Relativistic Doppler effect
From Academic Kids

In physics, the relativistic Doppler effect is change in the observed frequency of light due to the relative motion of source and observer when taking into account the Special Theory of Relativity.
The change in frequency when the observer and the source are receding from each other is given by:
<math>f_o = f_s \cdot \sqrt{\frac{1\frac{v}{c}}{1+\frac{v}{c}} }<math>
where
<math>f_o <math> = frequency observed
<math>f_s<math> = frequency of source
<math>v<math> = relative velocity, positive when object and observer are moving away
<math>c<math> = speed of light
In the Special Theory of Relativity, space and time are not absolute, and a speed is something like a rotation in spacetime, so that someone travelling at a different speed has a different point of view regarding which sets of points are simultaneous (at the same time point) or at the same point of space. The speed of light is a special, limiting sort of rotation. In some ways it is like an infinite quantity (using the arctanh function, where a rotation θ is <math>\theta \equiv \operatorname{arctanh} (v/c)<math>), which is interpreted as a constant in ordinary measurement. Since adding anything to infinity (in the special cases where this can be defined to make sense) is still infinity, the speed of a photon of light from a source that is moving towards or away from the observer is not affected by that relative motion.
This is counterintuitive if space and time are assumed to be absolute. Thinking of a speed as a sort of rotation can help.
However, while the speed of a photon does not change, a change of speed by the observer does correspond to a sort of rotation in which the time and space axes are different. An application of the Lorentz transformation shows that the effect is that the frequency, and therefore energy, of the photons is affected: photons from any approaching source are increased in energy and photons from a receding source are reduced in energy: this is more intuitively satisfying. It is similar to the change in kinetic energy of an object due to the thing or person throwing it moving relative to the observer. This is the relativistic Doppler effect.
You can never "run alongside" a photon: this does not seem too unreasonable; but, what seems less reasonable, when assuming absolute time (and space), is that no matter how fast you chase a photon it is always moving away from you at exactly the same speed; but you do have the consolation that the photon appear to have less energy the faster you chase it; but it will never quite have zero energy. Again, thinking of a speed as a sort of rotation can help.
In this article the term light is intended to mean any electromagnetic radiation, which is a particle that has no rest mass and which must always travel at the speed of light (when in a vacuum).
The article on electromagnetic radiation also gives the relationship between frequency and energy of a particle.