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Special Relativity

Observational Effects

In astronomy we see many objects traveling at close to the speed of light. The most common objects are astronomical jets from galaxies and x-ray binary systems. When an object moves at close to the speed of light, its appearance is altered by special relativity and the propagation time of light. For an object that is much smaller than its distance from an observer, which is always the situation in astronomy, the alteration is a simple rotation of the object, a Doppler shift and change in intensity of its emitted radiation, and an alteration of the rate at which events appear to occur. In some instances, the object appears to move faster than the speed of light.

The precise appearance of a relativistically-moving object depends on two parameters: the angle between the direction of motion and the observer (??), and the precise speed. Normally the speed is expressed as the Lorentz factor ??, which is defined in terms of the velocity v of the object and the speed of light c as ??2 = 1 - v2/c2.

Three effects of special relativity change an object's appearance. The measured length of an object in the direction of motion becomes shorter as an object's speed becomes greater; this effect is called length-contraction, and the effect goes as the object's length when at rest divided by ??. The measured time for a physical process to occur (nuclear decay, for instance) decreases as the speed increases; this effect is time dilation, and the amount of time that the event is lengthened by is proportional to ??. And the radiation emitted by an object is Doppler shifted, so that radiation emitted along the direction of motion is brighter and shifted to higher frequencies, radiation emitted against the direction of motion is dimmer and shifted to lower frequencies, and for radiation emitted at other angles, its brightness and Doppler shift falls somewhere in between these extremes.

The time of propagation of the radiation introduces angle-dependent effects that also alter the appearance of a relativistically-moving object. For emission in the direction of motion, the object closely follows the emitted radiation, which makes times appear contracted. The converse happens for emission into directions away from the direction of motion. The time-delay also affects the apparent shape of the object, because radiation from the farthest parts of an object has farther to travel. At a given instance, an observer sees radiation that is emitted at different times, with the radiation emitted from the most distant parts of the object emitted earliest. This shears the appearance of an object, with the farthest parts of the object appearing to lag the nearest parts.

Taking both relativistic effects and propagation effects into account, the length contraction and the distance shearing, an object moving a high velocity does not look contracted, but rotated. This is most easily seen when the object is a cube moving perpendicular to the line of sight, with the cube oriented square to us and to the direction of motion. The length of the cube's face that is towards us is contracted by 1/??. The back of the cube is visible to us, because the light from the far edge of the cube has farther to travel, so that it is located well back from the radiation emitted by the trailing edge of the cube's front face. The length of the back face appears to have the length v/c. Squaring these two factors and adding give 1, so the two factors can be interpreted as the cosine and sine of an angle; the cube therefore appears rotated. This effect occurs for all emission angles and all shapes as long as the size of the object is much smaller than the distance to the observer.

For a given velocity, there is a characteristic angle given by sin?? = 1/?? that determines when the characteristics of the object change with angle. This critical angle is called the beaming angle. At angles less than this, when the object is traveling nearly straight towards us, the object appears at its brightest, with a spectrum Doppler-shifted to the highest possible energy; it appears slight rotated, so that the forward part of the object is visible; and events within the object appear compressed in time, so that what appears to take seconds to an observer actually takes months to someone traveling with the object. As the viewing angle increases to values larger than this critical angle, each of these effects begin to reverse: the object becomes dimmer, with a Doppler shift that decreases with angle until at the angle sin?? = ( 2/[ ?? + 1 ] )1/2 it becomes a shift to lower energies; the object appears to rotate so that we no longer see the front, but instead we see the back; and the speed at which events occur increases until, at the same angle at which the Doppler shift reverses, events appear to take longer to an observer than they do to someone traveling with the object.

One effect, superluminal motion, deserves special mention. This effect is purely a propagation-time effect. For certain angles, an object moving at the speed of light will appear to be moving faster than the speed of light. The apparent velocity along the path of motion seen by an observer is v/( 1 - v cos ?? /c). Superluminal motion is seen for objects moving with velocity v > c/2 when the emission angle satisfies cos?? > c/v - 1. The maximum effect is seen for ?? = 0.

One needs distance markers to derive an apparent velocity along the path traveled by the object. In astronomy, we normally don't have markers. We do, however, often have a distance, and if we measure the rate at which the angle changes between a fast object and a reference object, we can derive a transverse apparent velocity. The transverse apparent velocity is equal to v cos ??/( 1 - v cos ?? /c). The angle at which the apparent velocity is its maximum is sin ?? = 1/??, the beaming angle. Superluminal motion is seen for a transverse apparent velocity when an object is moving with v > 2-1/2c and has an emission angle that satisfies sin 2?? > c2 ( 1 - v2/c2 )/v2.

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