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| [[File:Kruksal diagram.jpg|thumb|right|360px|Kruskal-Szekeres diagram. The quadrants are the black hole interior (II), the white hole interior (IV) and the two exterior regions (I and III). The solid 45° lines, which separate these four regions, are the [[event horizon]]s. The solid hyperbolas which bound the top and bottom of the diagram are the physical singularities. The dashed hyperbolas represent [[Contour line|contour]]s of the Schwarzschild radial coordinate, and the dashed lines through the origin represent contours of the Schwarzschild time coordinate. (Note that this diagram has used [[Einstein notation|a different convention]] for labeling the Kruskal-Szekeres coordinates.)]]
| | == heavy panting. == |
| In [[general relativity]] '''Kruskal–Szekeres coordinates''', named for [[Martin David Kruskal|Martin Kruskal]] and [[George Szekeres]], are a [[coordinate system]] for the [[Schwarzschild geometry]] for a [[black hole]]. These coordinates have the advantage that they cover the entire spacetime [[manifold]] of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity.
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| ==Definition==
| | Full of blood, the people lying there,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_42.htm オークリー 激安 サングラス], heavy panting.<br><br>'all shut up!!!' suddenly heard the roar, the original lie that's Leuca suddenly stood up,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_51.htm ランニング サングラス オークリー], stained with blood red in the face, he gazed downward sweep of the arena crowd of spectators at a glance,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_4.htm オークリー サングラス 価格], 'I not dead yet! '<br><br>then stood staring at the distant face with silvery white scales youth, and that youth face with silvery scales Leng Heng said: 'His Royal Highness,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_6.htm オークリー スポーツサングラス], before you have lost eight games in a row,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_3.htm サングラス オークリー 人気], I see you,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_47.htm オークリー ランニング サングラス], or directly under the ring of it. '<br><br>'you fart!'<br><br>Leuca roaring sound.<br><br>Boom,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_16.htm オークリーサングラス一覧]!<br><br>foot Yi Deng, the people rushed to teleport like that silvery scales directly in front of tens of young hands appeared out of thin air Liangbing hand ax, ax wielding which directly silvery white scales that step youth split in the past.<br><br>'Humph!' silvery scales of hundreds of young people leaving Disillusionment flashing moment, only to hear the thud,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_78.htm オークリーレーダーサングラス], silvery scales directly seize the youth |
| [[File:KruskalKoords.gif|thumb|right|360px|Kruskal-Szekeres diagram. Each frame of the animation shows a blue hyperbola as the surface where the Schwarzschild radial coordinate is constant (and with a smaller value in each successive frame, until it ends at the singularities).]] | | 相关的主题文章： |
| Kruskal–Szekeres coordinates are defined, from the [[Schwarzschild coordinates]] <math>(t,r,\theta,\phi)</math>, by replacing ''t'' and ''r'' by a new time coordinate V and a new spatial coordinate U:
| | <ul> |
| :<math>V = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)</math>
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| :<math>U = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)</math>
| | <li>[http://www.0668what.com/thread-53528-1-1.html http://www.0668what.com/thread-53528-1-1.html]</li> |
| for the exterior region <math>r>2GM</math>, and:
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| :<math>V = \left(1 - \frac{r}{2GM}\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)</math>
| | <li>[http://www.kaidajz.com/plus/feedback.php?aid=37 http://www.kaidajz.com/plus/feedback.php?aid=37]</li> |
| :<math>U = \left(1 - \frac{r}{2GM}\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)</math>
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| for the interior region <math>0<r<2GM</math>. Note ''GM'' is the [[gravitational constant]] multiplied with the Schwarzschild mass parameter, and this article is using [[natural units|unit]]s where ''[[Speed of light|c]]'' = 1.
| | <li>[http://www.gentoo.net/cgi-bin/archive.cgi http://www.gentoo.net/cgi-bin/archive.cgi]</li> |
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| It follows that the Schwarzschild ''r'', in terms of Kruskal–Szekeres coordinates, is implicitly given by:
| | </ul> |
| :<math>V^2 - U^2 = \left(1-\frac{r}{2GM}\right)e^{r/2GM}</math>
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| or using the [[Lambert W function]] as:
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| :<math>\frac{r}{2GM} = 1 + W \left( \frac{U^2 - V^2}{e} \right)</math>.
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| In these new coordinates the metric of the Schwarzschild black hole manifold is given by
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| :<math>ds^{2} = \frac{32G^3M^3}{r}e^{-r/2GM}(-dV^2 + dU^2) + r^2 d\Omega^2,</math>
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| written using the (− + + +) [[metric signature]] convention and where the angular component of the metric (the line element of the 2-sphere) is:
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| :<math>d\Omega^2\ \stackrel{\mathrm{def}}{=}\ d\theta^2+\sin^2\theta\,d\phi^2</math>
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| The location of the [[event horizon]] (''r'' = 2''GM'') in these coordinates is given by <math>V = \plusmn U\,</math>. Note that the metric is perfectly well defined and non-singular at the event horizon.
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| ==The maximally extended Schwarzschild solution==
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| The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates is defined for ''r'' > ''2GM'', and −∞ < ''t'' < ∞, which is the range for which the Schwarzschild coordinates make sense. However in this region, ''r'' is an analytic function of ''V'' and ''U'' and can be extended, as an analytic function at least to the first singularity which occurs at ''V^2-U^2=1''. Thus the above metric is a solution of Einstein's equations throughout this region. The allowed values are
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| :<math>-\infty < U < \infty\,</math>
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| :<math>V^2 - U^2 < 1.\,</math>
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| Note that this extension assumes that the solution is analytic everywhere.
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| In the maximally extended solution there are actually two singularities at ''r'' = 0, one for positive ''V'' and one for negative ''V''. The negative ''V'' singularity is the time-reversed black hole, sometimes dubbed a ''[[white hole]]''. Particles can escape from a white hole but they can never return.
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| The maximally extended Schwarzschild geometry can be divided into 4 regions each of which can be covered by a suitable set of Schwarzschild coordinates. The
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| Kruskal–Szekeres coordinates, on the other hand, cover the entire spacetime manifold. The four regions are separated by event horizons.
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| {| class="wikitable" style="margin: auto"
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| |-
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| ! I
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| | exterior region || <math>V^2 - U^2 < 0</math> and <math>U > 0</math> || <math>2GM < r</math>
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| |-
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| ! II
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| | interior black hole || <math>0 < V^2 - U^2 < 1</math> and <math>V > 0</math> || <math>0 < r < 2GM</math>
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| |-
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| ! III | |
| | parallel exterior region || <math>V^2 - U^2 < 0</math> and <math>U < 0</math> || <math>2GM < r</math>
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| |-
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| ! IV
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| | interior white hole || <math>0 < V^2 - U^2 < 1</math> and <math>V < 0</math> || <math>0 < r < 2GM</math>
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| |}
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| The transformation given above between Schwarzschild and Kruskal–Szekeres coordinates applies only in regions I and II. A similar transformation can be written down in the other two regions.
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| The Schwarzschild time coordinate ''t'' is given by
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| :<math>\tanh\left(\frac{t}{4GM}\right) =
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| \begin{cases}V/U & \mbox{(in I and III)} \\
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| U/V & \mbox{(in II and IV)}\end{cases}
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| </math>
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| In each region it runs from −∞ to +∞ with the infinities at the event horizons.
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| ==Qualitative features of the Kruskal-Szekeres diagram==
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| Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime. Chief among these is the fact that all radial light-like geodesics (the [[world line]]s of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in a Kruskal-Szekeres diagram (this can be derived from the metric equation given above, which guarantees that if <math>dU = \plusmn dV\,</math> then <math>ds = 0</math>).<ref>{{cite book | last = Misner | first = Charles W. | coauthors = Kip S. Thorne, John Archibald Wheeler | title = Gravitation | publisher = W. H. Freeman | year= 1973 | page = 835 | isbn = 978-0-7167-0344-0 }}</ref> All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the V coordinate) than 45 degrees. So, a [[light cone]] drawn in a Kruskal-Szekeres diagram will look just the same as a light cone in a [[Minkowski diagram]] in [[special relativity]].
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| The event horizons bounding the black hole and white hole interior regions are also a pair of straight lines at 45 degrees, reflecting the fact that a light ray emitted at the horizon in a radial direction (aimed outward in the case of the black hole, inward in the case of the white hole) would remain on the horizon forever. Thus the two black hole horizons coincide with the boundaries of the future light cone of an event at the center of the diagram (at U=0 and V=0), while the two white hole horizons coincide with the boundaries of the past light cone of this same event. Any event inside the black hole interior region will have a future light cone that remains in this region (such that any world line within the event's future light cone will eventually hit the black hole singularity, which appears as a [[hyperbola]] bounded by the two black hole horizons), and any event inside the white hole interior region will have a past light cone that remains in this region (such that any world line within this past light cone must have originated in the white hole singularity, a hyperbola bounded by the two white hole horizons). Note that although the horizon looks as though it is an outward expanding cone, the area of this surface, given by ''r'' is just <math>16\pi M^2</math>, a constant. Ie, these coordinates can be deceptive if care is not exercised.
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| It may be instructive to consider what curves of constant ''Schwarzschild'' coordinate would look like when plotted on a Kruskal-Szekeres diagram. It turns out that curves of constant r-coordinate in Schwarzschild coordinates always look like hyperbolas bounded by a pair of event horizons at 45 degrees, while lines of constant t-coordinate in Schwarzschild coordinates always look like straight lines at various angles passing through the center of the diagram. The black hole event horizon bordering exterior region I would coincide with a Schwarzschild t-coordinate of +∞ while the white hole event horizon bordering this region would coincide with a Schwarzschild t-coordinate of −∞, reflecting the fact that in Schwarzschild coordinates an infalling particle takes an infinite coordinate time to reach the horizon (i.e. the particle's distance from the horizon approaches zero as the Schwarzschild t-coordinate approaches infinity), and a particle traveling up away from the horizon must have crossed it an infinite coordinate time in the past. This is just an artifact of how Schwarzschild coordinates are defined; a free-falling particle will only take a finite [[proper time]] (time as measured by its own clock) to pass between an outside observer and an event horizon, and if the particle's world line is drawn in the Kruskal-Szekeres diagram this will also only take a finite coordinate time in Kruskal–Szekeres coordinates.
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| The Schwarzschild coordinate system can only cover a single exterior region and a single interior region, such as regions I and II in the Kruskal-Szekeres diagram. The Kruskal-Szekeres coordinate system, on the other hand, can cover a "maximally extended" spacetime which includes the region covered by Schwarzschild coordinates. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": any [[geodesic (general relativity)|geodesic]] path can be extended arbitrarily far in either direction unless it runs into a [[gravitational singularity]]. Technically, this means that a maximally extended spacetime is either "geodesically complete" (meaning any geodesic can be extended to arbitrarily large positive or negative values of its 'affine parameter',<ref>{{cite book | last = Hawking | first = Stephen W. | coauthors = George F. R. Ellis | title = The Large Scale Structure of Space-Time | publisher = Cambridge University Press | year= 1975 | page = [http://books.google.com/books?id=QagG_KI7Ll8C&lpg=PP1&pg=PA257#v=onepage&q&f=false 257] | isbn = 978-0-521-09906-6 }}</ref> which in the case of a timelike geodesic could just be the [[proper time]]), or if any geodesics are incomplete, it can only be because they end at a singularity.<ref>{{cite book | last = Hobson | first = Michael Paul | coauthors = George Efstathiou, Anthony N. Lasenby | title = General Relativity: An Introduction for Physicists | publisher = Cambridge University Press | year= 2006 | page = [http://books.google.com/books?id=5dryXCWR7EIC&lpg=PP1&pg=PA270#v=onepage&q&f=false 270] | isbn = 978-0-521-82951-9 }}</ref><ref>{{cite book | last = Ellis | first = George | coauthors = Antonio Lanza, John Miller | title = The Renaissance of General Relativity and Cosmology: A Survey to Celebrate the 65th Birthday of Dennis Sciama | publisher = Cambridge University Press | year= 1994 | pages = [http://books.google.com/books?id=8cAHaIVu6DYC&lpg=PP1&pg=PA26#v=onepage&q&f=false 26–27] | isbn = 978-0-521-43377-8 }}</ref> In order to satisfy this requirement, it was found that in addition to the black hole interior region (region II) which particles enter when they fall through the event horizon from the exterior (region I), there has to be a separate white hole interior region (region IV) which allows us to extend the trajectories of particles which an outside observer sees rising up ''away'' from the event horizon, along with a separate exterior region (region III) which allows us to extend some possible particle trajectories in the two interior regions. There are actually multiple possible ways to extend the exterior Schwarzschild solution into a maximally extended spacetime, but the Kruskal-Szekeres extension is unique in that it is a maximal, [[analytic function|analytic]], [[simply connected space|simply connected]] [[vacuum solution]] in which all maximally extended geodesics are either complete or else the [[scalar curvature|curvature scalar]] diverges along them in finite affine time.<ref>{{cite book | last = Ashtekar | first = Abhay | title = One Hundred Years of Relativity | publisher = World Scientific Publishing Company | year= 2006 | page = [http://books.google.com/books?id=8jzSKJAswMwC&lpg=PP1&pg=PA97#v=onepage&q&f=false 97] | isbn = 978-981-256-394-1 }}</ref>
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| ==Lightcone variant==
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| In the literature the Kruskal–Szekeres coordinates sometimes also appear in their lightcone variant:
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| : <math>\tilde{U} = V - U</math>
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| : <math>\tilde{V} = V + U,</math> | |
| in which the metric is given by
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| :<math>ds^{2} = -\frac{32G^3M^3}{r}e^{-r/2GM}(d\tilde{U} d\tilde{V}) + r^2 d\Omega^2,</math>
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| and ''r'' is defined implicitly by the equation
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| :<math>\tilde{U} \tilde{V} = \left(1-\frac{r}{2GM}\right)e^{r/2GM}.</math>
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| (some sources use an alternate notation where the regular Kruskal–Szekeres coordinates are labeled T and R instead of V and U, and the Kruskal-Szekeres lightcone coordinates are labeled u and v rather than <math>\tilde{U}</math> and <math>\tilde{V}</math>)<ref>{{cite book | last = Mukhanov | first = Viatcheslav | coauthors = Sergei Winitzki | title = Introduction to Quantum Effects in Gravity | publisher = Cambridge University Press | year= 2007 | pages = [http://books.google.com/books?id=vmwHoxf2958C&lpg=PP1&pg=PA111#v=onepage&q&f=false 111–112] | isbn = 978-0-521-86834-1 }}</ref>
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| These lightcone coordinates have the useful feature that outgoing [[Minkowski space#Causal structure|null]] [[geodesics]] are given by <math>\tilde{U} = \text{constant}</math>, while ingoing null geodesics are given by <math>\tilde{V} = \text{constant}</math>. Furthermore, the (future and past) event horizon(s) are given by the equation <math>\tilde{U} \tilde{V} = 0</math>, and curvature singularity is given by the equation <math>\tilde{U} \tilde{V} = 1</math>.
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| The lightcone coordinates derive closely from [[Eddington-Finkelstein coordinates]].<ref>MWT, Gravitation.</ref>
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| ==See also==
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| *[[Schwarzschild coordinates]]
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| *[[Eddington-Finkelstein coordinates]]
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| *[[Isotropic coordinates]]
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| *[[Gullstrand-Painlevé coordinates]]
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| ==References==
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| * {{cite book | title=Gravitation | last = Misner, Thorne, Wheeler | year=1973 | publisher = W H Freeman and Company| isbn=0-7167-0344-0}}
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| ==Notes==
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| {{reflist|2}}
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| {{DEFAULTSORT:Kruskal-Szekeres Coordinates}}
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| [[Category:Coordinate charts in general relativity]]
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| [[Category:Lorentzian manifolds]]
| |
heavy panting.
Full of blood, the people lying there,オークリー 激安 サングラス, heavy panting.
'all shut up!!!' suddenly heard the roar, the original lie that's Leuca suddenly stood up,ランニング サングラス オークリー, stained with blood red in the face, he gazed downward sweep of the arena crowd of spectators at a glance,オークリー サングラス 価格, 'I not dead yet! '
then stood staring at the distant face with silvery white scales youth, and that youth face with silvery scales Leng Heng said: 'His Royal Highness,オークリー スポーツサングラス, before you have lost eight games in a row,サングラス オークリー 人気, I see you,オークリー ランニング サングラス, or directly under the ring of it. '
'you fart!'
Leuca roaring sound.
Boom,オークリーサングラス一覧!
foot Yi Deng, the people rushed to teleport like that silvery scales directly in front of tens of young hands appeared out of thin air Liangbing hand ax, ax wielding which directly silvery white scales that step youth split in the past.
'Humph!' silvery scales of hundreds of young people leaving Disillusionment flashing moment, only to hear the thud,オークリーレーダーサングラス, silvery scales directly seize the youth
相关的主题文章：