Cosmology · Spacetime, curvature and cosmological models

Geometry in Friedmann–Lemaître models

Separate a cosmic time coordinate, an evolving scale factor and constant-curvature spatial geometry, then compare intrinsic circle measurements without an external embedding.

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

Identify the geometric assumptions and conventions of an FLRW background.
Compare circumference and radial distance in three supplied spatial-curvature models.
Distinguish a spatial slice’s curvature from spacetime curvature and from the dynamical equations for expansion.

Symmetries organize a background geometry

A Friedmann–Lemaître–Robertson–Walker background, usually shortened to FLRW, describes a spacetime organized into homogeneous and isotropic spatial slices. Its smooth background has a cosmic time coordinate and an evolving scale factor a(t). The symmetries apply to the spatial description at each such time, not to an absence of time evolution.

Observers at fixed comoving spatial coordinates follow the ideal background flow. Their proper time is cosmic time in this metric. This is a specified observer family; it is not a claim that every moving local clock has the same proper-time history.

The metric supplies geometric relationships. To predict a(t), a physical model must also specify gravity, component properties and conditions selecting a solution. The preceding expansion equations use general relativity for that dynamical step; the symmetry-based metric and the gravitational field equations play distinct roles.

Local structures, peculiar motions and perturbations require more detail than the smooth background. The background remains useful because it makes the assumed reference geometry explicit, allowing observations and corrections to be compared consistently.

Keep the length scale and coordinates distinct

Use dimensionless a(t), a fixed positive reference length R_ref and dimensionless radial coordinate χ. On a fixed cosmic-time slice, write the radial proper-distance element as aR_ref dχ. The same length factor multiplies the angular part of the spatial metric.

In a simply connected constant-curvature local model, define S_k(χ) = sin χ for k = +1, S_k(χ) = χ for k = 0, and S_k(χ) = sinh χ for k = −1. The symbol sinh denotes the hyperbolic sine. All functions here take dimensionless arguments.

An optional full metric expression is ds² = −c²dt² + (aR_ref)²[dχ² + S_k(χ)²(dθ² + sin²θ dφ²)]. The angular coordinates θ and φ are in radians. A first reading can use the distance and circle rules below without manipulating this notation.

For k = ±1, aR_ref is the curvature-radius magnitude of the spatial slice in this convention. For k = 0, R_ref is a fiducial coordinate length, not a finite curvature radius. The three local forms describe curvature choices; they do not by themselves list every possible global topology.

Compare distances measured within a slice

At one cosmic time, choose a circle in a two-dimensional geodesic section of the three-dimensional spatial slice. Its radial proper distance from the centre is D = aR_refχ, and its circumference is C = 2πaR_ref S_k(χ). These are intrinsic measurements within the specified geometry.

Compare that circumference with the Euclidean reference 2πD for the same radial proper distance. Their ratio is C/(2πD) = S_k(χ)/χ for χ > 0. The common length factor cancels, leaving a dimensionless comparison.

For the positive χ values in the table, positive curvature gives a smaller circumference than 2πD, zero curvature gives equality and negative curvature gives a larger circumference. This comparison concerns specified metric distances, not a circle drawn with a ruler on an arbitrary projection.

At χ = 1, the three ratios are about 0.84147, 1 and 1.17520. They are exact-model predictions rounded for teaching, not fitted measurements of the curvature of our Universe. The chosen interval 0 < χ ≤ 1 also avoids issues involving large circles or global identifications.

Intrinsic circumference relative to 2π times radial proper distance, at one cosmic time in the supplied local constant-curvature models
χ = D/(aR_ref)	Positive curvature: sin χ/χ	Zero curvature	Negative curvature: sinh χ/χ
0.25	0.98962	1	1.01045
0.5	0.95885	1	1.04219
1	0.84147	1	1.17520

Expansion changes lengths and curvature magnitude

Hold a comoving χ fixed and compare two slices whose scale factors differ by a factor of two. Both D and C double. Their ratio C/D remains the same because the same scale factor multiplies each length.

The instantaneous spatial sectional curvature is k/(aR_ref)². Doubling a therefore reduces the magnitude of a nonzero spatial curvature by a factor of four while preserving its sign. The coefficient K = k/R_ref² used in the earlier expansion equation is fixed; the slice’s curvature includes the additional a⁻² factor.

For small positive χ, all three circumference ratios approach 1. Curved geometry can therefore look nearly Euclidean in a sufficiently small spatial region. At χ = 0 the displayed ratio formula would be 0/0; its limiting value is 1, not a direct substitution of zero into a defined ratio.

Changing coordinate labels on a given slice does not change its intrinsic curvature or measured circumference-to-radius relation. A different spacetime foliation is a different construction and should not be confused with relabeling the same spatial slice.

At one cosmic time, three supplied constant-spatial-curvature models share positive length factor aR_ref. The horizontal axis is dimensionless radial distance χ = D/(aR_ref) from 0 to 1. The vertical axis is dimensionless circumference C/(2πaR_ref) from 0 to 1.2. The solid blue circle-marked curve is sin χ for positive curvature, the dotted pale square-marked straight line is χ for zero curvature, and the dashed orange diamond-marked curve is sinh χ for negative curvature. At χ = 1, the plotted values, also the circumference-to-Euclidean ratios C/(2πD), are about 0.84147, 1 and 1.17520. At χ = 0 all circumferences are zero; the ratio C/(2πD) is undefined there and has limit 1 as χ approaches zero. These are intrinsic circle and radial-distance relationships in a two-dimensional geodesic section of a three-dimensional spatial slice, not an external embedding or a literal surface picture of the Universe. Doubling a multiplies C and D by two without changing their ratio. For zero curvature, R_ref is a fiducial coordinate length rather than a finite curvature radius. The curves are exact teaching models on the stated interval, not fitted cosmic measurements or global topology determinations. Original Ark teaching model.

Keep three geometric questions separate

Spatial curvature refers to the geometry of a chosen cosmic-time slice. Four-dimensional spacetime curvature also concerns how time and spatial geometry fit together. A spatially flat FLRW slicing does not require the full spacetime to be flat.

An external curved-surface picture can be a visualization aid, but intrinsic curvature does not require a physical embedding space outside the Universe. The metric relationships between lengths and angles define what is being compared.

Global topology asks how the spatial pieces connect over the whole space. It is additional information beyond the local curvature form, as the next lesson’s flat periodic example demonstrates. A label such as positive, flat or negative curvature does not alone establish every global property or future fate.

When reading a geometric claim, identify the time slice, coordinate and length convention, distance definition and global assumptions. These details prevent an equation for one geometry from being combined silently with a diagram or parameter using another convention.

Worked reasoning

Cancel the common length factor

At χ = 1, what fraction of the Euclidean circumference 2πD occurs in the positive-curvature model?

Use C = 2πaR_ref sin χ and D = aR_refχ at the same cosmic time.
Divide to obtain C/(2πD) = sin χ/χ.
At χ = 1, the ratio is sin 1 ≈ 0.84147, about 84.15 percent of the equal-radial-distance Euclidean reference.

About 0.84147 of the Euclidean reference circumference in this specified positive-curvature model.

Try the reasoning

If a doubles while χ stays fixed, does C/D double?
Show the answer and reasoning
No. Both C and D double, so their ratio is unchanged.
Both metric lengths carry the same factor aR_ref.
Does a spatially flat slice establish that the full four-dimensional spacetime is flat?
Show the answer and reasoning
No.
Spatial-slice curvature and spacetime curvature are different geometric quantities.
Does the intrinsic circumference comparison require a physical embedding space outside the Universe?
Show the answer and reasoning
No.
The supplied spatial metric defines the measured length relationships without an external embedding.

What to carry forward

An FLRW metric states spatial symmetry, cosmic time and scale conventions; dynamics require additional physical equations.
Intrinsic circumference and radial distance distinguish local spatial-curvature models.
Spatial curvature, spacetime curvature and global topology answer different questions.

References

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