Derivation of Einstein – Cartan theory from general relativity page 1
March 17, 2016
R J Petti
146 Gray Street, Arlington, Massachusetts 02476 U.S.A.
E-mail: rjpetti@alum.mit.edu
Abstract . This article presents a derivation of Einstein–Cartan theory (EC) from general relativity. Part one derives translational holonomy around one Kerr mass and an integral version of torsion as holonomy/area with no additional assumptions or parameters. Part two computes the classical continuum limit of distributions of
Kerr masses that converge to a continuum with constant densities of mass, angular momentum (a.m.), and charge.
The limit generates torsion and the spin-torsion relationship of EC. This construction of curvature and torsion is equivalent to definition of curvature with Cartan forms on fiber bundles. The continuum limit assumes inequalities relating mass, a.m., loop radius, and charge. The inequality a 2 << r 2
restricts the derivation to scales larger than that of atoms. Advantages of EC include accommodating exchange of classical intrinsic and orbital a.m. (as in turbulence), and very likely providing a better classical limit for quantum gravity, where treatment of spin is more important than in classical physics. According to other research, EC generates inflationlike expansion in high density Friedmann cosmologies and eliminates singularities in Schwarzschild–Kerr models. arXiv:1301.1588v19 [gr-qc]
PACS codes: 04.20.Cv Fundamental problems and general formalism
04.50.Kd Modified theories of gravity
Keywords: general relativity, spin, Einstein–Cartan theory, affine torsion
Contents
1.
Introduction ............................................................................................................................................... 2
1.1.
Exchange of classical intrinsic and orbital angular momentum in general relativity ........................................ 3
1.2.
Mathematical approaches to validating Einstein–Cartan theory ....................................................................... 3
2.
Summary of the derivation....................................................................................................................... 4
3.
Mathematical tools .................................................................................................................................... 5
3.1.
Development of curves ...................................................................................................................................... 5
3.2.
Relation between development, curvature, and torsion for linear connections ................................................. 6
3.3.
Rotational holonomy in the Schwarzschild solution ......................................................................................... 7
3.4.
Correction of a numerical factor in the original derivation ............................................................................... 7
4.
Translational holonomy and surrogate torsion around a Kerr mass .................................................. 8
4.1.
Objective and summary ..................................................................................................................................... 8
4.2.
Translational holonomy of equatorial spacelike loops ...................................................................................... 8
4.3.
Translational holonomy of spacelike polar loops .............................................................................................. 9
4.4.
Surrogate torsion around a Kerr mass ..............................................................................................................10
5.
Continuum limit of a distributions of Kerr masses ............................................................................. 10
5.1.
Variables in the continuum model and inequality constraints on values of the variables ................................10
5.2.
Background curvature ......................................................................................................................................12
5.3.
Translational holonomy / area around a discrete rotating mass in the continuum limit ...................................12
5.4.
Comparison of results with Einstein–Cartan theory .........................................................................................13
6.
Three meanings of spin .......................................................................................................................... 14

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7.
Limitations of the derivation .................................................................................................................. 15
7.1.
The derivation includes only highly symmetric configurations of spinning matter .........................................15
7.2.
Limited rigor in the continuum limit process ...................................................................................................15
7.3.
Lack of empirical evidence for Einstein–Cartan theory ...................................................................................16
8.
Conclusion .............................................................................................................................................. 16
9.
Acknowledgements ................................................................................................................................ 16
Appendix A: Mathematical notation ............................................................................................................. 17
Appendix B: Connections, curvature and torsion on fiber bundles ......................................................... 18
B–1 Relationship between curvature and torsion .....................................................................................................18
B–2 Relation between development, curvature, and torsion for linear connections ................................................19
Appendix C: Proof of theorem 1 (translational holonomy of equatorial loops) ....................................... 20
C–1 Holonomic coordinate bases and orthonormal frames .....................................................................................20
C–2 Lie brackets of frame fields ..............................................................................................................................20
C–3 Connection coefficients ....................................................................................................................................21
C–4 Development of equatorial loop into Minkowski space ...................................................................................23
C–5 Translational holonomy of an equatorial loop developed into Minkowski space ............................................25
Appendix D: Proof of theorem 2 (translational holonomy of polar loops) ............................................... 26
Appendix E: Areas for Further Research ..................................................................................................... 26
E.1
Do black holes have central singularities? .......................................................................................................26
E.2
Is the “big bang” really a “big bounce”? ..........................................................................................................26
E.3
Is torsion the inflaton field of cosmic inflation? ...............................................................................................26
E.4
Is EC a better classical limit for quantum gravity?...........................................................................................26
E.5
Does torsion trace occur in virtual fields joining ends of fermion world lines? ...............................................27
E.6
Can including torsion improve the theory of turbulence? ................................................................................27
Appendix F: Other comments about the derivation .................................................................................... 27
Appendix G: Computer algebra validation of computations ..................................................................... 28
References....................................................................................................................................................... 48
1.
Introduction
In 1922 E. Cartan proposed extending GR by including affine torsion in the most natural way: do not force torsion to zero where it naturally appears in covariant derivatives and curvatures [Cartan 1922, Cartan 1923].
The resulting theory, known as Einstein–Cartan theory (EC), is the minimal extension of GR that resolves the problem of exchange of intrinsic and orbital angular momenta. The theory is minimal in that (a) when we drop the assumption that torsion vanishes, GR spontaneously becomes EC, and (b) torsion is zero except inside regions that contain spin density. Cartan tried to explain torsion to Einstein several times, after which Einstein confessed to Cartan in 1929 that he “didn’t at all understand the explanations you gave me [about the role of torsion in geometry]; still less was it clear to me how they might be made useful for physical theory”
[Debever 1979].
As the master theory of classical physics, general relativity (GR) has an outstanding flaw: it cannot describe exchange of classical intrinsic a.m. and orbital a.m. The simplest manifestation of this problem is that the
Einstein tensor in Riemannian geometry is symmetric, hence the momentum tensor must be symmetric.
However, classical continuum mechanics shows that the momentum tensor is non-symmetric during exchange of intrinsic and orbital a.m. This has not been seen as a serious problem because, arguably, there is no intrinsic a.m. in classical physics. However, the essence of classical turbulence is transformation of orbital a.m. to scales too small to be included in fluid models.
The late 20th century view of EC was that it requires inclusion of torsion without adequate justification, it is unsupported by empirical evidence (though it doesn’t conflict with any known evidence), it solves no significant problem, and it complicates gravitational theory with no compensating benefit. A review article

Derivation of Einstein
–
Cartan theory from general relativity 3 about EC in The Encyclopedia of Mathematical Physics [Trautman 2006] treats the theory as a viable but unproven speculation. In their summary of fifty years of research on gauge theories of gravity, Blagojevi ć and
Hehl regard EC as one of two viable gauge theories of gravitation that could replace (I prefer “extend”) GR
[Blagojevi ć and Hehl 2013].
1.1.
Exchange of classical intrinsic and orbital angular momentum in general relativity
We use the term “classical spin” to denote classical intrinsic a.m. Classical spin is of two kinds.
• The classical limit of quantum spin. This is important in classical models of the early universe because matter has a high density, and much of the matter consists of spin one-half fields. The field equations of
EC have quadratic torsion terms, so spin can be important even if mean spin density is zero.
• Orbital a.m. on scales smaller than the scale tracked by a classical model. For example, in classical statistical fluid mechanics, turbulence converts macroscopic orbital a.m. into a.m. at smaller scales than the classical models track [Monin and Yaglom 1971]. Energy that is transferred to a smaller scale than that of a classical model is modeled as internal energy. Torsion can do the same for subscale
A comprehensive theory of classical spacetime physics must be able, at least in principle, to describe exchange of classical spin and macroscopic orbital a.m.
GR accommodates models of matter that include classical spin and orbital a.m. GR’s law of conservation of a.m. can accommodate spin and orbital a.m. [Hehl 1976b]. These facts are sometimes interpreted as evidence that GR’s treatment of a.m. is complete. However, GR cannot accommodate exchange of classical spin and orbital a.m.: the Einstein tensor in Riemannian geometry is symmetric, so the momentum tensor in GR must be symmetric. Classical continuum mechanics shows that the momentum tensor is non-symmetric during exchange of intrinsic and orbital a.m.
1.2.
Mathematical approaches to validating Einstein–Cartan theory
Three mathematical arguments have been proposed to validate EC.
1.2.1.
Torsion and the Poincaré group model spin and spin-orbit exchange
The derivation of EC from a Lagrangian requires including gauge fields for translations. Sciama used a formulation of GR based on the Lorentz group and frame fields. By varying with respect to the frame fields, he derived torsion and the spin-torsion relationship of EC [Sciama 1962]. Kibble extended the structure group of GR to the Poincaré group and derived torsion and the spin-torsion relationship [Kibble 1962]. In both formulations, torsion can be nonzero, the momentum tensor can be non-symmetric, and EC can model exchange of classical spin and orbital a.m. In retrospect, neither retaining the Lorentz structure group and changing the representation to orthonormal frames, nor expanding the structure group to the Poincaré group, was a very convincing way to justify an extension of GR to EC except to a small minority of theorists.
The current view of the symmetry group of GR is that the theory is based on Cartan connections
[Kobayashi 1972], in which the symmetry group is a semidirect product of the Lorentz group and spacetime translations. This group fixes the origin in each affine fiber space, which the full Poincaré group does not
[Petti 2006].
1.2.2.
Agreement of linearized versions of Einstein–Cartan theory and general relativity
Adamowicz showed that GR plus a linearized classical model of matter with spin yields the same linearized equations for the time-time and space-space components of the metric as linearized EC [Adamowicz 1975].
He does not show that EC follows from GR plus classical spin. Indeed, he writes, “It is possible ‘a priori’ to solve this problem [of dust with intrinsic a.m.] exactly in the formalism of GR but in the general situation we have no practical approach because of mathematical difficulties.” Adamowicz’s conclusion about a.m. is at best incomplete. For the reasons discussed above, GR cannot accommodate exchange of classical spin an orbital a.m.

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1.2.3.
Einstein–Cartan theory can be derived from general relativity
This work discusses the third argument for EC: general relativistic models of rotating matter generate torsion and the spin–torsion relation of EC, in two parts. In Part 1, the geometry of an isolated Kerr mass generates an integral surrogate for affine torsion (translational holonomy per unit area of a loop). The derivation of translational holonomy is rigorous. In Part 2, the continuum limit of a fluid of Kerr masses generates affine torsion and the spin-torsion equation of EC.
One might argue that, even if torsion and the spin-torsion relationship of EC can be derived from GR, this does not imply that EC can be derived from GR. However EC is, by a wide margin, the least invasive and most attractive way known to extend GR with torsion.
• Where classical spin is zero, EC is identical with GR.
• If we remove GR’s ad-hoc assumption that torsion is zero, then GR spontaneously becomes EC.
• Riemann–Cartan geometry is a most natural extension of Riemannian geometry: it adds torsion
(translational curvature) while retaining the key properties of Riemannian geometry. Riemann–Cartan geometry extends the interpretation of Riemannian geometry as the continuum model of rotational line defects (disclinations) by adding translational line defects (dislocations).
• Blagojevi ć and Hehl consider EC to be one of two viable alternatives to GR among ten gravitational gauge theories [Blagojevi ć and Hehl 2013]. The other alternative that the authors consider to be viable, teleparallel gravitation theory, radically departs from GR by setting Riemannian curvature to zero and describing gravitation with torsion.
Because of the intimate relation between GR and EC, I recommend viewing EC as an extension of GR to include intrinsic a.m., not a replacement for GR.
The remainder of this paper is divided into these main sections: Section 2 Summary of the derivation;
Section 3 Mathematical tools; Section 4 Translational holonomy and surrogate torsion around a Kerr mass;
Section 5 Continuum limit of a distributions of Kerr masses; Section 6 Three meanings of spin; Section 7
Limitations of the derivation; Section 8 Conclusion. Appendices: Appendix A: Mathematical notation;
Appendix B: Connections, curvature and torsion on fiber bundles. Appendix C: Proof of theorem 1
(translational holonomy of equatorial loops); Appendix D: Proof of theorem 2 (translational holonomy of polar loops); Appendix E: Open Questions for Research; Appendix F: Other comments about the derivation;
Appendix G: Computer algebra validation of computations.
An early version of part one of this derivation was published in [Petti 1986]. This article contains significantly modified and computations of translational holonomy, a more complete description of the continuum limit, more and simpler explanations, discussion of limitations of the derivation, a computer algebra validation of the computations, and discussion of some areas of gravitational research where Einstein–Cartan theory is relevant.
2.
Summary of the derivation
The derivation of the field equations of EC from GR is similar to derivation of a differential field equation from a discrete model or an integral law in fluid mechanics or electrodynamics. The computation is the same as a derivation of equations for a special relativistic spin fluid from a distribution of discrete rotating masses, except that it includes the general relativistic gravitational field of the discrete masses. Alternatively, it is the same as a derivation of equations for a fluid model from a distribution of exterior Schwarzschild masses, except that the discrete masses have correlated rotations.
The computation consists of two parts.
(a) Part one: For one Kerr mass, compute the translational holonomy around a spacelike equatorial loop and a spacelike polar loop. (Section 4)
(b) Part two: compute torsion in the continuum limit of a distribution of Kerr masses.

Derivation of Einstein
–
Cartan theory from general relativity 5
• Form a sequence of arrays of discrete Kerr masses with correlated rotations, such that the sequence converges to a continuum of matter in which densities of mass and a.m. are constants.
• Enforce inequality constraints among mass, a
(intrinsic a.m./mass), loop radius, and electric charge that restrict validity to the exterior Kerr region and to classical physics. Take the continuum limit of the sequence of arrays. (Section 5)
• In the continuum limit, translational holonomy becomes affine torsion, the distribution of rotating masses becomes a spin fluid with a nonzero spin density, and the relation between spin and torsion is exactly as in EC.
The derivation is not a general derivation of EC because it uses a sequence of arrays of discrete masses for which the densities of mass and a.m. converge to constant. However, this configuration is the most basic case in all continuum field theories. In addition, the inequality constraints limit the generality of the derivation.
Affine torsion is most familiar as the antisymmetric part of the connection coefficients, or as the antisymmetrized covariant derivative [Kobayashi and Nomizu 1963; Bishop and Crittenden 1964;
Milnor 1963]. A deeper view of affine torsion is that it is translational curvature; it is analogous in many respects to Riemannian (rotational) curvature.
• Rotational holonomy measures the net rotation of a linear frame when the frame is parallel translated around a closed loop. Rotational curvature is the rotational holonomy per unit area surrounded by the loop, in the limit of very small loops.
• Translational holonomy measures the “failure-to-close vector” when a closed loop in the manifold with torsion is “developed” into a flat space. Translational curvature (torsion) is the translational holonomy per unit area surrounded by the loop, in the limit of very small loops.
For more discussion and drawings of curvature and torsion as affine defects, see [Petti 2001].
The deepest definition of curvature of all kinds is stated in terms of Cartan differential forms: it is the horizontal component of the exterior derivative of the connection 1-form on a principal fiber bundle; see Appendix B for details. The contravariant version of this definition is the limit of holonomy per unit area enclosed by a loop.
The contravariant definition of curvature is analogous to defining the derivative of a function as the limit of the slopes of successively smaller secants; or to defining a Lie bracket of vector fields as a limit of the failureto-close-vector per unit area of the commutator of two vector fields, as the area enclosed by the integral curves gets smaller. The limit of holonomy per unit area is curvature, not just an intuitive heuristic for curvature.
3.
Mathematical tools
For a summary of notational conventions, see Appendix A.
3.1.
Development of curves
The motivation for defining the development of curves is quite simple. We want to isolate the influence of local manifold geometry from the influence of acceleration upon the shape of a curve.
Define:
• Ξ
is a smooth (pseudo-) Riemannian manifold of dimension n
with local coordinates
ξµ
, metric g
µ ν
, and metric connection coefficients Γ
λ ν
µ (s) whose covariant differentiation is denoted by D .
•
C: [0,1] → Ξ
is a smooth curve in
Ξ
, with
C(0) = ξ0
and tangent vector field u
.
•
A(s): T
C(0)
Ξ → T
C(s)
Ξ is an isometry of the tangent space at
C(0)
with the tangent space at
C(s)
, by parallel translation along the curve C.
Lemma 1: The mapping
A(s)
satisfies the differential system

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(1) dA
µ
ν
(s) + u
λ
(s) Γ
λσ
µ
(s) A
σν
(s) = 0 ds with initial condition
A
µ
ν
(0) = kronecker_delta
µ
ν
.
The index µ refers to a vector basis at
C(s)
, while the index ν refers to a vector basis at
C(0)
. The proof of
Lemma 1 consists of parallel translating the vector along C from
C(0)
to
C(s)
.
Define similar structures for another manifold Ξ ’
:
• Ξ ’
is a flat smooth (pseudo-) Riemannian manifold of the same dimension as Ξ with local coordinates ξ '
µ , metric g'
µ ν
, and metric connection coefficients Γ '
λ ν
µ
(s)
whose covariant differentiation is denoted by
D’
.
•
C’: [0,1] → Ξ '
is a smooth curve in Ξ '
, with
C’(0) = ξ ’0
and tangent vector field u’
.
• A’(s): T
C’(0)
Ξ ’ → T
C’(s)
Ξ ’
is an isometry of the tangent space at
C’(0)
with the tangent space at C’(s) , by parallel translation along the curve
C’
.
Applying Lemma 1 to the curve on manifold Ξ ’
, the mapping
A’(s)
satisfies the differential system
(1’) dA’
µ
ν
(s)+ u’
λ
(s) Γ '
λσ
µ
(s) A’
σν
(s) = 0 ds with initial condition
A’
µ
ν
(0) = kronecker_delta
µ
ν
.
Choose an isometry of the tangent spaces at the initial points of the two curves,
L: T
ξΞ
→ T
ξ ’
Ξ '
.
Definition:
The development of the curve
C
in Ξ into the manifold Ξ ’
is the curve
C’: [0,1] → Ξ ’ defined by the differential system:
(2) D’u’ u’ = A’(s) L [A(s)] -1 D u
u with
C’(0) = ξ ’0
and u’(0) = L u(0)
.
Development of curve
C
maps the tangent vector u(s)
back to the point ξ
0
, then to a tangent vector at
ξ '0 ∈ Ξ '
, then parallel translates the vector from
C’(0)
to
C’(s)
. Equivalently, development of a curve maps the tangent vector u(s)
at
C(s)
directly to the vector u’(s)
at
C(s) in Ξ '
by parallel translating the mapping
L
along
C
from
C(0) to
C(s)
(which yields the mapping
A’(s) L [A(s)] -1
).
3.2.
Relation between development, curvature, and torsion for linear connections
Choose a point ξ ∈ Ξ and two vectors
V
and
W
at ξ . Construct a one-parameter family of closed loops
Ct(s) through
ξ
which are tangent at
ξ
to the plane of
V ^ W
, and which converge to
ξ
as t → 0
. Develop the curves
Ct
into a flat (pseudo-) Euclidean n manifold Ξ '
.
3.2.1.
Definition of curvature using rotational holonomy
If the developed curve does not close, close it with a short straight line. Parallel translate a vector basis around the curves
Ct
. Use the same pattern of rotations as in
Ct
to translate the vector basis along
C’t
.
Rotational holonomy is defined as the linear transformation gt
that maps the starting vector basis onto the ending vector basis that results from parallel translation around the loop
C’t
.
Conventional (rotational) curvature is defined as the rotational holonomy per unit area enclosed by the loop, in the limit as the area of the loop approaches zero.

Derivation of Einstein
–
Cartan theory from general relativity 7
(3) – R(V ^ W)= lim gt _ t → 0 area(t)
3.2.2.
Definition of affine torsion using translational holonomy
The developed curves
C’t
described above generally do not close. Translational holonomy is defined as the failure-to-close vector needed to close the developed curve,
C’ t
(1) – C’ t
(0) .
Affine torsion is defined as the translational holonomy per unit area enclosed by the loop, in the limit as the area of the loop approaches zero.
(4) – T(V ^ W) = lim (C’ t
(1) – C’ t
(0)) t → 0 area(t)
We now have the mathematical tools we need to create torsion out of GR, with no additional assumptions. We can generate rotational and translational holonomy around loops in a spacetime Ξ , and interpret them as integral surrogates for (rotational) curvature and torsion. By taking limits as the loops shrink to a point, we have a welldefined procedure for translating the integral concepts of holonomy into the differential concepts of curvature and torsion.
3.3.
Rotational holonomy in the Schwarzschild solution
As an introduction to the technique, we calculate the rotational holonomy of a spacelike circular orbit in the exterior Schwarzschild solution. Use coordinates so that the metric takes the form
(5) ds 2 = –(1–2 m/r) dt 2 + dr 2 + r 2 (d θ 2 + sin 2 ( θ ) d ϕ 2 )
1– 2 m/r
For a spacelike circular orbit of coordinate radius r
around the equator (
θ = π /2
), the rotational holonomy is spacelike and lies in the equatorial plane [the ( r , ϕ ) plane]. It is given by
(6) 2 π [1 – (1 – 2 m/r) ½ ]
In the limit where m << r
, the holonomy per unit area enclosed by the loop (using coordinate area, where the coordinate r directly reflects the area of the sphere r= constant) becomes approximately
(7) 2 m = 8 π
r 3 3 m
(4 π /3) r
3
The measure of area of the closed loop is that seen by an observer at spatial infinity. This expression is an integral surrogate for the sectional curvature
R ( r, ϕ
)
. Three of these terms provide a surrogate for the scalar curvature
(8) R = R ij ji (note that R
0ij0
= 0 )
With m/(4 π /3 r 3 )
approximating the mass density, we have recovered the source term for the field equations of GR: the field equations assign the value
8 π rho
(where rho
is the energy density in a fluid model of matter) to the scalar curvature.
3.4.
Correction of a numerical factor in the original derivation
The original derivation contains a discrepancy of a factor of two in the spin-torsion equation. This is due to introducing in an inconsistent way a factor of two into the definition of torsion in terms of connection coefficients. This relationship should be
(9) T
μν
λ
( ξ ) = Γ
μν
λ
( ξ ) – Γ
νμ
λ
( ξ ) whereas the original derivation inserted a factor of ½ on the right side of this equation.

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4.
Translational holonomy and surrogate torsion around a Kerr mass
4.1.
Objective and summary
The objective of Section 4 is to derive the translational holonomy of various closed loops around a Kerr mass.
The final result is that the only significant translational holonomy arises from an equatorial spacelike loop, and the translation is timelike and proportional to the a.m. of the mass.
The computations of translational holonomy use only the “exterior” Kerr solution: if the massive object is a black hole, the computations need not use the region near the event horizon; if the massive object is an extended body without a singularity, the computations do not use any properties of a hypothetical but currently undiscovered interior Kerr solution.
Section 4.2 computes the translational holonomy for equatorial loops. Section 4.3 shows that translational holonomy components for other loops are of sufficiently low order that they do not affect the continuum limit calculations that derive torsion from translational holonomy.
The subsequent calculations use the Kerr solution in Boyer-Lindquist coordinates [Misner, Thorne, and
Wheeler 1973].
(10) ds 2 = – ∆ [dt – a sin 2 ( θ ) d ϕ ] 2
ρ 2
+ sin 2 ( θ ) [(r 2 + a 2 ) d ϕ – a dt] 2 + ρ 2 dr 2 + ρ 2 d θ 2
ρ 2 ∆ where
0 ≤ ϕ ≤ 2 π and
0 ≤ θ ≤ π ,
(11) a := S/m
(12) ∆ := r 2 + a 2 – 2 m r + q 2
(13) ρ 2 := r 2 + a 2 cos 2 ( θ )
This form of the metric uses four orthogonal 1-forms, which define an orthonormal basis for the tangent vectors at each point of Ξ .
4.2.
Translational holonomy of equatorial spacelike loops
We shall calculate the translational holonomy of an equatorial loop of constant coordinate radius r
. Define an equatorial loop
(14) C(s) = (0
, r
,
π /2
,
2 π s), for 0 ≤ s ≤ 1
Develop this curve into a flat Minkowski space Ξ ’ with coordinates
(t’
, x’
, y’
, z’)
. Start the development at the point p’
with coordinates
(t’0
, x’0
, y’0
, z’0)
.
(15)
The map
L: T p
Ξ → T p’
Ξ ’
introduced in Section 3.1 is defined by the matrix of partial derivatives t r θ ϕ
┌───────────────── t’ │ c2 0 0 c1
│
│ x’ │
∂ (t’, x’, y’, z’) = │
∂ (t, r, θ , ϕ ) y’ │
│
│ z’ │
ν
0
ν
0
– c1
1
0
0
0
0
1
ν
0
– c2
ν
0

Derivation of Einstein
–
Cartan theory from general relativity 9 where
(16)
(17) c1:= a (– r – m + q 2 ) r 2 r c2:= – ∆ ½
r
(18) ν :=(c2 2 – c1 2 ) ½
Since
|c2| > |c1|
, ν is real. If a
, q << r
, then ν ≈
1 – m/r
≈
1
.
(19) ν 2 = 1 – 2 m + q 2 – 2 a 2 m + 2 a 2 q 2 – a 2 m 2 + 2 a 2 m q 2 – a 2 q 4 ) r r 2 r 3 r 4 r 5 r 6
Theorem 1 presents the translational holonomy of an equatorial loop in terms of these expressions:
(20) k1 = 2 π a (3 m r 3 — 2 q 2 r 2 + a 2 m r — a 2 q 2 )
ν r
4
( 21 ) k2 = ∆ ½ (r 4 – a 2 m r + a 2 q 2 )
ν 2 r
4
Theorem 1.
The development of the loop C into the Minkowski manifold Ξ ’
yields the curve
C(s) = (t’(s), x’(s), y’(s), z’(s)) which is given in Minkowski coordinates by t’(s) = k1 s + t’
0 x’(s) = k2 cos(2 πν s)+ x’
0 y’(s) = k2 sin(2 πν s) + y’
0 z’(s) = z’
0
C(s)
sweeps out a circular arc which closes in the Minkowski space, but does not close in the space of the
Kerr solution unless ν = 1. If m
, a
, q << r
, then ν ≈ 1 and the slope dt’/ds = k1
is proportional to a
.
Proof . The proof consists of finding an exact solution of the differential equation
(1’)
for development of the equatorial loop into Minkowski space. The computation uses connection coefficients that have spacetime indices for the first index (the direction of differentiation), and orthonormal frame indices for the second and third indices. See Appendix C for the connection coefficients and details of the computations. End of proof.
The translational holonomy (or “failure-to-close vector”) of the curve
C
is
C’(1) – C’(0)
.
(22)
┌
│
│ translational holonomy
= │
│
│
└ k2
- k1 ┐
│
( cos(2 π ν ) –
1) │ k2 sin(2 π ν )
0
│
│
│
┘
This result can be found on line (d72) in Appendix G.
4.3.
Translational holonomy of spacelike polar loops
Consider a loop which passes through both the north and south polar axes of a rotating object. In particular, consider the curve

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C(s) =

(0, r, 2 π s, 0),


(0, r, 2 π (1–s), π ), for 0 ≤ s for 0.5 ≤
≤ 0.5
s ≤ 1
Develop this curve into the Minkowski space Ξ ’
.
Theorem 2.
The translational holonomy of the polar loop C is zero.
Proof: See Appendix D.
Theorems 1 and 2 give us the translational holonomy for all closed spacelike curves of constant radius that surround a Kerr mass. The holonomy around any spacelike planar loop of constant radius is, to linear approximation, cos( θ )
times the holonomy of the equatorial loop, where theta is the angle between the axis of the rotating object and the spacelike normal to the plane of the loop.
4.4.
Surrogate torsion around a Kerr mass
We can use the translational holonomy computed for a single Kerr mass to compute an integral surrogate for its torsion. Torsion is the limit of translational holonomy around a loop divided by the area enclosed by the loop. We use as the area of an equatorial loop the area as seen by an observer at infinity: π r 2
. Therefore in the orthonormal field basis, translational holonomy/area for equatorial loop is on line (d77), Appendix G.
(23)
┌ - k1/( π r 2 ) ┐ translational holonomy/area
= │ k2 (cos(2 π ν ) – 1)/( π r 2 ) │
│
└ k2 sin(2 π ν )/( π r 2 )
0
│
┘
It is convenient to replace the frequency ν with the variable φ
= 1 –
ν ≈ 0
and to expand in terms of m
, r
, a
, and q
to get line (d78), Appendix G.
(24) trans’l holonomy
= │ area
┌
│
- 2 a (3 m r 3 – 2 q 2 r 2 + a 2
(1φ ) r 6
m r – a 2 q 2 )
│ │
│
(cos(2 π φ )-1)(r 4 – a 2 m r + a 2 q 2 )(r 2 – 2 m r + q 2 + a 2 ) 1/2 │
π (1φ ) 2 r 6
┐
│
│
│
│ sin(2 π φ
)(r 4 – a 2 m r + a 2 q 2 )(r 2 – 2 m r + q 2 + a 2 ) 1/2
│
│
│
└
π (1φ ) 2 r 6
0
│
│
│
┘
Starting with one Kerr solution in GR, we have derived translational holonomy and an integral surrogate for torsion, (translational holonomy)/(area of loop). All results obtained up to this point are derivable from the
Kerr solution without any additional assumptions.
5.
Continuum limit of a distributions of Kerr masses
The objective of section 5 is to demonstrate that a restricted class of classical, discrete GR models has a continuum limit with constant densities of mass and spin, torsion, and the spin-torsion relationship of EC.
5.1.
Variables in the continuum model and inequality constraints on values of the variables
Table 1 shows magnitudes of key quantities of some rotating systems in Planck units ( c = K = ħ
=
1/4 π ε
0
=
1
), in which the units of time, length and mass are
5.39106 ∙ 10-44 s, 1.616200 ∙ 10-35 m, 2.17651 ∙ 10-
8 kg, 1.875545956 ∙ 10-18 Coulomb
.

Derivation of Einstein
–
Cartan theory from general relativity
Table 1 mass ( m ) , radius, spin/mass ( a) , angular momentum ( J ) and electric charge ( q ) for some rotating systems
11
System
Electron
Proton
Sun
Earth/Sun
Galaxy m
4 ∙ 10 -23
1 ∙ 10 -20
1 ∙ 10 38
1 ∙ 10 38
6 ∙ 10 50 radius
2 ∙ 10 20
5 ∙ 10 19
1 ∙ 10 43
9 ∙ 10 45
3 ∙ 10 55
J
0.5
0.5
2 ∙ 10 75
2 ∙ 10 75
1 ∙ 10 101 a
1 ∙ 10 22
5 ∙ 10 19
2 ∙ 10 37
2 ∙ 10 37
2 ∙ 10 51 q
9 ∙ 10 -2
9 ∙ 10 -2
0
0
0
Specify a sequence of distribution of Kerr masses that converges to a continuum with constant mass density, spin/mass, and charge density.
• r
= average half-distance between centers of Kerr masses as seen by an observer at infinity= radius of equatorial loop used to compute holonomy.
The cross sectional area assigned to loop of radius r
around each Kerr mass (used to compute holonomy) is π r 2
, and the volume is
4 π /3 r 3
. These values underestimate the area and volume per vertex in a lattice with half–distance r
by factors of
( π r 2 )/(2 r) 2 ≈ 3/4
for area and
(4 π /3 r 3 )/(2 r) 3 ≈ 2
for volume. The area should be
(2r) 2
and the volume should be
(2r) 3
in a flat space. When we derive torsion as
(translational holonomy)/area, the area factor is not significant because both the numerator and denominator are multiplied by approximately the same factor, and the factors are close to 1.
Perhaps a better value is r = (6/ π ) 1/3
. This value makes the volume of the sphere of radius r equal to the volume of the cube in flat space that should be allocated to each Kerr mass. For the same reason, any adjustment to the value of r
to account for the curvature caused by the interaction of Kerr masses also makes no significant difference in this ratio, because we specified that r >> m
.
• md
= mass density in the continuum model. This density is approximately md = m/(4 π /3 r 3
).
Computations use mc
=
4 π /3 md
to prevent proliferation of factors of
4 π /3
. At the end, we switch to md
.
• qd
= electric charge density in the continuum model. This density is approximately qd = q/(4 π /3 r 3 )
.
Computations use qc
=
4 π /3 qd
to prevent proliferation of factors of
4 π /3
. At the end, we switch to qd
.
We restrict the values of the parameters for the reasons shown in Table 2.
Table 2 restrictions on parameters of the model
Restriction Rationale
Restrict the loop used to compute holonomy to the far exterior region of each Kerr mass. m << r mc r 2 << 1
This follows from m << r
and the definition mc = m/r 3
. qc
is net charge density, which is usually small due to the strength of electrostatic force. qc r 2 << mc qc enters the expression holonomy in the form qc 2 r 2 + mc . In Planck units,
1 C/m 3 ~ 2 10 -87
, r ~ 10 22
, 1 kg/m 3 ~ 2 10 -97
, so normally qc 2 r 2 << mc
.
a 2 << r 2
Virtually all of classical physics is described by the case a 2 << r 2
. As an extreme case, a distribution of 100% spin-polarized neutrons would have a ≈ 5 10 19
. Such a model

12 Petti
Restriction Rationale satisfies
|a| << r
if r > 10 22
, that is, r > 10 –12 m
. This is a very good approximation to a continuum limit in classical physics.
The first three restrictions are compatible with taking a continuum limit r → 0
. Assuming a ≠ 0
, the fourth restriction implies that the limit process cannot continue to r → 0
. The case a 2 > ≈ r 2
has a limit in which torsion
~ 1/r 2
as r
approaches the radii of atoms or elementary particles. This should not be surprising: the classical theory of electromagnetism fails to produce a stable model of an atom at these scales because the a.m. of electrons causes them to radiate energy. Also, classical fluids break initiate turbulent flow when a.m. is large compared to viscosity, as reflected in the Reynolds number.
These conditions restrict this derivation to classical physics at scales larger than those of atoms.
5.2.
Background curvature
Background curvature
Rb
due to the distribution of Kerr masses makes the area of the disk smaller by roughly a factor of
(1 – (r/Rb) 2 /6)
, which is negligible when r
is small compared to
Rb
.
We could in principle include the background curvature as follows. Start with a GR solution with continuum mass density md
and continuum spin density a md
. Gradually reduce the density of the continuum mass, while adding a distribution of small Kerr masses that keep the mass and spin densities of the model constant, except for density fluctuations on a small scale. We mention this to show that background curvature is not in principle an obstacle to the derivation. We will not pursue this idea further here.
5.3.
Translational holonomy / area around a discrete rotating mass in the continuum limit
We can restate translational holonomy in terms of the continuum densities mc
and qc
. In the continuum limit, terms that contain a factor of r
to a positive power disappear in each expression.
(25) ∆ = r 2 + a 2 – 2 mc r 4 + qc 2 r 6
(26)
(27)
(28)
(29) c1 = a (–1 – mc r 2 + qc 2 r 4 ) r c2 = – ∆ ½ = – (1 – 2 mc r 2 + a 2 + q 2 r 4 ) ½ r r 2 k1 = 2 π a (3 mc r 2 — 2 qc 2 r 4 + a 2 mc – a 2 qc 2 r 2 )
ν k2 = ∆ ½ (1
– a 2 mc + a 2 qc 2 r 2 )
ν 2
(30) ν 2 = (1 – 2 a 2 mc) +
(
2 a 2 qc 2 + 2 mc – a 2 mc 2 ) r 2 + qc 2 (1 + 2 a 2 mc 2
) r 4 – a 2 qc 4 r 6 )
The area of an equatorial loop as seen from infinity is π r 2
. So holonomy per unit area is
(31)
┌
│
- k1/ π r 2
┐
│
│ k2 (cos(2 π ν ) – 1)/ π r 2
│ translational holonomy
= │ area
│
│ k2 sin(2 π ν )/ π r 2
│
│
└ 0
│
┘
In equation (23), expand k1
and k2
, and replace m
and q
with the densities mc r 3
and qc r 3
to get (line (d82) in Appendix G):

Derivation of Einstein
–
Cartan theory from general relativity 13
(32) trans’l holonomy
= │ area
┌
│
2 π a (2 qc 2 r 4 + a 2 qc 2 r
1φ
2 – 3 mc r 2 – a 2 mc)
│ │
│
(cos(2 π φ )-1)(a 2 qc 2 r 2 – a 2 mc + 1)(qc 2 r 6 – 2 mc r 4 + r 2 + a 2 ) 1/2
│
(1φ ) 2
┐
│
│
│
│ sin(2 π φ )(a 2 qc 2 r 2 – a 2 mc + 1)(qc 2 r 6 – 2 mc r 4 + r 2 + a 2 ) 1/2
│
│
│
└
(1φ ) 2
0
│
│
│
┘
At this point, apply the restrictions on the variables listed in Table
2
.
• Set charge density qc = 0
to simplify the expressions. We shall not further pursue the case qc ≠ 0
.
• Assume r 2 << 1/mc
. This implies that
1 – φ ~ (1-2 mc r2) 1/2 ~ 1 – mc r 2
.
• Set cos(2 π mc r 2 ) = 1
and sin(2 π mc r 2 ) → 0
.
• Assume a 2 << r 2
by setting
3 r 2 + a 2 → 3 r 2
.
• Replace mc
with
4 π /3 md
, where md
is the correctly normalized mass density.
This yield the expression for translational holonomy/area (line (d88) in Appendix G):
(33)
┌ integral surrogate torsion for equatorial loop ≈ │
│
└
- 8 π a md
0
0
0
┐
│
│
┘
This result is expressed using the orthonormal frame field. When expressed in coordinate frames, the expression for surrogate torsion has the same expression (line (d89), Appendix G).
This result can be written in a form that exhibits its structure as
– 8 π spin/volume
.
(34)
┌ integral surrogate torsion for equatorial loop ≈ │
│
└
- 8 π a md/(4 π /3 r 3 ) ┐
0 │
0
0
│
┘
5.4.
Comparison of results with Einstein–Cartan theory
We want the holonomy per unit area for a loop of a radius
R
on the scale of the continuum model; so
R >> r
.
• The holonomy of the large loop is larger by a factor of
(R/r) 2
, compared to that of the small loop around a single small Kerr mass. We can segment the large disk into small regions (for example squares, which introduce some inaccuracy) whose boundaries cancel inside the big disk.
• The area surrounded by the large loop is larger by a factor of
(R/r) 2
, compared to the value for the small loop of radius r . This result is exact for a flat space and approximately true for a continuum model with a sufficiently large radius of background curvature, based on the estimate of the impact of background curvature above.
Therefore the translational holonomy per unit area of the loop of radius
R
is equal to that for a single small
Kerr mass surrounded by a small loop.
The field equations of EC are
(35)
(36)
G i
µ = 8 π K P i
µ
ST ij
µ = 8 π K Spin ij
µ

14 Petti where
G
and
P
are the Einstein tensor and momentum tensor respectively,
Spin
is the intrinsic a.m. (“spin”) tensor, and
ST
is the modified torsion tensor.
Using Theorem 1, Theorem 2, and the calculations in Section 5.2, the torsion and the spin density are related as in the spin–torsion field equation of EC.
The absence of the torsion trace terms in the heuristic result is due to the simplicity of the rotating mass in this calculation. The “in-plane” torsion in a discrete model corresponds to the torsion trace in the continuum limit.
The in-plane holonomy is present in the discrete calculation, and it vanishes in the passage to the continuum limit. The torsion trace also vanishes for Dirac fields. The torsion trace apparently describes more complex rotational moments than the first-order moments of the Kerr solution or of Dirac fields. The remaining parts of torsion (totally antisymmetric, torsion trace, and other components) are also not included in this derivation.
6.
Three meanings of spin
We should distinguish several types of tensor indices that are commonly considered to have spin. The distinction is based on differing geometric meanings of tensor indices. This distinction is illustrated by electromagnetic fields, which are commonly said to possess spin, but which do not generate affine torsion.
All tensor indices in continuum mechanics represent either conserved currents or spacetime directions or spacetime flux boxes through which current flows are measured using exterior derivatives (gradients, divergences, curls). Connection forms and curvatures also fit this pattern, though the spacetime boxes are used to measure holonomy, not flux.
When we formulate mechanics on fiber bundles, the distinction between flux boxes and currents translates into a distinction between base space tensor indices and fiber tensor indices. This is a natural distinction; for example, no one confuses base space and fiber indices in Yang-Mills gauge theories because the dimensionalities and symmetry groups of representation spaces differ for the two types of indices.
• “Base space indices” or “spacetime indices” (denoted here by lower case Greek letters λ , µ , ν , … ), represent spacetime configurations, mostly flux boxes through which current flow is measured. Derivatives with respect to these indices occur only in exterior derivatives or divergences (which are Hodge duals of exterior derivatives), or in Lie derivatives. These indices should not be covariant differentiated with any connection. However, it is often convenient in computations to differentiate such indices with respect to the Levi-Civita connection because this yields the same result for exterior derivatives and divergences as writing out the Hodge duals and using ordinary coordinate derivatives.
• “Fiber indices” (denoted by lower case Roman letters a , b , c … , for spinors by
A, B, C … , and for Yang-
Mills fields by
P, Q, R
… ) represent conserved currents. No derivatives are taken with respect to these indices. These indices are covariant-differentiated with the full connection including affine torsion and unitary connections. In classical spacetime physics, these indices represent momentum, or in antisymmetric pairs, a.m.
Here are some examples of the distinction between base space and fiber indices.
• Momentum
P a
µ has fiber index a
for momentum and spacetime index µ for a hypersurface through which flux is measured.
• Spin density
J ab
µ has an antisymmetric pair of fiber indices a
, b
for intrinsic a.m., and spacetime index µ for a hypersurface through which flux is measured.
• The electromagnetic potential i A
µ
and Yang-Mills potentials iA
P
Q µ
are connection coefficients with spacetime index µ for a spacetime direction, and an antisymmetric pair of indices
P
,
Q
for a rotation in unitary representation space. iA
µ
has an unwritten antisymmetric pair of indices for the unique plane in the representation space of group
U(1)
.

Derivation of Einstein
–
Cartan theory from general relativity 15
• The electromagnetic field i F
µν
(= exterior derivative of iA
µ
) and Yang–Mills fields iF
P
Q µν
are curvatures that represents rotational holonomy per unit area. They have an antisymmetric pair of spacetime indices µ ,
ν for the plane in which unitary curvature is measured. The antisymmetric pair of indices P , Q represents rotation in the unitary fiber space.
• Dirac spinor ψ A µ has one fiber index A for a direction in spin representation space; and spacetime index µ for a spacetime hypersurface.
Physicists usually count any tensor index whose symmetry group is a rotation group as contributing to spin.
We distinguish three basic types of spin.
(i) Spacetime spin is carried by antisymmetric pairs of fiber indices (denoted a
, b
, c
,
A
… in our notation);
These indices never represent base space (spacetime) directions or hypersurfaces.
(ii) In unitary gauge theories, gauge spin is carried by antisymmetric pairs of fiber indices of the gauge group
(denoted
P
,
Q
… in our notation). These indices do not generate spacetime spin.
(iii) Representation spin consists of any index that arises from representation of a rotation group that does not satisfy the requirements of the first two types of spin. For example, the electromagnetic potential i A
µ
has spin only in the sense that index
µ
has the symmetries of the Lorentz rotation group; it has no spacetime spin and it is not associated with torsion.
7.
Limitations of the derivation
7.1.
The derivation uses highly symmetric configurations of spinning matter
In its present form, the derivation applies only to classical arrays of discrete masses for which the densities of mass, momentum, a.m. (and electric charge) converge to constants. The restriction to a region with constant densities is not fundamental. A configuration with constant densities is the most basic case in all continuum field theories. Riemann and Lebesgue integrals can be represented by convergent sequences of areas under piecewise functions that have constant values on small regions; solutions of differential equations can be represented by Sobolev-convergent sequences of piecewise functions in finite element analysis [Strang 1973].
These facts indicate that a wide range of distributions of mass, a.m., and spin can be represented by convergent sequences of small regions built of Kerr masses (possibly including initial data such as intrinsic curvature of a
3-surface and a shift vector). Proving this result is beyond the scope of this work.
The derivation yields only torsion whose Burgers vector (metallurgists’ term for the translational holonomy, or failure-to-close vector) is orthogonal to the plane of the loop traversed. (Also, it is well known that classical limits of Dirac spinor fields yield antisymmetric torsion that fits well into EC.) The present derivation does not yield the torsion trace or the remainder of the torsion tensor. The author does not know of any classical model whose continuum limit includes other components of torsion.
7.2.
Limited rigor in the continuum limit process
Part 1 of the derivation (listed in section 2) – computation of translational holonomy around closed paths for one Kerr mass – is rigorous. (See section 4 and Appendix C).
Part 2 – computing continuum limits of a distribution of Kerr masses – is not completely rigorous in the way it takes limits (section 5). However, this step uses the same methods as would be used in a rigorous computation of mass density of a dust cosmological model derived from a distribution of exterior Schwarzschild solutions.
The restriction a 2 << r 2
prevents us from taking the limit r → 0, but this limit strays into smaller distance scales than those in classical physics.
The third step consists of interpreting of the equations derived in the second step.
One difference between continuum limits in Schwarzschild and Kerr dust models is that exact interior
Schwarzschild solutions are known, but not exact interior Kerr solution. The lack of a known exact interior
Kerr solutions is not a fundamental problem for the derivation, for two reasons. (a) Rigorous solutions of

16 Petti nonlinear field theories are routinely defined by convergent sequences of approximate solutions; a solution need not be an exact symbolic solution (as discussed in Appendix F, section (d)). (b) If GR did not have interior
Kerr solutions (exact symbolic solutions or solutions defined as limits of convergent approximations), then GR would be incapable of describing static, rotating cylindrically symmetric bodies that have zero moments at infinity except for total mass, charge and a.m.
The proper role of rigor in mathematics and physics was discussed by over a dozen mathematicians and physicists (most of them Field Medalists) in the Bulletin of the America n Mathematical Society [Jaffe 1993;
Atiyah 1994; Thurston 1994]. The consensus view was (a) rigor has an important role in assuring the reliability of results; (b) rigor in practice has degrees and is not a binary property; (c) the intellectual processes of innovation and of rigorous proof often differ; (d) an excessive focus on rigor in the innovation process would stifle innovation. Thurston conjectured that computerization will eventually raise standards of rigor of proof.
7.3.
Lack of empirical evidence for Einstein–Cartan theory
While absence of direct experimental evidence for EC is not a limitation of the derivation, it may limit the degree of certainty assigned to EC. Normally a physical theory is accepted if it is the simplest candidate that explains prior observations or makes new predictions that are verified empirically and that currently-accepted theories fail to explain.
The most promising places to seek experimental evidence for EC are places having high spin density, such as the early universe and the interior of neutron stars. Direct empirical evidence supporting EC may not be obtainable with current technology because torsion vanishes where spin density is zero, and because torsion forces are weak like gravitational forces.
I believe a deeper issue underlies the issue of empirical evidence. For most physicists, a mathematical derivation from accepted theories – even one that elegantly fixes a basic problem such treatment of spin-orbit coupling – is inadequate to tip the balance of credibility in favor of changes in basic theories without empirical evidence. In my opinion, this viewpoint does not adequately reflect that GR – perhaps the best of contemporary theories in this respect – is so restrictive that it is virtually impossible to include new terms that (a) are covariant, (b) extend without compromising mathematical foundations of the accepted theory, (c) have clear physical interpretations in terms of known phenomena, and (d) can be derived from the accepted theory.
8.
Conclusion
EC extends GR to describe exchange of intrinsic and orbital a.m., which fixes, in the least invasive way, the most outstanding problem of GR as the master theory of classical spacetime physics: GR does not support exchange of classical intrinsic and orbital a.m., as occurs in turbulent fluids. Part one computes the integral surrogate of torsion around one Kerr mass. Part two constructs a sequence of arrays of general relativistic solutions built from Kerr masses that converge to a classical continuum with constant densities of mass, momentum, a.m. and charge. The classical continuum limit yields torsion and the spin–torsion relation of EC, not GR. Except for GR, the only gauge theories of gravity that are compatible with all current empirical evidence are EC – and teleparallel gravity, which is a radical departure from GR.
9.
Acknowledgements
I would like to thank R K Sachs for nurturing my knowledge of relativity and an inquisitive attitude many years ago. Professor Jean Krisch suggested improvements to this paper. Jean de Valpine brought to my attention the correspondence between Einstein and Cartan. Some materials in part one (sections 4 and 5 and
Appendices B, C) in this article appear in [Petti 1986].

Derivation of Einstein
–
Cartan theory from general relativity 17
Appendix A: Mathematical notation
Ξ
T( Ξ )
GL(n) a smooth (pseudo-) Riemannian manifold of dimension n
with local coordinates ξµ , metric g
µν
, and metric connection 1-form
ω : TP( Ξ ,G) → L(G)
whose connection coefficients are denoted by Γ
λ ν
µ (s) and covariant differentiation is denoted by D . Denote a point as ξ ∈ Ξ . the tangent bundle of the manifold
Ξ linear frame group of the linear frame bundle of Ξ
Y a linear space
Rn
with linear coordinates yi
on which
GL(n)
acts on the left
P( Ξ ,GL(n))
the linear frame bundle over
Ξ
with structure group
GL(n)
.
The bundle projection is denoted
π : P → Ξ
.
B( Ξ ,GL(n),Y)
the bundle associated with
P( Ξ ,GL(n))
with fiber
Y
. The bundle projection is denoted
π : P → Ξ
A(n) affine group that is bundle-isomorphic to the affine frame bundle of Ξ
X an affine space
An
with affine coordinates xi
on which
A(n)
acts on the left
P( Ξ ,A(n)) a principal bundle over
Ξ
with structure group
A(n)
. We do not use the affine frame bundle of Ξ , because the affine frame bundle has a fixed solder form, and we shall vary the solder form.
B( Ξ ,A(n),X) the bundle associated with
P( Ξ ,A(n))
with fiber
X
.
Ξ ’ is a flat smooth (pseudo-) Riemannian manifold of the same dimension as
Ξ
with local coordinates
ξ '
µ , metric g'
µν
, and metric connection 1-form ω whose connection coefficients are denoted by
Γ '
λ ν
µ (s) and whose covariant differentiation is denoted by
D’
. Denote a point in Ξ ’
by ξ '
.
C: [0,1] → Ξ
a smooth curve in
Ξ
, with
C(0) = ξ0
and tangent vector field u
.
C’: [0,1] → Ξ '
is a smooth curve in Ξ '
, with
C’(0) = ξ ’0
. u’ tangent vector field along the curve
C’
.
L : T
ξΞ
→ T
ξ ’
Ξ
is a linear isometry between tangent spaces at
ξ ∈ Ξ
and
ξ ' ∈ Ξ '
.

18 Petti
Appendix B: Connections, curvature and torsion on fiber bundles
We shall assume that the reader is familiar with the basic results of the theory of connections on fiber bundles.
[Bishop and Crittenden 1964] and [Kobayashi and Nomizu 1963 and 1969]. The purpose of this section is to focus upon the relation between holonomy and curvature in the case of linear frame bundles. If the reader is not familiar with the theory of connections on fiber bundles, the rest of the paper should be coherent without this section.
B–1 Relationship between curvature and torsion
The most general and elegant definition of holonomy is given in terms of the theory of connections on fiber bundles. Let
P
be a principal bundle over Ξ with structure group
G
, endowed with a connection. Define a smooth curve C in Ξ starting and ending at the point ξ . Starting at a point p
0
in the fiber over ξ , form the horizontal lift of the curve
C
into
P
. The lifted curve will end at a point p
1
in the fiber over ξ . Then there is a unique element g ∈ G
such that
(37) p0 g = p1 .
By the equivariance condition in the definition of the connection, the element g is independent of the choice of p
0
, and depends only on the connection on
P
and the curve
C
in Ξ .
The group element g
is the holonomy of the loop
C
. If we have a linear representation of the structure group, then the holonomy g is transformation by which any basis of the representation space is changed upon parallel translation around
C
.
The most general and elegant definition of curvature is also given in terms of the theory of connections on fiber bundles. There are two equivalent ways to construct the curvature:
(i) Choose a point ξ in Ξ and two vectors V and W at ξ . Construct a one-parameter family of closed loops
Ct(s)
through ξ which are tangent at ξ to the plane of
V ^ W
, and which converge to ξ as t → 0
. Let gt
be the holonomy of the loop
Ct
. Then the curvature
R
is defined as
(38) – R(V ^ W) = lim gt _ t → 0 area(t) where area(t)
= area of the loop C t
in any smooth coordinate system whose coordinate vectors at ξ include
V
and
W
. The curvature depends only on the 2-form
V ^ W
, and not on the choice of the vectors
V
and
W
, or the curves
Ct
. The limit exists and is finite when appropriate smoothness conditions are imposed upon the connection.
This construction yields a simple intuitive interpretation of curvature: curvature is holonomy per unit area, in the limit where the area encircled by the loop goes to zero.
(ii) Let ω = the connection 1-form on the bundle
P
.
(39) ω : TP → L(G)
The curvature is the horizontal component of the 2-form d ω (exterior derivative of ω ):
(40) Ω ( ∙ , ∙ ) = d ω (Hor( ∙ ), Hor( ∙ )) where
Hor
is the horizontal projection map for tangents to the bundle
P
.
The horizontal 2-form Ω on P uniquely determines a Lie-algebra-valued 2-tensor field on Ξ . If the bundle is a linear frame bundle, then the Lie algebra components of the curvature can be identified with (1, 1) rotation tensor fields on Ξ . This gives a (1,3) tensor field
R
on Ξ , which is the curvature tensor usually used in Riemannian geometry.

Derivation of Einstein
–
Cartan theory from general relativity 19
The second definition of curvature is the customary one in differential geometry. The first one offers richer intuitive insights for the purposes of this paper.
We can derive the first definition of curvature from the second using the relation d ω ( X , Y ) = ½ ( X ω ( Y ) – Y ω ( X ) – ω ([ X , Y ])) which holds for all 1-forms ω and all smooth vector fields X and Y . If ω is the connection form and X and Y are the horizontal lifts of coordinate vector fields in Ξ , then we get
(41) d ω (X, Y) = – ½ ω ([X, Y])
For the X and Y specified, [X, Y] is vertical, and ω is a vertical projection operator (identifying the Lie algebra of the structure group with the vertical fiber).
[ X , Y ]
is defined by traversing an integral curve of
X
followed by those of Y, –X , and –Y , subtracting the coordinates of the initial location from those of the end point given by the integration process, and taking the limit, per unit area enclosed, as the lengths of integral curves traversed approach zero. When projected into Ξ , this construction for [X, Y] yields a family of closed loops at the point p in M , like the loops used in the first definition of curvature. The construction for
[X, Y]
amounts to parallel translation around these closed loops, and ω ([X, Y]) is the holonomy per unit area in the limit of small areas of the loops traversed. The conventional normalization for
R
is that
(42) R(V, W) = 2 d ω (X, Y) where V and W are projections in Ξ of the vectors
X
and
Y
tangent to
P
. Hence
R
is minus the holonomy per unit area.
B–2 Relation between development, curvature, and torsion for linear connections
Linear connections are characterized by two conditions:
(i) The structure group G is a subgroup of the linear automorphisms of Rn onto itself (commonly represented by
GL( n,R )
= group of n-by-n real matrices).
(ii) There is a solder 1-form :
TP → Rn
which is horizontal and equivariant. This form identifies a point p
in
P with a basis of tangents to Ξ at π (p)
.
The principal bundle is identifiable as a subbundle of the bundle of linear bases of the tangents of Ξ .
There are two ways to define the torsion of a linear connection:
(i) At a point ξ ∈ Ξ , form a family of loops
Ct(s)
which converge to p and are tangent to
V ^ W
, as in Section 3.2 above. Develop the curves Ct into a flat (pseudo-) Euclidean n manifold. The developed curves generally do not close. Torsion can be defined as
(43) – T(V ^ W) = lim (C’ t
(1) – C’ t
(0)) t → 0 area(t)
Thus, torsion is the amount by which the developed curve of a loop in Ξ fails to close, per unit area enclosed by the loop, in the limit as the enclosed area approaches zero. Torsion is infinitesimal translational holonomy per unit area. We can think of torsion as translational curvature. Indeed, in the theory of affine connections, where the structure group is a subgroup of the inhomogeneous automorphisms of Rn , torsion appears as the curvature components associated with the translations in the Lie algebra of the structure group.
(ii) Torsion can be defined as the horizontal part of the exterior derivative of the fundamental 1-form θ :
(44) Tor( ∙ , ∙ ) = d θ (Hor( ∙ ), Hor( ∙ ))

20 Petti
The horizontal 2-form Tor on B determines a unique (1, 2) tensor field
T
on Ξ , by transforming the
R n –valued index of Tor into tangent vectors of Ξ . T is the torsion tensor normally used in Riemann-
Cartan geometry.
The second definition of torsion can be used to derive the first definition in terms of translational holonomy, much the same as in the case of curvature. This construction requires use of the inhomogeneous linear group as the structure group.
Appendix C: Proof of theorem 1 (translational holonomy of equatorial loops)
C–1 Holonomic coordinate bases and orthonormal frames
A basis of the tangent space is called “holonomic” if and only if the Lie brackets of the basis vectorfields are zero; equivalently, if and only if there exist local coordinate systems that yield the basis vectors.
Throughout the entire article, we use the following symbols. a := s/m
∆ := r 2 + a 2 – 2 m r + q 2
ρ 2 := r 2 + a 2 cos 2 ( θ )
A := – 2 m r + q 2 c := cos( θ ), s := sin( θ )
Two bases will be used for the tangent vectors and covectors on the Kerr manifold. The first is the coordinate basis:
( ∂ / ∂ t, ∂ / ∂ r, ∂ / ∂ θ , ∂ / ∂ ϕ )
; with dual basis
(dt, dr, d θ , d ϕ )
.
The second basis is an orthonormal frame field. e
0
┌ ┐
= 1
│ ( r 2 + a 2 ) ∂ + a ∂
│
ρ
∆ ½ └
∂ t ∂ ϕ ┘ e
1
= ∆ ½ ∂
ρ ∂ r e
2
= 1 ∂
ρ ∂
θ
┌ e
3
= 1
│ a sin 2 (
ρ sin( θ )
└
┐
θ ) ∂ + a ∂
│
∂ t ∂ ϕ ┘
The dual basis of the orthonormal frame field is: e 0 = ∆ ½ [ dt – a sin 2 ( θ ) d ϕ ]
ρ e 1 = ρ dr
∆ ½ e 2 = ρ d θ e 3 = sin( θ ) [(r 2 + a 2 ) d ϕ – a dt]
ρ
C–2 Lie brackets of frame fields
A basis of the tangent space is called “anholonomic” if and only if the Lie brackets of the basis vector fields are non-zero. Orthonormal bases for tangents equivalently, if and only if there exists local coordinate systems that yield the basis vectors. Orthonormal frame fields normally have

Derivation of Einstein
–
Cartan theory from general relativity 21
The Lie brackets of the frame fields are specified frame brackets coefficients fb
as in the following formula.
[ei, ej] = fbij k e k i,j,k = 0,1,2,3
Note that fbij k =
fbji k. The non-vanishing frame bracket coefficients are given below.
Figure 1:
Frame bracket coefficients for the orthonormal frame fields a2 (r-m) c2 + m r2 r (q2 + a2)
∆
½
ρ 3
0
a2 c s
ρ 3
0 fb01 k =
0 fb02 k =
0
2 a r s
ρ 3
0
0 fb12 k =
0
a2 c s
ρ 3
∆
½ r
ρ 3
0 fb13 k =
0
0
∆
½ r
ρ 3
2 a ∆
½ c
ρ 3 fb23 k = 0
0
- (r2+a2) c
ρ 3 s
C–3 Connection coefficients
The most convenient way to write the connection coefficients is to use the coordinate basis for the direction of covariant differentiation (first index), and the frame field basis for the rotation of frames (second and third indices).
Connection coefficient = Γ kb a where k
is a coordinate index and a and b are frame field indices.

22 Petti
Below are the connection coefficients using the first spacetime coordinate basis (above) for the directions of covariant differentiation, and the orthonormal frame basis for fiber directions (the second and third indices).
Figure 2:
Connection coefficients for Kerr solution in mixed coordinate/frame field basis
0 – 1(m + r a)
ρ 2 ρ 2
0 0
(45)
Γ tb a =
–1(m + r a)
ρ 2 ρ 2
0
0
0
0
0
0
– a c A
ρ 4
0 0 a c A
ρ 4
0
— b →
0 0 0 a s r
ρ 2 ∆ 1/2
(46)
Γ rb a =
0
0
0 a 2 s c
ρ 2 ∆ 1/2 a 2 s c
ρ 2 ∆ 1/2
0
0
0 a s r
ρ 2 ∆ 1/2
0 0 0
0 0 0 a c ∆ 1/2
———————
ρ 2
(47) Γ
θ b a =
0
0
0 r ∆ 1/2
————
ρ 2
r ∆ 1/2
– ————
ρ 2
0
0
0 a c ∆ 1/2
——————
ρ 2
0 0 0

Derivation of Einstein
–
Cartan theory from general relativity 23
0
— a (a 2 (r-m) c 2
+ r (r 2 +mr+q 2 )) s 2
ρ 4
a s c ∆ 1/2
– —————————
ρ 2
0
(48)
Γ ϕ b a =
— a (a 2 (r-m) c 2
+ r (r 2 +mr+q 2 )) s 2
ρ 4
a s c ∆ 1/2
– —————————
ρ 2
0
0
0
0
0
r s ∆ 1/2
– ——————
ρ 2 c ┌
— │ a 2 s 2 ∆ –(r 2 +a 2 ) 2
ρ 4 └
┐
│
┘ r s ∆ 1/2
————————
ρ 2
c ┌
– — │ a
ρ 4 └
2 s 2 ∆ –(r 2 +a 2 ) 2
┐
│
┘
0
C–4 Development of equatorial loop into Minkowski space
The starting point and initial tangent of the equatorial loop used in Theorem 1 are given below.
(49) C
µ
( 0 ) =
( Kerr coords)
0 r
0
0
d C
µ
(0)/d σ =
(
Kerr coords)
0
0
0
2 π d Ca(0)/d σ =
( frames)
–2 π a ∆
½ r
0
0
2 π (r2+a2) r
(50) Accel
µ
( σ ) =
(Kerr cords)
–2 π a Δ
½ r
0
0
2 π a(r2+ a2) r
Accela( σ )=
(frames)
0
–4 π 2 ∆
½
(r4 - a2 m r + a2 q2) r4
0
0
=
0
–4 π
2 k 2 ν2
0
0
We want to use the mapping
A
µ
ν
( σ )
that parallel translates vectors at the start of the equatorial loop to any point on the loop with parameter value σ . The indices on
A
µ
ν
are orthonormal frame indices.
Lemma 1 states that
A( σ ) satisfies the differential system dA
µ
ν
(s)+ u
λ
(s) Γ
λσ
µ
(s) A
σν
(s) = 0 ds
In Kerr coordinates, on the specified equatorial loop, this equation is:
(51) dA
µ
ν
(s)+ 2 π Γ ϕσ
µ
(s) A
σν
(s) = 0 ds

24 Petti
The matrix
A( σ )
below integrates parallel translation for the chosen equatorial loop satisfies this equation with initial condition
A(0) = identity
.
Notation: Z
:= – r + q2/r – m cc := cos(2 π ν θ ),ss := sin(2 π ν θ )
(52)
A
µ
ν
( σ ) = a Z2 (1 – cc) + 1
ν
2 r4
– a Z ss
ν r2
– a Z ss
ν r2
(a2 Z2 – ∆ r2)(1–cc) + 1
ν
2 r4
0
0
– a ∆
½
Z (1-cc)
ν
2 r3
∆
½ ss
ν r
0 0 1 0 a ∆
½
(1 – cc)
ν
2 r3
– ∆
½ ss
ν r
0
1 – ∆ (1 – cc)
ν
2 r2
Pull vector
Accel( σ )
at
C( σ ) back to the starting point
C(0)
by multiplying by
A-1( σ )
.
4 π
2 a k2 ν (r2 + m r – q2) sin(2 π ν σ ) r3
(53) A-1( σ ) Accel( σ )=
– 4 π
2 k2 (r4(r2-2mr+q2) – a2(m r-q2)(2r2 + m r – q2)) cos(2 π ν σ ) r6
0
The next step is to choose a fixed isometry
L
µ
ν
between the tangent vectors at initial points
C(0)
in spacetime
Ξ and
C’(0)
in the development space Ξ '
. We have chosen the starting point and initial direction of the developed curve so that the solutions in Minkowski coordinates reflect the rotational symmetry of the problem.
– ∆
½
ν r
0
– 4 π
2 k2 ν ∆
½ sin(2 π ν σ ) r
0 a Z
ν r2
0 1 0 0
(54) L
µ
ν
=
– a Z
ν r2
0 0
∆
½
ν r
0 0 1 0
Since we use Minkowski coordinates
(t’, x’, y’, z’)
on the flat space Ξ ’
, parallel translation in local coordinates is trivial. The acceleration of curve
C( σ )
is mapped from
C( σ )
to
C(0)
to
C’(0)
to
C’( σ )
via that mapping
A’(s) L [A(s)] -1
, is:

Derivation of Einstein
–
Cartan theory from general relativity
The acceleration of the developed curve in Minkowski space is:
25
0
(55) Accel’( σ ) =
4 π
2 k2 ν
2 cos(2 π ν σ )
4 π
2 k2 ν
2 sin(2 π ν σ )
0
The initial condition d C’(0)/d σ is determined by the orthogonal map L, and the choice of the initial point
C’(0)
of the developed curve is arbitrary. We choose both of these to express the rotational symmetry.
The initial conditions for development of the equatorial loop into Minkowski space are:
-k1
(56) C’(0) = 0 d C’(0)/d σ =
2 π k2 ν
0
The development of the equatorial loop into Minkowski space is:
-k1
0
2 π k2 ν
0
-k1 σ
(57) C’( σ ) = k2 cos(2 π ν σ ) k2 sin(2 π ν σ )
0
C–5 Translational holonomy of an equatorial loop developed into Minkowski space
The translational holonomy (the “failure-to-close vector”) of the equatorial loop into Minkowski space is
C’(1) C’(0)
.
(58) C’(1) C’(0) =
This completes the proof of Theorem 1.
-k1 k2 (cos(2 π ν ) – 1) k2 sin(2 π ν )
0

26 Petti
Appendix D: Proof of theorem 2 (translational holonomy of polar loops)
For development of a curve in the θ direction extending from θ
0
to θ
1
, the development matrix
A
has the form
(59) where
│ cosh(F)
Aab = │ 0
│ 0
│ sinh(F)
0 0 cosh(G) –sin(G) sin(G) cos(G)
0 0 sinh(F) │
0 │
0 │ cosh(F) │
(60)
│ ┌ ┐ │ θ
1
F = │ – ∆ tanh –1 │ a sin( θ )
│ │
│ r
2 + a
2 └
( r
2 + a
2 )½
┘ │ θ
0
(61)
│
G = │ – ∆ tanh –1
│ r 2 + a 2
┌
│ r tan( θ )
┐
│
│
│
θ
1
└
(r 2 + a 2 )½
┘ │ θ
0
The tensor indices a
and b
refer to frame field directions. When θ
0
and θ
1
are multiples of π , both
F
and
G vanish, so the matrix
A
is the identity. Therefore the development of a closed loop in the θ direction is closed, so the translational holonomy is zero.
Appendix E: Open Questions for Research
E.1 Do black holes have central singularities?
Poplawski showed that in black hole models containing spinning matter, EC replaces singularities with continuous physics. Black holes have a minimum size of about 10 16 kg (10 24 m
Planck
) [Poplawski 2010b].
These results may alter or eliminate the “information paradox” of black holes, since matter does not fall into a singularity, and gradual evaporation of black holes is eliminated by the minimum mass result.
E.2 Is the “big bang” really a “big bounce”?
Poplawski analyzed closed Friedmann cosmological models based on EC that contain a classical Weyssenhoff spin fluid [Poplawski 2010a], and models in which spinning matter is described by spinor fields
[Poplawski 2012].
E.3 Is torsion the inflaton field of cosmic inflation?
At high densities, the torsion repulsive energy dominates behavior, resulting in a minimum radius and extreme rates of re-expansion. The “big bang” becomes the “big bounce.” These results suggest that quantized torsion is the inflaton field.
E.4 Is EC a better classical limit for quantum gravity?
Quantum spin has a more fundamental role in quantum theories than in classical physics. Therefore EC is likely to be a better classical limit of any reasonably correct quantum theory of gravitation; such a quantum theory likely will include either torsion or a quantum precursor of torsion.
The current consensus of researchers in loop quantum gravity (LQG) is that the classical limit of LQG must be GR. However, they have not been able to obtain GR as the classical limit of LQG.

Derivation of Einstein
–
Cartan theory from general relativity 27
E.5 Does torsion trace occur in virtual fields joining ends of fermion world lines?
The second Bianchi identify of geometry is satisfied by edge dislocations that join ends of screw dislocations. edge dislocations have translational holonomy in the plane of a loop surrounding the edge defect (“in-plane torsion”), and fermions have antisymmetric torsion (antisymmetrized out-of-plane torsion). Therefore it is quite reasonable to expect that virtual particles that join the ends of fermion lines contain torsion trace.
E.6 Can including torsion improve the theory of turbulence?
Energy transferred to scales smaller than those of a classical fluid model are represented as internal energy.
Should subscale a.m. (orbital a.m. transferred to smaller scales smaller than those of the classical fluid model) be modelled by torsion?
Appendix F: Other comments about the derivation
Below are additional comments in the derivation followed by this author’s responses. a) The derivation integrates through singularities.
If GR has an interior Kerr solution or a similar solution for rotating bodies (as I argue above it must), then no singularities occur in the derivation. Even if Kerr singularities are present, the derivation computes line integrals whose paths surround singularities, divided by areas as seen by an observer at infinity. The computations are virtually identical to those used to compute mass densities of dust models derived from a distribution of Schwarzschild solutions. See section E–2 for further discussion. b) Spin is a quantum phenomenon; it has no place in classical physics
Any classical model that has orbital a.m. on a smaller scale than the scale of the model (and that is exchanged with orbital a.m.) must include the density of subscale orbital a.m. The classical theory of spin fluids routinely models subscale orbital a.m. as intrinsic a.m., or classical spin. The essence of turbulence in statistical fluid mechanics is conversion of orbital a.m. into a.m. at smaller scales without classical limit. c) Equating spin density with torsion doesn't work; for example, the Maxwell field has spin 1, but it does not generate affine torsion in EC.
We must distinguish base space tensor indices and fiber tensor indices. All tensor indices that represent conserved currents are covariant differentiated, and spacetime boxes through which fluxes are measured are not covariant differentiated. This is similar to Yang-Mills theory, where physicists are accustomed to distinguishing base space indices and current indices. See Section 6 for a more explicit discussion. d) A distribution of Kerr solutions is not an exact solution in GR, so the computation proves nothing.
The best mathematical approach defines a solution of differential equations as a convergent sequence of approximate solutions in a Sobolev space (whose norm is L2 norm of derivatives of functions). For nonlinear PDEs, exact symbolic solutions are very special cases. An engineering-level introduction to
Sobolev norms in finite element analysis can be found in [Strang 1973]. e) Deriving torsion from a continuum limit of GR does not establish torsion as basic physics like GR.
All classical continuum field theories, including GR and quantum mechanics, are continuum limits of deeper theories. It makes no sense to single out EC as less fundamental because it is derived in this way.
To the contrary, the fundamental assumption of loop quantum gravity is that rotational and translational holonomy around discrete loops underlie rotational (Riemannian) curvature and translational curvature
(torsion).

28 Petti f) Deriving torsion and the spin-torsion relation from GR does not establish EC.
No theory – including Newtonian mechanics, special relativity, GR, and quantum field theory – can counter the possibility that it might eventually be superseded by a better theory. We should apply “Occam's Razor” that simplicity and fit with current evidence should temper skepticism. g) Deriving torsion from general relativity in this way precludes extension to larger gauge group.
The work herein constructively derives torsion from GR; it does not preclude extending the symmetry group to include conformal or other nonmetric symmetries. h) It is not possible to derive a non-symmetric connection from symmetric connection by a limiting process.
This statement is similar to asserting that Riemannian curvature cannot be derived as the continuum limit of Euclidean pieces. A cone can be made from a flat sheet in the shape of a circular sector with central angle 2 π – θ . The cone has a Dirac delta function of Riemannian curvature of magnitude θ at its apex and is flat elsewhere. The continuum limit of many such cones can yield a surface with Riemannian curvature.
Appendix G: Computer algebra validation of computations
Computer algebra system Macsyma 2.4.1a was used to verify most of the computations in this work
[Macsyma 1996]. This appendix is the script of that validation. As above, we use lower case Greek letters for coordinate indices and lower case Roman letters for fiber indices.
Outline of the Macsyma computer algebra script
1.
2.
3.
4.
5.
Geometry of the Kerr-Newman metric
Connection coefficients
Translational holonomy around and equatorial loop
Translational holonomy /area yields affine torsion
Spin density and spin-torsion relationship
Conversions between conventional tensor notations and Macsyma notations
Description
Covariant metric in coordinate basis
Contravariant metric in coordinate basis
Frame field
Inverse frame field
Covariant metric in frame basis
Contravariant metric in frame basis
Christoffel symbols in frame basis
Christoffel symbols in mixed coordinate-frame-frame basis
Conventional notation g
μν g
μν fri
μ ifr
μ i gij gij mcskji mcs
μ j i
Macsyma notation lg ug fr ifr lfg ufg mcs mcs_cff
Relationships g
μν
= invert(g
μν
) ifr
μ i = invert(fri μ
) gij = fri
μ
g
μν
frj
ν gij = ifr
μ i g μν
ifr
ν j mcs
μ j i = ifr
μ k mcskji

Derivation of Einstein
–
Cartan theory from general relativity 29
This Macsyma script automates the computation is to compute the translational holonomy of an equatorial loop around a charged rotating black hole. Original souroce:: "On the Local Geometry of Rotating Matter," by R. J.
Petti, Gen. Rel. Grav., vol 18, no. 5, pages 441-460 (1986). Updated version: http://arxiv.org/abs/1301.1588.
1.1 Metric and Frame Fields
Enter coordinates [t, r, θ , φ ], turn on frame fields, and set lowered frame metric LFGij = Minkowski metric.
(c1)
(d1)
(init_ctensor(), ratfac : true, dim : 4, ct_coords : [t, r, theta, phi], assume(m > 0),
cframe_flag : true, lfg : diag_matrix(-1,1,1,1) )
- 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Enter inverse frame field FRI i
µ
that defines the Kerr Newman metric.
If d/dx µ are coordinate basis fields and Ei are frame fields, then Ei = d/dx µ FR µ i .
Think of FRI as the “square root of the metric tensor” since the lowered metric LG µν = FRI i µ LFGij FRI j ν .
(c2)
(d2) fri : matrix(sqrt(\delta)*[1, 0, 0, -a*sin(theta)^2]/rho, [0, rho/sqrt(\delta), 0, 0],
[0, 0, rho, 0], sin(theta)*[-a, 0, 0, r^2+a^2] / rho)
ρ
∆
0
-
0 a sin (
θ
)
ρ
0 0 - a sin
2
(
θ
)
∆
ρ
ρ
0
∆
0 ρ
0
( r
2
+ a
2
0
)
sin (
θ
)
0 0
ρ
Define derivatives of ∆ and ρ with respect to the coordinates r and θ .
(c3)
(gradef(\delta, r, 2*(r-m)), gradef(rho, r, r/rho),
gradef(rho, theta, -a^2*sin(theta)*cos(theta)/rho), assume(rho>0))$

30 Petti
Define some transformation rules for expressions that appear frequently.
(c4)
(defrule(delta_def, r^2-2*m*r+a^2+q^2, \delta),defrule(delta_expand, \delta, r^2-2*m*r+a^2+q^2),
defrule(rho2_def1, r^2+(a*cos(theta))^2, rho^2), defrule(rho2_def2, a^2*sin(theta)^2-r^2-a^2, -rho^2),
defrule(rho2_expand, rho^2, r^2+(a*cos(theta))^2), defrule(rho_expand, rho, sqrt(r^2+(a*cos(theta))^2)),
defrule(r2_a2, a^2*sin(theta)^2+rho^2, r^2+a^2), defrule(mr_expand, r^2+a^2-\delta, 2*m*r-q^2),
disprule(delta_expand, rho2_expand))$ delta_expand ( defrule ) : ∆→ r
2
- 2 m r + q
2
+ a
2
(e4)
(e5) rho2_expand ( defrule ) : ρ
2
→ a
2
cos
2
( θ ) + r
Compute and display frame field FR i
µ , also lowered metric LG µν , raised metric UG ij .
2
(c6) (cmetric(), fr : apply1(factor(fr), rho2_def2)) r
2
+ a
2
ρ ∆
0 0 a sin
ρ
(
θ
)
(d6)
0
0 a
ρ ∆
ρ
∆
0
0
0
1
ρ
0
Re-express the metric tensor LG µν using transformation rule MR_EXPAND.
0
0
1
ρ sin (
θ
)
(c7) lg : apply1(lg, mr_expand)
-
∆ - a
2
sin
2
(
θ
)
ρ
2
(d7)
0
0 a sin
2
(
θ
)
(
∆ - r
2
- a
2 )
ρ
2
0 0 a sin
2
(
θ
)
(
∆ - r
2
- a
2 )
ρ
2
ρ
2
0
∆
0 ρ
2
0
0
0 - sin
2
(
θ
)
( a
2
sin
2
0
(
θ
)
∆ - r
4
- 2 a
2
r
2
- a
4 )
ρ
2
1.2 Frame Brackets
The Lie brackets of the frame fields were computed by command CMETRIC above.
Apply simplification rules to the frame brackets.
(c8)
(e8)
(for i thru dim do for j thru dim do for k thru dim do fb[i,j,k] :
apply1(trigsimp(apply1(fb[i,j,k],delta_expand,rho2_expand)), rho2_def1,delta_def),
cdisplay(fb) )$ fb
1, 2, 1
=
( a
2
r - a
2
m
)
cos
2
(
θ
) + m r
2
+
(
- q
2
- a
2 )
r
ρ
3
∆

Derivation of Einstein
–
Cartan theory from general relativity 31
(e9)
(e10)
(e11)
(e12) fb
1, 2, 4
=
2 a r sin ( θ )
ρ
3 fb
1, 3, 1
= - a
2
cos ( θ ) sin ( θ )
ρ
3 fb
2, 3, 2
= - a
2
cos ( θ ) sin ( θ )
ρ
3 fb
2, 3, 3
= - r ∆
ρ
3
(e13)
(e14)
(e15) fb
2, 4, 4
= - r ∆
ρ
3 fb
3, 4, 1
=
2 a cos (
θ
)
∆
ρ
3 fb
3, 4, 4
= -
( r
2
+ a
2 )
cos (
θ
)
ρ
3
sin (
θ
)
2.1 Connection coefficients with frame fields
Compute the connection coefficieints MCSkj i , with 1 index up and 2 down using frame fields.
(c16) christof(false)$
Apply simplification rules to the connection coeffiicents.
(c17)
(e17)
(e18)
(e19)
(e20)
(for i thru dim do for j thru dim do for k thru dim do
mcs[i,j,k] : apply1(trigsimp(apply1(mcs[i,j,k],delta_expand,rho2_expand)),rho2_def1,delta_def), cdisplay(mc s) )$ mcs
1, 1, 2
=
( a
2
r - a
2
m
)
cos
2
(
θ
) + m r
2
+
(
- q
2
- a
2 )
r
ρ
3
∆ mcs
1, 1, 3
= - a
2
cos (
θ
) sin (
θ
)
ρ
3 mcs
1, 2, 1
=
( a
2
r - a
2
m
)
cos
2
(
θ
) + m r
2
+
(
- q
2
- a
2 )
r
ρ
3
∆ mcs
1, 2, 4
= a r sin ( θ )
ρ
3

32
(e21)
(e26)
(e27)
(e28)
(e22)
(e23)
(e24)
(e25)
(e29)
(e30)
(e31)
(e32)
(e33)
(e34)
(e35) mcs
1, 3, 1
= - a
2
cos ( θ ) sin ( θ )
ρ
3 mcs
1, 3, 4
= a cos ( θ ) ∆
ρ
3 mcs
1, 4, 2
= - a r sin (
θ
)
ρ
3 mcs
1, 4, 3
= - a cos (
θ
)
∆
ρ
3 mcs
2, 1, 4
= - a r sin (
θ
)
ρ
3 mcs
2, 3, 2
= - a
2
cos (
θ
) sin (
θ
)
ρ
3 mcs
2, 4, 1
= - a r sin (
θ
)
ρ
3 mcs
3, 1, 4
= a cos (
θ
)
∆
ρ
3 mcs
3, 2, 3
= r ∆
ρ
3 mcs
3, 3, 2
= - r ∆
ρ
3 mcs
3, 4, 1
= a cos (
θ
)
∆
ρ
3 mcs
4, 1, 2
= - a r sin (
θ
)
ρ
3 mcs
4, 1, 3
= - a cos (
θ
)
∆
ρ
3 mcs
4, 2, 1
= - a r sin (
θ
)
ρ
3 mcs
4, 2, 4
= r ∆
ρ
3
Petti

Derivation of Einstein
–
Cartan theory from general relativity 33
(e36)
(e37) mcs
4, 3, 1
= - a cos ( θ ) ∆
ρ
3 mcs
4, 3, 4
=
( r
2
+ a
2 )
cos ( θ )
ρ
3
sin ( θ )
(e38) mcs
4, 4, 2
= - r ∆
ρ
3
(e39) mcs
4, 4, 3
= -
( r
2
+ a
2 )
cos (
θ
)
ρ
3
sin ( θ )
2.2 Connection coefficients with one coordinate index
Compute conection coefficients MCS µ j i in which the first index is a coordinate index (in the direction of covariant differentiation) and the second and third indices are frame indices (a rotation in the fiber space). Apply simplification rules.
(c41)
(array(mcs_cff,4,4,4),
for b thru 4 do for c thru 4 do for i thru 4 do
mcs_cff[i,b,c]: apply1(factor(sum(fri[a,i]*mcs[a,b,c],a,1,4)), rho2_def2,trigsimp, mr_expand) )$
Express these mixed coordinate / frame-field connection coefficients as matrices.
(c42)
(array(mcs_cffmat,4), for i thru 4 do (
mcs_cffmat[i] : genmatrix( lambda([b,c],mcs_cff[i,c,b]), 4,4),
if i <4 then (disp('mcs_cffmat[i]), disp(mcs_cffmat[i]))
else (disp(['mcs_cffmat[i], "first two columns"], submat(mcs_cffmat[i], [1,2,3,4], [1,2])),
disp(['mcs_cffmat[i], "last two columns"], submat(mcs_cffmat[i], [1,2,3,4], [3,4])))) )$ mcs_cffmat
1
0 - a
2
m cos
2
( θ ) - m r
2
+ q
2
r
ρ
4
0
- a
2
m cos
2
( θ ) - m r
2
+ q
2
r
ρ
4
0 0
0 0 0
0
0
- a cos ( θ )
(
∆ - r
ρ
4
2
- a
2 )
0 0 a cos ( θ )
(
∆ - r
ρ
4
2
- a
2 )
0

34 Petti mcs_cffmat
2
0 0 0 - a r sin ( θ )
ρ
2
∆
0 0 - a
2
cos ( θ ) sin ( θ )
ρ
2
∆
0
0
- a r sin ( θ )
ρ
2
∆ a
2
cos (
θ
) sin (
θ
)
ρ
2
∆
0
0
0
0
0 mcs_cffmat
3
0 0 0 a cos ( θ ) ∆
ρ
2
0 0 - r ∆
ρ
2
0
0 r ∆
ρ
2
0 0 a cos (
θ
)
∆
ρ
2
0 0 0
0
[ mcs_cffmat
4
, first two columns ]
- a
( a
2
( r - m ) cos
2
( θ ) + r
3
+ m r
2
- q
2
r
)
sin
2
( θ )
ρ
4
- a
( a
2
( r - m ) cos
2
( θ ) + r
3
+ m r
2
- q
2
r
)
sin
2
( θ )
ρ
4
- a cos ( θ ) sin ( θ ) ∆
ρ
2
0
0
0 r sin (
θ
)
∆
ρ
2

Derivation of Einstein
–
Cartan theory from general relativity
[ mcs_cffmat
4
, last two columns ]
- a cos (
θ
) sin (
θ
)
∆
ρ
2
0
0
0
- r sin (
θ
)
∆
ρ
2 cos (
θ
)
( a
2
sin
2
(
θ
)
∆ - r
4
- 2 a
2
r
2
- a
4 )
ρ
4
- cos (
θ
)
( a
2
sin
2
(
θ
)
∆ - r
4
- 2 a
2
r
2
- a
4 )
ρ
4
0
The frequency ν is defined so the curve forms a closed loop in the space with the Kerr masses.
(c43)
(c_1 : a/r^2*(-r-m+q^2/r), c_2 : - sqrt(\delta)/r,
nu_exp: sqrt(map('factor, multthru(ratsimp(apply1( c_2^2 - c_1^2, delta_expand))))),
apply(defrule, ['nu_expand, nu, nu_exp]), disprule(nu_expand))
35
(d43) nu_expand ( defrule ) : ν→ -
2 m r
+ q
2 r
2
-
2 a
2
m r
3
+ a
2 (
2 q
2
- m
2 ) r
4
+
2 a
2
m q
2 r
5
- a
2
q
4 r
6
+ 1
Define k1 and k2, which are used in the solutions for translational holonomy (the displacement vector) in the timelike and equatorial directions respectively.
(c44)
(e44)
(e45)
(e46)
(e47)
(defrule(k1_def, (3*m*r^3+a^2*m*r-2*q^2*r^2-a^2*q^2), k1*r^4*nu/(2*%pi*a)),
defrule(k2_def, (r^4-a^2*m*r+a^2*q^2), k2*r^4*nu^2/sqrt(\delta)),
defrule(k1_expand, k1, 2*%pi*a*(3*m*r^3+a^2*m*r-2*q^2*r^2-a^2*q^2)/(r^4*nu)),
defrule(k2_expand, k2, sqrt(\delta)*(r^4-a^2*m*r+a^2*q^2)/(r^4*nu^2)),
disprule(k1_def, k2_def, k1_expand, k2_expand) )$ k1_def ( defrule ) : 3 m r
3
- 2 q
2
r
2
+ a
2
m r - a
2
q
2
→ k1 ν r
4
2 π a k2_def ( defrule ) : r
4
- a
2
m r + a
2
q
2
→ k2 ν
2
r
4
∆ k1_expand ( defrule ) : k1 →
2 π a
(
3 m r
3
- 2 q
2
r
2
+ a
2
m r - a
2
q
2 )
ν r
4 k2_expand ( defrule ) : k2 →
( r
4
- a
2
m r + a
2
q
2 )
∆
ν
2
r
4

36 Petti
3.1 The equatorial loop
Define an equatorial loop at radius r around which to compute holonomy. Restrict analyisis to region where r >> m.
(c48) (assume(r > m), define(equator(sigma), [0 ; r ; %pi/2; 2*%pi*sigma]))
0 r
(d48) equator (
σ
) :=
2
2 π σ
Compute the tangent vector to the path in the coordinate frame.
(c49) ucoord : diff(equator(sigma), sigma)
(d49)
0
0
0
2 π
Express the tangent vector with orthonormal frames. FRI
µ j is the inverse frame field expressed as a matrix.
(c50) uframe : subst([rho=sqrt(r^2+a^2*cos(theta)^2), theta=%pi/2], fri) . ucoord
(d50)
2 π a ∆
-
0
0 r
2 π
( r
2
+ a
2 ) r
Compute the acceleration vector with orthonormal frames.
(c51)
(d51)
(sum(ucoord[k]* (mcs_cffmat[k] . uframe), k,1,4),
covdiffuuframe1 : ratsimp(subst([rho=sqrt(r^2+a^2*cos(theta)^2),theta=%pi/2], %%), m,q,\delta) )
-
4 π
2 ( r
0
4
- a
2
m r + a
2
q
2 )
∆
0
0 r
4
Express the acceleration in terms of k2. Define frequency ν so the loop returns to its starting point when σ = 1.
(c52) covdiffuuframe2 : apply1( covdiffuuframe1, k2_def)
(d52)
- 4 π
0
0
2
0
k2 ν
2

Derivation of Einstein
–
Cartan theory from general relativity 37
3.2 Matrix for parallel translation on an equatorial curve
Matrix A( σ ) parallel translates vectors from equator(0) to equator( σ ). It satisfies the matrix differential equation below, which is expressed in orthonormal frames. The initial condition is A(0) = identity.
(c53)
(d53) block([doallmxops:false, domxmxops:false],
'diff(genmatrix(a,4),sigma) + u4 . genmatrix('mcs_cff4,4) . genmatrix(a,4) = 0) u4 .
. mcs_cff4
1, 1 mcs_cff4
2, 1 mcs_cff4
3, 1 mcs_cff4
1, 2 mcs_cff4
2, 2 mcs_cff4
3, 2 mcs_cff4
1, 3 mcs_cff4
2, 3 mcs_cff4
3, 3 mcs_cff4
1, 4 mcs_cff4
2, 4 mcs_cff4
3, 4 mcs_cff4
4, 1 a
1, 1 a
2, 1 a a
1, 2
2, 2 a
1, 3 a
2, 3 mcs_cff4
4, 2 a a
1, 4
2, 4
+ a
3, 1 a
4, 1 a
3, 2 a
4, 2 a
3, 3 a
4, 3 a
3, 4 a
4, 4 mcs_cff4 a
1, 1 a
2, 1 a
3, 1 a
4, 1 a a a a
1, 2
2, 2
3, 2
4, 2
4, 3 a
1, 3 a
2, 3 a a
3, 3
4, 3 mcs_cff4
4, 4 a a a a
1, 4
2, 4
3, 4
4, 4
σ
= 0
Construct the solution matrix Amat.
(c54) mat : [0, c_1, 0,0 ; c_1, 0,0, c_2 ; 0,0,0,0 ; 0, -c_2, 0,0]
0 a



- r + r
2 q
2 r
- m



(d54) a



- r + r
2
0 q
2 r
- m



0
0
0 r
∆
0
0 -
0 0 r
∆
0
0
0

38 Petti
Here is the solution matrix Amat. displayed in two parts: colmumns 1,2 then columns 3,4.
(c55)
(amat(sigma) := ''(ident(4) + (1-cos(2*%pi*nu*sigma))*mat^^2/nu^2-sin(2*%pi*nu*sigma)*mat/nu),
disp("amat(sigma), columns 1 and 2", submat(amat(sigma), [1,2,3,4], [1,2])),
disp("amat(sigma), columns 3 and 4", submat(amat(sigma), [1,2,3,4], [3,4])) )$ amat(sigma), columns 1 and 2 a
2



- r + q
2 r
- a



- r + a



- r + q
2 r
- m



2
(
1 - cos ( 2 π ν σ ) )
ν
2
r
4
+ 1 q
2 r
- m



sin ( 2 π ν σ )
ν r
2
0
- m



(
1 - cos ( 2 π ν σ
) )
∆
ν
2
r
3
-
(
1 - cos ( 2 π a



- r +
π ν σ
) ) 


 q
2 r
- m



sin ( 2 π ν σ )
ν r
2 a
2



- r + r
4 q
2 r
ν
2
0
- m



2
-
- sin ( 2 π ν σ
)
∆
ν r
∆ r
2




+ 1 amat(sigma), columns 3 and 4
0 - a



- r + q
2 r
0
1
0 1 -
- m



(
1 - cos ( 2 π ν σ ) )
∆
ν
2
r
3 sin ( 2 π ν σ ) ∆
ν r
0
(
1 - cos ( 2 π π ν σ ) )
∆
ν
2
r
2
Verify that Amat(0) = identity.
(c56) amat(0)
(d56)
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Verify that Amat satisfies the defining equation for parallel translation.
(c57)
(d57)
(ratsubst(c_2^2 - c_1^2, nu^2 , subst(theta=%pi/2,
2*%pi*ratsubst(r^2+a^2*cos(theta)^2,rho^2,mcs_cffmat[4])) . amat(sigma)),
trigsimp(ratsimp( diff(amat(sigma), sigma) + %%, \delta)) = zeromatrix(4,4) )
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

Derivation of Einstein
–
Cartan theory from general relativity 39
The acceleration, pulled back to the starting point equator(0) by parallel translation, is:
(c58)
(d58)
(ratsubst(c_2^2 - c_1^2, nu^2, amat(-sigma) . covdiffuuframe2 ),
acceleration0 : factor(apply1(%%, delta_expand)) )
-
4 π
2
a k2 ν
( r
2
+ m r - q
2 )
sin ( 2 π ν σ
)



4 π r
3
2
k2
( r
6
- 2 m r
5
+ q
2
r
4
- 2 a
2
m r
3
+ 2 a
2
q
2
r
2
- a
2
* cos ( 2 π ν σ
)
m
2
r
2
+ 2 a
2
m q
2
r - a
2
q
4 ) 


0 r
6
-
4 π
2
k2 ν r
2
- 2 m r + q
2
+ a
2
sin ( 2 π ν σ
) r
3.3 Development into Minkowski space
Definition of "Development of a Curve": Let x(t) map [0, 1] into a Riemannian manifold M. Let tanmap be an isometry from the tangent space at x(0) to the tangent space at some point y(0) in Riemannian manifold N. A development of x(t) is a curve y(t) in N starting at y(0) with velocity [tanmap(A -1 (t) . x'(t))] and acceleration [tanmap(A -1 (t) . x''(t))]. (A is parallel translation in M.) y(t) reflects the acceleration of x(t) in M but not the curvature of M.
Define an isometry from the tangents at equator(0) in Kerr space to the tangents at a point in Minkowski space.
(c59) tanmap: [c_2/nu, 0,0, c_1/nu ; 0,1,0,0 ; -c_1/nu, 0,0, -c_2/nu ; 0,0, 1, 0]
(d59)
-
-
∆
ν r
0 a



- r +
ν r
2
0 q
2 r
- m



0 0
1 0
0 0
0 1 a



- r +
ν r
2
0 q
2 r
∆
ν r
0
- m



Tanmap depends on Kerr coordinate r and parameters a, m and q. Tanmap preserves the Minkowski metric LFG.
(c60) ratsimp(apply1( subst(c_2^2 - c_1^2, nu^2, tanmap^` . lfg . tanmap), delta_expand))
(d60)
- 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

40 Petti
Since Minkowski space is flat, parallel translation is trivial.
(c62)
(d62)
(yvec: genvector(lambda([k], funmake(concat('y,k-1),[sigma])),4),
eqn_flat: diff(yvec, sigma,2) = acceleration1) y0 ( σ )
σ σ y1 ( σ )
σ σ y2 (
σ
)
σ σ y3 ( σ )
σ σ
=
- 4 π
- 4 π
2
k2 ν
2
0
cos
2
k2 ν
2
sin ( 2 π ν σ
)
0
( 2 π ν σ )
The first order initial conditions are:
(c63) initcond_flat1 : diff(yvec, sigma) = factor(apply1(tanmap . uframe, delta_expand))
(d63) y0 (
σ
)
σ y1 (
σ
)
σ y2 (
σ
)
σ y3 (
σ
)
σ
=
-
2 π a
(
3 m r
3
- 2 q
2
r
2
+ a
2
m r - a
2
q
2 )
ν r
4
0
2 π r
2
- 2 m r + q
2
+ a
2 ( r
4
- a
2
m r + a
2
q
2 )
ν r
4
0
Restate the initial conditions in terms of coeffiicients k1 and k2.
(c64) initcond_flat2 : apply1(initcond_flat1, k1_def, k2_def, delta_def)
(d64) y0 (
σ
)
σ y1 ( σ )
σ y2 (
σ
)
σ y3 ( σ )
σ
=
- k1
0
2 π k2
0
ν
Apply the first order initial conditions.
(c65) for k thru 4 do atvalue(part(initcond_flat2,1,k,1), sigma=0, part(initcond_flat2,2,k,1))$
The initial point in Minkowski space is arbitrary. These initial values exploit symmetry about theaxis (x, y) = (0, 0).
(c66)
(map(lambda([yy,yval], atvalue(funmake(yy,[sigma]),sigma=0, yval)), [y0,y1,y2,y3], [0,k2,0,0]),
printprops([y0,y1,y2,y3], atvalue) )$ y0 ( 0 ) = 0 y0 ( @1 )
@1
[ @1 = 0 ]
= - k1 y1 ( 0 ) = k2 y1 ( @1 )
@1
[ @1 = 0 ]
= 0

Derivation of Einstein
–
Cartan theory from general relativity y2 ( 0 ) = 0 y2 ( @1 )
@1
[ @1 = 0 ]
= 2 π k2 ν y3 ( 0 ) = 0 y3 ( @1 )
@1
[ @1 = 0 ]
= 0
Find the solution to the equations for the developed curve.
(c67) odematsys(eqn_flat, yvec)
(d67)




 y0 ( σ ) y1 ( σ ) y2 (
σ
) y3 ( σ )
=
- k1 σ k2 cos ( 2 π ν σ ) k2 sin ( 2 π ν σ
)
0





Write the solution as a Macsyma function.
(c68) soln_flat(sigma) := ''(part(%,1,2))
(d68) soln_flat ( σ ) :=
- k1 σ k2 cos ( 2 π ν σ ) k2 sin ( 2
0
π ν σ )
Verify that the solution satisfies the differential equations for development in flat space.
(c69) diff(soln_flat(sigma), sigma,2) = rhs(eqn_flat)
(d69)
- 4 π
- 4 π
2
k2 ν
2
0
cos ( 2 π ν σ )
2
k2 ν
2
sin ( 2 π ν σ
)
0
=
- 4 π
0
2
k2 ν
2
cos
- 4 π
2
k2 ν
2
sin
( 2 π ν σ
( 2 π ν σ
)
)
0
Verify that the solution satisfies the first order initial conditions.
(c70) at(diff(soln_flat(sigma), sigma), sigma=0)
(d70)
- k1
0
2 π k2
0
ν
Evaluate the end point of the loop.
(c71) at(soln_flat(sigma), sigma=0)
(d71)
0 k2
0
0
41

42
3.4 Translational holonomy
The translational holonomy is defined as (final point - initial point) of the developed curve in Minkowski space.
(c72) kerr_holonomy1 : soln_flat(1) - soln_flat(0)
(d72) k2 cos
- k1
( 2 π ν ) - k2 k2 sin (
0
2 π ν
)
Frequency ν <~ 1, so define φ = 1 - ν >~ 0 to use in trig functions.Compute the gradiant of φ with respect to r.
(c73) (phi_exp : 1 - nu_exp, gradef(phi, r, ratsubst(phi, phi_exp, diff(phi_exp, r))), 'phi = phi_exp)
Petti
(d73) φ = 1 -
Substitute 1 - φ for ν in holonomy.
-
2 m r
+ q
2 r
2
-
2 a
2
m r
3
+ a
2 (
2 q
2
- m
2 ) r
4
+
2 a
2
m q
2 r
5
- a
2
q
4 r
6
+ 1
(c74) kerr_holonomy2 : subst(nu = 1-phi, kerr_holonomy1)
(d74) k2
- k1
( cos ( 2 π φ
) - 1
)
- k2 sin
0
( 2 π φ
)
Expand k1 and k2 to express holonomy in terms of m, r, a, q, and φ .
(c75)
(d75) kerr_holonomy3 : ratsimp(subst(nu=1-phi, apply1( kerr_holonomy2, k1_expand, k2_expand, delta_expand)), r)
2 π a
(
3 m r
3
- 2 q
2
r
2
+ a
2
m r - a
2
q
2 )
(
φ - 1 ) r
4
( cos ( 2 π φ ) - 1
)
r
2
- 2 m r + q
2
+ a
2 ( r
4
- a
2
m r + a
2
q
2 )
(
φ - 1 )
2
r
4
- sin ( 2 π φ ) r
2
- 2 m r + q
2
+ a
2 ( r
4
- a
2
m r + a
2
q
2 )
(
φ - 1 )
2
0
r
4
Assume |a| << r, m << r, q=0. In this case, φ ~ m/r ~ 0.
Set small quantities --> 0 where they are added to larger quantities, as determined by comparing powers of r.
(c76) kerr_holonomy4 : subst(3*r^2+a^2 = 3*r^2, factor(subst([q=0, phi=0], kerr_holonomy3)))
(d76)
-
6 π a m
0
0
0 r

Derivation of Einstein
–
Cartan theory from general relativity 43
3.5 Approximate torsion for an isolated Kerr mass
Approximate torsion of an isolated Kerr mass as translational holonomy per unit area of the equatorial loop.
(c77) kerr_approx_torsion1 : factor(kerr_holonomy2/ (%pi*r^2))
(d77)
- k1
π r
2 k2
( cos ( 2 π φ
) - 1
)
π r
2
- k2 sin ( 2 π φ )
π r
0
2
Express approximate Kerr torsion in terms of m, a, q, r, and φ .
(c78)
(d78) kerr_approx_torsion2 : factor(subst(nu=1-phi, apply1(kerr_approx_torsion1, k1_expand, k2_expand, delta_e xpand)))
2 a
(
3 m r
3
- 2 q
2
r
2
+ a
2
m r - a
2
q
2 )
( φ - 1 ) r
6
( cos ( 2 π φ
) - 1
)
r
2
- 2 m r + q
2
+ a
2 ( r
4
- a
2
m r + a
2
q
2 )
π ( φ - 1 )
2
r
6
- sin ( 2 π φ
) r
2
- 2 m r + q
2
+ a
2 ( r
4
- a
2
m r + a
2
q
2 )
π ( φ - 1 )
2
0
r
6
Assume |a| << r, m << r, q=0. In this case, φ ~ m/r ~ 0.
Set small quantities --> 0 where they aree added to larger quantities, as determined by comparing powers of r.
(c79) kerr_approx_torsion3 : subst(3*r^2+a^2 = 3*r^2, factor(subst([q=0, phi=0], kerr_approx_torsion2)))
(d79)
-
6 a m
0
0
0 r
3

44 Petti
4.1 Holonomy in the continuum limit
Up to this point, all results are exact solutions based on the exterior Kerr metric.
Consider a sequence of regular distributions of Kerr masses that approaches a continuum limit with constant densities of mass, angular momentum, and charge. Let r be the average half-distance between adjacent Kerr masses. As r becomes smaller, m and q decrease as r 3 , and a does not change. (Recall the m, a and q are properties of each Kerr mass.)
Perhaps a better value for r is (6/ π ) (1/3) x
(half-distance between Kerr masses). This value makes the volume of the sphere of radius r equal to the volume of the cube in flat space that should be allocated to each Kerr mass. When we derive torsion as (translational holonomy)/area, this extra factor makes no significant difference because the numerator and denominator are each multiplied by (approximately) the same factor. For the same reason, any adjustment to the value of r to account for the curvature caused by the interaction of Kerr masses also makes no significant difference in this ratio.
Express the quantities computed above in terms of md = m /(4 π /3r 3 ) = density of matter in continuum limit. Use mc = m/r 3 in computations.
a = (angular momentum)/mass ratio, which is unchanged in the continuum limit qd = (4 π /3r 3 ) = density of electric charge.in continuum limit. Use qc = q/r 3 in computations.
In Planck units, 1 kg/m 3 ~ 2*10 -97 . , so assume 0 < mc << 1.
(c80) (assume(mc >0, mc < 1), density_substitutions : [m = mc*r^3, q = qc*r^3])
(d80)
[ m = mc r
3
, q = qc r
3
Express φ in terms of continuum densities and store as a transformation rule.
]
(c81)
(d81)
(phi_expc: subst(density_substitutions, expand(phi_exp)),
apply(defrule, ['phi_expandc, phi, phi_expc]), disprule(phi_expandc)) phi_expandc ( defrule ) : φ→ 1
- - a
2
qc
4
r
6
+ 2 a
2
mc qc
2
r
4
+ qc
2
r
4
+ 2 a
2
qc
2
r
2
- a
2
mc
2
r
2
- 2 mc r
2
- 2 a
2
mc + 1
Express the approximate torsion in terms of mc, a, qc, r, and φ .
(c82)
(d82) cont_holonomy1 : ratsimp(combine(multthru(subst(nu=1-phi,
subst(density_substitutions, apply1(kerr_holonomy2, k1_expand, k2_expand, delta_expand))))), a, r, qc, mc)
2 π a
( qc
2 (
- 2 r
4
- a
2
r
2 )
+ mc
(
3 r
2
+ a
2 ) )
φ - 1
-
( cos ( 2 π π φ
) - 1
) (
- a
2
qc
2
r
2
+ a
2
mc - 1
)
qc
2
r
6
- 2 mc r
4
+ r
2
+ a
2
(
φ - 1 )
2 sin ( 2 π φ )
(
- a
2
qc
2
r
2
+ a
2
mc - 1
)
qc
2
r
6
- 2 mc r
4
+ r
2
+ a
2
(
φ - 1 )
2
0

Derivation of Einstein
–
Cartan theory from general relativity 45
Assume mc r 2 <<1 and a 2 << r 2 , in which case φ ~ mc r 2 ~ 0.
(c83)
(d83) cont_holonomy2 : subst(2*r^2+a^2 = r^2,ratsimp(subst([phi=0, 3*r^2 + a^2=3*r^2], cont_holonomy1), r, qc
))
2 π a r
2 ( qc
2
0
0
0
r
2
- 3 mc
)
We can normally assume qc 2 r 2 << mc: in Planck units, 1 C/m 3 ~ 2 x 10 -87 , 1 kg/m 3 ~ 2 x 10 -97 , and 10 22 ~ r .
The holonomy is proportional to the area of the loop (r 2 ).
(c84) cont_holonomy3 : subst(qc^2*r^2-3*mc=-3*mc, cont_holonomy2)
(d84)
- 6 π a mc r
0
0
0
2
Alte rnate cas e : Assume r 2 << a 2 (so we can take the limit r --> 0). This is not normal in classical physics.
(c85)
(d85) cont_holonomy2x : ratsimp(subst([3*r^2 + a^2=a^2, -2*r^4 - a^2*r^2= -a^2*r^2, phi=0], cont_holonomy1),
r)
2 π a
3 ( qc
2
0
0
0
r
2
- mc
)
Alte rnate cas e : Assume r 2 << a 2 and qc 2 r 2 << mc as above. Holonomy is proportional to a 3 mc, and not to the area of the loop.
(c86) cont_holonomy3x : subst(qc^2*r^2-mc=-mc, cont_holonomy2x)
(d86)
- 2 π a
3
0
0
0
mc
4.2 Torsion in the continuum limit
Compute approximate torsion of a single Kerr mass in the distrbution of Kerr masses as holonomy/( π r 2 ), where π r 2 ~ area of the loop as seen from infinity.
(c87) cont_torsion1 : cont_holonomy3 / (%pi*r^2)
(d87)
- 6 a mc
0
0
0

46 Petti
The definition of mc in terms of m omitted a factor of 4 π /3, so mc = md 4 π /3, where md = mass density.
Similarly, qc = qd 4 π /3, where qd = charge density.
(c88) cont_torsion2 : ratsimp(subst(mc = md *4*%pi/3, cont_torsion1))
(d88)
- 8 π
0
a md
0
0
4.3 Torsion in coordinate frame
Express the holonomy vector in coordinate frames [ d/dt, d/dr, d/d θ , d/d φ ] using frame field FR µ i .
(c89)
(d89) fr_mat : subst([theta=%pi/2, qc=0, r^2+a^2=r^2, -2*mc*r^4 + r^2 + a^2=r^2, a/r=0, a/r^2=1/r],
subst(density_substitutions, apply1(fr, delta_expand, rho_expand))) r
1 0 0 0
0 1 0 0
0 0
1 r
0
1
0 0
1 r
Express the holonomy vector in coordinate frames [ d/dt, d/dr, d/d θ , d/d φ ] using frame field FR µ i . Set a/r to zero.
(c90) coord_torsion1 : limit(fr_mat . cont_torsion2, r, inf)
(d90)
- 8 π a md
0
0
0
The Einstein-Cartan equation relating spin and torsion is:
(c91) torsion = 8*%pi * \k * "spin_density"
(d91) torsion = 8 π K spin_density
The spin density is a 3-index tensor : two (lower) indices specify the plane of rotation, and the third (upper) index specifies the plane in which flux of angular momentum is measured.
The sign of the angular momentum is negative because (a) spin density is an antisymmetric (0,3) tensor; (b) we are computing torsion as a (1,2) tensor with the timelike component contravariant; and (c) the sign of the time-time component of our metric is negative.
Using the volume as seen by an observer at infinity, the r-phi spin density of a Kerr mass for a neighborhood of radius r is:
(c92) kerr_spin_density : [- a*md ; 0; 0; 0]
(d92)
- a md
0
0
0

Derivation of Einstein
–
Cartan theory from general relativity
Using the continuum approximations for torsion and spin density, and K=1, we get :
(c93) coord_torsion1 = 8 * %pi * kerr_spin_density
(d93)
- 8 π
0
a md
0
0
=
- 8 π
0
a md
0
0
47

Derivation of Einstein – Cartan theory from general relativity page 48
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