The Arn Revew o Physs : 8 New Et Solutons or Stt Ally Symmetr Ensten Vuum Equtons Ahmd T. Al Froo Rhmn * nd Syeedul Islm Kng Adul A nversty Fulty o Sene Deprtment o Mthemts Jeddh Sud Ar nd Mthemts Deprtment Fulty o Sene Al-Ahr nversty Nsr Cty Cro Egypt Deprtment o Mthemts Jdvpur nversty Kolt West Bengl Ind The Ensten stt vuum solutons wth lly symmetry hve een onsdered. The symmetry group nlyss sovetor elds method redues the Ensten equtons to ordnry derentl equtons ODEs. These equtons n e solved nlytlly wth the d o Mthemt progrmm. Some symmetry trnsormtons nd mny new et smlrty solutons o Ensten vuum equtons re otned.. Introduton Stt solutons o Enstens equtons wth spe symmetry le spherl symmetry or ylndrl symmetry re tve re o reserh n generl reltvty. Smlr to the solutons wth spherlly symmetry the solutons wth ylndrl symmetry omnng trnslton long wth rotton round n s re well studed. These solutons re mportnt euse they n e onsdered s ppromtons to non-spherl nd etended mtter nd eld ongurtons. To study the osmologl models grvttonl ollpse o non-spherl mtter dstrutons nd the study o grvttonl wves ylndrl symmetry hs een used eetvely. In poneerng wor Lev-Cvt otned ylndrlly symmetr stt vuum soluton o Enstens eld equtons. These solutons hve the mportnt grvttonl nd osmologl mpltons. Due to omplted nture o the eld equtons n stt nd lly symmetr ongurtons very lttle studes hve een done so r. Ths t motvtes us to study the vuum solutons o ths sort. It wll e nterestng to otn solutons o Ensten vuum equtons wth stt l symmetry usng the symmetry group nlyss sovetor elds method. It s group o ontnuous trnsormtons tht leve gven mly o equtons nvrnt -. Most o the physl phenomen n vrous elds o sene re desred y nonlner prtl derentl equtons 7-. To study these physl phenomen one wll hve to solve the nonlner prtl derentl equtons PDEs. The Ensten equtons re hghly non lner n nture. We wll nd Ensten vuum equtons wth stt l symmetry usng the symmetry group nlyss sovetor elds method. Aordng to Ovsnnov the usul Le nntesml nvrne pproh ould s well e employed n order to onstrut symmetry groups. In ths pper we pply the so-lled symmetry nlyss method n the study o the stt vuum solutons wth lly symmetry. The mn dvntge o ths method s tht t trnsorms the nonlner PDEs nto system o ordnry derentl equtons ODEs. We now tht Ensten eld equtons re hghly nonlner n nture thereore symmetry nlyss method ould e useul to solve these nonlner derentl equtons. The smlrty solutons re very helpul to redue the ndependent vrles o the systems. As result one n employ ths method to nvestgte the et soluton o the eld equtons. sully smlrty solutons wll hnge the system o nonlner PDEs nto system o PDEs. We wll nd new lss o nvrnt solutons or the stt vuum Ensten eld equtons wth lly symmetr spe-tme. The pln o the the pper s s ollows. The Ensten eld equtons or stt lly symmetr spe-tme n generl orm re ntrodued n Se.. In Se. symmetry nlyss nd sovetor elds or Ensten eld equtons re otned. In Se. we nd mny new et solutons or Ensten eld equtons. The pper ends wth onluson n Se.. * rhmn@u.ernet.n
The Arn Revew o Physs : 8. Ensten Feld Equtons or Stt Ally Symmetr Felds The most generl stt non-rottng lly symmetr metr n e wrtten n the ylndrl-oordnte orm s ν ds dt ldφ e d d Where only nd the oordntes l nd ν re untons o nd φ orrespond to ylndrl oordntes. Moreover the metr Eqn. n the sene o grvtton redues to Mnows metr n the sme oordntes. We wll use the ollowng generl nterestng susttuton l Where s unton o nd only. The Ensten vuum equtons n e redued to the ollowng two nonlner PDEs n the smple orm s E E Also nowng nd rom Eqns. nd the unton ν n e otned y qudrtures o the ollowng equtons ν ν One o the smplest soluton o Ensten equton s whh leds to well nown orm o Weyl soluton or stt lly symmetr metrs. Ater the Shwrshld soluton the lly symmetr Weyl solutons hve een the most studed n lssl reltvty. The metr whh hs een nmed ter Shwrshld ws dsovered y hm nd Lev Cvt. An etensve dsusson o t s gven y Synge. Muh more reently some solutons o the Weyl metr nd ther nterpretton hve een dsussed y Bonnor n revew pper n 99 7 nd Stephn et l. n revew oo n. It s urous tht the Shwrshld soluton omes out s one o the prtulr solutons o Weyl. In smplest Weyl soluton or n solted system Curon otned the smple soluton o Eqn. n the orm 7 ep M 7 The metr otned y the soluton s lled Curon metr wth mss M nd qudrupole moment M. The soluton n 7 s the Newtonn potentl or spherl prtle ut the Curon soluton s derent rom tht o Shwrshld. It s r-eld s tht o mss t wth multpoles on t. Even smpler soluton o Eqn. or s the ylndrlly symmetr potentl σ 8 orgnlly onsdered y Lev Cvt. In Newtonn theory ths s the grvttonl potentl o n nnte unorm lne-mss σ eng the mss per unt length. There s useul soluton o Eqn. n the orm m m m m m m 9 m Where δ nd m re onstnts. The Weyl metr generted y Eqn. 9 s lled δ -metr nd s reerrng to n solted ody whh generles the Curon metr. The Curon metr s otned y lettng nd eepng m nte. The δ -metr ws dsovered y Drmos 8. Msr hoose prtulr soluton o Eqn. nmely: δ
The Arn Revew o Physs : 8 7 ep δ ot m m m Vorhees ntrodued nother soluton o Eqn. gven s δ m m m m m m The unton ln wth gven y one o Eqns. 9 nd nd orresponds to Newtonn potentl o rod o lner mss densty δ nd length m. The three metrs otned y Eqns. 9- re equvlent ny o them s lled δ -metr or γ -metr 7 nd ts eld hs mss δ m nd qudrupole moment m δ δ. In ll ses ove the uthors te the soluton o Eqn. or. In ths pper we te the generl orm o Weyl metr when to otn new et solutons or stt lly symmetr vuum equtons nd usng the geometr prolongton tehnque.. Symmetry Anlyss or Vuum Ensten Feld Equtons In order to otn n et smlrty solutons we use Le group nlyss theory on Eqns. nd. For ths we wrte rom the nvrne ondtons Pr V E E Where E re the Ensten eld Eqns. nd under study nd Pr s the seond prolongton o the vetor eld V. Epndng ths system wth the d o Mthemt progrm nd usng the orgnl Eqns. nd to elmnte nd nd settng the oeents o ll equtons n the system n Eqn. nvolvng nd vrous produts to ero gvng rse to the essentl set o determnng equtons o symmetres or Ensten vuum equtons s the ollowng: * ε O ε * ε O ε * ε O ε * ε O ε s the nntesml Le pont trnsormtons. We hve ssumed tht Eqns. nd re nvrnt under the trnsormtons gven n Eqn.. The orrespondng nntesml genertor sovetor eld s V u Where u nd u. The oeents nd re untons o nd to e determned
The Arn Revew o Physs : 8 8 u u Solvng the set o Eqns. ove wth the d o Mthemt progrm we otn the omponents o the generl sovetor eld Eqn. o Ensten vuum Eqns. nd n the orm s ln ln Where re rtrry onstnts nd nd re untons o nd stsyng the ollowng ondtons 7 The generl orm o the sovetor eld V ssoted wth Ensten vuum Eqns. nd n e wrtten s V ln ln 8 It s worth notng tht ths sovetor eld ontns our prmeters onstnts... nd two prmeters untons group o symmetres emedded n n nnte dmensonl Le group represented y two rtrry untons nd. Knowng the sovetor elds o Ensten eld equtons we re le to redue the orgnl PDEs to ODEs.. New Et Solutons o Ensten Vuum Equtons To nd the symmetry trnsormtons we susttute the omponents o sovetor eld 8 n the ollowng system o nonlner ODEs ln ln d d d d 9 The solutons o the system Eqn. 9 re non trvl. However n spel ses o on the sovetor elds Eqn. 8 we n nd mny symmetry trnsormtons. To nd sutle redutons we must put nd. nder ondton n Eqn. 7 the untons nd eome 7 Hene the system Eqn.9 te the orm 7 ln d d d d Now we shll study our ses desed elow. Cse. whle nd 7 re rtrry onstnts. Solvng the system o Eqns. we get the smlrty vrle nd smlrty trnsormtons s
The Arn Revew o Physs : 8 9 Θ Ψ Where s new ndependent vrle Θ nd Ψ re untons o whle nd 7 re rtrry onstnts. Then Eqns. nd under the trnsormtons redue to the ollowng system o ODEs Θ Θ ΘΨ ΘΘ ΘΘ Θ Ψ Ψ Ψ Ψ Where denotes derentton wth respet to. The soluton o Eqn. n e epressed n the orm Θ χ χ ep ζ Ψ ζψ dζ dχ Ψ Where nd re onstnts o ntegrton. Eqn. s n ODE or Ψ only nd depends on one prmeter. I we otn the soluton o Eqn. we n otn the soluton o Eqn. nd then we n nd the et soluton o Ensten eld equtons -. Here we wll study ll ses o s the ollowng: Cse... The soluton o Eqns. nd n ths se tes the orm Ψ tn Θ tn 7 Where nd re onstnts o ntegrton. Susttutng Eqns. nd 7 n Eqn. we otn solutons gven s tn 8 tn 9 l tn nd led to the soluton ν h tn tn ln tn Where h s onstnt o ntegrton. Now we n sy tht the metr oeents whh re represented y Eqns. 9 nd re the new et solutons o Ensten vuum equton -. Cse... The soluton o Eqns. nd n ths se tes the ollowng generl orm Ψ sn os tn tn Susttutng Eqns. 8 nd 9 n Eqn. we otn
The Arn Revew o Physs : 8 tn tn tn tn tn sn tn os Θ Where nd re onstnts o ntegrton. Susttutng nd n we otn the ollowng solutons tn sn tn os tn sn tn os tn tn tn tn Susttutng Eqns. nd n Eqn. we otn tn sn tn os tn tn tn tn l From nd we hve: tn sn tn os tn tn tn tn ep h ν 7 Where h s onstnt o ntegrton nd the relton must e stsed. Now we n sy tht the metr oeents whh re represented y Eqns. nd 7 re nother generl orm o et solutons o Ensten vuum equtons -. Cse. nd 7 re rtrry onstnts. Solvng the system o ODEs we otn the smlrty vrle nd smlrty trnsormtons s Ψ Θ ep 8 Where s new ndependent vrle smlrty vrle Θ nd Ψ re untons o whle 7 nd
The Arn Revew o Physs : 8 re rtrry onstnts. The Eqns. nd under the trnsormtons Eqn. 8 redue to the ollowng system o ODEs Ψ Ψ Ψ Where denotes derentton wth respet to. By ntegrtng Eqn. 9 we hve Θ Θ Θ Θ Ψ Θ Ψ 9 ~ ~ Θ Θ Θ ~ Ψ ep ~ ~ d Θ Θ Where s onstnt o ntegrton. I we susttute Eqn. n Eqn. we otn n ntegro-derentl equton o Θ only. Ths resultng equton s not esy to solve n generl se. Eqn. s tsel Eqn. nd the soluton o t s Eqn. when nd Eqn. when. Remr. In ths se we do not te euse t s spel se o the se when. Remr. There re mny hoes o whh yeld new et smlrty solutons o Ensten eld equtons - ut t s not esy to solve the Eqns. nd n these ses. So we my study these ses n detls n uture wor. Cse. 7 nd re rtrry onstnts. Solvng the system o ODEs n Eqn. we nd the smlrty vrle nd smlrty trnsormtons s ep Θ ep Ψ Where s new ndependent vrle Θ nd Ψ re untons o whle nd re rtrry onstnts. Eqns. nd under the trnsormtons Eqn. redue to the ollowng system o ODEs Where Θ ΘΘ Θ Θ Θ Ψ Θ Θ ΘΨ Ψ Ψ Ψ 7 s derentton y. The soluton o Eqn. n e epressed n the orm χ χ ζ Θ ep ep ep dζ dχ Ψ Ψ Ψ Where nd re onstnt o ntegrton. Eqn. s ODE or Ψ only nd depend on the one prmeter. I we otn the soluton o Eqn. we n otn the soluton o Eqn. nd then we n nd et solutons o Ensten eld equtons -. Here we wll study ll ses o s the ollowng: Cse... The soluton o Eqns. nd n ths se tes the orm Θ ep 7 Susttutng nd 7 n nd we nd ondton. So we wll study two ses. Cse.... The metr oeents n ths se tes the orm Ψ
The Arn Revew o Physs : 8 ep ep ep ep h l ν 8 Where h s onstnt o ntegrton. So the metr oeents n Eqn. 8 orm nother new et solutons o Ensten vuum equtons -. Cse.... The metr oeents n ths se te the orm: ep h l ν 9 Where h s onstnt o ntegrton. Now we n sy tht the metr oeents n Eqn. 9 re nother new et solutons o Ensten vuum equtons -. Cse... The soluton o Eqns. nd n ths se tes the ollowng generl orm: ep sn os Ψ ep tn tn sn os Θ Susttutng Eqns. nd n Eqn. we otn the solutons ep sn os ep tn tn sn os Susttutng nd n we otn
The Arn Revew o Physs : 8 l os sn tn ep tn From Eqns. nd we hve ep ν h os sn tn tn Where h s onstnt o ntegrton nd. Thereore the metr oeents whh re represented y Eqns. - orm nother new generl orm o et solutons o Ensten vuum equtons -. Ths soluton s desred s soltry wve soluton o Ensten eld equtons. Cse. nd 7 nd re rtrry onstnts. Solvng the system o Eqn. we nd the smlrty vrle nd smlrty trnsormtons s ep Θ ep Ψ ep Ψ re untons o whle nd re rtrry onstnts. Eqns. nd under the trnsormtons redue to the ollowng system o ODEs Θ Where Θ Θ Ψ Ψ Ψ Θ Ψ 7 Θ Ψ 7 8 s derentton y. The soluton o Eqn. 7 n e epressed n the orm Where s new ndependent vrle Θ nd Θ Θ Θ ~ Ψ ep d 9 Θ Θ Where s onstnt o ntegrton. I we susttute Eqn. 9 n Eqn. 8 we otn n ntegro derentl equton o Θ only. The resultng equton s not esy to solve n generl se. Eqn. 8 s tsel Eqn. nd the soluton o t s Eqn. when nd Eqn. when. In ths se we do not te euse t s spel se o the se when. However there re mny hoes o whh yeld new et solutons o Ensten eld equtons ut t s not esy to solve Eqns. nd n ths ses. So we my study ths se n detls n uture wor.. Conluson There re severl motvtons suh s grvttonl nd osmologl mpltons o lly symmetr Ensten vuum solutons ledng to ths wor. These solutons re the most mportnt euse they hve mny ppltons on the re o reserh
The Arn Revew o Physs : 8 suh s grvttonl wves grvttonl ollpse numerl reltvty et. We dedued mny new et solutons or Ensten vuum equtons or stt non-rottng lly symmetr elds wth the tehnque o the symmetry nlyss. Ths tehnque hs proved to e powerul one n the lst dedes n derent equtons nd ths new pplton omes to onrm ths stte. We ompletely solved the determnng equtons or the sovetor eld nd otned ll lnerly ndependent sovetor elds o Ensten vuum equtons or stt elds. We determned the sovetor elds symmetres ssoted wth Ensten vuum equtons orrespondng to -symmetr metr wth no rotton n generl orm o Weyl metr when. Also we ound mny new et smlrty solutons tht mght prove to e nterestng physlly. More generl solutons suh s non-sttonry ones n e onsdered n urther wors. Anowledgements FR wshes to thn the uthortes o the Inter-nversty Centre or Astronomy nd Astrophyss Pune Ind or provdng the Vstng Assoteshp. SI s lso thnul to DST Govt. o Ind or provdng nnl support under INSPIRE Fellowshp. Reerenes T. Lev-Cvt Rend. A. Lne 8 99. A. T. Al Phys. Sr. 79 9. A. T. Al Phys. Sr. 79 9. A. T. Al nd E. R. Hssn Appl. Mth. Comp. 7. A. T. Al F. Rhmn nd A. Mll Int. J. Theor. Phys. 97. A. T. Al Ast. Appl. Anl. 9897 8. 7 A. T. Al nd A. K. Ydv Int. J. Theor. Phys. 7. 8 A. T. Al A. K. Ydv F. Rhmn nd A. Mll Phys Srpt 89. 9 A. T. Al S. R. Mhmood nd A. K. Ydv Astr. Spe S. 9 9. S. K. Attllh M. F. El-Sgh nd A. T. Al Commun. Nonlner S. Numer. Smult. 7 7. G. W. Blumn nd S. Kume Symmetres nd Derentl Equtons n Appled Senes Sprnger New Yor 989. M. F. El-Sgh nd A. T. Al Int. J. Nonlner S. Numer Smult.. M. F. El-Sgh nd A. T. Al Commun. Nonlner S. Numer Smult. 78 8. M. F. El-Sgh A. T. Al nd S. El-Gnn Appl. Mth. nd Inorm. S. 8. K. S. Mehemer S. Z. Husseny A. T. Al nd R. E. Ao-Elhr Phys. Sr. 8 7. P. J. Olver Applton o Le Groups to Derentl Equtons n Grdute Tets n Mthemts Vol.7 seond edton Sprnger New Yor 99. 7 C. M. Curon Pro. London Mth. So. 77 9. 8 G. Drmos Memorl des Senes Mthemtques Guther-Vllrs Prs 9. 9 N. H. Irgmov Trnsormton Groups Appled to Mthemtl Physs D. Redel Dortreht 98. J. N. Islm Rottng Felds n Generl Reltvty Cmrdge nversty Press Cmrdge 98. M. Msr Pro. Ntl. Inst. S. Ind A 7 9. L. V. Ovsnnov Group Anlyss o Derentl Equtons trnslted y Y. Chpovsy Ed. W. F. Ames Adem Press New Yor-London 98. H. Stephn D. Krmer M. MCllum C. Hoenselers nd E. Herlt Et Solutons o Enstens Feld Equtons seond edton Cmrdge nversty Press Cmrdge. B. H. Vorhees Phys. Rev. D 9 97. H. Weyl Annl. Phys 7 97. J. L. Synge Reltvty: The Generl Theory North-Hollnd Pulshng Compny Amsterdm 9. 7 W B Bonnor Gen. Rel. Grv. 99. Reeved: August Aepted: Mrh