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Reference Ic/87/241 REFERENCE IC/87/241 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS IT -MODEL OF STRONG GRAVITY Ali El-Tahir INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/8T/2141 1, INTRODUCTION Gravitational theories based on Lagrangians which incorporate terms quadratic (or generally nonlinear) in the scalar curvature R had been put forward by many authors [l] during the last three decades. Some International Atomic Energy Agency of these authors [£] were motivated by the electromagnetic analogy and and endeavoured to construct a unified field theory. Different Lagrangian United Rations Educational Scientific and Cultural Organization approaches were made in which the quadratic R was introduced to play a significant INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS role especially in regions of strong gravity, Einstein's general theory of relativity (GR) which is based on the Hilbert variational principle, where the Lagrangian is identified with the scalar curvature, proved to he successful in the weak gravitational field. But in the strong-field limit it has tvo major setbacks. E - MODEL OF STRONG GRAVITY * First.it is isolated, in its geometrical picture, from the mainstream of physical theories hence it is not amenable to quantization or unification. Second, it is essentially a singular description whence it paradoxially predicts the gravitational collapse with its attendant formation of Ali El-Tahir ** "black holes", the evidence for the existence of which is rather weak. International Centre for Theoretical Physics, Trieste, Italy. On the other handjit is assumed from the electromagnetic theory, that the quantizibility and the singularity are incompatible in the sense that the singular features of classical equations usually disappear if the theory is subject to quantization. Thus attempts were made at constructing a nonsingular theory which will hopefully bridge the gap between GR and ABSTRACT quantum physics. Some of these attempts [3] in which quadratic [k] or nonlinear [5] Lagrangians are utilized proved to be successful by allowing A metric based an the simplest nonlinear Lagrangian which is nonlingular solution spacetimes [6], identified with the square of the scalar curvature R is considered. The region where the metric is dominated by this Lagrangian is assumed In a previous work [7] we explored the possibility of a nonsingular to be confined to the vicinity of the gravitational source where gravity metric based on a very general Lagrangian where a possible singularity can is most strong." Expressions for R and the metric were obtained as be averted by an appropriate choice of the Lagrangian structure. We also functions of the radial distance, exhibiting nonsingular solutions. considered a Lagrangian having the form, where a, B and y are constants. MIRAMARE - TRIESTE Requiring that as r •+ " , R(r) •> 0 we can assume that the linear September 1987 term in this Lagrangian will bring a dominant contribution to the metric at large distances, whereas the contribution due to the quadratic H-tem will be dominant in the proximity of the gravitational source. It is in this * '['0 be submitted for publication, ** Permanent addrnns: Department of Physics, University of Khartoum, -2- Khnrtoum, f;udan. being the affine connection. The semi-colon signifies the covariant proximity that gravity is strong and possibly quantum effects may be prevailing. Thus, singular features of spacetime metric are very much connected to the differentiations, i.e. choice of linear R Lagrangian while a quadratic R term can be thought (7) of as representing possible quantum effects. and In this present paper, being interested in strong gravity region,we ignore the diminishing influence on the metric by the linear R term of the (8) Lagrangian. Thus, we consider a special case of the form (1) by setting a = 1, 0 = y = O,viz. and the invariant D'Alerobertian is defined by (9) Such this form of Lagrangian and other quadratic forms have been strongly criticized "by Bicknell [8] and by Buchdahl [9] who regarded theories based We consider the following invariant proper time interval using spherical on these forms as nonviable! The points they raised against these Lagrangians coordinates, i.e. are. i) the metric based on them does not satisfy flat space Unit at , do) asymptotically large distances; which represents a static isotropic metric where we denoted ii) when matter term is incorporated to them this leady to disagreement with observations. Being confined to a region of strong gravity In the vicinity of the source, these two objections are considered nonvalid or at least irrelevant to our present situation. (11) 2. H -GRAVITATIONAL EQUATIONS IN THE STATIC ISOTEOPIC METRIC and fi 2 The following gravitational equations based on the Lagrangian ^L = R -0 for (12) can be obtained; and owing to [k] e» (13) where 2 and R are the metric tensor and the Ricci tensor respectively (IV UV By using (11) we calculate r for ]i,v,X = t,r,0,Q3 and the defined by following relation can be obtained: and A 37 r BdtrJ . (5) Also by contracting R by g we get with (15) =2 k^r + 3, (6) -U- -3- This yields R as a function of only r, i.e. By virtue of these equations we can obtain expressions for the metric (16) coefficients g = A.(r) and g, . = -B(r) in terms of r, R and & . rr ^^ This means that the covariant derivatives vill vanish for all values of x To achieve this we proceed as follows; We use (lit) In (20) to obtain other than r. He have (23) Also, from (19), one gets, and (17) (2!*) Flence, by (j) and (8) and the values of r one can obtain; Then by substituting E from (23) in (2k) we obtain uv ' h+B —kf Br.-^ A B Tifr+Zj • (25) •p R=s'h Inserting the expressions for — from [2k) into (25) gives (26) and and by (25) and (2M one gets >>/;i* = R>,m = -rjo k.-o, (18) R£ 6 rRB_R_£_A (27) We now return to Kq.(3) which constitutes a set of l6 differential equations out of which only 3 are independent, while all other components of H or, by eliminating § from the right-hand side of (27) by {2U) it yields, identically go to zero. Thus by (ll), (17) and (18) we will have, (28) B IRA \ R R r • (19) Moreover, since R = I and in view of (11), Eq.(29) will lead to (29) Hfr+ (20) and henc e (30) -£ (SI) Now by eliminating the term j-=r — by (26) we vill have (22) (31) Then using {2k) in this equation yields the following expressions for the metric coefficients g and g in terms of r, R(r) and R(r), i.e. rr tt -5- -6- The integration of (39) yields the following expression for the scalar curvature, i.e. (32) and by where t=- Btr) = -c r"V*A en (33) where C is an arbitrary constant. Furthermore, by (3), (ll), (l't), (15) and (19) we can calculate the vanishing combination, -b- which leads to with a, b, d, p, X, X and X being some constant parameters which may be, -1 generally, complex. (31*) AA = Finally the substitution of (39), and (ltl) in (32) and (33) yields expressions for the metrimetric coefficients g and g as functions Also, Eqs.(30) and (25) lead to of only the radial distance r. (35) If we assume that the curvature can "be constant, i.e. R = R , then, 3. CONCLUSION 2 in this case, Eqs.(3^) and (35) will respectively yield, The afore-resulted metric hased on E -Lagrangian, describes the strong gravitational field at finite differences from the gravitational source. Therefore, no statement can "be made about the flat-space limit. The R -metric (36) diminishes to zero at a relatively short distance from the gravitating centre, where, possibly, a linear Lagrangian could take over to secure the flatness of and space at asymptotically large distances. This metric exhibits no singular behaviour at Schwarzschild's radius or at any finite distance from the centre, (3T) and the constant parameters can be adjusted to avert any possible occurrance of singularity. The nature of these parameters, which are, generally, complex, where k and q are constants. Then assuming that within a small range of may be revealed if, perhaps, a quantum version of this model can be developed. r, > Being complex, they will cause the metric to have an imaginary part in the (38) strong fieia domain. This allows these parameters to be thought of as indicating a classical limit of quantum gravitational quantities. therefore by (33), (36) and (37) we obtain We introduce the concept of the completeness of R{r) to be an access to quantum gravity, the formulation of which, generally, includes (39) imaginary parts. By doing so, we were guided by the fact, that any correct classical theory must be quantizable. We expect the result ve obtained would where K is a new constant. revive interest in the gravitational theory based on quadratic Lagrangians of the scalar curvature and its contractions. -7- 11 a fl'j —X.. EEFERENCES ACKHOWLEDGMENTS The author is deeply grateful to Professor D.W. Sciama, Dr. J. Barrow [1] G. Stephenson, J. Phys. A: Math. Gen. 10,'2 (1977) 181. anil Dr. A.E.G. Fituart for useful suggestions. He would also like to thank [2] C. Lancsos, J. Math. Phys. 10, 6 (196l) 105T. Professor Abdus Salam, the International Atomic Energy Agency and UNESCO Cor hospitality at the International Centre for Theoretical Physics, Trieste, [3] C.tf. Kilmister, Les Theories Relativistes de la Gravitation. (Paris, Centre National de la Recherche Scientifique, 1962). [U] C.H. Yang, Phys. Rev. Lett. 33. U971*) Ui+3.
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