Jans-Newman-Wncour and Wyman solutons are the same 1 arxv:gr-qc/9701021v2 20 May 1997 K. S. Vrbhadra Theoretcal Astrophyscs Group Tata Insttute of Fundamental esearch Hom Bhabha oad, Colaba, Mumba 400005, Inda. Abstract We show that the well-known most general statc and sphercally symmetrc exact soluton to the Ensten-massless scalar equatons gven by Wyman s the same as one found by Jans, Newman and Wncour several years ago. We obtan the energy assocated wth ths spacetme and fnd that the total energy for the case of the purely scalar feld s zero. To appear n Int. J. Mod. Phys. A 1 Ths paper s dedcated to the memory of Professor Nathan osen. 1
Even before the general theory of relatvty was proposed, scalar feld has been conjectured to gve rse to the long-range gravtatonal felds[1]. Several theores nvolvng scalar felds have been suggested [2]. The subject of scalar felds mnmally as well as conformally coupled to gravtaton has fascnated many researchers mnds ([3] -[10]). Snce the last few years there has been a growng nterest n studyng the gravtatonal collapse of scalar felds and the nature of sngulartes n the Ensten-massless scalar (EMS) theory (see ef. [4] and references theren). There has been consderable nterest n obtanng solutons to the EMS as well as the Ensten-massless conformal scalar equatons. Jans, Newman and Wncour (JNW) [5] obtaned statc and sphercally symmetrc exact soluton to the EMS equatons. Wyman[6] further obtaned a statc sphercally symmetrc exact soluton to the EMS equatons. There s no reference to the JNW soluton n hs paper. Agnese and La Camera[7] expressed the Wyman soluton n a more convenent form. oberts[8] showed that the most general statc sphercally symmetc soluton to the EMS equatons (wth zero cosmologcal constant) s asymptotcally flat and ths s the Wyman soluton. The Wyman soluton s well-known n the lterature[9].in the present note we show that the Wyman soluton s the same as the JNW soluton, whch was obtaned almost twelve years ago. We further obtan the total energy assocated wth ths spacetme. We use geometrzed unts and follow the conventon that Latn (Greek) ndces take values 0... 3 (1... 3). The EMS feld equatons are j 1 2 g j = 8π S j, (1) where S j, the energy-momentum tensor of the massless scalar feld, s gven by S j = Φ, Φ,j 1 2 g j g ab Φ,a Φ,b, (2) and Φ ;, = 0. (3) 2
Φ stands for the massless scalar feld. j s the cc tensor and s the cc scalar. Equaton (1) wth Eq. (2) can be expressed as j = 8π Φ, Φ,j. (4) JNW[5] obtaned statc and sphercally symmetrc exact soluton to the EMS equatons, whch s gven by the lne element ds 2 = ( 1 a 1 + a + ) 1/µ dt 2 ( 1 a 1 + a + ) 1/µ d 2 ( 1 a ) 1 1/µ ( 1 + a + ) 1+1/µ 2 dω,(5) wth dω = dθ 2 + sn 2 θdφ 2, (6) and the scalar feld Φ = σ µ ln ( 1 a 1 + a + ), (7) where a ± = r 0 (µ ± 1)/2 (8) and µ = 1 + 16πσ 2. (9) r 0 and σ are two constant parameters n the soluton. r 0 = 0 gves the flat spacetme, whereas σ = 0 gves the lne element ( ds 2 = 1 + r ) 1 ( 0 dt 2 1 + r 0 ) ( d 2 3 1 + r 0 ) 2 2 dω, (10)
whch obvously represents the Schwarschld metrc. The JNW soluton can be put n a more convenent form, n coordnates t, r, θ, φ, by the lne element ( ds 2 = 1 r) b γ ( dt 2 1 r) b γ ( dr 2 1 r) b 1 γ r 2 dω (11) and the scalar feld Φ = q ( b 4π ln 1 b ), (12) r where γ = 2m b, b = 2 m 2 + q 2. (13) m and q are constant parameters. The radal coordnates and r are related through r = + a + (14) and the spacetme parameters r 0, σ and m, q are related through r 0 = 2m, σ = 1 q 16π m. (15) The soluton to the EMS equatons, expressed by Eqs. (11) (13), s exactly the wellknown Wyman soluton (see ef. [7]). Thus, Wyman dd not obtan a new soluton, but he redscovered the JNW soluton ndependently. The JNW soluton, n t, r, θ, φ coordnates, has a curvature sngularty at r = b. Garfnkle, Horowtz and Stromnger [10] obtaned a nce form of charged dlaton black hole soluton, characterzed by mass, charge, and a couplng parameter (whch controls the strength of the couplng of the dlaton to the Maxwell feld). 4
A partcular soluton of ths (puttng the electrc charge parameter zero) yelds the JNW (Wyman) soluton. Ths fact s not notced n ther paper. It s of nterest to obtan the energy assocated wth the JNW spacetme. The energymomentum localzaton has been a recalctrant problem snce the outset of the general theory of relatvty. Though, several energy-momentum complexes have been shown to gve the same and reasonable result (local values) for any Kerr-Schld class metrc as well as for the Ensten-osen spacetme, the subject of energy-momentum localzaton s stll debatable (see ef. [11] and references theren). However, the total energy of an asymptotcally flat spacetme s unanmously accepted. Usng the Ensten energy-momentum complex we frst obtan an energy expresson for a general nonstatc sphercally symmetrc metrc and then we calculate the total energy assocated wth the JNW spacetme. A general nonstatc sphercally symmetrc lne element s ds 2 = Bdt 2 Adr 2 Dr 2 ( dθ 2 + sn 2 θdφ 2), (16) where B = B (r, t),a = A (r, t), D = D (r, t). The Ensten energy-momentum complex (see efs. [11]) s Θ k = 1 16π H kl,l, (17) where H kl = H lk = g n g [ g ( g kn g lm g ln g km)],m. (18) The Ensten energy-momentum complex satsfes the local conservaton laws Θ k x k = 0, (19) where Θ k = g ( T k ) + ϑ k. (20) 5
T k s the symmetrc energy-momentum tensor due to matter and all nongravtatonal felds and ϑ k s the energy-momentum pseudotensor due to the gravtatonal feld only. The energy and momentum components are gven by P = 1 16π H 0α n α ds, (21) where ds s the nfntesmal surface element and n α s the outward unt normal vector. P 0 stands for the energy and P α stand for the lnear momentum components. As t s known that the energy-momentum complexes gve meanngful result only f calculatons are carred out n quas-mnkowskan coordnates, we transform the lne element n coordnates t, x, y, z (accordng to x = r sn θ cosφ, y = r sn θ sn φ, z = r cosθ). Then we calculate the requred components of H kl and these are B H0 01 = 2 A H 02 0 = 2 H 03 0 = 2 B A B A x (A D D r) r 2, y (A D D r) r 2, z (A D D r) r 2, (22) where the prme denotes the partal dervatve wth respect to the radal coordnate r. Usng the above n (21) we obtan the energy E = 1 2 B A r (A D D r). (23) Substtutng B = A 1 = (1 b/r) γ and D = (1 b/r) 1 γ we obtan the energy assocated wth the JNW spacetme. E total = m. (24) The total energy of the JNW spacetme s gven by the parameter m. We have repeated these calculatons usng other energy-momentum complexes and have found the same result 6
as gven by Eq. (24). The total energy of a purely scalar feld (.e. for m = 0) s zero. It s of nterest to nvestgate whether or not ths s true n general,.e. for any purely massless scalar feld. Acknowledgements Thanks are due to Professor P. C. Vadya for a careful readng of the manuscrpt. 7
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