Cosmology · Horizons and the observable Universe

Recession speed and locally measured speed

Use the exact empty Milne example to calculate a recession derivative above c while the same test observer has an inertial speed below c.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

State which distance and simultaneity convention defines a recession speed.
Calculate two distinct speed comparisons for one exact timelike worldline.
Explain why a distant recession derivative above c does not let matter locally overtake light.

A speed requires a distance and a time prescription

A local inertial observer compares nearby events using a local ruler and clock. A cosmological proper recession derivative compares distant comoving observers on successive cosmic-time slices. Both are meaningful when defined, but they are different constructions.

For comoving observers in an FLRW background, a same-cosmic-time proper separation D = a r has derivative dD/dt = HD when the length-valued comoving coordinate r is fixed. Distant values of HD can exceed c without defining a local material speed above c.

An explicit example makes the distinction more concrete than a slogan. The following empty Milne spacetime is a description of part of flat Minkowski spacetime using expanding cosmic slices. It is not a model fitted to the actual matter-filled Universe.

The example changes both the time slicing and the distant distance comparison. Its result should not be read as saying that every numerical difference between speeds is a mere mistake, or that the actual cosmological spacetime must be flat.

One geometry, two time slicings

Let T_M and X be one-dimensional inertial Minkowski time and position. Introduce positive Milne cosmic time t and dimensionless radial rapidity ζ by T_M = t cosh ζ and X = ct sinh ζ. Here c is the local speed of light.

The chart covers the future interior T_M > |X|/c, with T_M > 0. Its radial metric is ds² = −c²dt² + (ct)²dζ². The full Milne model has negatively curved spatial slices despite flat four-dimensional spacetime.

The null boundary of this coordinate domain is not an ordinary finite-ζ comoving point. Taking ζ to infinity is a limit, not placing a material observer with a finite coordinate on that boundary.

Along a t = constant radial slice, the proper separation from the origin to a test observer at fixed ζ ≥ 0 is D = ctζ. Thus the radial scale expands with H = 1/t. ζ is dimensionless rapidity here, not the length-valued comoving coordinate used in the earlier horizon calculations.

The same observer can have different distant speed comparisons

For fixed ζ, the Milne proper recession derivative is dD/dt = cζ. In the inertial coordinates the same observer follows X = cT_M tanh ζ, so its inertial velocity is dX/dT_M = c tanh ζ.

At ζ = 1, the recession derivative is c while the inertial speed is about 0.76159c. At ζ = 2, the recession derivative is 2c while the inertial speed is about 0.96403c. The two statements use different simultaneous slices and distance/time prescriptions.

The inertial worldline remains timelike for every finite ζ because |tanh ζ| < 1. A massive test observer does not overtake a local light ray. The value 2c belongs to the cosmic slice-distance derivative, not to a local material velocity.

Even at the same numerical cosmic time t, the distant observer event has inertial time T_M = t cosh ζ, while the origin event has inertial time t. These events are simultaneous in Milne cosmic time but are not simultaneous in that inertial frame.

Fixed-ζ test observers in the exact empty Milne example; all entries are dimensionless and no observed cosmic velocity is supplied
ζ	(dD/dt)/c	(dX/dT_M)/c	T_M/t	X/(ct)	D/(ct)
0.5	0.5	0.462117	1.127626	0.521095	0.5
1	1	0.761594	1.543081	1.175201	1
2	2	0.964028	3.762196	3.626860	2

An exact empty Milne test-observer example uses T_M=t cosh ζ and X=ct sinh ζ for t>0 within the future Minkowski interior T_M>|X|/c and T_M>0. The radial rapidity ζ is dimensionless, unlike the length-valued comoving coordinate in the earlier horizon examples. On a constant Milne cosmic-time slice D=ctζ and H=1/t, so the recession derivative divided by c is ζ. In the inertial coordinates the same fixed-ζ worldline has velocity dX/dT_M divided by c equal to tanh ζ. The graph displays only 0≤ζ≤2: dashed orange squares identify the recession derivative, solid blue circles the inertial velocity, and a dotted guide marks the reference value 1. At ζ=0.5 the pair is (0.5, approximately 0.462117), at ζ=1 it is (1, approximately 0.761594), and at ζ=2 it is (2, approximately 0.964028). The quantity above 1 is a distant cosmic-slice recession derivative, not a local material velocity above c. For every finite ζ the inertial worldline is timelike, with cosh²ζ−sinh²ζ=1, while locally measured photons remain null. Milne spatial slices have negative curvature although this four-dimensional spacetime is flat. The null chart boundary is a limiting surface, not an ordinary finite-ζ comoving observer. This exact empty example is not a fitted description of the actual matter-filled Universe and makes no current velocity, horizon or parameter measurement. Original Ark teaching model.

An invariant check separates physics from a label

At fixed ζ, differentiating the coordinate map gives dT_M = cosh ζ dt and dX = c sinh ζ dt. Substitution into −c²dT_M² + dX² yields −c²dt² because cosh²ζ − sinh²ζ = 1.

The worldline therefore has proper time t up to an additive constant along the declared comoving test observer. This invariant interval test confirms the timelike character independently of the large recession derivative.

A local photon measurement remains null in either coordinate system. Changing a distant simultaneity convention cannot turn this massive worldline into a light ray or allow a signal to travel outside the local causal cone.

One should also keep spatial curvature and spacetime curvature separate. The Milne chart demonstrates that an expanding cosmic scale factor and curved spatial slices need not imply curved four-dimensional spacetime. The actual Universe requires evidence and a matter-containing model beyond this example.

Translate a speed claim before comparing it with c

When a statement says that a distant galaxy recedes faster than light, first identify the distance and cosmic-time slicing used. A proper recession derivative above c does not assert that the galaxy passes a nearby observer at a local speed above c.

Special-relativistic local velocity addition applies to the velocity quantities defined in its own local inertial comparison. Inserting a distant HD directly without establishing the same comparison mixes definitions.

Likewise, a cosmological redshift does not supply an unqualified unique recession speed independently of the background and distance convention. The relation between a measured spectrum and a distant speed must state the model and definition.

The exact Milne calculation is a useful counterexample to a false universal claim about recession. It does not determine present cosmological parameters, horizon radii or the geometry of our actual Universe.

Worked reasoning

Calculate both speeds for one observer

For ζ = 2, compare Milne proper recession with the Minkowski inertial speed.

The Milne slice distance is D = ctζ = 2ct, so dD/dt = 2c.
The inertial worldline has X/(cT_M) = sinh 2/cosh 2 = tanh 2.
Thus dX/dT_M ≈ 0.964028c, below c. The invariant worldline interval remains −c²dt².

The same observer has recession derivative 2c and inertial speed about 0.964028c under the two stated comparisons.

Try the reasoning

May a distant cosmic recession derivative be inserted into a formula for two local inertial velocities without redefining the comparison?
Show the answer and reasoning
No.
The distance, clock and simultaneity prescriptions differ.
Does the test observer at ζ = 2 become a locally faster-than-light object?
Show the answer and reasoning
No.
Its inertial speed is c tanh 2 < c, and its worldline is timelike.
Does the expanding Milne scale require curved four-dimensional spacetime?
Show the answer and reasoning
No; this example is a chart of flat Minkowski spacetime.
Its spatial slices are curved. This does not establish that the actual cosmological spacetime is flat.

What to carry forward

Distant recession and local velocity use different comparisons.
The exact Milne example has arbitrarily large proper recession derivatives while finite-rapidity material worldlines stay timelike.
A speed statement is interpretable only after its distance, time and simultaneity conventions are specified.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.