arXiv:0704.0986v1  [gr-qc]  7 Apr 2007
On reference frames in spacetime and
gravitational energy in freely falling frames
J. W. Maluf ∗, F. F. Faria and S. C. Ulhoa
Instituto de F́ısica,
Universidade de Braśılia
C. P. 04385
70.919-970 Braśılia DF, Brazil
Abstract
We consider the interpretation of tetrad fields as reference frames
in spacetime. Reference frames may be characterized by an antisym-
metric acceleration tensor, whose components are identified as the
inertial accelerations of the frame (the translational acceleration and
the frequency of rotation of the frame). This tensor is closely related to
gravitoelectromagnetic field quantities. We construct the set of tetrad
fields adapted to observers that are in free fall in the Schwarzschild
spacetime, and show that the gravitational energy-momentum con-
structed out of this set of tetrad fields, in the framework of the telepar-
allel equivalent of general relatrivity, vanishes. This result is in agree-
ment with the principle of equivalence, and may be taken as a condi-
tion for a viable definition of gravitational energy.
PACS numbers: 04.20.Cv, 04.20.Fy
(*) e-mail: wadih@unb.br
http://arxiv.org/abs/0704.0986v1
1 Introduction
It is a long-established practice in physics to describe the gravitational field
by means of theories invariant under local Lorentz transformations. This is
the case of the Einstein-Cartan theory, for instance, or more generally of the
metric-affine approach to the gravitational field [1]. In the latter formulation,
the theory of gravity is considered as a gauge theory of the Poincaré group.
The motivation for addressing theories of gravity by means of local Lorentz
(SO(3,1)) symmetry is partially due to the impact of the Yang-Mills gauge
theory in particle physics and quantum field theory. Because of the local
SO(3,1) symmetry, it is possible to assert that in such theories “all reference
frames are equivalent”.
The investigation of metric-affine theories of gravity is important because
one might have to go beyond the Riemannian formulation of general relativity
in order to deal with structures that pertain to a possible quantum theory
of gravity. The relevance of the Poincaré group and its representations in
quantum field theory is well known. In spite of the above mentioned feature
of the local SO(3,1) symmetry, there is no physical reason that prevents
the possibility of considering theories of gravity invariant under the global
Lorentz symmetry.
One theory that exhibits invariance under global SO(3,1) symmetry is
the teleparallel equivalent of general relativity (TEGR) [2, 3, 4, 5, 6, 7, 8].
The Lagrangian density of the theory is invariant under local SO(3,1) trans-
formations up to a nontrivial, nonvanishing total divergence [9], and for this
reason the local SO(3,1) group is not a symmetry of the theory. (From a
different perspective, the TEGR may be considered as a gauge theory for the
translation group [10].) Because of the global SO(3,1) symmetry, we must
ascribe an interpretation to six degrees of freedom of the tetrad field. In the
TEGR two sets of tetrad fields that yield the same spacetime metric tensor
are physically distinct. Thus we should interpret the tetrad fields as refer-
ence frames adapted to ideal observers in spacetime. Therefore two sets of
tetrad fields that are related by a local SO(3,1) transformation yield the same
metrical properties of the spacetime, but represent reference frames that are
characterized by different inertial accelerations. In a given gravitational field
configuration, the Schwarzschild spacetime, say, a moving observer or an ob-
server at rest are described by different sets of tetrad fields, and both sets
of tetrads are related by some sort of SO(3,1) transformation. Of course the
proper interpretation of the translational and rotational accelerations of a
frame makes sense at least in the case of asymptotically flat spacetimes.
In this paper we carry out an analysis of the inertial accelerations of a
frame in the context of the TEGR. The inertial accelerations are represented
by a second rank antisymmetric tensor under global SO(3,1) transforma-
tions that is coordinate independent. This tensor can be decomposed into
translational and rotational accelerations (the latter is in fact the rotational
frequency of the frame). By considering the weak field limit we will see that
there is a very interesting relationship between the translational acceleration
and rotational frequency of the frame, and electric and magnetic fields, re-
spectively. This relationship is explicitly investigated in the context of the
Kerr spacetime. The translational acceleration and rotational frequency that
are necessary no maintain a static frame in the spacetime are closely related
to the electric field of a point charge and to the magnetic field of a perfect
magnetic dipole, respectively. The present analysis is very much similar to
the usual formulation of gravitoelectromagnetism.
We consider the four-velocity of observers that are in free fall (radially)
in the Schwarzschild spacetime and construct the reference frame adapted to
such observers. We show that the expression for the gravitational energy-
momentum that arises in the framework of the TEGR [4, 5, 7] vanishes,
if evaluated in this frame. This is a very interesting result that shows the
consistency of the above definition with the principle of equivalence. The
local effects of gravity are not measured by an observer in free fall, who
defines a locally inertial reference frame. In this frame the acceleration of
the observer vanishes (section 3), and therefore he cannot measure neither
the gravitational force exerted on him nor the mass of the black hole. Thus
in a freely falling frame the gravitational energy should vanish. The tetrad
field that establishes the reference frame of an observer in free fall is related
to other (possibly static) frames by a frame transformation, not a coordinate
transformation. For instance, it is possible to establish a transformation from
the freely falling frame to a frame adapted to observers that are asympotically
at rest in the Schwarzschild spacetime, out of which we obtain the usual value
for the total gravitational energy of the spacetime. We believe that viable
definitions of gravitational energy-momentum should exhibit this feature.
Notation: spacetime indices µ, ν, ... and SO(3,1) indices a, b, ... run from
0 to 3. Time and space indices are indicated according to µ = 0, i, a =
(0), (i). The tetrad field is denoted by ea µ, and the torsion tensor reads
Taµν = ∂µeaν − ∂νeaµ. The flat, Minkowski spacetime metric tensor raises
and lowers tetrad indices and is fixed by ηab = eaµebνg
µν = (− + ++). The
determinant of the tetrad field is represented by e = det(ea µ).
2 The field equations of the TEGR
Einstein’s general relativity is determined by the field equations. The latter
may be written either in terms of the metric tensor or of the tetrad field.
The TEGR is a reformulation of Einstein’s general relativity in terms of
the tetrad field. Sometimes the theory is also called “tetrad gravity” [9].
The tetrad field is anyway necessary to describe the coupling of Dirac spinor
fields with the gravitational field. The formulation of general relativity in
a different geometrical framework allows a new insight into the theory, and
this is precisely what happens in the consideration of the TEGR.
The Lagrangian density for the gravitational field in the TEGR is given
L = −k e (
T abcTabc +
T abcTbac − T aTa)− LM
≡ −k eΣabcTabc − LM , (1)
where k = 1/(16π), and LM stands for the Lagrangian density for the matter
fields. As usual, tetrad fields convert spacetime into Lorentz indices and vice-
versa. The tensor Σabc is defined by
Σabc =
(T abc + T bac − T cab) +
(ηacT b − ηabT c) , (2)
and T a = T b b
a. The quadratic combination ΣabcTabc is proportional to the
scalar curvature R(e), except for a total divergence [7].
The field equations for the tetrad field read
eaλebµ∂ν(eΣ
bλν)− e(Σbν aTbνµ −
eaµTbcdΣ
bcd) =
eTaµ . (3)
where eTaµ = δLM/δe
aµ. It is possible to prove by explicit calculations that
the left hand side of Eq. (3) is exactly given by 1
e [Raµ(e)− 12eaµR(e)]. The
field equations above may be rewritten in the form
∂ν(eΣ
aλν) =
e ea µ(t
λµ + T λµ) , (4)
where
tλµ = k(4ΣbcλTbc
µ − gλµΣbcdTbcd) , (5)
is interpreted as the gravitational energy-momentum tensor [7].
The Lagrangian density defined by Eq. (1) is invariant under global
SO(3,1) transformations of the tetrad field. As we asserted before, un-
der local SO(3,1) transformations the purely gravitational part of Eq. (1),
−k eΣabcTabc, transforms into −k eΣabcTabc plus a nontrivial, nonvanishing
total divergence [9]. The integral of this total divergence in general is non-
vanishing, unless restrictive conditions are imposed on the Lorentz transfor-
mation matrices.
The Hamiltonian formulation of the TEGR is obtained by first establish-
ing the phase space variables. The Lagrangian density does not contain the
time derivative of the tetrad component ea0. Therefore this quantity will
arise as a Lagrange multiplier. The momentum canonically conjugated to
eai is given by Π
ai = δL/δėai. The Hamiltonian formulation is obtained by
rewriting the Lagrangian density in the form L = pq̇ − H , in terms of eai,
Πai and Lagrange multipliers. The Legendre transform can be successfuly
carried out, and the final form of the Hamiltonian density reads [11]
H = ea0C
a + αikΓ
ik + βkΓ
k , (6)
plus a surface term. αik and βk are Lagrange multipliers that (after solving
the field equations) are identified as αik = 1/2(Ti0k + Tk0i) and βk = T00k.
Ca, Γik and Γk are first class constraints.
The constraint Ca is written as Ca = −∂iΠai+ha, where ha is an intricate
expression of the field variables. The integral form of the constraint equation
Ca = 0 motivates the definition of the total energy-momentum four-vector
P a [4],
P a = −
d3x∂iΠ
ai . (7)
V is an arbitrary volume of the three-dimensional space. In the configuration
space we have
Πai = −4keΣa0i . (8)
The emergence of total divergences in the form of scalar or vector densities
is possible in the framework of theories constructed out of the torsion tensor.
Metric theories of gravity do not share this feature. We note that by making
λ = 0 in eq. (4) and identifying Πai in the left hand side of the latter, the
integral form of eq. (4) is written as
P a =
d3x e ea µ(t
0µ + T 0µ) . (9)
In empty spacetimes and in the framework of black holes P a does represent
the gravitational energy-momentum contained in a volume V of the three-
dimensional space. Several applications to well known gravitational field
configurations support this interpretation.
3 Reference frames in spacetime
A set of four orthonormal, linearly independent vector fields in spacetime
establish a reference frame. Altogether, they define a tetrad field ea µ, which
allows the projection of vectors and tensors in spacetime in the local frame
of an observer.
Each set of tetrad fields defines a class of reference frames [12]. If we
denote by xµ(s) the world line C of an observer in spacetime (s is the proper
time of the observer), and by uµ(s) = dxµ/ds its velocity along C, we identify
the observer’s velocity with the a = (0) component of ea
µ. Thus uµ(s) =
µ along C. The acceleration aµ of the observer is given by the absolute
derivative of uµ along C,
De(0)
= uα∇αe(0) µ , (10)
where the covariant derivative is constructed out of the Christoffel symbols.
Thus ea
µ determines the velocity and acceleration along the worldline of an
observer adapted to the frame. Therefore a given set of tetrad fields, for which
µ describes a congruence of timelike curves, is adapted to a particular
class of observers, namely, to observers characterized by the velocity field
uµ = e(0)
µ, endowed with acceleration aµ. If ea µ → δaµ in the limit r → ∞,
then ea µ is adapted to static observers at spacelike infinity.
A geometrical characterization of tetrad fields as an observer’s frame can
be given by considering the acceleration of the frame along an arbitrary
path xµ(s) of the observer in spacetime. The acceleration of the frame is
determined by the absolute derivative of ea
µ along xµ(s). Thus, assuming
that the observer carries an orthonormal tetrad frame ea
µ, the acceleration
of the latter along the path is given by [13, 14]
µ , (11)
where φab is the antisymmetric acceleration tensor. According to Refs. [13,
14], in analogy with the Faraday tensor we can identify φab → (a,Ω), where
a is the translational acceleration (φ(0)(i) = a(i)) and Ω is the frequency
of rotation of the local spatial frame with respect to a nonrotating (Fermi-
Walker transported [12]) frame. It follows from Eq. (11) that
b = eb µ
= eb µ u
λ∇λea µ . (12)
Therefore given any set of tetrad fields for an arbitrary gravitational field
configuration, its geometrical interpretation can be obtained by suitably in-
terpreting the velocity field uµ = e(0)
µ and the acceleration tensor φab. The
acceleration vector aµ defined by Eq. (10) may be projected on a frame in
order to yield
ab = eb µa
µ = eb µu
α∇αe(0) µ = φ(0) b . (13)
Thus aµ and φ(0)(i) are not different accelerations of the frame.
The expression of aµ given by Eq. (10) may be rewritten as
aµ = uα∇αe(0) µ = uα∇αuµ =
, (14)
where Γ
αβ are the Christoffel symbols. We see that if u
µ = e(0)
µ represents
a geodesic trajectory, then the frame is in free fall and aµ = φ(0)(i) = 0.
Therefore we conclude that nonvanishing values of the latter quantities do
represent inertial accelerations of the frame.
In view of the orthogonality of the tetrads we write Eq. (12) as φa
−uλea µ∇λeb µ, where ∇λeb µ = ∂λeb µ − Γσλµeb σ. Now we take into account
the identity ∂λe
µ − Γσλµeb σ + 0ωλ b cec µ = 0, where 0ωλ b c is the metric
compatible, torsion free Levi-Civita connection, and express φa
b according
b = e(0)
µ( 0ωµ
a) . (15)
At last we consider the identity 0ωµ
b = −Kµ a b, where −Kµ a b is the
contortion tensor defined by
Kµab =
ν(Tλµν + Tνλµ + Tµλν) , (16)
and Tλµν = e
λTaµν (see, for instance, Eq. (4) of Ref. [7]; the identity is
obtained by requiring the vanishing of a general SO(3,1) connection ωµab, or
by direct calculation). After simple manipulations we finally obtain
φab =
[T(0)ab + Ta(0)b − Tb(0)a] . (17)
The expression above is clearly not invariant under local SO(3,1) trans-
formations, but is invariant under coordinate transformations. The values of
φab for a given tetrad field may be used to characterize the frame. We recall
that we are assuming the observer to carry the set of tetrad fields along xµ(s),
for which we have uµ = e(0)
µ. We interpret φab as the inertial accelerations
along xµ(s).
Two simple, straightforward applications of Eq. (17) are the following:
(i) The tetrad field adapted to observers at rest in Minkowski spacetime is
given by ea µ(ct, x, y, z) = δ
µ. We consider a time-dependent boost in the x
direction, say, after which the tetrad field reads
ea µ(ct, x, y, z) =
γ −βγ 0 0
−βγ γ 0 0
0 0 1 0
0 0 0 1
, (18)
where γ = (1 − β2)−1/2, β = v/c and v = v(t). The frame above is
then adapted to observers whose four-velocity is uµ = e(0)
µ(ct, x, y, z) =
(γ, βγ, 0, 0). After simple calculations we obtain
φ(0)(1) =
[βγ] =
1− v2/c2
, (19)
φ(0)(2) = 0 ,
φ(0)(3) = 0 ,
and φ(i)(j) = 0.
(ii) A frame adapted to an observer in Minkowski spacetime whose four-
velocity is e(0)
µ = (1, 0, 0, 0) and which rotates around the z axis, say, reads
ea µ(ct, x, y, z) =
1 0 0 0
0 cosω(t) − sinω(t) 0
0 sinω(t) cosω(t) 0
0 0 0 1
. (20)
It is easy to carry out the simple calculations and obtain
φ(2)(3) = 0 , (21)
φ(3)(1) = 0 ,
φ(1)(2) = −
and φ(0)(i) = 0. Together with the discussion regarding Eq. (14), the exam-
ples above support the interpretation of φa
b as the inertial accelerations of
the frame.
4 A freely falling frame in the Schwarzschild
spacetime
We will consider in this section a frame that is in free fall in the Schwarzschild
spacetime, namely, that is radially accelerated towards the center of the black
hole. We will take into account the kinematical quantities discussed the
preceeding section, in order to illustrate the construction of the tetrad field.
The spacetime is described by the line element
ds2 = −α−2dt2 + α2dr2 + r2(dθ2 + sin2 θdφ2) , (22)
where
α−2 = 1−
. (23)
Let us define the quantity β,
= (1− α−2)1/2 , (24)
which will be useful in the following.
An observer that is in radial free fall in the Schwarzschild spacetime is
endowed with the four-velocity [15]
, 0, 0
. (25)
The simplest set of tetrad fields that satisfies the condition
α = uα , (26)
is given by
eaµ =
−1 −α2β 0 0
β sin θ cosφ α2 sin θ cos φ r cos θ cosφ −r sin θ sinφ
β sin θ sinφ α2 sin θ sinφ r cos θ sinφ r sin θ cos φ
β cos θ α2 cos θ −r sin θ 0
. (27)
We recall that the index a labels the lines, and µ the columns. Since the
frame is in free fall the equation φ(0)(i) = 0 is satisfied. It is not difficult to
show that this set of tetrad fields also satisfies the conditions
φ(i)(j) =
[T(0)(i)(j) + T(i)(0)(j) − T(j)(0)(i)] = 0 . (28)
Three of the four conditions established by Eq. (26) are more relevant
for our purposes, namely, the three components of the frame velocity in
the three-dimensional space, ui = e(0)
i. Together with the three conditions
determined by Eq. (28), we have six conditions on the frame. We may assert
that these six conditions completely fix the structure of the tetrad field, even
though Eq. (28) has been verified a posteriori. Therefore Eq. (27) describes
a nonrotating frame in radial free fall in the Schwarzschild spacetime.
We will evaluate the gravitational energy-momentum out of the tetrad
field above, but will omit the details of the calculations which are alge-
braically long, but otherwise simple. The nonvanishing components of the
torsion tensor are
T001 = −β∂rβ (29)
T101 = −α2∂rβ
T202 = −rβ
T303 = −rβ sin2 θ
T212 = r(1− α2)
T313 = r(1− α2) sin2 θ .
The gravitational energy contained within a spherical surface of constant
radius is given by
P (0) = −
dSj Π
(0)j = 4k
dS1 e(e
001 + e(0) 1Σ
101) , (30)
where
Σ001 =
(g00g11g22T212 + g
00g11g33T313) , (31)
Σ101 = −
(g00g11g22T202 + g
00g11g33T303) .
We find that
e(e(0) 0Σ
001 + e(0) 1Σ
101) = r sin θ(α2 − 1− α2β2) (32)
= 0 ,
and therefore the gravitational energy contained within a surface of constant
radius as well as the total gravitational energy of the spacetime vanishes, if
evaluated in the frame of a freely falling observer. This is a very interesting
property of the whole formalism described in section 2. The vanishing of the
gravitational energy for freely falling observers is a feature that is consistent
with (and a consequence of) the principle of equivalence, since local effects of
gravity are not measured by observers in free fall. For other frames that are
related to Eq. (27) by a local Lorentz transformation we obtain nonvanishing
values of P (0). In particular, the total gravitational energy calculated out of
frames such that ea µ(t, x, y, z) → δaµ in the asymptotic limit r → ∞ is
exactly P (0) = m [4]. The latter tetrad field is adapted to observers at rest
at spacelike infinity. Thus the vanishing of gravitational energy in freely
falling frames shows that the localizability of the gravitational energy is not
inconsistent with with the principle of equivalence. The result given by Eqs.
(30-32) is a very good example of the frame dependence of the gravitational
energy definition (7).
It can be easily verified that the gravitational momentum components
P (1) and P (2) vanish in view of integrals like
0 dφ sinφ = 0 =
0 dφ cosφ,
whereas P (3) vanishes due to
0 dθ sin θ cos θ = 0.
It is important to remark that in general the vanishing of φab does not
imply the vanishing of P a. For an observer at rest at spacelike infinity the
total gravitational energy is nonvanishing, whereas for these observers we
have φab ∼= 0 (in the limit r → ∞; see next section).
5 Static frames in the Kerr spacetime
Another interesting application of the definitions of velocity and inertial ac-
celeration of a frame discussed in section 3 is the analysis of a static frame
in Kerr’s spacetime. The latter is established by the line element
ds2 = −
dt2 −
2χ sin2 θ
dφ dt+
dr2 (33)
+ρ2dθ2 +
Σ2 sin2 θ
dφ2 ,
with the following definitions:
∆ = r2 + a2 − 2mr , (34)
ρ2 = r2 + a2 cos2 θ ,
Σ2 = (r2 + a2)2 −∆a2 sin2 θ ,
ψ2 = ∆− a2 sin2 θ ,
χ = 2amr .
A static reference frame in Kerr’s spacetime is defined by the congruence
of timelike curves uµ(s) such that ui = 0, namely, the spatial velocity of the
observers is zero with respect to static observers at spacelike infinity. Since
we identify ui = e(0)
i, a static reference frame is established by the condition
i = 0 . (35)
In view of the orthogonality of the tetrads, the equation above implies e(k) 0 =
0. This latter equation remains satisfied after a local rotation of the frame,
ẽ(k) 0 = Λ
0 = 0. Therefore condition (35) determines the static char-
acter of the frame, up to an orientation of the frame in the three-dimensional
space.
A simple form for the tetrad field that satisfies Eq. (35) (or, equivalently,
e(k) 0 = 0) reads
eaµ =
−A 0 0 −B
0 C sin θ cosφ ρ cos θ cos φ −D sin θ sinφ
0 C sin θ sin φ ρ cos θ sinφ D sin θ cosφ
0 C cos θ −ρ sin θ 0
, (36)
with the following definitions
, (37)
χ sin2 θ
In the expression of D we have
Λ = (ψ2Σ2 + χ2 sin2 θ)1/2 .
We are interested in the calculation of φab given by Eq. (17), and for this
purpose it is useful to work with the inverse tetrad field ea
µ. It reads
sin θ sinφ − ρχ
sin θ cosφ 0
sin θ cos φ
sin θ sin φ
cos θ
cos θ cosφ 1
cos θ sinφ −1
sin θ
0 −ρψ
sin θ
sin θ
, (38)
where now the index a labels the columns, and µ the lines.
The frame determined by Eqs. (36) and (38) is valid in the region outside
the ergosphere. The function ψ2 = ∆ − a2 sin2 θ vanishes over the external
surface of the ergosphere (defined by r = r⋆ = m+
m2 − a2 cos2 θ; over this
surface g00 = 0), and we see that various components of Eqs. (36) and (38)
are not well defined over this surface. It is well known that it is not possible
to maintain static observers inside the ergosphere of the Kerr spacetime.
By inspecting Eq. (38) we see that for large values of r we have
µ(t, r, θ, φ) ∼= (0, cos θ,−(1/r) sin θ, 0) ,
µ(t, x, y, z) ∼= (0, 0, 0, 1) . (39)
Therefore we may assert that the frame given by Eq. (37) is characterized by
the following properties: (i) the frame is static, because Eq. (35) is verified;
(ii) the e(3)
µ components are oriented along the symmetry axis of the black
hole (the z direction). The second condition is ultimately reponsible for the
simple form of Eq. (36).
The evaluation of φab is long but straightforward, and for this reason
we will omit the details of the calculations. For convenience of notation we
define the vectors
r̂ = sin θ cosφ x̂+ sin θ sinφ ŷ + cos θ ẑ (40)
θ̂ = cos θ cos φ x̂+ cos θ sin φ ŷ− sin θ ẑ
which have well defined meaning as unit vectors in the asymptotic limit
r → ∞. We also define the three-dimensional vectors
a = (φ01, φ02, φ03) , (41)
Ω = (φ23, φ31, φ12) . (42)
We obtain the following expressions for a and Ω:
sin θ cos θ θ̂
, (43)
Ω = −
cos θ r̂+
sin θ ∂r
sin θ ∂θ
r̂ . (44)
The specific functional form of the vectors above completely characterize
the frame determined by Eq. (36). The determination of a and Ω is equiva-
lent to the fixation of six components of the tetrad field. Equations (43) and
(44) represent the inertial accelerations that one must exert on the frame
in order to verify that (i) the frame is static (condition (35)), and that (ii)
the e(3)
µ components of the tetrad field asymptotically coincides with the
symmetry axis of the black hole.
The form of a and Ω for large values of r is very interesting. It is easy to
verify that in the limit r → ∞ we obtain
r̂ , (45)
Ω ∼= −
2 cos θ r̂+ sin θ θ̂
. (46)
After the identificationsm↔ q and 4πma↔ m̄, where q is the electric charge
and m̄ is the magnetic dipole moment, equations (45) and (46) resemble the
electric field of a point charge and the magnetic field of a perfect dipole
that points in the z direction, respectively. These equations represent a
manifestation of gravitoelectromagnetism.
If we abandon the statical condition given by Eq. (35), an observer lo-
cated at a position (r, θ, φ) will be subject to an acceleration −a and to a
rotational motion determined by −Ω = ΩD, which is the dragging frequency
of the frame. Thus the gravitomagnetic effect is locally equivalent to iner-
tial effects in a frame rotating with frequency −ΩD, the latter having the
magnetic dipole moment structure given by Eq. (46). This is precisely the
gravitational Larmor’s theorem, discussed in Ref. [16].
The emergence of gravitoelectromagnetic (GEM) field quantities in the
context of the acceleration tensor φab presents no difference with respect to
the usual approach in the literature. Let us assume that tetrad field satisfies
the boundary conditions
ea µ ∼= δaµ +
ha µ , (47)
where ha µ is the perturbation of the flat space-time tetrad field in the limit
r → ∞, and that in this limit the SO(3,1) and spacetime indices acquire the
same significance. It is straightforward to verify that in this case we have
φ(0)(i) ∼= −∂i
, (48)
φ(i)(j) ∼= −
. (49)
Thus we identify
h00 , (50)
Ai = −
h0i . (51)
The identification above is equivalent to the one usually made in the litera-
ture, namely, Φ = (1/4)h̄00 and Ai = −(1/2)h̄0i [17], where h̄µν is the trace-
reversed field quantity defined by h̄µν = hµν−(1/2)ηµνh, and h = ηµνhµν . The
latter identification is made directly in the weak field form of the metric ten-
sor of a slowly rotating source. Assuming that h00 = 2Φ/c
2 and hij = δijh00,
where c is the speed of light (according to Eq. (1.4) of Ref. [17]), we obtain
h̄00 = 2h00, and therefore V = (1/4)h̄00. To our knowledge, the identification
of the GEM field quantities out of the tensor φab has not been addressed in
the literature so far.
6 Comments
Gravity theories invariant under the global SO(3,1) group are physically ac-
ceptable. The gravitational field equations determine the gravitational field,
not the frame. A given gravitational field configuration admits an infinity
of frames which in general are distinct from each other. We know that the
physical properties of a system are different in a static and in an accelerated
frame, for instance, and this feature should also hold in general relativity.
The gravitational energy-momentum that is defined in the realm of the
TEGR is frame dependent. This issue has been partially discussed before
in Refs. [7, 8], and also in Ref. [9]. This dependence is considered here to
be a natural property of the definition. The frame may be characterized by
the six components of the antisymmetric tensor φab, defined by Eq. (17),
which determine the translational acceleration and rotational frequency of
the frame, and which resembles the electric field of a point charge and the
magnetic field of a dipole, respectively, in the weak field limit of the Kerr
spacetime (in the consideration of a static frame).
In section 4 we have shown that the gravitational energy-momentum cal-
culated out of a frame that is nonrotating and freely falling in the Schwarzschild
spacetime vanishes. We expect this property to hold in the consideration of
a general spacetime geometry, in which case the analysis is somewhat more
complicated, because the frame is expected not to rotate with respect to a
Fermi-Walker transported frame. In general the construction of the latter
frame is not trivial.
It is clear that if the gravitational energy-momentum definition were in-
variant under local Lorentz transformations, we would not arrive at the result
of section 4, since the the value of P a on a three-dimensional volume V would
be the same for all frames, and presumably nonvanishing.
A common critique of the localizability of gravitational energy is that the
latter is unattainable because of the principle of equivalence. In this paper we
have seen that this is not the case. Definition (7) for the gravitational energy-
momentum yields the expected results both for observers asymptotically at
rest and for freely falling observers.
Acknowledgement
J. W. M. is grateful to G. F. Rubilar for helpful discussions on reference
frames. This work was supported in part by CNPQ (Brazil).
References
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by P. G. Bergmann and V. de Sabbata (Plenum, New York, 1980).
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Branco, Phys. Rev. D 65, 124001 (2002).
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[17] B. Mashhoon, Gravitoelectromagnetism: a Brief Review [gr-qc/0311030].

  We consider the interpretation of tetrad fields as reference frames in
spacetime. Reference frames may be characterized by an antisymmetric
acceleration tensor, whose components are identified as the inertial
accelerations of the frame (the translational acceleration and the frequency of
rotation of the frame). This tensor is closely related to
gravitoelectromagnetic field quantities. We construct the set of tetrad fields
adapted to observers that are in free fall in the Schwarzschild spacetime, and
show that the gravitational energy-momentum constructed out of this set of
tetrad fields, in the framework of the teleparallel equivalent of general
relatrivity, vanishes. This result is in agreement with the principle of
equivalence, and may be taken as a condition for a viable definition of
gravitational energy.

Introduction
It is a long-established practice in physics to describe the gravitational field
by means of theories invariant under local Lorentz transformations. This is
the case of the Einstein-Cartan theory, for instance, or more generally of the
metric-affine approach to the gravitational field [1]. In the latter formulation,
the theory of gravity is considered as a gauge theory of the Poincaré group.
The motivation for addressing theories of gravity by means of local Lorentz
(SO(3,1)) symmetry is partially due to the impact of the Yang-Mills gauge
theory in particle physics and quantum field theory. Because of the local
SO(3,1) symmetry, it is possible to assert that in such theories “all reference
frames are equivalent”.
The investigation of metric-affine theories of gravity is important because
one might have to go beyond the Riemannian formulation of general relativity
in order to deal with structures that pertain to a possible quantum theory
of gravity. The relevance of the Poincaré group and its representations in
quantum field theory is well known. In spite of the above mentioned feature
of the local SO(3,1) symmetry, there is no physical reason that prevents
the possibility of considering theories of gravity invariant under the global
Lorentz symmetry.
One theory that exhibits invariance under global SO(3,1) symmetry is
the teleparallel equivalent of general relativity (TEGR) [2, 3, 4, 5, 6, 7, 8].
The Lagrangian density of the theory is invariant under local SO(3,1) trans-
formations up to a nontrivial, nonvanishing total divergence [9], and for this
reason the local SO(3,1) group is not a symmetry of the theory. (From a
different perspective, the TEGR may be considered as a gauge theory for the
translation group [10].) Because of the global SO(3,1) symmetry, we must
ascribe an interpretation to six degrees of freedom of the tetrad field. In the
TEGR two sets of tetrad fields that yield the same spacetime metric tensor
are physically distinct. Thus we should interpret the tetrad fields as refer-
ence frames adapted to ideal observers in spacetime. Therefore two sets of
tetrad fields that are related by a local SO(3,1) transformation yield the same
metrical properties of the spacetime, but represent reference frames that are
characterized by different inertial accelerations. In a given gravitational field
configuration, the Schwarzschild spacetime, say, a moving observer or an ob-
server at rest are described by different sets of tetrad fields, and both sets
of tetrads are related by some sort of SO(3,1) transformation. Of course the
proper interpretation of the translational and rotational accelerations of a
frame makes sense at least in the case of asymptotically flat spacetimes.
In this paper we carry out an analysis of the inertial accelerations of a
frame in the context of the TEGR. The inertial accelerations are represented
by a second rank antisymmetric tensor under global SO(3,1) transforma-
tions that is coordinate independent. This tensor can be decomposed into
translational and rotational accelerations (the latter is in fact the rotational
frequency of the frame). By considering the weak field limit we will see that
there is a very interesting relationship between the translational acceleration
and rotational frequency of the frame, and electric and magnetic fields, re-
spectively. This relationship is explicitly investigated in the context of the
Kerr spacetime. The translational acceleration and rotational frequency that
are necessary no maintain a static frame in the spacetime are closely related
to the electric field of a point charge and to the magnetic field of a perfect
magnetic dipole, respectively. The present analysis is very much similar to
the usual formulation of gravitoelectromagnetism.
We consider the four-velocity of observers that are in free fall (radially)
in the Schwarzschild spacetime and construct the reference frame adapted to
such observers. We show that the expression for the gravitational energy-
momentum that arises in the framework of the TEGR [4, 5, 7] vanishes,
if evaluated in this frame. This is a very interesting result that shows the
consistency of the above definition with the principle of equivalence. The
local effects of gravity are not measured by an observer in free fall, who
defines a locally inertial reference frame. In this frame the acceleration of
the observer vanishes (section 3), and therefore he cannot measure neither
the gravitational force exerted on him nor the mass of the black hole. Thus
in a freely falling frame the gravitational energy should vanish. The tetrad
field that establishes the reference frame of an observer in free fall is related
to other (possibly static) frames by a frame transformation, not a coordinate
transformation. For instance, it is possible to establish a transformation from
the freely falling frame to a frame adapted to observers that are asympotically
at rest in the Schwarzschild spacetime, out of which we obtain the usual value
for the total gravitational energy of the spacetime. We believe that viable
definitions of gravitational energy-momentum should exhibit this feature.
Notation: spacetime indices µ, ν, ... and SO(3,1) indices a, b, ... run from
0 to 3. Time and space indices are indicated according to µ = 0, i, a =
(0), (i). The tetrad field is denoted by ea µ, and the torsion tensor reads
Taµν = ∂µeaν − ∂νeaµ. The flat, Minkowski spacetime metric tensor raises
and lowers tetrad indices and is fixed by ηab = eaµebνg
µν = (− + ++). The
determinant of the tetrad field is represented by e = det(ea µ).
2 The field equations of the TEGR
Einstein’s general relativity is determined by the field equations. The latter
may be written either in terms of the metric tensor or of the tetrad field.
The TEGR is a reformulation of Einstein’s general relativity in terms of
the tetrad field. Sometimes the theory is also called “tetrad gravity” [9].
The tetrad field is anyway necessary to describe the coupling of Dirac spinor
fields with the gravitational field. The formulation of general relativity in
a different geometrical framework allows a new insight into the theory, and
this is precisely what happens in the consideration of the TEGR.
The Lagrangian density for the gravitational field in the TEGR is given
L = −k e (
T abcTabc +
T abcTbac − T aTa)− LM
≡ −k eΣabcTabc − LM , (1)
where k = 1/(16π), and LM stands for the Lagrangian density for the matter
fields. As usual, tetrad fields convert spacetime into Lorentz indices and vice-
versa. The tensor Σabc is defined by
Σabc =
(T abc + T bac − T cab) +
(ηacT b − ηabT c) , (2)
and T a = T b b
a. The quadratic combination ΣabcTabc is proportional to the
scalar curvature R(e), except for a total divergence [7].
The field equations for the tetrad field read
eaλebµ∂ν(eΣ
bλν)− e(Σbν aTbνµ −
eaµTbcdΣ
bcd) =
eTaµ . (3)
where eTaµ = δLM/δe
aµ. It is possible to prove by explicit calculations that
the left hand side of Eq. (3) is exactly given by 1
e [Raµ(e)− 12eaµR(e)]. The
field equations above may be rewritten in the form
∂ν(eΣ
aλν) =
e ea µ(t
λµ + T λµ) , (4)
where
tλµ = k(4ΣbcλTbc
µ − gλµΣbcdTbcd) , (5)
is interpreted as the gravitational energy-momentum tensor [7].
The Lagrangian density defined by Eq. (1) is invariant under global
SO(3,1) transformations of the tetrad field. As we asserted before, un-
der local SO(3,1) transformations the purely gravitational part of Eq. (1),
−k eΣabcTabc, transforms into −k eΣabcTabc plus a nontrivial, nonvanishing
total divergence [9]. The integral of this total divergence in general is non-
vanishing, unless restrictive conditions are imposed on the Lorentz transfor-
mation matrices.
The Hamiltonian formulation of the TEGR is obtained by first establish-
ing the phase space variables. The Lagrangian density does not contain the
time derivative of the tetrad component ea0. Therefore this quantity will
arise as a Lagrange multiplier. The momentum canonically conjugated to
eai is given by Π
ai = δL/δėai. The Hamiltonian formulation is obtained by
rewriting the Lagrangian density in the form L = pq̇ − H , in terms of eai,
Πai and Lagrange multipliers. The Legendre transform can be successfuly
carried out, and the final form of the Hamiltonian density reads [11]
H = ea0C
a + αikΓ
ik + βkΓ
k , (6)
plus a surface term. αik and βk are Lagrange multipliers that (after solving
the field equations) are identified as αik = 1/2(Ti0k + Tk0i) and βk = T00k.
Ca, Γik and Γk are first class constraints.
The constraint Ca is written as Ca = −∂iΠai+ha, where ha is an intricate
expression of the field variables. The integral form of the constraint equation
Ca = 0 motivates the definition of the total energy-momentum four-vector
P a [4],
P a = −
d3x∂iΠ
ai . (7)
V is an arbitrary volume of the three-dimensional space. In the configuration
space we have
Πai = −4keΣa0i . (8)
The emergence of total divergences in the form of scalar or vector densities
is possible in the framework of theories constructed out of the torsion tensor.
Metric theories of gravity do not share this feature. We note that by making
λ = 0 in eq. (4) and identifying Πai in the left hand side of the latter, the
integral form of eq. (4) is written as
P a =
d3x e ea µ(t
0µ + T 0µ) . (9)
In empty spacetimes and in the framework of black holes P a does represent
the gravitational energy-momentum contained in a volume V of the three-
dimensional space. Several applications to well known gravitational field
configurations support this interpretation.
3 Reference frames in spacetime
A set of four orthonormal, linearly independent vector fields in spacetime
establish a reference frame. Altogether, they define a tetrad field ea µ, which
allows the projection of vectors and tensors in spacetime in the local frame
of an observer.
Each set of tetrad fields defines a class of reference frames [12]. If we
denote by xµ(s) the world line C of an observer in spacetime (s is the proper
time of the observer), and by uµ(s) = dxµ/ds its velocity along C, we identify
the observer’s velocity with the a = (0) component of ea
µ. Thus uµ(s) =
µ along C. The acceleration aµ of the observer is given by the absolute
derivative of uµ along C,
De(0)
= uα∇αe(0) µ , (10)
where the covariant derivative is constructed out of the Christoffel symbols.
Thus ea
µ determines the velocity and acceleration along the worldline of an
observer adapted to the frame. Therefore a given set of tetrad fields, for which
µ describes a congruence of timelike curves, is adapted to a particular
class of observers, namely, to observers characterized by the velocity field
uµ = e(0)
µ, endowed with acceleration aµ. If ea µ → δaµ in the limit r → ∞,
then ea µ is adapted to static observers at spacelike infinity.
A geometrical characterization of tetrad fields as an observer’s frame can
be given by considering the acceleration of the frame along an arbitrary
path xµ(s) of the observer in spacetime. The acceleration of the frame is
determined by the absolute derivative of ea
µ along xµ(s). Thus, assuming
that the observer carries an orthonormal tetrad frame ea
µ, the acceleration
of the latter along the path is given by [13, 14]
µ , (11)
where φab is the antisymmetric acceleration tensor. According to Refs. [13,
14], in analogy with the Faraday tensor we can identify φab → (a,Ω), where
a is the translational acceleration (φ(0)(i) = a(i)) and Ω is the frequency
of rotation of the local spatial frame with respect to a nonrotating (Fermi-
Walker transported [12]) frame. It follows from Eq. (11) that
b = eb µ
= eb µ u
λ∇λea µ . (12)
Therefore given any set of tetrad fields for an arbitrary gravitational field
configuration, its geometrical interpretation can be obtained by suitably in-
terpreting the velocity field uµ = e(0)
µ and the acceleration tensor φab. The
acceleration vector aµ defined by Eq. (10) may be projected on a frame in
order to yield
ab = eb µa
µ = eb µu
α∇αe(0) µ = φ(0) b . (13)
Thus aµ and φ(0)(i) are not different accelerations of the frame.
The expression of aµ given by Eq. (10) may be rewritten as
aµ = uα∇αe(0) µ = uα∇αuµ =
, (14)
where Γ
αβ are the Christoffel symbols. We see that if u
µ = e(0)
µ represents
a geodesic trajectory, then the frame is in free fall and aµ = φ(0)(i) = 0.
Therefore we conclude that nonvanishing values of the latter quantities do
represent inertial accelerations of the frame.
In view of the orthogonality of the tetrads we write Eq. (12) as φa
−uλea µ∇λeb µ, where ∇λeb µ = ∂λeb µ − Γσλµeb σ. Now we take into account
the identity ∂λe
µ − Γσλµeb σ + 0ωλ b cec µ = 0, where 0ωλ b c is the metric
compatible, torsion free Levi-Civita connection, and express φa
b according
b = e(0)
µ( 0ωµ
a) . (15)
At last we consider the identity 0ωµ
b = −Kµ a b, where −Kµ a b is the
contortion tensor defined by
Kµab =
ν(Tλµν + Tνλµ + Tµλν) , (16)
and Tλµν = e
λTaµν (see, for instance, Eq. (4) of Ref. [7]; the identity is
obtained by requiring the vanishing of a general SO(3,1) connection ωµab, or
by direct calculation). After simple manipulations we finally obtain
φab =
[T(0)ab + Ta(0)b − Tb(0)a] . (17)
The expression above is clearly not invariant under local SO(3,1) trans-
formations, but is invariant under coordinate transformations. The values of
φab for a given tetrad field may be used to characterize the frame. We recall
that we are assuming the observer to carry the set of tetrad fields along xµ(s),
for which we have uµ = e(0)
µ. We interpret φab as the inertial accelerations
along xµ(s).
Two simple, straightforward applications of Eq. (17) are the following:
(i) The tetrad field adapted to observers at rest in Minkowski spacetime is
given by ea µ(ct, x, y, z) = δ
µ. We consider a time-dependent boost in the x
direction, say, after which the tetrad field reads
ea µ(ct, x, y, z) =
γ −βγ 0 0
−βγ γ 0 0
0 0 1 0
0 0 0 1
, (18)
where γ = (1 − β2)−1/2, β = v/c and v = v(t). The frame above is
then adapted to observers whose four-velocity is uµ = e(0)
µ(ct, x, y, z) =
(γ, βγ, 0, 0). After simple calculations we obtain
φ(0)(1) =
[βγ] =
1− v2/c2
, (19)
φ(0)(2) = 0 ,
φ(0)(3) = 0 ,
and φ(i)(j) = 0.
(ii) A frame adapted to an observer in Minkowski spacetime whose four-
velocity is e(0)
µ = (1, 0, 0, 0) and which rotates around the z axis, say, reads
ea µ(ct, x, y, z) =
1 0 0 0
0 cosω(t) − sinω(t) 0
0 sinω(t) cosω(t) 0
0 0 0 1
. (20)
It is easy to carry out the simple calculations and obtain
φ(2)(3) = 0 , (21)
φ(3)(1) = 0 ,
φ(1)(2) = −
and φ(0)(i) = 0. Together with the discussion regarding Eq. (14), the exam-
ples above support the interpretation of φa
b as the inertial accelerations of
the frame.
4 A freely falling frame in the Schwarzschild
spacetime
We will consider in this section a frame that is in free fall in the Schwarzschild
spacetime, namely, that is radially accelerated towards the center of the black
hole. We will take into account the kinematical quantities discussed the
preceeding section, in order to illustrate the construction of the tetrad field.
The spacetime is described by the line element
ds2 = −α−2dt2 + α2dr2 + r2(dθ2 + sin2 θdφ2) , (22)
where
α−2 = 1−
. (23)
Let us define the quantity β,
= (1− α−2)1/2 , (24)
which will be useful in the following.
An observer that is in radial free fall in the Schwarzschild spacetime is
endowed with the four-velocity [15]
, 0, 0
. (25)
The simplest set of tetrad fields that satisfies the condition
α = uα , (26)
is given by
eaµ =
−1 −α2β 0 0
β sin θ cosφ α2 sin θ cos φ r cos θ cosφ −r sin θ sinφ
β sin θ sinφ α2 sin θ sinφ r cos θ sinφ r sin θ cos φ
β cos θ α2 cos θ −r sin θ 0
. (27)
We recall that the index a labels the lines, and µ the columns. Since the
frame is in free fall the equation φ(0)(i) = 0 is satisfied. It is not difficult to
show that this set of tetrad fields also satisfies the conditions
φ(i)(j) =
[T(0)(i)(j) + T(i)(0)(j) − T(j)(0)(i)] = 0 . (28)
Three of the four conditions established by Eq. (26) are more relevant
for our purposes, namely, the three components of the frame velocity in
the three-dimensional space, ui = e(0)
i. Together with the three conditions
determined by Eq. (28), we have six conditions on the frame. We may assert
that these six conditions completely fix the structure of the tetrad field, even
though Eq. (28) has been verified a posteriori. Therefore Eq. (27) describes
a nonrotating frame in radial free fall in the Schwarzschild spacetime.
We will evaluate the gravitational energy-momentum out of the tetrad
field above, but will omit the details of the calculations which are alge-
braically long, but otherwise simple. The nonvanishing components of the
torsion tensor are
T001 = −β∂rβ (29)
T101 = −α2∂rβ
T202 = −rβ
T303 = −rβ sin2 θ
T212 = r(1− α2)
T313 = r(1− α2) sin2 θ .
The gravitational energy contained within a spherical surface of constant
radius is given by
P (0) = −
dSj Π
(0)j = 4k
dS1 e(e
001 + e(0) 1Σ
101) , (30)
where
Σ001 =
(g00g11g22T212 + g
00g11g33T313) , (31)
Σ101 = −
(g00g11g22T202 + g
00g11g33T303) .
We find that
e(e(0) 0Σ
001 + e(0) 1Σ
101) = r sin θ(α2 − 1− α2β2) (32)
= 0 ,
and therefore the gravitational energy contained within a surface of constant
radius as well as the total gravitational energy of the spacetime vanishes, if
evaluated in the frame of a freely falling observer. This is a very interesting
property of the whole formalism described in section 2. The vanishing of the
gravitational energy for freely falling observers is a feature that is consistent
with (and a consequence of) the principle of equivalence, since local effects of
gravity are not measured by observers in free fall. For other frames that are
related to Eq. (27) by a local Lorentz transformation we obtain nonvanishing
values of P (0). In particular, the total gravitational energy calculated out of
frames such that ea µ(t, x, y, z) → δaµ in the asymptotic limit r → ∞ is
exactly P (0) = m [4]. The latter tetrad field is adapted to observers at rest
at spacelike infinity. Thus the vanishing of gravitational energy in freely
falling frames shows that the localizability of the gravitational energy is not
inconsistent with with the principle of equivalence. The result given by Eqs.
(30-32) is a very good example of the frame dependence of the gravitational
energy definition (7).
It can be easily verified that the gravitational momentum components
P (1) and P (2) vanish in view of integrals like
0 dφ sinφ = 0 =
0 dφ cosφ,
whereas P (3) vanishes due to
0 dθ sin θ cos θ = 0.
It is important to remark that in general the vanishing of φab does not
imply the vanishing of P a. For an observer at rest at spacelike infinity the
total gravitational energy is nonvanishing, whereas for these observers we
have φab ∼= 0 (in the limit r → ∞; see next section).
5 Static frames in the Kerr spacetime
Another interesting application of the definitions of velocity and inertial ac-
celeration of a frame discussed in section 3 is the analysis of a static frame
in Kerr’s spacetime. The latter is established by the line element
ds2 = −
dt2 −
2χ sin2 θ
dφ dt+
dr2 (33)
+ρ2dθ2 +
Σ2 sin2 θ
dφ2 ,
with the following definitions:
∆ = r2 + a2 − 2mr , (34)
ρ2 = r2 + a2 cos2 θ ,
Σ2 = (r2 + a2)2 −∆a2 sin2 θ ,
ψ2 = ∆− a2 sin2 θ ,
χ = 2amr .
A static reference frame in Kerr’s spacetime is defined by the congruence
of timelike curves uµ(s) such that ui = 0, namely, the spatial velocity of the
observers is zero with respect to static observers at spacelike infinity. Since
we identify ui = e(0)
i, a static reference frame is established by the condition
i = 0 . (35)
In view of the orthogonality of the tetrads, the equation above implies e(k) 0 =
0. This latter equation remains satisfied after a local rotation of the frame,
ẽ(k) 0 = Λ
0 = 0. Therefore condition (35) determines the static char-
acter of the frame, up to an orientation of the frame in the three-dimensional
space.
A simple form for the tetrad field that satisfies Eq. (35) (or, equivalently,
e(k) 0 = 0) reads
eaµ =
−A 0 0 −B
0 C sin θ cosφ ρ cos θ cos φ −D sin θ sinφ
0 C sin θ sin φ ρ cos θ sinφ D sin θ cosφ
0 C cos θ −ρ sin θ 0
, (36)
with the following definitions
, (37)
χ sin2 θ
In the expression of D we have
Λ = (ψ2Σ2 + χ2 sin2 θ)1/2 .
We are interested in the calculation of φab given by Eq. (17), and for this
purpose it is useful to work with the inverse tetrad field ea
µ. It reads
sin θ sinφ − ρχ
sin θ cosφ 0
sin θ cos φ
sin θ sin φ
cos θ
cos θ cosφ 1
cos θ sinφ −1
sin θ
0 −ρψ
sin θ
sin θ
, (38)
where now the index a labels the columns, and µ the lines.
The frame determined by Eqs. (36) and (38) is valid in the region outside
the ergosphere. The function ψ2 = ∆ − a2 sin2 θ vanishes over the external
surface of the ergosphere (defined by r = r⋆ = m+
m2 − a2 cos2 θ; over this
surface g00 = 0), and we see that various components of Eqs. (36) and (38)
are not well defined over this surface. It is well known that it is not possible
to maintain static observers inside the ergosphere of the Kerr spacetime.
By inspecting Eq. (38) we see that for large values of r we have
µ(t, r, θ, φ) ∼= (0, cos θ,−(1/r) sin θ, 0) ,
µ(t, x, y, z) ∼= (0, 0, 0, 1) . (39)
Therefore we may assert that the frame given by Eq. (37) is characterized by
the following properties: (i) the frame is static, because Eq. (35) is verified;
(ii) the e(3)
µ components are oriented along the symmetry axis of the black
hole (the z direction). The second condition is ultimately reponsible for the
simple form of Eq. (36).
The evaluation of φab is long but straightforward, and for this reason
we will omit the details of the calculations. For convenience of notation we
define the vectors
r̂ = sin θ cosφ x̂+ sin θ sinφ ŷ + cos θ ẑ (40)
θ̂ = cos θ cos φ x̂+ cos θ sin φ ŷ− sin θ ẑ
which have well defined meaning as unit vectors in the asymptotic limit
r → ∞. We also define the three-dimensional vectors
a = (φ01, φ02, φ03) , (41)
Ω = (φ23, φ31, φ12) . (42)
We obtain the following expressions for a and Ω:
sin θ cos θ θ̂
, (43)
Ω = −
cos θ r̂+
sin θ ∂r
sin θ ∂θ
r̂ . (44)
The specific functional form of the vectors above completely characterize
the frame determined by Eq. (36). The determination of a and Ω is equiva-
lent to the fixation of six components of the tetrad field. Equations (43) and
(44) represent the inertial accelerations that one must exert on the frame
in order to verify that (i) the frame is static (condition (35)), and that (ii)
the e(3)
µ components of the tetrad field asymptotically coincides with the
symmetry axis of the black hole.
The form of a and Ω for large values of r is very interesting. It is easy to
verify that in the limit r → ∞ we obtain
r̂ , (45)
Ω ∼= −
2 cos θ r̂+ sin θ θ̂
. (46)
After the identificationsm↔ q and 4πma↔ m̄, where q is the electric charge
and m̄ is the magnetic dipole moment, equations (45) and (46) resemble the
electric field of a point charge and the magnetic field of a perfect dipole
that points in the z direction, respectively. These equations represent a
manifestation of gravitoelectromagnetism.
If we abandon the statical condition given by Eq. (35), an observer lo-
cated at a position (r, θ, φ) will be subject to an acceleration −a and to a
rotational motion determined by −Ω = ΩD, which is the dragging frequency
of the frame. Thus the gravitomagnetic effect is locally equivalent to iner-
tial effects in a frame rotating with frequency −ΩD, the latter having the
magnetic dipole moment structure given by Eq. (46). This is precisely the
gravitational Larmor’s theorem, discussed in Ref. [16].
The emergence of gravitoelectromagnetic (GEM) field quantities in the
context of the acceleration tensor φab presents no difference with respect to
the usual approach in the literature. Let us assume that tetrad field satisfies
the boundary conditions
ea µ ∼= δaµ +
ha µ , (47)
where ha µ is the perturbation of the flat space-time tetrad field in the limit
r → ∞, and that in this limit the SO(3,1) and spacetime indices acquire the
same significance. It is straightforward to verify that in this case we have
φ(0)(i) ∼= −∂i
, (48)
φ(i)(j) ∼= −
. (49)
Thus we identify
h00 , (50)
Ai = −
h0i . (51)
The identification above is equivalent to the one usually made in the litera-
ture, namely, Φ = (1/4)h̄00 and Ai = −(1/2)h̄0i [17], where h̄µν is the trace-
reversed field quantity defined by h̄µν = hµν−(1/2)ηµνh, and h = ηµνhµν . The
latter identification is made directly in the weak field form of the metric ten-
sor of a slowly rotating source. Assuming that h00 = 2Φ/c
2 and hij = δijh00,
where c is the speed of light (according to Eq. (1.4) of Ref. [17]), we obtain
h̄00 = 2h00, and therefore V = (1/4)h̄00. To our knowledge, the identification
of the GEM field quantities out of the tensor φab has not been addressed in
the literature so far.
6 Comments
Gravity theories invariant under the global SO(3,1) group are physically ac-
ceptable. The gravitational field equations determine the gravitational field,
not the frame. A given gravitational field configuration admits an infinity
of frames which in general are distinct from each other. We know that the
physical properties of a system are different in a static and in an accelerated
frame, for instance, and this feature should also hold in general relativity.
The gravitational energy-momentum that is defined in the realm of the
TEGR is frame dependent. This issue has been partially discussed before
in Refs. [7, 8], and also in Ref. [9]. This dependence is considered here to
be a natural property of the definition. The frame may be characterized by
the six components of the antisymmetric tensor φab, defined by Eq. (17),
which determine the translational acceleration and rotational frequency of
the frame, and which resembles the electric field of a point charge and the
magnetic field of a dipole, respectively, in the weak field limit of the Kerr
spacetime (in the consideration of a static frame).
In section 4 we have shown that the gravitational energy-momentum cal-
culated out of a frame that is nonrotating and freely falling in the Schwarzschild
spacetime vanishes. We expect this property to hold in the consideration of
a general spacetime geometry, in which case the analysis is somewhat more
complicated, because the frame is expected not to rotate with respect to a
Fermi-Walker transported frame. In general the construction of the latter
frame is not trivial.
It is clear that if the gravitational energy-momentum definition were in-
variant under local Lorentz transformations, we would not arrive at the result
of section 4, since the the value of P a on a three-dimensional volume V would
be the same for all frames, and presumably nonvanishing.
A common critique of the localizability of gravitational energy is that the
latter is unattainable because of the principle of equivalence. In this paper we
have seen that this is not the case. Definition (7) for the gravitational energy-
momentum yields the expected results both for observers asymptotically at
rest and for freely falling observers.
Acknowledgement
J. W. M. is grateful to G. F. Rubilar for helpful discussions on reference
frames. This work was supported in part by CNPQ (Brazil).
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