Redshift from a Non-Stationary Metric

1. Understanding Redshift from a Non-Stationary Metric

The redshift arises because the wavelength of light is stretched as it propagates through a dynamically changing metric. The fundamental reason is that in General Relativity, light follows null geodesics, and the metric determines how these geodesics evolve over time.

The most common example of a non-stationary metric is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which describes an expanding or contracting universe:

ds² = -c² dt² + a²(t) (dr² / (1 - k r²) + r² dθ² + r² sin²θ dφ²)

2. Derivation of the Redshift Equation

The redshift z is defined as the relative change in wavelength:

z = (λ_observed - λ_emitted) / λ_emitted

or equivalently in terms of frequency:

z = (f_emitted - f_observed) / f_observed

Since light follows a null geodesic ds² = 0, the proper time interval for a comoving observer is:

dt / a(t) = constant

A photon emitted at time t_e and received at time t_o will experience a shift in wavelength due to the change in a(t). The key idea is that the number of wave crests remains constant, but the spatial separation between them changes as space expands.

Using the property that the frequency of light is inversely proportional to the scale factor:

f_observed / f_emitted = a(t_e) / a(t_o)

we define the cosmological redshift as:

z = (a(t_o) / a(t_e)) - 1

3. Special Cases

Small Redshifts (z ≪ 1)

For small z, we approximate the scale factor using the Hubble Law:

a(t) ≈ 1 + H₀ (t - t_o)

This gives the Doppler approximation:

z ≈ H₀ d / c

Large Redshifts (z ≫ 1)

At high redshifts, we need the full Friedmann equations to compute a(t), leading to:

1 + z = (a(t_o) / a(t_e)) = exp(∫_{t_e}^{t_o} H(t) dt)

where H(t) is the Hubble parameter.

4. Conclusion

A non-stationary metric, such as the expanding FLRW metric, leads to a redshift in light due to the stretching of spacetime. The redshift is directly related to the scale factor a(t), and the equation:

1 + z = a(t_o) / a(t_e)

is fundamental in cosmology, helping us measure the expansion history of the universe.