Most recent edit on 2008-09-21 02:37:28 by CharlesFrancis
Additions:
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.
Deletions:
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 spacetime curvaturePound-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.
Edited on 2008-09-21 02:34:41 by CharlesFrancis
Additions:
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 spacetime curvaturePound-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.
Deletions:
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 "<a href="http://www.teleconnection.info/rqg/BasicsOfCurvature#EquivalencePrinciple">spacetime» curvature</a>""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.
Edited on 2008-09-21 02:32:31 by CharlesFrancis
Additions:
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 "<a href="http://www.teleconnection.info/rqg/BasicsOfCurvature#EquivalencePrinciple">spacetime» curvature</a>""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.
Edited on 2008-08-13 02:51:57 by CharlesFrancis
Additions:
← Basics of Curvature ↑ →
Deletions:
← Basics of Curvature →
Edited on 2008-05-01 01:23:32 by CharlesFrancis
Additions:
Definition: A tangent chart is a chart which is also a tangent space.
Edited on 2008-05-01 01:16:02 by CharlesFrancis
Additions:
<img class=left alt="livingroom5.GIF" title="Convex mirror" src="images/curvature/livingroom5.GIF"></td><td><img alt="curvature-5" title="Lensed Geometry" src="images/curvature/dollarmag2.jpg"></td></table>""
Deletions:
<img class=left alt="livingroom5.gif" title="Convex mirror" src="images/curvature/livingroom5.GIF"></td><td><img alt="curvature-5" title="Lensed Geometry" src="images/curvature/dollarmag2.jpg"></td></table>
<<span class=math><b>Definition:</b> ; A <i>tangent chart</i> is a chart which is also a tangent space.""
Edited on 2008-04-22 14:06:00 by ErikAnderson
Additions:
<img class=left alt="livingroom5.gif" title="Convex mirror" src="images/curvature/livingroom5.GIF"></td><td><img alt="curvature-5" title="Lensed Geometry" src="images/curvature/dollarmag2.jpg"></td></table>""
Deletions:
<img class=left alt="livingroom5.GIF" title="Convex mirror" src="images/curvature/livingroom5.GIF"></td><td><img alt="curvature-5" title="Lensed Geometry" src="images/curvature/dollarmag2.jpg"></td></table>""
Edited on 2008-04-22 12:10:18 by CharlesFrancis
Additions:
 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. |
  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. |
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:
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.
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.
 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. |
 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.
 |
Definition: The space described by a geometry is a manifold.
Definition: A chart is a map on flat space of a region of a manifold.
Definition: An atlas is a collection of charts covering a manifold.
 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. |
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.
 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. |
Deletions:
 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. |
  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. |
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:
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.
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.
 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. |
 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.
 |
Definition: The space described by a geometry is a manifold.
Definition: A chart is a map on flat space of a region of a manifold.
Definition: An atlas is a collection of charts covering a manifold.
 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. |
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.
 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. |
Edited on 2008-03-26 23:22:51 by CharlesFrancis
No differences.
Edited on 2008-03-26 23:21:37 by CharlesFrancis
Additions:
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».
Definition: An affine connection is a rule which defines parallel vectors, when the vectors are defined at points separated by a small displacement.
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.
Deletions:
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 a connection».
Definition: A connection is a rule which defines parallel vectors, when the vectors are defined at different points.
On the geometry of the Earth’s surface, and in general relativity, the 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. An
affine connection» is one defined between vectors at nearby points of the manifold.
Definition: An affine connection is a connection defined between vectors at points separated by an infinitesimal displacement.
Edited on 2008-03-24 00:34:25 by CharlesFrancis
Additions:
Singularities
Tangent Space
Deletions:
Singularities
Tangent Space
Edited on 2008-03-23 01:09:22 by CharlesFrancis
Additions:
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 a connection».
On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between vectors at nearby points of the manifold.
Deletions:
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 is given mathematically by a connection».
On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Edited on 2008-03-23 00:36:35 by CharlesFrancis
Additions:
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 is given mathematically by a connection».
On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Definition: An affine connection is a connection defined between vectors at points separated by an infinitesimal displacement.
Deletions:
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.
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a connection» enables us to compare a tangent chart at one point of a manifold with a tangent chart at another point. On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Definition: An affine connection is a connection defined between tangent chart at points separated by an infinitesimal displacement.
Edited on 2008-03-23 00:28:57 by CharlesFrancis
Additions:
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.
Definition: A connection is a rule which defines parallel vectors, when the vectors are defined at different points.
Deletions:
Dynamical properties like momentum are represented by vectors associated with moving objects. 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.
Definition: A connection is a rule which tells us what happens when a means of measurement moves from one position to another.
Edited on 2008-03-21 04:38:55 by CharlesFrancis
Additions:
← Basics of Curvature →
A gentle introduction to ideas encapsulated in the mathematics of Riemannian Geometry.
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 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:
| Negative curvature: |
|
k increasing with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decreasing as r increases. |
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:
| Negative curvature: |
|
k accelerating with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decelerating as r increases. |
Definition: A singularity is a point where the second derivative of k is not defined..
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»
|
Definition: The space described by a geometry is a manifold.
Definition: A chart is a map on flat space of a region of a manifold.
Definition: 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. |
Definition: A tangent chart is a chart which is also a tangent space.
Parallel Displacement
Dynamical properties like momentum are represented by vectors associated with moving objects. 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.
Definition: A connection is a rule which tells us what happens when a means of measurement moves from one position to another.
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a connection» enables us to compare a tangent chart at one point of a manifold with a tangent chart at another point. On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Definition: An affine connection is a connection defined between tangent chart at points separated by an infinitesimal displacement.
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.
Basics of Curvature ↑ The Equivalence Principle →
Deletions:
← Concepts of General Relativity →
In general relativity, Einstein put his physical ideas on the nature of time and space, into the mathematical language of
tensors and
Riemannian geometry.
The Spacetime Manifold
According to the general principle, an observer anywhere can use the radar method to define locally Minkowski coordinates, but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.
Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called manifolds». Spacetime is described as a Lorentzian manifold». By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the Pound-Rebka experiment. In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys special relativity. Together with Einstein’s field equation, this determines a curved geometry which precisely accounts for Newton’s law of gravity.
Differentiability
A chart of spacetime is not a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, tables of data and formulae containing a numerical description. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.
In general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. It may be said that, in the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation.
General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.
Tangent Charts
In principle many forms of coordinates can be used in mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, general covariance automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in special relativity, using the radar method. In this case the chart is made on Minkowski spacetime, which has constant Minkowski metric, h. h is a non-physical metric, analogous to the metric of the paper on which a map is drawn. h does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a tangent chart, an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.
Definition: A tangent chart is defined by the radar method and has a constant, non-physical, Minkowski metric, equal to to the physical metric at an observer’s origin of coordinates.
Coordinate Time and Proper Time
In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point,
A, the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at
B, the origin of Beth’s coordinates. Alf and Beth both determine locally Minkowski coordinates. Alf’s time axis is denoted the 0-axis, and his space axes are labelled
1,
2,
3. Beth’s coordinates are denoted with primes,
0',
1',
2',
3'.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coodinates, denote the vector describing
t' seconds for a stationary object at the origin of Beth’s coordinates
τ = τi' = (t', 0, 0, 0).
t' is the actual amount of time measured by Beth using her own clock, and is known as
proper time. In Alf’s coordinates,
is found by
coordinate transformation. If Beth is stationary in Alf’s coordinates,
τ = τi' = (kt', 0, 0, 0), where
k is the
gravitational redshift factor.
Definition: Proper time is the amount of time for an object, as it would be measured on a local clock moving with that object.
Definition: Proper length is the length an object as it would be measured by an observer moving with that object.
Stationary Observers
 Alf and Beth are stationary with respect to each other and Beth is at a coordinate distance r from Alf as measured in Alf’s coordinates. Alf and Beth each measure the coordinate length of a short rod located at Beth’s origin, and aligned on an axis with Alf and Beth. Using radar, Alf determines that the rod has a coordinate length d, while Beth determines a length d' in her coordinates. Because the rod is at Beth’s origin, d' is the true length, or proper length, of the rod. |
 When Alf and Beth try to align their spacetime diagrams, maintaining synchronisation (lines of equal time are horizontal for both) and the constancy of the speed of light (light at 45°), they find a mismatch, because their clocks do not run at the same rate. In the diagrams, Beth’s clock runs faster than Alf’s by a factor k > 1; the arguments also hold when k < 1 and Beth’s clock is slower than Alf’s. Synchronisation requires that, for k > 1, Beth’s coordinates are enlarged by the factor k, so that if Beth’s coordinates are superimposed on Alf’s, and aligned at Beth, Alf appears at a coordinate distance r' = kr, displaced from his position in his own coordinates. |
 The diagrams for measurement of the coordinate length of the rod are superimposed. The proper time interval 2d' in Beth’s coordinates corresponds to a time 2kd' in Alf’s coordinates. The proper length d' in Beth’s coordinates appears with coordinate distance d = d'⁄k in Alf’s coordinates. | |
 Beth now turns the rod perpendicular to the axis from Alf. Beth calculates that the angle subtended in tangent space by the rod at Alf is θ' = d'/r'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod,
is equal to its proper length, measured locally by Beth.
|
The Spacetime Metric
The
metric,
gij, lowers the indices of
contravariant vectors in such a way that the inner product between vectors
x and
y is an
invariant,
In particular, the magnitude
|x| of the vector
x is invariant,
so that the metric is a means of determining magnitude of a vector in any coordinate system.
The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. General relativity allows that, in principle, Beth could be anywhere in Alf’s coordinate space. For each point,
x, where Beth could be, there is a different metric,
gij(x).
Definition: The metric field is defined, on a given coordinate system, by
The metric field is often simply called the metric. Care should be taken. Failure to distinguish the metric at a given coordinate from the metric field as a function of coordinate space can lead to confusion.
We have just seen that for quantities local to Beth, measured, or coordinate, time 2kd' in Alf’s coordinates corresponds to a proper time 2d', that coordinate distance d' ⁄ k corresponds to a proper distance d', and that angular distances are unchanged. Thus, a simple form for g can be given in spherical coordinates,
The Schwarzschild» solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
These forms of the metric, g, are coordinate dependent. The metric is a tensor quantity, and will be used in tensor equations. Tensor equations are covariant, so that if a tensor equation holds in one set of coordinates, then it will hold in any coordinates. In particular, a tensor equation will hold after a Lorentz transform of either Alf’s or Beth’s coordinates, so the choice of stationary observers places no limitation on generality.
The Affine Connection
The metric field is a measure of the distortion present in a chart. It is not sufficient to describe curvature — we have seen examples of distorted spaces, like the lensed and mirrored geometries, which are actually flat. To describe curvature requires a connection in addition to the metric field. Given the metric field, a connection describes the relationship between one set of coordinate axis, at x, say, and another set, at y.
In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between nearby points using parallel displacement in tangent space, and projecting back into the curved surface. This is in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through parallel transport, and is path dependent. A connection defined only through nearby points is called an affine connection. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.
Einstein was not satisfied that the affine connection used in general relativity is correct. He appears to have had two principle reasons for wanting to replace it. The first concerns his philosophical approach. The hole argument» gave him considerable trouble in his original formulation of general relativity. In general relativity, the metric is defined at all points in space. Empirically, spacetime coordinates can only be defined in the presence of matter. It is physically meaningless to define a metric when there is nothing to be measured, as in the vacuum of space. His later attempts at unified field theory were rooted in percieved inconsistencies in the treatment of electromagnetism. Such an inconsistency can be seen in the mismatch of Alf’s and Beth’s coordinates. Light emitted from the end of the short rod and seen by Beth does not behave in the same way as light emitted from the same point, in the same direction, and seen by Alf — if it did, Alf’s coordinate length would be altered by the same factor as his coordinate time rather than by the inverse factor, and space would have to be flat. Investigation of the inconsistency lead Einstein to the discovery of a different type of connection, a remote connection which he called distant parallelism, and which is also known as teleparallelism. The possibility of such a connection had previously been pointed out by Cartan», known as the best geometer in Europe, and who was the first to recognise the distinct roles of the connection and the metric in the description of curvature.
Interest in teleparallelism has been low; it is sometimes suggested in popular literature that most physicists believe that Einstein had lost the plot in these attempts at unified field theory. In my view, any physicists who believe such a thing have very little understanding of either general relativity or of the depth of Einstein’s thought and insight. Einstein was not successful in his attempts at unifying general relativity and classical electromagnetism using distant parallelism. This does not indicate that he was working on the wrong lines, but that he was ahead of his time. The teleconnection in relational quantum gravity uses teleparallelism in a more modern context, to unify general relativity with quantum electrodynamics. At the time when Einstein was working on unified field theory, quantum mechanics was in its infancy and had been given neither axiomatic formulation nor reasonable interpretation; the description of the photon in quantum electrodynamics did not exist until some years later.
Concepts of General Relativity ↑ Riemann Curvature →
Edited on 2008-03-21 04:38:11 by CharlesFrancis
Additions:
← Concepts of General Relativity →
In general relativity, Einstein put his physical ideas on the nature of time and space, into the mathematical language of
tensors and
Riemannian geometry.
The Spacetime Manifold
According to the general principle, an observer anywhere can use the radar method to define locally Minkowski coordinates, but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.
Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called manifolds». Spacetime is described as a Lorentzian manifold». By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the Pound-Rebka experiment. In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys special relativity. Together with Einstein’s field equation, this determines a curved geometry which precisely accounts for Newton’s law of gravity.
Differentiability
A chart of spacetime is not a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, tables of data and formulae containing a numerical description. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.
In general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. It may be said that, in the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation.
General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.
Tangent Charts
In principle many forms of coordinates can be used in mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, general covariance automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in special relativity, using the radar method. In this case the chart is made on Minkowski spacetime, which has constant Minkowski metric, h. h is a non-physical metric, analogous to the metric of the paper on which a map is drawn. h does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a tangent chart, an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.
Definition: A tangent chart is defined by the radar method and has a constant, non-physical, Minkowski metric, equal to to the physical metric at an observer’s origin of coordinates.
Coordinate Time and Proper Time
In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point,
A, the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at
B, the origin of Beth’s coordinates. Alf and Beth both determine locally Minkowski coordinates. Alf’s time axis is denoted the 0-axis, and his space axes are labelled
1,
2,
3. Beth’s coordinates are denoted with primes,
0',
1',
2',
3'.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coodinates, denote the vector describing
t' seconds for a stationary object at the origin of Beth’s coordinates
τ = τi' = (t', 0, 0, 0).
t' is the actual amount of time measured by Beth using her own clock, and is known as
proper time. In Alf’s coordinates,
is found by
coordinate transformation. If Beth is stationary in Alf’s coordinates,
τ = τi' = (kt', 0, 0, 0), where
k is the
gravitational redshift factor.
Definition: Proper time is the amount of time for an object, as it would be measured on a local clock moving with that object.
Definition: Proper length is the length an object as it would be measured by an observer moving with that object.
Stationary Observers
| Alf and Beth are stationary with respect to each other and Beth is at a coordinate distance r from Alf as measured in Alf’s coordinates. Alf and Beth each measure the coordinate length of a short rod located at Beth’s origin, and aligned on an axis with Alf and Beth. Using radar, Alf determines that the rod has a coordinate length d, while Beth determines a length d' in her coordinates. Because the rod is at Beth’s origin, d' is the true length, or proper length, of the rod. |
| When Alf and Beth try to align their spacetime diagrams, maintaining synchronisation (lines of equal time are horizontal for both) and the constancy of the speed of light (light at 45°), they find a mismatch, because their clocks do not run at the same rate. In the diagrams, Beth’s clock runs faster than Alf’s by a factor k > 1; the arguments also hold when k < 1 and Beth’s clock is slower than Alf’s. Synchronisation requires that, for k > 1, Beth’s coordinates are enlarged by the factor k, so that if Beth’s coordinates are superimposed on Alf’s, and aligned at Beth, Alf appears at a coordinate distance r' = kr, displaced from his position in his own coordinates. |
| The diagrams for measurement of the coordinate length of the rod are superimposed. The proper time interval 2d' in Beth’s coordinates corresponds to a time 2kd' in Alf’s coordinates. The proper length d' in Beth’s coordinates appears with coordinate distance d = d'⁄k in Alf’s coordinates. | |
| Beth now turns the rod perpendicular to the axis from Alf. Beth calculates that the angle subtended in tangent space by the rod at Alf is θ' = d'/r'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod,
is equal to its proper length, measured locally by Beth.
|
The Spacetime Metric
The
metric,
gij, lowers the indices of
contravariant vectors in such a way that the inner product between vectors
x and
y is an
invariant,
In particular, the magnitude
|x| of the vector
x is invariant,
so that the metric is a means of determining magnitude of a vector in any coordinate system.
The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. General relativity allows that, in principle, Beth could be anywhere in Alf’s coordinate space. For each point,
x, where Beth could be, there is a different metric,
gij(x).
Definition: The metric field is defined, on a given coordinate system, by
The metric field is often simply called the metric. Care should be taken. Failure to distinguish the metric at a given coordinate from the metric field as a function of coordinate space can lead to confusion.
We have just seen that for quantities local to Beth, measured, or coordinate, time 2kd' in Alf’s coordinates corresponds to a proper time 2d', that coordinate distance d' ⁄ k corresponds to a proper distance d', and that angular distances are unchanged. Thus, a simple form for g can be given in spherical coordinates,
The Schwarzschild» solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
These forms of the metric, g, are coordinate dependent. The metric is a tensor quantity, and will be used in tensor equations. Tensor equations are covariant, so that if a tensor equation holds in one set of coordinates, then it will hold in any coordinates. In particular, a tensor equation will hold after a Lorentz transform of either Alf’s or Beth’s coordinates, so the choice of stationary observers places no limitation on generality.
The Affine Connection
The metric field is a measure of the distortion present in a chart. It is not sufficient to describe curvature — we have seen examples of distorted spaces, like the lensed and mirrored geometries, which are actually flat. To describe curvature requires a connection in addition to the metric field. Given the metric field, a connection describes the relationship between one set of coordinate axis, at x, say, and another set, at y.
In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between nearby points using parallel displacement in tangent space, and projecting back into the curved surface. This is in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through parallel transport, and is path dependent. A connection defined only through nearby points is called an affine connection. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.
Einstein was not satisfied that the affine connection used in general relativity is correct. He appears to have had two principle reasons for wanting to replace it. The first concerns his philosophical approach. The hole argument» gave him considerable trouble in his original formulation of general relativity. In general relativity, the metric is defined at all points in space. Empirically, spacetime coordinates can only be defined in the presence of matter. It is physically meaningless to define a metric when there is nothing to be measured, as in the vacuum of space. His later attempts at unified field theory were rooted in percieved inconsistencies in the treatment of electromagnetism. Such an inconsistency can be seen in the mismatch of Alf’s and Beth’s coordinates. Light emitted from the end of the short rod and seen by Beth does not behave in the same way as light emitted from the same point, in the same direction, and seen by Alf — if it did, Alf’s coordinate length would be altered by the same factor as his coordinate time rather than by the inverse factor, and space would have to be flat. Investigation of the inconsistency lead Einstein to the discovery of a different type of connection, a remote connection which he called distant parallelism, and which is also known as teleparallelism. The possibility of such a connection had previously been pointed out by Cartan», known as the best geometer in Europe, and who was the first to recognise the distinct roles of the connection and the metric in the description of curvature.
Interest in teleparallelism has been low; it is sometimes suggested in popular literature that most physicists believe that Einstein had lost the plot in these attempts at unified field theory. In my view, any physicists who believe such a thing have very little understanding of either general relativity or of the depth of Einstein’s thought and insight. Einstein was not successful in his attempts at unifying general relativity and classical electromagnetism using distant parallelism. This does not indicate that he was working on the wrong lines, but that he was ahead of his time. The teleconnection in relational quantum gravity uses teleparallelism in a more modern context, to unify general relativity with quantum electrodynamics. At the time when Einstein was working on unified field theory, quantum mechanics was in its infancy and had been given neither axiomatic formulation nor reasonable interpretation; the description of the photon in quantum electrodynamics did not exist until some years later.
Concepts of General Relativity ↑ Riemann Curvature →
Deletions:
← Basics of Curvature →
A gentle introduction to ideas encapsulated in the mathematics of Riemannian Geometry.
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 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:
| Negative curvature: |
|
k increasing with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decreasing as r increases. |
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:
| Negative curvature: |
|
k accelerating with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decelerating as r increases. |
Definition: A singularity is a point where the second derivative of k is not defined..
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»
|
Definition: The space described by a geometry is a manifold.
Definition: A chart is a map on flat space of a region of a manifold.
Definition: 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. |
Definition: A tangent chart is a chart which is also a tangent space.
Parallel Displacement
Dynamical properties like momentum are represented by vectors associated with moving objects. 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.
Definition: A connection is a rule which tells us what happens when a means of measurement moves from one position to another.
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a connection» enables us to compare a tangent chart at one point of a manifold with a tangent chart at another point. On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Definition: An affine connection is a connection defined between tangent chart at points separated by an infinitesimal displacement.
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.
Basics of Curvature ↑ The Equivalence Principle →
Edited on 2008-03-19 03:38:37 by CharlesFrancis
Additions:
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a connection» enables us to compare a tangent chart at one point of a manifold with a tangent chart at another point. On the geometry of the Earth’s surface, and in general relativity, the 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. An affine connection» is one defined between tangent charts at nearby points of the manifold.
Definition: An affine connection is a connection defined between tangent chart at points separated by an infinitesimal displacement.
Deletions:
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a connection» enables us to compare a coordinate system, or chart, with an origin at one point of a manifold with one at another origin. On the geometry of the Earth’s surface, and in general relativity, the connection is given by parallel displacement, in accordance with physical experience, that it is meaningful to move a ruler parallel to itself. An connection defined between charts with near origins is affine»
Definition: An affine connection is a connection defined between charts having origins separated by an infinitesimal displacement.
Edited on 2008-03-19 03:18:19 by CharlesFrancis
No differences.
Edited on 2008-03-19 03:17:12 by CharlesFrancis
Additions:
A <i>coordinate system</i>, or <a href=http://mathworld.wolfram.com/CoordinateChart.html>chart</a><sup>»</sup»> 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 <a href=http://en.wikipedia.org/wiki/Geographic_coordinate_system>geographic</a><sup>»</sup»> (longitude-latitude) coordinates. An <a href=http://en.wikipedia.org/wiki/Atlas>atlas</a><sup>»</sup»> is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an <a href=http://en.wikipedia.org/wiki/Atlas_%28topology%29>atlas</a><sup>»</sup»> is a collection of charts covering a <a href=http://en.wikipedia.org/wiki/Manifold>manifold</a><sup>»</sup»>
A <i>coordinate system</i>, or <a href=http://mathworld.wolfram.com/CoordinateChart.html>chart</a><sup>»</sup»> 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 <a href=http://en.wikipedia.org/wiki/Geographic_coordinate_system>geographic</a><sup>»</sup»> (longitude-latitude) coordinates. An <a href=http://en.wikipedia.org/wiki/Atlas>atlas</a><sup>»</sup»> is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an <a href=http://en.wikipedia.org/wiki/Atlas_%28topology%29>atlas</a><sup>»</sup»> is a collection of charts covering a <a href=http://en.wikipedia.org/wiki/Manifold>manifold</a><sup>»</sup»>
Deletions:
A <i>coordinate system</i>, or <a href=http://mathworld.wolfram.com/CoordinateChart.html>chart</a><sup>»</sup»> 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 <a href=http://en.wikipedia.org/wiki/Geographic_coordinate_system>geographic</a><sup>»</sup>(longitude-latitude») coordinates. An <a href=http://en.wikipedia.org/wiki/Atlas» atlas</a><sup>»</sup> is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an <a href=http://en.wikipedia.org/wiki/Atlas_%28topology%29» atlas</a><sup>»</sup> is a collection of charts covering a <a href=http://en.wikipedia.org/wiki/Manifold» manifold</a><sup>»</sup>
Oldest known version of this page was edited on 2008-03-19 03:15:04 by CharlesFrancis []
Page view:
← Basics of Curvature →
A gentle introduction to ideas encapsulated in the mathematics of Riemannian Geometry.
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 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:
| Negative curvature: |
|
k increasing with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decreasing as r increases. |
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:
| Negative curvature: |
|
k accelerating with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decelerating as r increases. |
Definition: A singularity is a point where the second derivative of k is not defined..
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 a>» is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an a>» is a collection of charts covering a a>»
|
Definition: The space described by a geometry is a manifold.
Definition: A chart is a map on flat space of a region of a manifold.
Definition: 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. |
Definition: A tangent chart is a chart which is also a tangent space.
Parallel Displacement
Dynamical properties like momentum are represented by vectors associated with moving objects. 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.
Definition: A connection is a rule which tells us what happens when a means of measurement moves from one position to another.
Measurement is a physical construct, and is used to determine a coordinate system. Mathematically, a
connection» enables us to compare a coordinate system, or chart, with an origin at one point of a manifold with one at another origin. On the geometry of the Earth’s surface, and in general relativity, the connection is given by parallel displacement, in accordance with physical experience, that it is meaningful to move a ruler parallel to itself. An connection defined between charts with near origins is
affine»
Definition: An affine connection is a connection defined between charts having origins separated by an infinitesimal displacement.
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.
Basics of Curvature ↑ The Equivalence Principle →
