Can Gravity Probe B usefully constrain torsion gravity theories?
Can Gravity Probe B usefully constrain torsion gravity theories?
Éanna É. Flanagan, Eran Rosenthal
Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853
(Dated: draft of May 8, 2007; printed October 9, 2018 at 9:18)
In most theories of gravity involving torsion, the source for torsion is the intrinsic spin of matter.
Since the spins of fermions are normally randomly oriented in macroscopic bodies, the amount of
torsion generated by macroscopic bodies is normally negligible. However, in a recent paper, Mao et
al. (gr-qc/0608121) point out that there is a class of theories, including the Hayashi-Shirafuji (1979)
theory, in which the angular momentum of macroscopic spinning bodies generates a significant
amount of torsion. They further argue that, by the principle of action equals reaction, one would
expect the angular momentum of test bodies to couple to a background torsion field, and therefore
the precession of the Gravity Probe B gyroscopes should be affected in these theories by the torsion
generated by the Earth.
We show that in fact the principle of action equals reaction does not apply to these theories,
essentially because the torsion is not an independent dynamical degree of freedom. We examine
in detail a generalization of the Hayashi-Shirafuji theory suggested by Mao et al. called Einstein-
Hayashi-Shirafuji theory. There are a variety of different versions of this theory, depending on the
precise form of the coupling to matter chosen for the torsion. We show that for any coupling to
matter that is compatible with the spin transport equation postulated by Mao et al., the theory has
either ghosts or an ill-posed initial value formulation. These theoretical problems can be avoided by
specializing the parameters of the theory and in addition choosing the standard minimal coupling
to matter of the torsion tensor. This yields a consistent theory, but one in which the action equals
reaction principle is violated, and in which the angular momentum of the gyroscopes does not
couple to the Earth’s torsion field. Thus, the Einstein-Hayashi-Shirafuji theory does not predict a
detectable torsion signal for Gravity Probe B. There may be other torsion theories which do.
I. INTRODUCTION AND SUMMARY
A. Theories of gravity with torsion
General relativity (GR) is in good agreement with all
current experimental data from laboratory tests, the So-
lar System1 and binary pulsars [2]. However there is good
motivation to consider modifications and extensions of
GR: low energy limits of string theory and higher dimen-
sional models usually involve extra, long range universal
forces mediated by scalar fields, and in addition the ob-
served acceleration of the Universe may be due to a mod-
ification of GR at large distances. One may hope that
new and highly accurate experiments, such as Gravity
Probe B [3], will enable one to test for deviations from
One natural framework in which to generalize GR is
to allow the connection Γ
to be a nonsymmetric inde-
pendent dynamical variable instead of being determined
by the metric. The covariant derivative of a vector vµ is
defined in the usual way as
∇µvν = ∂µvν + Γνµλvλ. (1.1)
If one retains the assumption that the connection is met-
ric compatible, ∇µgνλ = 0, then it can be shown that the
connection is determined uniquely by the metric and the
1 An exception is the Pioneer anomaly [1], which remains contro-
versial.
torsion tensor
S λµν ≡ Γλ[µν]. (1.2)
One obtains
Γλµν =
−K λµν (1.3)
where the first term is the Levi-Civita connection deter-
mined by the metric and
K λµν = −S λµν − Sλνµ − Sλµν (1.4)
is called the contorsion tensor. A spacetime equipped
with a metric gµν and a torsion tensor is called a
Riemann-Cartan spacetime. The Riemann tensor R
of the full connection (1.3) is related to the usual Rie-
mann tensor R̃
of the Levi-Civita connection by
+ ∇̃ρK µλν − ∇̃λK
ρν +K
K σρν
−K µρσ K σλν , (1.5)
where ∇̃µ is the Levi-Civita derivative operator, and our
convention for the Riemann tensor is given by Eq. (B5).
The action for theories of gravity in this framework has
the generic form
S[gµν ,Γ
µν ,Ψ] = SG[gµν ,Γ
µν ] + Smatter[gµν ,Γ
µν ,Ψ] ,
(1.6)
where SG and Smatter are the gravitational and matter ac-
tions and Ψ collectively denotes the matter fields. There
is an extensive literature on theories of gravity of this
type; see the review articles [4, 5, 6, 7].
http://arxiv.org/abs/0704.1447v2
http://arxiv.org/abs/gr-qc/0608121
It is often useful to re-express these theories using the
tetrad formalism [8]. In this formalism the independent
variables are taken to be a tetrad of four linearly in-
dependent vector fields e µa (x), and a tetrad connection
ω abµ = −ω baµ defined by
~∇~ea~eb = e µa ω cµ b ~ec. (1.7)
Here tetrad indices a run from 0 to 3 and are
raised and lowered using the Minkowski metric ηab ≡
diag(−1, 1, 1, 1). The two sets of variables, gµν ,Γλµν and
e µa , ω
µ are related by
gµν = ηabe
ν , (1.8a)
Γλµν = e
eaν,µ + ω
, (1.8b)
where the dual basis of one-forms eaµ is defined by
e µa e
µ = δ
a. (1.9)
The action of the theory in terms of the tetrad variables
S[e µa , ω
µ ,Ψ] = SG[e
a , ω
µ ] + Smatter[e
a , ω
µ ,Ψ].
(1.10)
Normally the theory is invariant under local Lorentz
transformations Λ ba = Λ
a (x) of the tetrad
e µa → Λ ba e
, (1.11a)
ω aµ b → Λac Λ db ω cµ d − Λac,µΛ cb , (1.11b)
together with the corresponding transformations of any
fermionic matter fields.
There are three different categories of theories involv-
ing torsion:
• Theories in which torsion is an independent dynam-
ical variable and the field equations for torsion are
algebraic, for example the Einstein-Cartan theory
[4] in which the gravitational action is proportional
to the Ricci scalar. In these theories the torsion
vanishes in vacuum.
• Theories in which the torsion tensor is an indepen-
dent dynamical variable and a propagating degree
of freedom, for example Refs. [4, 6, 7, 9, 10, 11, 12].
The source for torsion is the tensor
δSmatter
δω abµ
For the standard, minimal coupling of torsion to
matter, this tensor is a measure of density of funda-
mental or intrinsic spin, and thus is very small when
averaged over macroscopic distances in unpolarized
matter [4, 6]. (For non-standard couplings it is con-
ceivable that this tensor could be non-negligible.)
2 The connection (1.8b) is automatically metric compatible by
virtue of the antisymmetry of ω abµ on a and b.
• Theories in which the torsion is not an indepen-
dent dynamical variable, but is specified in terms
of some other degrees of freedom in the theory, for
example a scalar potential [13, 14] or a rank 2 ten-
sor potential [7, 15, 16].
A particular special case of the third category are the
so-called teleparallel theories [17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32]. In these theories the
only dynamical variable is the tetrad e µa , the tetrad con-
nection ω abµ is not an independent variable. In addition
the local Lorentz transformations (1.11a) are not a sym-
metry of the theory. The tetrad therefore contains 6 extra
physical degrees of freedom which are normally gauged
away by the local Lorentz symmetry. In linear perturba-
tion theory about flat spacetime these extra degrees of
freedom act like a antisymmetric, rank 2 tensor poten-
tial for the torsion; see Sec. II below for more details. In
teleparallel theories the torsion is defined to be
S λµν =
e λa (e
ν,µ − eaµ,ν) . (1.12)
The form of this equation is invariant under coordinate
transformations but not under the local Lorentz transfor-
mations (1.11a). It follows from this definition and from
Eq. (1.8b) that the tetrad connection ω abµ vanishes, and
it follows that the Riemann tensor
Rabµν ≡ eaλebσRλσµν
= ω abν ,µ − ω abµ ,ν + ω aµ c ω cbν − ω aν c ω cbµ (1.13)
also vanishes. Thus in teleparallel theories the curvature
(1.5) of the full connection vanishes, and so on the right
hand side of Eq. (1.5) the contorsion terms must can-
cel the curvature term. Hence, if the spacetime metric
is close to that predicted by general relativity, so that
R̃ ∼ M/r3 at a distance r from a mass M , then the
torsion must be of order S ∼ K ∼ M/r2. Therefore,
teleparallel theories generically predict a non-negligible
torsion for macroscopic, unpolarized bodies, unlike con-
ventional torsion theories.
B. Constraining torsion with Gravity Probe B
The prevailing lore about torsion theories has been
that they are very difficult to distinguish from general rel-
ativity, since the torsion generated by macroscopic bod-
ies is normally negligibly small for the reasons discussed
above [7, 33]. However, a recent paper by Mao, Tegmark,
Guth and Cabi (MTGC) [34] points out that teleparal-
lel theories are an exception in this regard. They sug-
gest that Gravity Probe B (GPB) might be an ideal tool
to probe such torsion theories. In particular they argue
that since the angular momentum of macroscopic bod-
ies generates torsion, one would expect that the angular
momentum of test bodies such as the GPB gyroscopes
would couple to the Earth’s torsion field, by the princi-
ple of “action equals reaction”.
MTGC also review the literature on the equations of
motion and spin precession of test bodies in torsion the-
ories [14, 16, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. They
argue that because there is some disagreement in this lit-
erature, and because the precise form of the coupling of
torsion to matter is not known, it is reasonable to assume
that test bodies fall along geodesics of the full connection
(called autoparallels), and to assume that the spin of a
gyroscope is parallel transported with respect to the full
connection. They introduce a theory of gravity called
the Einstein-Hayashi-Shirafuji (EHS) theory, a general-
ization of an earlier teleparallel theory of Hayashi and
Shirafuji [18], and compute the constraints that GPB
will be able to place on this theory for their assumed
equations of motion and spin transport.
In this paper we re-examine the utility of GPB as a
probe of torsion gravity theories. We agree with the
general philosophy expressed by MTGC that the pre-
cise form of the coupling of torsion to matter is some-
thing that should be tested experimentally rather than
assumed a priori. However, while it is conceivable that
there could exist couplings which would predict a de-
tectable torsion signal for GPB, we show that teleparallel
theories and the EHS theory in particular do not.
We start by discussing the action equals reaction prin-
ciple. This appears to be a robust and very generic ar-
gument, indicating that the angular momentum of a test
body should couple to torsion in theories where spinning
bodies generate torsion. However, in fact there is a loop-
hole in the argument, and in particular it does not apply
to the EHS theory, as we show in detail in Sec. IV be-
low. The nature of the loophole can be understood using
a simple model. Consider in Minkowski spacetime the
following theory of two scalar fields Φ1 and Φ2 and a
particle of mass m
S = −
(∇Φ1)2 + (∇Φ2)2
dλ(m+ qΦ1)
. (1.14)
Here λ is a parameter along the worldline and q is a scalar
charge. In this theory the particle generates a Φ1 field but
not a Φ2 field, and correspondingly it feels a force from
the Φ1 field but not the Φ2 field, in accordance with the
action equals reaction idea. However, consider now the
theory in non-canonical variables Φ̃1 = Φ1, Φ̃2 = Φ1+Φ2.
In terms of these variables the particle generates both a
Φ̃1 field and a Φ̃2 field, but feels a force only from the
Φ̃1 field, in violation of action equals reaction. Thus, we
see that the action equals reaction principle can only be
applied to the independent dynamical variables in the
theory, which diagonalize the kinetic energy term in the
action. In the EHS theory, the torsion and metric are
not independent dynamical variables; see Sec. IV below.
C. The Einstein-Hayashi-Shirafuji theory
In the remainder of this paper we examine in detail
the EHS theory suggested by MTGC. In this theory the
defining relation (1.12) between torsion and tetrad for
teleparallel theories is replaced by
S λµν =
e λa (e
ν,µ − eaµ,ν) , (1.15)
where parameter σ lies in the range 0 ≤ σ ≤ 1. For σ = 1
this reduces to the teleparallel case, while for σ = 0 the
torsion tensor vanishes. Thus, the EHS theory interpo-
lates between GR at σ = 0, in which the torsion van-
ishes but the Riemann tensor is in general different from
zero, and the Hayashi-Shirafuji teleparallel theory[18] at
σ = 1, in which the Riemann tensor vanishes but the
torsion tensor is in general different from zero.
The only dynamical variable in the gravitational sector
of this theory is the tetrad e µa , and, as for the teleparallel
theories, the theory is generally covariant but not invari-
ant under the local Lorentz transformations (1.11a) of
the tetrad. The action for the theory can be written as
S[e µa ,Ψ] = SG[e
a ] + Smatter[e
a , ω
µ ,Ψ], (1.16)
where Ψ denotes the matter fields and, in the second
term, ω abµ denotes the tetrad connection obtained from
the torsion tensor (1.15) via Eqs. (1.3), (1.4) and (1.8b).
The matter action Smatter is not specified by MTGC, so
there are different versions of the EHS theory depend-
ing on the form of the coupling to torsion chosen in this
matter action. The gravitational action is given by
a ] =
a1tµνλt
µνλ + a2vνv
ν + a3aνa
(1.17)
Here a1, a2 and a3 are free parameters with dimensions
of mass squared (we use units with ~ = c = 1), and the
tensors tµνλ, v
ν and aν are defined to be the irreducible
pieces of the torsion tensor (1.15), but with the factor of
σ removed:
vµ = σ
−1S λµλ , (1.18a)
σ−1ǫµνρσS
σρν , (1.18b)
tλµν = σ
−1Sν(µλ) +
(gνλvµ + gνµvλ)−
gλµvν .(1.18c)
Also in Eq. (1.17) g denotes the metric determinant,
where gµν is given in terms of the tetrad by Eq. (1.8a).
For fixed a1, a2 and a3, this gravitational action is in-
dependent of the parameter σ; this parameter enters the
theory only through the dependence of the matter action
on the torsion tensor (1.15).3
3 The parameters defining the theory are therefore (a1, a2, a3, σ).
MTGC use a different set of parameters (c1, c2, c3, κ, σ) which
are not all independent. The relation between the two sets of
parameters can be derived using the identity of Appendix B and
is a1 = σ
2c1 − 4/(3κ), a2 = σ
2c2 + 4/(3κ), a3 = σ
2c3 − 3/κ.
In Appendix B we show that the gravitational action
can be rewritten in terms of the Ricci scalar R({}) of the
Levi-Civita connection as
a ] =
−g [d1R({}) + 4d2vνvν + 9d3aνaν ] ,
(1.19)
where d1 = −3a1/8, d2 = (a1 + a2)/4, d3 = a3/9− a1/4.
We shall refer to the three-dimensional space parameter-
ized by (d1, d2, d3) as the gravitational-action parameter
space.
As mentioned above, there are different versions of the
EHS theory, depending on the form of the matter action
Smatter chosen; MTGC do not specify a matter action.
Consider now what is required in order to predict the
signal seen by GPB. The experiment consists of an Earth-
orbiting satellite carrying four very stable gyroscopes,
and the measured quantity is the time dependence of the
angles between the spins of the gyroscopes and the direc-
tion to a fixed guide star. To compute this quantity in
an arbitrary Riemann-Cartan spacetime, it is sufficient
to know the equations of motion and of spin transport
for a spinning point particle4. These equations can be
computed in principle for any matter action. MTGC as-
sume that the matter action is such that the trajectory
of the spinning point particle is either an autoparallel (a
geodesic of the full connection) or an extremal (a geodesic
of the metric), and that its spin is parallel transported
with respect to the full connection.
D. Requirements necessary to ensure physical
viability of the theory
In this paper we constrain the parameter space of the
EHS theory by imposing a set of physical requirements.
To simplify the analysis we first linearize the EHS theory
with respect to a flat torsion-free spacetime, and then im-
pose physical requirements on the linearized theory. The
linearized theory is completely characterized by two ten-
sor fields: a symmetric field hµν , and an antisymmetric
field aµν . In terms of these fields the torsion tensor is
given by S λµν = σ(h
[ν,µ]
− 2aλ
[ν,µ]
)/2, and the metric is
given by gµν = ηµν + hµν .
We impose three types of requirements. First, we re-
quire that the theory have no ghosts, i.e., that the Hamil-
tonian of the theory be bounded from below. This re-
quirement rules out most of the three-dimensional pa-
rameter space of the gravitational action SG. The re-
maining viable subdomain of the parameter space con-
sists of two intersecting two-dimensional planes.
Second, we further require that the theory have a well
posed initial value formulation. This means that if the
4 One also needs to know the trajectories of photons, but these are
determined by gauge invariance to be just the null geodesics of
the metric, as in GR [4].
physical degrees of freedom are specified on an initial
spacelike hypersurface, the future evolution of these de-
grees of freedom is uniquely determined. Now, for many
theories some of the degrees of freedom are nonphysical
and are associated with a gauge symmetry. For exam-
ple, in classical electrodynamics the field equations for
the vector potential Aµ are invariant under the gauge
transformation Aµ → Aµ + ϕ,µ. These field equations
therefore do not predict a unique evolution for the vector
potential, and correspondingly consist of a set of under-
determined partial differential equations. Nevertheless,
classical electrodynamics has a well posed initial value
formulation, because the degrees of freedom whose evo-
lution cannot be predicted are pure gauge.
A similar situation arises in linearized EHS theory.
There, the gravitational action is invariant under certain
symmetries of the dynamical variables (not diffeomor-
phisms), and correspondingly the field equations form a
set of an underdetermined partial differential equations.
Therefore the theory can have a well posed initial value
formulation only if the undetermined degrees of freedom
are pure gauge. This can be the case only if the matter
action is also invariant under the symmetries. However,
the equations of motion and spin precession postulated
by MTGC do not respect these symmetries, and they
should inherit such a property from the matter action.
We conclude therefore that the initial value formulation
is ill-posed. This argument applies in most of the re-
maining portion of the parameter space of the theory.
The subdomain that is not excluded by this argument
and by the requirement of no ghosts consists of a single
line in the three dimensional space.
Third, in this remaining subdomain, the linearized
gravitational action reduces to that of GR; it depends
on the tetrad only through the metric gµν . In particu-
lar this means that the torsion tensor is completely un-
determined in vacuum, and so for a generic matter ac-
tion, the motion of test bodies cannot be predicted. The
corresponding inconsistency of the Hayashi-Shirafuji the-
ory in this limit has been previously discussed in Refs.
[18, 23, 25, 26, 27, 29, 30, 31]. This inconsistency is
avoided if one constructs a special matter action in which
the unpredictability of the torsion tensor is associated
with a gauge symmetry of the theory, in this case the
predications of the EHS theory coincide with the predic-
tions of GR.
Finally, our argument that the initial value formulation
is ill posed can be evaded by modifying the coupling of
the torsion tensor to matter in the theory. Rather than
postulating the equations of motion and spin precession
used by MTGC, we instead assume that the coupling
of torsion to matter is the standard, minimal coupling
described in Refs. [4, 6]. For this coupling, the matter
action is invariant under the symmetries of the gravita-
tional action discussed above, in a portion of the parame-
ter space, and so the theory has a well posed initial value
formulation in which the undetermined degrees of free-
dom are interpreted as gauge degrees of freedom. This
Autoparallel Extremal Standard Matter Coupling
Sector of parameter space
D0 = {d2 6= 0, d3 6= 0} Ghosts Ghosts Ghosts
D1 = {d2 6= 0, d3 = 0} I.V.F. I.V.F. I.V.F.
D2 = {d2 = 0, d3 6= 0} I.V.F. I.V.F. Consistent but no GPB torsion signal
(action=reaction violated)
D3 = {d2 = 0, d3 = 0} Inconsistent Inconsistent/GR Inconsistent
TABLE I: A summary of the status of the Einstein-Hayashi-Shirafuji theory [34] in different sectors of its parameter space.
The rows of the table are these different sectors; the parameters d2 and d3, which appear in the gravitational part of the
action, are defined in Eq. (1.19) in the text. There are different versions of the Einstein-Hayashi-Shirafuji theory depending on
the precise form chosen of the coupling of the torsion tensor to matter fields. These different versions are the columns of the
table. “Autoparallel” means that it is assumed that the matter coupling is such that freely falling bodies move on geodesics of
the full connection, while “Extremal” means they move on geodesics of the Levi-Civita connection determined by the metric.
These were the two cases considered by Mao et al. [34]. “Standard Matter Coupling” means the standard, minimal coupling
of torsion to matter fields [4, 6], which in general gives rise to motions of test bodies that is neither autoparallel nor extremal.
The meanings of the various entries in the table are as follows. “Ghosts” means that some of the degrees of freedom in the
theory are ghostlike at short distances, signaling an instability that rules out the theory. “I.V.F.” means that the theory does
not have a well posed initial value formulation, and so is ruled out. “Inconsistent” means that the theory does not predict
the value of the torsion tensor, so the motion of test bodies cannot be predicted, while “GR” means that the theory reduces
to general relativity. Finally “no GPB torsion signal” means that there is no torsion-induced coupling between the Earth’s
angular momentum and that of the Gravity Probe B gyroscopes; there is only a coupling between the fundamental spins of the
Earth’s fermions and of those in the gyroscopes, which gives a negligible signal as those spins are randomly oriented.
is the interpretation suggested in the original paper by
Hayashi and Shirafuji [18]. For this case, we again ex-
amine the linearized theory for the fields hµν and aµν .
We find that hµν satisfies the same equation as the met-
ric perturbation in GR, while aµν satisfies a wave equa-
tion (with a suitable choice of gauge) whose source is
obtained from the intrinsic spin density of matter. As
discussed earlier, this implies that for a macroscopic ob-
ject for which the spins of the elementary particles are
not correlated over macroscopic scales, aµν will be neg-
ligible. Hence the spacetime of the linearized theory is
completely characterized by the metric alone, and so its
predictions coincide with those of GR5 and there will be
no extra signal in GPB.
A summary of the status of the EHS theory in various
different cases discussed above is given in Table I.
To summarize, there are no cases in which the EHS
theory gives a detectable torsion signal in GPB. However,
it is nevertheless possible that other torsion theories in
the other categories discussed in Sec. I A, with suitable
choice of matter coupling, could predict a detectable sig-
nal. Various possibilities for non-minimal couplings are
discussed by Shapiro [6]. It would be interesting to find
a torsion theory that predicts a detectable torsion signal
for GPB; such a theory would be an example to which
the theory-independent framework developed by MTGC
5 This is despite the fact that the torsion tensor is generically
nonzero. The torsion tensor is not an independent degree of
freedom in this limit, it is given in terms of the metric by the
first term in Eq. (2.3b).
(a generalization of the parameterized post-Newtonian
framework to include torsion) could be applied.
E. Organization of this paper
This paper is organized as follows. In Sec. II we derive
the dynamical variables and the action of the linearized
EHS theory. In Sec. II B we study the action for the an-
tisymmetric field aµν , temporarily setting the symmetric
field hµν to zero. We show that this theory has ghosts
on a subdomain of the parameter space. Section II E and
Appendix A extend this result to the complete linearized
theory, including the symmetric field hµν , thereby ruling
out a subdomain of the parameters space. We then focus
on the complementary subdomain and show that it is in-
variant under certain symmetries. Section III reviews the
necessary requirements for a well posed initial-value for-
mulation. In Sec. III A we use these requirements to rule
out a subdomain of the parameter space that has an ill-
posed initial value formulation. The remaining portion
of the parameter space is discussed in Sec. II F
Finally, section IV considers the EHS theory with the
standard matter-torsion coupling. In a certain portion of
parameter space this theory has a well-posed initial value
formulation and no ghosts, but we show that the devi-
ations of its predictions from those of GR are negligible
for unpolarized macroscopic bodies. Final conclusions
are given in Sec. V.
II. LINEARIZATION ABOUT FLAT,
TORSION-FREE SPACETIME OF THE
EINSTEIN-HAYASHI-SHIRAFUJI THEORY
A. Action and variables of linearized theory
To linearize the EHS theory we first decompose the
tetrads e µa and dual one-forms e
µ into background
tetrads and one-forms and perturbations:
e µa = b
a + δc
a , (2.1a)
eaµ = b
µ + δe
µ. (2.1b)
We assume that the background tetrads b µa are constants
for which the metric (1.8a) is the Minkowski metric,
gαβ = ηαβ , and for which the torsion (1.15) is vanishing,
S λµν = 0. Thus, to zeroth order, the spacetime is flat
and torsion-free. Throughout this paper we will work to
leading order in the tetrad perturbations. Hereafter, un-
less we explicitly state otherwise, Greek indices are raised
and lowered with ηµν and Latin indices with ηab.
From the definition (1.9) of the dual basis applied to
both the full tetrads and the background tetrads, we find
that δeaµb
a = −δc νa baµ. Thus we can take δeaµ to be
the fundamental variable of the theory. We next convert
this quantity into a spacetime rank 2 tensor using the
background tetrad, and take the independent symmetric
and antisymmetric pieces. This yields the definitions
hµν ≡ 2δeb(µb
ν)ηab, (2.2a)
aµν ≡ δeb[µb
ν]ηab. (2.2b)
The formulae (1.8a) and (1.15) for the metric and torsion
now yield
gµν = ηµν + hµν , (2.3a)
S λµν =
hλ[ν,µ] − 2a
[ν,µ]
. (2.3b)
The linearized gravitational action SlinearG can now be ob-
tained by substituting the expressions (2.3) for the met-
ric and torsion into the action (1.19) and expanding to
quadratic order in hµν and aµν . The resulting action can
be written schematically in the form
SlinearG = SS [hµν ] + SC [hµν , aαβ ] + SA[aµν ] , (2.4)
where SS is a quadratic in the symmetric tensor hµν ,
SA is quadratic in the antisymmetric tensor aµν , and SC
contains the cross terms.
B. The antisymmetric term in the action
We now focus on the antisymmetric term SA in the ac-
tion, ignoring for the moment the other two terms. For
this theory we derive two kinds of results. First, we con-
strain the parameter space (d1, d2, d3) by imposing the
requirement of no ghosts, and second we derive symme-
try transformations under which the action SA (with spe-
cific parameters) is invariant. Later in Sec. II E we will
extend some of these results to the complete linearized
action SlinearG .
The antisymmetric action SA[aµν ] is constructed from
the antisymmetric field aµν in Minkowski spacetime, and
consists of terms that are product of derivatives of aµν ,
of the form (∂λaµν)
2. Actions of this type also arise in
gravitational theories with a non-symmetric metric and
have been extensively studied. See Refs. [45, 46] for a
discussion of the existence of ghosts in theories of this
type.
There are only three linearly independent terms of the
form (∂λaµν)
2, namely
aµ ,σσ , aµν,λa
µν,λ , aµν,λa
µλ,ν .
From these terms one can construct only two functionally
independent actions, since the identity
a ,σµσ − aµλ,σaµσ,λ = ∂λ(aµλa ,σµσ − aµλ,σaµσ) (2.5)
shows that a linear combination of the terms is a diver-
gence which can be converted to a surface term upon
integrating and thereby discarded. Therefore, the most
general action of this type can be written as
d4x[d2a
aµσ,σ + d3a
a∗µσ,σ] , (2.6)
where a∗µν ≡ −ǫµνρσaρσ/2 (here ǫµνρσ is the Levi Civita
tensor of a flat spacetime), and d2,d3 are free parameters.
Indeed an explicit calculation using Eqs. (1.18), (1.19)
and (2.3) shows that SA is given by the expression (2.6),
where the parameters d2 and d3 are those defined after
Eq. (1.19). Next, we specialize to a particular Lorentz
frame, and rewrite the action in terms of the vectors E
and B defined by Ei = a0i and Bi =
ǫijkajk, where
i, j, k run from 1 to 3. This gives
aµσ,σ = (Ė+∇ ×B)2 − (∇ ·E)2 , (2.7)
where an overdot denotes differentiation with respect to
time. The corresponding expression for a
a∗µσ,σ has
the same form, except for the substitutions E → B and
B → −E.
We now consider the case where both d2 and d3 are
nonzero. In this case Hamiltonian density corresponding
to the action (2.6) takes the form
(πB + 2d3∇×E)2 + d3(∇·B)2 − d3(∇×E)2
(πE − 2d2∇×B)2 + d2(∇·E)2 − d2(∇×B)2 ,
(2.8)
where πE and πB denote the momenta conjugate to E
and B.
C. Ghosts
We now constrain the values of the coefficients d2 and
d3 by demanding that
3x be bounded from below.
Let us start by considering the coefficient d3, and sup-
pose first that d3 < 0. At a given point in space, keeping
the values of E, B and πE fixed, we can make HA arbi-
trarily negative by choosing πB to be arbitrarily large.
The same is clearly true for the integrated Hamiltonian
3x. Next suppose that d3 > 0. By fixing B, ∇ ·E,
πE and adjusting the value of πB to keep the value of
πB + 2d3∇×E fixed, we can make HA arbitrarily neg-
ative, this time by choosing an arbitrarily large value for
∇×E. By applying analogous considerations to d2 we
reach the conclusion that the theory defined by HA has
ghosts on the domain
D0 = {(d1, d2, d3)| d2 6= 0, d3 6= 0} (2.9)
in parameter space. Similar analyses can be found in
Refs. [45, 46] for non-symmetric gravity theories, and in
Ref. [47] for teleparallel gravity theories.
D. Symmetries
We now consider the domain in parameter space not
excluded by the above analysis, which consists of the two
2D regions
D1 = {(d1, d2, d3)| d2 6= 0, d3 = 0} , (2.10a)
D2 = {(d1, d2, d3)| d2 = 0, d3 6= 0} , (2.10b)
together with the line
D3 = {(d1, d2, d3)| d2 = 0, d3 = 0} . (2.10c)
The antisymmetric action SA vanishes identically on D3,
so in this subsection we will not consider D3 any further.
We are interested in the symmetries of the antisym-
metric action SA on the domains D1 and D2. From the
formula (2.6) for the action we see that these two do-
mains are isomorphic to one another under the duality
transformation
aµν → a∗µν or E → B, B → −E. (2.11)
Therefore it is sufficient to focus on one of the domains,
say D1. From Eqs. (2.6) and (2.7), the antisymmetric
action on this domain is
SA |D1 =
d2[(Ė+∇×B)2 − (∇ · E)2] d4x . (2.12)
Consider now the initial value problem for E and B.
Suppose that we are given sufficient initial data on some
constant time hypersurface, and that we wish to deter-
mine the time evolution of E(t) and B(t) using the action
(2.12). Note that this action is independent of the longi-
tudinal part of B, which means that the evolution of this
longitudinal part can be prescribed arbitrarily, indepen-
dent of the initial data. Therefore, the field equations for
E and B must form a set of an underdetermined partial
differential equations. As discussed in the introduction,
this can only be consistent if the undetermined degrees
of freedom can be interpreted as being gauge degrees of
freedom.
The action (2.12) is invariant under the symmetry
aµν → aµν + ǫµναβχα,β , (2.13)
where χα(x) is an arbitrary vector field. This can be seen
from the fact that the action (2.12) is given by the first
term in Eq. (2.6), which depends on aµν only through its
divergence a ,νµν . Similarly on the domain D2 the anti-
symmetric action (2.6) is invariant under the symmetry
aµν → aµν + χ[µ,ν]. (2.14)
These are the symmetries that are responsible for the in-
determinacy in the evolution equations. We will study
in later sections the conditions under which these sym-
metries can be interpreted as gauge symmetries, thus al-
lowing the theory to have a well posed initial value for-
mulation. As discussed in the introduction, the gauge
symmetry interpretation requires the matter action to
be invariant6 under the symmetries (2.13) and (2.14).
E. The complete linearized action
Up to now we have ignored the pieces SS and SC of
the complete linearized action (2.4), and have studied
a reduced theory depending only on the antisymmetric
field aµν described by the action SA alone. We showed
that this reduced theory has ghosts if both d2 and d3
are nonzero, i.e., on the domain D0. In Appendix A this
result is generalized to the complete linearized theory, in-
cluding the symmetric field hµν , showing that the com-
plete theory also has ghosts in the domainD0. Essentially
we show that that whenever the Hamiltonian
3x is
unbounded from below, then the corresponding Hamilto-
nian of the full linearized theory is also unbounded from
below.
The symmetries (2.13) and (2.14) of the reduced the-
ory on the domains D1 and D2 also generalize to the
complete linearized theory. This can be seen as follows.
Since the symmetries only involve the antisymmetric field
aµν , the only additional term in the complete action (2.4)
whose invariance needs to be checked is the cross term
6 A general discussion of the problems that arise when the gravi-
tational action is invariant under a symmetry not shared by the
matter action can be found in Leclerc [31].
SC [hµν , aρσ]. This cross term can be written as
hαβ,γaµν,ρP
αβγµνρ d4x ,
where Pαβγµνρ is a tensor constructed from the
Minkowski metric. Integrating by parts and discarding a
surface term yields
SC = −
hαβaµν,ργP
αβγµνρ d4x .
At least two of the indices on aµν,ργ must be contracted
with one another, and since aµν is antisymmetric it fol-
lows that only divergence terms of the form a σµσ, γ can
appear. These divergence terms are invariant under the
symmetry (2.13).
For the symmetry (2.14), we compute explicitly the
cross term SC specialized to the domain D2. Since
d2 = 0, the only term in the general action (1.19) that
can contribute to this cross term is the aνa
ν term in-
volving the square of the axial piece of the torsion; the
Ricci scalar term depends only on hµν . Using the defini-
tion (1.18b) of this axial piece together with the formula
(2.3b) for the torsion in terms of hµν and aµν gives
λνρσaνσ,ρ. (2.15)
Since this depends only on the antisymmetric field aµν it
does not generate any cross terms, and we conclude that
the cross term SC vanishes identically on the domain
D2. Therefore the complete action is invariant under the
symmetry (2.14) on D2.
F. The general relativity limit of the gravitational
action
So far we have considered the domains D0, D1 and D2
of the gravitational action parameter space. We now fo-
cus on the remaining domain D3. From the definitions
(2.10c) and (1.19) we find that in this domain the grav-
itational action reduces to that of general relativity, so
it is invariant under local Lorentz transformations of the
tetrad e µa → Λ ba (x)e
. This invariance guarantees that
the left hand side of the Euler-Lagrange equation
eaµgρν =
δSmatter
eaµgρν ,
is a symmetric tensor [8]. Consistency now requires the
right hand side to be symmetric. However, the torsion
tensor is not invariant under local Lorentz transforma-
tions. Therefore, a generic matter action that couples
between the torsion tensor and matter fields would break
the local Lorentz symmetry. This matter action produces
a nonsymmetric right hand side for the Euler-Lagrange
equation, thereby rendering the theory inconsistent. The
corresponding inconsistency of the Hayashi-Shirafuji the-
ory in this limit has been previously discussed in Refs.
[18, 23, 25, 26, 27, 29, 30, 31]. The inconsistency is
avoided if the matter action is invariant under local
Lorentz transformations. In this case the torsion ten-
sor is undetermined by the EHS theory, and the theory
reduces to GR.
III. INITIAL VALUE FORMULATION OF THE
THEORY
In this section we focus on the domains D1 and D2 that
are not ruled out by the existence of ghosts, and examine
in more detail the conditions under which the theory on
these domains has a well posed initial value formulation.
Suppose that we specify as initial data {aµν , hαβ} and
{ȧµν , ḣαβ} on some initial constant time hypersurface,
and ask whether the evolution of the fields for all sub-
sequent times is uniquely determined. Now the action
(2.6) is invariant under certain symmetries which allow
us to generate new solutions that correspond to the same
initial data. These symmetries consist of, first, diffeomor-
phisms xµ → xµ−ξµ(x) under which the fields transform
hµν → hµν + 2ξ(µ,ν) , (3.1a)
aµν → aµν + ξ[ν,µ] , (3.1b)
and second, the symmetries (2.13) on D1 and (2.14) on
D2. By invoking one of these transformations in a space-
time region to the future of the initial data hypersurface,
we can generate new solutions for the field-equations that
correspond to the same initial data. Therefore the evo-
lution of the fields hµν and aµν cannot be uniquely pre-
dicted. For the theory to have a well-posed initial value
formulation it is necessary that all of these transforma-
tions correspond to gauge symmetries, which means that
all observables should remain invariant under these trans-
formations. Then the failure of the theory to uniquely
predict hµν and aµν is merely associated with the unpre-
dictability of nonphysical degrees of freedom.
We now focus on the observables that will be mea-
sured by the GPB experiment. If we use the equations
of motion and spin transport for test bodies postulated
by MTGC, then these observables are not invariant un-
der the symmetries (2.13) and (2.14), as we now show.
Thus the initial value formulation is ill posed for these
postulated equations of motion.
A. Observations with Gravity Probe B
We focus on one of the four GPB gyroscopes, and
represent it as a particle with trajectory zα(τ) where
τ is proper time, and with spin sα(τ). We let the 4-
momentum of the photons from the distant fixed guide
star be kα. Let θ be the angle, as measured in the frame
of the gyroscope, between its spin and the direction to
the guide star. Then we have
cos θ =
~s · ~k
(~u · ~k)
, (3.2)
where ~s·~k = gµνsµkν . This can be seen from the formulae
for these vectors in the rest frame of the gyroscope: ~u =
(1,0), ~k = ω(1,−n) and ~s = (0, s), where n is a unit
vector in the direction of the star.
The equations of motion and spin precession postu-
lated by MTGC are
= 0 ,
= 0 , (3.3)
where uµ = dzµ/dτ is the 4-velocity and
≡ uµ∇µ (3.4)
is the covariant derivative operator along the worldline
with respect to the full connection ∇µ. In other words,
the particle travels on a geodesic of the full connection
(an autoparallel curve). MTGC also consider the possi-
bility that the equation of motion is
≡ uν∇̃νuµ = 0, (3.5)
where ∇̃ν is the Levi-Civita connection determined by
the metric, so that the particle travels along a geodesic of
the metric (an extremal curve). As mentioned by MTGC
this possibility is theoretically inconsistent since by Eqs.
(3.3) the orthogonality of uµ and sµ is not maintained
during the evolution. Nevertheless, we shall also con-
sider this possibility below. Finally, as mentioned above,
photons follow null geodesics of the metric:
kµ∇̃µkν = 0. (3.6)
We now apply the operator D/Dτ to the formula (3.2)
for the angle θ, and use the autoparallel equations of
motion (3.3) together with ∇µgµλ = 0. This gives
d[cos θ]
(~u · ~k)
~s · ~k
~u · ~k
. (3.7)
The measurable accumulated change in cos(θ) in the in-
terval τ1 → τ2 is therefore
∆[cos(θ)] =
(~u · ~k)
~s− ~s ·
~u · ~k
(3.8)
Next, we examine how the change in angle (3.8) trans-
forms under the symmetry transformations (2.13) and
(2.14). For this purpose it is sufficient to consider the
motion of the gyroscope in a flat, torsion-free spacetime,
since we are working to linear order. We use Lorentzian
coordinates where
= 0, which implies that
for an initially static gyroscope uµ, sµ and kµ are all
constants, so that ∆[cos(θ)] = 0.
Consider now the effect of the transformations (2.13) or
(2.14). Denoting the transformed quantities with primes
we find that
= 0, Γ
µν = 0 + δΓ
From Eqs. (3.3) we find that z′µ(τ ′) = zµ(τ) + δzµ(τ)
and s′µ(τ ′) = sµ(τ) + δsµ(τ), where both δz(τ) and δsµ
are O(χ). It turns out that the precise expressions for
these quantities are not required for our calculation. No-
tice that for a fixed distant star the field kµ(x) (before the
transformation) is approximately constant in a neighbor-
hood of the gyroscope, in the sense that kµ,ν = 0. Now,
Eq. (3.6) implies that kµ(x) remains invariant under the
transformations. Therefore the derivative of k′µ(x) along
z′(τ ′) is given by
u′αk′β . (3.9)
Substituting Eq. (3.9) and s′µ(τ
′) into Eq. (3.7), and re-
taining only terms which are O(χ) (so we can drop the
distinction between τ and τ ′) we obtain
d[cos(θ′)]
(~u · ~k)
−~s ·
~u · ~k
. (3.10)
From Eqs. (1.3), (1.4) and (2.3) the change δΓ
in the
connection coefficients is
= σηβλǫ
µλρσχσ,ρα , (3.11)
for the symmetry (2.13) and
= (σ/2)(χ
) (3.12)
for the symmetry (2.14).
We substitute these transformation rules into Eq.
(3.10) and then substitute the result into Eq. (3.8). Re-
calling that χα are arbitrary functions of the coordinates,
we find that by invoking the transformations (2.13) or
(2.14) we can set ∆[cos(θ)] to have an arbitrary value.
Thus, the observable angle is not invariant under the
symmetry transformations, and hence they cannot be in-
terpreted to be gauge transformations. [In particular this
implies that the matter action must be non-invariant].
Consequently the initial value formulation of the theory
is ill posed.
We now repeat this analysis for extremal worldlines
satisfying Eq. (3.5). Equation (3.10) acquires an addi-
tional term
− ~s ·
(~u · ~k)2
~k · D~u
= − ~s ·
(~u · ~k)2
on the right hand side. In addition the change to ex-
tremal worldlines alters the quantity δuµ, but this does
not appear in the final formula (3.10). As before we find
that the initial value formulation is ill posed.
Finally, MTGC also consider the possibility of an ex-
tremal worldline together with the following equation for
the spin precession
= 0. (3.13)
Here the antisymmetric tensor sαβ is related to the parti-
cle spin through sµ = ǫµνρσuνsρσ. This relation guaran-
tees that the orthogonality condition sµuµ = 0 is satisfied
throughout the motion of the particle.7 By examining the
transformation of ∆[cos(θ)] under the symmetries (2.13)
and (2.14) for an extremal worldline for which the law for
the spin precession is given by (3.13), we find as before
that the initial value formulation is ill posed.
IV. EINSTEIN-HAYASHI-SHIRAFUJI THEORY
WITH STANDARD MATTER COUPLING
The analysis so far suggests that in order to obtain a
consistent theory, one needs to choose a matter action
which is invariant under the symmetries (2.13) or (2.14)
of the gravitational action. This would allow those sym-
metries to be interpreted as gauge symmetries and allow
the theory to have a well posed initial value formula-
tion. In this section we show that the standard, mini-
mal coupling of matter to torsion [4, 6] does respect the
symmetry (2.14), and so one does obtain a consistent
linearized theory on the domain D2 by choosing this cou-
pling.8 For the special case of σ = 1, this matter action is
the one suggested in the original Hayashi-Shirafuji paper
[18]. Following the analysis of Hayashi and Shirafuji, we
show that the predictions of this linearized theory coin-
cide with those of linearized GR for macroscopic sources
with negligible net intrinsic spin.
The standard Dirac action in an Einstein-Cartan
spacetime is SD =
−gLD, where
e µa (ψ̄γ
aDµψ −Dµψγaψ)−mψ̄ψ , (4.1a)
Dµ = ∂µ −
σbcgρνe
b ∇µe ρc , (4.1b)
∇µe ρc = ∂µe ρc + Γρµνe νc , (4.1c)
σbc =
[γb, γc], (4.1d)
7 As noted by MTGC, normally one also demands a “transver-
sality” condition sαβu
β = 0. However, by Eqs. (3.5) and (3.13)
this condition can not be maintained along an extremal world-
line. This also implies that the norm of sµ is not constant along
the evolution.
8 Note that this implies in particular that the standard matter
coupling does not predict either extremal curves or autoparallel
curves for the motions of test bodies, since those cases are not
invariant under the symmetry (2.14).
and γb are Dirac matrices with the representation given
in Ref. [48] that satisfy γaγb + γbγa = −2ηab. Also ψ̄
denotes the adjoint spinor defined by ψ̄ = ψ†γ0, where †
denotes Hermitian conjugation. The torsion tensor enters
this action only through the covariant derivative in Eq.
(4.1b). As is well known, this action can be written as
the usual torsion-free action together with a coupling of
the fermion to the axial piece (1.18b) of the torsion tensor
[6]. Using the definitions (1.4) and (1.18) we obtain
e µa (ψ̄γ
aD̃µψ − D̃µψγaψ)−mψ̄ψ
aµψ̄γ
5γbψ . (4.2)
Here D̃µ = ∂µ− iσbcgρνe νb ∇̃µe ρc /4, ∇̃µe ρc = ∂µe ρc +
e νc , and γ
5 = iγ0γ1γ2γ3, where we use the convention
ǫ0123 =
Now the first line in the Dirac action (4.2) depends
only on the metric, and in particular it is independent of
aµν , so it is trivially invariant under the transformation
(2.14). The second line depends on aµν only through the
axial piece aµ of the torsion, which is given by Eq. (2.15),
and which is also invariant under (2.14). Therefore the
entire action (4.2) is invariant under the symmetry.
We conclude that on the domain D2 we have a consis-
tent theory with a well posed initial value formulation,
in which the symmetry (2.14) is a gauge symmetry. The
self-consistency of this theory for σ = 1 was previously
discussed by Leclerc [32]. From Eqs. (1.19) and (2.10b),
the complete action for the theory is
S[eaµ,Ψ] =
R({}) + 9d3aµaµ
+SD[e
µ, ψ, ψ̄] + . . . , (4.3)
where κ̂ = 1/(2d1). Here the ellipses denote additional
terms in the standard model of particle physics that are
not coupled to the torsion; the only term that couples
to the torsion is the Dirac action for the fermions (the
“minimal coupling” scheme of Refs. [4, 6]).
The linearized equation of motion for eaµ obtained
from this action gives equations for hµν and aµν :
�h̄µν + h̄
ρ(µ,ν)
ηµν h̄
= κ̂T(µν), (4.4a)
�aµν + 2a
ρ[µ,ν]
T[µν] . (4.4b)
Here � = ηαβ∂α∂β , h̄µν ≡ hµν − ηµνh ρρ , Tµν is the non-
symmetric energy-momentum tensor defined by
T νµ ≡
δSmatter
eaµ, (4.5)
and Tµν is its leading order term in a perturbative ex-
pansion (i.e. Tµν is independent of hµν and aµν). The
source terms in Eqs. (4.4) obey the conservation laws
= 0 and T
= 0, by virtue of the invariance
of the matter action under the transformations (3.1) and
(2.14). By using these transformations we can impose
the gauge conditions h̄ ,νµν = 0 and a
µν = 0, thereby
reducing the field equations to wave equations:
�h̄µν = −2κ̂T(µν), (4.6a)
�aµν =
T[µν] . (4.6b)
The first of these is the usual linearized Einstein equa-
tion.
Next we examine the antisymmetric piece of the stress
energy tensor which acts as a source for aµν in Eq. (4.6b).
One can show [18] that this antisymmetric piece is related
to the divergence of the spin density tensor by
T[βα] = σb
ab ,µ
, (4.7)
where the subscript b νc indicates evaluation at the back-
ground values of the tetrad. Recall that the spin density
is defined by
δSmatter[e
ν ,Ψ]
δω abµ
where in this definition the matter action is considered
to be a functional of the independent variables efα , ω
and Ψ. The matter action can be brought to this de-
sired form by substituting Dµ = ∂µ − i4σ
bcgρµω
the expression for LD in Eq. (4.1a). Relation (1.7) guar-
antees that this expression for Dµ equals to our original
expression in Eq. (4.1b).
Equations (4.6b) and (4.7) imply that aµν is sourced
only by the intrinsic spin density of matter. As we have
discussed, integrating Eq. (4.6b) for a macroscopic object
for which the spins of the elementary particles are not
aligned over a macroscopic scale gives a negligible aµν
[18], and consequently the predictions of the linearized
theory coincide with those of GR. Thus there is no extra
torsion-related signal predicted for GPB for this theory.
The lack of an experimental signature of torsion may
seem strange, since the torsion tensor is non-vanishing
even in the limit where one can neglect intrinsic spin.
As discussed in the introduction, the explanation is that
the torsion is not an independent dynamical degree of
freedom. More specifically, to linear order, macroscopic
bodies give rise to a metric perturbation hµν in the same
way as in GR, and then the torsion is simply defined to
S λµν =
hλ[ν,µ] (4.8)
[c.f. the first term in Eq. (2.3b)]. This definition has no
dynamical consequence, since only the axial piece (1.18b)
of torsion couples to matter, and the expression (4.8) has
no axial piece.
V. CONCLUSIONS
Preliminary results from Gravity Probe B will be an-
nounced in April 2007. The primary scientific goals of
the experiment are to verify the predictions of general
relativity for geodetic precession and for dragging of in-
ertial frames [3]. However the mission is also potentially
useful as a probe of modifications of general relativity.
One class of theories of gravity that GPB could poten-
tially usefully constrain are theories involving a dynam-
ical torsion tensor. Mao et. al. suggested a particular
class of torsion theories that they argued would predict a
measurable torsion signal for GPB [34]. We have shown
that this particular class of theories is internally consis-
tent in only a small region of its parameter space, and
in that consistent region does not predict any signal for
GPB. There may exist other torsion theories which could
be usefully constrained by GPB. It would be interesting
to find such theories.
Acknowledgments
This research was sponsored in part by NSF grant
PHY-0457200.
APPENDIX A: GHOSTS IN THE LINEARIZED
THEORY
In this Appendix we show that the Hamiltonian of the
complete linearized EHS theory (2.4) is unbounded below
whenever the corresponding Hamiltonian of the antisym-
metric term (2.6) is unbounded from below. This allows
us to deduce the existence of ghosts in the complete the-
ory from their existence in the reduced theory (2.6).
It is sufficient to show this property for the case of finite
dimensional, quadratic dynamical systems. We consider
a system with N degrees of freedom whose Lagrangian
and Hamiltonian are given by
q̇mq̇nKmn − V (qm) , (A1)
pmpn(K
+ V (qm) . (A2)
Here qm denotes the generalized coordinates and pm =
Kmnq̇n denotes the conjugate momenta, where the in-
dices m,n run from 1 to N, and Kmn = Knm. Recall
that the antisymmetric action SA was obtained by ig-
noring some of the dynamical variables of the linearized
EHS theory. In a discrete theory this corresponds to
writing a reduced Lagrangian Lr that depends only on
some of generalized coordinates qi, where i runs from 1
to M , M < N . The reduced theory has the following
Lagrangian and Hamiltonian
q̇iq̇jkij − U(qi) , (A3)
p̃ip̃j(k
+ U(qi) . (A4)
Here p̃i = kij q̇j , where kij = Kij for i, j = 1...M and
U(q1, ....qM ) = V (q1....qM , 0..0), (A5)
i.e., the potential U(qi) is obtained from V (qm) by setting
qm = 0 for m =M + 1...N .
Now suppose that the potential term U(qi) of the re-
duced Hamiltonian Hr is unbounded from below. It fol-
lows immediately from the definition (A5) that V (qm) is
also unbounded from below, and so the complete Hamil-
tonian (A2) is unbounded from below.
Suppose next that the kinetic term of the reduced
Hamiltonian Hr is unbounded from below. By virtue of
Eq. (A4) this means that at least one of the eigenvalues
of (k−1)ij must be negative. Denoting the eigenvalues of
the matrix kij by λ(i), 1 ≤ i ≤ M , we find that there
exists an eigenvalue l for which λ−1
< 0. This implies
that there exists an M dimensional eigenvector w
i for
which w
j kij = λ(l) < 0. We now construct an N di-
mensional vector defined by ηm = (w
1 ...w
, 0...0). By
definition, this vector satisfies ηmηnKmn = λ(l) < 0, im-
plying that Kmn has a negative eigenvalue. This means
that the complete Hamiltonian (A2) is unbounded from
below.
APPENDIX B: ALTERNATIVE FORM OF
GRAVITATIONAL ACTION
In this Appendix we derive identity
−gR({})d4x =
tµνλt
ν − 6aνaν
, (B1)
where R({}) is the Ricci scalar of the Levi-Civita con-
nection, and tµνλ, v
µ and aµ are the irreducible pieces
(1.18) of the torsion tensor with the factor of σ removed.
Combining this identity with the formula (1.17) for the
gravitational action of the EHS theory yields the alter-
native form (1.19) of that action.
The idea is to introduce a new torsion tensor
S̄ λµν ≡
e λa (e
ν,µ − eaµ,ν) . (B2)
This is just the torsion tensor (1.15) of the EHS theory
but specialized to σ = 1, i.e., it is the torsion tensor of
the Hayashi-Shirafuji teleparallel theory[18]. From Eqs.
(1.15) and (1.18) it is related to the fields tµνλ, v
µ and
aν by
S̄νµλ =
(tλµν − tλνµ) +
(gλµvν − gλνvµ) + ǫλµνρaρ .
We denote the corresponding metric compatible con-
nection by Γ̄αβγ and the corresponding Riemann tensor
by R̄
. This Riemann tensor vanishes identically by
virtue of the definition (B2), as we discussed in the intro-
duction. The Ricci scalar R̄ ≡ gβδR̄αβαδ also vanishes,
which implies
−gR̄d4x = 0 . (B4)
We now substitute into Eq. (B4) the formula
R̄αβγδ = Γ̄
δβ,γ − Γ̄αγβ,δ + Γ̄αγµΓ̄
− Γ̄αδµΓ̄
, (B5)
together with barred versions of Eqs. (1.3) and (1.4).
This gives
−g[R({})−2∇̃αK̄β αβ −K̄
+K̄β αµ K̄
µ ] = 0
Here ∇̃α denotes the the Levi-Civita derivative opera-
tor. Discarding the total derivative term and using the
decomposition (B3) together with the barred version of
Eq. (1.4) now yields the desired identity (B1).
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