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2008

Chern-Simons modified as a torsion theory and its interaction with fermions

Stephon Alexander Haverford College

Nicolas Yunes

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Repository Citation Chern-Simons Modi ed Gravity as a Torsion Theory and its Interaction with Fermions with Nico Yunes Phys. Rev D77: 124040, 2008

This Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected]. PHYSICAL REVIEW D 77, 124040 (2008)

Chern-Simons modified gravity as a torsion theory and its interaction with fermions

Stephon Alexander and Nicola´s Yunes Institute for Gravity and the Cosmos, Department of , The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 11 April 2008; published 24 June 2008) We study the tetrad formulation of Chern-Simons (CS) modified gravity, which adds a Pontryagin term to the Einstein-Hilbert action with a -dependent coupling field. We first verify that CS modified gravity leads to a theory with torsion, where this is given by an antisymmetric product of the tensor and derivatives of the CS coupling. We then calculate the torsion in the far field of a weakly gravitating source within the parameterized post-Newtonian formalism, and specialize the result to Earth. We find that CS torsion vanishes only if the coupling vanishes, thus generically leading to a modification of gyroscopic precession, irrespective of the coupling choice. Perhaps most interestingly, we couple fermions to CS modified gravity via the standard Dirac action and find that these further correct the . Such a correction leads to two new results: (i) a generic enhancement of CS modified gravity by the Dirac equation and axial fermion currents; (ii) a new two-fermion interaction, mediated by an axial current and the CS correction. We conclude with a discussion of the consequences of these results in particle detectors and realistic astrophysical systems.

DOI: 10.1103/PhysRevD.77.124040 PACS numbers: 04.50.Kd, 04.20.Fy, 04.40.Nr

CS term renders a candidate holomorphic ground state I. INTRODUCTION wavefunction invariant under large gauge transformations A quantum gravitational theory that is mathematically of the Ashtekar variables [12]. The CS correc- consistent, predictive, and in agreement with all experi- tion, is also related to the Immirzi parameter of loop mental data is a holy grail of physics. Many extensions of , which determines the spectrum of quan- (GR) have been proposed since its in- tum geometrical operators [13,14]. ception, most of which have not passed the test of time and CS modified gravity proposes an extension to GR by increasingly more accurate experiments (see e.g. [1] for a adding a parity-violating, Chern-Pontryagin term to the current review). Recently, however, two competing para- Einstein-Hilbert action, multiplied by a spacetime- digms have arisen that hold the promise to unify GR with dependent coupling scalar [15]. This theory modifies the quantum theory: [2–4] and loop quantum GR field equations by adding a new cottonlike C-tensor, gravity [5–7]. which is composed of derivatives of the Ricci tensor and Although both of these extensions are technically theo- the dual to the Riemann. Additionally, the equations of retically incomplete, there has been a recent effort to study motion for the scalar field provide a new Pontryagin con- its predictability [8,9]. Because of the intrinsic complexity straint that preserves diffeomorphism invariance. The of these theories, such efforts have been traditionally lim- structure of the C-tensor allows the modified theory to ited or model dependent [10]. Recently, however, these preserve some of the classical solutions of GR, such as theories have advanced enough that predictions can be the Schwarzschild, the Friedmann-Robertson-Walker, and made and one generic and unavoidable low-energy limit the line elements [15,16]. of both theories has been discovered: Chern-Simons (CS) Although some classic GR solutions are preserved in CS modified gravity. modified gravity, parity violation is inherent in the modi- In string theory, the absence of a CS term in the action fied theory, leading to possibly observable effects. One leads to the Green-Schwarz anomaly, which requires can- such effect is amplitude birefringence [15,17], which leads cellation to preserve unitarity and quantum consistency. In to a distinct imprint that could be detectable through most perturbative string theories (e.g. type IIB, I, heterotic) gravitational wave observations [18]. Birefringent gravita- with four-dimensional compactifications, the Green- tional waves have actually been successfully employed to Schwarz mechanism requires the inclusion of a CS term propose an explanation to the leptogenesis problem [19,20] [11]. In fact, this term is induced in all string theories due and could also leave an imprint in the cosmic-microwave to duality symmetries in the presence of Ramond-Ramond background [21–23]. Another consequence of CS modified scalars or D-instanton charges [3,11]. Even in heterotic gravity is modified precession, which has been studied in M theory the CS term is required through the use of an the far-field limit [24,25], leading to a weak bound on the anomaly inflow. CS scalar with LAGEOS [26]. Recent investigations have In , the CS term arises as a natural also concentrated on spinning solutions [27,28], extension to the Hamiltonian constraint. In particular, the as well as black hole perturbations [29], both of which have

1550-7998=2008=77(12)=124040(11) 124040-1 Ó 2008 The American Physical Society STEPHON ALEXANDER AND NICOLA´ S YUNES PHYSICAL REVIEW D 77, 124040 (2008) been seen to be corrected in CS modified gravity. For The fermion enhancement effect arises as a consequence further studies of these and related issues see e.g. [30– of the Dirac equation in fermion-extended CS modified 39] and references therein. gravity. Because of the inclusion of fermions, a new field In this paper, we study CS modified gravity within the equation arises (the Dirac equation), which couples deriva- first-order or (see e.g. [40] for a review). tives of the Dirac to the connection, which now In this formalism, one rewrites the action in terms of a contains both a symmetric, torsion-free part and a torsion- tetrad and a generalized connection that needs not be full piece. In this way, the torsion tensor, and thus, the CS torsion free. One then varies the action with respect to correction, are sourced by derivatives of the Dirac spinor these fields to obtain the and the so- through the Dirac equation. Such a result implies that all called second Cartan structure equation, which in GR CS corrections are magnified in physical scenarios where reduces to the torsion-free condition. CS modified gravity, fermionic currents are large. however, leads to a torsion-full condition, where the tor- We conclude with a discussion of the consequences of sion tensor is proportional to an antisymmetric product of these two new results. On the one hand, the new two- the Riemann tensor and partial derivatives of the CS scalar. fermion interaction could potentially lead to observables, We first compute the torsion tensor in the far field of a related to fermion processes. Particle accelerators, how- weakly gravitating body within the parameterized post- ever, are unlikely to see this correction, since the Ricci Newtonian (PPN) formalism for a generic CS scalar [41– scalar vanishes in the neighborhood of the Solar System, 46]. We find that the torsion tensor is proportional to thus annihilating the modification. On the other hand, the contractions of the Levi-Civita symbol, derivatives of the fermionic enhancement effect renders the modified theory CS scalar, and derivatives of the Newtonian and PPN even more appealing, since CS corrections would then be vector potentials. This tensor is evaluated around Earth naturally enhanced in several realistic astrophysical sce- and found to generically persist, unless the CS scalar field narios, such as pulsars, merging neutron stars, and super- vanishes identically, thus reducing CS modified gravity to novae, perhaps even leading to stronger bounds of CS GR. The nonvanishing of the CS torsion tensor generically modified gravity. leads to a modified frame-dragging effect and gyroscopic The remainder of this paper presents further details and precession. The results found here thus provide great theo- calculations of the results mentioned above and it is di- retical motivation for studies of generic torsion theories vided as follows. Section II reviews the tetrad formalism in and their effect in Solar System experiments similar to GR and establishes notation; Sec. III reformulates CS [47]. modified gravity in the tetrad formalism and finds the After investigating the torsion tensor, we concentrate on torsion tensor of the modified theory; Sec. IV computes the inclusion of fermions in CS modified gravity, since the torsion tensor in the far field of a weakly gravitating these are known to also lead to torsion (see e.g. [13,48]). body, later specializing the result to fields around Earth; We find that indeed the torsion tensor is now given by the Sec. V adds fermions to the modified theory, derives the sum of the CS torsion and a new fermion-induced term, fermionic enhancement effect, and calculates the new fer- which depends on the axial fermion current. The fermion- mion interactions; Sec. VI discusses the implications of extended torsion tensor can then be used to obtain two new these results in astrophysical scenarios and particle detec- results: a new two-fermion interaction and a fermionic tors; Sec. VII concludes and points to future research. enhancement of CS modified gravity. We use the following conventions: commas stand for Interaction terms are common in torsion-full theories. partial derivatives @a ¼ ;a; parenthesis and square- For example, Riemann-Cartan theory leads to a four- brackets in index lists stand for symmetrization AðabÞ ¼ fermion interaction term, mediated by the axial current. 1=2ðAab þ AbaÞ and antisymmetrization A½ab ¼ These interactions are computed by inserting into the 1=2ðAab AbaÞ, respectively; uppercase Latin letters action the full connection: a torsion-free, symmetric part fA; B; ...g stand for internal indices, lowercase Latin letters (the Christoffel connection) plus a certain linear combina- at the beginning of the alphabet fa; b; ...;hg stand for tion of components of the torsion tensor (the contorsion spacetime indices, while those in the middle of the alpha- tensor). In the fermion-extended version of CS modified bet fi; j; ...g stand for spatial indices only. The order sym- gravity, we find that the interaction term consists of three bol OðAÞ stands for terms of order A, and we use geometric contributions: a new two-fermion term, a modified four- units, such that G ¼ 1 ¼ c. fermion term, and a new six-fermion term. The four- fermion interaction is in fact similar to that found in II. FIRST-ORDER FORMALISM IN GR Riemann-Cartan theory, also suppressed by a factor of G, the gravitational constant. The six-fermion interaction is In this section, we review the first-order formalism of further suppressed by a factor of G2. The two-fermion GR and establish notation, following mainly [40]. Let us process, however, is G-independent and mediated by de- then consider a 4-dimensional M with an asso- rivatives of the CS scalar, the axial fermion current, the ciated 4-dimensional metric gab. Let us further introduce at I Ricci scalar, and the Ricci tensor. each point on the manifold a tetrad ea, such that the metric

124040-2 CHERN-SIMONS MODIFIED GRAVITY AS A TORSION ... PHYSICAL REVIEW D 77, 124040 (2008) I J Z can be written as gab ¼ eaebIJ, with IJ the Minkowski 4 abcd I J KL SEH ¼ d x~ e e F ; (2.7) metric. Internal and spacetime indices are raised and low- 4 IJKL a b cd ered with IJ and gab, respectively. abcd where ~ is the Levi-Civita symbol and abcd ¼ Let us now introduce the spacetime and , I J K L c J eae ec e IJKL is the Levi-Civita volume form. We here A and A , which given a mix tensor gbI satisfy b d ab aI depart slightly from the conventions of [40] by not adding c J DakbI ¼ @akbI þ Aab kcI þ AaI kbJ; (2.1) an extra factor of 2 in the action, which is a matter of convention. Equation (2.7) can be derived by using the where Da is a generalized . The torsion identity 2 e tensor is defined via D½aDbf ¼ Tab Def, for some scalar b c de function f, thus satisfying F ¼ ½deFbc ; (2.8) e : 2 e Tab ¼ A½ab : (2.2) and [49] abcd 4 ½c d The requirement that the spin connection be torsion free is abef ¼ e f ; (2.9a) simply A c ¼ 0 and that it be compatible with the inter- pffiffiffiffiffiffiffi ½ab ~abcd 4 ½c d abef ¼þ ge f ; (2.9b) nal metric IJ is equivalent to AaðIJÞ ¼ 0. ~abcd ~ 4 ½c d The generalized covariant derivative can be shown to be abef ¼þ e f ; (2.9c) compatible with the tetrad, thus satisfying which will be extremely useful in a later section. I 0 Daeb ¼ : (2.3) Let us now obtain the field equations of the theory by varying the Lagrangian density with respect to the tetrad This relation then implies that the spacetime and spin and the connection: the field equations and the second connections are related via Cartan structure equation. Variation with respect to the e I 1 I K I 1 I tetrad yields Aab ¼ðeeÞ AaK eb ðeeÞ @aeb; (2.4) ~ abcd J KL 0 which is simply a change of . Sometimes these rela- IJKLebFcd ¼ ; (2.10) tions are referred to as the ‘‘tetrad postulate,’’ which we since the tensor depends only on the connection. discuss further in the appendix. When the spacetime and Equation (2.10) constitutes the field equations, which is a spin connections satisfy Eq. (2.4), then the spacetime generalization of the Einstein field equations for a generic, connection is given by the sum of the not necessarily torsion-free connection. and the (provided the spin connection is Variation with respect to the connection is a bit more torsion-full). The contorsion tensor shall be discussed later, complicated. Let us begin by rewriting the variation of the but it is essentially constructed from the torsion tensor. curvature tensor as With this generalized covariant derivative and connec- KL 2 KL e KL tions we can now define the generalized curvature Fcd ¼ D½cAd Tcd Ae : (2.11) through the failure of commutativity of the generalized covariant derivatives. One can show that Before we vary this Lagrangian density with respect to the connection, it is convenient to integrate by parts the first J 2 J J FabI ¼ @½aAbI þ½Aa;AbI ; (2.5a) term to find F d ¼ 2@ A d þ½A ;A d; (2.5b) Z abc ½a bc a b c 4 abcd I J KL SEH ¼ d x½2D ð~ e e ÞA 4 ½c IJKL a b d where the anticommutator is shorthand for e ~abcd I J KL þ Tcd IJKLeaebAe : (2.12) ½A ;A J :¼ A KA J A KA J; a b I aI bK bI aK KL (2.6) We can now vary this action with respect to Ae and d : e d e d ½Aa;Abc ¼ Aac Abe Abc Aae : demand that the variation vanishes to find 2 ~abce I J e ~abcd I J Note that if the connection is metric compatible and torsion Dcð IJKLeaebÞ¼Tcd IJKLeaeb: (2.13) free (i.e. if it is the Christoffel connection), then the curva- ture tensor is simply the Riemann tensor. The left-hand side of this equation vanishes because the Let us now rewrite the Einstein-Hilbert action in terms generalized covariant derivative is tetrad compatible and of these new variables. Note, however, that we wish to thus Eq. (2.13) is simply the torsion-free condition of GR, work with the trace of the generalized curvature tensor, and T e ¼ 0: (2.14) not the Ricci scalar, since these two quantities are not cd necessarily equivalent in the presence of torsion. The In this case, then, the generalized connection reduces to the Einstein-Hilbert (EH) action is given by the well-known Christoffel one and the field equations to the Einstein expression equations.

124040-3 STEPHON ALEXANDER AND NICOLA´ S YUNES PHYSICAL REVIEW D 77, 124040 (2008) III. FIRST-ORDER FORMALISM OF CS MODIFIED a abcd e 1 f 1 l f K ¼ bf ð2Rcde 3 ce dl Þ; (3.6) GRAVITY where we again follow the conventions of [40] and define In this section we shall present a pedagogical introduc- the Riemann tensor via R f ¼ 2@ f þ 2 l f. tion to CS modified gravity in the second-order formalism cde ½c de ½cje dl We then find that the CS action in first-order form is simply and derive its first-order former. This section will thus both   serve as an introduction to the modified theory, which was Z 1 1 ¼ 4 ~abcd J I K I originally proposed in second-order form, and as a basis to SCS 2 d x vaAbI 2 FcdJ 3 AcJ AdK ; establish the CS notation of this paper. (3.7) CS modified gravity [15] postulates the following action pffiffiffiffiffiffiffi [50]: where we used that abcd ¼ð1= gÞ~abcd and we have replaced spacetime by internal indices, since these are fully S ¼ SEH þ SCS (3.1a)   contracted. Z ffiffiffiffiffiffiffi 1 4 p ? Let us now vary the first-order CS action with respect to SCS ¼ d x g þ RR ; (3.1b) 4 the tetrad and the connection. The field equations remain the same as in GR, namely, Eq. (2.10), because the CS where SEH is given in Eq. (2.7) and we follow the con- action does not depend on the tetrad. We then find that the ventions of [28]. The quantity is here the so-called CS field equations of CS modified gravity are similar to those scalar, which serves as a spacetime coordinate-dependent of GR, provided the connection and the curvature tensor coupling function. In principle, one should include a ki- are the generalized ones. netic and a potential term for the scalar field in the CS The second structure equation is a bit more difficult to action, but we shall ignore these here since they do not derive. Let us then first perform a general variation of the contribute to torsion. The Chern-Pontryagin term is defined CS modified action in first-order form to find via Z 4 1 ~abcd J I ? : cdef a b S ¼ d xva ðAbI FcdJ EÞ; (3.8) RR ¼ 2 R befR acd; (3.2) 4 with Rabcd the Riemann tensor. The parity-violating nature where of CS modified gravity is encoded in the Levi-Civita E ¼ A JF I 2A KA IA J 2A KA IA J tensor. Note here that CS modified gravity is intrinsically bI cdJ 3 cJ dK bI 3 cJ dK bI 2 2 J K I 4-dimensional, which is different from the þ 3AbI AcJ AdK : (3.9) 1-dimensional theory that goes by a similar name. KL Before decomposing CS modified gravity in first-order Upon variation with respect to Aa and contraction with form, it is convenient to slightly rewrite the action. Let us the Levi-Civita symbol, the above term identically van- then integrate by parts to obtain ishes and we are left with Z Z pffiffiffiffiffiffiffi SCS 4 4 a ¼ d x~abcdv F : (3.10) SCS ¼ d x gvaK : (3.3) KL b cdKL 2 Aa 4 We here neglect any boundary terms since [51] has shown, Combining Eq. (3.10) with the variation of the Einstein- within the second-order formulation, that CS modified Hilbert action with respect to the spin connection we find gravity indeed leads to a well-posed boundary value prob- the second structure equation, namely, lem, through the addition of boundary counterterms. The ~abcdT eeI eJ ¼ ~ebcdv F ; (3.11) CS velocity and CS acceleration are defined via IJKL cd a b b cdKL : which agrees with [53] up to conventional prefactors. va ¼ra ¼ @a; (3.4a) Let us now attempt to isolate the torsion tensor in CS : vab ¼ravb ¼rarb; (3.4b) modified gravity. Equation (3.11) is in principle a differ- ential equation for the torsion tensor, since the generalized where ra is the covariant derivative operator associated curvature tensor contains derivatives of the contorsion. c a with the Christoffel connection ab. The quantity K is the Following [13], we can parameterize the full connection so-called Pontryagin current, which in four is via given by A IJ ¼ ! IJ þ C IJ; (3.12) a : abcd e f 2 l f a a a K ¼ ð@c þ 3ce Þ; (3.5) bf de dl IJ where !a is a torsion-free, symmetric connection that a ? 2 IJ and satisfies raK ¼ RR= [52]. depends only on the tetrad and Ca is the contorsion We can now write the CS action in first-order form. The tensor. The contorsion is related to the torsion via Pontryagin current can be written in terms of the Riemann T ce ¼ 2C IKe ; (3.13) tensor as ab cI ½a bK

124040-4 CHERN-SIMONS MODIFIED GRAVITY AS A TORSION ... PHYSICAL REVIEW D 77, 124040 (2008) where the factor of 2 comes from our definition of the Rabcd ¼ hd½a;bc hc½a;bd þ Oð4Þ: (4.2) torsion tensor (see Appendix B). We can thus schemati- cally rewrite Eq. (5.4)as With this linearized Riemann tensor we find that the CS torsion tensor becomes ~abcd e I J ~ebcd IJKL Tcd eaeb ¼ vbRcdKL½! 1 ebcd 2 T n ¼ nbefv h þ Oð4Þ: (4.3) þ ~ vbHcdKL½@T; !T; T ; cd 2 b e½c;df (3.14) Henceforth, we shall work to leading order in the torsion tensor and consistently drop remainders of Oð4Þ and where RabIJ½! is the standard IJ 2 higher. We can decompose the torsion tensor into temporal that depends on !a only, while HabIJ½@T; !T; T repre- sents all other terms in the generalized curvature tensor that and spatial components to find are at least linear in the torsion tensor. The solution to this 0 1 0ijk equation to linear order in the CS velocity is simply Tcd ¼ 2 vihj½c;dk; (4.4a) i 1 0ijk 1 0ijk T ¼ v0h þ v ðh0 h 0Þ; n 1 nbef O 2 cd 2 j½c;dk 2 j ½c;dk k½c;d Tcd ¼4 vbRcdef þ ðvÞ ; (3.15) (4.4b) n 1? nb O 2 Tcd ¼4 Rcd vb þ ðvÞ : where we remind the reader that Latin indices in the middle n ð2Þ n One can easily check that inserting Tcd ¼ Tcd into of the alphabet fi; j; ...g stand for spatial indices only. We ð2Þ n ð1Þ n ð1Þ n Eq. (3.14), where Tcd ¼ Tcd þ , Tcd is the recognize these terms as flat-space curls and cross products first-order solution given in Eq. (3.15) and is undeter- of the metric perturbation and the CS velocity. Note that mined, forces to be at least quadratic in va. the CS acceleration does not contribute to the torsion tensor. Let us now specialize the torsion tensor to a specific IV. CS TORSION IN THE FAR FIELD source. In GR and in the PPN formalism, the metric The torsion tensor found in the previous section has an perturbation can be written to first nonvanishing order as intriguing form, resembling the wedge-product of the Riemann tensor and the CS velocity. In this section we h00 ¼ 2U; h0i ¼4Vi;hij ¼ 2Uij; (4.5) study the structure of the torsion tensor in the far field. We begin by considering its functional form in the PPN for- where U is the Newtonian potential and Vi is a PPN vector malism and finish with a discussion of this tensor around potential. In general, the gravitomagnetic sector of the Earth. metric contains two independent vector potentials, but in most cases of interest, these vector potentials are identical. A. CS torsion in the PPN formalism For example, for a single stationary source at rest, these potentials are Let us then begin by rewriting the as a m m linear combination of flat space and a metric perturbation ~ j k U ¼ ;Vi ¼ 2 ijka n ; (4.6) gab ¼ ab þ hab. Let us further work in the PPN formal- r 2r ism, where different components of the metric perturbation i i are assumed to be of the following orders: h00 ¼ Oð2Þ, where m is the mass of the body, a ¼ J =m is the specific i i h0i ¼ Oð3Þ, hij ¼ Oð2Þ. In this section, the notation OðAÞ , n ¼ x =r is a unit vector, and r is the stands for terms of order A, where is the perturbation distance from the center of the body to a field point. Note parameter of PN theory: the strength of the gravitational that since r_ ¼ 0, all time derivatives of the metric vanish. field (i.e. an expansion in G) or the speed of particles (i.e. The metric perturbation presented above, however, is not an expansion in 1=c). Note then that time derivatives are a solution to the CS modified field equations. For the case 0 0 0 smaller by an order of relative to spatial derivatives. We where va ¼ðv0; ; ; Þ, such a solution can be constructed shall not review the PPN formalism in detail here, but by adding a term in the shift to the standard PPN metric instead we refer the reader to [41–46]. [24,25]: We can now construct the Riemann tensor to leading 2 ~ order in the metric perturbation. Let us restrict attention to h0i ¼ v0ijkVj;k: (4.7) a quasi-Cartesian , such that a b 2 2 2 2 Note, however, that these CS corrections to g generically abdx dx ¼dt þ dx þ dy þ dz . We then find that ab add corrections to the torsion tensor that are at least qua- d 2 d O 4 Rabc ¼ @½a bc þ ð Þ; (4.1) dratic in the CS velocity, and thus, we shall neglect them. We can now find all nonzero components of the torsion c and, following the conventions of [40], ab ¼ tensor in CS modified gravity in terms of PPN potentials, cd ð1=2Þg ½2gdða;bÞ gab;d and then namely:

124040-5 STEPHON ALEXANDER AND NICOLA´ S YUNES PHYSICAL REVIEW D 77, 124040 (2008) 0 0ijk 3Ma T0l ¼ viVj;kl; (4.8a) t 2 cos sin Ttr ¼ 5 ð vrr þ v Þ; (4.11a) 0 0ik 2r T ¼ U v ; (4.8b) ln ½n ;l k i 3Ma i 0ijk 1 0ijk t sin cos ¼ þ Tt ¼ 4 ðvrr v Þ; (4.11b) T0l v0Vj;kl 2 vjU;lk; (4.8c) 2r i 0ij 0ijk T ¼ U v0 2v V ; (4.8d) 3Ma ln ½l ;nj j ½l;nk T t ¼ v cos; (4.11c) t 2r4 where we have assumed the source is at rest and the CS 3Ma T ¼ v cos; (4.11d) velocity is generic. The expressions presented above are t 2r4 t generically valid for any weakly gravitating system in the 3Ma T r ¼ v cos; (4.11e) far field. r 2r4 3Ma T ¼ v cos; (4.11f) B. CS torsion around Earth r 2r4 r Let us now specialize the above analysis to bodies 3Ma T ¼ ðv r cos þ v sinÞ; (4.11g) orbiting Earth and use the following line element: r 2r5 r 3Ma     sin 2 cos 2M 2M T ¼ 4 ðvrr v Þ; (4.11h) ds2 ¼ 1 þ dt2 þ 1 þ dr2 2r r r 4Ma and all remaining components can be either obtained by sin2dtd þ r2d2 þ r2sin2d2; (4.9) symmetry or vanish to this order. r Torsion will affect motion through the where fr; ; g are spherical polar coordinates. This line Papapetrou equations, which then leads to modified pre- element agrees both with the PPN formalism described cession relative to the GR prediction. We have checked that above, as well as with the far-field linearization of the the only possible way for all components of the torsion Kerr line element. We shall here treat both M=r 1 and tensor to vanish is for the CS acceleration to identically 0 0 a=r 1 as independent perturbation parameters. For vanish va ¼ . Even when vr ¼ v ¼ v ¼ , the so- bodies orbiting Earth, these quantities satisfy M=r called canonical choice for va, there are still six nonvan- Oð1010Þ and a=r Oð107Þ. ishing torsion tensor components. Clearly then, even for 0 We can now compute the components of the torsion noncanonical velocities where only vt ¼ , there are still tensor by inserting the line element of Eq. (4.9) into ten nonvanishing components of this tensor. We can con- Eq. (3.15). Alternatively, we could have used Eqs. (4.8a)– clude that torsion and modified precession are inherent to (4.8d) and the PPN potentials of the previous section, but CS modified gravity irrespective of the choice of coupling the calculation is simplified if instead of quasi-Cartesian parameter. coordinates we use spherical polar coordinates. We used The torsion tensor presented here could be used to Maple [54] to find that the largest nonvanishing compo- calculate the change in the precessional angular frequency nents of the torsion tensor are of gyroscopes orbiting Earth. Such a calculation is natu- rally interesting because it could be compared to Solar M v System experiments, such as Gravity Probe B, and thus T t ¼ ; (4.10a) r 2r3 sin lead to a bound on the CS velocity. Such a test was first M proposed by Alexander and Yunes [24,25], who computed T r ¼ ðv þ 3v asin2Þ; (4.10b) t 2r3 sin t this angular frequency in canonical CS modified gravity. M Smith, et al. [26] extended this analysis and placed the first T ¼ v sin; (4.10c) actual bounds on the canonical choice of CS velocity. t 2r3 r M Unfortunately, such Solar System bounds tend to be rather T r ¼ v sin; (4.10d) weak due to the feebleness of the gravitational force in the t 2 3 r Solar System. M 2 3 sin2 Ttr ¼ 5 ð v þ vta Þ; (4.10e) The modified angular velocity of precession has not yet 2r sin been calculated for a generic choice of CS velocity. Mao M sin T r ¼ ð2v r2 þ 3v aÞ; (4.10f) et al. [47] have considered Solar System tests of a re- 2r3 t stricted class of torsion theories. This class is parameter- M sin 2 ized by torsion tensors that are stationary, spherically or T ¼ ðv r þ 3v aÞ; (4.10g) r 2r5 t axially symmetric, and parity preserving. The torsion ten- M v sor associated with CS modified gravity generically breaks T ¼ t ; (4.10h) r 2r3 sin parity unless one concocts a CS velocity that is parity violating, such as the flat-space curl of some other field. followed in magnitude by Even then, the explicit appearance of the CS velocity in the

124040-6 CHERN-SIMONS MODIFIED GRAVITY AS A TORSION ... PHYSICAL REVIEW D 77, 124040 (2008) torsion tensor tends to break spherical or axial symmetry. 1 1 T n ¼ nbefv R n Je þ OðvÞ2: (5.5) Thus, the torsion tensor considered here is more general cd 4 b cdef 4 cde 5 than that considered in [47]. Nonetheless, the analysis The torsion tensor can be manipulated slightly and presented in this paper provides a solid motivation for written like a fermion term with a modified current. For the study of the effect of generic torsion theories on this purpose, let us express the Riemann tensor in terms of Solar System experiments, similar in spirit to that of its 4-dimensional irreducible decomposition [47]. A careful examination of generic torsion theories 1 and Solar System experiments is, however, beyond the Rabcd ¼ Cabcd þ ga½cRdb gb½cRda 3Rga½cgdb; scope of this paper. (5.6)

where Cabcd is the . The torsion tensor can then V. CS MODIFIED GRAVITYAND FERMIONS be written as 1 1 In this section, we study the inclusion of a minimally n nb 6 nbf nbef Tcd ¼ ð cdvbR þ ½cRdfÞ vbCcdef coupled fermion term to CS modified Lagrangian. Let us 12 4 1 then consider the fermion-extended CS action n Je: (5.7) 4 cde 5 S ¼ SEH þ SCS þ SD (5.1a) For conformally flat spaces with , we can 1 Z 4 pffiffiffiffiffiffiffi use R ¼ g R=4 and C ¼ 0 to simplify the torsion S ¼ d x gði IeaD þ c:c:Þ; (5.1b) ab ab abcd D 2 I a into 1 where the Einstein-Hilbert action SEH was given in n ¼ n e Tcd 4 cdeS5; (5.8) Eq. (2.7), while the CS action SCS was presented in Eq. (3.1b). Equation (5.1b) is nothing but the massless where we have defined the CS extended axial current Dirac action [55], where c.c. stands for complex conjuga- Se ¼ Je þ veR: (5.9) tion, is a Dirac spinor, and I are gamma matrices. Note 5 5 6 that the first-order formalism is essential for the inclusion The torsion tensor is related to the contorsion via of fermions in the theory, since Dirac live naturally Eq. (3.13), which can be inverted to find in SUð2Þ. Therefore, covariant derivatives associated with n ¼ 1½ n þ 2 n the Dirac action are not the usual SOð3; 1Þ ones, but instead Ccd 2 Tcd T ðcdÞ : (5.10) D : 1 4 IJ are given by a ¼ @a ð = ÞAa IJ , where we One can check that the antisymmetrization of Eq. (3.12)in here follow the sign conventions of [13]. its first two indices with Eq. (5.10) for the contorsion Let us now find the field equations and the second returns the definition of torsion in Eq. (2.2)[56]. Using structure equation of the fermion-extended CS modified the torsion tensor found in Eq. (5.5), the contorsion be- gravity. The variation of the action with respect to the comes   tetrad leads to 1 1 C n ¼ ?R nbv þ 2?Rn bv þ n Je : ~ abcd J KL 8 a cd 8 cd b ðcdÞ b cde 5 IJKLebFcd ¼ GT I; (5.2) (5.11) which is nothing but the Einstein equations with a generic connection in the presence of a source, given by Dirac The fermionic part of the contorsion found here agrees fermions. with that found by [13,57] in the limit as the Immirzi The second structure equation can be obtained by vary- parameter tends to infinity and the vanishes. ing the action with respect to the connection. Following Once the torsion has been computed, one can reinsert it [13] and using the identity into the full action to obtain the equations of motion. We rewrite the full connection as the sum of a symmetric, I ½J K IJKL 2 I½J K bc ¼i 5L þ ; (5.3) torsion-free part !a and the contorsion, as in Eq. (3.12). Each contribution to the action then takes the we find that the requirement that the variation of the action form S ¼ S½!þS½C, where the first term is completely vanishes implies independent of the contorsion and the second term leads to e contorsion-induced interaction terms. We find that ~ abcd T eeI eJ ¼ ~ebcdv F eeI JQ; IJKL cd a b b cdKL I KLQ 5 contorsion-dependent contributions to the action are given by the following: (5.4)  Z 1 Q : Q ½ ¼ 4 bð2 a Þ where J5 ¼ 5 is the axial fermion current and SEH C d xe 16 J5 v Rab vbR where the of the tetrad vanishes. Once more,  3 we can invert Eq. (5.4) to leading order in the CS velocity a þ J5 J5a ; (5.12) to find an expression for the torsion: 32

124040-7 STEPHON ALEXANDER AND NICOLA´ S YUNES PHYSICAL REVIEW D 77, 124040 (2008) for the Einstein-Hilbert part; equation (in this case, for a massless fermion), namely  aD 0 Z 1 a ¼ . Splitting the connection into torsion-free ½ ¼ 4 b½2 a þ Þ SD C d xe 16 J5 v Rab vbR and torsion-full pieces, one can easily show that the  Dirac equation becomes 3 a J J5 ; (5.13) 16 5 a a ð!Þ 1 a KL M Da ¼ 4eMCa KL ; (5.17) for the Dirac part; and ð!Þ Z  where Da stands for the covariant derivative associated 4 va a with the torsion-free connection only. SCS½C¼ d xe L ½! @J; ! @J ! 8 The Dirac equation can then be thought of as sourcing 1 the contorsion tensor, and thus, the CS correction. One þ ð a 2 a b Þ 16 J5 vaR J5 v Rab could insert the contorsion tensor found in Eq. (5.11)to  1 1 find an explicit equation for the relation between the abcd a b þ va J5d@cJ5b ðvaJ5 ÞðJ5 J5bÞ ; torsion-free Dirac equation and the fermion and CS torsion 64 2562 tensor. In essence, this equation implies that the CS effects (5.14) will be enhanced in spacetime regions where the momen- for the CS part, where La½... stands for a contraction of tum of Dirac fermions is large. Several realistic astrophys- derivatives of the axial current with the torsion-free con- ical scenarios exist where such an enhancement should be nection. Interestingly, the CS contribution to the Einstein- present, but we shall discuss these in the next section. Hilbert and Dirac parts of the action identically vanish, 2 yielding the standard J5 interaction of Riemann-Cartan VI. PHYSICAL IMPLICATIONS theory in the presence of minimally coupled fermions: In this section, we comment and theorize on some of the 3 Z 4 a results found in the previous sections and their consequen- SEH½CþS ½C¼ G d xeJ J5 : (5.15) D 2 5 a ces. As we have seen, CS modified gravity can be mapped The CS contribution to the action, however, adds new to a torsion theory, where the torsion tensor is proportional parity-violating interactions and the full action in to the CS velocity. Torsion then leads to a new and unsup- Riemann normal coordinates (! d ¼ 0) reduces to pressed interaction term in the Dirac action, which repre-  ac sents a two-fermion process, mediated by the spacetime Z 1 ½ ¼ 4 ð a 2 a b Þ curvature and the CS velocity. S C d xe 16 J5 vaR J5 v Rab   Can such modified fermion interaction be detected in 3 1 particle detectors? The four-fermion process is suppressed a abcd 2 G J5 J5a 6 vaJ5d@cJ5b by a factor of the gravitational constant, so its detectability  is questionable. The two-fermion interaction, however, is 2 2 a b þ G ðvaJ5 ÞðJ5 J5bÞ : (5.16) not suppressed by such a factor and it is precisely where the main CS correction resides. This interaction does depend CS modified gravity has introduced a new parity- on the Ricci tensor and scalar, which are close to zero in the violating interaction that is not suppressed by Newton’s neighborhood of the Solar System and might again sup- gravitation constant and can be in fact enhanced by the CS press the effect. Nonetheless, such suppression might be velocity. This new interaction [first line in Eq. (5.16)] is a overcome if the CS effect is enhanced by fermion currents. two-fermion process that couples both to the CS velocity as Are there any realistic physical scenarios where this well as to the spacetime curvature through the Ricci tensor enhancement would actually occur? Such scenarios would and scalar. CS modified gravity also modifies the standard require a large fermion current, which in essence implies four-fermion interaction that is common to Riemann- large changes in fermion density. Dynamical compact Cartan theory with minimally coupled Dirac fermions. stars, such as neutron stars and white dwarves, in fact The modification consists of the addition of an antisym- possess a large fermion density, since they are supported metric product of the current and its derivative [second line against gravitational collapse via electron or neutron in Eq. (5.16)]. This interaction, however, is suppressed by degeneracy pressure. Moreover, these systems can be dy- one factor of G. Moreover, a new 6-fermion interaction is namical, spinning rapidly, vibrating quasinormally, quak- produced, but this one is further suppressed by a factor of ing or accreting mass from a binary companion. Slightly G2. more hypothetical sources, such as quark or strange stars Before concluding, let us discuss one further equation of [58,59], would lead to even stronger enhancements since motion contained in fermion-extended CS modified grav- their fermion density is even larger. ity. Such an equation arises because the Dirac action has an Any fermionic compact object that undergoes a violent additional degree of freedom: the Dirac spinor. Variation of change in its multipolar structure will also possess large the action with respect to the Dirac spinor yields the Dirac number density gradients. A few examples of such events

124040-8 CHERN-SIMONS MODIFIED GRAVITY AS A TORSION ... PHYSICAL REVIEW D 77, 124040 (2008) are double neutron star or neutron star-black hole binary torsion when minimally coupled to Riemann-Cartan the- mergers and supernovae. In all such systems, one of the ory. We indeed found that the torsion tensor was now binary components tidally disrupts and then either collides composed of the CS torsion piece, together with a new or is swallowed by the black hole horizon, leading to a term that depends on the fermion axial current. The torsion large change in fermion number density. In the cosmologi- tensor was then used to reconstruct the full connection via cal context, the big bang event, as well as inflation, also the contorsion, which when inserted into the full action led unavoidingly lead to large fermion currents, where fermi- to a new and unsuppressed two-fermion interaction. ons accelerate violently. Although this interaction is not suppressed by the gravita- Many of the sources described here are also target tion constant, it does depend on the Ricci tensor and scalar. sources for gravitational wave detection [60]. Periodic This dependence might make its detection with particle sources, such as pulsars, as well as inspiraling compact detectors on Earth difficult but might also enhance its binaries of various types, are preferred systems for gravi- effect in the neighborhood of compact sources. tational wave detection, because these are precisely pro- We then concentrated on the Dirac equation, which was duced by a changing multipolar structure. Moreover, the shown to source the CS torsion tensor. This enhancement CS effect is not only enhanced by large fermion currents, effect depends both on the fermion axial current and de- but also for systems whose Riemann tensor becomes large. rivatives of the Dirac spinors, amplifying all CS effects We then see that even for binary black hole mergers, where when large changes in fermion density are present. We there are no fermion currents in play, the CS modification discussed the astrophysical implications of this enhancing might play an important role. In this sense, the interplay mechanism and found several interesting sources where between gravitational wave detection and CS modified such an effect could be observed. Examples of such gravity might be important in the future [18]. sources include binary neutron star mergers, accreting The CS modification to GR is naturally enhanced in a neutron stars with their associated supernovae, inflationary plethora of realistic astrophysical scenarios. All that is cosmology, and the big bang. Not surprisingly, we found required for such an enhancement is the existence of large that the enhancing mechanism seems to be maximized for fermion currents, which are inherent in the natural evolu- the same type of sources preferred by gravitational wave tion of fermionic compact objects. We shall not quantify detection: compact sources with large changes in their the enhancement effect further in this paper, since this task multipolar structure. is extremely system dependent, and instead, we relegate it Open questions still remain in the context of CS modi- to future work [61]. fied gravity. One such question is that of well posedness as a boundary value problem. Although this issue has already VII. CONCLUSION been satisfactorily settled in [51] within the second-order We have studied CS modified gravity in the tetrad formalism, a similar study in the first-order formalism is formalism. We rewrote the CS modified action in terms still absent. An analogous story must, of course, exist in of a tetrad and a generalized connection and varied it with first-order form, and thus, one could construct the first- respect to these fields to obtain the modified field equations order version of any required boundary terms and counter- and the second structure equation. In doing so, we found terms required by the variational principle to make the that the torsion-free condition of GR does not hold in CS modified theory well posed. modified gravity, where now the torsion tensor is propor- Other future work could concentrate on studying the tional to an antisymmetric product of the Riemann tensor enhancing effect further, in connection with some specific and derivatives of the scalar field. astrophysical scenario. For example, one could study jet We investigated this torsion tensor in the far field of a production in rapidly rotating neutron stars or pulsars with weakly gravitating object within the PPN framework. We the CS correction. Alternatively, one could numerically found that the torsion tensor is proportional to the contrac- investigate some simplified supernovae models in the pres- tion of derivatives of the Newtonian and PPN vector po- ence of a CS correction. tentials with the Levi-Civita symbol and derivatives of the Another possible avenue of future research could be the CS scalar. This torsion was shown not to vanish for any numerical study of nonlinear CS modified gravity. For nontrivial choice of the CS scalar, thus suggesting that example, one could model the merger of binary black holes torsion is unavoidable in CS modified gravity. Such torsion in the modified theory to determine how waveforms will generically lead to a modification of the frame- change as a function of the CS scalar. Perhaps most inter- dragging effect of GR, as well as a change in the precession esting is the numerical evolution of the merger of black of gyroscopes. In this way, this paper provides great mo- hole–neutron star systems, since here both the fermionic tivation for the study of the effect of generic torsion enhancement and the curvature enhancements should be theories on Solar System experiments. present. Many of these open questions shall be answered in We then focused on the addition of fermions to the the near future, hopefully shedding new light on some of modified theory, since these are known to also produce the mysteries of the low-energy limit of quantum gravity.

124040-9 STEPHON ALEXANDER AND NICOLA´ S YUNES PHYSICAL REVIEW D 77, 124040 (2008) ACKNOWLEDGMENTS which vanishes if and only if the connection is purely antisymmetric on its internal indices, i.e. A ¼ 0.We We would like to thank Ben Owen for his continuous aðIJÞ support. We are also indebted to Daniel Grumiller and see then that spacetime metric compatibility is equivalent Andrew Randono, who provided essential help and clar- to internal metric compatibility. We also clearly see that the ifications with some calculations. We would also like to tetrad postulate does not automatically force metric thank Abhay Ashtekar, Roman Jackiw, and Frans Pretorius compatibility. for useful comments and suggestions. S. A. acknowledges The generalized connection can be written in terms of the support of the Eberly College of Science. We both the Christoffel connection and the contorsion tensor. One acknowledge the support of the Center for Gravitational can simply show that the metric compatibility requirement A ¼ 0 A ¼ Wave Physics, which is funded by the National Science aðIJÞ together with the torsion-free condition ðabÞc 0 Foundation under Cooperative Agreement No. PHY 01- lead uniquely to Aabc ¼ abc. When the torsion-free 14375. Some of the calculations presented in this paper condition is dropped, this is not the case anymore. In this were carried out with the symbolic manipulation software case, the generalized connection is a linear superposition of Maple, in combination with the tensor package GRTensorII the Christoffel connection and the so-called contorsion [54]. tensor, which can be constructed as linear superpositions of the torsion tensor.

APPENDIX A: BASICS OF THE FIRST-ORDER FORMALISM APPENDIX B: TORSION AND CONTORSION We here review the so-called tetrad postulate following In this appendix we review the definition of torsion and Carroll [49]. This postulate forces the covariant derivative its relation to the contorsion tensor, thus establishing fur- ther notation. Let us then consider first the torsion-free of the tetrad to vanish. Let us then begin by considering the IJ case, where the connection is simply given by Aa ¼ generalized covariant derivative of some vector X: IJ !a . The tetrad postulate then establishes that the trans- D D b a b b c a X ¼ð aX Þdx @b ¼ð@aX þ AabX Þdx @b: formation between the spin and the spacetime connection (A1) IK K I I !a eb ¼ @aeb þ !ab e: (B1) Let us now rewrite this quantity with internal indices, namely, Furthermore, the antisymmetrization of the tetrad postulate on the spacetime indices yields I a DX ¼ðDaX Þdx e^I; @ eI ¼ ! IKeK; (B2) b d I b I d I J b d a ½a b a b ¼½eI X @aed þ eI ed@aX þ Aa JedeI X dx @b: 0 (A2) where !½ab ¼ by the torsion-free condition. Equation (B2) establishes that the torsion-free spin con- Comparing both expressions and requiring that they be nection is nothing but the transformed Christoffel equal we find the constraint connection. c c I J c I Let us now consider the torsion-full case, where we split Aab ¼ eI @aeb þ ebeI Aa J; (A3) IJ IJ the full connection Aa into a torsion-free piece !a and a IJ or equivalently torsion-full part Ca . We further assume that the torsion- free piece satisfies the tetrad postulate with respect to the I ¼ I cð J Þ1 ð J Þ1 I Aa J ecAab eb eb @aeb: (A4) torsion-free covariant derivative, thus once more rendering IJ Equation (A4) shows clearly the character of the trans- !a the transformed Christoffel connection. The tetrad formation. From these equations, one can trivially derive postulate with respect to the full covariant derivative yields that the transformation law for the contorsion D I 0 C ceI ¼ C IKe : (B3) aeb ¼ : (A5) ab c a bK Note that this relation was achieved without ever requiring Antisymmetrizing the lower two spacetime indices in c : metric compatibility or torsion freeness. Thus, the tetrad Eq. (B3) and using the definition of torsion Tab ¼ postulate is an independent requirement that must always 2 c A½ab leads to the relation between contorsion and tor- hold and unequivocally leads to the vanishing of the co- sion: variant derivative of the tetrad. c I 2 IK The tetrad postulate, however, does not necessarily re- Tab ec ¼ C½a ebK: (B4) quire metric compatibility. One can easily show that The inversion of the torsion-contorsion relation then yields D I JD agbc ¼ ebec aIJ; (A6) explicitly Eq. (5.10).

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