The general relationship between recessional velocity \(\left(v_{r}\right)\) and redshift \((z)\) is \(v_{r}=c z .\) This simple relationship fails, however, for very distant galaxies with large redshifts. Explain why.

The relativistic Doppler effect comes into play when we observe objects moving at velocities close to the speed of light. The classical Doppler effect, used for sound waves, isn't sufficient here because it doesn't consider the effects of relativity.

In relativity, the observed frequency and wavelength of light alter with velocity due to time dilation and length contraction. This modification in light behavior is what we call the relativistic Doppler effect. A high recession velocity, which we denote as \(v_r\), alters how we perceive the redshift \(z\) of a galaxy. Even a small increase in velocity near the speed of light causes substantial changes in redshift.

The mathematical expression for the relativistic Doppler effect is: \[ v_r = c \frac{(1+z)^2 - 1}{(1+z)^2 + 1} \] This formula is crucial because it helps us measure velocities of far-off galaxies more accurately when they exhibit large redshifts.

Cosmic expansion refers to the continuous growth of the Universe. This expansion was first proposed by Edwin Hubble, who noticed that galaxies are moving away from us and each other, implying that the Universe is expanding.

This ongoing stretching of space means that the light from distant galaxies shifts towards longer wavelengths, a phenomenon known as redshift.

The further away a galaxy is, the faster it seems to be receding from us. This relationship is described by Hubble's Law, which states that the recessional velocity \(v_r\) of a galaxy is proportional to its distance \(D\). However, as distances become enormous, the impact of cosmic expansion combined with high velocities ventures into the relativistic domain. This is the reason the simple relationship \( v_{r} = c z \) becomes oversimplified for very distant galaxies.

Cosmic expansion not only stretches wavelengths, thus increasing redshift \(z\), but also augments the apparent velocity, necessitating the use of relativistic formulas to describe high-speed motion accurately.

High redshift galaxies are those that display significant redshifts, indicating they are extremely distant and receding from us at high speeds. The high redshift values mean that these galaxies are from the early Universe, providing valuable information about its history and expansion.

For these galaxies, simple linear relations between velocity and redshift don’t hold because redshifts involve more than just the ordinary Doppler effect. They reflect the Universe's expansion and necessitate incorporating relativistic physics.

When we observe high redshift galaxies, we see them as they were billions of years ago. This helps astronomers understand the conditions and processes of the early Universe. Using the relativistic Doppler effect formula, we derive more precise velocities for these galaxies, ensuring our understanding of cosmic expansion is accurate.

Hence, recognizing and accurately interpreting high redshift is essential for cosmology, as it influences our comprehension of how the universe has expanded and evolved over billions of years.