IL NUOVO CIMENTO VOL. 112 B, N. 11 Novembre 1997 NOTE BREVI
Comment on “Classical solutions of two-dimensional
dilaton gravity revisited”
H. C. DHARA
Benupalchak High School
Shibnarayanchak, U.N. Pur, Via M. Hat, Howrah 711 410, W.B., India
(ricevuto il 7 Luglio 1997; approvato il 16 Settembre 1997)
Summary. — In a recent paper, Agnese and Camera (Nuovo Cimento B, 110 (1995)
109) have obtained a set of equations for a bosonic closed string in two dimensions containing coupling of gravity and dilaton field. The present note completely solves these equations.
PACS 04.50 – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.
In a recent paper, Agnese and Camera [1] have obtained the following equations for a bosonic closed string in two dimensions containing coupling of gravity and dilaton field:
n 91n8(n82l8)22n8 F840 , (1) F92 (n81l8) F840 , (2) C 4 exp [ 2 l] 1F8(F82n8) 40 , (3)
where the metric is of the form
gtt4 2exp [ 2 n(x) ] , gxx4 exp [ 2 l(x) ] ,
and the action is
S 4 1
2 p
d2xk2g exp [22 F]
(
R 14(˜F)22 C
)
, C is the central charge and the tachyon field is assumed to be zero.The above authors have discussed solutions of these equations for l 1n40. But in fact these equations can be completely integrated as follows.
H.C.DHARA
1556
Equation (2) is readily integrated to give F84K1e(n 1l),
(4)
where K1is a constant of integration.
Putting eq. (4) in eq. (1), one can easily integrate eq. (1) to get n 84el(K
1en1 K2e2n) ,
(5)
where K2is another constant of integration.
From eqns. (4) and (5) one gets dF
dn 4
K1e2 n
K1e2 n1 K2
which on integration gives
K3e2 F4 K1e2 n1 K2,
(6)
where K3is another constant of integration.
Puttings eqs. (4) and (5) into eq. (3), one gets C 44K1K2.
(7)
Hence eqs. (1)-(3) have been reduced to eqs. (5), (6) and (7), where K1, K2and K3are
constants of integration.
Obtaining explicit solutions of eqs. (5)-(7) is trivial.
R E F E R E N C E S
