1727: stars trace small ellipses over a year, of semi-major axis 20.5 arcseconds, because the Earth’s velocity round the Sun changes direction. That is the same formula at $\beta = 10^{-4}$, where it reduces to $\Delta\theta \approx \beta\sin\theta$ and needs no relativity at all — Bradley explained it with a Newtonian argument about the ship and the rain, which is a change of frame of the plainest kind.

What relativity changes is not the small-$\beta$ answer but the structure. The Newtonian derivation adds the observer’s velocity to the light’s and gets a resultant of the wrong magnitude, which does not matter at $10^{-4}$ and matters entirely at $0.9$. And the Newtonian derivation gives no aberration at all for light arriving from directly ahead or behind, which is right, while giving the wrong dependence in between.

Four powers #

The remapping of directions has a consequence for brightness that is larger than the remapping itself.

Consider a source that radiates isotropically in its own frame. In the laboratory, four separate things happen to what arrives from a given direction, and each contributes one factor of $\delta = 1/\gamma(1 - \beta\cos\theta)$:

• each photon’s energy is multiplied by $\delta$;
• photons arrive $\delta$ times more often;
• the solid angle they occupy is compressed, which is two powers, since a solid angle is an angle squared.

So the received intensity carries $\delta^4$. Front against back the ratio is

$$\left(\frac{\delta_{\text{fwd}}}{\delta_{\text{back}}}\right)^4 = \left(\frac{1+\beta}{1-\beta}\right)^2,$$

which runs away long before $\beta$ does.

Two conclusions follow that are worth stating separately.

It is worth noticing which of the four powers survive a change of question. If what is measured is the number of photons arriving rather than the energy, one power drops and the ratio is $\delta^3$. If the source is a surface whose own emission is being resolved rather than a point, the solid-angle factors partly cancel against the source’s apparent size and the surface brightness carries $\delta^4$ while the flux carries $\delta^3$. Each version is the same transformation with a different quantity carried through it, and the exponent is a piece of bookkeeping rather than a law.

A source pointed away is not dimmed but deleted. At 0.99c the backward intensity is four ten-thousandths of the isotropic value, so a source that would be comfortably detectable at rest is undetectable pointing away.

Any collection of such sources is a biased sample. If jets point in random directions, the ones an observer sees are overwhelmingly the ones pointed nearly at them, by a factor that is a large power of $\gamma$. The population an instrument records and the population that exists differ by that factor, and correcting for it is the whole difficulty of counting such objects. That is a statement about the sample rather than about the sources — a different kind of conclusion from anything else in this essay, and one the arithmetic forces.

The dipole that measures the Earth’s own velocity #

Bradley’s measurement has a modern successor that uses the whole sky at once, and it gives the Earth’s velocity with respect to a frame nobody chose.

The cosmic microwave background is very nearly the same temperature in every direction. Very nearly: it is warmer on one side of the sky and cooler on the other, by about three and a third millikelvin on two and three quarter kelvin, in a pattern that is a clean dipole. The interpretation is the one this essay is about — an observer moving through a uniform radiation field sees it blueshifted ahead and redshifted behind, by a fractional amount $\beta\cos\theta$ to first order.

Reading the amplitude gives $\beta = 1.23\times10^{-3}$, a speed of 370 kilometres a second, in a direction that can be quoted to a fraction of a degree. That is the solar system’s motion with respect to the frame in which the background looks isotropic — a frame defined by the matter of the universe rather than by any local reference, and the closest thing to a preferred frame that exists.

The measurement is Bradley’s experiment with three changes. The velocity is a thousand times larger, the source is the whole sky rather than one star, and — the useful part — there is no need to wait six months and compare, because the pattern is a static feature of the sky whose amplitude is read directly.

There is also a second-order signature, and it has been detected. The same boost that shifts the temperature also aberrates the directions, so the fine structure of the background is very slightly magnified toward the direction of motion and demagnified away from it. That is aberration at $\beta = 10^{-3}$ acting on a pattern of hot and cold spots, and separating it from the far larger temperature dipole is a matter of the two having different signatures across angular scales. It is the same map as this essay’s, applied to a sky rather than to a source.

The same cone, made by an accelerating charge #

The forward cone is not confined to what a traveller sees. A charge that radiates while moving fast has its own emission beamed into the same cone, for the same reason and by the same transformation.

That is why synchrotron radiation is a searchlight, and it is a case where the beaming is not an inconvenience to be corrected for but the entire reason the source is useful. An electron going round a ring at $\gamma \approx 10^4$ radiates into a cone of half-angle $10^{-4}$ radians, tangent to its path, so a detector fixed in the laboratory sees a flash lasting only as long as that cone sweeps past — which is far shorter than the orbital period, and which is why the emitted spectrum extends to frequencies enormously higher than the orbital frequency. The short pulse is the aberration cone doing the work; the underlying radiation is the Larmor formula’s and knows nothing about it.

Underneath all of it the invariant structure is untouched. A boost tilts the axes of a spacetime diagram and leaves the light cone exactly where it was, which is the geometric statement that light travels at $c$ in every frame. Aberration is what that tilting does to the directions of null lines within the cone — and no amount of tilting moves a null line off it, which is why the effect redistributes light without changing its speed.

Counting what is in the cone #

The half-sky angle is easy to state and easy to underrate, so it is worth turning into a count.

The fraction of the sky lying inside a cone of half-angle $\theta$ is $(1 - \cos\theta)/2$. At rest, the forward hemisphere is half the sky. Boosted to $\beta$, that same half arrives inside $\arccos\beta$, which occupies

$$\frac{1 - \beta}{2}$$

of the observer’s sky. At 0.99c that is half a per cent — so half of everything visible has been squeezed into a two-hundredth of the field of view, an areal compression of a hundred.

The compression is not uniform across the cone, which is why the map matters and a single angle does not. Differentiating, the solid-angle magnification is $\delta^2$, so the very centre of the forward cone is compressed by $\gamma^2(1+\beta)^2 \approx 4\gamma^2$ — two hundred at $\beta = 0.99$ — while the edge of the cone is barely compressed at all.

What makes those speeds expensive is the energy curve, and the numbers are worth having. Reaching 0.5c costs 15 per cent of the rest energy; 0.9c costs 129 per cent; 0.99c costs 609 per cent. Every effect in this essay is steep in $\beta$ near one, and the cost of getting there is steeper still — which is the practical reason aberration is met in the radiation from electrons rather than in the view from a ship.

What a traveller actually sees #

Two further effects have to be added before the picture is what a camera would record, and both are frequently omitted.

The colour goes with the angle. The Doppler factor varies across the field, so the forward cone is not merely crowded but blue, and the sky far off-axis is red. At $\beta = 0.99$ the forward shift is a factor of 14, which takes the whole visible band into the far ultraviolet and brings the infrared into view; a traveller sees a different spectrum, not merely a brighter one, and what a photon carries scales with its frequency so the momentum delivered scales with it too.

A moving object looks rotated, not contracted. Light from the far side of an object left earlier than light from the near side, so an object passing at high speed presents its back to the observer. That is the Terrell–Penrose effect, and it means length contraction is not something a photograph shows, though it is entirely real in the sense that matters.

Neither effect was noticed for half a century. Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed, because the calculation everyone did was the contraction of simultaneous positions rather than the arrival of light — and simultaneity is exactly the thing that has stopped being frame-independent. It is an unusually clean example of a correct result being drawn wrongly for fifty years, by people who had the transformation in front of them and had asked the wrong question of it.

The mechanism under both the crowding and the Doppler shift is one number. A clock’s tick is longer in a frame it is moving in, because the light inside it has further to go, and that same factor is the $\gamma$ in the Doppler formula and the $\gamma$ in the cone’s angle. Everything here comes from one construction applied to a direction instead of to a duration.

The brightness that gives the speed away #

The bias argument above is a statement about a population and there is a sharper one about individual sources, which is what established that relativistic beaming was happening at all.

A source of radio waves has a brightness temperature — the temperature a thermal emitter would need to be at to look that bright at that frequency and that angular size. For an incoherent synchrotron source there is a ceiling on it: above about $10^{12}$ kelvin the electrons producing the radiation scatter it themselves, losing energy so fast that the source cools itself in far less than a light-crossing time. A source cannot sit above that limit and persist.

Observed compact radio sources sit far above it, at $10^{14}$ kelvin and beyond, when their brightness and angular size are taken at face value.

The resolution is that the face value is wrong, and beaming is what makes it wrong. A source approaching at $\delta$ has its flux multiplied by a large power of $\delta$ and its variability timescale — which is what the angular size is inferred from — divided by $\delta$, so the inferred brightness temperature is too high by a factor of $\delta^3$ or so. Requiring the intrinsic value to sit below the limit puts a lower bound on $\delta$, and for the extreme sources that bound is ten or more.

What makes the argument satisfying is that it was a prediction before it was a measurement. The bound on $\delta$ from brightness temperatures was derived first, and it implied bulk motion at a Lorentz factor of ten in the emitting material; the apparent superluminal motion later observed in the same sources requires the same thing, by a completely independent route. Two constraints on one quantity, from a brightness and from a proper motion, agreeing.

One factor, two unknowns #

There is a limitation in all of this that is worth naming, because it is why the sources are so hard to characterise.

Everything an observer measures — the flux enhancement, the compressed variability timescale, the apparent superluminal speed — depends on the source’s velocity and its angle to the line of sight only through the Doppler factor $\delta$ and, for the apparent motion, through one further combination. Two unknowns, and observations that mostly constrain one number.

So a source with $\delta = 10$ might be moving at $0.995c$ at five degrees to the line of sight, or at $0.999c$ at fifteen. The two have different intrinsic luminosities by a large factor and different intrinsic lengths, and no amount of the same kind of measurement separates them.

Breaking the degeneracy takes something with a different dependence. Seeing both a jet and its counter-jet — one approaching, one receding — gives a ratio that depends on the angle differently from the flux, and the counter-jet is usually beamed into invisibility, which is the difficulty. Measuring the same source at two epochs far enough apart to see the geometry change gives another. And a source whose jet bends provides several angles at once.

The general shape of the problem is common enough to be worth recognising. A set of measurements that all depend on the same combination of two parameters constrains that combination beautifully and the parameters not at all, and adding more of the same measurements does not help. What is needed is one measurement with a different functional dependence, and finding it is usually the whole experiment.

Where the model stops #

The source is a point and the motion is uniform. An extended source has different parts moving at different angles to the observer and beamed by different factors, so the image is distorted as well as brightened. An accelerating source has a $\delta$ that changes during the emission, and the change happens while the light is in flight rather than at the moment it was emitted, so what arrives is a mixture of factors from different parts of the source’s history. Neither correction is small for anything with structure.

The intensity argument assumes a flat spectrum. The bolometric intensity carries $\delta^4$; a measurement in a band carries $\delta^{3+\alpha}$, with $\alpha$ the spectral index, because the band samples a different part of the source’s spectrum once it is shifted. Quoting the fourth power for a band measurement is a standard error and can be wrong by a factor of several.

The medium is empty. Everything here is a vacuum transformation. In a medium there is a preferred frame — the medium’s — and light does not travel at $c$, so the aberration formula is modified and Cherenkov radiation becomes possible.

A medium would add something light does not have, and the contrast is worth drawing out. A source outrunning its own waves in a medium leaves a cone behind it whose half-angle is set by the ratio of the two speeds — a real, frame-independent structure, because the medium picks out a frame in which it is at rest. Nothing of the kind happens for light in vacuum at any speed, since no source can outrun it. The forward cone of this essay is a property of the observer’s frame and not of anything left behind in the world.

What the pictures cannot show #

The angle map is one-dimensional: it plots polar angle against polar angle, and says nothing about the azimuth, which is unchanged. The area compression is the visible consequence and it is the derivative of this map, which no plot of the map itself displays.

The beaming figure plots intensity against angle for a source that is isotropic in its own frame. Nothing distinguishes that from a source that is anisotropic and slow, which is the observational difficulty in the real cases: a beamed isotropic emitter and a directional one look identical from one direction, and the only way to separate them is to see the same source from two.

And no figure here shows time. The whole essay is about a steady state, and the most dramatic consequence of beaming — that a source’s variability appears faster than it is, by another factor of $\delta$ — is a statement about arrival times that a plot against angle cannot carry. It is also the observation that makes beaming testable rather than merely plausible: a source whose brightness changes faster than its own light-crossing time is either smaller than it looks or moving toward the observer, and the second is far easier to arrange than the first.

Where the ladder goes next #

This ladder has gone from a pitch that changes on approach through a shift that survives at right angles to a transformation acting on directions rather than frequencies. Each rung has taken the same Lorentz transformation and applied it to a different component of the same object.

The rung above is to stop treating the frequency and the direction separately. Both are parts of one four-vector, and the whole of aberration and Doppler shift together is the statement that a boost mixes its time part with its space part — which is the geometry of a spacetime diagram applied to light rather than to worldlines.

The habit worth carrying: when a transformation is presented as acting on one quantity, look for the others it must be acting on at the same time. A rule that changes frequencies and leaves directions alone would violate the invariance it was derived from, and the fact that both change together is usually more informative than either change on its own.