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The Development of Non-Euclidean Geometry

Euclidean geometry incorporates the parallel postulate, that parallel lines can be extended indefinitely always the same distance apart, but this assumes that the measurements made in one place are relevant to measurements made elsewhere. For centuries people had tried to prove the parallel postulate but Gauss» was the first to realise that it might not actually be true. As part of his employment in cartography, mapping the surface of the Earth using measurements made on the surface of the Earth, Gauss developed mathematics to describe relationships between measurements within curved surfaces». This was the beginning of the study of non-Euclidean geometry» and the origin of the term curvature» as it is used to describe spacetime. We will need to recognise that this use of the word curvature is rather different from the common idea of a curved surface.

Gauss developed mathematics for two dimensional curved surfaces embedded in three-space, and he also recognised that it is not necessarily the case that 3-space is everywhere Euclidean. He even had surveyers under his direction take measurements between peaks in the Andes to establish whether the angles of a triangle add to 180° (they did).

Unknown to each other, János Bolyai» and Nikolai Lobachevski» each developed non-Euclidean geometries, but the major step was made by Riemann», who generalised two dimensional non-Euclidean geometry and found a mathematical structure for dealing with general geometries in any number of dimensions. Riemannian geometry is certainly one of the greatest achievements in the history of mathematics. For it, and his numerous other achievements, many of them also of a difficult nature, Riemann gets my vote as the greatest mathematician in history, ahead of both Gauss and Newton».

Riemann considered himself physicist, more than mathematician — not many agreed, perhaps largely because so little of what he did was understood by others. He certainly hoped to gain insight into gravity from the study of non-Euclidean geometry. In this, of course, in the absence of relativity, he was not successful, but when Einstein found himself faced with the same problem, his friend Grossman» directed him to Riemann’s work. The main part of general relativity is the synthesis of Riemannian geomentry with special relativity.

Non-Euclidean geometry in n-dimensions was developed by generalising the geometry of the surface of the Earth to arbitrary curved surfaces, and to an arbitrary number of dimensions. This page introduces principle ideas in non-Euclidean geometry visually, with special reference to mapping the Earth’s surface. In Non-Euclidean Geometry, these ideas are applied more generally.

Intrinsic and Extrinsic Curvature

In many ways the term curvature is misleading when applied to spacetime. There are two types of curvature, known as intrinsic and extrinsic curvature. Extrinsic curvature is the familiar concept of curvature, visible curvature, or the shape of a two dimensional surface as it appears to us in three dimensional space. Intrinsic curvature refers to the geometrical properties of a space found from measurements taken within the space.

Intrinsic and extrinsic curvature are quite distinct ideas. A geometry can have intrinsic curvature but no extrinsic curvature, and one can have extrinsic curvature but no intrinsic curvature. The curvature of spacetime is intrinsic, because it refers to geometrical properties defined by measurement within spacetime. Extrinsic curvature has no meaning for spacetime, because there is no outside space to look at it from. To introduce the concept of intrinsic curvature we look at some simple examples.

Sphere:  When you construct two equal lines of longitude, on a sphere AD and BC, you find that CD is less than AB, so the parallel postulate does not hold. By the same token, the circumference of a circle is less than 2πr, where r is radial distance measured in the surface of the sphere. Cylinder:  A cylinder has extrinsic curvature, but no intrinsic, or Gaussian, curvature, because it can be made by rolling up a piece of paper without changing distance relationships between points. The circumference of a circle drawn on the paper is still 2πr when the paper is rolled into a cylinder.

Cone:  Like the cylinder you can make a cone by rolling flat paper. A circle enclosing the apex has circumference less than 2πr, a circle anywhere else has circumference equal to 2πr. The cone is intrinsically flat except at the apex, where the geometry has a singularity. World map:  A map of the world has intrinsic curvature, but no extrinsic curvature. It is drawn on flat paper, and does not look curved, but distances described by the map are geographical. It has exactly the same geometry as the world itself, and geometrically approximates a sphere.

Lenses and mirrors:  The image under a magnifying glass is curved according to any normal definition; it has extrinsic curvature. But the distance between two points in the image is measured by a ruler, and is the same as the original. The geometry is intrinsically flat. The same is true of an image in a curved mirror.

Curvature of Spacetime

There is no absolute distance scale. In practice coordinates are always defined with respect to reference matter within a neighbourhood of an observer. Distances are established locally by measuring matter relative to other matter. If all matter locally were to “shrink”, including ourselves and all the matter contained our measuring apparatus, we would have no means to detect the fact. Our fundamental length scales would “shrink” in direct proportion, and it would make no difference to the numerical results of local distance measurements. In the absence of such a difference, “shrinking” is meaningless. To talk about shrinking we need to make a comparison between the coordinates we define here and now, using here and now clocks and rulers, and the coordinates defined somewhere else.

The implication is that, if you hold up a stick, and a friend stands some distance away and measures its length by some triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if you bring the stick to him and he measures its length directly. Of course, in this instance, we could not detect a difference between the two measurements, but the effects of non-Euclidean geometry can be measured, for example in the Pound-Rebka experiment». In special relativity, distance measurements are reduced to measurements of time. The statement that spacetime is not flat is precisely equivalent to the statement that identical clocks at different positions do not necessarily measure time at the same rate.

The parallel postulate breaks down precisely because distance is defined by local comparison. It holds within the geometry of a piece of paper, because we can take a ruler and slide it across the paper, and thereby compare the distance between parallel lines at one place and the corresponding distance at another. On the scale of astronomical distances, no such sliding of rulers is possible — or if it is, we cannot say that the ruler itself does not “expand” or “shrink” in the process. Special relativity describes measurements of time and position, using physical measurement to set up a reference frame and coordinates within a neighbourhod of an observer. According to the general principle of relativity another observer at a distant point can carry out exactly the same kind of measurements, and set up coordinates in exactly the same way, but special relativity does not provide us with a way to compare the coordinates set up by one observer with those of a second, remote observer. The absence of direct comparison between distant coordinate systems creates the possibility that the parallel postulate does not hold and that spacetime is non-Euclidean.

Special relativity assumed that the redshift factor, k, depends only on the magnitude of relative velocity. Local laws of physics would not be affected if k also depends on relative position. Because of the dependency of distance measurements on time, this will result in “shrinking” or “expansion” of one observer’s coordinates compared to the other’s, separate from, and in addition to, special relativistic, velocity dependent, time dilation.

Positive and Negative Curvature

Intrinsic curvature is characterised by geometric properties. For example we can characterise curvature by comparison with the parallel postulate. If we measure the distance AB = d between any two points, and measure two equal distances AD = BC = h, perpendicular to AB, then we have no prior reason to assume that AB is equal to DC, as measured by an observer at D. So, DC = kAB = kd where k is a non-constant scale factor. This leads to a broad characterisation:

Negative curvature:		k increasing with h.
Zero curvature:		k constant as h increases.
Positive curvature:		k decreasing as h increases.

The parallel postulate is a not an ideal criterion on which to base a categorisation of geometry, because curvature is better understood at a point, not on a line. A more natural, characterisation of a geometry is found by considering the length of an arc, CD, of a circle of radius, r, subtended by an angle, θ, at the origin, O. Use a very small angle, θ, and drop perpendiculars of equal length from CD to a base line, AB, through the origin O. Then, in Euclidean geometry the length of CD is rθ, almost equal to AB. But in general the length of CD is krθ, and the value of k characterises the geometry according to the relationships:


Saddle:  k increases with r. So, curvature is negative.









Sphere:  k decreases as r increases. Curvature is positive.

Singularities

If in a small region the value of the circumference of a circle approaches 2πr, the geometry locally approximates a flat geometry. It has a flat tangent space (red). If there is no unique tangent space the geometry has a singularity, as at the apex. The practical implication is that, in general relativity, a singularity is a point where we do not know how to formulate the laws of physics.

Differentiation» is used to determine whether k is increasing or decreasing, but in a small region of the surface, the geometry approaches that of the tangent space, in which k = 1 is constant. k has a stationary point». At a stationary point, one cannot tell from the first derivative whether k is increasing or decreasing. It is necessary to look to the second derivative. Then the characterisation of curvature at a point is more accurately stated:

Tensor Curvature

The characterisation of curvature as positive or negative is simplistic in the general case. In general k may have dependencies on both position and direction. In two dimensions we need a quantity which can describe two vector directions to fully describe curvature. This will be a quantity with two vector indices, i.e. a tensor. In three dimensions, three indices are needed. The general description of curvature in spacetime requires a tensor with four vector indices. Curvature is most generally described by the Riemann curvature tensor.

Charts or Coordinate Systems

We can produce flat maps of regions of the Earth’s surface. By a map, we mean that we define a function from the points of the surface to the points of the map. The function must have sensible properties. The smaller the region being mapped, the less distortion we find in the map. A coordinate system, or chart» is a map such that the coordinates in the flat space are the same as the coordinates used to describe the geometry. For example, an equirectangular projection uses equal spacing between lines of longitude and latitude. It is a chart for geographic» (longitude-latitude) coordinates. An atlas» is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an atlas» is a collection of charts covering a manifold»

The Metric

Typically, charts are subject to scaling distortions; distance on the map is not proportional to the real distance between points. A metric is a function which undoes local scaling distortions and returns real distances as determined by measurement. This only works for distances short enough that the difference between a curved space and a flat space is not noticeable. In a two dimensional space, like the surface of the Earth, the metric is represented as a 2 × 2 matrix. In spacetime a 4 × 4 matrix will be needed. For reasons to do with Pythagoras’ theorem the components of the matrix are proportional to squared distances.

Tangent Space

The idea of a tangent space for two dimensional surfaces embedded in 3-space is clear. When we discuss the geometry of spacetime, we dispense with the idea of embedding in higher dimensonal space, but we retain the notion that a tangent space a flat space which meets the surface at a point, and which has geometrical properties identical to those of the surface within a small enough neighbourhood of the point. Thus a chart in which the scale is 1:1 at X is a tangent space at X. For example, an equirectangular projection, scaled such that cartographical distances on the projection are 1:1 with geographical distances at the equator, is a tangent space at any point on the equator.

Parallel Displacement

Dynamical properties like momentum are represented by vectors associated with moving objects. A vector is loosely described as an object with magnitude and direction, and may be represented as an arrowed line. In a curved surface, a vector must be defined at a particular position. Translating a vector on a curved surface does not generally make sense, but we can translate one in tangent space. In a small neighbourhood, the surface is indinstinguishable from the tangent space. So, for a small enough distance between X and Y, we can translate a vector in tangent space at X and project it to an indistinguishable vector at Y. This is parallel displacement.

In the lensed and mirrored spaces, scaling distortions do not imply that the space is curved. The metric is not suffient on its own to say whether a space is intrinsically curved. We also need to know what happens when a ruler (more generally a system of measurement) moves from one point to another. This requires a definition of parallel, and is given mathematically by an affine connection».


On the geometry of the Earth’s surface, and in general relativity, the affine connection is given by parallel displacement, in accordance with physical experience that it is meaningful to move a ruler parallel to itself over short distances.

Parallel Transport

Parallel transport means repeating parallel displacement for small distances along all path. Observe that the result of parallel transport depends on the path taken. The red vector at W, at the equator, points due North. Under parallel transport along the equator(green) to E, at a longitude 90° east of W, it continues to point due North. But if we parallel transport it to the North pole, N, then turn right and parallel transport it back to the equator, we arrive at E with the vector pointing due east.

Geodesic Motion

In a flat space, a straight line can be defined by translating a vector in the direction in which it is pointing. The same idea applies in curved space. A geodesic» is defined by parallel transport of a vector along its own axis. Since this is true in any number of dimensions and bodies always move in the direction of their velocity vector, straight line motion will be replaced by geodesic motion in curved spacetime.

It is easy to see that the shortest distance between two points is a geodesic, because it is indistinguishable from a straight line in any small neighbourhood, so that it is made up of shortest distances. The converse is not true. The long route round the equator from W to E is also a geodesic. There are an indefinite number of geodesics between two points at opposite ends of a diameter on a sphere.

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