""

""

In a static geometry, an observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance, r*, from A is defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the spacetime metric has g_{00} = k^{2}. |

An observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance, r*, from A is defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the spacetime metric has g_{00} = k^{2}. |

""

""

An observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance, r*, from A is defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the spacetime metric has g_{00} = k^{2}. |

An observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance from A is the so called “tortoise” coordinate, r*, defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the spacetime metric has g_{00} = k^{2}. |

This result is usually found by solving [[rqgravity.net/Gravitation#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[rqgravity.net/Gravitation#TheSchwarzschildSolution Schwarzschild solution]]. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]], Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

This result is usually found by solving [[rqgravity.net/Gravitation#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[Gravitation#TheSchwarzschildSolution Schwarzschild solution]]. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

This result is usually found by solving [[rqgravity.net/Gravitation#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[Schwarzschild Schwarzschild solution]]. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

This result is usually found by solving [[rqgravity.net/Gravitation#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it simplifies (slightly) the solution of the the field equation. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

This result is usually found by solving [[rqgravity.net/GeneralRelativity#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it simplifies (slightly) the solution of the the field equation. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

This result is usually found by solving [[rqgravity.net/GeneralRelativity/#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it simplifies (slightly) the solution of the the field equation. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

""

""""Using *r* as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length *l* at B, perpendicular to AB is *l / r*, as it would be in flat space. This is the defining condition for *Schwarzschild coordinates*.""

This result is usually found by solving [[Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it simplifies (slightly) the solution of the the field equation. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf. |

This result is usually found by solving [[Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it simplifies (slightly) the solution of the the field equation. Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf. |

""Using

Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

""Using *r* as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length *l* at B, perpendicular to AB is *l / r*, as it would be in flat space. This is the defining condition for *Schwarzschild coordinates*.""

Using *r* as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length *l* at B, perpendicular to AB is *l / r*, as it would be in flat space. This is the defining condition for *Schwarzschild coordinates*.

<<""**Definition:** The *Schwarzschild radial coordinate* is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :"" <<

""

""

<<""**Theorem:** Schwarzschild coordinates in vacuum have spacetime metric given by:""

""where*k* and *k* are functions of positon, *x*, and *k*(*x*) is the redshift of light from the origin to *x*."" <<

<<""

""

In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2}. |

<<""

""where

Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates. In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2}. |

<<""

""where

""where *k* and *k* are functions of position, *x*, and *k*(*x*) is the redshift of light from the origin to *x*."" <<

Using [[http://en.wikipedia.org/wiki/Cartesian_coordinate_system Cartesian space coordinates]],

""

""

Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates. In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2}. |

Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates. In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2} |

""

""

""

.""

Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf. |

""

Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates. In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2} |

Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf. |

""

Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates. In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2} |

""where *k* and *k* are functions of positon, *x*, and *k*(*x*) is the redshift of light from the origin to *x*."" <<

""""

""where*k* and *k* are functions of positon, *x*, and *k>/i>(**x*) is the redshift of light from the origin to *x*.*"" << *

Using Cartesian space coordinates,

""where

Using Cartesian space coordinates,

""where

Using cartesian space coordinates,

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""orthogonal"". In this case the metric is diagonal (as seen in ""Schwarzschild coordinates"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a ""tensor field"", that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a ""metric field"", which is a function of coordinate space, with the ""metric"" at a given position.

""""

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""orthogonal"". In this case the metric is diagonal (as seen in ""Schwarzschild coordinates"", and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field 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, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""*x*"", say, and another set, at ""*x* +*dx*"", where ""*dx*"" is a small displacement, such that we can meaningfully describe a vector at ""*x*"" as being parallel to one at ""*x* +*dx*"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

""""

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""orthogonal"". In this case the metric is diagonal (as seen in ""Schwarzschild coordinates"", and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field 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, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""

""""

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, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""

Page was generated in 0.1260 seconds