Why distance needs a time and a definition

A traveller's journey does not freeze the endpoints

Imagine a signal sent between two markers whose separation changes while the signal travels. The separation when it left, the separation when it arrived, and its speed multiplied by elapsed time answer different questions. They need not be equal. A map of the endpoints at one time is not a record of every point along the signal's journey.

Cosmic distances require a similar discipline. We receive light from an emission event, then use a cosmological model to relate that event to a source's location on a surface of equal cosmic time. A surface of equal cosmic time is a chosen set of events assigned the same time in the ideal homogeneous model. It is not a photograph taken simultaneously by our telescope.

The physical questions are precise once the definitions are supplied. How far apart were source and observer at emission in the model? What separation does the model assign at reception? What length equals c times the lookback interval? Asking simply 'How far away?' can hide the choice among them.

Keep a comoving label and a physical separation separate

For ideal markers moving with the large-scale cosmic flow, proper separation can be written D(t) = a(t)χ. Here t is cosmic time, a(t) is a dimensionless scale factor normalized to a(t₀) = 1 at reception, and χ is a constant comoving separation with units of length. D(t) measures separation along the specified cosmic-time slice in this background model.

Comoving coordinates factor out the average expansion, so the same ideal pair keeps its χ while D changes with a. They are labels adapted to a model, not physical rods that prevent motion. A galaxy with peculiar motion—motion relative to the average flow—need not keep exactly the same comoving position. Bound systems also require their own dynamics.

Under the pure cosmological-redshift interpretation, 1 + z = 1/aₑ, where aₑ is the emission scale factor. Therefore Dₑ = aₑχ and D₀ = χ give D₀/Dₑ = 1 + z for the ideal comoving pair. This compares two assigned separations at different times. It does not give χ without a model of the intervening expansion and the light path.

One synthetic model gives three different lengths

Choose a spatially flat homogeneous model whose scale factor grows linearly with cosmic time: a(t) = t/t₀. Let t₀ = 10 Gyr and let the received signal have cosmological redshift z = 1. This is a teaching model, not a fit to the real Universe. Its value is that every number below follows from one consistent expansion history.

Since aₑ = 1/(1 + z) = 0.5, linear expansion gives tₑ = 5 Gyr. The lookback interval is 5 Gyr, so c times that interval is 5 billion light-years, abbreviated Gly. This is the light-travel length for the model's cosmic-time interval.

Accounting for the changing scale factor along the light path gives a comoving separation χ = c t₀ ln 2 ≈ 6.931 Gly. The symbol ln means natural logarithm; ln 2 ≈ 0.6931. At reception, a₀ = 1, so proper separation is also 6.931 Gly. At emission, aₑ = 0.5, so proper separation was about 3.466 Gly.

The three lengths—3.466, 5.000 and 6.931 Gly—are not competing estimates of one undefined distance. They are different defined quantities in the same model. Nothing in this calculation asks light to exceed its local vacuum speed. The changing large-scale separation is part of the geometry being described.

Optional: calculate the comoving light path

You can retain the distinction in the table without doing calculus. To see where χ comes from, divide each small locally traversed length c dt by the scale factor at that time. The positive comoving path-length increment is |dχ| = c dt/a(t). Here dt is a small positive interval of cosmic time, and |dχ| is the magnitude of the corresponding comoving-coordinate change along a radial light path in the stated model. The radial coordinate measured outward from the observer decreases for incoming light; the absolute-value bars keep the summed path length positive.

Adding those increments from emission to reception gives χ = ∫[c/a(t)] dt, with limits tₑ and t₀. The integral is a continuous sum. For a(t) = t/t₀, substitute 1/a(t) = t₀/t. Then χ = c t₀ ∫(1/t) dt = c t₀ ln(t₀/tₑ). With tₑ = t₀/2, the logarithm is ln 2.

Check dimensions. The scale factor and the logarithm of the time ratio are dimensionless. c t₀ has dimensions of length. A logarithm of a time by itself would be ill-defined; the expression uses the dimensionless ratio t₀/tₑ. Substitution gives 10 Gly × 0.693147 ≈ 6.93147 Gly.

In the nonexpanding limit with a constant scale factor normalized to 1, the same continuous sum reduces to χ = c(t₀ − tₑ). That is why the familiar nearby stationary-distance rule works in its own regime. Expansion changes the relationship by changing a during propagation; the local value of c is unchanged.

A brightness distance is another definition again

Cosmologists also define luminosity distance from how received flux relates to emitted luminosity, and angular-diameter distance from how a known transverse size relates to its observed angle. Expansion affects photon energies, arrival rates and apparent angular sizes, so these operational distances need not equal either proper separation in the table.

Those definitions are useful because a telescope measures arriving light and angles. A source's intrinsic brightness or size, its evolution, dust and calibration enter the inference. The later cosmological-distance chapter develops the formulas and their relationships. At this stage, the essential habit is to ask which distance a quoted number means rather than silently interchanging them.

A cosmological model also separates the observable region from the total Universe. Even a carefully calculated horizon distance is an observational boundary within the model; it does not measure a physical wall or the total spatial extent. The horizons chapter will distinguish different causal limits.

Make a cosmic length reproducible

A useful distance statement identifies its type, redshift, model and time convention where relevant. For our synthetic example, '6.931 Gly present proper separation in the specified flat coasting model' says what was calculated. 'The light travelled 6.931 billion years' would replace the computed 5-Gyr lookback interval with the wrong quantity.

The purpose of definitions is to connect numbers with physical questions. Once the labels are clear, several distance values can agree with one coherent account, and a genuine disagreement becomes easier to recognize and test.

Worked reasoning

Try the reasoning

1. In the synthetic coasting example, which is larger: the reception-time proper separation or c times lookback time? Does this imply a local light speed greater than c?

   Show the answer and reasoning

   The proper separation is larger, about 6.931 Gly versus 5 Gly. It implies no such local speed violation.

   The quantities use different definitions in a changing geometry. A simultaneous model separation is not a path length obtained by freezing the endpoints.

2. For an ideal comoving pair with cosmological z = 2 and a₀ = 1, what ratio D₀/Dₑ follows? Does that supply either absolute distance?

   Show the answer and reasoning

   The ratio is 3, and neither absolute distance follows from it alone.

   1 + z = a₀/aₑ = D₀/Dₑ. A ratio does not fix the comoving separation or expansion history.

3. Why is ln(t₀/tₑ) meaningful in the example while ln(5 Gyr) is not a properly defined physical expression?

   Show the answer and reasoning

   The time ratio is dimensionless, whereas the numerical logarithm of a dimensional time depends on the arbitrary unit chosen.

   Changing from years to seconds must not change a physical prediction. A dimensionless ratio avoids that ambiguity.

What to carry forward

• A cosmic distance needs a definition and, for proper separation, a specified time slice.
• Comoving labels remove average expansion; proper separations retain it.
• c times lookback time is generally different from present separation in an expanding model.
• Different distance measures can be consistent values from the same physical history.