Generalized Sparling-Thirring form in the Brans-Dicke theory

Abstract

The definition of the Sparling-Thirring form is extended to the Brans-Dicke
theory. By writing the Brans-Dicke field equations in a formally Maxwell-like
form, a superpotential and a corresponding pseudo energy-momentum form are
defined. The general energy expression provided by the superpotential in the
Jordan frame is discussed in relation to the corresponding expression in the
Einstein frame. In order to substantiate its formal definition, the generalized
Sparling-Thirring form is used to calculate the energy for the spherically
symmetric vacuum solution in the Brans-Dicke theory.

Full text

arXiv:1410.4134v1 [gr-qc] 15 Oct 2014
Generalized Sparling-Thirring form in the Brans-Dicke theory
Ahmet Baykal1, ∗and ¨
Ozg¨ur Delice2, †
1Department of Physics, Faculty of Arts and Sciences,
Ni˘gde University, Bor Yolu, 51240 Ni˘gde, Turkey
2Department of Physics, Faculty of Arts and Sciences,
Marmara University, 34722 ˙
Istanbul, Turkey
(Dated: October 20, 2014)
Abstract
The definition of the Sparling-Thirring form is extended to the Brans-Dicke theory. By writing
the Brans-Dicke field equations in a formally Maxwell-like form, a superpotential and a corre-
sponding pseudo energy-momentum form are defined. The general energy expression provided by
the superpotential in the Jordan frame is discussed in relation to the corresponding expression
in the Einstein frame. In order to substantiate its formal definition, the generalized Sparling-
Thirring form is used to calculate the energy for the spherically symmetric vacuum solution in the
Brans-Dicke theory.
PACS numbers: 4.50.Kd, 04.20.Cv, 04.20.Fy
∗abaykal@nigde.edu.tr
†ozgur.delice@marmara.edu.tr
1

I. INTRODUCTION
According to the equivalence principle, the gravitational field, as described by Einstein
as the geometry of spacetime, can locally be transformed away at a given point so that a
curved spacetime looks locally like a flat Minkowskian at an infinitesimal level. Conversely,
an accelerated frame in flat spacetime can emulate gravity [1]. Consequently, one can argue
that the equivalence principle renders any definition of a local energy density to be non-
tensorial. This line of thought lead to the definition of pseudo energy-momentum forms that
depend on a chosen frame. In particular, for an asymptotically flat spacetime, a working
definition of energy-momentum can be provided. A well-known example of such a definition
of pseudo-energy momentum form and the superenergy, in the framework of the general
relativity theory (GR), is the Sparling-Thirring 2-form and the associated pseudo-energy
momentum forms [1–3]. In GR, the Sparling-Thirring form is introduced to define the total
energy and it is related to the definition of energy by Arnowitt-Deser-Misner (known as the
ADM energy in short). The Sparling-Thirring form can be related to other definitions of
energy, such as the well-known Landau-Lifschitz energy [2, 4–6].
The constructions of a pseudotensor using field equations of gravity and the definitions
involving the Noether symmetries have previously been discussed in connection with the
quasilocal quantities in [7–11]. In a completely different approach, but using a similar
mathematical notation and set up, the Bell tensor is derived from a variational principle in
the context of a particular tensor-tensor model of gravitation [12]. More recently, conserved
Noether quantities were also studied in the context of the motion of spinning particles
in the BD theory [13]. A general formulation of generalized Bianchi identities relative to
conformally related frames and corresponding conservation laws in various modified gravity
models have been discussed in [14].
The scalar-tensor theory of gravitation proposed by Brans and Dicke [15] almost half a
century ago is still one of the popular theories of gravity which received rekindled interest in
the applications in particular to the cosmological dynamics as possible alternatives to dark
energy and dark matter [16, 17]. The dynamical equivalence between f(R)-type modified
gravity models to the scalar tensor theories, introduced in [18, 19] for particular modified
gravity models, also increased the interest to these theories [20–22]. For a treatment of this
equivalence in the differential form calculus see [23]. Any modified theory of gravity should
2

compete with the GR on the observational front at cosmological and solar system scale.
On the theoretical side, a prediction of such a modified theory is to be compared to those
of GR. The conserved currents for BD theory has been studied by applying the Lagrange-
Noether machinery [24–26]. More recently, using the exterior algebra of differential forms,
and in a more general context of Noether’s theorem, a new conserved charge is defined as
an application in [27]. In an earlier work, the Landau-Lifshitz superpotential is extended to
BD theory [28].
In the approach making use of the field equations, the definition of a pseudo-tensor can be
applied to any geometrical theory that follow from any Lagrangian once the field equations
formulated in terms of the tensor-valued differential forms. For example, a generalized
Sparling-Thirring form was previously defined [29] for the dimensionally continued Euler-
Poincare form
LEP = Ωab ∧ · · · ∧ Ωcd ∗eab ∧ · · · ∧ ecd,(1)
where LEP can be considered as natural generalization of Einstein-Hilbert Lagrangian
LEH =1
2κ2Ωab ∧ ∗eab,(2)
leading to second order field equations (in the notation is introduced in the following section).
In this generalization, an exact differential is separated out from the field equations following
from (1) using the definition of the curvature 2-forms as in the case of the Einstein field
equations.
In the present work, the Sparling-Thirring construction is extended to the BD scalar-
tensor theory. The BD scalar field is incorporated into the definition of Sparling-Thirring
form, as a consequence of the vanishing torsion constraint on the connection 1-form obtained
by the constrained first order formalism to the BD theory Lagrangian.
The organization of the paper is as follows. In the next section, after briefly discussing
the notation of the tensor-valued forms and the algebra of differential forms, the BD field
equations are obtained by the variation of the BD action via a first order formalism. The use
of the modern language of differential forms has several advantages over tensorial treatment,
in that the differential forms are natural integrands on manifolds and allows one to use
Stokes’ theorem to convert volume integrals to flux integrals. Moreover, the use of differential
forms also allow one to define energy for asymptotically flat spacetimes without introducing
an approximation scheme. The definition of energy superpotential is therefore requires
3

the BD field equations to be formulated relative to an orthonormal coframe. Due to this,
we express the well known BD field equations relative to an orthonormal coframe in this
section. In section 3, following the construction of Sparling-Thirring form in GR, the BD
field equations in this form is used to define a superpotential and a corresponding energy
momentum form which involves the nonminimally coupled dynamical BD scalar field as
well. In accordance with the equivalence principle, the resulting expressions for energy
and momentum depends on a chosen frame as in GR. Since the definition of BD pseudo-
energy-momentum form is particularly suitable to discuss the mathematical expressions in
Jordan frame and conformally related Einstein frames, we obtain relevant expressions in the
Einstein frame as well in section 4. In section 5, the energy definition is used to define the
total mass of the spherically symmetric and asymptotically flat vacuum BD solution. The
paper ends with general remarks regarding to some possible energy definitions in related
modified gravity theories in line with the discussion below.
II. BD FIELD EQUATIONS
The notation used for the algebra of differential forms is adopted from [30]. The calcula-
tions in this paper will be carried out relative to a set of orthonormal basis coframe 1-forms
{ea}for which the metric reads g=ηab ea⊗ebwith ηab = diag(−+ ++). The set of basis
frame fields is {Xa}and the abbreviation iXa≡iais used for the contraction operator with
respect to the basis frame field Xa.∗denotes the Hodge dual operator acting on the basis
forms and ∗1 = e0∧e1∧e2∧e3is the oriented volume element. The abbreviations of the
form ea∧ · · · ∧ ec∧ · · · ≡ ea··· c··· for the exterior products of basis 1-forms are extensively
used for the convenience of the notation. The first structure equations of Maurer-Cartan
read
Θa=Dea=dea+ωab∧eb= 0,(3)
with the vanishing torsion two form, i. e., Θa= 0. Dis the covariant exterior derivative
operator, acting on tensor-valued forms, and a suitable definition and its relation to covariant
derivative can be found for example in [1]. The curvature 2-form Ωabwith Ωab=1
2Rabcdecd
satisfies the second structure equation of the Maurer-Cartan
Ωab=dωab+ωac∧ωcb.(4)
4

The discussion below will be based exclusively on the original BD Lagrangian among other
scalar-tensor theories. The approach to the definition of a total energy in BD theory below
involves surface integrals of pseudo-tensorial quantities, therefore it is convenient to use the
algebra of differential forms defined on pseudo-Riemannian manifolds that are already built
for integration.
Expressed in terms of differential forms, the total Lagrangian 4-form for the original BD
Lagrangian with matter fields included has the form
Ltot. =LBD[φ, ea, ωab] + 8π
c4Lmatter[g, ψ],(5)
where the gravitational part in the so-called Jordan frame is given by
LBD =φ
2Ωab ∧ ∗eab −ω
2φdφ ∧ ∗dφ, (6)
and the matter part Lmatter[g, ψ] is assumed to be independent of the BD scalar and the
spinor fields. The gravitational coupling constant Gin GR is replaced by a dynamical
scalar field φ−1with a corresponding kinetic term for the scalar field. ωis the free BD
parameter and GR limit is recovered in the ω7→ ∞ limit. For the vacuum solutions or
for when the matter energy momentum tensor has a non vanishing trace, however, things
are more complicated and the above limit may not give [31–33] the GR limit, unless the
arbitrary parameters are fixed by some physical grounds such as post Newtonian extension
for asymptotically flat solutions [34].
In the general framework of first order formalism for gravity, the independent gravita-
tional variables can be taken as the set of basis coframe 1-forms {ea}and the connection
1-forms {ωab}. The local Lorentz invariance forbids any gravitational action to have explicit
dependence on {ωab}and the first order derivatives of deaand dωabenters into a gravita-
tional Lagrangian with local Lorentz symmetry by the tensorial quantities such as Θaand
Ωab. The minimal coupling prescription for the matter fields implies that deaand dωaboccurs
only in the gravitational sector in the presence of the matter fields. On the other hand, the
BD scalar field φcouples to the metric nonminimally. Because of the nonminimal coupling,
the BD scalar field is dynamical even in the absence of the kinetic term for it.
An essential ingredient of the discussion is the formulation of the BD equations in terms
of exterior forms which then allows one to write the conservation laws in differential and
integral forms as well easily with the help of Stokes’ theorem with considerable technical
5

advantage over the methods using tensor components. Hence, a derivation of BD equations
in the desired form using exterior algebra of forms using the Lagrange multiplier method is
presented below.
The vanishing torsion constraint can be implemented into the variational procedure by
introducing Lagrange multiplier 4-form term
LC=λa∧Θa,(7)
to the original Lagrangian form LBD, where the Lagrange multiplier 2-form λais a vector-
valued 2-form imposing the dynamical constraint Θa= 0. The Lagrangian Lext. =
Lext.[φ, ea, ωab, λa, ψ] for the extended gravitational part then has the explicit form
Lext. =LBD [φ, ea, ωab] + LC[ea, ωab, λa].(8)
The total variational derivative of Lext. with respect to independent variables can be found
as
δLext.=δφ 1
2R∗1−ω
2φd∗dφ +ω
2φ2dφ ∧ ∗dφ+δea∧φ
2Ωbc ∧ ∗eabc +Dλa+ω
φ∗Ta[φ]
+δωab ∧1
2D(φ∗eab)−(ea∧λb−eb∧λa)+δλa∧Θa,(9)
up to an omitted boundary term. The energy-momentum 3-form of the scalar field ∗Ta[φ] =
Ta
b[φ]∗ebhas the explicit expression
∗Ta[φ]≡1
2[(iadφ)∗dφ +dφ ∧ia∗dφ].(10)
Assuming that there are no spinor fields to couple to the connection, the independent
connection equations read
D(φ∗eab)−(ea∧λb−eb∧λa) = 0.(11)
These equations can be considered as the defining equation for the Lagrange multiplier 2-
forms λaand they can uniquely be solved for the Lagrange multiplier 2-forms by calculating
its contractions by taking the constraint Θa= 0 into account. One finds that
λa=∗(dφ ∧ea).(12)
Consequently, using the expression (12) for the Lagrange multiplier form in the metric
field equations induced by the coframe variational derivative δLext./δea≡ − ∗ Ea= 0 in (9)
6

read
∗Ea=φ∗Ga−D∗(dφ ∧ea)−ω
φ∗Ta[φ],(13)
where Ea≡Eabebis vector-valued 1-form. In the presence of the matter fields, the BD field
equations take the form
∗Ea−8π
c4∗Ta[ψ] = 0,(14)
where ∗Ta[ψ] stands for the energy-momentum form for the matter field ψderived from
the variational derivative of Lmatter [g, ψ] with respect to the basis coframe 1-forms. ∗Ta[ψ]
depends on the metric tensor as well and therefore in the notation used here, it may also
be appropriate to state the dependence by writing it in the form ∗Ta[ψ, ea]. However, the
metric dependence is surpassed for simplicity in the case that the metric dependence is
obvious from the context in the discussions relative to a given coframe.
Although it is customary to write the BD equations (13) by dividing it with φ, the
Lagrange multiplier and the ωφ−1∗Ta[φ] terms are considered to be on the “geometry”
side of the BD field equations. As a consequence of the diffeomorphism invariance of the
BD Lagrangian, it follows from the corresponding Noether identity that D∗Ea= 0 [35].
Alternatively, in a more straightforward way, and with the help of the identities
D(φ∗Ga) = dφ ∧ ∗Ga,
D2∗(dφ ∧ea) = Ωab∧ ∗(dφ ∧eb) = dφ ∧ ∗Ra,(15)
Dω
φ∗Ta[φ]=ω
2φ(iadφ)d∗dφ −1
φdφ ∧ ∗dφ,
it is possible to arrive at the identity
D∗Ea=−1
2(iadφ)ω
φd∗dφ −ω
φ2dφ ∧ ∗dφ +R∗1,(16)
as expected. The right-hand side vanishes identically provided that the field equation for
the BD scalar is satisfied since the terms on the right-hand side inside the bracket are
proportional to the field equations for the BD scalar given below. This consideration is in
line with the well-known case of the Einstein-massless scalar field equations, Ra= (iadφ)dφ,
from which the field equations for φfollows from the Bianchi identity. Consequently, one has
D∗Ta[ψ] = 0, from which one can derive geodesic postulate for test particles by introducing
the matter energy momentum of ideal fluid [30]. As it is well-known, this is not the case in
7

the conformally related Einstein frame, where scalar field couples nonminimally to a matter
field.
On the other hand, the field equation for the BD scalar that follows from the variational
derivative
δLBD/δφ = 0 is given by
ω d ∗dφ −ω
φdφ ∧ ∗dφ +φ R ∗1 = 0.(17)
The BD scalar couples to the matter energy momentum through the last term in (17). Then,
by combining the scalar field equation with the trace of the metric equations, the equation
for the BD scalar reduces to
d∗dφ =8π
c4
1
2ω+ 3T[ψ]∗1,(18)
for which the trace T[ψ]≡Ta
a[ψ] of the matter energy-momentum tensor act as the source
term. As pointed out in the invariance of LBD discussion above, the reduced field equation
(18) follows from Bianchi identity for the BD field equations together with its trace.
III. DEFINITIONS OF THE SUPERPOTENTIAL AND THE PSEUDO-ENERGY-
MOMENTUM FORMS
The definitions of the superpotential for the BD theory can be given in terms of the
original Sparling-Thirring 2-form in GR. Thus it is appropriate to recall the construction
of Sparling-Thirring form[1–3]. The Einstein 3-form is defined in terms of the following
contraction
∗Ga=−1
2Ωbc ∧ ∗eabc,(19)
where 1-form Gacan be defined in terms of the Einstein tensor components as Ga≡Gabeb.
The definition (19) is suitable to separate out an exact differential out of Einstein tensor
regardless of the presence of any matter Lagrangian. Explicitly, by inserting the second
structure equation (4) into the right-hand side of (19), probably technically in the most
straightforward way, one finds
∗Ga=d∗Fa+∗ta,(20)
with the Sparling-Thirring 2-form
∗Fa=−1
2ωbc ∧ ∗eabc,(21)
8

and the gravitational energy-momentum (pseudo-tensor) 3-form
∗ta≡1
2(ωbc ∧ωad∧ ∗ebcd −ωbd ∧ωdc∧ ∗eabc).(22)
The classical definitions of the Sparling-Thirring superpotential and the corresponding
energy-momentum pseudo-tensor above can be related to well-known expressions for other
superpotentials [2]. In particular, the components tab of the pseudo-tensorial object ∗ta=
tab∗ebare symmetrical relative to a coordinate basis [1].
The definitions (21) and (22) then help to split ∗Ea, in an equally straightforward way
as in the GR case above, in the following Maxwell-like form for BD theory
∗Ea=d∗ Fa+∗T a=8π
c4∗Ta[ψ],(23)
with the generalized definitions of the superpotential
∗ Fa≡φ∗Fa− ∗(dφ ∧ea),(24)
and the corresponding energy-momentum form
∗ T a=φ∗ta−dφ ∧ ∗Fa−ωab∧ ∗(dφ ∧eb)−ω
φ∗Ta[φ],(25)
for the superpotential and the pseudo-energy-momentum forms respectively for the BD
vacuum case. A conserved pseudo 4-current then follows from the equations (23) as the
simple consequence of the differential identity d2≡0. The expression on the right-hand
side of (25) is pseudo tensorial and it involves connection 1-forms which can be transformed
away at a given point in accordance with the equivalence principle.
The formulation of the field equations in the form of a conservation law in terms of
differential forms allows one to rewrite the conserved quantities as flux integrals by making
use of Stokes’s theorem. By defining total energy-momentum pseudo form
∗τa≡8π
c4∗Ta[ψ]− ∗T a,(26)
a conserved 4-current Pa
BD then can be expressed as the following flux integral
Pa
BD ≡ZU
∗τa=ZU
d∗ Fa=Z∂U
∗Fa,(27)
with ∂U as the boundary of a three-dimensional submanifold U⊂Mon a pseudo-
Riemannian manifold M.
9

The BD superpotential ∗F acan be put in a simplified form analogous to the Sparling-
Thirring form as
∗ Fa=−1
2Λbc ∧ ∗eabc,(28)
with a modified connection 1-form denoted by Λbc =−Λcb that is given explicitly by the
following form
Λbc ≡φ ωbc +1
2(ebicdφ −ecibdφ).(29)
By definition, 1-form Λbc incorporates the constraint on the connection 1-form resulting from
nonminimal coupling of the BD scalar in a peculiar way.
The BD superpotential ∗F adefined in (28) naturally involves BD scalar which carries dy-
namical degrees of freedom and the scalar contribution to the superenergy definition results
from the vanishing torsion constraint term on the independent connection. Consequently,
the dynamical coupling constant in BD theory also incorporated into the total energy by
definition through the Lagrange multiplier term and at the same time the construction is
guided by the mathematical structure of the field equations formulated in terms of differ-
ential forms. By construction, the conserved quantities require the field equations to be
satisfied.
In parallel to the original definition of the total energy in terms of Sparling-Thirring form
for an asymptotically flat geometry in GR, one can define the total energy as the temporal
component of conserved 4-current EBD ≡P0
BD as
EBD ≡ZS2
∞
∗F0=−1
2ZS2
∞φωjk +1
2(ejikdφ −ekijdφ)∧ ∗e0jk ,(30)
where the flux integral is over two dimensional sphere with infinite radius denoted by S2
∞
evaluated at a constant value of t. As for the other classical pseudo energy-momentum
and superpotential forms, the 4-momentum definition (27) depends on the frame in which
it is computed [7]. In the same way as the superpotentials are calculated in GR for an
asymptotically flat spacetime, the BD superpotential is to be calculated in a coordinate
system that is most nearly Minkowskian as well.
In contrast to the previous energy-momentum pseudo-tensors for scalar-tensor theories
in the literature, the use of exterior forms in defining conserved currents in BD theory is
technically straightforward which amounts to a judicious arrangement of the terms in the
field equations. This arrangement of the field equations not only separates the higher order
10

terms but also singles out the leading terms that are linear in the derivatives of the field
variables.
IV. JORDAN FRAME VS. EINSTEIN FRAME
The definition of the total energy in BD theory above involves no approximation scheme
exercised in other approaches, for example linearization of the field equations around a
suitable background solution [37, 38]. The corresponding definition in the Einstein frame
below allow one to discuss the energy definition relative to Einstein frame in a straightforward
manner.
It is a well-known fact that the conformal scaling of the metric components gµν (relative
to a coordinate coframe with Greek indices) by
gµν 7→ ˜gµν =φ gµν,(31)
brings the BD Lagrangian into the Einstein frame with a new scalar field coupled non-
minimally to matter fields. The conformal transformation (31) is equivalent to the scaling
of the coframe basis 1-forms as [39]
ea7→ ˜ea=φ1/2ea.(32)
Furthermore, the interior product operators of the frames are related by iXa=φ1/2i˜
Xa,
whereas the invariant volume forms are related by ∗1 = φ−2˜
∗1. Consequently, the connection
1-forms transform as
ωab 7→ ˜ωab =ωab −1
2φ(ebiadφ −eaibdφ).(33)
The conformal transformation (32) brings BD Lagrangian into the so-called Einstein frame
with a minimally-coupled massless scalar field αas
˜
LBD =1
2˜
Ωab ∧˜
∗˜eab −1
2dα ∧˜
∗dα, (34)
up to an omitted closed form. The scalar field αis related to φby
α= (ω+ 3/2)1/2ln φ. (35)
The metric field equations that follow from the scaled Lagrangian can be obtained as
˜
∗˜
Ga−˜
∗˜
Ta[α]−e−2bα˜
∗˜
Ta[˜g, ψ] = 0,(36)
11

where the conformally scaled metric is defined as ˜g=e−2bαgand the constant bis defined
in terms of the BD parameter ωas
b=2
2ω+ 31/2
.(37)
In the Einstein frame the gravitational part assumes the familiar form while coupling to the
matter becomes nonminimal while in the Jordan frame coupling to the matter is through
a dynamical BD scalar field. As a consequence, contrary to a matter energy-momentum
forms relative to the Jordan frame, a matter energy-momentum is not covariantly constant
in the Einstein frame. On the other hand, the construction of the expression for the BD
superpotential is not modified by a matter field Lagrangian coupled to the BD Lagrangian.
The familiar Sparling-Thirring superpotential form can be adopted for the field equations
(36) defined relative to the Einstein frame. The above definition of BD superpotential (28)
facilitates the expressions in the Jordan frame. By applying the conformal transformation
used above to the generalized Sparling-Thirring form ∗F ain the Jordan frame
∗ Fa=−1
2φωbc +1
2(ebicdφ −ecibdφ)∧ ∗eabc
=−1
2φ˜ωbc ∧ ∗eabc,(38)
where ˜ωab is given by (33). Then one immediately finds
Λab =φ˜ωab.(39)
Finally, by using the defining relations for the conformal transformations (32), one ends up
with the remarkable result
∗ Fa=φ1/2˜
∗˜
Fa,(40)
where ˜
Fais the Sparling-Thirring form in the Einstein frame of the form ˜
Fa=−1/2 ˜ωbc ∧
˜
∗˜eabc similar to (21). This equation relates the Sparling-Thirring form in the Einstein frame
and the generalized BD superpotential defined in the corresponding Jordan frame.
V. GRAVITATIONAL ENERGY FOR A SPHERICALLY SYMMETRIC
VACUUM METRIC
In testing the validity of the formal definition of a conserved charge that makes explicit
use of the field equations, an exact solution to the field equations provides a valuable tool
12

in explicit applications. In the discussion below, a particular spherically symmetric, static,
vacuum solution to the BD theory will be taken into account. In the so-called isotropic
coordinates, the metric for the spherically symmetric vacuum solution to the BD theory has
the form [15]
ds2=−f2(r)dt ⊗dt +h2(r)δijdxi⊗dxj,(41)
with the explicit forms of the metric functions given by
f(r) = eα0r−B
r+B1/λ
,(42)
h(r) = eβ01 + B
r2r−B
r+B(λ−C−1)/λ
.(43)
The metric functions depend only on the function rdefined by r2=δij xixjin terms of the
the spatial coordinates {xi}, for i, j = 1,2,3. α0, β0, B are integration constants and the
remaining constants Cand λare related further by
λ2≡(C+ 1)2−C(1 −1
2ωC).(44)
As a function of r, the BD scalar is given by
φ=φ0r−B
r+BC/λ
.(45)
The solution above, known as the Brans-I solution in the literature, is not the only
spherically symmetric, static, vacuum solution to the BD theory. There are four classes of
such solutions that are usually named as Brans I-IV solutions [40]. However, the Brans-II
solution is not an independent solution and can be obtained by a complex substitution of
the parameters Cand λby making use of the Brans-I solution [41]. The Brans-III and
Brans-IV solutions are not independent of each other as well [42]. These particular solutions
require the BD parameter ωto take negative values (ω < −3/2) which implies the violation
of the weak energy condition. Moreover, by relating the parameters of the Brans-I solution
to the BD parameter ω, it was shown in [41] that only the Brans-I solution can describe
the gravitational field exterior to a nonsingular, spherically symmetric object obtained by
matching conditions in the weak field regime. Henceforth, only the Brans-I solution given
by Eqs. (41)-(45) are taken into the account in the application below.
13

The explicit expressions for the Levi-Civita connection 1-forms relative to the orthonormal
coframe defined by e0=fdt and ek=h dxkare given by
ω0j=f′
fr xjdt, (46)
ωjk =−h′
hr xjdxk−xkdxj,(47)
where ′stands for the derivative with respect to r. The expansion of connection 1-forms
into the associated coordinate basis coframe forms in (46)-(47), rather then the orthonormal
coframe is convenient in order to evaluate the flux integral in what follows.
Moreover, with the assumption φ=φ(r), one finds that the modified connection 1-form
Λjk has the explicit expression
Λjk =−(φh′+1
2hφ′)1
rxjdxk−xkdxj.(48)
Consequently, the expression in (48) can be used in the definition (27) to calculate the
total energy EBD as
EBD =−lim
r→∞
1
2ZS2
r
ǫ0ijk Λij ∧ek
= lim
r→∞
1
2ZS2
r
ǫijk Λij ∧ek,(49)
where ǫ0ijk =−ǫ0ijk and ǫijk stands for the permutation symbol on the spatial submanifold
t= constant. By considering (48), the total energy becomes
EBD =−lim
r→∞ ZS2
r
φ1/2(φ1/2h)′ǫijk
xi
rdxj∧dxk.(50)
The integration measure above can be put into a form convenient for the spherical sym-
metry by recalling the basic geometrical formula for S2on Euclidean space R3. On a three
dimensional Euclidian space R3, the unit S2can simply be defined by setting r= 1. In terms
of the Cartesian coordinates {xi}, one has r2=δij xixj. By differentiating this expression
and taking the Hodge dual one finds
⋆ rdr =1
2ǫijk xidxj∧dxk,(51)
where ⋆is the Hodge dual in R3. Restriction of the expression on the right-hand side to S2
r
gives the volume element on S2
rwhich is conveniently taken to be r2dΩ with dΩ standing
for an infinitesimal element of the solid angle. Eventually, the energy expression becomes
EBD =−lim
r→∞ ZS2
r
2φ1/2(φ1/2h)′r2dΩ.(52)
14

For φ∼φ0=constant, the expression on the right-hand side of Eq. (52) becomes propor-
tional to the corresponding expression in GR (see, for example, the expression given in [1])
up to a constant.
Evaluating (52) for the Brans-I solution defined by Eqs. (41)-(45) above, one finds
EBD = 8πMBD =8π
λB(2 + C)φ0eβ0.(53)
The expression on the right-hand side contains five parameters, but due to the relation (44),
only four of them are free.
Let us compare this expression with corresponding expression of GR. By considering the
linear expansion of the static vacuum BD solution given in (41) and the scalar field (45), and
subsequently matching it to a static Newtonian source, the parameters B, C, λ, φ0, β0can
be expressed in terms of the corresponding GR parameters together with the BD parameter
ωas [15, 41]
B=λM
2, C =−1
ω+2 , β0= 0,(54)
λ=2ω+3
2ω+4 1/2, φ0=2ω+4
2ω+3
1
G.(55)
Note that in our notation we have set G= 1.Putting these into (53), we have found that
the Einstein limit of the total energy is given by
EE= 8πM. (56)
It is possible to obtain this result in a more straightforward way as well. The choice C=
0, λ = 1 brings the Brans-I solution to the corresponding one in GR, which yields the result
(EBD )lim ω7→∞ = 16πBE(57)
in the Einstein limit defined by ω7→ ∞. In the expression above, BEis a constant and the
particular choice BE=M/2 yields the correct GR limit.
VI. CONCLUDING REMARKS
Both Abbott-Deser [36] and Deser-Tekin [37] charge definitions make essential use of the
field equations as well as the Killing symmetries of the background solution. For example, the
Deser-Tekin charges are constructed by linearizing the field equations around a background
15

solution (flat Minkowski space in our case above). Subsequently, a conserved quantity is
expressed as a flux integral in the background by making use of the linearized equations.
The construction of the generalized Sparling-Thirring superpotential is achieved in the same
spirit as the construction of a Deser-Tekin charge and the use of exterior forms and Stokes’s
theorem allows one to obtain flux integrals on background spacetime in a practically useful
form see, for example, the construction in [38] by making use the Killing vector fields of
the background spacetime. As for the BD example above, the energy in the Deser-Tekin
approach is simply obtained by linearizing the generalized superpotential ∗Fa.
As another important technical point, note that the construction of the BD superpotential
is insensitive to a possible potential term for the scalar field. The definition also covers the
special case where the kinetic term for the BD scalar is absent and therefore the construction
can be applied to more general f(R) models since they are known to have scalar-tensor
equivalent models with a potential term for the scalar field. To facilitate a comparison with
these theories, let us consider the simplest modification of the Einstein-Hilbert Lagrangian
form
L=1
2f(R)∗1,(58)
with f(R) assumed to be an arbitrary differentiable and nonlinear function of the scalar
curvature R. Particular forms of the function fare studied with different motivations, for
example, arising from cosmological applications.
The metric field equations for vacuum that follow from the modified Lagrangian above
take the form [43]
f′∗Ga−D∗(df′∧ea) + 1
2(Rf′−f)∗ea= 0.(59)
By comparing the field equations with the BD field equation (13) above one can see that
the field redefinition f′≡φwhich entails the potential term V(φ) by the Legendre trans-
formation
V(φ)≡R(φ)f′(R(φ)) −f(R(φ)),(60)
for the scalar field φ. Consequently, in terms of the scalar field φ, the scalar-tensor equivalent
equations for (59) become
φ∗Ga−D∗(dφ ∧ea) + 1
2V(φ)∗ea= 0,(61)
16

which are in fact the vacuum BD equations with ω= 0 and with the potential term for
scalar field. The form of the field equations (59) is not very common in the vast literature
involving various f(R) models and reader is referred to [44] in relating (59) to the more
familiar coordinate expression. In either form, the metric equations allow one to define
gravitational energy in parallel to the BD case presented above immediately. By using the
form of the field equation given in (59), one finds the total energy by the flux integral
Ef(R)≡−1
2Z∂U f′ωbc +1
2f′′ (ebicdR −ecibdR)∧ ∗e0bc.(62)
For the simple case f(R) = Rthe above formula gives back the Sparling-Thirring form up
to a constant multiple. For the next simple case with f(R) = R+αR2the leading term
is the second term on the right-hand side and the above formula reduces up to a constant
multiple to the energy definition of Deser-Tekin for a particular subcase of general quadratic
curvature gravity in four dimensions [37]. It follows by definition that the energy has the
expression for the R2term is of the form
ER2∼Z∂U
∗(dR ∧e0) (63)
for asymptotically flat solutions [45]. Further scrutiny of the generalized Sparling-Thirring
forms for modified gravity models in relation to the definitions of the Deser-Tekin charges
will be taken up elsewhere.
[1] W. Thirring Classical Mathematical Physics: Dynamical Systems and Field Theories 3rd edn.
(Springer-Verlag, New York, 2003)
[2] W. Thirring W, R. Wallner, 1978 The use of exterior forms in Einstein’s gravitation theory,
Revista Brasileira de Fisica 8, 686-723 (1978)
[3] G. A. J. Sparling, Twistors, spinors and the Einstein vacuum equations, University of Pitts-
burg, Preprint (1984)
[4] L. B. Szabados, Quasi-Local energy-momentum and angular Momentum in General Relativity,
Liv. Rev. Rel. 2009-4 (2009)
[5] I. M. Benn, 1987 Conservation laws in arbitrary spacetimes, Ann. Inst. Henri Poincar´e, Section
AXXXVII, 67-91 (1987)
17

[6] A. Waldyr, Jr. Rodrigues, The nature of gravitational field and its legitimate energy-
momentum tensor, Rep. Math. Phys. 69, 265-279 (2012)
[7] L. L. So, J. M. Nester, H. Chen Energy-momentum density in small regions: the classical
pseudotensors, Class. Quantum Grav. 26, 085004 (2009)
[8] J. M. Nester, General pseudotensors and quasilocal quantities, Class. Quantum Grav. 21,
S261-S280 (2004)
[9] Y. -N. Obukhov, G. F. Rubilar, 2006 Invariant conserved currents in gravity theories with
local Lorentz and diffeomorpism symmetry, Phys. Rev. D 74, 064002 (2006)
[10] Y. -N. Obukhov, G. F. Rubilar, Covariance properties and regulatization of conserved currents
in tetrad gravity, Phys. Rev. D 73 124017 (2006)
[11] Y. -N. Obukhov, G. F. Rubilar, Invariant conserved currents: diffeomorpism and local Lorentz
symmetries, Phys. Rev. D 76, 124030 (2007)
[12] T. Dereli, R. W. Tucker, On the energy-momentum density of gravitational plane waves, Class.
Quantum Grav. 21, 1459-1465 (2004)
[13] D. A. Burton, R. W. Tucker, C. H. Wang, Spinning particles in scalar-tensor gravity Phys.
Lett. A 372, 3141-3144 (2008)
[14] T. Koivisto, Covariant conservation of energy momentum in modified gravities, Class. Quan-
tum Grav. 23, 4289-4296 (2006)
[15] C. Brans, R. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. D
125, 925-935 (1961)
[16] V. Faraoni, Cosmology in scalar-tensor gravity (Kluwer Academic Publishers, Dordrecht, 2004)
[17] Y. Fujii, K. Maeda, The scalar-tensor theory of gravitation (Cambridge Monographs on Math-
ematical Physics, Cambridge, 2007)
[18] B. Whitt, Fourth-order gravity as general relativity plus matter, Phys. Lett. B 145, 176-178
(1984)
[19] P. Teyssandier, Ph. Tourrenc, The Cauchy problem for the R+R2theories of gravity without
torsion, J. Math. Phys. (N. Y.) 24, 2793-2799 (1983)
[20] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, Modified gravity and cosmology, Phys. Rep.
513, 1-189 (2012)
[21] S. Nojiri, S. D. Odintsov, Unified cosmic history in modified gravity: From f(R) theory to
Lorentz non-invariant models, Phys. Rep. 505, 59-144 (2010)
18

[22] T. P. Sotiriou, V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451-497 (2010)
[23] A. Baykal, O. Delice, 2013 Multi-scalar-tensor equivalents for modified gravitational actions,
Phys. Rev. D 88, 084041 (2013)
[24] H. B. Hart, Conserved vectors in scalar-tensor gravitational theories, Phys. Rev. D 11, 960-962
(1975)
[25] H. B. Hart, Conservation laws and symmetry properties of scalar-tensor gravitational theories,
Phys. Rev. D 5, 1256-1262 (1972)
[26] D. Barraco, V. Hamity, The energy concept and the binding energy in a class of scalar-tensor
theories of gravity, Class. Quantum Grav. 11, 2113-2126 (1994)
[27] Y. -N. Obukhov, G. F. Rubilar, Invariant conserved currents for gravity, Phys. Lett. B 660,
240-246 (2008)
[28] Y. Nutku, 1969 The energy-momentum complex in the Brans-Dicke theory, ApJ 158, 991-996
(1969)
[29] M. Dubois-Violette, J. Madore, Conservation laws and integrability consitions for gravitational
and Yang-Mills field equations, Comm. Math. Phys. 108, 2013-223 (1987)
[30] I. M. Benn, R. W. Tucker, An introduction to spinors and geometry with applications in
physics, (IOP Publishing Ltd., Bristol, 1987)
[31] C. Romero, A. Barros, Does the Brans-Dicke theory of gravity go over to general relativity
when ω7→ ∞?, Phys. Lett. A 173, 243-246 (1993)
[32] N. Banerjee, S. Sen, Does Brans-Dicke theory always yield general relativity in the infinite ω
limit?, Phys. Rev. D 56, 1334-1337 (1997)
[33] V. Faraoni, Illusions of general relativity in Brans-Dicke gravity, Phys. Rev. D 59, 084021
(1999)
[34] A. Bhadra, K. K. Nandi, ωdependence of the scalar field in Brans-Dicke theory, Phys. Rev.
D64, 087501 (2001)
[35] F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman, Metric affine gauge theory of gravity:
Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys.
Rept. 258, 1-171 (1995)
[36] L. F. Abbott, S. Deser 1982 Stability of gravity with a cosmological constant, Nucl. Phys. B
195, 76-96 (1982)
[37] S. Deser, B. Tekin, Gravitational energy in quadratic curvature gravities, Phys. Rev. Lett. 89,
19

101101 (2002)
[38] H. Cebeci, ¨
O Sarıo˘glu, B. Tekin, Negative mass solutions in gravity, Phys. Rev. D 73, 06402
(2006)
[39] H. Cebeci, T. Dereli, Axi-dilaton gravity in D≥4 dimensional space-times with torsion, Phys.
Rev. D 71, 024016 (2005)
[40] C. H. Brans, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. D 125,
2194-2201 (1962)
[41] A. Bhadra, K. Sarkar, On static spherically symmetric solutions of the vacuum Brans-Dicke
theory, Gen. Relativ. Gravit. 37, 2189-2199 (2006)
[42] A. Bhadra, K. K. Nandi, Brans type II-IV solutions in the Einstein frame and physical inter-
pretation of constants in the solutions, Mod. Phys. Letts. A 16, 2079-2089 (2001)
[43] A. Baykal, ¨
O Delice, A unified approach to variational derivatives of modified gravitational
actions, Class. Quantum Grav. 28, 015014 (2011)
[44] A. Baykal, Variational derivatives of gravitational actions, Eur. Phys. J.-Plus 128, 125 (2013)
[45] A. Baykal, Energy definition for quadratic curvature gravities, Phys. Rev. D 86, 127501 (2012)
20