Metric-affine approach to teleparallel gravity

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Metric-affine approach to teleparallel gravity

Yu. N. Obukhov*and J. G. Pereira

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil

共Received 2 December 2002; published 27 February 2003兲

The teleparallel gravity theory, treated physically as a gauge theory of translations, naturally represents a particular case of the most general gauge-theoretic model based on the general affine group of spacetime. On the other hand, geometrically, the Weitzenbo¨ck spacetime of distant parallelism is a particular case of the general metric-affine spacetime manifold. These physical and geometrical facts offer a new approach to teleparallelism. We present a systematic treatment of teleparallel gravity within the framework of the metric-affine theory. The symmetries, conservation laws and the field equations are consistently derived, and the physical consequences are discussed in detail. We demonstrate that the so-called teleparallel GR-equivalent model has a number of attractive features which distinguishes it among the general teleparallel theories, although it has a consistency problem when dealing with spinning matter sources.

DOI: 10.1103/PhysRevD.67.044016 PACS number共s兲: 04.50.⫹h, 03.50.Kk, 04.20.Jb I. INTRODUCTION

Theories of gravity based on the geometry of distant par-allelism关1–7兴 are commonly considered as the closest alter-native to general relativity 共GR兲 theory. Teleparallel gravity models possess a number of attractive features both from the geometrical and physical viewpoints. Teleparallelism natu-rally arises within the framework of the gauge theory of the spacetime translation group. Translations are closely related to the group of general coordinate transformations which un-derlies GR. Accordingly, the energy-momentum current rep-resents the matter source in the field equations of teleparallel gravity, as in GR.

Geometrically, teleparallel models are described by the Weitzenbo¨ck spacetime. The latter is characterized by a trivial curvature and nonzero torsion. The tetrad 共or a co-frame兲 field is the basic dynamical variable which can be treated as the gauge potential corresponding to the group of local translations. Then the torsion is naturally interpreted as the corresponding gauge field strength. As a result, a gravi-tational teleparallel Lagrangian is straightforwardly con-structed, in a Yang-Mills manner, from the quadratic torsion invariants.

Teleparallel gravity belongs naturally to the class of the metric-affine gravitational theories 共MAG兲. Quite generally

关8兴, MAG can be understood within the framework of the

gauge approach for the affine group, a semidirect product of the general linear group and the translation group. The cor-responding gauge field potentials are the linear connection and the coframe, whereas the curvature and the torsion turn out to be the gauge gravitational field strengths. In addition, the metric represents one more fundamental field with the nonmetricity as the corresponding field strength.

From the gauge-theoretic point of view, teleparallel grav-ity is distinguished among the other MAG models by reduc-ing the affine symmetry group to the translation subgroup. In geometric language, teleparallelism arises from the general

metric-affine spacetime after we impose two constraints by putting curvature and nonmetricity equal to zero. In this sense, teleparallelism can be treated, both physically and geometrically, as a particular case of the general MAG theory.

In the present paper we will study teleparallel gravity within the MAG approach. A similar analysis was previously developed in 关6兴 for the Poincare´ gauge approach. We will consider the general class of teleparallel models which are characterized by three dimensionless coupling constants. Af-ter recalling some basic facts about MAG in Sec. II, we derive the field equations of the general teleparallel theory in Sec. III. Then in Sec. IV we show that the teleparallel gravity model can be reduced to an effective Einstein theory with some modified energy-momentum current as a source. When the coupling constants have specific values 共5.1兲, the exact equivalence between general relativity and teleparallel grav-ity is established for spinless sources. However, in Sec. V we demonstrate certain consistency problems for the sources with spin. Among the general teleparallel models, there is a class of theories with the proper conformal properties. We discuss the definition of conformal symmetry in teleparallel-ism in Sec. VI. Finally, we analyze the spherically symmetric solutions for general teleparallel gravity coupled to the elec-tromagnetic field. We pay special attention to the behavior of the 共Riemannian兲 curvature and torsion invariants. This as-pect was not studied in detail before. The formulation of the spherically symmetric problem and its preliminary analysis is given in Secs. VII–IX. A number of new solutions is ob-tained, both charged and uncharged. In Sec. X we present the class of conformally flat solutions that generalizes the solu-tions of Bertotti and Robinson. The corresponding geometry appears to be regular from both the Riemannian curvature and the teleparallel torsion points of view. The complete set of uncharged solutions is presented in Sec. XI. This is done for all possible choices of the three coupling constants of teleparallel gravity. We analyze in Sec. XII the correspond-ing spacetime geometries, and find that the generic solutions are no black holes. In Sec. XIII we show that only in the above mentioned general relativity limit, i.e., for the specific values of the coupling constants 共5.1兲, we recover the *On leave from Department of Theoretical Physics, Moscow State

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Schwarzschild and the Reissner-Nordstro¨m black hole con-figurations. This fact can be considered as an argument fa-voring the choice of the rigid structure of the teleparallel Lagrangian with the fixed values of the three coupling con-stants. The analysis of the torsion invariants reveals, how-ever, that even for the case of the teleparallel equivalent of the general relativity, the horizon of a black hole represents a singular surface for the torsion. This latter fact was not no-ticed in the literature previously.

Our basic notations and conventions are those of 关8兴. In particular, the signature of the metric is assumed to be (⫺,

⫹,⫹,⫹). Spacetime coordinates are labeled by the Latin

indices, i, j , . . .⫽0, . . . ,3 共for example, dxi), whereas the Greek indices, ␣,␤, . . .⫽0, . . . ,3, label the local frame components 共for example, the coframe 1-form ␽␣). Along with the coframe 1-forms ␽␣, we will widely use the so-called ␩-basis of the dual coframes. Namely, we define the Hodge dual such that␩ª*1 is the volume 4-form. Further-more, denoting by e_␣ the vector frame, we have that ␩_␣

ªe␣c␩⫽*␽␣, ␩␣␤ªe␤c␩␣⫽*(␽␣⵩␽␤), ␩␣␤␥

ªe␥c␩␣␤, and␩␣␤␥␦ªe␦c␩␣␤␥. The last expression is thus the totally antisymmetric Levi-Civita tensor.

II. PRELIMINARIES: METRIC-AFFINE APPROACH In this section we recall some basic facts concerning metric-affine geometry in four dimensions. For a more de-tailed discussion in arbitrary dimensions, see关8兴. The metric-affine spacetime is described by the metric g_␣␤, the coframe 1-forms ␽␣, and the linear connection 1-forms⌫_␤␣. These are interpreted as the generalized gauge potentials, while the corresponding field strengths are the nonmetricity 1-form Q_␣␤⫽⫺Dg_␣␤, and the 2-forms of torsion T␣⫽D␽␣ and curvature R_␤␣⫽d⌫_␤␣⫹⌫_␥␣⵩⌫_␤␥.

In our analysis of the teleparallel gravity we will heavily use the results related to the irreducible decompositions of the fundamental geometric and physical objects in MAG. The details about this method can be found in 关8兴 and 关9兴. Here, we will mainly need to recall the irreducible parts of the torsion 2-form T␣ which are defined by

(2)_T␣_ª1 3␽ ␣_{⵩T, with Tªe} ␣cT␣, 共2.1兲 (3)_T␣_ª⫺1 3*共␽ ␣_{⵩P兲, with P}_ª_*_共T␣_⵩_␽ ␣兲, 共2.2兲 (1)_T␣_ªT␣_⫺(2)_T␣_⫺(3)_T␣_. _共2.3兲 In tensor components, the torsion 2-form

T␣⫽1 2Ti j

␣_dxi_⵩dxj_⫽1 2T␮␯

␣_␽␮_⵩_␽␯ _共2.4兲 gives rise, through the above decomposition, to the vectors of trace and axial trace of torsion:

e_␣cT⫽T_␮␣␮, e_␣cP⫽1 2T

␮␯␭_␩

␮␯␭␣. 共2.5兲

Given the metric g_␣␤, the Christoffel connection⌫˜_␤␣ is defined as a unique connection with vanishing torsion and nonmetricity: D˜␽␣⫽0 and D˜ g_␣␤⫽0. The Riemannian op-erators and geometrical objects, constructed from the Christ-offel connection, will be denoted by a tilde. The general affine connection can always be decomposed into Riemann-ian and post-RiemannRiemann-ian parts,

⌫␤␣⫽⌫˜␤␣⫹N␤␣, 共2.6兲 where the distortion 1-form N_␣␤ can be expressed in terms of torsion and nonmetricity as

N_␣␤⫽⫺e[␣cT␤]⫹ 1 2共e␣ce␤cT␥兲␽ ␥_⫹共e [␣cQ␤]␥兲␽␥ ⫹1₂Q_␣␤. 共2.7兲

The MAG Lagrangian of the gravitational field is con-structed from the forms of curvature, torsion, and nonmetric-ity. In关9兴 the method of irreducible decompositions was ap-plied to the most general quadratic Lagrangian, revealing the possibility of reformulating the MAG field equations as an effective Einstein equation. Here we will use the results of

关9兴 to obtain an analogous reformulation of teleparallel

grav-ity.

III. FIELD EQUATIONS

In the teleparallel theory, we have two geometrical con-straints:

R_␤␣⫽d⌫_␤␣⫹⌫_␥␣⵩⌫_␤␥⫽0, 共3.1兲 Q_␣␤⫽⫺Dg_␣␤⫽⫺dg_␣␤⫹⌫_␤␥g_␣␥⫹⌫_␣␥g_␣␥⫽0.

共3.2兲

These equations mean that there is a distant parallelism in spacetime. The result of a parallel transport of a vector does not depend on the path. Moreover, the lengths and angles are not changed during a parallel transport.

One may wonder, what actually we do gain when we ‘‘embed’’ teleparallel gravity into the MAG theory. After all, Eq. 共3.1兲 may be solved trivially by performing a linear transformation of the frame and connection

e_␣→L␤_␣e_␤, ⌫_␣␤→L␥_␣⌫_␥␦L⫺1␤_␦⫹L⫺1␤_␥dL␥_␣.

共3.3兲

In view of Eqs.共3.1兲,共3.2兲, it is always possible to choose the matrix L␤_␣ of the linear transformation in such a way that the transformed local metric becomes the standard Minkowski, g_␣␤⫽o_␣␤ªdiag(⫺1,⫹1,⫹1,⫹1), whereas the transformed local connection becomes trivial:⌫_␣␤⫽0. Then the共co兲frame is left over as the only dynamical variable. We will call such a choice a Weitzenbo¨ck gauge. The visible

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disadvantage of such an approach is the resulting rigid struc-ture of the tetrad field which cannot be altered at one’s wish. To a great extent, this is contrary to the very spirit of rela-tivity theory, which treats all coordinates and frame systems on an equal footing basis. By embedding teleparallelism into MAG, we remove the unwarranted rigidity of the tetrad field and return to the far more physically natural situation, where we can choose a reference and coordinate system at our best convenience.

Now we can formulate teleparallel gravity as the MAG model with the constraints 共3.1兲,共3.2兲. Correspondingly, the Lagrangian of the teleparallel theory quite generally can be written as V⫽ 1 2␬T ␣_⵩_*

冉

兺

I⫽1 3 aI(I)T␣

冊

⫹ 1 2␮ ␣␤_⵩Q ␣␤⫺␯␣␤⵩R␣␤. 共3.4兲

Here a1,a2,a3 are the three dimensionless coupling con-stants, and the gravitational coupling constant is, as usual,

␬⫽8␲G/c3, with G the Newton’s gravitational constant. In-serting Eqs. 共2.1兲–共2.4兲 into the first term in Eq. 共3.4兲, we find T␣⵩*

冉

兺

I_⫽1 3 aI(I)T␣

冊

⫽共c1T␮␯␣T␮␯␣⫹c2T␮␣␮T␯␣␯ ⫹c3T␮␯␣T␣␮␯兲␩, 共3.5兲 where the new coefficients are the following combinations of the original coupling constants:

c1⫽ 2a1⫹a3 3 , c2⫽ 2共a2⫺a1兲 3 , c3⫽ 2共a3⫺a1兲 3 . 共3.6兲

Variation of the teleparallel action with respect to the Lagrange multipliers ␮␣␤ and ␯␣_␤ yields the constraints

共3.1兲 and 共3.2兲 of the geometry of distant parallelism. The

Lagrangian共3.4兲 is invariant under the redefinition

␮␣␤_→_␮␣␤_⫹D_␰␣␤_, _共3.7兲

␯␣

␤→␯␣␤⫹D␹␣␤⫺␰␣␤ 共3.8兲 of the Lagrange multiplier fields. Here,␰␣␤⫽␰␤␣is an arbi-trary symmetric 2-form, whereas␹␣_␤ is an arbitrary 1-form. The proof of the invariance is based on the three Bianchi identities of MAG. Indeed, we have for the additional␹term arising from Eq.共3.8兲 in the Lagrangian

共D␹␣

␤兲⵩R␣␤⫽d共␹␣␤⵩R␣␤兲⫹␹␣␤⵩DR␣␤. 共3.9兲 Being a total differential, the first term in the right-hand side can be discarded. The last term vanishes in view of the Bi-anchi identity DR_␣␤⬅0. As for the␰terms which appear in the Lagrangian when we make the transformation共3.7兲,共3.8兲, we find 1 2共D␰ ␣␤_兲⵩Q ␣␤⫹␰␣␤⵩R␣␤⫽ 1 2d共␰ ␣␤_⵩Q ␣␤兲⫺ 1 2␰ ␣␤ ⵩共DQ␣␤⫺2R(␣␤)兲. 共3.10兲

Again, as a total derivative, the first term can be neglected, whereas the last term vanishes by virtue of the Bianchi iden-tity DQ_␣␤⬅2R(␣␤). As a result, the field equations derived from Eq.共3.4兲 will determine the Lagrange multipliers only up to the ambiguity imposed by the symmetry 共3.7兲,共3.8兲.

The gravitational field equations are derived from the total Lagrangian V⫹Lmatby independent variations with respect to the coframe ␽␣ and connection ⌫_␤␣ 1-forms. The corre-sponding so-called first and second field equations read

DH_␣⫺E_␣⫽⌺_␣, 共3.11兲

DH␣_␤⫺E␣_␤⫽⌬␣_␤. 共3.12兲 The covector-valued forms appearing in the left-hand side of Eq. 共3.11兲 are given by

H_␣ª⫺ ⳵V ⳵T␣⫽⫺ 1 ␬*

冉

I

兺

_⫽1 3 aI(I)T␣

冊

, 共3.13兲 E_␣ªe_␣cV⫹共e_␣cT␤兲⵩H_␤⫽1 2关共e␣cT ␤_兲⵩H ␤⫺T␤ ⵩共e␣cH␤兲兴. 共3.14兲

In the last expression we have taken into account that V⫽

⫺1

2T␣⵩H␣, in view of Eqs.共3.13兲 and 共3.4兲.

The matter sources in the right-hand sides of 共3.11兲 and

共3.12兲 are the canonical energy-momentum current and the

canonical spin current. For the minimally coupled matter field⌿A( p-form, in general兲, we have explicitly

⌺␣ª⳵ Lmat

⳵␽␣⫽e␣cLmat⫺共e␣cD⌿A兲⵩

⳵Lmat ⳵D⌿A⫹共e␣c⌿ A_兲 ⵩⳵Lmat ⳵⌿A , 共3.15兲 ⌬␣ ␤ª⳵ Lmat ⳵⌫␣␤⫽共ᐉ ␣ ␤AB⌿B兲⵩ ⳵Lmat ⳵D⌿A. 共3.16兲

We will assume that the matter field⌿A realizes some rep-resentation of the Lorentz group, and then ᐉ_␣␤⫽⫺ᐉ_␤␣ are the corresponding generators of infinitesimal Lorentz trans-formations. Accordingly, the hypermomentum current re-duces to the spin current,

⌬␣␤⫽␶␣␤⫽⫺␶␤␣. 共3.17兲 The tensor-valued forms appearing in the left-hand side of the field equation 共3.12兲 are given by

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M␣␤ª⫺2 ⳵V ⳵Q_␣␤⫽␮ ␣␤_, _共3.18兲 H␣_␤ª⫺ ⳵V ⳵R_␣␤⫽␯ ␣ ␤, 共3.19兲 E␣_␤ª⫺␽␣⵩H_␤⫺M␣_␤⫽⫺␽␣⵩H_␤⫺␮␣_␤. 共3.20兲

Substituting all this into Eq.共3.12兲, we find the equation

␮␣

␤⫹D␯␣␤⫽⫺␽␣⵩H␤⫹⌬␣␤. 共3.21兲 The left-hand side of this equation is evidently invariant un-der the transformations共3.7兲,共3.8兲. Correspondingly, the field equation 共3.21兲 offers maximum of the possible: It deter-mines the gauge invariant piece of the Lagrange multipliers, namely␮␣_␤⫹D␯␣_␤, in terms of the spin current ⌬␣_␤ and of the translational field momentum H_␣.

It is important to notice that the Lagrange multipliers␮␣␤ and ␯␣_␤ decouple from the first field equation 共3.11兲. As a result, technically we can simply discard the second field equation 共3.12兲 because it merely determines the Lagrange multipliers, whereas the dynamics of the gravitational field is governed by Eq.共3.11兲.

IV. EFFECTIVE EINSTEIN EQUATION

In关9兴 it was demonstrated that a certain quadratic metric-affine Lagrangian has very special properties. This Lagrang-ian reads V(0)⫽ 1 2␬

再

⫺R␣␤⵩␩ ␣␤_⫺(1)_T␣_⵩_*(1)_T ␣⫹2(2)T␣⵩*(2)T␣ ⫹1 2 (3)_T␣_⵩_*(3)_T ␣⫹(2)Q␣␤⵩␽␤⵩*(1)T␣⫺2(3)Q␣␤ ⵩␽␤_⵩_*(2)_T␣_⫺2(4)_Q ␣␤⵩␽␤⵩*(2)T␣⫹ 1 4 (1)_Q ␣␤ ⵩*(1)_Q␣␤_⫺1 2 (2)_Q ␣␤⵩*(2)Q␣␤⫺ 1 8 (3)_Q ␣␤ ⵩*(3)_Q␣␤_⫹3 8 (4)_Q ␣␤⵩*(4)Q␣␤⫹共(3)Q␣␥⵩␽␣兲 ⵩*_共(4)_Q␤␥_⵩_␽ ␤兲

冎

. 共4.1兲

The definition of the four irreducible parts of the nonmetric-ity (J)Q_␣␤, J⫽1, . . . ,4 is given in 关8,9兴; we do not need them here though because the linear and quadratic non-metricity terms are zero anyway, in view of the teleparallel constraint共3.2兲.

The gravitational gauge field momenta for the Lagrangian

共4.1兲 are given by H_␣(0)ª⫺⳵V (0) ⳵T␣ ⬅⫺ 1 2␬N ␮␯_⵩_␩ ␣␮␯, H(0)␣␤ª⫺⳵ V(0) ⳵R_␣␤ ⫽ 1 2␬ ␩ ␣ ␤, 共4.2兲

and the corresponding field equations coincide completely with Einstein’s equations of general relativity:

DH_␣(0)⫺E_␣(0)⬅ 1 2␬R˜

␮␯_⵩_␩

␣␮␯, DH(0)␣␤⫺E(0)␣␤⬅0.

共4.3兲

We can use this fact and identically rewrite the teleparallel Lagrangian 共3.4兲 as the sum

V⫽⫺a1V(0)⫹Vˆ, 共4.4兲 where Vˆ⫽ 1 2␬共␣2T ␣_⵩_*(2)_T ␣⫹␣3T␣⵩*(3)T␣兲⫹ 1 2␮ ␣␤_⵩Q ␣␤ ⫺␯␣ ␤⵩R␣␤, 共4.5兲

and the new ‘‘shifted’’ coupling constants are defined by

␣2⫽a2⫹2a1, ␣3⫽a3⫹ 1

2a1. 共4.6兲 Correspondingly, the field momenta can be rewritten in the form M␣␤⫽⫺a1M(0)␣␤⫹Mˆ␣␤, H␣⫽⫺a1H_␣(0)⫹Hˆ␣, H␣_␤⫽⫺a1H(0)␣␤⫹Hˆ␣␤, 共4.7兲 E_␣⫽⫺a1E␣ (0)_⫹Eˆ ␣, E␣␤⫽⫺a1E(0)␣␤⫹Eˆ␣␤. 共4.8兲

We easily find that Mˆ␣␤ª⫺2⳵Vˆ /⳵Q_␣␤⫽M␣␤, and Hˆ␣_␤ª

⫺⳵Vˆ /⳵R_␣␤⫽H␣_␤, whereas Hˆ_␣ª⫺ ⳵V ˆ ⳵T␣⫽⫺ 1 ␬*共␣2(2)T␣⫹␣3(3)T␣兲 ⫽⫺₃1_␬关␣2*共␽␣⵩T兲⫹␣3␽␣⵩P兴. 共4.9兲 Here, we explicitly used the irreducible decompositions共2.1兲 and 共2.2兲 of the torsion, in terms of the trace 1-form T and the axial trace 1-form P. Then, with the help of the identities

共4.3兲, one can transform the field equations 共3.11兲 and 共3.12兲

of MAG into

⫺a1 2 R˜

␮␯_⵩_␩

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␮␣

␤⫹D␯␣␤⫽⫺␽␣⵩Hˆ␤⫹⌬␣␤.

共4.11兲

Accordingly, we can view the teleparallel field equations as the Einstein general relativity theory共4.10兲 with the effective energy-momentum current ⌺_␣eff⫽(1/⫺a₁)(⌺_␣⫺DHˆ_␣⫹Eˆ_␣). Recall also that Eˆ_␣⫽e_␣cVˆ⫹(e_␣cT␤)⵩Hˆ_␤ and DHˆ_␣⫽D˜ Hˆ_␣

⫺N␣␤⵩Hˆ␤. Now, substituting Eqs.共4.9兲 and 共2.7兲, we find explicitly DHˆ_␣⫺Eˆ_␣⫽␣2 3␬

冋

⫺␩␣␤⵩D˜共e ␤_cT兲⫹_*_T ␣⵩T⫺ 1 2␽␣⵩P⵩T ⫺1 2e␣c共T⵩*T兲

册

⫹ ␣3 3␬

冋

␽␣⵩dP⫺2T␣⵩P⫹P ⵩e␣c*P⫹ 1 2e␣c共P⵩*P兲

册

. 共4.12兲 The effective Einstein equation共4.10兲 contains symmetric and antisymmetric parts. It is convenient to consider them separately. The antisymmetric piece is extracted by taking the interior product of e␣c with Eq. 共4.10兲. Taking into ac-count Eq.共4.12兲, the result reads

2␣₃d P⫺␣₂共*dT⫹P⵩T兲⫹e_␣c共␣₂*T␣⵩T⫺2␣₃T␣⵩P兲

⫽3␬e␣c⌺_␣. 共4.13兲

We can now subtract the antisymmetric part from Eq.共4.10兲, which technically means substituting d P from the above equation into the effective Einstein equation. As a result, we finally obtain ⫺a1 2 R˜ ␮␯_⵩_␩ ␣␮␯⫽␬

冉

⌺␣⫺ 1 2␽␣⵩e ␤_c⌺ ␤

冊

⫹␣₃2

冋

␩␣␤ ⵩D˜_共e␤_cT兲⫺1 2␽␣⵩*dT⫺*T␣⵩T ⫹1 2␽␣⵩e␤c共*T ␤_⵩T兲⫹1 2e␣c共T⵩*T兲

册

⫹␣3 3

冋

2T␣⵩P⫺␽␣⵩e␤c共T ␤_⵩P兲⫺P ⵩e␣c*P⫺ 1 2e␣c共P⵩*P兲

册

. 共4.14兲 Analogously to the separation of the antisymmetric part, we can also extract the trace of the effective Einstein field equation. For this purpose, we multiply Eq. 共4.14兲 with ␽␣

⵩ from the left, and we find

␣2共d*T⫹T⵩*T兲⫹ ␣3 3 P⵩*P⫽␬␽ ␣_⵩⌺ ␣⫺a1R␩. 共4.15兲

Here, RªR_␣␤␤␣ is the curvature scalar, and we used the identities: ␽␣⵩*T_␣⫽⫺*T, ␽␣⵩␽␤⵩␩_␮␯⫽2(␦_␮␣␦_␯␤

⫺␦␯␣␦␮␤)␩. Other useful formulas are e␣c*␺⫽*(␺⵩␽␣) for any form␺; also␽␣⵩␩_␤⫽␦_␤␣␩, ␽␣⵩␩_␮␯⫽␦_␯␣␩_␮⫺␦_␮␣␩_␯, and␽␣⵩␩_␮␯␭⫽␦_␮␣␩_␯␭⫹␦_␯␣␩_␭␮⫹␦_␭␣␩_␮␯.

V. GENERAL RELATIVITY LIMIT: A PROBLEM WITH SPINNING MATTER?

When the coupling constants are chosen as

a1⫽⫺1, a2⫽2, a3⫽ 1

2, 共5.1兲

we find from Eq.共4.6兲 that all␣I⫽0. 关In terms of the tensor reformulation 共3.5兲, the relations 共3.6兲 yield c1⫽⫺21,c2

⫽2, and c3⫽1. These are the well-known values of the teleparallel equivalent of GR, as used in关3–5兴, e.g.兴 Conse-quently, Vˆ⫽0, and thus Hˆ_␣⫽0 and Eˆ_␣⫽0. The teleparallel field equations共4.10兲 reduce to the general relativity theory, except for the fact that the physical source in the right-hand side of Eq. 共4.10兲 is not the ‘‘metrical,’’ but the canonical energy-momentum current ⌺_␣. Thus one should be careful when the matter field has nontrivial spin.

In order to check the consistency of the teleparallel theory, we first notice that Eq.共4.11兲 forces the spin current to satisfy

␶␣␤⫽D␯[␣␤]. 共5.2兲

Consequently, it must be conserved:

D␶_␣␤⫽DD␯[␣␤]⫽0. 共5.3兲 As the next step, we multiply the Einstein equation 共4.10兲 from the left by ␽_␤. Then, making an antisymmetrization, we find that the antisymmetric piece of the energy-momentum current must vanish:

␽[␤⵩⌺␣]⫽0. 共5.4兲

This is a consequence of the symmetry of the Einstein tensor which is equivalent to the condition R˜␮␯⵩␽[␤⵩␩␣]␮␯⫽0. The same conclusion is obtained directly from Eq.共4.13兲, in which␣I⫽0 leads to the vanishing left-hand side. Recalling now the angular momentum conservation law,

␽[␤⵩⌺␣]⫹D␶␤␣⫽0, 共5.5兲 we conclude that the spin current is separately conserved, in full agreement with Eq. 共5.3兲.

Summarizing, the teleparallel gravity can consistently couple either to a spinless matter or to a matter with a con-served spin tensor. Since, for example, the Dirac spinor field does not have such properties, the teleparallel description of gravity is not applicable to that case.

‘‘New general relativity’’

The ‘‘new general relativity’’ 关1,2,12–14兴 model is de-fined by choosing the coupling constants as follows:

(6)

a1⫽⫺1, a2⫽2, a3⫽” 1

2. 共5.6兲

With this choice, the above inconsistency problem is avoided.

Indeed, now we have ␣₃⫽” 0, and hence Eq. 共4.13兲 re-duces to the meaningful equation for the axial trace of tor-sion:

d P⫺e_␣c共T␣⵩P兲⫽ 3␬ 2␣3

e␣c⌺_␣. 共5.7兲

The right-hand side, which represents the antisymmetric part of the energy-momentum current, can be nontrivial now.

A different type of inconsistency which arises for this model was first noticed by Kopczyn´ski 关6兴 who has shown the existence of an ‘‘extra symmetry’’ of the Lagrangian which deforms the coframe 共without touching the connec-tion兲 in such a way that the axial trace remains invariant. Such a symmetry makes the theory physically nonpredictable because torsion is not determined uniquely by the field equa-tions. Later, Nester 关7兴 clarified that point by establishing conditions under which such a hidden symmetry can arise.

VI. CONFORMAL TRANSFORMATIONS IN TELEPARALLEL GRAVITY

Conformal transformation in gravity is usually understood as the scaling of the line element

ds2→⍀2ds2, or equivalently, gi j→⍀2gi j. 共6.1兲 The conformal factor⍀ can be an arbitrary function of the spacetime coordinates. In teleparallel gravity, the above transformation is naturally realized as the scaling of the cof-rame:

␽␣_→⍀_␽␣_. _共6.2兲

In terms of the components ␽␣⫽␽_i␣dxi, the conformal transformation reads

␽i␣→⍀␽i␣, and accordingly gi j⫽␽i␣␽␤jg␣␤→⍀ 2_g

i j.

共6.3兲

The local metric, which in the case of the orthonormal frames is equal to the Minkowski metric g_␣␤⫽o_␣␤ªdiag (⫺1,⫹1,⫹1,⫹1), is not affected by the scaling. The con-formal transformation of the coframe共6.2兲 induces the scal-ing of the volume 4-form, ␩ª␽0ˆ⵩␽1ˆ⵩␽2ˆ⵩␽3ˆ→⍀4␩. The dual frame vectors e_␣ evidently transform to ⍀⫺1e_␣ under Eq. 共6.2兲.

It is natural to assume that the local linear connection

⌫␤␣ is conformally invariant. This is what we encounter in the realization of conformal symmetry in the Poincare´ gauge gravity 关15–17兴; see also the discussion in 关18兴. Then we conclude that the Weitzenbo¨ck conditions共3.1兲 and 共3.2兲 are not changed by the conformal transformation. Consequently,

Eq. 共6.2兲 leaves the fundamental structure of the teleparallel gravity untouched, and it describes a map between the dif-ferent teleparallel models.

Unlike curvature and nonmetricity, the torsion 2-form

共which plays the role of the gauge field strength in the

teleparallel theory兲 transforms in a nontrivial way under the conformal scaling. Under the action of Eq.共6.2兲, we find

T␣⫽D␽␣→⍀T␣⫹d⍀⵩␽␣. 共6.4兲 As a result, the torsion trace 1-form changes as T→⍀T

⫺3d⍀, and hence only the second part of torsion 共2.1兲 has

the nontrivial transformation property

(2)_T␣_→⍀(2)_T␣_⫹d⍀⵩_␽␣_, _共6.5兲 whereas the two other irreducible parts of torsion, given by Eqs. 共2.2兲 and 共2.3兲, transform covariantly:

(1)_T␣_→⍀(1)_T␣_, (3)_T␣_→⍀(3)_T␣_. _共6.6兲 Accordingly, the class of models共3.4兲, with a vanishing cou-pling constant a2⫽0, will display the proper conformal be-havior in the sense that the teleparallel Lagrangian is re-scaled as

V→⍀2_V. _共6.7兲

Notice that, since the action is changed, it is therefore not conformal invariant.

We can use the general Lagrange-Noether machinery 关8兴 developed for MAG to derive the corresponding properties of an arbitrary model with a proper conformal behavior. Let us consider a model with the most general Lagrangian L

⫽L(␺,D␺,g_␣␤,␽␣,⌫_␣␤,T␣) for an arbitrary matter field␺ interacting with the teleparallel gravitational field. Under the action of the above defined 共infinitesimal兲 conformal trans-formation, the Lagrangian changes by

␦L⫽␦␽␣⵩⌺_␣⫹␦␺⵩⳵L ⳵␺⫹d

冋

␦␽␣⵩ ⳵L ⳵T␣⫹␦␺⵩ ⳵L ⳵D␺

册

. 共6.8兲

Here, we took into account the conformal invariance of the metric and connection, ␦g_␣␤⫽0 and ␦⌫_␣␤⫽0. Suppose now that the matter field and the Lagrangian have the proper conformal behavior in the sense that, under the infinitesimal rescaling (⍀⫽1⫹␻), we have

␦␺⫽k␻␺, ␦L⫽ᐉ␻L, 共6.9兲 with the numbers k andᐉ giving the conformal weight of the matter field and of the Lagrangian, respectively. Substituting this into Eq. 共6.8兲, we find the two identities:

␽␣_⵩ ⳵L ⳵T␣⫹k␺⵩ ⳵L ⳵D␺⫽0, 共6.10兲 ᐉL⫺␽␣_⵩⌺ ␣⫹␽␣⵩D ⳵ L ⳵T␣⫺T ␣_⵩ ⳵L ⳵T␣

(7)

⫺D␺⵩_⳵⳵L

D␺⫺k

␦L

␦␺⫽0. 共6.11兲

These arise, as usual 关8兴, by considering the terms propor-tional to␻ and d␻ separately.

Specializing now to the case of the purely gravitational Lagrangian L⫽V, which does not depend on␺ and has the conformal weight ᐉ⫽2 关see Eq. 共6.7兲兴, we obtain the two identities ␽␣_⵩ ⳵V ⳵T␣⫽0, 2V⫽␽ ␣_⵩ ⳵V ⳵␽␣⫹T␣⵩ ⳵V ⳵T␣. 共6.12兲 Turning to the teleparallel model 共3.4兲 under consideration, we can verify explicitly that these identities are indeed valid for the case a2⫽0 共note that H␣⫽⫺⳵V/⳵T␣).

Every extra invariance of the Lagrangian means that there is a certain arbitrariness in the classical solutions of the field equations. In particular, the above analysis shows that the solutions of the teleparallel models with a2⫽0 can only be determined up to an arbitrary scale factor ⍀ of the coframe field. Correspondingly, in order to have a predictable telepar-allel theory, we will mainly confine ourselves to the class of Lagrangians with a2⫽” 0.

VII. SPHERICAL SYMMETRY AND GEOMETRIC INVARIANTS

Let us now proceed with the analysis of the classical so-lutions of the general teleparallel model. As a first step, we naturally turn our attention to the compact object configura-tions, and specifically to the case of spherical symmetry. It is worthwhile to note that such a study was never performed in full generality for the models with the three arbitrary cou-pling constants a1,a2,a3. A partial analysis was done in关2兴. As usual in the study of exact solutions, we have two complementary aspects. The first one concerns the conve-nient choice of the local coordinates and of the correspond-ing ansatz for the dynamical fields. The second aspect is to provide the invariant characterization of the resulting geom-etry. Roughly speaking, the choice of an ansatz helps to solve the field equations more easily, whereas the invariant de-scription provides the correct understanding of the physical contents of a solution.

We look for a spherically symmetric solution with the line element

g⫽⫺A2dt2⫹B2共dr2⫹r2d␪2⫹r2sin2␪d␾2兲. 共7.1兲 The two functions A⫽A(r) and B⫽B(r) depend on the ra-dial variable r. It is, however, not so trivial to come up with the ansatz for a tetrad. A convenient choice reads

␽0ˆ_⫽Adt, _␽1ˆ_⫽Bdr, _␽2ˆ_⫽Brd_␪_, _␽3ˆ_{⫽Br sin}_␪_d_␾_.

共7.2兲

In addition, for the共non-Riemannian兲 connection we choose

⌫21⫽⫺⌫12⫽⫺d␪, ⌫31⫽⫺⌫13⫽⫺sin␪d␾,

A B

冊

⬘

册

⬘

. 共7.9兲

The components of this 2-form represent the Weyl tensor, W˜␣␤⫽1

2C␮␯␣␤␽␮⵩␽␯. The Weyl quadratic invariant thus reads

W˜_␣␤⵩*W˜␣␤⫽W 2

3 ␩, 共7.10兲

and consequently we can consistently use Eq. 共7.9兲 for the description of the resulting geometry.

Besides the Riemannian Weyl tensor, the spacetime ge-ometry is naturally characterized by the quadratic invariants of the torsion. For the spherically symmetric configurations

共7.4兲, we have explicitly T_␣⵩*T␣⫽ 1 B2

冋冉

A

⬘

A

冊

2 ⫺2

冉

B_B

⬘

冊

2

册

␩. 共7.11兲 These two invariants—the Riemannian curvature and the quadratic torsion—provide the sufficient tools for under-standing the contents of the classical solutions.

VIII. COUPLED GRAVITATIONAL AND ELECTROMAGNETIC FIELDS

As we discovered above, the material sources with spin may lead to certain inconsistencies in the framework of teleparallelism. Correspondingly, in order to be on the safe side, we will limit ourselves to the case of the spinless matter when the teleparallel gravity appears to be totally applicable. Of all the possible spinless matter, the Maxwell field clearly represents a very interesting and physically important case.

Accordingly, in our study of the spherically symmetric solutions, we will investigate the case when matter is repre-sented by the electromagnetic field. The spin current of the electromagnetic field is trivial, whereas its energy momen-tum reads

⌺␣⫽

␭

2关共e␣cF兲⵩*F⫺共e␣c*F兲⵩F兴. 共8.1兲 Here, the electromagnetic vacuum constant 共the ‘‘vacuum impedance’’兲

␭⫽

冑

␧0

␮0

, 共8.2兲

is defined in terms of the electric ␧0 and magnetic␮0 con-stants of the vacuum.

Using the standard spherically symmetric ansatz for the electromagnetic potential 1-form,

A⫽ f A␽

0ˆ_{⫽ f dt,} _共8.3兲

with f⫽ f (r), we have for the field strength

F⫽dA⫽⫺ f

⬘

AB␽

0ˆ_⵩_␽1ˆ_, _*_F_⫽ f

⬘

AB␽

2ˆ_⵩_␽3ˆ_. _共8.4兲 Then, we derive the Maxwell equation:

d*F⫽ 1 r2B3 d dr

再

r2B A f

⬘

冎

␽ 1ˆ_⵩_␽2ˆ_⵩_␽3ˆ_⫽0. _共8.5兲 On the other hand, a direct computation yields for the left-hand side of the teleparallel gravitational field equations

_共a 1⫺a2兲 A

⬘

A ⫺共a1⫹2a2兲 B

⬘

B

册冎

冊

2 ␽0ˆ_⵩_␽1ˆ_⵩_␽2ˆ_. _共8.16兲

(9)

In the presence of a nontrivial electromagnetic field, we need, in addition to the above geometric invariants, an in-variant description of the matter source configuration. As it is well known, there are two invariants of the Maxwell field. For the spherical ansatz共8.4兲, one of the invariants is trivial, F⵩F⬅0, whereas the other reads

F⵩*F⫽⫺

冉

f

⬘

AB

冊

2

␩. 共8.17兲

IX. ANALYZING THE FIELD EQUATIONS

The Maxwell equation 共8.5兲 can be straightforwardly in-tegrated. This yields

f

⬘

⫽ qA

r2B, 共9.1兲

with q an integration constant. Its value is determined by the total electric charge Q of the source which is calculated as usual from the integral over the 2-sphere around the source:

Q⫽

冕

S₂␭ *F⫽

冕

S₂␭ f

⬘

AB␽ 2ˆ_⵩_␽3ˆ_⫽

冕

S₂␭q sin␪ d␪⵩d␾ ⫽4␲␭q. 共9.2兲

Inserting Eq.共9.1兲 into Eq. 共8.17兲, we find for the Maxwell invariant

F⵩*F⫽⫺ q 2

r4B4␩. 共9.3兲 From Eqs. 共8.6兲–共8.9兲 and Eqs. 共8.13兲–共8.16兲 we find, after making use of Eq.共9.1兲, and of some simple rearrange-ments: U0⫽⫺ 6␬␭q2 r4B2 , U1⫽ 3␬␭q2 r4B2 , U2⫽0. 共9.4兲 Using Eq.共8.10兲 in Eq. 共9.4兲, we obtain the equation

B ⫽ k₁⫺3␬␭q f r2_AB , 共9.6兲 with k1 an integration constant. Analogously, from Eqs.

共8.12兲 and 共9.4兲 we find another first integral: 共a1⫺a2兲 A

⬘

B ⫽ k2 r3AB, 共9.7兲

with k2 a second integration constant. Let us introduce a new variable by

␸ªk1⫺3␬␭q f . 共9.8兲

Differentiating this and using Eq.共9.1兲, we find the equation

␸

⬘

⫽⫺3␬␭q

2_A

r2_B . 共9.9兲

Suppose a1a2⫽” 0 共the special case a1a2⫽0 will be consid-ered separately兲. Then, combining Eqs. 共9.6兲 and 共9.7兲, we finally get the system of first order equations

A

⬘

A⫽

1 9a1a2r2AB

冋

共a1⫹2a2兲␸⫺2共2a1⫹a2兲 k2 r

册

, 共9.10兲 B

⬘

B⫽ 1 9a1a2r2AB

冋

共a1⫺a2兲␸⫺共4a1⫺a2兲 k2

r

册

.

共9.11兲

Together with Eq. 共9.9兲, these equations comprise a system of three first order ordinary differential equations for the three unknown functions A,B, f . As an immediate conse-quence, the sum of Eqs.共9.10兲 and 共9.11兲 yields

共AB兲

⬘

⫽ 1

9a₁a₂r2

冋

共2a1⫹a2兲␸⫺共8a1⫹a2兲 k2

r . 共10.4兲

Substituting this, together with Eq.共10.1兲, into Eq. 共9.9兲, we get

a₀⫽ k2

3␬␭q2. 共10.5兲

Furthermore, the integration of Eq. 共10.3兲 yields

B2⫽␬␭q 2 a2r2

⫹k3. 共10.6兲

The new integration constant is fixed to be equal to zero k₃

⫽0, which follows from Eq. 共9.13兲 after all the above is

substituted. Thus, we obtain finally the conformally flat so-lution A2⫽ k₂2 9␬␭q2a₂r2, B 2_⫽␬␭q 2 a₂r2 . 共10.7兲 In order to understand this spacetime geometry, we have to compute the corresponding Riemannian curvature 共recall that the total Riemann-Cartan curvature is identically zero兲. A direct calculation yields

R ˜0ˆ1ˆ_⫽ a2 ␬␭q2␽ 0ˆ_⵩_␽1ˆ_, _R_˜2ˆ3ˆ_⫽⫺ a2 ␬␭q2␽ 2ˆ_⵩_␽3ˆ_, _共10.8兲

with all other components of the curvature 2-form trivial. As we see, the curvature is everywhere regular, and the Riemann tensor has the double-duality property,

R

˜_␣␤␥␦_⫽1 4␩

␥␦␳␴_␩

␣␤␮␯R˜␳␴␮␯. 共10.9兲 The Riemannian Weyl tensor vanishes identically, as it is clearly seen from Eq. 共7.8兲. This is consistent with the fact that the metric is conformally flat. Geometrically, the result-ing spacetime is the direct product of the two 2-dimensional spaces of constant curvature, a hyperbolic space and a sphere. A solution of that type was originally described by Bertotti and Robinson 关10,11兴 in the framework of general relativity theory.

In the teleparallel gravity, we have torsion as the basic field variable. It is straightforward to see that in the general-ized Bertotti-Robinson solution 共10.7兲, torsion is also con-stant with the quadratic invariant共7.11兲 given by

T_␣⵩*T␣⫽⫺ a2

␬␭q2␩. 共10.10兲 Note that the constant magnitude of the torsion is again de-scribed by the same combination of the coupling constants which determine the constant Riemannian curvature 共10.8兲 of that solution.

The electromagnetic invariant 共9.3兲, when using Eq.

共10.7兲, demonstrates that the source is likewise represented

by the ‘‘constant’’ electromagnetic field configuration:

F⵩*F⫽⫺

冉

a2

␬␭q

冊

2

␩. 共10.11兲

Vanishing coupling constant a1Ä0

Before we proceed to consider the general case, we have to analyze the special case when a1a2⫽0; see Sec. IX. When a₂⫽0, we have the conformally covariant theory, and consequently, the conformal factor of the metric remains un-determined. We will not consider such models which lack physical predictability.

When, however, a₁⫽0, the pair of equations 共9.6兲 and

共9.7兲 reduce to the system

A

⬘

A ⫹2 B

⬘

B⫽⫺ ␸ a₂r2AB, 共10.12兲 A

⬘

A ⫹2 B

⬘

B⫽⫺ k2 a2r3AB . 共10.13兲

This system yields the explicit electromagnetic function

␸⫽k_r2. 共10.14兲

Substituting this into Eq.共9.9兲, we find the proportionality of the metric functions:

(11)

B⫽3␬␭q 2 k2

A. 共10.15兲

This brings us then back to the results of the previous sub-section.

XI. UNCHARGED SOLUTIONS

From now on, we will confine our attention to the generic teleparallel models with a1a2⫽” 0. Let us first obtain the con-figurations with zero charge, q⫽0. Then, from Eq. 共9.8兲 we have ␸⫽k1, and Eq.共9.12兲 can be immediately integrated:

AB⫽ 1

9a1a2r

冋

⫺共2a1⫹a2兲k1⫹共8a1⫹a2兲 k₂ 2r

册

⫹k3.

共11.1兲

When we substitute this into Eq.共9.14兲, we find the value of the new integration constant explicitly:

k3⫽

共a1⫹2a2兲k1 2 36a1a2k2

. 共11.2兲

A. Special case: 8a1¿a2Ä0

Suppose the coupling constants satisfy

8a1⫹a2⫽0. 共11.3兲

Then, Eqs.共11.1兲 and 共11.2兲 yield

AB⫽ k1 12a1

冉

5k1 8k2⫺ 1 r

冊

. 共11.4兲

r0⫽8k2/5k1, A0B0⫽5k1 2_/96k

2a1.

This spacetime is not a black hole. The curvature tensor is singular at r⫽0 and at r⫽r₀, so although the metric com-ponent g00⫽⫺A2 vanishes at the finite radius r⫽r0, this is

not a horizon. Indeed, substituting A and B into Eq.共7.8兲, we get the Riemannian Weyl curvature:

W⫽⫺r0共48r 3_⫺136r2_r 0⫹110rr0 2_⫺25r 0 3_兲 4r6B₀2e5r0/2r共r 0/r⫺1兲3/2 . 共11.8兲

It is easy to see that it diverges at r⫽r0. Analogously, for the torsion invariant共7.11兲, we find

T_␣⵩*T␣ ⫽r0 2_共172r4_⫺564r3_r 0⫹687r 2_r 0 2_⫺370rr 0 3_⫹75r 0 4_兲 16r8B₀2e5r0/2r共r 0/r⫺1兲7/2 ␩. 共11.9兲

Thus, from both Riemannian and teleparallel viewpoints, the resulting geometry is singular at r⫽0 and at r⫽r0.

B. General case: 8a1¿a2Ä”0

When 8a1⫹a2⫽” 0, we can use Eq. 共11.2兲 to rewrite Eq.

共11.1兲 as AB⫽k1 2_共8a 1⫹a2兲 18a1a2k2

冉

k2 k1r ⫺␣ 2

冊冉

k2 k1r ⫺␤ 2

冊

, 共11.10兲 where we have introduced the constant parameters

␣⫽2共2a1⫹a2兲⫹

冑

⫺18a1a2 8a1⫹a2

, 共11.11兲

␤⫽2共2a1⫹a2兲⫺

冑

⫺18a1a2 8a1⫹a2

. 共11.12兲

In particular, we can easily see that

␣⫹␤ 2 ⫽ 2共2a1⫹a2兲 8a₁⫹a₂ , ␣␤ 4 ⫽ a1⫹2a2 2共8a₁⫹a₂兲. 共11.13兲

From Eqs.共11.11兲,共11.12兲 we conclude that the product a1a2 must be negative. Substituting Eq.共11.10兲 into Eq. 共9.10兲, we find the equation

1⫺␤ . 共11.16兲

(12)

The integration constants are originally related by A0B0

⫽k1 2

(8a1⫹a2)/18a1a2k1. However, taking into account the possibility of scaling both the time and the radial coordinate by two arbitrary factors, the constants A0 and B0 can have any real value. In order to simplify the notation, it will be convenient to introduce m⫽k2/k1.

Performing a differentiation in Eq.共7.8兲, we find the non-trivial Riemannian Weyl curvature:

W⫽ mw r6_B 0 2

冉

m r⫺ ␣ 2

冊

2␣⫺4

冉

m r⫺ ␤ 2

冊

2␤⫺4 , 共11.17兲

where we denoted the polynomial

wª1 8兵32m 3_共_␣_⫹_␤_兲共_␣_⫹_␤_⫺1兲⫹4m2_r_关共_␣_⫹_␤_{兲„3⫺2共}_␣ ⫹␤兲⫺16␣␤…⫹12␣␤兴⫹2mr2_关8_␣␤_共2_␣␤_⫹_␣_⫹_␤_⫺1兲 ⫺共␣⫹␤兲2_兴⫹3r3_␣␤_共_␣_⫹_␤_⫺4_␣␤_兲_其 ⫽ ⫺3a2 共8a1⫹a2兲2 关a1共r⫺4m兲共3r2⫺8mr⫹8m2兲⫹2a2共r ⫺m兲2_{共3r⫺8m兲兴.} _共11.18兲

Analogously, for the quadratic torsion invariant 共7.11兲, we find T_␣⵩*T␣⫽m 2_T r6B0 2

冉

m r ⫺ ␣ 2

冊

2␣⫺4

冉

m r⫺ ␤ 2

冊

2␤⫺4 ␩, 共11.19兲

where we have another polynomial defined as

Tªm2_关3共_␣_⫹_␤_兲2_⫹8共1⫺_␣_⫺_␤_{兲兴⫹2mr关共}_␣_⫹_␤_兲2_⫺共_␣_⫹_␤_兲 ⫻共2⫹3␣␤兲⫹4␣␤兴⫹r2_关共_␣_⫹_␤_兲2_/2_⫹_␣␤_„3_␣␤_⫺2共_␣ ⫹␤兲…兴⫽ 12 共8a1⫹a2兲2 关a1 2 共r⫺4m兲2_⫹2a 2 2 共r⫺m兲2_兴. 共11.20兲

XII. COUPLING CONSTANTS, INVARIANTS, AND SINGULARITIES

The values of the constants ␣ and ␤ are crucial for un-derstanding the spacetime geometry of the solutions obtained above, since they determine the behavior of the metric func-tions共11.15兲 and 共11.16兲. A straightforward analysis of Eqs.

共11.11兲,共11.12兲 shows that we can simplify those formulas to

the following equivalent ones:

␣⫽2共a1⫹

冑

⫺2a1a2兲 4a1⫹

冑

⫺2a1a2 , 共12.1兲 ␤⫽2共a1⫺

冑

⫺2a1a2兲 4a1⫺

冑

⫺2a1a2 . 共12.2兲

Moreover, one can eliminate

冑

⫺2a1a2 from Eqs.

共12.1兲,共12.2兲, and express one constant directly in terms of

the other:

␣⫽4₅⫺5_⫺4␤_␤, or equivalently, ␤⫽4⫺5␣

5⫺4␣. 共12.3兲 From these simple derivations, we can immediately establish a number of important consequences. Since a1a2⬍0 共in or-der to have real solutions兲, we have to analyze the two cases for the original coupling constants:共a兲 a1⬎0,a2⬍0, and 共b兲 a1⬍0,a2⬎0.

For a1⬎0,a2⬍0, we see from Eq. 共12.1兲 that␣ is posi-tive. Moreover,

1

2⬍␣⬍2 for all a1,a2. 共12.4兲 At the same time, Eq.共12.2兲 shows that

0⬍␤⬍1

2 for a1⬎⫺2a2, 共12.5兲

␤⬎2 for a1⬍⫺a2/8, 共12.6兲

␤⬍0 for ⫺a2/8⬍a1⬍⫺2a2. 共12.7兲 For a1⫽⫺2a2, we see that ␤⫽0, whereas when a1⫽

⫺a2/8, we return to the exceptional case considered in Sec. XI A共then,␤ formally diverges兲.

For a1⬍0,a2⬎0, we analogously find from Eq. 共12.2兲 that now ␤ is positive,

1

2⬍␤⬍2 for all a1,a2, 共12.8兲 whereas Eq. 共12.1兲 yields

0⬍␣⬍1

2 for a1⬍⫺2a2, 共12.9兲

␣⬎2 for a1⬎⫺a2/8, 共12.10兲

␣⬍0 for ⫺2a2⬍a1⬍⫺a2/8. 共12.11兲 For a₁⫽⫺2a₂, we see that ␣⫽0, whereas when a₁⫽

⫺a2/8, we again obtain the exceptional case considered in Sec. XI A共then␣ becomes infinite兲.

It is important to find the zeros of the metric coefficient g00⫽⫺A2, since these values determine the position of a possible horizon. For the solutions under consideration, A always has one or two zeros: 共a兲 at r⫽2m/␣ when ␣⬎0, and/or 共b兲 at r⫽2m/␤ if␤⬎0. The function 共11.15兲 would not have zeros only in the case when both ␣⭐0 and␤⭐0, but this never happens according to the above analysis.

In order to decide whether a zero of g00corresponds to a horizon, we have to study the behavior of the curvature and torsion invariants at that value of the radial coordinate. As it

(13)

is clear from Eq.共11.17兲, the Riemannian curvature diverges at the zeros of A, unless the polynomial共11.18兲 also vanishes there. Hence, it is important to find the value of the polyno-mial共11.18兲 at the zeros of the function A. A direct substitu-tion yields w共2m/␣兲⫽2m3共␣⫺␤兲22␣ 2_⫺3_␣_⫹1 ␣2 , 共12.12兲 w共2m/␤兲⫽2m3_共_␣_⫺_␤_兲22␤ 2_⫺3_␤_⫹1 ␤2 .

Since ␣ can never be equal to ␤ 共for any nonzero coupling constants a₁,a₂), we conclude from Eq. 共12.12兲 that the polynomial w can have common zeros with A only when either ␣ or ␤ is equal to 1 or 1/2. Equations 共12.1兲,共12.2兲 show that this is possible only when 2a1⫹a2⫽0 共then either

␣ or␤ is equal to 1兲, or when one of the two constants a1or a2 vanishes.

Analogously, for the torsion quadratic invariant we find

T共2m/␣兲⫽m2_共_␣_⫺_␤_兲23␣ 2_⫺4_␣_⫹2 ␣2 , T共2m/␤兲⫽m2_共_␣_⫺_␤_兲23␤ 2_⫺4_␤_⫹2 ␤2 . 共12.13兲

As one can easily see, these quantities are nonvanishing for any choice of the constants ␣,␤. Correspondingly, the tor-sion invariant is always singular at the zeros of g00 in all teleparallel gravity models.

A. General relativity limit: Schwarzschild black hole As we have mentioned in Sec. V, the specific choice共5.1兲 of the coupling constants gives rise to a teleparallel model which is called the teleparallel equivalent of GR. Accord-ingly, when a₁⫽⫺1 and a₂⫽2, we find from Eqs.

共11.11兲,共11.12兲 that

2 , 共12.16兲

and consequently, the Riemannian Weyl curvature is de-scribed by

W⫽⫺ 6mr 3

B₀2共m⫹r/2兲6. 共12.17兲 The most important observation is that the Riemannian cur-vature is thus regular at the zero r⫽2m of the metric func-tion A(r), which means that we have a horizon here. The resulting geometry then describes the well known Schwarzs-child black hole with the horizon at r⫽2m. The singularity at r⫽⫺2m is the usual point-mass source singularity at the origin.

Since we are dealing with teleparallel gravity, it is neces-sary also to analyze the behavior of torsion. A direct substi-tution of Eq.共12.14兲 into Eqs. 共11.19兲,共11.20兲 yields

T_␣⵩*T␣⫽m

2_共3r2_⫺8mr⫹8m2_兲r2

B₀2共m⫹r/2兲6共m⫺r/2兲2 ␩. 共12.18兲 As we see, the torsion invariant diverges not only at the origin r⫽⫺2m, but also at the Schwarzschild horizon r

⫽2m. Unlike the ‘‘regularizing’’ effect of the polynomial 共12.16兲 in the curvature invariant, the analogous polynomial

in the numerator of Eq.共12.18兲 does not help to remove the singularity of the torsion invariant.

We emphasize this fact as the main difference between the standard general relativistic description of the Schwarzschild black hole and its teleparallel counterpart. The horizon is a regular surface from the viewpoint of the Riemannian geom-etry, but it is singular from the viewpoint of teleparallel grav-ity.

B. No black holes in teleparallel gravity?

Returning to the case of the general teleparallel Lagrang-ian, we may ask if a 共class of兲 model exists with a specific choice of the coupling constants for which the above solu-tions describe a black hole. As we see from Eq. 共11.17兲, the curvature is singular at both zeros of the metric function A, i.e., at r⫽2m/␣ and at r⫽2m/␤, when

␣⬍2 and ␤⬍2. 共12.19兲

Correspondingly, recalling the results of the beginning of the current section, the teleparallel Lagrangians with 兩a₁兩

⬎兩a2兩/8 do not have spherically symmetric solutions de-scribing black holes.

There is, in principle, a possibility that the configurations with

␣⬎2 or ␤⬎2 共12.20兲

may turn out to be black holes. Note that both ␣ and ␤ cannot be simultaneously greater than 2, as was shown above in this section. When these conditions are satisfied, the cur-vature becomes regular either at r⫽2m/␣ or at r⫽2m/␤. However, the problem is that the corresponding surface (r

⫽2m/␣ or r⫽2m/␤) would have an infinite area. This is different from what we usually expect from a horizon sur-face.