If a light source is moving away from us, does the light arrive moving slower?

If an object is moving away from us at half the speed of light and shining a light back at us, is the light that hits us going half the speed of light relative to us?
A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left. Image credit; wikimedia user TxAlien, CC A-SA 3.0

A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left. Image credit; wikimedia user TxAlien, CC A-SA 3.0

It is not!

One of the fundamental principles of relativity is that the speed of light in a vacuum is always a constant. It doesn’t matter where you’re standing or how fast you’re moving, you should always observe light in space to move at the same speed.

That’s not to say that light is unaffected by the motion of whatever is giving off the light! To understand what’s going on here, it’s easier to think of light as a wave than it is to think of it as a series of particles.

We’re all familiar with the change in tone of an ambulance’s siren as it passes us. It seems higher pitched as it moves towards us, and then as it passes us and moves away from us, the siren drops in tone. The speed of the sound wave going through the air hasn’t changed, but because the source is moving, we perceive a change in pitch. This is the Doppler effect.

Imagine touching the surface of a pool of water so that you create one ripple in the surface. It will move smoothly away from where you touched it in all directions. If you keep tapping the same place in the water, you’ll get a series of concentric rings. But if instead of tapping the same place, you move your finger over a little bit, your second ring is offset from the first one. The inner ripple of water will be closer to the outer ripple on one side, and further from the other side. Depending on how fast you move your hand between your first and second taps, this effect can be more or less obvious - the faster you move your hand, the more crushed together the ripples will be on one side, and the further apart they get on the other. 

The same thing can happen to light waves. While light is always moving away from its source at the same speed, if your source is moving at a significant fraction of the speed of light relative to whoever’s watching, the wavelength (or color) of the light will change dramatically.

Let’s use your example: 50% of the speed of light. If that object is shining a pure blue light, as blue as the human eye can detect (about 400 nm), if it’s moving away from us at half the speed of light, the light I would observe would be shifted to longer wavelengths to such a degree that it would be an extremely deep red (690 nm).

The color spectrum rendered into the sRGB color space using a gray background to preserve the actual colors. The numbers are wavelength, in nanometers. Image credit: wikimedia user Spigget, CC A-SA 3.0

The color spectrum rendered into the sRGB color space using a gray background to preserve the actual colors. The numbers are wavelength, in nanometers. Image credit: wikimedia user Spigget, CC A-SA 3.0

This is the fundamental idea behind using a redshift as a measure of distance.  The further away an object is, the faster it's moving away from us, and the redder its light has become. So if we can measure that shift, we can figure out how far away it must be.

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