Contents

Grids, Sensors, and Signals

The idea of a sensor grid was introduced earlier as a way of making measurements on moving objects, and recording their histories as they pass. Sensors make only local measurements on local events. That's the simplest way of making measurements of position and time. In this way, it's possible to simply express the fundamental ideas of relativity.

Earlier, we defined a signal as any physical object (matter, radiation) that moves from one place to another. This section focuses on radiant signals only (or just signals for short). The most common radiant signal is light (electromagnetic radiation). In addition to making measurements on passing objects, sensors can also make measurements by detecting radiant signals. The best examples of this come from astronomy. It's usually not possible to travel to distant astronomical objects, so astronomers are left with the only source of data left to them: incoming signals hitting their detectors. It's true that the ultimate source of the signal is distant, but the detection event itself is a local measurement on the signal as it arrives at the local sensor. The only difference is that the signal usually refers back to a distant, non-local object.

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Astronomers make use of all the incoming radiant signals they can get their eager astronomer hands on:

These signals:

Signals: Energy And Direction

Many different quantities can be measured using a signal. They can be divided into two categories.

First, quantities related to the energy carried by the signal:

Second, quantities related to the direction of the signal:

Most of these quantities are here listed as apparent quantities. This is to emphasize that they are parochial, not invariant, and change from one grid to another during a boost. Using the word apparent has one goal: to help you avoid slipping back into non-relativistic ways of thinking when reasoning about high-speed objects.

Signals travelling at the speed limit (such as light) must arrive at the detector by travelling on the detector's past light cone. Quantities related to direction can be found by looking at the intersection of the detector's past light cone with the history of the emitter.

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When an object can be approximated as a point, then its history is a curve or a line. When an object has a measurable size, then it's said to be an extended object, and it can't be approximated as a point. To represent an extended object, you can represent it as a collection of points that form a volume in space. In space-time, the history of the object is the collected histories of all those points, taken as a group. That collection makes up a history tube in space time. It corresponds to the volume of points changing position over time.

The diagram above shows the past light cone of a detector intersecting the history tube of an extended object, at one moment in time according to the detector's wristwatch. You can see that, in general, the intersection events are the same as the emission events of the light signal, corresponding to light emitted from the surface of the extended object. The emission events don't all share the same ct value: they aren't simultaneous. This is because the light cone slices across the history tube at a 45° angle. However, the light signal detection events are indeed simultaneous, since the light signals all meet up at the same event.

So, the image of an extended object seen in a detector is made up of light that was emitted at different times. This is simply caused by the differences in travel-time for the light emitted from the different parts of the object.

When the object isn't moving with respect to the detector, the differences in travel-time don't make much difference. When the object is moving, and when the differences in travel-time are significant, then the differences show up as distortions in the detected image. The larger the differences in travel-time, the greater the distortions.

Photon Signals

The remainder of Part II is specific to the most common kind of radiant signal: light, or electromagnetic radiation. Electromagnetic radiation is carried by photons, massless particles that always travel at the speed limit (through empty space). Like any radiant signal, a photon carries a certain amount of energy in a certain direction. Usually, you're interested in exactly those two things - the photon's energy and its direction of motion. Sometimes, the polarization can be of interest too. The energy of a photon is proportional to its frequency. The energy/frequency of a photon is continuous. It's traditional to divide up these continuous values into named ranges or chunks. In order from the lowest to the highest energy/frequency, these chunks are: It's important to note that the energy and the direction of motion of a photon are both parochial, and change from one grid to another. A photon appearing as a gamma-ray photon in one grid can appear, say, as a microwave photon in a second grid, which is boosted with respect to the first grid. It's important to keep this in mind. We investigate below how the energy and direction of motion change during a boost.

A photon is represented in a diagram much like an event on the light cone. The only difference is that axes are relabelled to represent energy and direction, not the usual ct and x,y,z:

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Since (ct,x,y,z) isn't being displayed here, it's not correct to say that this is a space-time diagram. However, the diagram has the exact same geometry as a space-time diagram, with a light cone, boost transformations, and so on. The only difference is that the axes are labelled differently. That's all.

Photons are represented by a line from the origin to a spot somewhere on the future light cone. Let's call this line the photon vector. (The official name is propagation vector, or wave vector.) The energy of a photon corresponds to how high up the light cone the photon vector climbs. The higher it climbs up the light cone, the higher the energy. The photon vector is a 4-vector. Just like event coordinates, the photon vector has 3 space parts and 1 time part. The space parts define the direction in which the photon is moving, while the time part defines the energy of the photon. For the photon vector, the higher the photon energy:

In the diagram above on the left, the blue arrow indicates a photon with higher energy than the green arrow, since it climbs higher up the light cone. Here, we make use of various colors for photon vectors, to show their relative energy differences:

Events On The Light Cone With Constant ct

Both the frequency and direction of motion of a photon are parochial items. How do they change during a boost? Well, let's go back to space-time events for a moment. If you apply the boost transformation to events that: then you can see also, at the same time, exactly how the energy and direction of a photon change during the same boost.

The selected events appear in a circle centered on the ct axis. This animation shows how these events get boosted. As you can see, the usual toothpaste effect still applies: some events are squeezed toward the origin, while others are stretched away from it.

The important point here is that the boost transformation for these events on the light cone can be interpreted as applying to photon vectors as well. The reason this works is because, as mentioned above, the photon vector is a 4-vector. So, the boost transformation for events also defines how the 4 parts of the photon vector change during a boost, in the same way it does for events. It's the exact same math, and the exact same geometry. So, the boost transformation for these events shows how to calculate the change in frequency and direction of photons.

The circle around the ct axis corresponds to photon vectors having the same energy, but different directions. The generated ellipse corresponds to how these photon vectors appear in the boosted grid.

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The ellipse in the animation shows two important effects:

Apparent Direction: Aberration

The sensors in a grid can detect and measure the direction of motion of photons coming from a distant object (such as a star). The direction is measured relative to some selected direction, usually taken as one of the xyz axes. How is a photon's direction of motion affected by a boost transformation?

In a boosted grid, the photon's direction of motion is deflected towards the direction of relative motion of the two grids. This effect is called the aberration of light. In the ultra-relativistic case, the stars are seen to be bunched up towards the direction of motion of the detector. The aberration of light was discovered by accident by James Bradley in 1729. Aberration is an interesting example of an effect that could have easily been predicted before being observed, but wasn't.

Here's the aberration formula:

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In the above diagram, P and P' denote the direction of motion of the photon in the two grids. It's important to note that the direction of motion of the photon is opposite to the direction in which the light is seen by the detector (S and S'). If you wish, you can instead define θ as being 180° minus the θ shown above, but then the formula changes slightly:

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The effect of aberration can be equivalently seen as:

As you can see from the animations, the amount of concentration in the forward direction can become quite high. As a measure of the amount of concentration, this site uses the idea of the half-sky radius, meaning the angular radius from the direction of motion that encloses what would be half the sky in an unboosted grid.

Apparent Frequency: Doppler Effect

Any process that repeats itself is called periodic. Some examples: The time it takes for the process to repeat is called the period, measured in seconds. Periodic behavior is usually measured using the inverse of the period, called the frequency. The unit of frequency is 1/second, which is also called hertz. The shorter the period, the higher the frequency. All of the above examples can be measured in hertz:
It's important to understand that, from the point of view of physics, all of the above measurements are in hertz. That's because 'ticks', 'rotations', 'waves' and so on are dimensionless. They're dimensionless since they represent counts of things, and counts of things are pure numbers, without any physical unit.

Periodic processes can send out periodic signals. The signal is usually electromagnetic radiation moving at the speed limit. These signals can be detected by the sensors in a grid. In this way, each sensor can measure the frequency of the signals received at its own location. There are two cases, depending on whether the signal source is moving with respect to the sensor grid:

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The change in frequency for the detected signal from a moving signal source is called the Doppler effect. Here's an animation showing the effect for motion along the x axis. The Doppler effect has two underlying causes, which can work in opposing ways:

A change in frequency is often described as:

Doppler Factor

The change in frequency seen in the Doppler effect is given by a number called the Doppler factor. The Doppler factor is usually given the symbol D. Here's the formula for D (note that D is dimensionless, just like β and Γ):

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The Doppler factor D is a bit trickier than β and Γ, since it depends on two things, not just one:

Of course, Γ also appears in the formula for D, but Γ depends only on β, so it's not independent.

For each β, there's always a θ for which D=1, and the effect passes from blueshift to redshift (or vice versa). Here, the angle (from the direction of motion) for which D=1 is called the neutral angle. The neutral angle is always less than or equal to 90°. Objects at the neutral angle have D=1, and are neither blueshifted nor redshifted.

This animation is an excellent way to explore how D changes with β and θ. Note that for the case of θ=90° (the case of transverse motion), D is just the inverse of the warp factor Γ. Here's a second animation that combines aberration and the Doppler effect in a single diagram.

The frequency of a signal seen by a detector is the frequency-at-rest multiplied by the Doppler factor D. The Doppler factor appears again and again in formulas related to waves and light. Some examples:

ItemVaries AsItemVaries As
Any FrequencyDPlane Wave Energy Density (J m-3) D2
Photon EnergyDPlane Wave Energy Flux Density (J s-1 m-2)D2
Photon FrequencyDTemperature of black body radiationD
Photon Wave NumberDSolid angleD-2
Photon WavelengthD-1Angular sizeD-1
Plane Wave E,H field intensitiesDDetected energy per sec per steradD4
Aberration sin θD-1Emitted energy per sec per steradD3

Apparent Brightness: Beaming

There are different ways of measuring how bright an object is. One fundamental measure is the total amount of energy received per unit area on a detector, per unit time, over all frequencies. This is called the energy flux density. During a boost, the energy flux density varies as the square of the doppler factor D. For a given object, the exact way in which the brightness changes with D depends on: The increase in brightness depends on D raised to some power. Because of this, the increase in brightness can become very high. This is called relativistic beaming. The section on astrophysical jets contains more about relativistic beaming.

For the brightness of stars, astronomers usually measure brightness using a logarithmic scale called stellar magnitude. A bright star may have magnitude 1.0, while a dim star, barely visible to the human eye, might be of magnitude 6.0. The magnitude depends on the logarithm of the energy flux density (per hertz). For example, a difference of 5 magnitudes means a brightness difference of 100 times. For historical reasons, stellar magnitude actually increases as the brightness decreases (which is a bit confusing).

The radiation from stars is often roughly approximated as blackbody radiation, which is characterized by a temperature T expressed in Kelvins. The temperature T corresponds to the surface temperature of the star. The temperature also corresponds to the color of the star: cooler stars are reddish-orange, hotter stars are blue-white. During a boost, the temperature T is multiplied by the Doppler factor D, and the color also changes to match the new temperature.

For a star of surface temperature T, here's an approximate formula (McKinley, Doherty, 1979) for the change in a star's brightness during a boost, expressed in stellar magnitudes, which depends on both D and T (expressed in Kelvins):

Here's a diagram that illustrates the above formula. It shows the change in magnitude for various spectral classes of stars. The spectral class corresponds to the star's surface temperature T.

The Appearance of a Sky Full of Stars

In movies and television, science fiction animations often show a spacecraft travelling through a field of stars. How accurate are those depictions? Do they agree with the rules of relativity? No, they don't. If you were travelling through space with high β in the neighbourhood of the Sun, the basic character what would you see is shown by this interesting simulation. It accounts for:

In summary, as β increases from zero,

After you reach top speed, you wouldn't see "star trails" as you travel through space, as often depicted in science fiction. Space is simply too empty and vast. Even at ultra-relativistic speeds, the chance of a rapid fly-by of a star is very, very small. These star trails are created by animators to simply give an impression of rapid motion. They've used physics that's appropriate for showing rapid motion on the surface of the Earth, but not for interstellar space. If you wanted to see an actual fly-by of a star, you would have to plan it precisely, and aim towards a specific target.

Here are two interesting graphs showing the total number of visible stars, their total relative brightness, and how they change with increasing β. These diagrams were created using 2,539,802 stars taken from the Tycho-2 star catalog. The limiting magnitude of 5.0 was selected to correspond to viewing the stars through a spacecraft window. As in the simulation, the direction of motion is towards Polaris. Notice the very strong cut-off at the highest speeds. This is because most of the starlight is blueshifted out of the visible range. Here, for the brightness unit, a single star of magnitude 0.0 has 1 unit of brightness.

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Here's a large image created using the Tycho-2 star catalog. It shows the stars brighter than magnitude 5.0, at β=0.993, and it uses the stereographic projection. As in the simulation, the direction of motion is towards Polaris. The image file is large, and it may take your browser a few minutes to render it. It's best to download the image file directly, and open it in some tool other than a browser. The image differs from the simulation in these ways:

These changes were made to let you better appreciate the overall density of stars. To create such images on your own, using the Tycho-2 star catalog, you can use this desktop software, created explicitly for that purpose.

Apparent Angular Size

Extended sources of light, such as a galaxy or a star cluster, have an angular size, the number of degrees it appears to extend across the sky. Point-like sources of light, such as a star, don't have a measurable apparent angular size.

Because of small differences in the amount of aberration between different directions, the angular size of an extended source of light changes during a boost. During a boost, a (small) angular size is divided by the corresponding Doppler factor D. For example, an object appearing at 15° from the direction of motion, with β=0.6, will have its angular size reduced by D(0.6,15°)=1.9027 with respect to the same measurement in a grid with β=0. (The Doppler factor is applied here after accounting for aberration.)

This implies a curious effect. For a travelling spacecraft, items in the forward view will appear to have a smaller angular size, and will seem compressed, and apparently more distant, compared with the appearance from another spacecraft which is not moving in that direction. Conversely, items in the rearward view will appear magnified, and apparently closer, since D will be less than 1.

You can see this effect using the star field simulation, once for β=0 and once for β=0.5:

(There are small distortions in the chart because it projects the celestial sphere onto the flat screen. But in this case, the distortions are pretty small.)

Apparent Outline

A sensor can take an image of an object, using the incoming light. That image will have an outline. Does that outline change during a boost? In general, yes. In a boosted grid, the outline of an image will be distorted from the image seen in an unboosted grid. The cause is the differences in light travel-time mentioned earlier.

Spherical objects are an exception to the rule: the image of a spherical object will always have a circular outline in every grid, no matter what the boost.

There's a second exception as well. When an object has a small angular size as seen from the detector, then:

Apparent Velocity

By itself, a signal doesn't carry any distance information. However, distance information can be available through other means. (For instance, through triangulation using two detectors.) If a distance is available, then it can be combined with the direction, to deduce an apparent velocity (symbol b). Canvas not supported. Please upgrade your browser.

The apparent velocity b is usually split into two parts:

In an inertial grid, the true velocity of an object is extracted from the event data gathered by multiple sensors, strung out along the object's history. Here, we're measuring the apparent velocity b using direction data from a single sensor (located far from the emitter), combined with an independent meausurement of the distance. Remember that the apparent direction comes from the geometric intersection in space-time of the detector's past light cone and the emitter's history. This is another example of a geometric point that doesn't obey the speed limit rule. The apparent velocity of an object is not at all the same as its true velocity. (To emphasize this, it's best to use a different symbol b instead of β.)

Notice the scissors effect in the following diagrams for radial motion. At high β, the history of an approaching emitter can be at a very small angle with respect to the detector's light cone. Geometrically, it's sort of like the closing of a pair of scissors. As time passes, the intersection point will sweep across the emitter's history. The higher β, the faster the sweeping speed of the intersection point. The same sort of effect is seen even when the motion is not purely radial.

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Here's an animation that demonstrates the scissors effect.

The scissors effect means that, in the limiting case, no early warning of an approaching threat is possible if the threat moves towards you at an ultra-relativistic speed. In such cases, the news of the approaching disaster arrives just before the disaster itself.

In general, there's both radial and transverse motion. Here's an animation that shows how both parts of the apparent velocity change with β and θ (the angle the motion makes with the line of sight). It uses these formulas:

Astrophysical Jets

The above effects regarding apparent speed have actually been observed numerous times, in astrophysical jets. These jets explode out of the centers of galaxies at high speeds. They can also occur on a smaller scale, around black holes behaving as microquasars.

The speeds of the material in the jets can be very high, approaching β=1. The transverse motions of blobs in a jet can actually be tracked over the course of a few years (or even a few days, in the case of microquasars). When the distance to the source of the jet is known, then the apparent transverse speed can be calculated from the year-to-year transverse motion. Sometimes the apparent speed of a jet is found to be more than 1 light-year per year.

Here's a series of images of a radio galaxy named 3C111. It shows a blob of matter emitted from the core in late 1996. It moves (downwards, as shown here) about 15 light-years in under 4 years:

This is not a violation of the speed limit rule, since, as stated above, the apparent speed is the speed of a geometric intersection point in space-time.

Here's an animation that lets you explore these jets.

Note that the direction of the jet in space is random. If the jet happens to point roughly towards the Earth, then the scissors effects mentioned above take place. You have to use your imagination a bit with these blobs, since the information in an image is only 2-dimensional, but the motion is 3-dimensional. The emitted blob is not travelling only in the downward direction here. It's direction of motion is mostly towards you. It's important to understand that.

Due to beaming, the apparent brightness of a jet depends on the geometry. If a jet is pointed towards the Earth, the brightness of the jet is increased. If a jet is pointed away from the Earth, its brightness decreases.

Other examples of jets:

In general, it's difficult for astronomers to find the distances to objects. It's interesting to note that these astrophysical jets provide a new, relativistic way of measuring distances.