Motion, gravity and cosmological redshift

The ratio does not carry a label naming its cause

Imagine an identified reference pattern received at wavelengths 1.10 times its reference values. Its measured redshift is z = 0.10. That calculation compares wavelengths. It does not yet tell us whether the source is moving locally, lies in a different gravitational environment, or is being described across an expanding cosmological background.

To interpret the shift, specify the emitter, the receiver, their clocks and motions, and a spacetime model connecting the events. The same numerical ratio need not imply the same physical history. An explanation becomes testable when it predicts observations beyond the original ratio.

Motion, gravity and cosmological expansion are descriptions within a common relativistic framework. Dividing a total measured shift into labelled contributions can depend on the chosen observers and model. The categories below are useful controlled situations, not three unrelated laws that can be assigned uniquely to every photon without assumptions.

Relative motion changes the measured frequency

Consider source and receiver in an approximately flat region, with specified relative motion along the line of sight. Recession produces a redshift; approach produces a blueshift. Their measurements differ even though each locally measures vacuum light to travel at c. The change concerns wavelength and frequency, not one observer catching light that locally travels more slowly.

For a small purely radial speed compared with c, the Doppler approximation is z ≈ vᵣ/c, or vᵣ ≈ cz. Here vᵣ is positive for recession in this convention. For a synthetic z = 0.001 dominated by this local motion, rounded c gives vᵣ ≈ 300 km/s. The approximation assumes the local Doppler interpretation and neglects terms important at larger speeds.

Do not extend that shortcut to arbitrary z or to cosmological separation rates. At high speeds, relativistic clock and motion effects matter. Even motion perpendicular to the line of sight can affect a relativistic frequency comparison. The introductory along-the-line-of-sight example is not a complete account of all moving-observer cases.

A galaxy's peculiar velocity is its motion relative to the chosen average cosmic flow. It can shift spectral features in addition to the background expansion contribution. For nearby objects, treating every measured redshift as pure expansion can therefore distort a distance inference.

Compare observers at different gravitational potentials

In a stationary weak gravitational field, light emitted by a stationary source deeper in the potential and received by a stationary observer higher up is gravitationally redshifted. The reverse comparison is blueshifted. Stationary here means that the observers remain at fixed positions in that static description; their relative motion is not the explanation being isolated.

Each observer uses a local clock to measure the radiation frequency. General relativity relates those clock comparisons across the gravitational environment. Saying that light climbs out and loses energy is a useful description relative to the stated observers; it should not be turned into a claim that the photon slows below c in a local vacuum measurement.

Spectra from compact stars can be used to study this effect, but motion and calibration also enter the observation. Separating a gravitational contribution can require orbital information, an instrument calibration and a model of the emitting region. A quoted 'equivalent velocity' may simply express the size of a redshift in speed units; it need not describe the star physically receding at that speed.

Expansion compares the geometry at emission and reception

In an ideal homogeneous expanding model, source and receiver following the cosmic flow satisfy 1 + z = aₒ/aₑ. The dimensionless scale factor a describes how the background's comoving separations change. Subscripts o and e refer to observation and emission. With aₒ = 1, a cosmological z = 1 corresponds to aₑ = 0.5.

This means that an ideal comoving separation at emission was half its observation-time proper separation. It is not a statement that the total Universe had a measured diameter of some particular length. It also does not describe bound atoms or a galaxy automatically expanding by that factor.

The endpoint ratio does not determine the full intervening expansion history. Many scale-factor functions can connect the same values at different times. Inferring a lookback interval or a cosmic distance requires that history, together with the relevant distance definition. A cosmological redshift should not be converted into a unique local special-relativistic velocity merely by naming a Doppler formula.

Use additional observations to separate the accounts

When a specified model supplies successive compatible wavelength factors, multiply the factors 1 + z. For teaching, suppose those factors are 1.10 and 1.01. The total is 1.111, giving z = 0.111. Adding the two finite redshifts would instead give 0.110. Products automatically include the extra 0.001 cross term.

This arithmetic does not independently measure the contributions. Their physical decomposition must use consistently defined frames and a suitable model. A total observed shift alone does not tell us which two factors to insert.

A local orbit predicts time-varying motion that can be checked with repeated spectra and orbital data. A gravitational interpretation can be compared with independently constrained masses, radii and source motions. A cosmological interpretation is assessed across source populations and complementary evidence about distances and expansion. Calibration or line-identification errors must be tested alongside the physical accounts.

The discipline is to attach each conclusion to its actual evidence. A secure spectral ratio can coexist with uncertainty about its components. More data can improve the interpretation without changing what the original measurement meant.

Worked reasoning

Try the reasoning

1. What assumptions accompany the estimate vᵣ ≈ cz?

   Show the answer and reasoning

   A low-speed, local, predominantly radial Doppler interpretation with other contributions appropriately handled.

   The redshift definition alone establishes none of these conditions. Cosmological z is not universally a local radial velocity divided by c.

2. For a pure cosmological z = 2 with aₒ = 1, what is aₑ?

   Show the answer and reasoning

   1/3.

   Use aₑ = 1/(1 + z). This ratio supplies no absolute separation or travel time without further information.

3. Can one accurately measured wavelength ratio, without other information, uniquely separate gravitational, motion and cosmological contributions?

   Show the answer and reasoning

   No.

   Several physical histories can produce the same ratio. Distinguishing them requires extra data and a consistently specified model.

What to carry forward

• A redshift is a measured wavelength comparison before it is an explanation.
• Motion, gravitational clock comparisons and expansion require different stated conditions.
• Finite compatible shift factors multiply; their redshifts do not generally add.
• Independent observations test an interpretation beyond the original spectral ratio.