arXiv:1804.04434v3 [gr-qc] 17 Oct 2018 Acknowledgments 6 Conclusions equ Einstein-Cartan-Dirac 5 the of limit Non-relativistic 4 equations Einstein-Dirac the of limit Non-relativistic 3 equations Einstein-Cartan-Dirac The Preliminaries: 2 Introduction 1 Contents 3 [email protected] o-eaiitclmto isenCra-ia equatio Einstein-Cartan-Dirac of limit Non-relativistic wnn Khanapurkar Swanand . osrit ntemti ...... equations . Einstein-Dirac . . the . . . of . limit . . (NR) . . Non-Relativistic metric . . the on . . 3.5 tensor Constraints ansatz momentum . above energy the . the 3.4 with Analyzing equation metric Dirac the the 3.3 and Analyzing spinor the for 3.2 Ansatz 3.1 E-mail: ED qain,aanrcvrn h Schr¨odinger-Newto the limit analys recovering our this again generalize deriving equations, we sph (ECD) while torsion, to of literature, naturally assumption the couples field the in work relaxes earlier analysis an Our non-rela equations. the Dirac as Schr¨odinger-Newton equation the derive We 1 ninIsiueo cec dcto n eerh(IE) Pun (IISER), Research and Education Science of Institute Indian 3 ainlIsiueo ehooyKraaa(IK,Surathkal (NITK), Karnataka Technology of Institute National 4 2 aaIsiueo udmna eerh(IR,Mma 400005, Mumbai (TIFR), Research Fundamental of Institute Tata ninIsiueo ehooyMda IT) hna 006 Indi 600036, Chennai (IITM), Madras Technology of Institute Indian 1 [email protected] , 4 [email protected] eidrP Singh P. Tejinder 1 ra Pradhan Arnab , Abstract 1 T µν ihteaoeast ...... ansatz above the with 4 st h Einstein-Cartan-Dirac the to is equation. n 2 ic h pno h Dirac the of spin the Since . eatDhruv Vedant , iitclmto h Einstein- the of limit tivistic rclsmer,md in made symmetry, erical ations , 2 108 India. 411008, e 705 India 575025, [email protected] India. a...... 3 and , . . . . ns 14 13 12 11 10 6 3 2 9 7 6 , 7 Appendix 14 7.1 Form of the Einstein tensor evaluated using the generic metric upto second order 14 7.2 Constraints on the metric due to the asymptotic flatness condition ...... 15 [1] 7.2.1 Constraint on gµν ...... 15 [2] 7.2.2 Constraint on gµν ...... 16 7.3 MetricandChristoffelsymbols...... 17 7.4 Tetrads ...... 17 o 7.5 Riemannian part of the spin connections (γ(a)(b)(c))...... 18 7.6 Einsteintensor ...... 19 7.7 Analysis of the components of the metric EM tensor ...... 19 7.7.1 Analysis of kT0µ ...... 19 7.7.2 Analysis of kTµν ...... 20 7.8 Generic components of Tµν ...... 20

1 Introduction

The Schr¨odinger-Newton [SN] equation has been proposed in the literature as a model for investigating the effects of self-gravity on the motion of a non-relativistic quantum particle [1, 2, 3, 4] (specifically as a model for gravitational localization of macroscopic objects). It is a nonlinear modification to the Schr¨odinger equation with a Newtonian gravitational potential φ: ∂ψ(r, t) ~2 i~ = 2ψ(r, t)+ mφ ψ(r, t), (1) ∂t −2m∇ where the self-gravitating potential φ is assumed to be classical and obeys the semi-classical Poisson equation 2φ =4πGm ψ 2. (2) ∇ | | The coupled system of the above two equations in integro-differential form is given by ~2 2 ∂ψ(r, t) 2 2 ψ(r′, t) 3 i~ = ψ(r, t) Gm | | d r′ψ(r, t), (3) ∂t −2m∇ − r r Z | − ′| and is known as the Schr¨odinger-Newton [SN] equation. There are two broad (complemen- tary) viewpoints under which the SN equation has been dealt with in the literature, amongst others. In one of them, it is considered as a hypothesis and the ways to falsify it are studied through theoretical and (or) experimental considerations, e.g. the localization of wave-packets for macroscopic objects [5], with a gravitationally induced inhibition of quantum dispersion. The second approach focuses on whether the SN equation can be understood as a consequence of the known principles of physics. It is viewed as a model for self-interaction of matter waves. Notable work in this context [6] shows that the SN equation is the non-relativistic limit of the Einstein-Klein-Gordon system and the Einstein-Dirac system for a spherically symmetric space- time. Our present paper follows the second approach. We relax the assumption of a spherically symmetric space-time made in [6] and obtain the SN equation as the non-relativistic limit of the Einstein-Dirac equations. Since the spin of the Dirac field couples naturally to torsion, we also study the Einstein-Cartan-Dirac equations, and obtain it’s non-relativistic limit. These equa- tions are a special case of the Einstein-Cartan-Sciama-Kibble theory [7, 9, 10, 8, 12, 11, 13, 14], which we will henceforth refer to as the Einstein-Cartan theory. The plan of the paper is as follows. In Section 2 we describe the Einstein-Cartan-Dirac equations. Section 3 is the central part of the paper - the non-relativistic limit of the Einstein- Dirac equations is derived here. We first describe the ansatz for the Dirac state and for the

2 metric, which is used to derive the non-relativistic limit. We then describe in detail the non- relativistic expansion for the Dirac equation, and for the energy-momentum tensor. It is then shown that the non-relativistic limit of the Eistein-Dirac equations is the Schr¨odinger-Newton equation, as expected. In Section 4, the non-relativistic limit of the Einstein-Cartan-Dirac equations - which include the torsion of the Dirac field - is derived. It is shown that torsion does not contribute in the non-relativistic limit, and once again we obtain the Schr¨odinger- Newton equation. Conclusions are presented in the next section, while the detailed Appendix gives calculations of the geometric variables such as metric, connection and curvature, as well as the energy-momentum tensor, for the ansatz used in this paper. The present paper is part of a series of our works [15, 16, 17, 18, 19, 20, 21] which inves- tigate the role of torsion in microscopic physics, and the motivation for including torsion in the Einstein-Dirac equations. The fundamental motivation comes from noting that given a relativistic point mass m, Einstein equations as well as the Dirac equation both claim to hold for it, irrespective of the numerical value of the mass. This is because there is no mass scale in either of the system of equations, but of course both cannot hold for all masses. Only from experiments we know that Einstein equations hold for macroscopic masses, and Dirac equation for small masses. But how large is large, and how small is small? There has to be an under- lying dynamics with an inbuilt mass scale, to which the Dirac equation and Einstein equation are small mass and large mass approximations, respectively. The search for this underlying dynamics is aided by the fact that general relativity has Schwarzschild radius as a funda- mental length (depending linearly on mass) and Dirac equation has Compton wavelength as fundamental length (depending inversely on mass). This strongly suggests that the underlying theory should have one unified length, and also that it should include torsion, which domi- nates over curvature for small masses, because in this domain spin dominates mass. We have developed such curvature-torsion models, and investigated what physical role torsion might play in the modified Dirac equation. It is in this spirit that in this paper we are studying the non-relativistic limit of the Eistein-Cartan-Dirac equations, to look for signatures of torsion.

2 Preliminaries: The Einstein-Cartan-Dirac equations

The antisymmetric part of the affine connection, 1 Q µ =Γ µ = (Γ µ Γ µ), (4) αβ [αβ] 2 αβ − βα is called torsion. The affine connection is related to the Christoffel symbols by

µ Γ µ = K µ, (5) αβ αβ − αβ   µ µ µ µ µ where Kαβ is the contorsion tensor, and is given by Kαβ = Qαβ Q αβ + Qβ α. For a matter field ψ, which is minimally coupled to gravity− and torsion,− the action is given by [8] 1 S = d4x√ g (ψ, ψ,g) R(g,∂g,Q) , (6) − Lm ∇ − 2k Z h i where k = 8πG/c4. The first and the second term on the right hand side correspond to contribution from matter and gravity respectively. Varying the action with respect to ψ (matter

3 field), gµν (metric) and Kαβµ (contorsion), the following field equations are obtained: δ(√ g ) − Lm =0, (7) δψ δ(√ gR) δ(√ g ) − =2k − Lm and (8) δgµν δgµν δ(√ gR) δ(√ g ) − =2k − Lm . (9) δKαβµ δKαβµ Eq. (7) yields the matter field equation on a space-time with torsion. The right hand side of Eq. (8) is related to the metric energy-momentum tensor Tµν, while the right hand side of Eq. (9) is associated with the spin density tensor Sµβα. Equations (8) and (9) together give the Einstein-Cartan field equations: Gµν = k Σµν , (10) T µβα = k τ µβα. (11) Gµν is the asymmetric Einstein tensor constructed from the asymmetric connection. Σµν is the canonical energy-momentum tensor (asymmetric) constructed from the metric energy- momentum tensor (symmetric) and the spin density tensor. In Eq. (11), T µβα is the modified torsion (traceless part of the torsion tensor); it is algebraically related to Sµβα on the right hand side. On setting the torsion to zero, the field equations of general relativity are recovered. For a Dirac field (ψ), the matter lagrangian density is given by i~c = (ψγµ ψ ψγµψ) mc2ψψ. (12) Lm 2 ∇µ −∇µ −

We denote a Riemannian space-time by V4 and a space-time with torsion by U4. Minimally coupling a Dirac field on U4 leads to the Einstein-Cartan-Dirac (ECD) theory. The spinors µ µ are defined on V4 and U4 using tetrads. We usee ˆ = ∂ as the coordinate basis, which is covariant under general coordinate transformations. Spinors (defined on a Minkowski space- time) on the other hand are associated with basis vectors which are covariant under local Lorentz transformations. To this aim, we define at each point on the manifold, four orthonormal basis fields (tetrad fields)e ˆi(x), one for each i value. The tetrad fields satisfy the relation i i µ i eˆ (x)= eµ(x)ˆe , where the transformation matrix eµ is such that

(i) (k) eµ eν η(i)(k) = gµν. (13)

(i) The trasformation matrix eµ facilitates the conversion of the components of any world ten- sor (which transform according to general coordinate transformations) to the corresponding components in a local Minkowski space (these latter components being covariant under local Lorentz transformations). Greek indices are raised and lowered using the metric gµν, while the Latin indices are raised and lowered using η(i)(k). Parentheses around indices is a matter of convention. We adopt the following conventions for the remainder of the paper: Objects with Greek indices (world indices), e.g. α,ζ,δ, transforms according to general • coordinate transformations and are raised and lowered using the metric gµν. Objects with Latin indices within parentheses (tetrad indices), e.g. (a) or (i), transform • according to local Lorentz transformations and are raised and lowered using η(i)(k). Latin indices without parentheses, e.g. i, j, b, c refer to objects in Minkowski space, which • transform according to global Lorentz transformations.

4 In general 0, 1, 2, 3 refer to world indices while (0), (1), (2), (3) refer to tetrad indices. •

{} represents the covariant derivative with the Christoffel connections ( ), while •denotes ∇ the total covariant derivative. {} ∇

Commas (, ) refer to partial derivatives and semicolons (; ) to the Riemannian covariant • derivative, which implies (;) and {} are the same for tensors. For spinors, (; ) involves ∇ a partial derivative and the Riemannian part of the spin connection.

Just as the affine connection Γ facilitates parallel transport of geometrical objects with world (Greek) indices, the spin connection (γ) does so for anholonomic objects (those with Latin indices). The affine connection Γ has two parts - Riemannian ( ) and torsional (con- (k)(i) {} (i)(k) structed from the contorsion tensor Kµ ), similarly, the spin connection (γµ ) has two o (i)(k) parts - Riemannian (denoted by γµ ) and torsional (constructed from the contorsion tensor (k)(i) Kµ ). These quantities are interrelated by

γ (i)(k) = γo (i)(k) K (k)(i) and (14) µ µ − µ γ (i)(k) = e(i)eν(k)Γ α eν(k)∂ e(i) µ α µν − µ ν α (15) = e(i)eν(k) K (k)(i) eν(k)∂ e(i). α µν − µ − µ ν   Using equations (14) and(15) , the Riemannian part of the spin connection can be expressed entirely in terms of the Christoffel symbols and the tetrads as [13]

α = eα e γo (k)(i) + eα ∂ e(i). (16) µν (i) ν(k) µ (i) µ ν   One can thus define the covariant derivative for spinors as 1 ψ = ∂ ψ + γo γ[(b)γ(c)]ψ (on V ) (17) ;µ µ 4 µ(b)(c) 4 1 1 and ψ = ∂ ψ + γ0 γ[(b)γ(c)]ψ K γ[(b)γ(c)]ψ. (on U ) (18) ∇µ µ 4 µ(c)(b) − 4 µ(c)(b) 4 The explicit form of the matter Lagrangian density is obtained by substituting equations (17) and (18) in Eq. (12). The Dirac equation is then given by Eq. (7): mc iγµψ ψ =0 (on V ) (19) ;µ − ~ 4 i mc and iγµψ + K γ[(a)γ(b)γ(c)]ψ ψ =0. (on U ) (20) ;µ 4 (a)(b)(c) − ~ 4 The gravitational field equations are obtained using equations (8) and (12):

8πG G ( )= T (on V ) (21) µν {} c4 µν 4 8πG 1 8πG 2 and G ( )= T g SαβλS . (on U ) (22) µν {} c4 µν − 2 c4 µν αβλ 4   The metric EM tensor (symmetric) is defined by

i~c T = Σ ( )= ψγ¯ ψ + ψγ¯ ψ ψ¯ γ ψ ψ¯ γ ψ . (23) µν (µν) {} 4 µ ;ν ν ;µ − ;µ ν − ;ν µ h i 5 Equations (19) and (21) are the governing equations for the Einstein-Dirac theory. The spin density tensor is obtained form the matter Lagrangian density (12): i~c Sµνα = − ψγ¯ [µγνγα]ψ. (24) 4 Using equations (24) and (9), Eq. (20) simplifies to the Hehl-Datta equation [8, 10] which together with Eq. (22) and the relation between the modified torsion tensor and the spin density tensor, constitutes the field equations for the Einstein-Cartan-Dirac theory: 8πG 1 8πG 2 G ( )= T g SαβλS , (25) µν {} c4 µν − 2 c4 µν αβλ   8πG T = K = S and (26) µνα − µνα c4 µνα 3 mc iγµψ =+ L2 ψγ5γ ψγ5γ(a)ψ + ψ. (27) ;µ 8 P l (a) ~

Here, LP l is the Planck length. The Lorentz signature, diag(+, -, -, -) is used througout the paper. The gamma matrices are represented in the Dirac basis, which happens to be the matrix representation of Clifford algebra Cl1,3[R]: I i I i 0 2 0 i 0 σ 5 i i j k l 0 2 i i 0 σ γ = β = ,γ = i ,γ = ǫijklγ γ γ γ = and α = βγ = i . 0 I2 σ 0 4! I2 0 σ 0  −  −      (28)

3 Non-relativistic limit of the Einstein-Dirac equations

3.1 Ansatz for the spinor and the metric Ansatz for the Dirac spinor: We expand ψ(x, t) as ψ(x, t)= eiS(x,t)~ (which can be done for any complex function of x and t). S can be expressed as a perturbative power series in √~ or (1/c), to obtain the semi-classical and the non-relativistic limit respectively. The scheme for obtaining the non-relativistic limit has been employed by Kiefer and Singh [22]. Giulini and Großardt in their work [6] construct a new ansatz with the parameter √~/c as follows:

2 n ic r ∞ √~ ψ(r, t)= e ~ S( ,t) a (r, t), (29) c n n=0 X   where S(r, t) is a scalar function and an(r, t) is a spinor field. We use this ansatz in our present work. Ansatz for the metric: We express a general metric as a perturbative power series in the parameter √~/c, similar to the expansion for the spinor: n ∞ √~ g (r, t)= η + g[n](r, t), (30) µν µν c µν n=1 X   [n] where gµν (x) are metric functions indexed by n. In the non-relativistic scheme, gravitational potentials are weak and cannot produce velocities comparable to c. Hence, we assume the [0] leading order function to be the Minkowski metric, gµν (x)= ηµν . The generic power series for the tetrads, spin coefficients and Einstein tensor are then given by n n ∞ √~ ∞ √~ eµ = δµ + eµ[n], γ = γ[n] , (31) (i) (i) c (i) (a)(b)(c) c (a)(b)(c) n=1   n=1   X n X n ∞ √~ ∞ √~ e(i) = δ(i) + e(i)[n] and G = G[n], (32) µ µ c µ µν c µν n=1 n=1 X   X   6 µ[n] (i)[n] [n] [n] [n] where e(i) , eµ , γ(a)(b)(c) and Gµν are functions of the metric gµν and its derivatives.

3.2 Analyzing the Dirac equation with the above ansatz

We first separate the spatial and the temporal part of the Dirac equation on V4 (Eq. 19). (a) (a) ν µ ν µ µ [Note that γ ψ;(a) = eµ e(a)γ ψ;ν = δµγ ψ;ν = γ ψ;µ]. mc iγµψ ψ =0 (33) ;µ − ~ i i mc iγ0∂ ψ + γ(0)γo γ[(b)γ(c)]ψ + iγα∂ ψ + γ(j)γo γ[(b)γ(c)]ψ ψ =0. (34) ⇒ 0 4 (0)(b)(c) α 4 (j)(b)(c) − ~ (0) (0) Multiplying both sides by e0 γ c, we get ic ic mc2 i∂ ψ + γo γ[(b)γ(c)]ψ + ic e(0)eα α(a)∂ ψ + e(0)α(j)γo γ[(b)γ(c)]ψ e(0)β ψ =0. t 4 0(b)(c) 0 (a) α 4 0 (j)(b)(c) − 0 ~ (35) Using the series expansion for the tetrads and the Riemannian part of the spin connection (equations (31) and (32)), we keep terms of the order c2, c and 1, and neglect terms of the 1 order n with n 1. This is sufficient for obtaining the equation obeyed by the leading order c ≥ spinor term, a0. We thus obtain i√~ i√~ o[1] [(b) (c)] √~ ~ (j) o[1] [(b) (c)] i∂tψ + γ0(b)(c)γ γ ψ + icα. ψ + i E. ψ + α γ(j)(b)(c)γ γ ψ 4 ∇ ∇ 4 (36) mc2 mc β ψ β e(0)[1]ψ βme(0)[2]ψ =0, − ~ − √~ 0 − 0

~ (0)[1] (1) 1 [1] (a) (0)[1] (2) 2 [1] (a) (0)[1] (3) 3 [1] (a) where E = e0 α + e(a) α , e0 α + e(a) α , e0 α + e(a) α . We now ! evaluate eachh term of Eq. (36) by substitutingi h the spinori ansatzh (29): i Term 1 n ic2S ∞ √~ +i∂ ψ = i∂ e ~ a t t c n n=0 h X   i 2 3 n ic S c ∞ √~ ~ = e Sa˙ n 1 + ia˙ n 3 . (37) ~3/2 c − − − n=0 X   h i Term 2

2 n i√~ o[1] i√~ o[1] ic S ∞ √~ + γ γ[(b)γ(c)]ψ =+ γ γ[(b)γ(c)] e ~ a 4 (0)(b)(c) 4 (0)(b)(c) c n n=0 h X   i 2 3 n ic S c ∞ √~ i√~ ~ o[1] [(b) (c)] = e γ γ γ an 3 . (38) ~3/2 c 4 (0)(b)(c) − n=0 X   h i Term 3 n ic2S ∞ √~ icα. ψ = ic α −→ e ~ a ∇ −→ · ∇ c n n=0 h X   i 2 2 n ic S c ∞ √~ ~ = ic−→α e i−→San + −→an 2 · ~ c ∇ ∇ − n=0 h X    i 2 3 n ic S c ∞ √~ ~ √~ √~ = e −→α −→San + i −→α −→an 2 . (39) ~3/2 c − · ∇ · ∇ − n=0 X   h i 7 Term 4

n ic2S ∞ √~ +i√~E.~ ~ ψ = i√~E.~ ~ e ~ a ∇ ∇ c n n=0 h X   i 2 3 ic S c ∞ ~ = e √~E.~ ~ S an 1 + i√~E.~ ~ an 3 . (40) ~3/2 − ∇ − ∇ − n=0 X   Term 5

2 n i√~ o[1] i√~ o[1] ic S ∞ √~ + α(j)γ γ[(b)γ(c)]ψ =+ α(j)γ γ[(b)γ(c)] e ~ a 4 (j)(b)(c) 4 (j)(b)(c) c n n=0 h X   i 2 3 n ic S c ∞ √~ i√~ ~ (j) o[1] [(b) (c)] = e α γ γ γ an 3 . (41) ~3/2 c 4 (j)(b)(c) − n=0 X   h i Term 6

2 2 2 n mc mc ic S ∞ √~ β ψ = β e ~ a − ~ − ~ c n n=0 X   2 3 n ic S c ∞ √~ ~ = e (βman 1). (42) − ~3/2 c − n=0 X   Term 7

2 n mc (0)[1] mc (0)[1] ic S ∞ √~ β e ψ = β e e ~ a − √~ 0 − √~ 0 c n n=0 h X   i 2 3 n ic S c ∞ √~ ~ (0)[1] = e βm e0 an 2 . (43) − ~3/2 c − n=0 X   h i Term 8

2 n (0)[2] (0)[2] ic S ∞ √~ βme ψ = βm e e ~ a − 0 − 0 c n n=0  h X   i 2 3 n ic S c ∞ √~ ~ (0)[2] = e βme0 an 3. (44) − ~3/2 c − n=0 X   We thus obtain

2 3 n ic S c ∞ √~ (0)[1] ~ √~ ~ ˙ √~ ~ ~ √~ ~ e ~α. S an S + βm + E. S an 1 + i ~α. βm e0 an 2 ~3/2 c − ∇ − ∇ − ∇ − − n=0 " X         i√~ i√~ √~ ~ ~ o[1] [(b) (c)] (j) o[1] [(b) (c)] (0)[2] + i∂t + i E. + γ(0)(b)(c)γ γ + α γ(j)(b)(c)γ γ βme0 an 3 =0. ∇ 4 4 − ! − # (45)

At the leading order (n=0), we get S =0, (46) ∇ which implies that the scalar S is a function of time only, i.e. S = S(t). The Dirac spinor is a 4-component object which can be written as an = (an,1, an,2, an,3, an,4). We split it into

8 > < ˙ two-component spinors, an =(an,1, an,2) and an =(an,3, an,4). At order n=1, we get S + βm + √~E.~ ~ S = 0. Since ~ S = 0, this implies ∇ ∇ ˙ > (m + S)a0 =0, (47a) and (m S˙)a< =0, (47b) − 0 < > which satisfies either S = mt with a0 =0 or S = +mt with a0 = 0. The wave function at ±imc2t − ~ this order is ψ = e a0, which corresponds to particles of positive energy (lower sign) and < negative energy (upper sign), at rest. We restrict ourselves to S = mt and a0 = 0, i.e. the positive energy solutions. It is implicitly assumed that the two cases− (positive and negative energies) can be studied separately. We digress at this point and analyze the energy-momentum tensor.

3.3 Analyzing the energy momentum tensor Tµν with the above ansatz The dynamical energy momentum tensor is given by Eq. (23). We analyze all the sixteen components of Tµν : 1) kT00 (with the indices of the gamma matrices raised): ~ 4iπG ¯ 0 c o [(i) (j)] ¯ c o [(i) (j)] ¯ 0 kT00 = 4 ψγ ∂tψ + [γ0(i)(j)γ γ ]ψ ∂tψ + [γ0(i)(j)γ γ ]ψ γ ψ (48) c " 4 − 4 #     4iπG~ ∞ √~ c = 1+ ne0[n] ψγ¯ (0) ∂ ψ + [γo γ[(i)γ(j)]]ψ (49) c4 c (0) t 4 0(i)(j) − n=1 "  X
   ¯ c o [(i) (j)] ¯ (0) ∂tψ + [γ0(i)(j)γ γ ]ψ γ ψ . 4 #   Substituting the spinor ansatz (Eq.29) in Eq.(48), we obtain a series expansion for kT00. At the leading order we get

n m ∞ √~ ∞ √~ 4iπG ˙ kT00 = an† iSam +a ˙ m 2 c2 c c −  n=0    m=0   h i n X X m (50) ∞ √~ ∞ √~ ∞ ˙ 1 + iSan† a˙ n† 2 am + O , c − − c cn n=0 n=0 n=3  X   h i X    X   with (n + m = 0), i.e.

∞ 4πGi > > > > 1 kT = i( m)a †a + i( m)a †a + O (51) 00 c2 − 0 0 − 0 0 cn n=3   X   8πGm a> 2 ∞ 1 = | 0 | + O . (52) c2 cn n=3 X   2) kT0µ (µ =1, 2, 3): ~ 2iπG ¯ 1 o [(i) (j)] ¯ 1 o [(i) (j)] kT0µ = 4 cψγ0 ∂µψ + [γµ(i)(j)γ γ ψ] + cψγµ ∂0ψ + [γ0(i)(j)γ γ ψ] c " 4 4     (53) ¯ 1 o [(i) (j)] ¯ ¯ 1 o [(i) (j)] ¯ c ∂µψ + [γµ(i)(j)γ γ ψ] γ0ψ c ∂0ψ + [γ0(i)(j)γ γ ψ] γµψ . − 4 − 4 #     9 1 Terms containing the spin coefficients (γµ(i)(j)) are of the order c3 or higher and hence do not 1 contribute at the order c2 . Rest of the terms are analyzed in appendix section 7.7.1, and are 1 shown to have no contribution at the order c2 . Hence, ∞ 1 kT = O . (54) 0µ cn n=3 X   3) kTµν (µ, ν =1, 2, 3): ~ 2iπG ¯ 1 o [(i) (j)] ¯ 1 o [(i) (j)] kTµν = 3 + ψγµ ∂νψ + [γν(i)(j)γ γ ψ] + ψγν ∂µψ + [γµ(i)(j)γ γ ψ] c " 4 4     (55) ¯ 1 o [(i) (j)] ¯ ¯ 1 o [(i) (j)] ¯ ∂ν ψ + [γν(i)(j)γ γ ψ] γµψ ∂µψ + [γµ(i)(j)γ γ ψ] γνψ . − 4 − 4 #     1 Once again, terms containing the spin coefficients (γµ(i)(j)) are of the order c3 or higher and 1 hence do not contribute at the order c2 . Rest of the terms are analyzed in appendix section 1 7.7.2, and are shown to have no contribution at the order c2 . Hence,

∞ 1 kT = O . (56) µν cn n=3 X   Results of the order analysis of the EM tensor, summarized by equations (52),(54) and (56) imply

T00 T00 1 | | 1, | | 1, and k T00 O 2 i, j (1, 2, 3). (57) T0i ≪ Tij ≪ | | ∼ c ∀ ∈ | | | |   Owing to Einstein’s equations, the same relations then hold for the components of the Einstein tensor, i.e.

G00 G00 1 | | 1, | | 1, and G00 O 2 i, j (1, 2, 3). (58) G0i ≪ Gij ≪ | | ∼ c ∀ ∈ | | | |   3.4 Constraints on the metric 1 In section 3.3 we showed that G O 2 while all the other components of G are | 00| ∼ c µν of higher order. For a generic metric ansatz, Gµνhas been calculated in appendix section 7.1. At this point we make an important assumption – the metric field is asymptotically flat. This leads to the following constraints on the metric components (proved in appendix section 7.2):

[1] 1) Gµν =0( µ, ν) together with the condition of asymptotic flatness of the metric, leads ∀ to the following results (proved in appendix section 7.2.1): g[1] =0, eµ[1] =0, e(i)[1] =0, and γ[1] =0 i, j, k, µ, ν (0, 1, 2, 3). (59) µν (i) µ (i)(j)(k) ∀ ∈ [2] [2] 2) Gµν = 0 (except for µ = ν = 0) leads to the following constraint: gµν = F (r, t)δµν for some field F (r, t) (proved in appendix section 7.2.2).

The full metric is then given by [n] [n] [n] [n] 10 0 0 F 0 0 0 g00 g01 g02 g03 n [n] [n] [n] [n] ~ ∞ √~ 0 1 0 0 0 F 0 0 g10 g11 g12 g13 gµν(r, t)=  − +   (r, t)+   (r, t), 0 0 1 0 c2 0 0 F 0 c [n] [n] [n] [n] n=3 g20 g21 g22 g23 −      [n] [n] [n] [n] 0 0 0 1  0 0 0 F  X g g g g   −     30 31 32 33       (60)

10 [2] [2] [2] [2] where g00 = g11 = g22 = g33 = F (r, t). The above metric has been employed to calculate other objects (tetrads, spin coefficients, etc.) in appendix sections 7.3, 7.5, 7.4 and 7.6.

3.5 Non-Relativistic (NR) limit of the Einstein-Dirac equations Dirac equation: In section 3.2 we analyzed Eq. (45) for n = 0 and n = 1. Using the results of sections 7.3 - 7.5, Eq. (45) can be further simplified to

2 3 n ic S c ∞ √~ mF (r, t) ~ ˙ √~ e (S + βm) an 1 + ia˙ n 3 + i −→α −→an 2 β an 3 =0. ~3/2 c − − − · ∇ − − 2 − n=0   h i X (61) At order n =2, Eq. (61) gives us ˙ > > S +m 0 a1 √~ 0 −→σ −→ a0 ˙ < i · ∇ < =0. (62) 0 S - m a1 − σ −→ 0 ! a0     −→ · ∇   The first of these two equations is trivially satisfied. The second equation yields a relation < > between a1 and a0 : i√~ σ −→ a< = − −→ · ∇ a>. (63) 1 2m 0 At order n = 3, we get ˙ > > S +m 0 a2 √~ 0 −→σ −→ a1 ˙ < i · ∇ < 0 S - m a2 − σ −→ 0 ! a1     −→ · ∇   mF (r,t) > i∂t 2 0 a0 − r =0, (64) − 0 i∂ + mF ( ,t) a< t 2 !  0  which comprises of two equations. Using Eq. (63), the first of these two equations gives us ∂a> ~2 m~F (r, t) i~ 0 = 2a> + a>. (65) ∂t −2m∇ 0 2 0 Einstein’s equations: The Einstein tensor has been evaluated in appendix section 7.6. Equat- ing G00 to kT00 we get ~ 2F (r, t) ∞ 1 8πGm a> 2 ∞ 1 ∇ + O = | 0 | + O . (66) c2 cn c2 cn n=3 n=3 X   X   At the leading order, this gives us 8πGm a> 2 2F (r, t)= | 0 | . (67) ∇ ~ ~F (r,t) Recognizing the quantity 2 as the Newtonian potential φ, we obtain the Schr¨odinger- > 2 Newton system of equations (mφ gravitational potential energy and m a0 mass density) as the NR limit of the Einstein-Dirac→ equations: | | → ∂a> ~2 i~ 0 = 2a> + mφ(r, t)a> and (68) ∂t −2m∇ 0 0 2φ(r, t)=4πGm a> 2 =4πGρ(r, t) (69) ∇ | 0 | > ~2 > 2 ∂a0 2 > 2 a0 (r′, t) 3 > = i~ = a Gm | | d r′a , (70) ⇒ ∂t −2m∇ 0 − r r 0 Z | − ′| the physical picture for which has already been discussed in section 1. This completes the derivation of the Schr¨odinger-Newton equation as the non-relativistic limit of the Einstein- Dirac equations.

11 4 Non-relativistic limit of the Einstein-Cartan-Dirac equa- tions

We shall now employ the WKB type ansatz of the previous section to the case when torsion is included. It is to be noted that the torsion of the Dirac field can be expressed directly in terms of the Dirac spinors. Once the substitution of the torsion tensor has been done in terms of the Dirac spinors, the non-linear Dirac equation no longer makes any reference to torsion. Similarly, in the Einstein-Cartan field equations, the contribution coming from torsion can be expressed in terms of the Dirac state. Thus the Einstein-Cartan-Dirac system is a coupled differential system for the metric and the Dirac state - just like the Einstein-Dirac system is - only, the non-linear terms are now different. Thus the WKB ansatz used earlier can be directly used in the presence of torsion as well. The Dirac equation on U4 (also known as the Hehl-Datta equation) is given by (Eq. (27)) 3 mc iγµψ L2 ψγ5γ ψγ5γ(a)ψ ψ =0. (71) ;µ − 8 P l (a) − ~ We have already analyzed the first and the last term on the left hand side using the ansatz for the spinor (29) and the metric (60). The second term arises due to torsion and makes the equation non-linear. We evaluate this term similar to the other two (section 3.2). Multiplying (0) (0) the middle term by e0 γ c (as was done in section 3.2 to obtain (35) from (34)), we get

2 (0) 3c 3c ic S ~F (r, t) ∞ 1 e γ(0) L2 ψγ5γ ψγ5γ(a)ψ = l2 e ~ 1+ + O − 0 8 P l (a) − 8 P l 2c2 cn h n=3  i n X l l ∞ ~ ∞ ~ ∞ ~ √ 5 √ 5 (a) √ a† γ γ a γ γ a , c n (a) c l c m  n=0     l=0     m=0    X X X (72) which simplifies to

2 3 n ic S c ~F (r, t) ∞ 1 3G ∞ √~ ~ 5 5 (a) e 1+ + O an† 1 iγ γ(a)an2 jγ γ an3 k , (73) ~3/2 2c2 cn 8 c − − − " n=3 # n1,n2,n3=0 X    X    where n = n1 + n2 + n3. This term modifies Eq. (61) as follows:

2 3 n ic S c ∞ √~ mF (r, t) ~ √~ e m an 1 + ia˙ n 3 + i −→α −→an 2 βman 1 β an 3 ~3/2 c − − · ∇ − − − − 2 − n=0  " X n (74) ∞ ~ 3G √ 5 5 (a) + an† 1 iγ γ(a)an2 jγ γ an3 k =0, 8 c − − − n1,n2,n3=0 #  X    where n = n + n + n and i + j + k = 5, with i n , j n and k n . Further, 1 2 3 ≤ 1 ≤ 2 ≤ 3 i, j, k, n1, n2, n3 (0,1,2,3,4,5). The non-linear term contributes only at order n=5 and higher. ∈ > As a result, the analysis for n =0, 1, 2 and 3 (considered in section 3.5) holds good. Thus a0 > 2 ~ r satisfies the Schr¨odinger equation, i.e. i~ ∂a0 = ~ 2a> + m F ( ,t) a>. ∂t − 2m ∇ 0 2 0 1 2 αβλ Einstein’s equations on U4 read: Gµν ( ) = kTµν 2 k gµνS Sαβλ. Gµν ( ) and Tµν have already been analyzed in section 3.5{}. The second− term on the right hand{} side, i.e. 1 k2g SαβλS , involves a contraction of the spin density tensor (24). We consider only − 2 µν αβλ

12 the first term in the series expansion of the metric, because the other terms together with the coupling constant are of orders not relevant for the NR limit. We thus obtain

2 2~2 ∞ ∞ ∞ ∞ ∞ ∞ 1 2 αβλ 2π G 0 [c a b] 0 1 − k g S S = g a† γ γ γ γ a† γ γ γ γ n = O , 2 00 αβλ − 00 c6 k m [c a b] m cn m=0 n=0 n=6 NX=0  Xk=0 Xl=0  X X  X  

(75)

1 which implies that this additional term does not contribute at the order c2 on the right hand side of Eq. (25). Hence, we once again recover Poisson’s equation. Thus the Schr¨odinger- Newton equation also happens to be the NR limit of the ECD theory, which implies that torsion does not contribute at the leading order.

5 Conclusions

While the non-relativitic limit of the Einstein-Dirac equations for a self-gravitating Dirac field has been calculated by Giulini and Großardt [6], we relax the assumption of a spheri- cally symmetric metric in our present work. The Schr¨odinger-Newton equation is obtained as the non-relativistic limit for a general metric, by considering a perturbative series in the √~ parameter c , for the spinor, the metric and other relevant quantities. This scheme for ob- taining the non-relativistic limit follows the WKB like expansion given by Giulini and Großardt [6]. The Einstein-Cartan-Dirac equations provide an elegant system for coupling matter to the geometry of space-time, where torsion arises due to the spin of the Dirac field. The non-relativistic limit of this system of equations (derived in section 4) yields the Schr¨odinger- 1 Newton equation, at the leading order of the parameter c . This suggests that torsion does not manifest itself at this order.   The effect of torsion in the higher order corrections to the Schr¨odinger-Newton equation, can be obtained from the Einstein-Cartan-Dirac equations, by considering a WKB type expansion for the spinor and other relevant quantities, as was done in the present work. However, in this paper, we have restricted ourselves to the analysis at the leading order, which gives us the non-relativistic limit. A similar prescription may also be employed to obtain the higher order corrections to the Schr¨odinger equation (starting from Dirac’s equation) and Newton’s equation for gravitation (starting from Einstein’s equations). The Einstein-Cartan-Dirac equations with the unified new length scale [19] provide for the possibility of a solitonic solution which interpolates between a black hole and a Dirac fermion. This is one of the primary motivations for us to study this system of field equations. The search for such solutions has been attempted in [21] and further work is in progress. One could well ask if Derrick’s theorem [23] could compel such solitonic solutions to be unstable. The theorem suggests that stationary localized solutions to non-linear wave equations such as considered here are unstable. In the present situation however, the inclusion of torsion [which has a dispersive effect] makes it more plausible to achieve a stable balancing solution where the dispersive aspect due to torsion balances the collapse aspect due to gravity. Moreover, a way out of Derrick’s no-go theorem is that the sought for solitonic solutions are periodic in time, rather than time-independent. Such solutions were actually reported by us in [21]. Rigorously speaking, the so-called Vakhitov-Kolokolov stability criterion [24] provides a precise condition for the linear stability of a periodic solitary wave solution. This requirement continues to hold for the Einstein-Cartan-Dirac equations as well.

13 6 Acknowledgments

SK would like to thank the Indian Institute of Science Education and Research, Pune and the Department of Science and Technology, Govt. of India for the SHE-INSPIRE fellowship. AP and VD would like to thank the Visiting Students’ Research Program 2017, organized by the Tata Institute of Fundamental Research (TIFR), Mumbai, under which a part of this work was accomplished. SK, AP and VD would like to express their sincere gratitude to the Department of Astronomy and Astrophysics, TIFR, for its hospitality.

7 Appendix

7.1 Form of the Einstein tensor evaluated using the generic metric upto second order The ansatz for the metric is given by (Eq. (30))

n ∞ √~ g (x)= η + g[n](x). µν µν c µν n=1 X   To the second order, the metric and its inverse is then given by

√~ ~ ∞ 1 g = η + g[1] + g[2] + O and (76) µν µν c µν c2 µν cn n=3     X   √~ ~ ∞ 1 gµν = ηµν gµν[1] [gµ[1]gβν[1] + gµν[2]]+ O . (77) − c − c2 β cn n=3     X   We evaluate Christoffel symbols, Riemann curvature tensor, Ricci tensor and the scalar curvature to obtain the Einstein tensor Gµν up to the second order as follows:

√~ ~ G ( )= G[1] ( )+ G[2] ( ), (78) µν {} c µν {} c2 µν {} where,     1 1 G[1] ( )= g[1] , g[1] = g[1] η g[1], g[1] =(ηµν g[1]), (79) µν {} −2 µν ij µν − 2 µν µν 1 1 G[2] ( )= g(2) + f(g[1]), g[2] = g[2] η g[2] and g[2] =(ηµνg[2]). (80) µν {} −2 µν µν ij µν − 2 µν µν [1] In Eq. (80), ‘f’ is a function of gµν , which is given by

1 f(g[1])= 2∂λg[1]∂ g[1] 2∂λg[1]∂ g[1] ∂ gλ[1]∂ gρ[1] ∂ gλ[1]∂ gρ[1]+ µν −4 ν λµ − λ µν − ρ ν µ λ − ρ ν λ µ h [1] ρ[1] [1] ∂ gλ[1]∂ρg + ∂ gλ[1]∂ g + ∂ gλ[1]∂ gρ[1] ∂ gλ[1]∂ρg ρ ν λµ ν ρ µ λ ν ρ λ µ − ν ρ λµ 1 i 2∂λg[1]∂ g[1] 2η ∂λg[1]∂ g[1] ∂ gλ[1]∂ gρ[1] ∂ gλ[1]∂ gρ[1] −8 ν λµ − µν λ − ρ ν µ λ − ρ µ λ ν h [1] ρ[1] +∂ gλ[1]∂ρg + ∂ gλ[1]∂ g + ∂ gλ[1]∂ gρ[1] ∂ gλ[1]∂ρg . ρ µ λν µ ρ ν λ µ ρ λ ν − ν ρ λµ[1] i G happens to be the same as G ( ) for V . For U on the other hand, G ( ) is the µν µν {} 4 4 µν {} Riemannian part of Gµν (a symmetric tensor constructed from the Christoffel symbols).

14 7.2 Constraints on the metric due to the asymptotic flatness con- dition [1] 7.2.1 Constraint on gµν

[1] From the analysis of section 3.3 one can argue that all the components of Gµν are zero, [1] [1] which implies g = gµν = 0, from Eq. (79) for µ = ν (off-diagonal terms). Gravitational µν 6 waves are the non-trivial solutions to this equation. However, they do not respect asymptotic [1] flatness. We are therefore obliged to consider the trivial solution, i.e. gµν = 0 for the off- diagonal terms. In order to evaluate the diagonal terms, we consider the following general form of the metric:

[1] f1 0 0 0 [1] [1]  0 f2 0 0  gµν = [1] . (81) 0 0 f3 0  [1]  0 0 0 f   4  Hence,  

f [1] + f [1] + f [1] + f [1] g¯[1] = 1 2 3 4 , (82) 00 2 f [1] + f [1] f [1] f [1] g¯[1] = 1 2 − 3 − 4 , (83) 11 2 f [1] + f [1] f [1] f [1] g¯[1] = 1 3 − 2 − 4 and (84) 22 2 f [1] + f [1] f [1] f [1] g¯[1] = 1 4 − 2 − 3 . (85) 33 2 Using the above equations, we get

f [1] + f [1] + f [1] + f [1] g¯[1] =  1 2 3 4 =0= f [1] + f [1] + f [1] + f [1] =0, (86) 00 2 ⇒ 1 2 3 4 f [1] + f [1] f [1] f [1] g¯[1] =  1 2 − 3 − 4 =0= f [1] + f [1] = f [1] + f [1], (87) 11 2 ⇒ 1 2 3 4 f [1] + f [1] f [1] f [1] g¯[1] =  1 3 − 2 − 4 =0= f [1] + f [1] = f [1] + f [1] and (88) 22 2 ⇒ 1 3 2 4 f [1] + f [1] f [1] f [1] g¯[1] =  1 4 − 2 − 3 =0= f [1] + f [1] = f [1] + f [1]. (89) 33 2 ⇒ 1 4 2 3 Equations (87), (88) and (89) imply

f [1] = f [1] = f [1] = f [1] + c , (90) 2 1 ⇒ 2 1 1 f [1] = f [1] = f [1] = f [1] + c and (91) 3 1 ⇒ 3 1 2 f [1] = f [1] = f [1] = f [1] + c . (92) 4 1 ⇒ 4 1 3

The constants c1,c2 and c3 must be zero, as any one of them not being equal to zero would violate the condition of asymptotic flatness. Hence, Eq. (86) implies, 4f [1] =0= f [1] = 0 1 ⇒ 1 (wave solutions and non-zero constants also satisfy the equation, but do not respect asymptotic flatness). Hence, f [1] =0 i, which in turn implies i ∀ g[1] =0 µ, ν. (93) µν ∀ 15 [2] 7.2.2 Constraint on gµν

[2] From the analysis of section 3.3 one can argue that the off-diagonal components of Gµν are  [2]  [2] zero. This implies, gµν = gµν =0(µ = ν), from Eq. (79). Following the same arguments of [2] 6 section 7.2.1, gµν =0(µ = ν) is the only allowed solution to the above equation. Once again, we consider the following6 general form of the metric in order to evaluate the diagonal terms:

[2] f1 0 0 0 [2] [2]  0 f2 0 0  gµν = [2] . (94) 0 0 f3 0  [2]  0 0 0 f   4  Hence,  

f [2] + f [2] + f [2] + f [2] g¯[2] = 1 2 3 4 , (95) 00 2 f [2] + f [2] f [2] f [2] g¯[2] = 1 2 − 3 − 4 , (96) 11 2 f [2] + f [2] f [2] f [2] g¯[2] = 1 3 − 2 − 4 and (97) 22 2 f [2] + f [2] f [2] f [2] g¯[2] = 1 4 − 2 − 3 . (98) 33 2 At the second order, all the components of the Einstein tensor are zero, except for the ‘00’ component. This implies

f [2] + f [2] + f [2] + f [2] g¯[2] =  1 2 3 4 = f [2] + f [2] + f [2] + f [2] =0, (99) 00 2 ⇒ 1 2 3 4 6 f [2] + f [2] f [2] f [2] g¯[2] =  1 2 − 3 − 4 = f [2] + f [2] = f [2] + f [2], (100) 11 2 ⇒ 1 2 3 4 f [2] + f [2] f [2] f [2] g¯[2] =  1 3 − 2 − 4 = f [2] + f [2] = f [2] + f [2] and (101) 22 2 ⇒ 1 3 2 4 f [2] + f [2] f [2] f [2] g¯[2] =  1 4 − 2 − 3 = f [2] + f [2] = f [2] + f [2]. (102) 33 2 ⇒ 1 4 2 3 Equations (100), (101) and (102) imply

f [2] = f [2] = f [2] = f [2], (103) 2 1 ⇒ 2 1 f [2] = f [2] = f [2] = f [2] and (104) 3 1 ⇒ 3 1 f [2] = f [2] = f [2] = f [2]. (105) 4 1 ⇒ 4 1 The absence of constants in the above equations follows from the arguments of section 7.2.1. [2] [2] [2] [2] Using equations (103), (104) and (105), we get f1 = f2 = f3 = f4 = F (r, t), hence, F (r, t)0 0 0 0 F (r, t)0 0 g[2] = . (106) µν  0 0 F (r, t) 0   0 0 0 F (r, t)    

16 7.3 Metric and Christoffel symbols The metric defined in equation (60) is of the form

~F (r,t) 1+ c2 0 0 0 ~F (r,t) 2 ∞ 1  0 1+ c 0 0  gµν = − ~F (r,t) + O n (107) 0 0 1+ 2 0 c c n=3  − ~F (r,t)  X    0 0 0 1+ 2   − c   ~F (r,t)  1 c2 0 0 0 − ~F (r,t) 2 ∞ 1 µν  0 1 c 0 0  and g = − − ~F (r,t) + O n . (108) 0 0 1 2 0 c c n=3  − − ~F (r,t)  X    0 0 0 1 2   − − c    Christoffel connections: For the above metric the non-zero Christoffel connections are

~∂ F (r, t) ∞ 1 Γ0 = µ + O , 0µ 2c2 cn n=3 X   ~∂ F (r, t) ∞ 1 Γµ = µ + O and (109) 00 2c2 cn n=3 X   ~∂ F (r, t) ∞ 1 Γµ = − µ + O , µµ 2c2 cn n=3 X   where µ runs from 1 to 3 (spatial coordinates only). It is worth noting that the zeroth and 1 first order terms in c are absent in Eq.(109). The other non zero Christoffel connections   1 are of order three and higher in c , which we do not mention here.   7.4 Tetrads Tetrads were introduced in section (2). For the metric defined by

~ ~ 2 F (r, t) 2 2 F (r, t) 2 dS = 1+ 2 c dt 1 2 dr , (110) " c # − " − c # the tetrad fields over the entire manifold are given by

1 1 1 1 2 2 2 2 1 ~F ~F ~F ~F eˆ(0) = 1+ 2 ∂t, eˆ(1) = 1 2 ∂x, eˆ(2) = 1 2 ∂y ande ˆ(3) = 1 2 ∂z. c c ! − c ! − c ! − c ! (111)

The transformation matrices (defined in Eq. (13)) which relates the world components with the anholonomic components are given by

17 ~F (r,t) 1+ 2c2 0 0 0 ~F (r,t) 0 1 2 0 0 ∞ 1 (i)  2c  eµ = − ~F (r,t) + O n , (112) 0 01 2 0 c 2c n=3  − ~F (r,t)  X    0 0 01 2   − 2c   ~F (r,t)  1 2c2 0 0 0 − ~F (r,t) 2 ∞ 1 µ  0 1+ 2c 0 0  e(i) = ~F (r,t) + O n , (113) 0 0 1+ 2 0 c 2c n=3  ~F (r,t)  X    0 0 01+ 2   2c   ~F (r,t)  1+ 2c2 0 0 0 ~F (r,t) 2 ∞ 1  0 1+ 2c 0 0  eν(k) = − ~F (r,t) + O n and (114) 0 0 1+ 2 0 c 2c n=3  − ~F (r,t)  X    0 0 0 1+ 2   − 2c   ~F (r,t)  1 2c2 0 0 0 − ~F (r,t) 2 ∞ 1 ν(k)  0 1 2c 0 0  e = − − ~F (r,t) + O n . (115) 0 0 1 2 0 c 2c n=3  − − ~F (r,t)  X    0 0 0 1 2   − − 2c    o 7.5 Riemannian part of the spin connections (γ(a)(b)(c)) Using the relation between Christoffel connections and tetrad transformation matrices (Eq.(16)), the Riemannian part of the spin connections (defined by Eq.(14)) are obtained as follows:

~F ~ 1+ 2 ∞ ~ ~ 2 ∞ o ∂0F 2c 1 o ∂iF F/2c 1 γ(0)(0)(0) = − + O , γ(i)(0)(0) = − + O , 2c2  ~F  cn 2c2 ~F cn 1 2 n=3 1+ 2 n=5 − 2c X     2c X    ~F    ~ 1+ 2 ∞ ~ o ∂iF 2c 1 o ∂iF 1 γ(0)(i)(0) = − + O , γ(0)(0)(i) = , 2c2  ~F  cn 2c2 ~F 1 2 n=3 1+ 2 − 2c X   2c ~ ~ 2  ∞  ∞ o ∂iF F/2c 1 o o 1 γ(i)(i)(i) = + O , γ(i)(i)(0) = γ(i)(0)(i) =+ O , 2c2 ~F cn cn 1+ 2 n=5 n=3 2c X   X   ~∂ F ∞  1 ∞ 1 γo = − 0 + O , γo = γo = γo =+ O , (0)(i)(i) 2c2 cn (0)(i)(j) i0j ij0 cn n=3 n=3 X   X   ~F ~ 1 2 ∞ ∞ o ∂0F − 2c 1 o o o 1 γ(i)(j)(j) = − + O and γ(i)(j)(k) = γ(i)(j)(i) = γ(j)(j)(i) =+ O . 2c2  ~F  cn cn 1+ 2 n=3 n=3 2c X   X     (116)

The torsional part of the spin connections (defined by Eq.(14)) manifests itself as a non- linear term in the Hehl-Datta equation. This term being completely expressible in terms of the Dirac spinor, is evaluated using the spinor ansatz while deriving the non-relativistic limit of the ECD system of equations.

18 7.6 Einstein tensor [1] In this section, we aim to evaluate the Einstein tensor. Gµν has already been shown to be [1] [1] [2] zero. Since gµν is zero, f[gµν ] (defined in Eq. (80)) is also zero. Using gµν (defined in section [2] (7.2.2)), Gµν (Eq.80) is evaluated as follows: 1 1 G[2] = g[2] where g[2] = g[2] η (ηαβh ), now (117) µν −2 µν µν µν − 2 µν αβ

~F (r,t) 10 0 0 c2 0 0 0 ~F (r,t) ~ µν 0 -1 0 0 0 c2 0 0 2 F (r, t) η hµν =    ~ r  = − , (118) 0 0 -1 0 F ( ,t) 2 0 0 c2 0 c  0 0 0 -1   ~F (r,t)     0 0 0 c2      thus G = 0 for µ = ν. The diagonal components are given by  µν 6

~ ~ 2 ~ 2 1 [2]  ∂t F (r, t) F (r, t) G00 = g00 = 2 F (r, t)= 4 + ∇ 2 (119) −2 −c " − c c # and G = 0 because g[2] =0 for α (1, 2, 3). (120) αα αα ∈ Thus, 2F (r, t)000 ∇ ~ 0 000 ∞ 1 Gµν =   + O . (121) c2 0 000 cn n=3  0 000 X       7.7 Analysis of the components of the metric EM tensor This section contains the calculations and proofs for some of the results used in section 3.3.

7.7.1 Analysis of kT0µ

After excluding the terms containing the spin coefficients γµ(i)(j), kT0µ (Eq. (53)) is given by

~ 2iπG ¯ 0 ¯ µ ¯ 0 ¯ µ kT0µ = 4 cψγ ∂µψ cψγ ∂0ψ c∂µψγ ψ + c∂0ψγ ψ (122) c " − − # n ~ ∞ √~ 2iπG 0[n] ¯ (0) ¯ (0) = − 3 1+ e(0) ψγ ∂µψ ∂µψγ ψ (123) c c " − #  n=1    X n 2iπG~ ∞ √~ + 1+ eµ[n] ∂ ψγ¯ (a)ψ ψγ¯ (a)∂ ψ . c4 c (a) t − t n=1 " #  X    The first term on the right hand side of Eq. (123) is n 2iπG~ ∞ √~ = a† ∂ a ∂ a† a (n = n + n ) c3 c n1 µ n2 − µ n1 n2 1 2 n=0 X     ∞ 1 = O . (124) cn n=3 X   19 While the second term is

n m ∞ √~ ∞ √~ 2iπG (a) ˙ = an† α iSam +a ˙ m 2 c2 c c −  n=0     m=0   h i Xn X m ∞ √~ ∞ √~ ∞ ˙ (a) 1 + iSan† a˙ n† 2 α am + O c − − c cn n=0 n=0 n=3  X   h i  X    X   ∞ 4πGm (a) 1 = (a†α a )+ O c2 0 0 cn n=3 X   4πGm 0 σ(a) a> ∞ 1 = a> 0 † 0 + O c2 0 σ(a) 0 0 cn "     n=3
X   ∞ 1 = O . (125) cn n=3 X   Hence, there is no contribution at the second order.

7.7.2 Analysis of kTµν

After excluding the terms containing the spin coefficients γµ(i)(j), kTµν (Eq. (55)) is given by

~ 2iπG ¯ µ ¯ ν ¯ µ ¯ ν kTµν = 3 ψγ ∂νψ ψγ ∂µψ + ∂ν ψγ ψ + ∂µψγ ψ c " − − # n ~ ∞ ~ 2iπG √ µ[n] (a) (a) † † = 3 1+ e(a) ψ α ∂νψ ∂ν ψ α ψ c c " − #  n=1    X n ~ ∞ ~ 2iπG √ ν[n] (b) (b) † † + 3 1+ e(b) ∂µψ α ψ ψ α ∂µψ c c " − #  n=1    Xn ~ ∞ ~ 2iπG √ µ (a) µ (a) ν (b) ν (b) = e a† α ∂ a e ∂ a† α + e a† α ∂ a e ∂ a† α a c3 c (a) n1 ν n2 − (a) ν n1 (b) n1 µ n2 − (b) µ n1 n2 n=0 X     ∞ 1 = O . (126) cn n=3 X   Hence, there is no contribution at the second order.

7.8 Generic components of Tµν Using the spin connections of section (7.5), we analyze the metric energy-momentum tensor (Eq.(23)), whose components are given on the following page.

20 ¯ ¯ ¯ ¯ ¯ ¯ ¯ 2ψγ0(∂0ψ ψγ0∂1ψ + ψγ1(∂0ψ ψγ0∂2ψ + ψγ2(∂0ψ ψγ0∂3ψ + ψγ3(∂0ψ

1 0 α α 0 1 0 α α 0 1 0 α α 0 1 0 α α 0  + [γ00αγ γ + γ0α0γ γ ]ψ) + [γ00αγ γ + γ0α0γ γ ]ψ) + [γ00αγ γ + γ0α0γ γ ]ψ) + [γ00αγ γ + γ0α0γ γ ]ψ)  4 4 4 4  ¯ 1 0 α ¯ ¯ 1 0 α ¯ ¯ 1 0 α ¯ ¯ 1 0 α   (∂0ψ + [γ00αγ γ ∂1ψγ0ψ (∂0ψ + [γ00αγ γ ∂2ψγ0ψ (∂0ψ + [γ00αγ γ ∂3ψγ0ψ (∂0ψ + [γ00αγ γ   − 4 − − 4 − − 4 − − 4   α 0 ¯ α 0 ¯ α 0 ¯ α 0 ¯   +γ0α0γ γ ]ψ)2γ0ψ +γ0α0γ γ ]ψ)γ1ψ) +γ0α0γ γ ]ψ)γ2ψ) +γ0α0γ γ ]ψ)γ3ψ)       ¯   ψγ1(∂0ψ     1 0 α α 0   + [γ00αγ γ + γ0α0γ γ ]ψ) ψγ¯ ∂ ψ + ψγ¯ ∂ ψ ψγ¯ ∂ ψ + ψγ¯ ∂ ψ   4 2(ψγ¯ ∂ ψ ∂ ψγ¯ ψ) 1 2 2 1 1 3 3 1   1 1 1 1 1 ¯ ¯ ¯ ¯   ¯ ¯ 0 i − ∂2ψγ1ψ ∂1ψγ2ψ ∂3ψγ1ψ ∂1ψγ3ψ   +ψγ0∂1ψ (∂0ψ + [γ00αγ γ − − − −   − 4   i 0 ¯ ¯   +γ0α0γ γ ]ψ)γ1ψ ∂1ψγ0ψ  i~c  −  Tµν =  

21  ¯  4  ψγ2(∂0ψ   1   + [γ γ0γα + γ γαγ0]ψ)   00α 0α0 ψγ¯ 2∂1ψ + ψγ¯ 1∂2ψ ψγ¯ 2∂3ψ + ψγ¯ 3∂2ψ   4 2(ψγ¯ ∂ ψ ∂ ψγ¯ ψ)   1 ¯ ¯ 2 2 2 2 ¯ ¯   ¯ ¯ 0 i ∂1ψγ2ψ ∂2ψγ1ψ − ∂3ψγ2ψ ∂2ψγ3ψ   +ψγ0∂2ψ (∂0ψ + [γ00αγ γ − − − −   − 4   i 0 ¯ ¯   +γ0α0γ γ ]ψ)γ2ψ ∂2ψγ0ψ   −     ¯   ψγ3(∂0ψ     1 0 α α 0   + [γ00αγ γ + γ0α0γ γ ]ψ) ψγ¯ ∂ ψ + ψγ¯ ∂ ψ ψγ¯ ∂ ψ + ψγ¯ ∂ ψ   4 3 1 1 3 3 2 2 3 2(ψγ¯ ∂ ψ ∂ ψγ¯ ψ)   1 ¯ ¯ ¯ ¯ 3 3 3 3   ¯ ¯ 0 i ∂1ψγ3ψ ∂3ψγ1ψ ∂2ψγ3ψ ∂3ψγ2ψ −   +ψγ0∂3ψ (∂0ψ + [γ00αγ γ − − − −   − 4   i 0 ¯ ¯   +γ0α0γ γ ]ψ)γ3ψ ∂3ψγ0ψ   −     (127)  References

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