International Relativity

Redshift and the Doppler Effect

When we analyze the light spectrum from distant galaxies, we observe a redshift—a shift in the emitted wavelength toward the red end compared to nearby galaxies. This is often explained by the Doppler effect, suggesting galaxies are receding from each other, implying an expanding universe.

Who is receding from whom? Is the other galaxy moving away from us, or are we moving away from it? And at what speed? In special relativity, it doesn’t matter—the formulas yield the same result. The relativistic Doppler formula for recession (replacing v with -v) gives:

1 + z = λ/λ₀ = f₀/f, where z is the redshift parameter.

From the relativistic Doppler effect, we can derive recession velocity from z:

Hubble originally related distance to z, not velocity: z ≈ H_z × L. Modern plots use z and apparent magnitude (brighter = lower magnitude) instead of distance, as luminosity decreases exponentially with distance.

Distance measurement: Based on observed luminosity (e.g., from supernovae). Lower brightness means greater distance—but this is the light-travel distance, not current or emission-time distance.

Galaxy Recession or Space Expansion?

Observations show galaxies receding, but is it the galaxies moving through space, or space itself expanding? The homologous expansion paradigm (from the 1930s) favors the latter: space stretches, dilating photon wavelengths and causing redshift.

In this view, no Doppler effect occurs—z + 1 is simply the scale factor by which space has expanded since emission. If z + 1 = 2, wavelengths doubled, meaning 1 cm became 2 cm.

The Scale Factor a(t)

In cosmology, the scale factor a(t) describes relative distances over time, normalized to a(t₀) = 1 today. Thus:

1 + z = a(t₀) / a(t) = 1 / a(t) ⇒ a(t) = 1 / (1 + z)

For z = 1, space doubled during light travel, so a(t) = 0.5. Focus on z-distance relations, not velocities.

Has Expansion Always Been at the Same Rate? Cosmological Models

Expansion history shapes universe models:

1. Static (Einstein, pre-Hubble): Constant a(t), balanced by cosmological constant (repulsive “vacuum pressure”).
2. Constant rate: Linear a(t) growth.
3. Decelerating: Gravity slows expansion; could lead to Big Crunch. Einstein-de Sitter is critical balance.
4. Accelerating (Inflationary): a(t) grows exponentially, driven by dark energy. Matches 2025 data: Universe expanded slower in past, now accelerating (JWST confirms early galaxies challenge pure ΛCDM).
5. Variable: E.g., early deceleration then acceleration. Ongoing “Hubble tension” (H₀ ~67 km/s/Mpc from CMB vs. ~73 from supernovae) suggests evolving dark energy.

JWST updates refine these, showing no major deviations yet.

Why Doesn’t the Sun-Earth Distance Increase with Expansion?

Local gravity dominates: Bound systems like our Solar System resist cosmic expansion. Expansion affects large scales (>100 Mpc); below that, gravitational binding wins. Think of it as spacetime curvature overriding the global stretch.