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However, if you choose a master clock and from it you define a coordinate system over a region of space, as we do in practice, so that time is defined everywhere within that coordinate system by reference to time on the master clock, then it is perfectly possible that another, identical, clock, placed at a different position from the master clock, will tick at a different rate from the master clock.
In practice we observe that identical remote clocks do not keep time. We are by now all familiar with satnav systems, which provide location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more [[http://en.wikipedia.org/wiki/Global_Positioning_System GPS]] (Global Positioning System) satellites. The position of the satnav device is determined by differences in the time taken for signals to reach the device from the different satellites. For this to work the satellites must contain extremely accurate atomic clocks kept synchronised to the GPS master clock. It is found the satellite based clocks require a gravitational correction due to height of the clocks above the surface of the earth, as predicted by general relativity.
Deletions:
However, as we do in practice, you choose a master clock and from it you define a coordinate system over a region of space, so that time is defined everywhere within that coordinate system by reference to time on the master clock, then it is perfectly possible that another, identical, clock, placed at a different position from the master clock, will tick at a different rate from the master clock.
In practice we observe that remote clocks do not keep time. We are by now all familiar with satnav systems, which provide location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more [[http://en.wikipedia.org/wiki/Global_Positioning_System GPS]] (Global Positioning System) satellites. The position of the satnav device is determined by differences in the time taken for signals to reach the device from the different satellites. For this to work the satellites must contain extremely accurate atomic clocks kept synchronised to the GPS master clock. It is found that the correction to the satellite based clocks includes a part due to height of the clocks above the surface of the earth, as predicted by general relativity.
No differences.
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""The Fundamental Insight of General Relativity""
====""""The Fundamental Insight of General Relativity====
General relativity is based on a simple insight. If you, the room you are in, and everything in your immediate environment, were to increase in size, and clocks were to go slower such that the speed of light remains the same, there would be no way that you could tell the difference without looking outside of your environment.
In fact one needs to be a little careful about what one means by “increase in size” and “clocks go slower”. The metre is ultimately defined by the method of measuring it relative to matter, and the second is ultimately defined by the rate at which a clock ticks. A metre rule cannot grow longer relative to itself, and a clock cannot go slow relative to itself, so if you define time and duration relative to a local ruler and local clock, then the units of distance and time which they measure are fixed by definition, and they cannot grow larger or smaller.
However, as we do in practice, you choose a master clock and from it you define a coordinate system over a region of space, so that time is defined everywhere within that coordinate system by reference to time on the master clock, then it is perfectly possible that another, identical, clock, placed at a different position from the master clock, will tick at a different rate from the master clock.
By thinking about how we measure things, Einstein established the possibility that this could happen, and demonstrated in the theory of general relativity that gravity can be explained if it does happen. In fact he showed a little more, that the existence of gravity means that it is necessarily the case that identical clocks separated by a distance do not necessarily keep time with each other.
In practice we observe that remote clocks do not keep time. We are by now all familiar with satnav systems, which provide location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more [[http://en.wikipedia.org/wiki/Global_Positioning_System GPS]] (Global Positioning System) satellites. The position of the satnav device is determined by differences in the time taken for signals to reach the device from the different satellites. For this to work the satellites must contain extremely accurate atomic clocks kept synchronised to the GPS master clock. It is found that the correction to the satellite based clocks includes a part due to height of the clocks above the surface of the earth, as predicted by general relativity.
Distance is defined locally from the rate of clocks and the speed of light. It follows that if the rate of clocks varies from position to position, relative to a master clock, then the length of rulers must also vary from position to position. As a result, parallel lines, extended indefinitely, do not necessarily remain at the same distance apart. In other words one of the axioms of Euclidean geometry (the parallel postulate) does not hold in the real world.
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"" 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:
""| 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. |
""
Additions:
A coordinate system, or chart is a map such that the coordinates in the flat space are the same as the coordinates used to describe the geometry. For example, an equirectangular projection uses equal spacing between lines of longitude and latitude. It is a chart for geographic (longitude-latitude) coordinates. An atlas is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an atlas is a collection of charts covering a manifold
"" The idea of a tangent space for two dimensional surfaces embedded in 3-space is clear. When we discuss the geometry of spacetime, we dispense with the idea of embedding in higher dimensonal space, but we retain the notion that a tangent space a flat space which meets the surface at a point, and which has geometrical properties identical to those of the surface within a small enough neighbourhood of the point. Thus a chart in which the scale is 1:1 at X is a tangent space at X. For example, an equirectangular projection, scaled such that cartographical distances on the projection are 1:1 with geographical distances at the equator, is a tangent space at any point on the equator. |
""
""| Parallel 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:
A coordinate system, or chart is a map such that the coordinates in the flat space are the same as the coordinates used to describe the geometry. For example, an equirectangular projection uses equal spacing between lines of longitude and latitude. It is a chart for geographic (longitude-latitude) coordinates. An atlas< is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an atlas is a collection of charts covering a manifold
""| The idea of a tangent space for two dimensional surfaces embedded in 3-space is clear. When we discuss the geometry of spacetime, we dispense with the idea of embedding in higher dimensonal space, but we retain the notion that a tangent space a flat space which meets the surface at a point, and which has geometrical properties identical to those of the surface within a small enough neighbourhood of the point. Thus a chart in which the scale is 1:1 at X is a tangent space at X. For example, an equirectangular projection, scaled such that cartographical distances on the projection are 1:1 with geographical distances at the equator, is a tangent space at any point on the equator. |
""
""| Parallel 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. |
""
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""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.
Deletions:
""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 class="ext".
Additions:
""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 class="ext".
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}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.
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
Deletions:
""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.
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}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.
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»
Additions:
[[http://en.wikipedia.org/wiki/Non-Euclidean_geometry 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.
"" 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.
Deletions:
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 [[NonEuclideanGeometry Non-Euclidean Geometry]], these ideas are applied more generally.
""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.
Additions:
Additions:
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}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.
Deletions:
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}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.
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