A school triangle has angles adding up to 180°. Now imagine a triangle with three right angles. It can exist when its sides lie on a spherical surface. The addition is sound: the geometry in which we apply the rule has changed.
Space describes possible locations of objects and the spatial relations between them. Curvature means that relations between measured distances and angles differ from Euclidean geometry. In general relativity, the physical account of gravity involves the joint geometry of space and time. To get a feel for this, we will start with an easily imagined surface.
Coordinates give addresses; geometry gives distances
To locate something in a room, we can measure three numbers: its distance from one wall, another wall and the floor. This uses three spatial dimensions. The numbers depend on our choice of origin and directions. Answering what rulers between objects will actually measure also requires rules for distance.
On graph paper, those rules are familiar. Two points are connected by a straight segment; Pythagoras’ theorem works for a right triangle. Redrawing the coordinate grid with wavy lines does not change the lengths of real segments. An unusual map alone establishes no curvature.
In physics, geometry is tested through measurements. The mathematical idea of a space also applies to a two-dimensional surface, along which movement has two independent directions. Einstein Online’s entry on space distinguishes this broader use from familiar three-dimensional space.
How to make a triangle with three right angles
Imagine a perfectly round globe without mountains. Walk from its North Pole along a meridian to the equator. Follow the equator for a quarter of a full circle: 90° in longitude. Return to the pole along another meridian.
Both meridians meet the equator at right angles. Their angle at the pole is also 90°. The sum is therefore 270°. These angles are measured between directions on the surface, not between lines in a flat picture.
Our sides follow great circles, intersections of the sphere with planes through its centre. The shorter arcs between our vertices are the shortest routes on the surface. They are examples of geodesics, locally the straightest possible lines of a given geometry. A shortcut through the globe is excluded: we are studying its surface.
Three vertices, two geometries
In a plane
The sides are straight segments. Angles: A = 45°, B = 45°, C = 90°.
45° + 45° + 90° = 180°
On a spherical surface
A is the North Pole; B and C lie on the equator, 90° of longitude apart. Each of the three angles is 90°.
90° + 90° + 90° = 270°
The difference between 270° and 180° shows why the plane’s rule cannot simply be transferred to a sphere. On a small patch of a large sphere, the deviation is much smaller, which is why a small courtyard can be mapped almost as a flat area. Einstein Online’s definition of curvature connects such deviations with geometry.
Branch · limits of the analogyOptional · safe to skip
Why is rolled paper a special case?
Roll a sheet into an open cylindrical tube without stretching or crumpling it. It looks bent from outside. Yet a small drawn triangle, measured along the paper, keeps its lengths and angles. The sheet can be unrolled without stretching.
Its intrinsic geometry is locally flat: its Gaussian curvature is zero. A sphere has different intrinsic geometry; a flat map cannot cover it everywhere without distortion. Examples 2.13–2.14 in Cornell’s surface geometry material compare the plane, sphere and cylinder.
“Locally” matters: a tube also closes around one direction, so its global properties differ from those of an infinite sheet. This example separates intrinsic curvature from an object’s external appearance.
Curvature can be found without looking from outside
An imaginary inhabitant of a spherical surface could measure triangle sides and angles without seeing the globe from outside. Comparing the measurements with plane geometry would suffice. Thus “curved” need not mean “bent into some external space”.
We used a third dimension to picture the sphere, but its intrinsic geometry can be described mathematically without it. AEI’s discussion of geometry and geodetic precession explains this distinction. Likewise, defining curvature in three-dimensional space requires no assumption of an external room with another spatial dimension.
The limit of the analogy matters: the globe’s surface has two dimensions; familiar space has three. The figure makes no claim that the whole Universe is a sphere’s surface. It illustrates a way of thinking: geometry is defined through relations that can be checked from within.
Branch · calculationOptional · safe to skip
What did we calculate for the sphere?
Call the difference between the sum of three angles and 180° the spherical excess. For our triangle:
E = (270)^(∘) − (180)^(∘); = (90)^(∘).
In radians, E = π/2. Eight identical triangles cover the sphere. For radius R, the area of one is:
A = (4π(R)^(2))/(8) = (π(R)^(2))/(2).
Then E/A = 1/(R)^(2), the Gaussian curvature of an ideal sphere. Its unit is inverse length squared, such as m⁻². Doubling R reduces it by a factor of four. This large triangle keeps the same angles while its area grows fourfold.
We are calculating a property of a specified mathematical sphere. This is no measurement of the Universe’s curvature and no universal instruction to divide an angle by an area for arbitrary spacetime geometry.
Gravity’s geometry also includes time
Return to physics. Describing a fall or a flight requires saying where an object is and when. In general relativity, such events form spacetime. Its geometry specifies how distances, time intervals and light paths relate.
Mass, energy, momentum and pressure are linked to this geometry. A small freely moving body, with no thrust, drag or other nongravitational force, follows a spacetime geodesic. This is a more precise idea than a ball pulled down a rubber funnel by Earth’s gravity. Space is no rubber substance; describing gravity requires no additional external “down”. AEI’s elementary account of gravity’s geometry connects motion with spacetime.
A physical manifestation of curvature is a tidal effect: nearby freely falling bodies can approach or separate from one another. A small freely falling lift removes the sensation of weight, but cannot eliminate such differences throughout an extended region. AEI’s explanation of weightlessness and curvature develops this point.
Spatial curvature and spacetime curvature are different concepts. A picture of a “spatial pit” cannot account for every gravitational effect. Nor does vacuum guarantee flatness: spacetime outside an idealised spherical, nonrotating star can be curved even with no matter at the selected location. A technical example is the exterior Schwarzschild solution in David Tong’s lectures, §6.1.
A geometrical account can be tested with observations
Light also travels according to spacetime geometry. Near a massive body, its path can bend so that a distant galaxy appears as an arc, several images or a ring. This is gravitational lensing. On 10 February 2025, ESA published Euclid observations of a ring around NGC 6505. The ring’s light came from a more distant galaxy; the nearer galaxy’s gravity altered its path to the telescope.
This is a testable effect of gravity on light. The ring alone determines neither the geometry of the entire Universe nor whether space literally resembles the sphere in our figure.
Our opening question now has a concrete answer: geometry can differ from flat geometry, and measurements made within it can reveal this. Gravity requires a joint description of space and time. Try comparing two accounts of the Moon’s free fall around Earth: force in the Newtonian model and geodesic motion in the relativistic one. They use different models to describe the same physical motion.