**What about distance?**

The reader may have wondered that ever since we started to discuss Redshift, we have only discussed how to find (radial) velocity, not distance. So why is redshift discussed in this EBook which is about measuring distance? The answer is that we needed this basic introduction to both optical and cosmological redshift to discuss the principal difficulties of measuring distance in really deep space and even to illustrate that the very concept of “distance” needs more scrutiny.

The naïve approach to derive distance from redshift is to use Hubble’s Law in reverse. From the measured redshift we can calculate velocity with the Doppler formula (corrected for relativistic effects) and with Hubble’s Law we can then work backwards to find the distance. This is quite inaccurate for various reasons.

**Nearby**

Hubble’s law describes the expansion of space but it ignores any local relative motion between the source and us as observers. Such motion is often dubbed “peculiar” motion. If we are really “close” (let’s say nearby galaxies) we can largely ignore the expansion of space and only observe peculiar motion. An example is the blue shift of the Andromeda galaxy. In that situation we can only use standard candles we discussed before to find distance. Redshift will only give radial velocity.

**Mid-range**

At larger distances this becomes blurred by the expansion of space but we still may see the effect of peculiar motion. This is a particularly difficult distance range in which we must both model peculiar motion and cosmological expansion.

**Far away**

At very large distances we only see the expansion. In the latter case Hubble’s law describes this, but then we cannot ignore relativist effects. The simple Doppler formula is not good enough. Objects that are moving with respect to an observer at high velocity (in comparison to the speed of light) exhibit time dilation due to this relative motion. Time passes more slowly on those objects and their atoms emit radiation with lower frequency. This would hold for relative motion in any direction, not only radial velocity. Another red shift is predicted by general relativity. Strong gravitational fields can cause time dilation. This is called **gravitational red shift**. This effect is generally small but becomes significant e.g. near a black hole.

**We need a Cosmological Model**

Cosmological and gravitational redshift cannot be explained by the Doppler effect, they require a cosmological theory that formulates the expansion of the Universe. Developing such a theory is the prime focus in modern Cosmology.

In that context, if we want to measure distance, we need to raise the question **what do we mean by distance in an expanding Universe?**