Cosmology · Spacetime, curvature and cosmological models

Spatial curvature is not the whole topology

Compare an infinite flat plane with a locally flat periodic square to see how global identifications change distances and total size without adding a physical edge.

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

Distinguish local metric curvature from global spatial identifications.
Calculate a shortest route on a specified flat periodic square.
Explain why a local observation or finite observable region does not alone determine the whole spatial topology.

Curvature and topology supply different information

A local metric tells how nearby distances and angles are measured. Curvature describes intrinsic geometric properties of that metric. Topology concerns how the space connects globally, including possible identifications and loops that cannot be continuously shrunk to a point.

Two spaces can have the same local curvature while differing in total size or global routes. A locally flat metric therefore does not, by itself, prove that a space is infinite. Additional global assumptions are needed.

Use a two-dimensional mathematical analogy to make this distinction calculable. It is not a claim that the Universe is a surface or that a picture of a square supplies a complete three-dimensional cosmological model.

The comparison holds the local Euclidean distance rule fixed. It changes the global identifications, allowing the different conclusion to be traced to topology rather than to a change in local curvature.

Pair the coordinate seams

First consider an infinite Euclidean plane. Now consider a second space represented by a square of side 10L₀, where L₀ is a positive length unit. Identify opposite sides: (x, y) is equivalent to (x + 10L₀, y) and to (x, y + 10L₀).

The result is a flat square two-torus, meaning a torus with the quotient metric inherited from the Euclidean plane. Its local distance rule remains Euclidean. The paired edges are coordinate seams, not material walls or physical boundaries.

The familiar doughnut embedded in three-dimensional Euclidean space has the same topology but a different metric. It generally has varying intrinsic curvature. Its visible bending should not be imported into the locally flat quotient used here.

A path crossing a paired seam continues smoothly in the actual quotient space. A drawing splits its representation between two sides of the square, but no object makes a physical jump through an external space at the seam.

The shortest global route can change

Place P at (L₀, 5L₀) and Q at (9L₀, 5L₀). On the infinite plane their separation is 8L₀ along the horizontal direction. The straight route across the drawn square has that same length.

On the periodic quotient, Q is also represented by Q′ = (−L₀, 5L₀), since shifting x by 10L₀ gives the same point. The straight distance from P to this equivalent image is 2L₀.

In the square drawing, that short route travels one L₀ from P to the left seam and another L₀ from the paired right seam to Q. Its total is 2L₀. The drawing’s direct route of 8L₀ is still an allowed route, but it is not the shortest one on the quotient.

An optional general rule minimizes √[(Δx + 10mL₀)² + (Δy + 10nL₀)²] over integers m and n. Different image choices represent different global routes. Here the smallest horizontal difference has magnitude 2L₀ and the smallest vertical difference is zero, establishing the stated shortest length.

Two mathematical spaces with the same local Euclidean metric and the same displayed point coordinates
Space	Local curvature	Shortest P-to-Q distance	Total area
Infinite Euclidean plane	Zero	8L₀	Infinite
Flat square two-torus, period 10L₀ each way	Zero	2L₀	100L₀²

A small spatial region can look the same

Around the position (5L₀, 5L₀), choose a disk of radius 3L₀. This is smaller than half the shortest period, 5L₀, so its local geometry is an ordinary Euclidean disk in either space. Its area is 9πL₀², about 28.27L₀².

The periodic space has finite total area 100L₀², but that disk alone does not reveal the global identification. The plane and quotient can therefore agree on this local window while disagreeing on total area and long routes.

A three-dimensional analogue identifies opposite faces of a locally Euclidean cube. If each period is 10L₀, the fundamental volume is 1,000L₀³. This supplies a mathematical example of compact locally flat space; it is not an observational determination of our Universe’s topology.

Compact identifications can also reduce global rotational symmetries. The flat square quotient’s preferred shortest-loop directions should not be used to claim full global isotropy simply because its local curvature is constant.

A two-dimensional mathematical analogy compares the infinite Euclidean plane with a flat square quotient of period 10L₀ along both coordinates. The plot uses equal x/L₀ and y/L₀ units from 0 to 10. Matching purple upward arrows pair left and right coordinate seams; matching green rightward arrows pair top and bottom seams. These are identifications, not physical walls. Circle P is at (1, 5) and square Q at (9, 5). The dotted orange direct route is 8L₀ long, which is the shortest separation on the infinite plane. On the quotient it remains allowed but is longer than the solid blue route: one L₀ from P to the left seam and one L₀ from the paired right seam to Q, totalling 2L₀. The two drawn blue segments represent one smooth quotient-space path, not a physical jump through an external space. A dotted circle centred at (5, 5) has radius 3L₀, below the injectivity radius 5L₀, and the same Euclidean geometry in either space. The plane has infinite area, while the paired square has area 100L₀² and still has zero local curvature. The square is a coordinate representation, not the surface of the Universe or an embedded doughnut. The local disk is not a computed cosmological horizon and does not determine the global topology. Original Ark teaching model.

Finite visibility is a separate question

The radius-3L₀ disk is a chosen spatial window in the analogy, not a calculated cosmological horizon. A real observable region depends on light propagation and the assumed cosmic history, so its extent must be calculated separately.

A finite region accessible to observations does not by itself establish either finite or infinite total spatial extent. Nor does a projection boundary on a sky map supply a physical edge. These are distinct limitations and geometric constructions.

Detecting a proposed global identification would require observational consequences and a model of what signals could reach the observer. The small mathematical example makes no current observational limit or repeated-image detection claim.

When interpreting a curvature measurement, state the global topology assumptions separately. Local curvature information can be valuable without settling every possible long route, total volume or identification beyond the region probed.

Worked reasoning

Use an equivalent image to find the short route

With P = (1, 5)L₀ and Q = (9, 5)L₀ on the supplied period-10L₀ flat square, what is their shortest separation?

The drawn direct horizontal separation is 8L₀.
Replace Q by its equivalent image Q′ = (−1, 5)L₀. The Euclidean separation from P is then 2L₀.
Other image choices do not give a smaller horizontal magnitude or a negative vertical contribution, so the minimum is 2L₀. The paired seam is not a wall.

The shortest quotient-space distance is 2L₀, compared with 8L₀ on the infinite plane.

Try the reasoning

Does zero local spatial curvature force infinite total area or volume?
Show the answer and reasoning
No.
The flat periodic square is a compact mathematical counterexample in two dimensions, with an analogous periodic three-dimensional construction.
May the visible curvature of a familiar embedded doughnut be assigned to the flat quotient metric?
Show the answer and reasoning
No.
The topology can agree while the metrics and intrinsic curvature differ.
Does the specified Euclidean radius-3L₀ disk determine whether the whole space is the plane or the periodic quotient?
Show the answer and reasoning
No.
That local disk has the same metric geometry in both spaces; their global identifications differ.

What to carry forward

Local metric curvature does not determine the whole global topology.
Paired coordinate seams can produce compact flat space without material boundaries.
A local region, a sky-map boundary and a cosmological observable horizon are different constructions.

References

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