DeSitter

From GRwiki

link nfs underground 2 no cd patch download dream theater ringtone ib 12 bad boy records redroof inn cheap las vegas wedding package credit agricole nord est service en ligne free music downloads without a membership grand theft auto chinatown wars ds ms walk in winnipeg ashley jordan password torrent britney spears blow job movie new zealand jobs naturist massage executive search recruiting sitemap scary video clip prank popped cherries socrates commit suicide top elinor miriam whites husband lexapro and adderall xr funky monkey polar express grade six the trade in canada simon mall gift card pavilion movie brooklyn arts crafts mission library furniture airline tickets to orlando florida payment protection insurance webmap stock photo airline offers military discounts mini stock race cars for sale blinkie maker age of empire 2 crack no cd aladdin usb key crack bone beads site acronis true image 7.0 crack getting approved for a home loan domain malesupersite.com prince of persia warrior within alternate ending video car rental denver painting business design card saturday jobs acquire capital boxers showing jobs in austin montgomery sheet metal men wearing jock straps fantasy girl oldham job center softube metal amp maryland eye care teen hitchikers settlement law the stock market for dummies small business grant forms real estate ,com pearl harbor bombing map bargain tours and travels pampered chef cookie molds sitemap cable speed test bit torrents ibm laptop video drivers canon printer driver amplitube crack serial public speaking video indiana elnora drugs legal real amateur couples agco division part scented decorative candles testing ignition coil albuquerque nissan dealers alien deep movie budget rent a car corporate office carrollton texas hawaiian wedding dresses ameture hand jobs the dump northstar legion mts calling card for india cobb premier portable grill sitemap ruben studdard associated financial group corporate office pattern recognition movie job in aberdeen stacy keibler bra and panties bean bag toss american airlines flight schedules seek average income sample application job letter for teacher real estate in pace, fl citizens bank not your typical bank natural cycle winnipeg the saddle club willowbrook stables graduation announcements webmap narcotic thrust i like it music video indian movie songs mp3 london, ontario jobs cleaning, housekeeper debeka jobs canadian online travel agents busty mature blow jobs amisha patel network administrator job in memphis tn drugs mushrooms stag scrap metal association licensed nova nurse practical scotia american airline flight status irish business news today british columbia accommodations medicare card arizona motorcycle license diosesmx trade hack free download statin jamaica payday advance equipment heat industrial transfer rocky movie photograph map 100 top boys and girls names nfs porsche 2000 crack addiction drug rehab developer sites lee county fl pretty woman and ringtone world bank total investment in yanacocha url shawna robinson site just want lots of money i don't care about clever i don't care about funny truststore keystore http romantic scenes in movies la harpe kansas new persian movies alicia richter alcohol drug poster ati hdtv component video adapter kit page shatec courses arraylists vb.net protein phosphorylation drug discovery world summit body glamour photography buy azithromycin symptoms of celiac disease acoustica beatcraft cracks virgin card online windows server 2008 enterprise edition oem key preffered mutual insurance company emmett till discover e car insurance bon iver blood bank download parkway oakland movie paul gilbert video a simple motors work buying a rental car class a driver job opportunity billing service software pennslyvania state board of nusing case a montemarano index nursery nurse examination board itty bitty titty committee be a home based mortgage underwriter car rental in miami araxis merge crack alaska vacation tour packages meezan internet banking skull tatoos workout music predator drone videos iraq letter of acknowledgement of debt trans union credit bureau decorative garden edging metal white a4desk crack train day union station washington dc url make money google adsense aqa exam board cauldron imperia invest panasonic professional video cameras tweety e cards how to score bejeweled sheet music for begginers on the key board free lego knights of the round table movie pioneer telephone kingfisher ralph wiggum grease we go together map site new port richey, fl mortgage book airline tickets application torrent site swiss chalet number pack tout compris credit agricole ancient egyptian work and entertainment sample letters to ca franchise tax board domain http amature tug job foremost signature insurance co amazing grace inspirational poem auto rental in las vegas av mix master crack travel asia luxury travel student travel international arjona de musica ricardo todo video y credit card merchant service metal video links concession stand apartment rentals on beacon hill boston ma apple dvd studio pro 3.0 crack deutsche bank private banking secrets in lace pause pretty rickys video format hard drive suffolk county national bank lil wayne cash money best airline fares cheap workers comp ins for auto obdy shops kl 133m video driver lauren ferrara sonoma sheriff wholesale umbrella doll nvidia fx5200 drivers download latest dvd movie stock car challenge unsubsidized fed loan application radisson hotel shanghai real estate investor software national city bank in cleveland bellagio hotel las vegas calgary and airport and hotel phuonghong.com video dpo jobs online goldmark credit union paparazzi videos webmap money magazine 50 best jobs in america wells fargo bank in rushville indiana oriental trading graduation items brisbane jobs index us tourist visa invitation letter sample fiji work clothes suppliers caldwell bankers real estate free egyptian math work sheets capital murder vs 1st degree murder richmond county georgia board of education borland c builder v6.0 downloadable enterprise trial crack woolworths gift cards apple vacation cruises sky venture punjabi movie songs cash flow volatility and prediction of future cash flows letter request for visa issuance top car rental agencies in the orlando international airport hollenbeck girl a2 visa need to get green card alaska airline credit card digital color printing case prefabbricate legno prezzi pacers piston brawl video board of nursing in illinois application introduce yourself letter in new place of work stock off selenium electron cloud taringa black metal site prom night movie camui gackt (server or administrator or help or desk or client or clinic or is or it or analyst or user or coordinator or support) work domain crosby stills nash and young work experience with childcares matisyahu video code candy wrapper software koa las vegas nv contaminant removal domain disney majic cruise discount becoming a certified travel agent hawaii dental naples florida luxury vacation home rentals condo vacation rentals in daytona florida policeman of the world aspen comic wallpaper artificial flower garland nissan ohio sherwood laptop video upgrade slow motion ringtone mona lisa sunset thomas cumslut knox county occupational license Four-dimensional deSitter spacetime satisfies the Einstein equations in vacuum with a positive cosmological constant: $G_{\mu\nu} + \Lambda g_{\mu\nu} = 0$ . deSitter spacetime is the surface

$-(T)^2 + (X)^2 + (Y)^2 + (Z)^2 + (W)^2 = \ell^2$

where $\Lambda \equiv 3/\ell^2$ , in the five-dimensional flat spacetime

$ds^2 = - (dT)^2 + (dX)^2 + (dY)^2 + (dZ)^2 + (dW)^2$

The surface can be covered with coordinates $t$ , $\chi$ , $\theta$ and $\phi$ , defined by

$\begin{array}{rcl} T & = & \ell \sinh(t/\ell) \\ X & = & \ell \cosh(t/\ell) \cos\chi \\ Y & = & \ell \cosh(t/\ell) \sin\chi \cos\theta \\ Z & = & \ell \cosh(t/\ell) \sin\chi \sin\theta \cos\phi\\ W & = & \ell \cosh(t/\ell) \sin\chi \sin\theta \sin\phi \end{array}$

In these coordinates the deSitter metric is

$ds^2 = - dt^2 + \ell^2 \cosh^2(t/\ell)\, d\chi^2 + \ell^2 \cosh^2(t/\ell)\, \sin^2\chi\, (d\theta^2 + \sin^2\theta \, d\phi^2)$

[edit] Static coordinates

Let

$\begin{array}{rcl} x_0 & = & \sqrt{\ell^2 - r^2} \sinh(t/\ell) \\ x_1 & = & \sqrt{\ell^2 - r^2} \cosh(t/\ell) \\ x_2 & = & r \, \sin\theta\, \sin\phi \\ x_3 & = & r\, \sin\theta \, \cos\phi \\ x_4 & = & r \,\cos\theta \end{array}$

The deSitter metric is

$ds^2 = -(1 - r^2/\ell^2) dt^2 + \frac{1}{(1 - r^2/\ell^2)} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2 )$

These coordinates cover only the strip $-\ell \leq x_4 \leq \ell$ of the full manifold. Let $r = \ell \sin\chi$ and the metric becomes

$ds^2 = -\cos^2\chi\, dt^2 + \ell^2 dr^2 + \ell^2\sin^2\chi (d\theta^2 + \sin^2\theta \, d\phi^2 )$

[edit] Conformally flat coordinates

Let

$\begin{array}{rcl} x_0 & = & \ell \sinh(t/\ell) + e^{t/\ell} X_i X_i /(2\ell)\\ x_1 & = & \ell \cosh(t/\ell) - e^{t/\ell} X_i X_i /(2\ell)\\ x_2 & = & e^{t/\ell} X_1 \\ x_3 & = & e^{t/\ell} X_2 \\ x_4 & = & e^{t/\ell} X_3 \end{array}$

where $X^i X^i \equiv (X_1)^2 + (X_2)^2 + (X_3)^2$ . The metric is

$ds^2 = -dt^2 + e^{2t/\ell} (dX_1^2 + dX_2^2 + dX_3^2)$

Only the region $x_0 + x_1 \geq 0$ is covered by these coordinates.

Retrieved from "http://grwiki.physics.ncsu.edu/wiki/DeSitter"

DeSitter

From GRwiki

[edit] Static coordinates

[edit] Conformally flat coordinates

Views

Personal tools

Navigation

Search

Toolbox