The Einstein Cartan theory:
the meaning and consequences of torsion
(Teoria Einsteina Cartana: znaczenie i konsekwencje torsji)
Paweł Laskoś-Grabowski
Master s thesis
Advisor: Professor Jerzy Kowalski-Glikman
University of Wrocław
Faculty of Physics and Astronomy
Theoretical physics
August 11, 2009
Abstract
In the following paper we focus on one of the oldest extensions to general rela-
tivity, the Einstein Cartan theory. We explain the physical sense of involved quan-
tities, stressing the meaning of torsion. We discuss the possibilities of measuring
torsion-related phenomena. Finally, we derive and solve the Mathisson Papapetrou
equations in few simple cases and provide explanations of the results.
W poniższej pracy skupimy się na jednym z najstarszych rozszerzeń ogólnej
teorii względności  teorii Einsteina Cartana. Omówimy i wyjaśnimy znaczenie
podstawowych wielkości, podkreślając znaczenie pojęcia torsji. Rozważymy moż-
liwości pomiaru zjawisk związanych z torsją. Na koniec wyprowadzimy równanie
Mathissona Papapetrou, rozwiążemy je w pewnych prostych przypadkach i zinter-
pretujemy otrzymane wyniki.
Contents
1 Introduction 5
2 The Einstein Cartan theory 7
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 From the ECT to traditional general relativity . . . . . . . . . . . . 8
2.3 The geometrical meaning of curvature and torsion . . . . . . . . . . 10
2.4 Measuring curvature and torsion . . . . . . . . . . . . . . . . . . . . 12
2.5 Torsion and the principle of equivalence . . . . . . . . . . . . . . . . 15
3 The Mathisson Papapetrou equation 17
3.1 Derivation of the Mathisson Papapetrou equation . . . . . . . . . . 17
3.2 Flat, torsioned spacetime . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Nonspinning particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Derivation of the equations of motion (a simple case) . . . . . . . . . 21
3.5 Graphical presentation of the results . . . . . . . . . . . . . . . . . . 24
4 Summary 31
Bibliography 33
3
Chapter 1
Introduction
General relativity is a theory that adopts a revolutionary formalism to simplify ex-
planation for physical phenomena, mainly those related to gravitational interaction.
The novelty lies in the idea, that the presence of mass in space curves its geom-
etry and in this way (and not by means of a separate gravitational force) affects
other masses movement. Gravitational force is no longer present in mathematical
description of events.
However, as should be expected, this  simplification actually only relocates
much of the troubles to another part of the formalism. Mathematical structure we
are working in is no longer simple Euclidean (or rather Minkowski) space(-time), but
is instead a manifold  more general object, which only locally may be described
as flat space. Notions such as  the shortest path connecting two points or  parallel
vectors originating in two points lose their natural meaning in this setting. Even
a vector at a point requires now additional care: a  straight arrow cannot belong
to a curved surface of the manifold, so at every point of the manifold we define
a (Euclidean) tangent space of vectors at this point.
If we notice that even partial derivatives with respect to coordinates lose tenso-
rial properties, it is clear that we are in need of a handful of mathematical entities
that will allow us to describe the world correctly and soundly. Among those are con-
nection, covariant derivatives, and curvature tensor (field), which, roughly speaking,
describes (to some extent) geometry of the manifold at given point. A complete set
of definitions, as well as a calculation that gives more intuitive understanding to the
notion of curvature, is presented later on.
We may now recall that also in geometry of curves in three-dimensional space
there is a quantity called  curvature (which actually is the same notion, but applied
to the case of somewhat degenerated, one-dimensional manifolds) and it doesn t
describe local geometry of the curve in full. There is also a  second-order curvature ,
namely torsion, and an entity of this name is also defined for higher-dimensional
manifolds. Among the assumptions of  basic general relativity we find that torsion
5
6 CHAPTER 1. INTRODUCTION
is always set to zero. Yet, there exist interesting extensions of the theory, in which
this constraint is lifted, such as the Einstein Cartan theory (ECT), to which we
turn in the next section.
Physical motivation to study theories with torsion is the fact that basic general
relativity does not include effects related to particle spin, partially because Einstein
published general relativity theory in 1916 and spin was not discovered before 1922
by Stern and Gerlach, described by Pauli in 1924 and incorporated into quantum
theory in 1927 8 by Pauli and Dirac. Nonetheless, as early as 1922 �lie Cartan
published his paper [Cart22], which describes foundations of a theory of spin-
torsion coupling. His input was initially overlooked by physicists, to be  rediscovered
in the 1950s independently by Dennis Sciama and Tom Kibble [Trau06].
After introducing the building blocks of the Einstein Cartan theory, we present
the way in which it encapsulates traditional general relativity. We introduce basic
intuitions on the two key quantities, curvature and torsion, and also discuss theo-
retical possibilities of their measurement. We also elaborate on the relation between
torsion and one of the foundations of all general relativistic theories, the principle
of equivalence.
Our last point of interest is the Mathisson Papapetrou (MP) equation [Math37]
 an equation of motion for a particle with spin in a spacetime with torsion. We
derive the equation and solve it in a chosen simple case to show that results may
differ significantly from those inferred within the regime of classical GR.
Chapter 2
The Einstein Cartan theory
2.1 Preliminaries
The symmetry group of traditional general relativity is the Lorentz group of local
rotations and boosts. In special relativity, however, the group of (global) symmetries
is the full Poincar� group, being a product of the Lorentz group and the translation
group. The Einstein Cartan theory extends this symmetry group to local trans-
formations, so we introduce two one-form gauge fields: a matrix-valued connection
(associated with rotations) and a vector-valued tetrad (associated with translations)
[Meis02]:
ab
�ab = �� dx�, ea = ea dx�. (2.1)
�
Latin indices correspond to the tangent space, Greek indices  to the manifold.
Under local Lorentz transform � connection and curvature transform according
to the following:
� � = ���T - (d�)�T , (2.2)
e e = ��e (2.3)
where � is the flat metric, � = diag(-, +, +, +).
a
Under infinitesimal translation transformation rules are as follow:
� � = � (2.4)
e e = e + d + �� (2.5)
Two-forms of curvature and torsion have the following definitions and transfor-
7
8 CHAPTER 2. THE EINSTEIN CARTAN THEORY
mation laws (under Lorentz transform):
Rab = d�ab + �a '" �cb, (2.6)
c
a
T = dea + �a '" eb, (2.7)
b
R R = ��R��T , (2.8)
T T = ��T. (2.9)
By differentiating definitions (2.6) and (2.7) we obtain two following identities,
first of which is named after Bianchi:
dRab + �a '" Rcb - Ra '" �cb = 0 (2.10)
c c
a b
dT + �a '" T - Ra '" eb = 0. (2.11)
b b
2.2 From the ECT to traditional general relativity
We will now show how the previous section relates to the usual metric formalism of
gravity theories, and how to recover traditional GR from the above notions.
The central object of GR is the metric tensor. It relates to the Einstein Cartan
formalism as follows:
g�� = eaeb �ab, (2.12)
� �
so it may be called  a square of the tetrad field . It is obviously symmetric.
We ll now see that under three assumptions, GR will be recovered from the
Einstein Cartan theory, i.e. the curvature two-form will become equivalent to Rie-
mann curvature tensor. The assuptions are:
1. the tetrad is invertible,
2. the tetrad is covariantly constant,
3. the torsion vanishes.
(Note that the first two are usually taken into account even in non-GRT calculations
in the ECT regime.) The meaning of the third is obvious, the first says that in
computation we may use the symbol e with raised Greek index (the Latin one may
be raised or lowered via � anyway) such that the  matrix ea� is the inverse of
ea�. It also implies invertibility of the metric, which is a basic assumption of GR.
Mathematical formulation of the second assumption reads
a
���ea
"�ea + ��beb - �� = 0 (2.13)
� � �
�
where � is the affine connection (different from the spin connection �), an object
necessary to define parallel transport of a vector on a manifold.
2.3. THE GEOMETRICAL MEANING OF CURVATURE AND TORSION 9
Let us now take a look at the components of torsion form.
a a a
���ea ���ea.
T�� = "�ea - "�ea + ��cec - ��cec = �� - �� (2.14)
� � � � � �
If the left-hand side is to vanish, then the affine connection is symmetric in the lower
indices. Its components are then called the Christoffel symbols and denoted without
�
tilde; in general case the difference � - � is called the defect or contorsion tensor,
in this paper denoted with . One may recall, that this is always assumed in GR.
We ll now multiply the equation (2.13) by ea�:
a
ea�("�ea) + ��beb ea� - �� eaea� = 0, (2.15)
� � �� �
a
ea�("�ea) + ��beb ea� - �� eaea� = 0. (2.16)
� � �� �
In the second equation we have just swapped � "! �. We ll now add them sidewise
using (2.12); the �-term will vanish due to antisymmetry of � in Latin indices.
"�g�� - �� g�� - �� g�� = 0. (2.17)
�� ��
This means that the metric is also covariantly constant  what could be expected,
as its  square root (the tetrad) is constant too. If we perform cyclic substitutions
of the indices,
"�g�� - �� g�� - �� g�� = 0, (2.18)
�� ��
"�g�� - �� g�� - �� g�� = 0, (2.19)
�� ��
then from (2.17)+(2.18)-(2.19) we will easily read the form of the Christoffel sym-
bols:
2�� g�� = "�g�� + "�g�� - "�g��, (2.20)
��
1
�� = g��("�g�� + "�g�� - "�g��), (2.21)
��
2
which is precisely the same obtained in general relativity. Additionally, from
�
"�(g��g��) = "��� = 0 (2.22)
one may find that the  upper index version of (2.17) reads
"�g�� + �� g�� + �� g�� = 0. (2.23)
�� ��
As � is a function of g (and thus of e), from (2.13) we may also express � in
terms of e.
ab
�� = �� eaeb� - eb�"�ea (2.24)
�� � �
Inserting this result into the expression for a component of curvature form,
ab ab ab a cb a cb
R�� = "��� - "��� + ��c�� - ��c�� , (2.25)
yields finally the desired result
ab
e�eb�R�� = -("��� - "��� + �� �� - �� �� ) = R� . (2.26)
a �� �� �� �� �� �� ���
10 CHAPTER 2. THE EINSTEIN CARTAN THEORY
2.3 The geometrical meaning of curvature and torsion
The meaning of curvature is quite easy to grasp. Let our manifold be the surface
of planet Earth and define (quite naturally) parallel transport of a vector along a
section of great circle to preserve the angle between the vector and the circle. Let
us take a vector at point of 0ć% latitude and longitude, pointing northward. If we
now parallel transport it along the equator to 90ć% (say eastern) longitude, then to
the north pole, and then along the 0ć% meridian back to the starting point, it points
to the west (see fig. 2.1). This is precisely the consequence of nonzero curvature 
vectors change direction when parallel transported along closed paths.
Because in general a manifold need not have constant curvature like a sphere,
mathematical formulation must be local and involve infinitesimal displacements. Let
�
us take two infinitesimal four-vectors ��, and consider a vector V at point x�. We
�
transport it to x� +�� + along the sides of a parallelogram, one time via the point
x� +�� (where it is equal to V�) to obtain final value W�, and second time via x� + �
(where it is equal to V ) to obtain W (see fig. 2.2). The following calculation shows
that the difference W - W� is nonzero.
� 
V�� = V - V �� (x)��, (2.27)
�
�  �
V � = V - V �� (x) , (2.28)
�
�
W� = V�� - V��� (x + �) � (2.29)
�
� 
= (V - V �� (x)��)
�
 �
- (V - V � (x)��)(�� (x) + "��� (x)��) � (2.30)
��
� �
�  �  �
H" V - V �� (x)(�� + ) - V ("��� (x) - �� (x)�� (x))�� , (2.31)
� ��
� �
�  �  �
W � H" V - V �� (x)(�� + ) - V ("��� (x) - �� (x)�� (x)) ��. (2.32)
� ��
� �
Because � is not constant, we obtain its values at the points x+�, x+ by the Taylor
expansion around x. The approximation arises because we drop terms of nonlinear
order in �, . Finally we arrive at (where �, R are taken at x)
�

W � - W� = V ("��� - �� �� - "��� + �� �� )�� � (2.33)
� �� � ��
� �
 �
= V R� �� . (2.34)
��
The difference is thus proportional to the initial vector V and the displacements , �,
where the  proportionality constant is the local value of curvature.
To acquire intuition about torsion, consider two (infinitesimal) vectors V, W at
point x�. If we transport each of the vectors along the other, namely obtain W at
� �
x� + V and V at x� + W , the  tips v, w of the two resulting vectors won t meet.
Similarly to the previous case, the difference is proportional to the value of torsion
2.3. THE GEOMETRICAL MEANING OF CURVATURE AND TORSION 11
Figure 2.1: A vector parallel-transported around a path on the Earth s surface
changes its direction. This is the consequence of a nonzero curvature.
Figure 2.2: Due to the curvature, transporting a vector by infinitesimal values of
�, depends on the order of transports.
12 CHAPTER 2. THE EINSTEIN CARTAN THEORY
Figure 2.3: A parallelogram, made up of two vectors and their parallel transports
with respect to each other, fails to close in the presence of torsion, which is a measure
of such failure.
at x (see fig. 2.3).
� �  �
V = V - V �� W , (2.35)
�
� �  �
W = W - W �� V , (2.36)
�
� � � �  �
v� = x� + W + V = x� + W + V - V �� W , (2.37)
�
� � � �  �
w� = x� + V + W = x� + V + W - W �� V , (2.38)
�
�
 �  �
(v - w)� = W V (�� - �� ) = T W V . (2.39)
� � �
One may initially conclude that in a world with torsion paths never close when
expected, and for example one cannot even perform the aforementioned  test for
curvature, as the two paths, along which we transport our test vector, end up in
different points. Actually, if V was transported along transported vector �, and
V�  along transported vector , they would indeed end up in different points
corresponding to v, w from torsion  test . This is clearly not the case. Moreover,
even with vanishing torsion, the tips of V , W meet in v = w, which is different (by
the term proportional to �) from x + V + W , corresponding to x + � + from the
curvature test.
2.4 Measuring curvature and torsion
In principle, virtually any experiment confirming general relativity may be also
treated as a measurement of curvature  because GR is a theory of curvature.
2.4. MEASURING CURVATURE AND TORSION 13
Nevertheless, we will focus on the issue in slightly greater detail.
The first idea is to physically recreate the situation depicted in fig. 2.2, by
sending two gyroscopes along two different paths to one destination. With axes of
rotation initially aligned parallel, the final angle between them would be proportional
to the curvature in a manner presented above. In some cases, where one can pick
a closed geodesic (a line such that parallel transport along it preserves its tangent
vector), like an orbit around Earth or Sun, the experiment may be simplified to
employ only one gyroscope sent along that geodesic.
Another method of curvature measurement relies on a concept of geodesic too.
As stated above, parallel transport along a geodesic doesn t change the vector tan-
gent to it, thus a covariant derivative of the tangent vector in direction of itself is 0,
or
"łł = 0 (2.40)
Ł
Ł
where ł(�) is the parameterisation of the curve and dot indicates derivative with re-
spect to �, which need not have the meaning of proper time. In coordinate language,
the above easily translates to
ł� + �� łł� = 0 (2.41)
� Ł Ł
�
which is the form most commonly called the geodesic equation.
Now we may consider two neighbouring geodesics separated (at some point) by
a vector ��. Because curvature is proportional to a commutator of two covariant
derivatives in the following sense [Wald84]:
R� x� = (""� - "�")x� (2.42)
��
and the relative acceleration of particles following the two neighbouring geodesics,
"ł"ł�� = łł�""���, (2.43)
Ł Ł
Ł Ł
is shown to obey the following [MTW73]:
"ł"ł�� + ["�, "ł]ł = 0 (2.44)
Ł
Ł Ł Ł
we may finally write out the geodesic deviation equation:
"ł"ł�� = -R� ł�ł��. (2.45)
Ł Ł
Ł Ł
��
Because in principle ł, � and the derivative on the left-hand side are measurable, cur-
Ł
vature may be calculated from motion of two particles along neighbouring geodesics.
Repeating the experiment for different pairs of geodesics, one may recover all com-
ponents of the curvature tensor.
There is another, less theoretical, method of measuring the curvature, described
in [MTW73]  a gravity gradiometer. A gradiometer consists of two arms, con-
nected in the middle by a torsional spring, with four equal masses attached at the
14 CHAPTER 2. THE EINSTEIN CARTAN THEORY
ends. Without the presence of other bodies the arms are perpendicular. They re-
main perpendicular, when acceleration is applied to the whole system, or in presence
of a mass distant or extended enough to produce a uniform gravitational field. In
a curved spacetime, however, a net torque is exerted on the spring and measured
by an electronic device. During the measurement, gravimeter rotates with constant
angular frequency; from the amplitude and phase of relative oscillations of its arms
certain components of the curvature tensor may be inferred.
An attempt to devise a method of torsion measurement may start out from
a general matter Lagrangian [Stoe85]:
LM a" LM (g��, "(n)g��, � , "(n)� , �A, "(n)�A), (2.46)
�� ��
where �A are the matter fields in general. As usual, the stress-energy tensor is
defined as
2 �(gLM )
T�� = , (2.47)
g �g��
where g = - det g��, and similiarly a new quantity, the spin density tensor is
introduced:
2 �(gLM )
S� � = . (2.48)
�
g ����
The conservation equation corresponding to the general Lagrangian reads
1 1
"�T�� = "� [� S�]�� - � �S� � + S� �"�� (2.49)
� � � ��
�� 2 2
where " denotes covariant derivative with respect to the Christoffel symbols (and
not the full affine connection). One immediately sees that any influence of the torsion
vanishes either if there is simply no torsion (present here disguised as the contorsion
tensor) or the spin density tensor vanishes (S = 0). In any of these cases,  classical
conservation law is recovered:
"�T�� = 0. (2.50)
A profound conclusion then follows: only particles with spin can be affected by
torsion. Orbital angular momentum, or generally rotational motion of any kind, is
of metric nature and is governed by the Christoffel part of the connection. If so,
in general case the only measurements of torsion may rely on particles with spin.
Two ideas then come: torsion-induced spectral line broadening (as torsion influences
energy of an electron of a radiating atom), and worldline deviation of two sibling
particles of opposing spin (similar to the geodesic deviation described above). Both
are theoretically feasible and practically unrealistic  they either produce effects
far below any presently detectable values, or (in the second case only) would require
a rigid, extended object consisting of particles with aligned spin, like a macroscopic
spin gyroscope.
2.5. TORSION AND THE PRINCIPLE OF EQUIVALENCE 15
In case of specific theories, however, one may use their distinct features to
discover other relations. In the case of the Einstein Cartan theory one of these
features is of course the Poincar� invariance. Thus, another conservation law may
be derived from invariance against coordinate transformations [Hehl71]:
"�T�� + 2T�� �T�� + 2T�� �T�� = S���R����, (2.51)
where, as we see, spin couples to curvature only and even nonspinning particles will
be affected by torsion. (Later on we ll see the same effect in the MP equation.) In
the semiclassical approximation (where p�, u�, s�� are momentum density, velocity
and spin density of a particle, respectively)
T�� = p�u�, (2.52)
S��� = s��u�, (2.53)
�
we arrive at a relation (where P , S�� are total values of momentum and spin of
a particle, respectively, and are defined as volume integrals of p�, s��)
� � �
V + �� P u� + � P u� = R� S��u�. (2.54)
�� �� ���
�
In the case of vanishing torsion, spin and P " u�, this is the geodesic equation.
In general case, however, in principle it allows us to measure all components of the
torsion tensor, similarly to the geodesic deviation equation.
Another result on the ground of the Einstein Cartan theory was shown in
[AdTr75], but with restriction to certain (ie. fully antisymmetric) types of tor-
sion.
2.5 Torsion and the principle of equivalence
Most of the theories of gravity acknowledge the postulate known as the principle of
equivalence, which may be formulated as in [MTW73, p. 207]:
The laws of physics are the same in any local Lorentz frame of curved
spacetime as in a global Lorentz frame of flat spacetime.
The local Lorentz frame at event P is defined as  the closest thing there is to a
global Lorentz frame at that event , so g��(P ) = ���, g��,ą(P ) = 0. Because of
nonvanishing curvature, which contains derivatives of the Christoffel symbols (and
thus second derivatives of g), further derivatives of g cannot be set to zero.
Elsewhere in [MTW73, p. 250] we may read:
Other [not torsion-free] types of covariant derivatives, as studied by
mathematicians, have no relevance for any gravitation theory based on
the equivalence principle.
16 CHAPTER 2. THE EINSTEIN CARTAN THEORY
This statement is not accurately true. Of course, the Lorentz (or, as it is sometimes
called, normal) frame as defined above doesn t exist in the general case, for now the
affine connection doesn t depend only on the metric (or, more specifically, on the
tetrad), but also on the spin connection (2.13). The latter may be nontrivial, ie.
may not be gauged away via Lorentz transformations.
Extensions of the principle of equivalence beyond the GRT have been proposed
since [Heyd75], [HHKN76]. Most general construction, valid not only for nonvan-
ishing torsion, but also nonmetricity Q��� = "�g��, may be found in [Hart95].
The procedure starts from an arbitrary orthonormal basis {Xi}P of the space
tangent to the manifold at event P . One then can, for each i, take geodesic Ci
passing through P , whose tangent vector at P is Xi, and extend them in some
neighbourhood Ui of P . Then there is a neighbourhood U of P such that U ą" Ui,
in which the Ci do not intersect; thus, frame {Xi}P may be unambiguously extended
along all Ci by parallel transport, and then  arbitrarily to the whole U.
�ijXk,
In general, the connection coefficients are defined as "XiXj = �k and
due to the construction "Xj = 0, thus in this frame the connection vanishes at
P . Additionally, the frame is orthonormal, and these two conditions correspond to
� = 0, g = �  properties of Lorentz frames. We may conclude by noting that there
exists a coordinate system x� on M such that at P there is Xi = "xi. Notably,
the connection expressed with respect to x� (instead of Xi) would not vanish, even
at P , when torsion is present. Anyway, this construction allows us to say that the
Einstein Cartan theory is still compatible with the equivalence principle [Ilie96].
Chapter 3
The Mathisson Papapetrou
equation
3.1 Derivation of the Mathisson Papapetrou equation
We will now carefully derive the Mathisson Papapetrou equation. We will work in
a representation that identifies a particle with the element of the connected Poincar�
group:
ę!
P+ = R4 � Lę! = {(z, �)} (3.1)
+
Now z is the four-vector of particle s location and � is related to (what we will call)
momentum and spin in the following way:
pa = m�a0 (3.2)
1
Sab�ab = ��12�-1 a" -iS (3.3)
2
a b a b
where m is particle s mass, (�ab)cd = -i(�c �d - �d�c), and  is a constant.
We will start out with the Lagrangian [BMSS83]:

L = paeaż� + i Tr(�12�-1D� �) + field part. (3.4)
�
2
The  field part will be dropped in our calculations, because it doesn t contain neither
z nor �. The covariant derivative with respect to proper time of the particle acts
here as follows:
ac
Ł
(D� �)a = �a + ż��� �cb (3.5)
b b
First we will perform variation with respect to �. If we express the infinitesimal
change by the (matrix) parameter as
�� = i � �� (3.6)
17
18 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
then we obtain variation of �-1 from the following:
0 = �(��-1) = (��)�-1 + � ��-1 = i � � ��-1 + � ��-1 (3.7)
��-1 = -i�-1 � � (3.8)
We define Jab = eaż�pb and perform variation of the first term:
�
�(paeaż�) = meaż���a0 = mieaża( � �)ab�b (3.9)
� � � 0
= -i( � �)baJab = -i Tr( � � J) (3.10)
The second term gives the following input:

�(. . .) = i �(Tr(�12�-1D� �)) (3.11)
2
 
= i Tr(�12 � -i�-1( � �)D� �) + i Tr(�12�-1D� (i � ��)) (3.12)
2 2

= i Tr(�12 � -i�-1( � �)D� �)
2
 
+ i Tr(�12�-1i(D� ( � �))�) + i Tr(�12�-1i( � �)D� �) (3.13)
2 2
i i
= Tr(i��12�-1D� ( � �)) = Tr(SD� ( � �)) (3.14)
2 2
Putting everything together we obtain the following:
i .
�L = -i Tr(J � �) + Tr(S( � �) + S[ż���, � �]) (3.15)
2
i
a" -i Tr(J � �) + Tr(-` � � + Sż��� � � - ż���S � �) (3.16)
2
1
= -i Tr((J + D� S) � �) (3.17)
2
What we mean by the equivalence sign in the second step, is the equality of integrals
of expressions on both sides. We use it in our derivation because we keep in mind
that we want to use the Hamilton s principle, so the object we are working on is
actually the integral of �L. We supress the integration sign for simplicity; the  a"
notation will be used once more.
The equation of motion associated with � is thus the spin precession equation:
Jab - Jba + (D� S)ab = 0 (3.18)
1
Because � is antisymmetric, it is sound to consider antisymmetric part of J + D� S
2
here.
Variation with respect to z leads to slightly more complicated calculations:

�L = paea�ż� + paż�z�"�ea + i Tr(�12�-1(�ż��� + ż�z�"��)�) (3.19)
� 
2
1
= paea�ż� + paż�z�"�ea + Tr(S(�ż��� + ż�z�"��)) (3.20)
� 
2
. 1
a" -(paea) �z� + paż�z�"�ea + Tr(-`�� - S�� + Sż"��)�z� (3.21)
Ł
� 
2
3.2. FLAT, TORSIONED SPACETIME 19
� �
If we denote with Jab = Jab - Jba, then from (3.18) we see that ` = -J -
[ż�, S]. After inserting this into our expression, we obtain:
�L . 1
�
= -(paea) + paż"�ea + Tr(J�� + [ż�, S]�� - S�� + Sż"��) (3.22)
Ł
� 
�z� 2
Now we will use the following identities:
"�� dz
�� = = "��ż (3.23)
Ł
"z d�
Tr([ż�, S]��) = Tr(żS��� - Sż���) = Tr(żS[��, �]) (3.24)
ba ab ab
�
Tr(J��) = (Jab - Jba)�� = -2Jab�� = -2eżpb�� (3.25)
a
Then our expression looks as follows:
�L .
ab
= -(paea) + paż"�ea - eżpb��
�  a
�z�
1
+ Tr(żS[��, �] - żS"�� + żS"��) (3.26)
2
In the last part we spot the expression for components of the curvature two-form:
R� = "�� - "�� + [��, �] (3.27)
In the first term we apply the expansion:
.
(paea) = Waea + pa
a = (D� pa - ż�ab pb)ea + paż"ea (3.28)
� � �  � �
Applying this, we arrive at our final result:
�L 1
= -eaD� pa + ż Tr(SR�)
�z� � 2
ab
+ ż�ab pbea - paż"ea + paż"�ea - eżpb�� (3.29)
 � �  a
1
a a
= -eaD� pa + ż Tr(SR�) + paż("�ea - "ea + ��beb - �beb ). (3.30)
�  �  �
2
a
The last bracket contains a component T� of the torsion two-form. Thus the
equation of motion is the desired Mathisson Papapetrou equation:
1
a
(D� pa)ea = pażT� + ż Tr SR�� (3.31)
�
2
3.2 Flat, torsioned spacetime
We will now turn to an explicit calculation in a special case of spacetime with
vanishing curvature and nonzero torsion. Let us first try to draw a few conclusions
from the condition
R = d� + � '" � = 0. (3.32)
20 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
From (2.8) we see that if curvature vanishes for some value of connection field,
then it remains zero after any Lorentz transform. Thus any connection that provides
zero curvature can be gauged away, because vanishing spin connection (trivially) pro-
vides zero curvature. Using (2.2) we can infer the general form of all such connection
fields:
0 = � = ���T - (d�)�T (3.33)
���T = (d�)�T (3.34)
�� = d� (3.35)
� = �-1 d� (3.36)
Under this condition the torsion will fulfil the following:
�T = �(de + � '" e) = �(de + �-1 d� '" e) = � de + d� '" e = d(�e) (3.37)
This means that �T is a constant one-form:
d(�T ) = d2(�e) = 0 (3.38)
Finally we conclude that in the flat spacetime the torsion is equal to constant
one-form, left-multiplied by the inverse of an arbitrary local Lorentz transform:
T = �-1 C. (3.39)
3.3 Nonspinning particle
We now turn to the simple case of a particle that has no spin. In this case, the spin
precession equation (3.18) reduces to a simpler form:
Jab - Jba = 0 (3.40)
eaż�pb = eb ż�pa (3.41)
� �
eaż�
pa
�
= (3.42)
eb ż� pb
�
The last form, in which we don t use the Einstein convention, means that the two
four-vectors are colinear, so there exists a function  that fulfils
pa = eaż� (3.43)
�
A simple argument shows that  is actually constant and equal to particle mass
m.
-m2 = �abpapb = 2�abeaeb ż�ż� = 2g��ż�ż� (3.44)
� �
Now the affine parameter may be chosen so that along the worldline of the particle
the norm of ż equals -1. Then indeed  = m.
3.4. DERIVATION OF THE EQUATIONS OF MOTION (A SIMPLE CASE) 21
3.4 Derivation of the equations of motion (a simple case)
First we further simplify the left-hand side of the MP equation under the assumptions
S = 0, pa = eaż� (the constant m factor is omitted, as it can now be cancelled from
�
both sides of the MP equation).
.
(D� pa)ea = (Wa + ż�ab pb)ea = ((ea�ż�) + ż�ab eb�ż�)ea (3.45)
�  �  �
= (ea�z� + 
a�ż� + �ab eb�ż�ż)ea (3.46)
�
 �
= z� + ż�(ż"ea� + �ab eb�ż)ea (3.47)
�
 �
= z� + ż�ż("ea� + �ab eb�)ea. (3.48)
�
 �
The term in the last parentheses is the covariant derivative of a tetrad compo-
nent, Dea�. On the right-hand side of the MP equation we have a term involving
torsion, which is an anti-symmetrized variant of this quantity. The whole term reads
pażTa� = eaż�ż(D�ea - Dea�) = ż�ż(eaD�ea - eaD�ea�) (3.49)
� � 
The indices �,  in the last term were interchanged, so that when both sides are
put together and rearranged, we obtain terms with the indices swapping places in
cyclic manner:
z� = ż�ż(eaD�ea - eaD�ea� - eaDea�). (3.50)
�
�  �
We ll now show that even the simplest nontrivial example yields complicated
equations. Let us take vanishing spin connection (so that covariant derivative is
replaced by partial derivative everywhere) and a diagonal tetrad that differs from
the trivial tetrad only in one component:
i
e0 = 1 + ąz1, e0 = ei = 0, ei = �j (3.51)
0 i 0 j
with constant factor ą, where lower index is curved ( Greek ) and upper is tangent
( Latin ).
We immediately see that for � = 2, 3 the right-hand side vanishes: only the
terms containing "1e0 are nontrivial. In the first term we have "�, in the second 
0
"�e�� and the last vanishes because here also a = � = �. Thus, z2,3 = 0.
�
For � = 1, the second and third terms will vanish for the same reason. The
first, however, will remain:
z1 = ż0ż0e0"1e00 (3.52)
�
0
Conversely for � = 0, the first term vanishes ("0). The remaining two survive
and are of identical form e0"1e00. The index labels are different here, but they are
0
contracted by the żż prefactor, so we may write
z0 = -2ż0ż1e0"1e00 (3.53)
�
0
22 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
We must keep in mind that e00 = -e0 and then
0
z0 = 2ż0ż1(1 + ąz1)ą (3.54)
�
z1 = -ż0ż0(1 + ąz1)ą (3.55)
�
All the complications arise, when we try to lower the indices on ż. In order to do
this, we need to find the form of the metric tensor g�� = eaeb �ab. It is  almost trivial
� �
 diagonal, with only one component different from ą1, namely g00 = -(1 + ąz1)2
(and correspondingly, g00 = -(1 + ąz1)-2). Clearly, ż1 = ż1, but
. .
ż0 = (g0�z�) = (g00z0) = -(1 + ąz1)-2ż0 + 2(1 + ąz1)-3z0ąż1 (3.56)
= -(1 + ąz1)-3(ż0 + ąz1ż0 - 2ąz0ż1). (3.57)
Plugging it into our equations (with z1 = z1) we obtain
z0 = -2ą(1 + ąz1)-2(ż0 + ąz1ż0 - 2ąz0ż1)ż1 (3.58)
�
z1 = -ą(1 + ąz1)-5(ż0 + ąz1ż0 - 2ąz0ż1)2. (3.59)
�
We may try to solve these equations by assuming that ą is small and negligible
in the orders higher than one. This assumption would reduce the system to
z0 = -2ąż0ż1 (3.60)
�
z1 = -ą(ż0)2. (3.61)
�
We now define x = ż0, y = ż1 and multiply first of the equations by x:
1 .
(x2) = �x = -2ąx2y (3.62)
2
Ź = -ąx2. (3.63)
We can now combine these two into one that contains y only:
.
� = -4ąŹy = -2ą(y2) , (3.64)
Ź = -2ąy2 - �. (3.65)
There are four solutions to this equation, depending on the sign of coefficients
ą, �:
1
y = - � = 0; (3.66)
2ą� + ł
�
y = tan( 2ą�� + ł) ą, � < 0; (3.67)
2ą
�
y = - tan( 2ą�� + ł) ą, � > 0; (3.68)
2ą
-�
y = - tanh( -2ą�� + ł) ą > 0 > �; (3.69)
2ą
-�
y = tanh( -2ą�� + ł) � > 0 > ą. (3.70)
2ą
3.4. DERIVATION OF THE EQUATIONS OF MOTION (A SIMPLE CASE) 23
For solutions (3.69), (3.70) we obtain an always-negative function under the
square root in the (following from (3.63)) relation
Ź
x = ą - . (3.71)
ą
For the remaining solutions we obtain the following form of x:
�/ą
x = ą " � = 0, (3.72)

cos( 2ą�� + ł)
"
"
2
x = ą = ą 2y � = 0. (3.73)
2ą� + ł
Finally, we may integrate the functions above, to recover the relation of z0,1 to
proper time. For � = 0 we obtain

1 1
"
z0 = ą ln " + tan( 2ą�� + ł) + C, (3.74)
2ą cos( 2ą�� + ł)
1
z1 = ln cos( 2ą�� + ł) + D, (3.75)
2ą
and for � = 0:
1
"
z0 = ą ln |2ą� + ł| + C, (3.76)
2ą
1
z1 = - ln |2ą� + ł| + D. (3.77)
2ą
The solutions should not diverge in the limit ą 0. In case � = 0 the limit of
both z0,1 is infinite (constant divided by zero) unless ł = ą1. Then the limits exist,
as proven by the l H�pital s rule:
2�
ln |2ą� + ł| 0 2�
2ą� +ł
lim = = lim = , (3.78)
ą0 ą 0 ą0 1 ł
"
z0 ą 2� + C, (3.79)
z1 -ł� + D. (3.80)
In case � = 0 we also start out with posing further conditions on gamma. Again,

the logarithms should be zero when ą = 0:
1 + sin ł
= ą1 and cos ł = ą1 (3.81)
cos ł
what implies ł = 0 (" Ą. Next, we try to apply the l H�pital rule again (where
"
= 2ą�� + ł):
1+sin
ln
0 cos cos2 + sin + sin2 1
cos
"
lim = = lim = � 2�� � (3.82)
ą0 ą 0 ą0 1 + sin cos2 2 ą
� � �/2�
= lim = ą (3.83)
ą0
2ą cos 0
24 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
The way to avoid this infinity is to impose a constraint on �. Independent from �,
it may be a function of ą. Because of the equality that follows from (3.63), (3.65):
�
x2 - 2y2 = (3.84)
ą
which, under ą 0 would fail to hold when � is constant and left-hand side remains
finite, as x, y stay  physical , we may declare
�(0) = 0 (3.85)
�
lim = � (0) = b < " (3.86)
ą0 ą
where prime denotes differentiation with respect to ą. Our limits may now be
evaluated to
1+sin
"
ln
1 � + ą�
cos
lim = lim � 2� � " (3.87)
ą0 ą cos 2 ą�
ą0
"
� � ą 2 b�
= lim + � = (3.88)
ą0 cos ą � cos ł
"
ln | cos | 0 sin � � ą
lim = = lim - � " + � = 2b� tan ł = 0 (3.89)
ą0 ą 0 ą0
cos
2 ą �
"
z0 ą b� + C (3.90)
z1 D (3.91)
We have proven that even in the case of vanishing curvature, particle without
spin and constant torsion one obtains highly complicated equations that may be
solved analytically only under many additional assumptions. Moreover, task of
constructing a tetrad that would produce simple equations of motion (eg. constant
content of the parentheses in (3.50)) seems equally hard.
3.5 Graphical presentation of the results
To demonstrate the results graphically, we employ the Maple computer algebra
system to plot numerical solutions of the full equations (3.58), (3.59). We supress
two spatial dimensions z2,3, because they are governed by simple linear functions of
� and are independent from z0,1.
We present plots depicting worldlines of two particles on the z0,1 plane for
different values of ą. One particle, plotted in black, departs from (0, 0) with velocity
vector (ż0, ż1) = (1, 0) and moves for 10 units of �. From the point it reaches, it
departs with new velocity (0, 1), again moving for 10 units of �. The other particle,
plotted in red, departs from the same initial point, but first with velocity (0, 1).
From the point it reaches in "� = 10, it departs with velocity (1, 0) and moves for
"� = 10.
3.5. GRAPHICAL PRESENTATION OF THE RESULTS 25
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.1: Reference plot for ą = 0.
As such setting is devised to model the  torsion test experiment from section
2.3, the paths fail to close, as expected (obviously except the reference case of ą = 0).
Other cases we have chosen are: ą = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5.
In the last three cases, the value of "� needed to be set to 7, 4 and 1, respectively,
because for bigger values the test particles did reach a singularity of the metric tensor
at the z1 = -ą-1 line. Also worth noticing is the fact that the first segment of the
path of the second particle is a straight line of length "� in all cases  because the
right-hand sides of equations (3.58), (3.59) vanish when z0, ż0 = 0.
26 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.2: Plot for ą = 0.0001.
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.3: Plot for ą = 0.0005.
3.5. GRAPHICAL PRESENTATION OF THE RESULTS 27
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.4: Plot for ą = 0.001.
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.5: Plot for ą = 0.005.
28 CHAPTER 3. THE MATHISSON PAPAPETROU EQUATION
10
8
6
z0
4
2
0
0 2 4 6 8 10
z1
Figure 3.6: Plot for ą = 0.01.
8
6
z0
4
2
0
0 2 4 6
z1
Figure 3.7: Plot for ą = 0.05 ("� = 7).
3.5. GRAPHICAL PRESENTATION OF THE RESULTS 29
4
3
z0
2
1
0
-1 0 1 2 3 4
z1
Figure 3.8: Plot for ą = 0.1 ("� = 4).
1
0,8
0,6
z0
0,4
0,2
0
-0,4 -0,2 0 0,2 0,4 0,6 0,8 1
z1
Figure 3.9: Plot for ą = 0.5 ("� = 1).
Chapter 4
Summary
This paper provides a focus on certain aspects of gravity theories going beyond
general relativity, and yet encompassing the traditional GRT as a special case. What
is perhaps most interesting, we are  at least theoretically  allowed to construct
a theory of gravitational interactions, that does not cross the principle of equivalence
and still extends beyond the commutative geometry.
Noncommutativity, or the fact that combining translations depends on the or-
der of combination, is represented in the Einstein Cartan theory by the quantity
of torsion. Qualitative effect of torsion on geometry is shown in the  failed par-
ellelogram of fig. 2.3. The idea is reproduced in the plot of numerical solution
of equations of motion, for sufficiently small (thus,  approaching the infinitesimal )
values of displacement, see fig. 3.9.
The Einstein Cartan theory turned out to be both interesting and mathemat-
ically complicated. In the Mathisson Papapetrou equation, probably one of the
simplest general equations of motion derived in this regime, the spin surprisingly
couples to the curvature of space, while the torsion affects momentum. Unfortu-
nately, it yields complicated and analytically untractable systems of ordinary differ-
ential equations even when all involved parameters are set to zero or some simple
form.
31
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