BLACK HOLE SOLUTIONS WITH TORSION By Shaun Andrew Culham B. Sc. (Honours Math and Physics) University of British Columbia. Vancouver, B . C . , 1993 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR T H E DEGREE OF M A S T E R OF SCIENCE in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1999 © Shaun Andrew C u l h a m , 1999 In presenting this thesis i n partial fulfillment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the L i b r a r y shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics T h e University of B r i t i s h C o l u m b i a 6224 A g r i c u l t u r a l R o a d Vancouver, B C , C a n a d a V 6 T 1Z1 Date: Abstract Classical general relativity theory is torsion free. However, since general relativity and general relativity w i t h torsion are experimentally indistinguishable at this time, it is important to explore the consequences of torsion through its physical effects. We do so by considering an important class of static spherically symmetric solutions, analogs of the Schwarzschild solutions which are at the foundation of most weak limit tests of General Relativity. We first show that in general spherically symmetric black holes w i t h torsion exist; these are the torsion analogs of the Bowick et al. [30] black hole solutions w i t h axionic charge. We next search for more general solutions w i t h non-vanishing torsion field exterior to the black hole. We specialize to spherically symmetric static metrics and spherically symmetric static spatial torsion fields. We find that i n certain cases such a solution exists; however falloff conditions on the torsion greatly restrict the form of these solutions. 11 Table of Contents Abstract ii Table of Contents 1 2 3 4 iii Introduction 1 1.1 Overview 1 1.2 Historical Overview 2 1.3 T h e Effects of Torsion 5 Development of General Relativity with Torsion 8 2.1 The Covariant Derivative Operator 8 2.2 T h e Curvature Tensor 11 2.3 T h e Einstein Equation 15 Solving the Einstein E q u a t i o n with a Special F o r m of Torsion 17 3.1 Special form of T b 17 3.2 De R h a m Cohomology Groups 19 3.3 Schwarzschild Black Holes w i t h Torsion Charge 21 3.4 A Generalizing Theorem 22 a c Solving the Einstein E q u a t i o n with a General F o r m of Torsion 25 4.1 General form of T h 25 4.2 Solving the Einstein Equation: Case A a c iii 30 5 4.3 Solving the Einstein Equation: Case B 34 4.4 Solving the Einstein Equation: Case C 36 4.5 Solving the Einstein Equation: Case D 39 Conclusion 43 Bibliography 45 A C a l c u l a t i o n s for C a s e A 48 B C a l c u l a t i o n s for C a s e B 55 C C a l c u l a t i o n s for C a s e C 60 D C a l c u l a t i o n s for C a s e D 67 E The Christoffel symbols 72 iv Chapter 1 Introduction 1.1 Overview T h i s thesis concerns a general relativity theory that includes torsion. T h e torsion tensor is defined by the equation (V V a f c - V V ) f = T where / is a scalar function and V b a a c a f t V J = 0, is a covariant derivative operator. (1.1) B y way of introduction I shall give a short explanation of why torsion was excluded i n the early development of general relativity theory. I shall then review some of the effects of torsion discussed i n the present literature. In the body of this thesis we shall look for black hole solutions w i t h torsion. T h i s was motivated by the observations of Bowick et al. [30] and work done in my undergraduate thesis [31] which showed that the Schwarzschild black hole can carry axionic charge. D u n c a n et al. [32] go further and suggest that there is a large class of models i n which torsion may be viewed as an axion and hence that one may find black hole solutions w i t h torsion. M y approach is not to identify torsion as an axion as done i n D u n c a n et al. [32], or to attribute dynamics to the torsion field as done i n Einstein-Cartan theory but rather to ask the basic question what are the geometries of spacetime allowing any general torsion field. T h e answer to this question is important as it provides a general constraint on possibilities for such 1 Chapter 1. Introduction 2 solutions, independent of a dynamical mechanism for generating torsion. Therefore, it w i l l apply to a l l theories that contain torsion. To begin, I shall show how the torsion tensor arises naturally from the basic properties of the covariant derivative operator. N o assumptions are made about spin acting as the source for this torsion. I shall then derive the curvature and Einstein tensors w i t h torsion. T h e torsion parts of the Einstein tensor are separated out so that i n the absence of any other sources they may be viewed as the stress energy tensor contributions of a torsion field i n the Einstein equation. T o solve this equation, we shall first look at the special case where the torsion is totally antisymmetric, analogous to the axion case, and then prove a general theorem about solutions of this form. Next we shall look at a more general case of torsion field that is static, spherically symmetric, w i t h purely spatial components. T h e problem of finding a solution w i t h this more general torsion w i l l be first broken down into different cases where separate parts of the torsion tensor, corresponding to different representations of spherical symmetry, are assumed to be zero. F i n a l l y the most general case when a l l terms are considered together w i l l be treated. We w i l l find that the resulting equations can not generally be solved algebraically. We w i l l then analyze the large distance w i t h a series expansion. We w i l l find that although solutions do exist for certain cases, there are only a few. T h i s is because the weak boundary condition that the torsion approach constant behavior as the distance scale goes to infinity greatly restricts the number of solutions significantly. 1.2 Historical Overview Torsion was first introduced into the study of general relativity by C a r t a n [1] i n 1922. Over the years, these ideas have evolved and at present the majority of the literature Chapter 1. Introduction 3 concerning torsion involves the Einstein-Cartan-Sciama-Kibble theory, or EinsteinC a r t a n theory for short. T h i s theory has been reviewed extensively by Hehl et al. [2] [3] [4] [5] [6] and is characterized by spin angular momentum acting as the source for torsion i n the same way that matter is the source for gravitational curvature. Historically, torsion is incompatible w i t h some definitions of the equivalence principle [7] [8]. Weinberg [9], for example, formulates the equivalence principle as the statement that at every space-time point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently region of the point in question, erated Cartesian small the laws of nature take the same form as in unaccel- coordinate systems in the absence of gravitation. T h e existence of a locally inertial reference frame depends on the connection associated w i t h the covariant derivative operator being induced to vanish, which can only happen if torsion is identically zero [10]. We define the connection, C J, a dx^ C J ( X ) by d y^ 2 o Y d ^ h W = ( W i t h this definition if can be shown that if we change from our x different coordinate system £ M 1 ' 2 ) coordinates to a the connection obeys the following transformation equations: <9£ dx 7 a dx dC T 1 dx 2 p Obviously the connection does not transform like a tensor and hence is not classified as one. T h e question that the equivalence principle raises is whether or not a transformation can be found that w i l l induce the connection to vanish. Consider the transformation C = ^ + \cj(x* - x (p))(x? a - x\p)) (1.4) Chapter 1. Introduction We see that £ ( p ) = x (p) 7 1 4 and -<T ^ 1 * -5* ^ 1 Substituting these into equation 1.3 we have dx 2 ± 1 o E q u a t i o n 1.4 can be rearranged so that we have - c(P))(e - f(p)) - o ( \ t - m ) ^ = e - \cjr 3 (i-5) W i t h this equation we can now write dx"f sz-cj(e-e(p))-o(\f-f( )\ ), 2 P and d 2 X d^df \ = C } -o(f-ttp)){a ) Thus i f C 1 is symmetric then a °(Q/3) — °a)3 and putting this result i n equation 1.3 we have Cjttl)=Cj([\ )-Cj{f\ )=Q. v v Thus we see that it is indeed possible to transform away the connection as long as the connection is symmetric. However a general connection is not necessarily symmetric, and i n the absence of any other conditions on the connection this statement does not hold. Torsion is by definition the non-symmetric part of the connection, Tj = -(CJ-Ce:), (1-6) Chapter 1. Introduction 5 which, using equation 1.3 we see, transforms as <9£ dx 7 like a tensor. a dx T In this sense torsion is incompatible w i t h the equivalence principle. For Einstein theory without torsion the condition that the connection is symmetric is embodied by the metric compatibility condition: V <? a 6c = 0 (1.8) Thus considering gravity theory w i t h torsion field amounts to relaxing this condition on the connection. 1.3 1 T h e Effects of Torsion Test particles obey the geodesic equation dV^ dx»d*> dr ^~d7~oV-^ x _ 2+C ( Since torsion is defined to be the antisymmetric part of the connection, C^, L 9 ) the path of a test particle is not changed by torsion which makes it difficult to design experiments that detect torsion. Test particles w i t h vector quantities such as spin, w i l l however, be effected. T h e directional derivative of a vector, f) P v uVv a b b =u — a + Cju v\ a • (1.10) is for example sensitive to the non-symmetric part of the connection. T h e Stanford gyroscope experiment [11], designed to measure the precession of a gyroscope i n orbit *As an aside, it can be noted that in the context Einstein-Cartan theory, which is not dealt with in this thesis, the author P. von der Heyde's [10] does present arguments leading to a different interpretation of the principle of equivalence. Chapter 1. Introduction 6 around the earth, was considered as a candidate for measuring a torsion effect [12], but it has since been shown that the experiment w i l l not be sensitive enough to detect a precession due to torsion [13]. T h e authors Hayashi et a l . [14] and P e i [15] investigate the spin precession of elementary particles w i t h intrinsic spin, moving i n a spacetime w i t h torsion. T h e first authors use the W K B approximation to derive their equations of spin precession, while the latter assume a general functional form of the Lagrangian and use variational methods. In both cases the spin precession equations differ from those found i n general relativity without torsion. Jurgen Audretsch [16] has used the W K B l i m i t of a noniterated Dirac equation i n R i e m a n n - C a r t a n spacetime to show that, in principle, torsion may be detected by measuring the spin precession of a massive s p i n - | particle or by measuring its orbit i n a Stern-Gerlach type experiment. Mario Seitz [17] [18] has looked at the charged massive P r o c a field, or spin-1 test particle, i n a spacetime w i t h curvature and torsion. A g a i n , using the W K B approximation, the equations of motion of the spin vector are derived and it is shown that, i n contrast to the Dirac test particle, the P r o c a field is sensitive to a l l irreducible parts of the torsion tensor. Ricardo Spinosa [19] has applied the same analysis as Seitz to the spin| test particles and discusses the possibility of measuring torsion using this field and spin precession experiments. Other torsion effects include the rotation of the plane of polarization i n photons [20]. Wolf [21] suggests that in the presence of a strong magnetic field such as that found i n the atmosphere of a pulsar, photon dispersion can take place due to the interaction of the electro-magnetic field and torsion and this might possibly serve as an astrophysical test for torsion. T h e authors de Sabbata et a l . [22] suggest the possible rotation of the polarization plane of polarized neutron beams when torsion is introduced i n a gravitational field. T h e general theory of the influence of spacetime torsion on neutron interference experiments is presented by Jurgen Audretsch and Clause L a m m e r z a h l [23]. J . A n a n d a n and B . Lesche [24] show that using the phase Chapter 1. Introduction 7 shifts i n neutron interference due to the gravitational field containing torsion they can, in principle, place an upper limit on torsion of about 1 0 ~ m . 7 _ 1 Various cosmological consequences of torsion have also been discussed. Andrzej T r a u t m a n [25], for example, has conjectured that spin and torsion may avert the singularities of gravitational collapse and cosmology. De Sabbata [26] shows a connection between torsion and elementary particle physics i n the early universe. J . Tafel [27] has studied homogeneous models of the Universe filled w i t h a spinning fluid i n framework of the Einstein-Cartan theory of gravitation. T . Fujishiro et al. [29] discuss the possible role of torsion i n the universe i n the context of cosmic string theory and H . Nieh [28] suggests that torsion may play a part i n the search for a successful quantum theory of gravity. Chapter 2 Development of General Relativity with Torsion 2.1 T h e Covariant Derivative Operator Following W a l d , [33] we define a covariant derivative operator V on a manifold M a as a map which takes each tensor field A G T(k, I) to a new tensor field which is an element of T(k, I + 1). In addition it must possess the following four properties: 1. Linearity: For a l l A , 5 G T{k, I) and a and j3 G 71, V (aA -- \ .. a c v /?B -"V. ) fll + a bi 6| = aV A - " ... + 8V B - V... fll ai a c 6l 6j j a CJ 6| 2. Leibnitz rule: For a l l A G T(k, /), B G T(k, I), v [A----v. £ '' V..J CR e = C 6 V [^- * ,.. f l e 6 ]B c l 6 l C t d l ... d r + A f l l - * ... f l 6 l 3. C o m m u t a t i v i t y w i t h contraction: For all A G T(k, I), Y7 / Aai — c—a k \ _ 8 y7 Aai--c—a k V [B c l 6 | e C ( d l ... ] d r 9 Chapter 2. Development of General Relativity with Torsion 4. Consistency w i t h the notion of tangent vectors as directional derivatives on scalar fields: For a l l / G T and all t e V a p *(/) ^ a f = In Einstein's General Relativity theory we also have the "torsion free" condition that for a l l f eJ , 7 (V V a where T° ab - V V ) / = T VJ = 0 c b 6 a ab (2.11) is the torsion tensor. For the purposes of this thesis we shall not impose this condition and shall look at the resulting changes i n General Relativity. T o that end let us consider two arbitrary covariant derivatives V and V . B o t h obey the above a a conditions except that we do not require that they be torsion free. T h e difference of these two operators w i l l be linear for a l l A, B e T(k, I) and a and /3 G 7Z by property 1 for the covariant derivative. Using the Leibnitz property and the fact that the operators must agree on scalars i n order to be consistent w i t h property 4 above we have (V -V )(M) f l a = ( V / ~ V f)u = 0+ a a + f(V U A b B V LO ) A B / ( V - V H a (2.12) a The importance of this result is that it shows that ( V — V ) defines a tensor of type a a (1, 2). T o see this explicitly we first note that at the point p the quantities V co and a b V c j ; , depend on how co changes as one moves away from p. T h e quantity ( V — V )c<;;, a b a a however only depends on the value of uj at p. To prove this we consider two different b covectors u> and ui such that, b b (u) - oj ) \ = b b p 0. Given this condition we can find smooth functions, f , which vanish at p and smooth a covectors e such that: b (u - uj ) = b b f e°. a 10 Chapter 2. Development of General Relativity with Torsion If we now apply E q u a t i o n 2.12 we get = V (/ e = /a(V -V K V (u -u; )-V (Cj -cu ) a b b a b b a a a 6 )-Va(/ e?) a a a = o, since each of the f a = 0 at p. Rearranging the left hand side of the above equation we get our final result ( V - V )uj = ( V - V H a a b a • a We have shown that (V — V ) defines a map of covectors at p to tensors of type a Q (0, 2) at p. Thus (V - V > for some C c ab = C uj , (2.13) c a 6 ab c e T(l,2). If we let T c ab be the torsion tensor defined by V then a W = T = c ab VJ (v v -v v )/ a 6 6 = v v / +C a = b a VJ + (C T VJ c ab - c ab c ab - (V V / + 6 a C V f) c ba c C )VJ c ba T h i s gives us the result that T c ab =T c a 6 + (CV - C ) c (2.14) 6a If we set V a =d = 5 a ( Chapter 2. where the ^ Development of General Relativity with Torsion 11 are just the standard coordinate basis vectors, we can rearrange equa- tion 2 . 1 3 and write W co = d u - C u . c a b a b ab (2.15) c C o m p a r i n g this to the standard result from General R e l a t i v i t y where we write S7 cvp = d (j/3 - TJUJ a a (2.16) 7 using the Christoffel symbol T J, we see that by the conventions used here C J = T J. A W h e n we set V A A = d and work i n a coordinate basis the torsion tensor, T , vanishes c a a ab since p a r t i a l derivatives commute. T h i s , taken together w i t h equation 2 . 1 4 , allows us to write Tp (T p = a ~ FgJ)- a (2-17) W h i c h is to say that when working in a coordinate basis the components of the torsion tensor may be expressed in terms of the skew symmetric components of the Christoffel symbols. 2.2 T h e Curvature Tensor We shall now consider the curvature tensor R f defined i n Penrose and Rindler[34] by ab ( V V a b - V f c V In terms of our covariant derivative V Rabc t d C T W )t c - a a = (VaV = d ab c R t d c abc we write 6 - V 6 V a - f V )t C ab d In order to express this i n terms of our covariant derivative V / € T such that / = t w and write b b 0 = (V -V )/ a a (2.18) c a we must first choose a Chapter 2. Development of General Relativity = (V - V )t W = [(V -V )t ]w = \{y -v )t with 12 Torsion B a a B + b a a b b a a b + t c }u b a t (V -V )w c a b ac b T h i s implies the desired result (V - V )t = -Cjf b a a (2.19) We can now begin to expand equation 2.18. First consider f Vt c = d ab c f (V t -C t ) c d ab = lT + c d e c ce (C -C )}(V t -C t ) c ab c ab d ba d c (2.20) e ce Now expand the first term on the right hand side of equation 2.18. v v</ = v v t + c d a a b = vt c ab d c - C Vt d a c b V {V t -C t ) d a +C b (V t c ab d c b - Cj?) - C {V t d d c a - c b C H) e (2.21) be B y switching the a and b indexes i n this equation we can write the second term on the right hand side of equation 2.18 as v \/ t = d b a V (V t -C t ) d b ac + C (V t c - C f) d ba d c a - C (V t d c d ce bc - CJf) c b (2.22) P u t t i n g all of these results together we finally rewrite equation 2.18 as R t d c abc = (v v - v v )t Q + (c b b c i d ba c - Cji?) /si dsi c si dsi c \ j . e l a c be ~ bc ° a e ) ~ U U U + v (c t ) d - C )(V t c ab + - va{cb f) d a l 1 rp ab d b - C Vt d c ac V c ac b + C Vt d bc cry ±d • rp csi d+e ^ + ab ^ce 1 1 c a Chapter 2. Development of General Relativity with 13 Torsion -(C -C )(V t -C t ) c c ab E> d ba d e c ce i rp d+c + (CjC de c/~i fry + d s~t XI &\+ c (2.23) - C CJ)f c d be b w i l l vanish and equation 2.23 If we again set V = d then the curvature tensor, R , d a C a abc yields r> Y7 s~i d d -K-abc — Va<^bc \j s~i d s~i b ^ a c ~~ ° a ~~ e d/~i e °fcc y , s~i e d/~i ~+~ W e ° « c rp ~ e/~~i d tc\ ^ec ab 1 T a k i n g components of this equation i n a coordinate basis where C J = T J and setting a T c ab a — 0 we have R py ~ a ^aCpJ a — a a S p OL — V' pC ^ — C fCp* p ry , p -f 1 5 p Q £ 1 + a e p p 1 Q/ 1 ~~ . ^ ^ 0 * — r ^ / r ep gl <5 Cp C * 6 e p ep 1- Q 7 £ 7 l ^ a 5 e + r ^ r ^ + r^r^J 5 a 7 - rjr-^ + (2.25) T h e last two terms cancel w i t h the forth and eighth terms. Since T ° = 0 we have ab TJ = rj - =o (2.26) which results i n the cancellation of the t h i r d and sixth terms. T h i s leaves us w i t h the classical equation for the components of the curvature tensor i n general relativity. Namely RapJ — dpT ^ + T ^Vp^ — Tg/r Q 7 = a = R p^ a a (2.27) For the purposes of this thesis we would like to express the general curvature of equation 2.24 i n terms of a torsion free covariant derivative, V a and this classical torsion free curvature, R pJ 5 • To begin we expand equation 2.24 using our previous definitions a -xP-y r> M-abc d — U o r~\ d a^bc , f~i °oe df~i e ° 6 c s-i — °ab e/~i °ec d — es~< d °ac ° 6 e Chapter 2. Development of General Relativity si d b^ac d si ~~ with dsi e ° a c i °6e 14 Torsion si esi d . si ° e c "+" esi d ° a e °6a si dsi e I si dsi e rp esi d U ^ae 6 c "+" W e ° a c ~~ ab ° e c ~ 1 After the cancellation of a few terms we are left w i t h r? a d rt-abc — si ab ~ si d , si °b^ac (si e \ ab ~ ~ Since T ,a d °a^bc U dsi ae si e\si d ba ) ec ~ U U e si ^bc U 1 dsi ~~ 6 e e ° a c u rp esi d ab ° e c = 0 equation 2.14 implies that T = c ab - { C a b - C c b a (2.28) ) Plugging this result into the above equation cancels the last two terms so that = 9C Rabf a - dC d bc b +C C d d ac ae - C CJ e (2.29) d bc b The connection can be separated into its symmetric and antisymmetric parts by writing Cb i^Pab = a C °) + + ba ^{C — a b C °) ba 2 ab Substituting this into our equation for the curvature we have p ^-abc d F ^ 1 r) ~~ a u L a p be b u ~ \(d T a ~ d i ac L p ' d p ae 1 1 - dT ) d b \(^ae T d e b c + d p be d ac a T T e bc be p L + \{T T ; d bc e d ae b - T TJ d be 1 e ac - T TJ) d be - TJT ) d be (2.30) The first four terms make up the standard torsion free curvature while the remaining terms result when torsion is present. T h i s expression w i l l be simplified further. Since V a is torsion free we have Vu a b = du a b - T uj ab e (2.31) Chapter 2. Development of General Relativity with Torsion 15 which implies that VT a = d bc dT a d - TJTJ bc - TJT +r T d D ae BE e bc (2.32) Switching the a and b indices i n this equation we also have V rp O d bJ-ac rp d p — b- ac (J L erp ba 1 1 d p ec 1 be erp i d ae 1 ' p drp be 1 e (ry q q \ ac 1 These two equations can be rearranged so that we have dT a = vT d a bc + rjTbe d + TJTJ d bc - r a % e (2.34) e and a rp d °bJ-ac X7 rp d —Vb-l-ac . p + erp ba 1 1 d ec p 1 + 1 be erp ae 1 d ~ p drp be 1 1 ac e (n nr\ (2.dDJ These last two equations can be substituted into equation 2.30 and after cancellation of terms our curvature is finally written as r> ^abc d r> — rL ^ fry d a6c rp ^^ a b c +\(T T *-T TJ) d ae 2.3 d bc be d Y7 T b a c d\ ' (2.36) T h e Einstein Equation The Einstein tensor is defined as Gab where R ab = Rab — ^QabR (2.37) is the R i c c i tensor defined as Rab = Raeb 6 (2.38) and R is the R i c c i scalar defined as R = R a a (2.39) Chapter 2. Development of General Relativity with Torsion 16 W i t h torsion present the R i c c i tensor is R ab =R ab - ^(VaT - V T e e e 6 e a b ) + \{TjT * - T T *) d d d (2.40) ab and the R i c c i scalar is R = R - \vdTe * + -TjTr d (2.41) We have lost two terms i n the R i c c i scalar because contraction across the antisymmetric components of the torsion tensor is zero. O u r Einstein tensor is fi ^ab — fi ^ab 1 - „ fvj rp e ^{VaJ-eb - \ g a b { - \ ^ „ v 7 ~ T Vl e e\ T a b ) + i 1 - drp e (T (i a e 1 d - b 1 T d 1 e drp 1 a b + -JJTD de d ) e\ (2.42) In the absence of any outside sources the Einstein equation is G ab =0 (2.43) or 1 f\h ab - rp e V 7 -^{Va-l-eb _ V e -\gab{\v T de d T h i s is the equation that we w i l l solve. rp e \ ab ) ~ J 1 frp ^K 1 ae - - T J T D rp drp e 1 db ~ 1 de drp 1 ab e\ ) (2.44) Chapter 3 Solving the Einstein Equation with a Special Form of Torsion 3.1 Special form of T& a c Bowick et al. [30] found a new axionic black hole solution. The same approach w i l l be used here to find black hole solutions to the Einstein equations w i t h torsion. We shall consider the special case where the torsion tensor is totally antisymmetric. Switching to the language of differential forms we define a 3-form, T = T dx a aPl A dx A dx , 13 (3.1) 7 and a potential 2-form, B = B dx A dx a (3.2) 13 aP such that T = dB where d is an operator called the exterior derivative. (3.3) If we let Q ( M ) be the set of P all p-forms on M , where M is Minkowski space time and p is any integer such that 0 < p < 4, then the exterior derivative is defined as d: W{M) -*tt (M) p+l 17 (3.4) Chapter 3. Solving the Einstein Equation such that if A G Q?(M) and 77 G Q (M), 18 0 < p, q < 4 then q (i) with a Special Form of Torsion d(X + r]) = dX + drj (3.5) (ii) d(X A 77) = dX A 7] + ( - 1 ) A A cfry) (3.6) (iii) ddX = 0 VA G Q ( M ) , 0 < p < 3 (3.7) (iv) dX = 0 VA G (3.8) (v) P P ft (M) 4 V/ G df = fi°(M) (3.9) It should be noted that 0 ° ( M ) is just the set of all infinitely differentiable functions on M . T h e dx a encountered in (3.1) are called basis 1-forms u . a UJ° = dt to = dx 1 UJ = dy a; = dz 2 3 Explicitly (3.10) T h e wedge product, A , is defined as A : (Q (M), Q ( M ) ) -> Q (M) P 9 such that if A G fi (M) and rj G Q, (M) p q A A TJ = then (-l) riAX pq = (3.11) p+q p + g<4 (3.12) 0 p + q>4 (3.13) F r o m which is follows that A A A = 0 and for our 1-forms to 01 u> A LO^ = —cu^ A u> a a (3.14) We also see that a 0-form wedged w i t h a 1-form results i n a 1-form. We shall now show there exists a special case where T = 0 and we have a Schwarzschild black hole solution w i t h non-zero torsion charge. Chapter 3. 3.2 Solving the Einstein Equation with a Special Form of Torsion 19 De R h a m Cohomology Groups We shall be using De R h a m cohomology groups when we put torsion charge on a black hole. To begin we make a few definitions. Let M n be a smooth n-dimensional manifold. D e f i n i t i o n 1 A p-form to G Q (M ) is closed if du = 0. D e f i n i t i o n 2 A p-form to G £l (M ) is exact if to = d'j for some 7 G p n p D e f i n i t i o n 3 Ker{Vt (M ) p 4 n n Q (M )) p+1 n VL ~ (M ). p l n is the set of all to G Q (M ) such that to is is the set of all to G Q (M ) such that to is p n closed. D e f i n i t i o n 4 J m ^ ' f M ™ ) ) -4 Q (M )) p n p n exact. D e f i n i t i o n 5 The p De Rham cohomology group is the quotient group defined by th H P { M n ) KermM-)^^\M-)) = Im(Qp- (M ) l H°(M ) = Ker(Q°(M ) n 4 n A p -4 n Qp(M )) n Q (M )) l n cohomology group can be thought of the set of a l l equivalence classes of closed th p-forms. We can denote an equivalence class by u> = co + dT where aT is used to denote Im(Q - (M ) p l 4 n The p th l p n A particular element co' G Q would be LO' = to + dj where x - OJ if U\ = toi + dj for some 7 G 2 fi (M ). p_1 n De R h a m cohomology group can be characterized as being t r i v i a l or non t r i v i a l . To say that H (M ) p class i n H (M ) p 7 G n We would say that u 7 G VL - (M ). p tt (M )). n fi (M ). p_1 n n is t r i v i a l , denoted H (M ) p n = 0, means that the only equivalence So that if u G 0 then to = 0 + dj is 0 = 0 + dT. In other words H (M ) p n = dj for some = 0 means that every closed p-form is exact. Chapter 3. Solving the Einstein Equation with a Special Form of Torsion A n application of this can be seen i n electromagnetism. Since H (R ) 2 20 = 0 we have the A theorem: riF = 0 ^ F = dA (3.15) which is just the generalization of the familiar theorem i n vector calculus vxi W h e n we say that H (M ) p p i = vy. is non trivial, denoted H (M ) n there exists LO 6 Q (M ) o^ = p n ^ 0, we are saying that such that co is closed but not exact. In working out our torsion n black hole solutions we shall need to show that we have a non t r i v i a l cohomology group. In proving this we shall appeal to a couple of theorems. Before these theorems can be stated a few more definitions need to be made. D e f i n i t i o n 6 Given two smooth manifolds M n —> N smooth F : M n x [0,1] ->• N m and the smooth maps f, g : m such that F(x, 0) = f(x) D e f i n i t i o n 7 Given two smooth manifolds m and N n then f is homotopic to g, denoted f ~ g, if and only if there exists a m alent to N , M denoted M n ~ N, M n and F(x, 1) = and N then M m n if and only if for f : M m n we have fg ~ 1 and gf ~ 1 where 1 is the identity N m g(x). is homotopy equivand g : N m map. We can now state the following theorems: T h e o r e m 1 If M ~ N T h e o r e m 2 If M is a compact orientable n-manifold n n m then H (M ) p « n H (M ) n n H (N ). p / 0 m without boundary then -> M n with a Special Form of Torsion 21 The last theorem is easy to prove, just let e be the volume n-form on M such that Chapter 3. Solving the Einstein Equation n J e — V where V is the non zero volume of the manifold. Mn H (M ) n = 0 then there exists 7 <E Q ~ (M ) n n l If we assume that such that e = dj. n T h e Generalized Stokes' theorem now tells us that since M n 3.3 has no boundary. T h i s contradicts J e = V so H (M ) n Mn 7^ 0. n Schwarzschild Black Holes with Torsion Charge If T = 0 then the torsion tensor, T , is zero and the Einstein equations 2.44 have abc the standard Schwarzschild black hole solution. T h i s solution has topology J J x 5 . 2 F r o m the above theorems we know that since K x S 2 H (S ). 2 2 H (S ) 2 2 Since S 2 ~ S 2 2 we have if (5R 2 2 2 x S ) 2 is a compact orientable 2-manifold without boundary we know that ^ 0. W h i c h means that there exists a closed 2-form on S 2 that is not exact. E x p l i c i t l y let B = (3.16) 4irr 2 where e is the induced volume 2-form on the 2-spheres of spherical symmetry i n the Schwarzschild solution. Since there are no 3-forms on S 2 T = dB = 0. we automatically have (3.17) Also .2 Qtor 9 (3.18) Chapter 3. Solving the Einstein Equation with a Special Form of Torsion 22 where q tor is the torsion charge carried by the Schwarzschild black hole. Thus we are led to the following solution for a static black hole of mass M and torsion charge q ' tor ds = - ( 1 - — )dt + (1 - —)~ dr r r 2 2 l g _ + r (d9 2 2 2 + s i n 6d<p ) 2 (3.19) 2 Qtor(_ ( 3 2 Q ) 4.7rr 2 T h i s torsion charge on the black hole could, i n principle, be detected i n the way suggested by Bowick et al.[30] w i t h an A h a r o n o v - B o h m like experiment measuring interference using a string world sheet nontrivial i n H (R 2 2 x S ). It should be noted that 2 if H (^R. x S ) was t r i v i a l then any closed potential 2-form B we chose would be exact 2 2 2 and there would be a 1-form 7 such that B = dj. T h e Generalized Stokes theorem would then give I = / cty = /" 7 = 0 Js Js Jas Thus we see that a t r i v i a l second cohomology group implies zero torsion charge. B 2 3.4 A Generalizing 2 2 Theorem The results i n the last section used the fact that 3.16 was a generator of the second cohomology group a n d its explicit formula to.construct torsion solutions. Also, the explicit realization of (3.16) used the spherical symmetry of the metric. T h i s means that the technique used can not be generalized directly. For example, one would not be able to find a multiple black hole solution w i t h torsion using the technique i n the last section because such solutions are not spherically symmetric. However, using De R h a m cohomology directly a very general family of solutions to the generalized Einstein equations w i t h torsion can be given. T h e other generalization is that the spacetime w i l l not be required to be static but only stationary. T h i s means that the spacetimes considered include the previous case but also include spacetimes without Chapter 3. Solving the Einstein Equation with a Special Form of Torsion 23 the time reversal symmetry. Examples of spacetimes without time reversal symmetry included spacetimes with rotating objects such as stars and black holes. T h e o r e m 3 Given any spacetime M a stationary spacetime which satisfies G b = a 87iT b and H (M) 2 a ^ 0 then there is torsion solution with G b = 87rT;> and torsion a dB = T with B any non trivial element of a H (M). 2 First, the spacetime is stationary i f the metric can be written i n the form ds = -V dt 2 2 + 2N dtdx 2 i i + h dx dx { j i3 (3.21) where the index i only runs from 1 to 3 and none of the components depend on t. The metric (3.21) reduces to a static metric when i t has the extra symmetry that it is invariant under t —, —t, or equivalently that Ni is identically zero. For any constant value of the time t, the level surface is a 3-manifold E and because there is no time dependence i n the metric (3.21), this spacetime is globally K x S . Furthermore, any field defined on E for given value of t w i l l not change at some other time ti. 0 The proof of this theorem is as follows: First, K x E ~ E because the tractable. is con- T h i s means that H (^R. x E ) « H (E) by Theorem 1 for D e R h a m coho2 2 mology. Let (3 be a non-trivial element of H (E). 2 T h e n (3 is closed but not exact by definition. Since (5 is defined on E , it does not depend on t. Furthermore, it has no component i n the time direction because it is defined entirely i n terms of E . Thus, it can be extended to the spacetime 3? x E trivially so that it has no time components. Namely, defining the components of a 2-form B i n the following way B^ = ^for i,i/ = l 2 3 A J ) (3.22) and zero otherwise. T h i s is the pullback of (3 v i a the projection m a p of 3? x E onto E. F r o m its definition, dB = 0 and it is not exact. Since i t B is not exact i t w i l l Chapter 3. Solving the Einstein Equation with a Special Form of Torsion 24 have nontrivial torsion charge when integrated over a 2-surface. Finally, T = d B = 0. Therefore, one still has a solution to the Einstein equations when the torsion charge is added. Observe that the theorem could have been stated i n terms of the spatial topology alone because the spacetime is $Rx E . Thus, the condition that H (M) ^ 0 is equivalent 2 to H (Y) 2 ^ 0. Finally, an example of the new solutions given by the above theorem is given. In general there are multiple black hole solutions to the Einstein equations which are stationary and have the explicit form ds 2 = -e {dt + A^dx") + e~ {dx + dy + dz ) 2U 2 2 2U 2 2 2 (3.23) where U and A^ represent the potentials for a n black hole solution w i t h electromagnetic charge and angular momentum [35] [36]. T h e spatial topology outside n black holes is just that of 5R minus n balls. T h i s 3-manifold has non-trivial second coho3 mology because one can not contract any 2-sphere which surrounds one of the voids produced by the removal a ball from K . Therefore, it follows that we have solutions 3 to the Einstein equations w i t h n torsion black holes. Chapter 4 Solving the Einstein Equation with a General Form of Torsion 4.1 General form of The Schwarzschild solution to the Einstein equation is found using the assumption of spherical symmetry. W i t h this i n m i n d we wish to find the most general form of the T abc tensor using the unit norm basis covectors r , 0 , and cp that is spatial, spherically a a a symmetric and antisymmetric on the first two indexes. 9 a = Va a n d (p = a a- — x If we choose r a = z a then A simple rotation in the xy plane that does not change the length of the vectors is accomplished by the transformation 6 — a cos X9 + sin \<p a (4.1) a 4> = — sin X6 + cos Xcp a a a r =r a (4.2) (4.3) a T h i s can be inverted so that 6 = cos X6 - sin X4> a a a 4> = sin X6 + cos a a 25 A0 a (4.4) (4.5) Chapter 4. Solving the Einstein Equation with a General Form of Torsion 26 r = f a (4.6) a If our tensor is form invariant under this rotation it w i l l be rotationally symmetric and since we can perform this rotation around the r = x a and r a a = y axes as well, our a tensor w i l l have the desired spherical symmetry. To begin we w i l l start w i t h the r vector which is trivially invariant under our a rotation. If we choose = rrr Tabc a b (4.7) c we w i l l not have the necessary antisymmetry on the first two indexes. If we now try building our tensor using two r vectors we w i l l have something of the form a Tabc = (r 0 Tabc = a (4.8) O r )r ~ b a b c or {r <t>b ~ a (4.9) ^ rb)r a c Neither of these has the desired spherical symmetry on its own, nor does a linear combination of the them. O u r next attempt w i l l contain only one r a vector. means our tensor w i l l be made up of the following pieces: #a</>6 — sin X(p ) (sin X9 = (cos X9 — cos A sin X8 9 a a a b + cos 2 b — sin X9 cp a \(f) ) + cos b b 2 — sin A cos X(p 9 a b X(j) (j) a b and 9b<t>a — 9a,9b ( A6*(, c o s — sin = cos A sin X9 9 = (cos = cos b — sin X9 a 2 a X9 9 a b — A^) + (sin cos 2 Xc} ) (cos a X9 + a cos X9b<f>a — b cos A sin X9 (j> a sin — sin X9 b — Xcfia) 2 X(p 9 b a — sin A cos Xc/) (fi b a X<p ) b sin A cos X(p 9 a b + sin 2 X(f> (fi a b This Chapter 4. Solving the Einstein Equation with a General Form of Torsion (pa^b = (sin X9 + cos A</>) (sin X9 + = sin + cos A sin X(p 9 a a 2 + sin A cos X9 9 a b b X9 (p a b cos X(f> ) b a + cos b 2 X<f) <f) a b L o o k i n g at these terms and how they transform we see that the combination 9 (pb a ~ (paO = b (cos A + s i n 2 2 - (cos A + s i n 2 X)9 cj) a b 2 X)(j> 9 a b is form invariant under our rotation so it w i l l process the desired symmetry. A l s o the combination 9 9b a + fiafib = (cos A + s i n A)(9 # + (sin A + cos = 9 8 2 2 2 a a A)</>0 a 6 }> 4> + b 2 6 a b has the desired form. T h i s means that the T abc T = abc {9 <t> a b tensor could be of the form: - (f> e )r a b (4.10) c The symmetry properties of the other indexes i n the T tensor are not fixed but to abc maintain rotational symmetry the only possibilities are that they are antisymmetric as well. N a m e l y Tabc = M b ~ (pa9b)r + (0 (pc c b - <j>b0c)r (4.11) ~ <PaO )r ~ (0 (pc a c a T h i s case has already been investigated when we considered T abc b = f(r)e . abc Another possibility is that our tensor contains terms such as: T abc = r {9 9 a b c + M c ) ~ r {9 9 b a c + <p <p ) a c (4.12) which has the needed antisymmetry i n the first two indexes. A n y other combination of terms w i t h this form w i l l not process this needed antisymmetry. 27 Chapter 4. Solving the Einstein Equation with a General Form of Torsion 28 O u r final possibility is that our tensor does not contain any r terms at a l l . T h e a 9 6 6 a b c and 4)a4>b4>c terms are excluded by the need for antisymmetry across the first two indexes. T h i s leaves terms of the form: (d ±(p d ){e + <p ) (4.i3) (e e ±<p <p ){6 + (p ) (4.i4) a(Pb a b c c and a b a b c c Terms w i t h (9a<f>b - (4.15) 4>aO ) b in them w i l l be excluded since they are invariant under rotation and no linear combinations of the 6 and 4> are. Similarly terms w i t h C c (e 0b + <Pa<l>b) (4-16) a in them are excluded. T h i s leaves 0 {0 (t> + <f> 0 ) M W c + a b c b c - 9 (9 cf> b a c + <j> e ) a (4.17) c and <}>bO ) c cp {9 <j> b a + c (4.18) <j> 9 ) a c which are not form invariant. A l s o 0 (0 9 a b c + Mc) - 0 (9 9 + Mc) - 4>b{8a0c b a c + <i> <p ) (4.19) + Mo) (4-20) a c and <t>a{9 0c b which is not form invariant. Thus there is no combination of just the 9 and (fi vectors a a w i t h the desired form. P u t t i n g this a l l together we see that the most general form of Chapter 4. Solving the Einstein Equation with a General Form of Torsion 29 the T bc tensor is of the form a T abc = + B(r)(Q (pb- A{r)e bc a +C(r)[r (9 9 a = A(r)eabc <pa6b)r a b c + c <p </> ) b + B(r)e r ab c c - r (9 9 b a + c + 2C(r)r h c [a b] <p (p )] a c (4.21) Chapter 4. 4.2 Solving the Einstein Equation with a General Form of Torsion 30 Solving the Einstein Equation: Case A The equation we wish to solve is 1 ab — 2^ fry rp Q Y7 T e eft v -\gab{\v T d e a d e e 1 e \ ^ (T > b J drp ae - -T T d ce c e d 1 db e rp 1 de ) drp 1 ab e\ ) (4.22) To do this we shall break the problem into 4 separate cases. In Case A we w i l l assume a form similar to the found i n the last chapter w i t h T abc where e abc = A(r)e (4.23) abc is totally antisymmetric and spatial: d (4.24) y/^giaB-yS] (4-25) tabc = Zabcdt In a coordinate basis e«/? 5 = 7 where g = det || g || aP (4.26) and [aj3^S\ is the totally antisymmetric symbol. In particular e ^ = e 4 m + Vsin0 (4.27) Calculating the components of the terms on the right hand side is rather lengthy and is done in A p p e n d i x A . T h e results are that the tt component equation is given by \(l-e- 2 n ) + 2-e- 2 n ) - 2-e" 2n = -A . (4.28) = -A . (4.29) 2 T h e rr component equation is 1(1 _ e 2 n 2 Chapter 4. Solving the Einstein Equation with a General Form of Torsion 31 The 99 and (p(p component equations are the same and given by, e- (m + m - mh + — --) 2n 2 = -A . (4.30) 2 The last equation is the 9<fi component, A + 2Am = 0. (4.31) We very quickly see that the 9<f> component equation has the solution A = a e- . (4.32) 2m 0 We are left w i t h three equations i n two unknown functions. T h e left hand side of these equations are identical to those found i n the Schwarzschild case. T h e right hand sides are nonzero and can be thought of as the contributions made to the stress energy tensor for the torsion field. T h e second and third equations can be equated to each other but they can not be equated to each other as is the case w i t h Schwarzchild. T h i s means that a l l three equations must be solved simultaneously. These equations are also non-linear which makes them very difficult to solve. M a n y attempts were made to reduce the equations into a single equation i n one variable without success. In the end an algebraic solution to the three remaining equations could not be found. A different approach was needed, so based on the known Schwarzschild behavior the following series expansion solutions were investigated no e 2m e = m + -^ + ^ 0 -*. = „ 0 1 + ^ 1 - + + + --- = E^ i=0 ' . . . = f3 A s a simplification we can choose a scale such that m (4-33) (4.34) 0 = 1. W e w i l l also use the expansions e-^ = l - ( t ~ ) 1=1 ' +( t ^ ? - - - 1=1 ' (4-35) Chapter 4. Solving the Einstein Equation with a General Form of Torsion 32 - [1-(E5) +( E T ) i=l = m ' - - 1 E - ^ 2 i=l ' i=l (4-36) ' ~E7?rr) + ETirf) —IS 2 i=l z ' i=l ' i=l r>+1 ' + ^ I - I E ^ +IE^) ----]^^^ ^ 2 i=i ' z 1 2n 0 77" i 0 i=i 0 0 77 (4-37) ' \2 „(l-(Er=j) + (Efi3) -"0 -1 e / / V ^ N \ / V ^ , 2 1 n 1 i=i ' °° 77- 0 77- 0 °° (4-38) —777- i[i-(E^i) + (E£j) -"]E7^ 2 («9) The Maple software package was used to substitute the series expansions into the component equations which were first multiplied by r . After dropping a l l terms of 2 order r~ and smaller the tt component equation is 2 —3 3 3 —air + -alrn-ir + [1 - n 2 - a , ( 3 m - 2m )] 1 1 - 2 m - 2 m i ( m ? - m ) ) - + 0 ( —) = 0. 2 Q 3 --ao(4mim 2 2 2 3 (4.40) 2 The r r component equation is —1 3 ~^ o a 1 + 2 o a r 2 -ia (4mim 2 2 m i r + [1 - - 2m 3 2 - ^ « o ( 3 w - 2m )] + [ n m i - n 2 2 m i (m? - m ))]^ + 2 0 0(1) = 0. x (4.41) and finally, the 99 = <f>4> component equation expands to ~^ l + \ l i a r2 a -^a (4m m 2 1 4 " [\ l(^ i m r m a ~ 2 m 2)] + [^(fiomi - n i ) - 2 m - 2 m ( m - m ) ) ] - + 0(1) 2 2 3 x 2 7* = 0. (4.42) 7"^ In order for these three equations to be true for a l l r we must have the coefficients in each term i n each series equated to zero. L o o k i n g at the leading term i n the tt component expansion we see that an must be zero. T h i s greatly simplifies the remaining Chapter 4. Solving the Einstein Equation with a General Form of Torsion 33 equations and we see from the coefficients left i n the tt component that no = 1. If we look at the r r component we see that m\ = n\. T h i s solution satisfies the last component equations and is looking like the standard Schwarzschild solution. Next we considered the series expansion out to order r~ . After the terms of order 2 r - 3 and greater were dropped we again see from the leading term of the rr component that a = 0. T h i s simplifies things so that the tt component is 0 ( l - n ) + n - + O ( - ) = 0. 0 (4.43) 2 Similarly, the rr component equation is (1 - n ) + ( n m 0 0 x n )x r + [ ( 2 n m - n + m i ( n i - n m i ) ] — + O ( ^ ) = 0. 0 2 2 (4.44) 0 and the 88 = (pep component equation expands to 1. .1 A . n - ( n m i - n j - + \-n [m-y ) 2 r 2 no 3 1 In + 2 n m - n - -n m\ + -m n +- — 4 4 2n u 0 x 1 2 0 2 2 Q x 1 ) K + ° ( ^ ) = °- x 0 r ( 4 2 ) r A g a i n , looking at the coefficients from the tt component we see that n now we see that n 4 5 6 = 0. T h e rr component gives us m,\ = n\ and m 2 0 = 1 but = 0. These conditions satisfy the remaining component equation and we have found the standard Schwarzschild solution. Chapter 4. Solving the Einstein Equation with a General Form of Torsion 4.3 34 Solving the Einstein Equation: Case B In Case B we w i l l assume that the torsion tensor is of the form T = B(r){9 (j> - <f> 9 )r = B(r)e r abc a b a b c ab (4.46) c and substitute this into equation 4.22. F i n d i n g the components of equation 4.22 is a very long calculation and is done i n A p p e n d i x B . T h e final result is that equation 4.22 has only five nonzero components. Canceling the metric factors we have the tt component J ^ _( (l l-_e --2 n ) + 2^^ e e 2 n ) + 2 n 2 = 0, (4.47) = 0. (4.48) and the rr component e1 ((1l -- ee -" * )) -- 22" * 2 n 2 2 n The 99 and (p(p component equations are equal and are given by e~ (m + m - rah H 2n )= 0 2 r (4.49) r Lastly, the 9(p component, B + 2B(m + h + -) = 0 . (4.50) r T h e first three equations are the same as those found i n the Schwarzschild problem. T h e solution can be found using the first two equations alone, the t h i r d is automatically satisfied if they are satisfied. T h e solution is ds = - ( 1 - — 2 r )dt + (1 - —)~ dr 2 l r 2 + r (d8 + s i n 2 2 2 8d(p ) 2 (4.51) If the first two equations are subtracted from one another we get m + h = 0. T h i s can be substituted into the 9(p component equation which reduces to B + 2B- = 0 Chapter 4. Solving the Einstein Equation with a General Form of Torsion 35 r dB * Jir r dr = ~ 2 h => \n(B(r)) = - 2 1 n ( r ) + T =* B(r) = r l where T is some scalar function. T h i s gives the final result of a standard Schwarzschild spacetime w i t h a torsion tensor of the form: T Tabc = ^ o.b c- e r (4.52) T h i s torsion tensor can be thought of as an extra field, comparable to an electric or magnetic field, present i n the background Schwarzschild spacetime. Chapter 4. Solving the Einstein Equation with a General Form of Torsion 4.4 36 Solving the Einstein Equation: Case C Assuming that the torsion tensor is of the form Tabc = C{r)[r (9 e + 4 b4> )-r (e 9 = 2C(r)r h a b [a c ) c b a c + <f> 4> )] a c (4.53) b]c the components of equation 4.22 as calculated i n A p p e n d i x C are: the tt component \(l-e- = 2 n ) + 2-e- 2n l[C(-rh + h + -) + C + l-C e }, 2 Z T 2n (4.54) ZJ the rr component [ i_(l_ -2n _ ^ -2n e ) 2 e ] l ( -n+ -)-C+ -C e }, 2 [C Z 2 (4.55) 2n m and the 99 — cfxp component e (m + m - mn H r = hc(m + h) + C + \c e \. 2 ZJ 2n r ) (4.56) ZJ We have three equations i n three unknown functions. A g a i n we are faced with a set of nonlinear coupled differential equations that must be solved simultaneously. N o t surprisingly, an algebraic solution could not be found for these equations, so a series expansion was investigated. We use the same series expansions as i n case A with the addition of t=o r and C = t ^ i=l ' (4-58) Chapter 4. Solving the Einstein Equation with a General Form of Torsion The component equations where multiplied by r 2 37 and then the series expansions were substituted in using Maple. After dropping all terms of order r~ 2 and smaller we see that the leading coefficient of the tt component yields the equation -1 r — ^ 2 n 2 = 0 (4.59) ' V 0 T h i s gives us the result that c = 0 which greatly simplifies the tt component, 0 1- n - . 1 c? 1 CiC h c - -c m 2 no 4 0 2 1 1 2 no the rr component, 3 c? 1 — n — 2ci 0 2n 1 „CiC h n m i — n\ — 3c H — c \ m \ — 3 4 . n 0 0 2 2 0 and the 99 = cpcfi component, 1 1 c? 2 4 no r 1 , 1 N 2 4 , ni no • L o o k i n g at the constant coefficient of the 99 = cpcp component we see that either c\ = 0 or c i = 2 n . If we now look at the rr constant coefficient we see that if c\ = 0 then 0 n 0 = 1, and i f C\ = 2 n we get the result { n 0 0 = j j , C\ = ^ } . T h e first solution is consistent w i t h the constant coefficient of the tt component but the second is not and is eliminated. If we now substitute the { n = 1, C\ = 0} solution into the remaining 0 coefficients we see that mi = n x and c = 0 and we have recovered the Schwarzschild 2 case again. If the series is extended out to order r~ 2 then i n the tt component the r~~ 2 coefficient yields 2c + n 3 2 = 0. (4.63) Chapter 4. Solving the Einstein Equation with a General Form of Torsion For the rr component, the r~ 2 coefficient gives us m - n - 4 c = 0. 2 2 (4.64) 3 The corresponding 66 equation is 2 m - n + ^ c = 0. 2 2 (4.65) 3 T h i s set of three equations i n three unknowns has the solution { m = n 2 Thus we see that to order r~ 2 2 = c = 0}. 3 the only solution we have is Schwarzschild w i t h C — 0. 38 Chapter 4. Solving the Einstein Equation with a General Form of Torsion 4.5 39 Solving the Einstein Equation: Case D Now we shall consider the most general form of the torsion tensor, = Tabc A(r)e +B(r)(9 <p -(f) O )r abc a +C(r)[r (8 8 a = b A(r)e + c b ab b c + B(r)e r abc b c - r {9 0 + <p 4>c)} <f> <p ) b a a c a + 2C(r)r h . c [a (4.66) b]c In calculating the components of equation 4.22 we can reuse some of the results from our previous cases. T h e three derivative terms w i l l act linearly so we can just add up the results from our previous cases. T h e three quadratic terms w i l l have new cross terms and these are calculated i n A p p e n d i x D . T h e final result is that the tt component is 2_(l-e- ) + 2 V " 2 n = \[C(-m 2 + h+^) + C+^C e } 2 2n + ^(2ABe + 3A ). (4.67) + \(2ABe 4 + A ). (4.68) 2n 2 The rr component is [^(1 - e- ) - 2^e~ ] 2n = hc(m-n 2n + -) r 2 -C+\c e ] 2 2n 2n 2 2 The 96 = (f>(f> component is r —On f •• . 9 . . [e (m + m - ran H M rn z = -[C{m + h)^C+ -C e ] l l 2 2n r rh.. ) r + -^A (4.69) 2 A n d lastly, the 9cf> component is _ i e m + \c(A B + 2B(m + h + ^ ) } - -\e"r 2 + V g n6 i [ + B)e (A + 2Am)e~ 2n m + n r 2 sin 9e 2n + B + 2B(rh sin 9[A + 2Ara] = 0 + h + - ) - C ( A + B)e 1 2n = 0 (4.70) Chapter 4. Solving the Einstein Equation with a General Form of Torsion 40 We have four equations i n five unknown functions so it is pointless to look for an algebraic solution. In an attempt to find a possible solution the unknown functions were expanded i n series using the same expansions as i n the previous cases w i t h the addition of: °° (]. r i=0 ' i . 0 — jn. 0 0 Z _ _ / i+l ' i=l r ' ' h. 0 i—i 0 i ' The component equations were first multiplied by r i+l ' i=l i=0 ' r — oh. 0 r and then the series expansions 2 were substituted i n w i t h the aid of Maple. After dropping a l l terms of order r~ and 2 smaller the leading order coefficient equations were examined to see i f we could get any simplifications as was seen i n Case C . T h e r coefficient of the 09 = (p(p component 2 leads to (n a + c ) = 0 2 (4.71) 2 0 We know that n 0 is strictly greater than zero which means the only solution to the above equation is a = 0 and c = 0. T h i s result greatly reduces the number of terms 0 0 in the expanded component equations. The tt component is - ^ - r + K l - n o ) - ^ + b ° a 2 + c ? ) + + 0 { 1 r ] = 0 ( 4 7 2 ) The rr component equation is W + b„a + 3c?) + ^ 1 ] 2 The 99 = 4>(p component equation expands to + O(i) = 0 (4.73) Chapter 4. Solving the Einstein Equation with a General Form of Torsion 41 Lastly, the 9<p component equation is (2b n 0 r + [bi - n ai + b ( n G l + bl) 0 _ l ( n 0 C l ( + mi) Q 0 ) + b^L] + 0(1) = o . ng r (4.75) boC2 0 L o o k i n g at the r coefficients we see that we have either bo = 0 or c\ = 2no and ai = 0. L o o k i n g at the 6 = 0 case we get the following four equations from the constant 0 coefficients: (l_ n o ) _^ 2__L a ( G l / j l + c 2 ) ) ( (1 - n ) - 2ci - \a\ - -^-{a b + 3 c ) , 4 zno x 7 6 ) (4.77) 2 0 4 x and 6 — n ai——ci(ai n X 0 + bi). (4.79) 0 T h i s is a system of four equations i n the four unknowns, no, a i , bi, C i , and has three solutions. T h e first is the no = 1 and ai = bi = C i = 0 case, which just leads to Schwarzschild. T w o other solutions are: 4 — 4 2 { n = - 1 , ai = - , 6 i = — , c = - } , (4.80) -4 { n = - 1 , cn = — , 6 (4.81) 0 x and 0 1 = 4 2 - , ci = - } . These lead to the exact solutions, ds = -dt - dr + r (d# + sm 9d<f) ), T 2 2 2 2 2 2 abc —4 = ^a r b c 4 +^ a b c 4 + —r h , (4.82) 4 + —r h . (4.83) [a b]c and ds = -dt - dr + r (d9 + sin 9d(p ), 2 2 2 2 2 2 T abc 4 = ^e a 6 r c 4 - —e abc [a b]c Chapter 4. Solving the Einstein Equation with a General Form of Torsion 42 However, since these solutions have a non-Lorentzian metric they w i l l be regarded as unphysical and discarded. T h e case c\ = 2 n and a i = 0 has no solutions. We next 0 considered the r ~ order terms. M a k i n g use of the results we get for the higher order 2 terms, { n = 1, a = b = c = a i = h = c = 0}, 0 0 0 0 (4.84) x we have the tt component 1 1 c a - + [n - - c ( 2 2 1 1 3 1 1 + m) - -cl - -a b - -a\ + 2c ] — + 0 ( —) = 0 (4.85) m i 2 2 3 T h e rr component equation is ( m i — ni — 3 c ) — h [2m — n — mAmi 2 r 2 + ^ c ( m i + m) - ^ c - -a b 2 - \a\ l 2 2 2 ni) — 2 - 4 c ] l + 0(1) = 0 3 (4.86) T h e 99 = 4>4> component equation expands to ( - ( m i - n i ) + c ) - + [—(mi - n i ) + 2 m - n - - m 2 1 -mm! 1 + -n 2 c 2 2 x + 1 3 X 1 - m) - - ( a + c ) + - c ] - + O ( - ) + 2 2 3 = 0 (4.87) Lastly, the 0</5 component equation expands to 1 -2a 2 2 2 +a { 2 b - [—(mi - n i ) + c ( a + 6 ) mi + 2ni) + 3 a + & ]1 3 3 2 2 + 0(1) = 0 (4.88) L o o k i n g at these equations it is clear that a = c = 0 and m i = n\. T a k i n g the first 2 2 three component equations we can solve for the three unknowns and get, m 2 = n = c = 0 2 3 (4.89) Notice that no conditions are placed on b which allows for the solution i n Case B . T h e 2 only other solution is the standard Schwarzschild w i t h A = B = C = 0. Chapter 5 Conclusion A s long as experiments do not eliminate the possibility of small torsion terms then general relativity and relativity theories involving torsion w i l l remain experimentally indistinguishable, and torsion theories w i l l be of interest. In classical general relativity, black holes are objects that can exhibit very few physical quantities to outside observers: mass, charge, magnetic charge, axionic charge, and angular momentum. We have shown that torsion charge should be added to this list and proved the existence of a whole class of stationary torsion black wholes. The torsion charge present in these solutions could be detected i n the way that Bowick et al.[30] suggest using an A h a r o n o v - B o h m like experiment. We also showed that if a spherically symmetric and spatial form is assumed for the torsion tensor, the only black hole solution has a torsion of the form: T (, = a c ^a bfca Since e abc = e[ i,r ], the first solution is just a special a c case of this solution w i t h T = 0. The non zero torsion present i n this solution can be thought of as an extra field present in the Schwarzschild spacetime. In looking for a series solution to our Einstein equations we only considered series out to order r ~ , 2 and d i d not find any new solutions. B u t , given the cosmological scale, any solution w i t h apparent long range behavior would be experimentally undetectable. A s for the solution we d i d find, it could be detected using the various spin precession methods [14] [15] or rotation of the plane of polarization experiments suggested by other authors 43 Chapter 5. [20] [21] [22]. Conclusion Bibliography [I] E . C a r t a n . "Sur une generalisation de l a notion de courbure de R i e m a n n et les espaces a torsion." Comptes Rendus, 174, 593 (1922). [2] F . W . Hehl, P. von der Heyde, and G . D . Kerlick. "General relativity w i t h spin and torsion: Foundations and prospects." Reviews of M o d e r n Physics, V o l . 48, N o . 3, 393 (1976) [3] F . W . Hehl. " S p i n and torsion i n general relativity: I. Foundations." General R e l ativity and G r a v i t a t i o n , V o l . 4, N o . 4, 333 (1973) [4] F . W . Hehl. " S p i n and torsion i n general relativity: II. Geometry and field equations." General R e l a t i v i t y and Gravitation, V o l . 5, N o . 5, 491 (1974) [5] F . W . Hehl, P. von der Heyde, and G . D . Kerlick. "General relativity w i t h spin and torsion and its deviations from Einstein's theory." Physical Review D , V o l . 10, No. 4, 1066 (1974) [6] A . Trautman. " O n the structure of the Einstein-Cartan equations." Symposia M a t h e m a t i c a 12, 139 (1973) [7] Gockeler, M . and Schucker, T . Differential Geometry, gauge theories, and gravity Cambridge University Press, 1987. (pg. 68) [8] Misner, C . W . , Thorne, K . S . , and Wheeler, J . A . Gravitation W . H Freeman and Company, 1973 (pg. 1067) [9] Weinberg, S. Gravitation and Cosmology John W i l e y and Sons, 1972 (pg. 68) [10] P. von der Heyde. " T h e equivalence principle i n the U4 theory of gravitation." Lettere al Nuovo Cimento, V o l . 14, N o . 7, 250 (1975) [II] C M . W i l l . Theory and Experiment in Gravitational Physics Cambridge University Press, 1981. [12] P . B . Yasskin and W . R . Stoeger. "Propagation equations for test bodies w i t h spin and rotation in theories of gravity w i t h torsion." Physical Review D , V o l . 21, N o . 8, 2081 (1980) 45 Bibliography- AG C M Zhang. "Some constraints on torsion detection physics." International Journal of M o d e r n Physics, A8, 5095 (1993) K . Hayashi, K . Nomura, and T . Shirafuji. " S p i n Precession i n spacetime w i t h torsion", Progress of Theoretical Physics, V o l . 84, N o . 6, 1085 (1990) C F . P e i . "General equations of motion for test particles i n spacetime w i t h torsion" International Journal of Theoretical Physics, V o l . 29, N o . 2, 161 (1990) J . Audretsch. " D i r a c electrons i n spacetimes w i t h torsion: Spinor propagation, spin precession, and nongeodesic orbits." Physical Review D , V o l . 24, N o . 6, 1470 (1981) M . Seitz. " P r o c a field i n a spacetime w i t h curvature and torsion." Classical and Q u a n t u m Gravity, 3, 1265 (1986) M . Seitz. " P r o c a test field i n a spacetime w i t h torsion." Classical and Q u a n t u m Gravity, 4, 473 (1987) R . Spinosa. " S p i n - | field i n spacetime w i t h torsion." Classical and Q u a n t u m G r a v ity, 4, 1799 (1987) I.V. Y a k u s h i n . "Torsion and interaction of polarized photons." Journal, 34, 1090 (1990) Soviet Physics C. Wolf. " P h o t o n dispersion as a consequence of the interaction of torsion w i t h an external magnetic field." Nuovo Cimento 91B, 231 (1986) V . de Sabbata, P.I. P r o n i n , and C . Sivaram. "Neutron interferometry i n gravitational field w i t h torsion." International Journal of Theoretical Physics, V o l . 30, No. 12, 1671 (1990) J . Audretsch and C . Lammerzahl. "Neutron interference: general theory of gravity, inertia and spacetime torsion." Journal of Physics, bf A 1 6 , 2457 (1983) J . A n a d a n and B . Lesche. "Interferometry i n a spacetime w i t h torsion." Lettere al Nuovo Cimento, V o l . 37, N o . 11, 391 (1983) A . Trautman. " S p i n and torsion may avert gravitational singularities." Nature Physical Science, V o l . 242, N o . 5, 7 (1973) V . de Sabbata. " T h e importance of spin and torsion i n the early universe." Nuovo Cimento A , V o l . 107A, N o . 3, 363 (1993) J . Tafel. " A class of cosmological models w i t h torsion and spin." A c t a Physica Polonica, V o l . B6, N o . 4, 537 (1975) Bibliography 47 [28] H . T . Nieh. "Possible role of torsion i n gravitational theories." In Proceeding of the T h i r d M a r c e l Grossmann Meeting on General Relativity. Science Press and N o r t h - H o l l a n d P u b l i s h i n g Company, 1983 [29] T . T . Fujishiro, M . J . Hayashi, and S. Takeshita. " W h a t is the role of torsion i n the universe?" Progress i n Theoretical Physics, Supplement N o . 110, 179 (1992) [30] M . J . B o w i c k , S.B.Giddings, J.A.Harvey, G . T . H o r o w i t z , and A.Strominger. " A x ionic black holes and an A h a r o n o v - B o h m effect for strings." Phys. R e v . Lett. 61, 2823 (1988). [31] S.A. C u l h a m . " A x i o n i c Black Holes." Undergraduate thesis. University of B r i t i s h C o l u m b i a 1993 [32] M . J . Duncan, N . Kaloper, and K . A . Olive " A x i o n hair and dynamical torsion from anomalies." Nuclear Physics B 387, 215-235 (1992) [33] W a l d , R . M . General Relativity T h e University of Chicago Press, 1984 [34] Penrose, R . and Rindler, W . Spinors and space-time Cambridge University Press, 1984 [35] Papapetrou, A . Proc. Roy. Irish A c a d . A 51, 191 (1947) [36] Kramers, D . , Stephani, H . , M a c D a l l u m , M . , and Herlt, E . Exact Solutions of Einstein's Field Equations Cambridge University Press, 1980 Appendix A Calculations for Case A Assuming that the torsion tensor is of the form = A(r)e T (A.l) abc abc we w i l l solve the equation 1 r~l ^ab — e f\J rp r,\ v rp v a J-eb - \ 9 a b { \ V d T e 1 d e e - ^ (rp e \ ab J -T ^K ae T ») 1 d c e d drp 1 db rp e 1 de drp 1 ab e \ ) (A.2) The terms on the right hand side The right hand side of equation A . 2 consists of six terms. Each of these w i l l be considered separately. Term 1: T e eb = A{r)e t =0 e (A.3) T h i s means that \ ^ a T e e b = 0 Term 2: jv T e e a f t = -\v A(r)e e 48 e ab (A.4) Appendix A. Calculations for Case A 49 = -\K ^eA(r) + = -\K VeA(r) + e b e F a ° cb e A(r)V e } e e ab A(r)[d e e e ab ^eh ° ac * + ^ec ab °]\ 6 e e &e (A-5) T h i s term has only antisymmetric components. Term 3: e drp ^ T i db ' ^ ae L A(r)e A(r)e aed rp ^ ^ — 1 e drp ae rp — ^ bd 1 rp de aed-L b d e b ~A (r)e A(r)e 2 ade b -\A (r)h 2 ab where Kb = g b + t t a a (A.6) b T h i s term has only diagonal components. Term 4: -Jde X b = \Mr)e e A(r)e d de e ab = 0 (A.7) Term 5: -±g a b V T d d = e e = Ag a b V A(r)e d 0 e d e (A.8) Term 6: We already have the result from Term 3 that \Tae Tdb d e = \A (r)h 2 ab (A.9) 50 Appendix A. Calculations for Case A This allows us to write \^\Tce % c e = \9a \A {r)h 2 b = \g A c 2 c ab (A.10) This term has only diagonal components The Components We shall look at the symmetric components first and only terms 3 and 6 have symmetric components. The tt component: We get contributions from term 6 to give - 9uA = -\e A 2 2m (A.ll) 2 4 The rr component: We get contributions from terms 3, and 6 to give 1 --A (r)h 3 + 2 rr -g A (r) 2 rr = --^A {r)g + ^g A (r) = e -^A 2 2 rr 2n rr (A.12) 2 The 69 component: We get contributions from terms 3 and 6 to give -^A (r)h g + ^gg A (r) 2 2 e The 4>4> component: = -±A (r)g 8 = r -A 2 0 2l 2 e + lg A (r) 2 gg (A.13) 51 Appendix A. Calculations for Case A We get contributions from terms 3 and 6 to give -\A (r)h + -^g A (;r) 2 2 H = -\A (r)g = r sm 9^A H + 2 H 2 2 - g A (r) 2 A H (A.14) 2 The antisymmetric components of our equation all come from term 2. T e r m 2 involves five subterms. There are six possible antisymmetric component equations that need to be evaluated. The rt component Since the epsilon tensor is totally spatial all of the subterms are zero except the fourth subterm i n term 2. We have -^V A(r)6 e r t e = \AY e J 7 tt r 1 2" = 0 (A.15) The 9t component A g a i n all the subterms are obviously zero except the fourth subterm i n term 2. -\v A(r)e * e et = = \AY \ J e 2A\T i e ^ + T ' 'e j ] r ( = e t )t t er r rt ef ) 0 (A.16) The (fit component: A g a i n all the subterms are obviously zero except the fourth subterm i n term 2. 1 oA[Y e^ + T 2' T et = 0 6 r ep e rt ( r 0 (A.17) 52 Appendix A. Calculations for Case A The rO component: ^ X ; ^ = + A{r){d 6 * -l ^ A{r) [tr9 e e er 1 70 fc e6» 1 rf ' t r6 e-y 1 r6 JJ t - ^ e / V ^ ( r ) + A(r)[r3^/ r — r 7 e r e — r e 7 0 7 e 6 ) e e r 7 + r£ e e e< r e ^]] 0 (A.18) The rep component: -\v*Tr* e = ~\{t ^ e er 7</> = e r7 ~ r £7 r 7 er e e 7<?i JJ r — Tp e 7 + T e r 7 e ^ ]] t eg e r( ) A{r)[d {e ,e9 ) 6e e v 6r e r4> €( ~\[0 + _ e r — r rci = = A{r)[d e ^ -\[e /VeA(r)+A(r)[de6 —T = A{r) + r e f r rr 0ti -\A{r)[de{e , g )-T e g^] ee r e ~A{r)[d {-e r m+n e H re4> sin 6r~ ) 2 2 e ZJ -(-sm6 = = cos 0) (e r -(r)[-e 0 ZJ m+n m + n 2 sin 0) ( r ~ s i n " 0)] cos 0 + e 2 m + n 2 cost?] (A.19) Appendix A. Calculations for Case A 53 The 9(p component: J V J V = A{r) + A{r)[d e ^ t ~ r , t e e6 = — e 7 e _ p r9 r0 1 r — r 00 fc fc p 1 6 66 r4> r 00 0r 0 fc r m+n 2 e4>r t +[r*r * + r „ . + r - \ [{e r > rr r v 1 r _ ^<^ ^ 7 e( A(r)[d (e g ) r 6 e f ]] £ ej rr 1 = + T e 9l -\[to<t*9 d A{r) + _ r e + r^ ^ j e ^ ] 6 r 6 r r sin 9)e~ A + A[d ( e V sin 8e~ ) m + 2n 2n r Z --(e r r m+n sm6)e- 2 --(e r r m+n - (-re- )(-e r 2n 2n 2n m+n m m + V sin 0) - l] 2n e n m m 2 n 2 sin 9(m - h) + m n sin 0(m + h) + e --e - r sm6[A 2 m n e"2rsm6 sin 0 - e - r sin 0 - e - r sin 9 - e " r sin 9 n +e - r m 2 2 2 r - - [ e - r s i n t L 4 + A[e - r -e - r = sin 6) - sin" 0 2 + [m + n + - + - ] ( e r r n 2 2n 2 m sin9)r~ 2 sm6)e- 2 - ( - r sin 6e- )(-e r = m+n m _ n n 2 r sin 0]] + 2Am] 2 (A.20) The Final Result The final result is that equation A.2 has only four nonzero components. Cancelling the metric factors we have the tt component: l(l-e- «) 2 2V "= ^ 2 + 2 (A.21) Appendix A. Calculations for Case A 54 The rr component: 1(1 The 99 = epep - e~ ) 2n - 2™e~ 2n = -A l 2 (A.22) component: - (m 2n e + m - r h h + - - - ) = -A r r 4 2 2 (A.23) The 9ep component: A + 2Am = 0 (A.24) Appendix B Calculations for Case B The equation we wish to solve is fry s~i ^ab — rp 2^ a e v-7 e ~ b V , -\aab{\v T d e l rp ab d e e e\ ) ~ ^ \ -\T 1 (rp drp a e 1 e db rp ~ 1 drp de 1 ab e\ ) (B.l) c e For simplicity we w i l l first assume that the torsion tensor is of the form = B(r){ea4>b - (p 6 )r = B(r)e r Tabc a b c ab (B.2) c The terms on the left hand side The spherically symmetric metric has the line element ds = -e dt 2 2m + e dr 2 2n 2 + r (d9 + sin 0d<p ) 2 2 2 (B.3) Given this metric it is a standard result [8] that the components of the left hand side of equation B . l i n a coordinate basis are the tt component: 1 G = -e [^(l 2m tt 77 ~ e~ ) + 2-e~ ] 2n 2n (B.4) The rr component: G rr = e [l(l 2 n - e- ) - 2™e- ] 2n 55 2n (B.5) 56 Appendix B. Calculations for Case B The 99 component: G = r [e- {m + m - m n + 2 ee 2n --)] 2 r (B.6) r The 4>(p component GM = r s i n 9[e- (m + m - mh + — - - ) ] 2 2 2n (B.7) 2 r r The terms on the right hand side The right hand side of equation B . l consists of six terms. Each of these w i l l be considered separately. Term 1: T = B(r)e r e eb e =0 = 0 eb (B.8) This means that ^VJV (B.9) Term 2: -V T l e e ab = -Uj B(r)e r e e ab 1 [e r V B{r) e 2 ab + + B(r)r [d e - e -\[e r V B(r) e ab + B{r)r V e e e ab e e e ab -r e }+B(r)e T r } c eb e ac ab Term 3: ^ n n dqn db 1 e '}i F drp r ~ ^ = = 1 ae 1 e bd ^L T r ~~ ^ aed B(r)e r B(r)e r d ae 0 e ab T e e c ea cb (B.10) c ec This term has only antisymmetric components. ' ^~a.e. B(r)e V r ] d b e Appendix B. Calculations for Case B 57 Term 4: -T % de = \B(r)e r B(r)e r e d b = 0 e de ab (B.ll) Term 5: \ g a b V d T * d e - V B{r)ey l = l9ab = d 0 (B.12) T e r m 6: We already have the result from T e r m 3 that 1 ^T T d a e e d b = 0 (B.13) T h i s allows us to write \ d a b \ T d c e T c e d = 0 (B.14) T h e Components L o o k i n g at the above results we see that there are no symmetric terms on the right hand side. The antisymmetric components of our equation a l l come from term 2. T e r m 2 involves five subterms. There are six possible antisymmetric component equations that need to be evaluated. T h e Christoffel symbols i n a coordinate basis for our metric are used i n evaluating these equations and are available i n A p p e n d i x E . T h e rt component: Since the epsilon tensor is totally spatial a l l of the subterms are zero except for the fourth subterm of term 2. We have -\V T € H ' = = \Br*T et 0 Vr (B.15) Appendix B. Calculations for Case B 58 The 9t component: A g a i n a l l the subterms are obviously zero except for the fourth subterm of term 2. 4^ 6 = \BrT = 0 " V et (B.16) The cpt component: A g a i n all the subterms are obviously zero except for the fourth subterm of term 2. -fa*' = = ^ T e t % 7 0 (B.17) The r9 component: The only term not obviously zero is the third subterm of term 2. - \ V e T e r e = l = \Br T % = -Br*T e 7 er ie r rr e 0 (B.18) The rep component: The only term not obviously zero is the third subterm of term 2. = 0 (B.19) Appendix B. Calculations for Case B 59 The 9(p component: -\VeT £ 94> = -±[w°V B(r) = _ A[ ™+V 2 £ e -T g e ~[e m + n + e i ne e m + n 2 r = -\e 2 r 2 sin 9) 2 r ] + B(r)(e r m+n f l 0 sinOB + B[e r 2 m+n s i n 9 i l - I T m + n e6 B[d {e r B + r r - T e<p m+n s e*- r /e r = + B(ry[d e e + sin 9)T 2 e er sm9(m + h) + 2 e 2rsin9 m+n m + h + - + -]} T T sin6[B + 2B(rh + h +-)] r T (B.20) The Final Result The final result is that equation B . l has only five nonzero components. Canceling the metric factors we have the t t component: ^(l-e- = 0 (B.21) — (1 - e~ ) - 2—e~ = 0 (B.22) 2 n ) +2 V 2 n The rr component: 1 ,„ _o„, 2n ~rh 2n r The 99 = (j)(f) component: e- (m + m -mh+ 2n 2 Tfh TL r r )= 0 (B.23) The 94> component: B + 2B(m + h+ -) = 0 r (B.24) Appendix C Calculations for Case C Starting with the assumption that the torsion tensor is of the form = C ( r ) [ r ( ^ c + W c ) - r 4 U + Wc)] T a abc = 2C(r)r h , [a (Cl) b]c where for example i n a coordinate basis ^H> = 9u = r ^ 2s i n 2 (C- ) 2 we w i l l solve the equation Gb a 2^ ^ a — eb ~ ^eTab £ -\gab{\v T d d e e ) & — ^ (T ae - -J ce ^db ^ T d de % «) T a b (C.3) The terms on the right hand side The right hand side of equation C 3 consists of six terms. Each of these w i l l be considered separately. Term 1: T e eb = 2C(r)r h f [e b = -C(r)r h b 60 e e = -2C(r)r b (CA) 61 Appendix C. Calculations for Case C This means that \v T * a = eb -\v 2C(r)r a = b -C(r)V r -r V C(r) a = -C(r)d r a b a b + C(r)r r -r V C(r) (C.5) c b ab c b a This term has only symmetric components. Term 2: ±V 2C(r)r h { e [a b = I ( V C ( r ) r V - V C(r)r V) e = a e b -(r h V C{r) + C(r)h V r - r h V C(r) - C(r)h V r l e a b b e b a + e e e a a e a e e b - C(r)r V h ) e e C(r)r V h e b b e a (C.6) This term has only antisymmetric components. Term 3: 1 T-i ^ ae = ^ rp drp e db l ~ ^ l drp e ae 1 bd ^LT T de 1 ~ ^ aed-L b C(r)r h C(r)r h ^ d [a = e]d [b \c {r)[r {e 9 2 a e a + <f> <f> ) - r (6 6 + fafa)] e d [r (9 9 + fV ) - r (9 9 d e e d e b = b e a d + 4>b \c\r)r r a b This term has only symmetric components. Term 4: -Jde Xb e = -C(r)r C(r)r h f e [a b Term 5 -- g V T * d 4 ab d e = -g V C{r)r l d ab d =0 (C.7) Appendix C. Calculations for Case C = \ga [C{r)V r + = l9a {C(r)T r d b d d b r V C{r)} d d + c dc r \7 C(r)} d d T h i s term has only symmetric components. Term 6: We already have the result from Term 3 that ~T T d ae = e db \c\r)r r a b T h i s allows us to write \g \T ab % ce c = e \g \c\r)r r< ab c T h i s term has only symmetric components The Components We shall look at the symmetric components first. The tt component: We get contributions from terms 1,5, and 6 to give C(r)T r V + \g [C{r)Y r tt r tt dc + r V C(r) + d d = Ce ^ -^me - \e [C(m + n + -) + C + = -le [C(-m 2 + h+-)+C r + -C e ] 2 2 m 2n 2m 2m l 2 \c\r)r c \c e ] 2 2n 2n The rr component: We get contributions from terms 1,3,5, and 6 to give -C(r)d r r r + C{r)V r r rr r - r V C{r) r r + ^C (r)r r 2 r r 63 Appendix C. Calculations for Case C L[C(m + = + h+-) + 2 C+l-C e } 2 r -C2he + Che 2n 2 - Ce 2n + 2n \c e 2 + -e [C(m + h+-) + C + l = 2 T \e [C(m-h+-) An \c e \ 2n ZJ 2n 2n Z -C + \c e ] 2n 2 T Z (C.12) 2n Z The 99 component: We get contributions from terms 1,5, and 6 to give C{r)T r + \gee[C{m + h + H) + C + -C e ] T ee l r = C{-re~ )e = -r [C(m 2 2n 2n + \r [C(m 2 2n + h+-) + C + \c e ] f z 2 2 Z 2n + h) + C+\c e ] 2 2 2 (C.13) 2n The 4>(f) component: We get contributions from terms 1,5, and 6 to give C{r)V r + \g„[C{m r H = r C(-re- + h+^) + C+ 2 2n 2n + \r s i n 9[C(m + h+-) + C + 2 \c e ] 2 T Z = 2 sin 6) e 2n 2 2 2n Z \r s i n 9{C(m + h) + C + \c e ] z z 2 -C e ] l 2 (0.14) 2n The antisymmetric components of our equation all come from term 2. Term 2 involves six subterms. Subterm 2.1 and 2.4 w i l l be of the form V f l C ( r ) , or V ^ C ^ r ) , both of which are zero. We now only have 4 subterms. In components our equation looks like \v 2C{r)r hfi t = [a \(C(r)h;\7 r e -C(r)h:V r e p a + - C(r)r V h/ a £ C(r)rpV K) e Appendix C. Calculations for Case C 64 x[C(r)Vfcr«-r r ] 7 M 7 + C ( r ) r [ 9 V - r / V + e7 % r e Q - T /r ] - C(r)hj[d r e 71 e p e - ccr)^^ K - r 7 V +r 7 eQ e e7 v]] (c.is) The partial terms we see i n this expression w i l l involve either h dgr , h^d^, e d r dgh , 6 e or dfjthj', all of which are zero so these terms drop out and we are left with: Jv 2C(r)r V e [a 1 -[-C(r)Vf /r 2"' £ 7 + C(r)r [-r /V + ,7 V I r e Q + C(r)Vr /r e 7 -o(r)r [-r ^ fl 6Q e 7 + r V]] (C16) e e7 There are six possible component equations that need to be evaluated. The rt component: Looking at our expression C.16 we see that the first, t h i r d and fourth lines immediately drop out. T h i s leaves \v 2C(r)r h^ t {r -[C(r)r [-r rh^ = 1 = \[C(r)r {-T = 0 r r e et + T n V - I V V } } Vll (C.17) The dt component:. Here the first, second and fourth lines of equation C.16 are obviously zero. T h i s leaves lv 2C(r)r h f £ [e t = \[C{r)h <T 6 7 et r ] 7 Appendix C. Calculations for Case C 65 = \[C{r)h T W ] 6 e = 8t r 0 (C.18) The (pt component: Here the first, second and fourth lines of equation C.16 are again obviously zero. T h i s leaves ^V 2C(r)r^f e = ^[C(r)Vr \ ] = \[C(r)h*T^r ] = 0 £( ' r (C.19) T h e r6 component: Here the t h i r d and fourth lines of equation C.16 are obviously zero. T h i s leaves \v 2C{r)r hfi t [r = \{-C(r)h T e 7 er r 7 + C(r)r [-r ^V + e V]] r £ r = \\-C{r)h Y /r e e e + C(r)r {-T r = 7 r V de " I * h/ + r„ 0 V l 0 (C.20) T h e rep component: Here the t h i r d and fourth lines of equation C.16 are obviously zero. T h i s leaves \v 2C{r)r h£ e [r = ^[-C(r)Vr /r £ 7 + C(r)r [-r /V + e r r = e V]] \[-C(r)h,%/r r + C(r)r [-r /h r = 7 0 e e d - r„ V + »X*ti r (C.21) 66 Appendix C. Calculations for Case C The 6(f) component: Here the second and fourth lines of equation C.16 are obviously zero. T h i s leaves ^ 2C(r)r h^ e ^[-C(r)Vr/r = [e 7 +C(r)Vr /r ] • e = 7 \[-C{r)hfc^r r +C{r)h Y r ] 9 r e = H r 0 (C.22) We may now write l\7 2C(r)r h f e [a = 0 b (C.23) The Final Result The final result is that equation C.3 has only four nonzero components. Cancelling the metric factors we have the tt component: l ( l - e - " ) + 2-e2 = \[C{-m + h+-) 2 n + C+ \c e ] 2 2n (C.24) The rr component: l{C(m-h 2 + -) -C r + lc e ] 2 2n 2 (C.25) The 99 — <f)(f) component: \e (m + m - mh + — r 2n --)] r 2 i [C{rh + h) + C + \c e \ Z Z 2 2n (C.26) Appendix D Calculations for Case D Now considering the most general form of the torsion tensor Tabc = A{r)e +B{r)(9 (j)b-<f>aOb)r abc a +C(r)[r (6 6 a = b c + </></>) - r (6 6 + <f> <f> )] c 6 c b a c a c A(r)eabc + B(r)eabr + 2C{r)r h , c [a (D.l) b]c we w i l l solve G = ab 1 ~ -(V T a 6 e 6 —V T e e a 6 1 ) — -(T d ae T e db d —T T de ) e ab -\9ab(\v T --T (D.2) de d e ce T h e terms on the right hand side The right hand side of equation D . 2 consists of six terms. T h e derivatives i n terms 1,2, and 5 w i l l act linearly so we can use our previous results to calculate them. We need only consider terms 3,4, and 6 here. Term 3: 1 1 rp ^ x a e drp 1 db T 1 e ae 1 drp ^ bd ~ = ^(B(r)e r 1 ae d (B(r)e r d rp _rp e ~ + A(r)e aed + A{r)e e b de b 67 de ^aed-Lb + 2C (r)r h ) [a e]d + 2C (r)r h ^ ) d [b e Appendix D. Calculations for Case D 68 This w i l l expand to potentially nine terms. Looking at each of them i n t u r n we have the B B term: ^B(r)e r B(r)e r d ae d = e b 0 (D.3) The B A term: -B(r)e r A(r)e l ae d de b = - -B(r)A(r)(9 <j> - (p 9 )(9 ^ - cp 9 )g = ~B(r)A(r){6 6 + (p 4> )g = ~B(r)A(r)h e l a e e a a b e b a b b rr rr (DA) 2n ab The B C term: - B{r)e r 2C(r)r h ^ = d A ae d [b ^B(r)C(r)(9 cp a - <p 9 )r e a e d (r (9 9 + ct> (p ) - r (9 6 + &(/>*)) d e d e d e b b = -- B(r)C{r){e 4> -(pa9e){9 9 e b = - -B(r)C(r){9 <p - (f> O )g = ~B(r)C(r)e e e A a l a b a b + M )g e rr rr (D.5) 2n ab The A B term: ^A(r)e B(r)e r d aed e b = ~A(r)B(r)(6 <p = ~B{r)A{r){9 9 = -- B(r)A(r)h e a - <p 9 )(9 ^ - <p 9 )g d d a a d b rr + (p (p )g b a b rr (D.6) 2n 4 d b ab The A A term: -^A(r)e A(r)e aed de b = -±A(r)e A(r)e ade -\A\r)h ab de b (D.7) Appendix D. Calculations for Case D 69 The A C term: -A{r)e 2C(r)r h^ l e aed [b ^A(r)[(9 <f> - 4> 9 )r + (6 <f> - (f> 9 )r - (9 <f> - <f> 9 )r ] a e [C(r)r„(9 9 d a e d e d + fy*) - r (9 9 + e d e b --^A{r)C{r){9 (j) a b e d a a d a d e </><f)] 6 - <j) 9 )g a b rr --A{r)C{r)e e (D.8) 2n ab The C B term: ±2C(r)r h B(r)e r d [a = e]d \{C(r)r (9 9 a e b e + (f) <f> )-r (9 9 + <j) <f) )] d e d e a d a d B(r)(9 4> - (f> 9 )r d d b = e b ^B(r)C(r)(9 cp -<j 9 )grr a = b )a b \B(r)C{r)e e (D.9) 2n ab T h e C A term -2C(r)r h A(r)e [a e]d ^[C(r)r (9 9 a de b e + <f> j> ) - r (9 9 + <f> <j> )] d e< d e a d a d A(r)[(9 <p - <j, e y + (9 <j> - <f> 9 )r - {9 </> - <f> 9 )r ] d d b d e b d e e b b e d b ^C{r)A{r)(9 (f> - 4> 9 )g r a b a b r ]c(r)A(r)e e (D.10) 2n ab T h e C C term hc(r)r h 2C(r)r h ^ d [a e]d [b 70 Appendix D. Calculations for Case D = \c (r)[r (9 6 + (f> cp ) - r (6 6 + fafa)] 2 a [n(9 e d = e e d e d e + </><V) - r (obe d e e a d + <^ )] (D.n) e \c {r)r r (D.12) 2 a b P u t t i n g this a l l together we have ~-T T = - -B{r)A(r)h e e d ae l db + \c r r - -A {r)h 2n l ab (D.13) 2 2 a ab b Term 4: ^ r e d = -C{r)r {B{r)+A{r))e r* = lc(r)(B(r) + A(r))e e (D.14) e ai l e ab 2n ab Term 6: We already have the result from Term 3 that ~\T T d ae e db = -{-B{r)A{r)h e l - A (r)h 2n + Crr) 2 ab (D.15) 2 ab a b c + C r r ) T h i s allows us to write -\9ao\T T ™ d ce d (-S(r)A(r)^e "-A (r)A = -^ = ~g {-2BAe 2 a 6 -3A 2n 2 ab 2 c 2 + Ce ) 2 c c (D.16) 2n The Final Result L o o k i n g at terms 3 and 6 we see that only two extra symmetric terms are created. Term 5 adds one extra antisymmetric term. Gathering our results from cases A , B , and C we have the tt component: -(l-e= -\C(-m 2 2 n ) + 2-e~ 2n + h + -) + C+ -C e ] r 2 2 2n + -(2ABe + 2n 4 3A ) 2 (D.17) Appendix D. Calculations for Case D 71 The rr component: [l(l-e--)-2^e--] l[C(m - n + - ) - C + \c e } + \{2ABe 2 2 r 2n 2n 2 4 + A) 2 (D.18) The 99 — 4>(p component: r —277 / [e •• - 2 • rn • (m + m — mn H ^ [ C ( m + n) + C + Tl. _ JI r r lc7V"] + i A (D.19) 2 The 9(f) component: - i 2 m e + \c{A + V sin 0[£ + 2 B ( m + n + - ) ] - -e - r 7* 2 m + B)e r m+n 2 sin 0 e 2n n 2 sin 9[A + 2Am] = 0 B + 2B(m + h + - ) + (A + 2Am)e~ r 2n - C(A + B)e 2n = 0 (D.20) Appendix E The Christoffel symbols In a coordinate basis the Christoffel symbols for a static, spherically symmetric metric can be calculated using I V A = ^ ip)i9u\ + dug^x ~ d g^) (E.l) x The spherically symmetric metric has the line element ds = -e dt 2 2m + e dr 2 2n + r (d6 + sin 8d<p ) 2 2 2 (E.2) 2 so that i n a coordinate basis the metric has components: 9tt = ~e , g 2m = e , g 2n rr = r, g 2 es = r sin 6 2 H (E.3) 2 and tt g = _ -2m^ e rr g = -2n^ e 69 = g ^ g H = r ^ s m ^Q (E.4) The nonzero Christoffel symbols for the spherically symmetric metric are: t_dm T p 1 tt r _ 2(m-n)^ — dr J _dn p _l ,^1 e r _ _ r p e -2n r i I \ / = -sinflcosfl H 1 r _ ~ 4>_ CQiB • 2 a -2n ' (E.5) 72
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Black hole solutions with torsion Culham, Shaun Andrew 1999
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Title | Black hole solutions with torsion |
Creator |
Culham, Shaun Andrew |
Date Issued | 1999 |
Description | Classical general relativity theory is torsion free. However, since general relativity and general relativity with torsion are experimentally indistinguishable at this time, it is important to explore the consequences of torsion through its physical effects. We do so by considering an important class of static spherically symmetric solutions, analogs of the Schwarzschild solutions which are at the foundation of most weak limit tests of General Relativity. We first show that in general spherically symmetric black holes with torsion exist; these are the torsion analogs of the Bowick et al. [30] black hole solutions with axionic charge. We next search for more general solutions with non-vanishing torsion field exterior to the black hole. We specialize to spherically symmetric static metrics and spherically symmetric static spatial torsion fields. We find that in certain cases such a solution exists; however falloff conditions on the torsion greatly restrict the form of these solutions. |
Extent | 2384057 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085119 |
URI | http://hdl.handle.net/2429/9332 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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