Proper, comoving and light-travel distances

One source can support several distance questions

How far away is a galaxy? A meaningful answer must say which distance is being reported. The separation on a chosen present cosmic-time slice, the separation at emission and c times the elapsed light-travel time are different constructions in an expanding geometry.

A proper radial separation is measured along a specified same-cosmic-time spatial slice in the background model. Assigning a present-time distance to a distant source requires that model; the light received now is not a direct photograph of every point on the present slice.

A comoving separation factors out the smooth background expansion for endpoints following the background flow. In a spatially flat model with a_ref = 1, let D_C be their fixed radial comoving separation. Their proper radial separation at cosmic time t is D_prop(t) = a(t)D_C.

This equation describes comoving background endpoints. It does not make the dimensions of a bound local ruler grow with a, or require every galaxy’s peculiar motion to vanish. Those physical details need their own account.

Use one ideal model for every comparison

Choose a spatially flat, expanding general-relativistic model containing only pressureless matter, with no radiation or cosmological constant. Write a(t) = (t/t₀)²ᐟ³, normalized to a(t₀) = 1. The positive reference time t₀ sets the unit; no measured cosmic age is being supplied.

Consider a comoving source that emits the received light at t_e = t₀/8. At emission its scale factor is a_e = (1/8)²ᐟ³ = 1/4. The background redshift is therefore 1 + z = 1/a_e = 4, giving z = 3.

For a radial light ray in this flat background, the comoving separation is D_C = c∫ from t_e to t₀ of dt/a(t). This formula uses the changing scale factor along the light path; it is not simply c times the coordinate-time interval.

Setting u = t/t₀ gives D_C = ct₀∫ from 1/8 to 1 of u⁻²ᐟ³du = 3ct₀[1 − (1/8)¹ᐟ³] = 1.5ct₀. The length ct₀ is an explicitly positive model unit. Only the stated positive-time interval is needed for this calculation.

The comoving separation stays fixed while proper separation changes

The same emitter and receiver have comoving separation 1.5ct₀ throughout this ideal background account. Their same-time proper separation at emission is a_eD_C = (1/4)(1.5ct₀) = 0.375ct₀.

At reception, a(t₀) = 1, so their present-slice proper separation is 1.5ct₀. Equality between the current proper number and the comoving number follows from the chosen normalization. A different reference normalization would require consistent corresponding coordinates.

The intermediate rows follow the same pair of comoving positions. They are spatial-separation calculations on different cosmic-time slices, not the receiver taking simultaneous distant measurements at each listed time.

Both the source history and the meaning of the distance must be kept fixed when making this comparison. Comparing a proper distance at one epoch with an unlabelled distance at another can turn a consistent model into an apparent numerical contradiction.

Light-travel length uses elapsed cosmic time

The elapsed cosmic time from emission to reception is t₀ − t_e = 7t₀/8. Multiplying by c gives the light-travel length c(t₀ − t_e) = 0.875ct₀. It answers a time-based question using the specified cosmic-time coordinate.

For this one signal, the emission-slice proper separation is 0.375ct₀, the light-travel length is 0.875ct₀ and the reception-slice proper separation is 1.5ct₀. The values differ because their definitions differ, although their units are the same.

A freely propagating photon is locally measured to travel at c in the appropriate local inertial description. That statement does not make c times elapsed cosmic time equal to every cosmological distance definition. The global geometry and time slices remain part of the calculation.

Nor is the light-travel length a present-time path drawn across a fixed static space. The calculation spans an evolving background. It should not be used as an automatic present separation, spatial radius or horizon without the corresponding additional definition.

Carry the convention into the reported number

A distance report should identify the quantity, reference epoch, background model, source motion assumptions and units. A redshift alone is not an exact model-free distance conversion. Peculiar motions and observational uncertainties add further questions in real applications.

For the flat normalized example, present proper and radial comoving separations share a numerical value. In curved spatial geometry, radial and transverse comoving distances also need to be distinguished; they are not interchangeable merely because both are called comoving.

The toy deliberately avoids an inference about an origin, a horizon or the full size of the Universe. Its finite interval is sufficient to show why one source can have several valid associated lengths.

Use the explicit question before choosing a formula: separation on which time slice, a comoving coordinate distance under which normalization, or elapsed travel time in which time coordinate? The definition makes the calculation interpretable.

Worked reasoning

Try the reasoning

1. Does D_C change for the supplied comoving endpoint pair as a grows from 1/4 to 1?

   Show the answer and reasoning

   No.

   D_C factors out the background expansion in the stated normalization; their proper separation changes instead.

2. Must c times lookback time equal the present-slice proper separation?

   Show the answer and reasoning

   No.

   The supplied model gives 0.875ct₀ for the former and 1.5ct₀ for the latter.

3. Is a model-assigned present separation a direct simultaneous photograph of the distant source now?

   Show the answer and reasoning

   No.

   We receive earlier emitted light and use a background history and distance definition to assign the present-slice quantity.

What to carry forward

• A cosmological distance needs a definition, epoch and geometric convention.
• Proper separation scales with a for the stated comoving pair, while its comoving separation stays fixed.
• Light-travel length and present proper separation are different quantities in an evolving background.