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.
Astronomers make use of all the incoming radiant signals they can get their eager astronomer hands on:
First, quantities related to the energy carried by 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.
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.
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:
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:
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.
The ellipse in the animation shows two important effects:
There are 3 angles in this situation:
In a boosted grid, the photon-direction is deflected away from the boost-direction. Equivalently, the detector-direction is deflected towards the boost-direction. 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.
There are two versions of the aberration formula, one for the photon-direction, and one for the detector-direction. In the formulas below, the prime denotes the boosted grid.
For the photon-direction:
For the detector-direction:
The effect of aberration can be equivalently seen as:
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:
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:
The Doppler factor D is a bit trickier than β and Γ, since it depends on two things, not just one:
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:
|Item||Varies As||Item||Varies As|
|Any Frequency||D||Plane Wave Energy Density (J m-3)||D2|
|Photon Energy||D||Plane Wave Energy Flux Density (J s-1 m-2)||D2|
|Photon Frequency||D||Temperature of black body radiation||D|
|Photon Wave Number||D||Solid angle||D-2|
|Photon Wavelength||D-1||Angular size||D-1|
|Plane Wave E,H field intensities||D||Detected energy per sec per sterad||D4|
|Aberration sin θ||D-1||Emitted energy per sec per sterad||D3/Γ|
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.
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.
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:
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:
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:
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.
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:
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: