Cosmology · Spacetime, curvature and cosmological models

Why cosmology uses a relativistic description

Use events, light cones and a specified travelling clock to see why cosmic distances, times and gravitational histories require an explicit spacetime geometry.

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

Classify three supplied event separations using an explicit flat-spacetime convention.
Calculate proper time for a specified straight inertial path without assigning it to every path.
Explain why a local inertial description does not supply a single global description of an expanding cosmological model.

A measurement connects events and observers

A telescope records light at a particular place and time. The light was emitted at another event. Relating those two events requires more than a spatial ruler: one must also specify clocks, observers and the way signals travel between them.

An event is a location in space together with a time. Coordinates label events, while a spacetime metric specifies intervals and the local rules for comparing lengths and times. Different coordinates can describe the same physical events without producing different measured results.

Special relativity supplies the flat-spacetime description. General relativity extends the framework to dynamical spacetime geometry and relates that geometry to energy and momentum. A cosmological model uses its assumptions about gravity and contents to connect a large-scale geometry with possible histories.

Newtonian approximations remain useful in suitable regimes. The need for a relativistic cosmic description does not make every elementary calculation require tensor calculus. It does mean that a familiar distance or a universal-looking time should not be extended to the whole Universe without checking its definition and domain.

Choose units and a sign convention

For a concrete comparison, use one space dimension of exact Minkowski spacetime, with the other two spatial coordinates zero. Work in one specified inertial frame. Let L be one light-second, defined as c times one SI second, so L = 299,792,458 m exactly.

Plot x/L horizontally and ct/L vertically. Both axis coordinates are dimensionless and use the same numerical scale. The origin event A has coordinates (ct/L, x/L) = (0, 0). A one-second frame-time interval corresponds to a vertical change of one in ct/L.

Choose the minus-plus sign convention. The squared interval from A is s²/L² = −(ct/L)² + (x/L)². In this convention a negative value is timelike, zero is null and a positive value is spacelike. The sign convention must be stated; another overall sign convention reverses the signs while preserving the physical categories.

This finite endpoint formula applies to the supplied flat-spacetime example. It is not a formula for arbitrary distant endpoints in a curved expanding universe. In that setting the metric and the relevant path or distance construction must be specified.

A light cone separates causal possibilities

Take B = (5, 3), C = (3, 3) and D = (2, 3), listing ct/L first. For B, the squared interval is −25 + 9 = −16, so the separation is timelike. A massive travelling clock can connect A to B along a suitable future-directed path.

For C, the interval is −9 + 9 = 0. A light signal emitted at A in the positive x direction reaches C along the future light cone. With equal axis scales, the light-cone boundaries are the lines ct/L = |x/L|.

For D, the interval is −4 + 9 = 5. The event exists as a specified point in this diagram, but a causal signal starting at A cannot reach D under this model. A spacelike point is not itself a proposal for an object travelling faster than light.

These categories concern pairs of events. They differ from a same-time recession-rate relation between widely separated comoving positions in a cosmological coordinate description. A value of HD for distant positions must not be inserted into this local inertial picture as an ordinary object speed without its geometric interpretation.

Chosen events relative to A in one-dimensional Minkowski spacetime, using the minus-plus interval convention
Event	ct/L	x/L	s²/L²	Separation from A
B	5	3	−16	Timelike
C	3	3	0	Null
D	2	3	5	Spacelike

A travelling clock measures its own proper time

Now specify the straight inertial path from A to B. Its coordinate speed is Δx/Δt = 3c/5, below c. The proper time recorded by a clock following that path is √(5² − 3²) seconds = 4 s, while the chosen inertial frame assigns a coordinate-time interval of 5 s.

The four-second result belongs to this specified straight timelike path. Proper time in general depends on the path, so assigning the same result to every possible accelerated route between A and B would be incorrect. An endpoint subtraction alone does not replace a clock-path calculation.

No massive travelling clock is assigned to the A-to-C null path or the A-to-D spacelike separation. In particular, a photon does not have a physical inertial rest frame whose clock can be treated like the travelling clock used for B.

An optional coordinate check boosts to the inertial frame moving at 3c/5. With β = 3/5 and γ = 5/4, use ct′/L = γ(ct/L − βx/L) and x′/L = γ(x/L − βct/L). B becomes (4, 0), so the same straight clock is at rest in that frame and its elapsed time is again 4 s. The coordinates change while the interval remains −16L².

An exact one-dimensional Minkowski teaching model uses one declared inertial frame, signature (−,+), and L = c × 1 second. The horizontal coordinate x/L spans −4 to 4 and the vertical coordinate ct/L spans 0 to 6; both use exactly 60 pixels per unit. The origin event is A = (ct/L, x/L) = (0, 0). Dashed orange future null rays satisfy ct/L = |x/L| and have 45-degree slopes in these equally scaled coordinates. A solid blue straight timelike path reaches circle B = (5, 3). Diamond C = (3, 3) lies on the right null ray. Square D = (2, 3) is outside the future light cone. Their squared intervals from A are −16, 0 and +5 in units L². The particular straight inertial A-to-B path has proper time 4 seconds, whereas B has coordinate time 5 seconds in this frame. The clock value depends on the specified timelike path. C and D are not assigned massive-clock proper times; D is an event, not a faster-than-light trajectory. No causal signal from A reaches D. This is an exact flat toy, not the entire expanding Universe; the endpoint flat formula is not asserted unchanged across arbitrary curved spacetime. Original Ark teaching model.

Local simplicity does not remove global geometry

In general relativity, a freely falling local inertial description can simplify the metric near an event. It does not remove tidal curvature across an arbitrary finite region or guarantee that one inertial coordinate chart describes the entire cosmological spacetime.

Cosmological geometry therefore needs a stated model for its spatial slices, time coordinate and evolving scale. Light paths and distant clock comparisons are then calculated within that geometry. The homogeneous and isotropic background considered next is one important approximation with explicit conditions.

An increasing coordinate distance alone does not prove spacetime curvature. Coordinates can also describe expansion-like motions in flat spacetime. Physical curvature and causal or clock predictions require the metric and invariant comparisons, rather than a coordinate label by itself.

For a first reading, retain the concrete example: events B and C lie in different causal categories, and a specified clock records 4 s rather than the frame interval of 5 s. The mathematical return path supplies the general interval, metric and proper-time machinery when a problem requires it.

Worked reasoning

Classify an event, then specify the clock

For B = (ct/L, x/L) = (5, 3), what is the interval category and the time recorded along the straight inertial path from A?

Calculate s²/L² = −5² + 3² = −16, which is timelike in the stated convention.
Specify the straight inertial A-to-B path; its speed is 3c/5.
The clock records √(25 − 9) s = 4 s. This does not assign the same proper time to every other path.

B is timelike from A; the specified straight travelling clock records 4 s.

Try the reasoning

Which supplied event lies on the future light cone of A?
Show the answer and reasoning
C = (3, 3).
Its squared interval is −9 + 9 = 0, and its time coordinate is positive.
Must a travelling clock between A and B record the chosen frame-time interval of 5 s?
Show the answer and reasoning
No.
The straight clock in this example records 4 s. Coordinate time and proper time have different definitions.
May the finite Minkowski endpoint formula be used unchanged between arbitrary distant events in a curved cosmological spacetime?
Show the answer and reasoning
No.
The curved-spacetime metric and the specified path or distance construction are required.

What to carry forward

Relativistic descriptions connect events, observers, signal paths and clocks through a metric.
The interval category is invariant even when inertial coordinates change.
A local inertial frame does not eliminate finite-region tidal geometry or establish one global cosmic inertial frame.

References

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