06.09 Special relativity and cosmology

Module 06 — Heat and Particles · quiz worth 17 pts

The big idea Special relativity connects measurements made in different inertial frames, while cosmology uses redshift, the cosmic microwave background, and primordial element abundances to test the history of an expanding universe.

Name the frame first

Relativity questions become manageable when you say where the clock or object is at rest.

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Symbols and reference values

SymbolMeaningUnit or value
v Relative speed m/s
c Vacuum speed of light 2.998 × 10⁸ m/s
γ Lorentz factor dimensionless
Δτ Proper time s
L₀ Proper length m
CMB temperature Present blackbody temperature about 2.725 K
1 Mpc One megaparsec about 3.26 million light-years
Before you start Review square roots, ratios, scientific notation, and graph proportionality. Relativity calculations also require you to name the observer's inertial frame.

Two relativity postulates

First, the laws of physics have the same form in every inertial frame; there is no preferred frame of absolute rest. Second, every inertial observer measures the same vacuum speed c for light, independent of source or observer motion. These statements require space and time measurements to transform together.

The same light clock has a vertical path in its rest frame and a longer diagonal path in the platform frame as the clock moves right; both observers measure light at c, so Δt is greater than proper time Δτ.
The two inertial frames use the same laws and both measure c. Because the moving-frame light path is longer, their time and distance coordinates differ.

Proper quantities and frames

Proper time Δτ is measured by one clock present at both events. Proper length L₀ is measured in the object's rest frame. An observer who sees the clock moving measures a longer interval Δt = γΔτ. An observer who sees the object moving measures a shorter dimension parallel to its motion, L = L₀/γ. Perpendicular dimensions are unchanged. These are consistent frame-dependent measurements, not merely a visual illusion.

Relativity formulas

Lorentz factor
γ=11−v2/c2\gamma=\frac{1}{\sqrt{1-v^2/c^2}}

γ is dimensionless and is at least 1.

Time dilation
Δt=γΔτ\Delta t=\gamma\Delta\tau

Δτ is proper time on the clock moving with the events.

Length contraction
L=L0γL=\frac{L_0}{\gamma}

Only the dimension parallel to relative motion contracts.

Rest energy
E0=mc2E_0=mc^2

m is invariant mass; total energy and momentum depend on frame.

Hubble-Lemaître relation
v≈H0dv\approx H_0d

For nearby galaxies in the smooth expansion, recession rate is approximately proportional to distance.

Solve a relativity problem

Name the frames

State which object or clock is at rest in each frame.

Find the proper quantity

Proper time uses one co-located clock; proper length belongs to the object's rest frame.

Calculate γ

Use v/c as a dimensionless ratio.

Choose multiply or divide

Moving clocks have a longer coordinate interval: multiply proper time by γ. Moving parallel lengths are shorter: divide proper length by γ.

Check the direction

The result must approach the classical value as v becomes small compared with c.

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Worked relativity examples

Clock moving at 0.60c

  • Relative speed. v=0.60cv=0.60c
  • Proper time. Δτ=8.0 s\Delta\tau=8.0\,\text{s}

Find the time interval in the frame where the clock moves.

  1. Calculate the Lorentz factor.
    γ=11−0.602=1.25\gamma=\frac{1}{\sqrt{1-0.60^2}}=1.25
  2. Dilate the proper time.
    Δt=(1.25)(8.0 s)\Delta t=(1.25)(8.0\,\text{s})

Δt = 10.0 s.

One-meter rod moving at 0.70c

  • Proper length. L0=1.00 mL_0=1.00\,\text{m}
  • Relative speed. v=0.70cv=0.70c

Find the measured length parallel to motion.

  1. Find γ.
    γ=11−0.702=1.400\gamma=\frac{1}{\sqrt{1-0.70^2}}=1.400
  2. Contract only the parallel dimension.
    L=1.001.400 mL=\frac{1.00}{1.400}\,\text{m}

L = 0.714 m. A perpendicular one-meter dimension would remain one meter.

From expansion to the hot Big Bang

The Big Bang model describes an early hot, dense state followed by expansion of space; it is not an explosion from one location into preexisting empty space. General relativity extrapolated too far reaches a limit where present theory is incomplete, so a physical singularity should not be presented as an experimentally established object. The universe's current age is about 13.8 billion years.

Light waves stretched to longer red wavelengths as space expands between galaxies.
Cosmological redshift is evidence that space has expanded; it is not a chemical explosion into empty space.

Three evidence lines

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Relativity and cosmology traps

There is no preferred absolute-rest frame

Relativity compares inertial frames without declaring one universal frame motionless.

Invariant mass does not increase with speed

Modern usage keeps rest mass invariant. Total energy and momentum increase with speed.

Contraction is directional

Only the dimension parallel to relative motion contracts.

Redshift is not the only evidence

The CMB and primordial light-element abundances independently test the hot early-universe model.

Relativity and cosmology practice

For relativity, label the frame. For cosmology, name the observation and the prediction it tests.