PHYSICAL REVIEW D 67, 024025 共2003兲
Self-force via a Green’s function decomposition
Steven Detweiler and Bernard F. Whiting
Department of Physics, P.O. Box 118440, University of Florida, Gainesville, Florida 32611-8440
共Received 1 March 2002; published 21 January 2003兲
The gravitational field in a neighborhood of a particle of small mass ␮ moving through curved spacetime is
naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through O( ␮ ).
One part is an inhomogeneous field which looks like the ␮ /r field tidally distorted by the local Riemann tensor.
The other part is a homogeneous field that completely determines the self-force of the particle interacting with
its own gravitational field, which changes the worldline at O( ␮ ) and includes the effects of radiation reaction.
Surprisingly, a local observer measuring the gravitational field in a neighborhood of a freely moving particle
sees geodesic motion of the particle in a perturbed vacuum geometry and would be unaware of the existence
of radiation at O( ␮ ). In the light of all previous work this is quite an unexpected result.
DOI: 10.1103/PhysRevD.67.024025
PACS number共s兲: 04.20.⫺q, 04.25.Nx, 04.30.Db
INTRODUCTION
The principle of equivalence in general relativity implies
that a freely moving particle, of infinitesimal mass and size,
follows a geodesic of spacetime 关1兴. For a small but finite
mass ␮ , the particle perturbs the spacetime geometry at
O( ␮ ). The interaction of the particle with this perturbation
of the metric is called the ‘‘self-force’’ which changes the
worldline at O( ␮ ) and includes the effects of radiation reaction.
A local observer measuring the gravitational field in a
neighborhood of the particle, with no a priori knowledge of
the background geometry, sees a combination of the background metric plus its O( ␮ ) perturbation caused by the particle. Local measurements, in a neighborhood of the particle,
cannot distinguish these two specific contributions. Nevertheless, we show below that the combined metric in the
neighborhood of the particle can be uniquely decomposed
into two distinct parts: 共i兲 the tidally distorted ␮ /r piece of
the perturbation of the metric, which is an inhomogeneous
solution of the perturbed Einstein equations at O( ␮ ), and 共ii兲
the sum of the background metric and the remainder of the
metric perturbation, which together are a homogeneous solution of the Einstein equations through O( ␮ ).
The self-force is shown to result in geodesic motion of the
particle, through O( ␮ ), in the homogeneous perturbed metric, 共ii兲 above. With only local measurements, the observer
has no means of distinguishing the homogeneous perturbation from the background metric through O( ␮ ). As the particle moves along a geodesic of this perturbed vacuum geometry, the observer sees no effect which one would be
compelled to interpret as radiation reaction at O( ␮ ). In this
precisely defined fashion, we extend the understanding of the
principle of equivalence in general relativity: The orbit of the
particle is a geodesic through O( ␮ ), in a locally measurable,
vacuum gravitational field.
HISTORICAL BACKGROUND
In Dirac’s 关2兴 analysis of the self-force for the electromagnetic field of a particle in flat spacetime, he used the
conservation of the stress-energy tensor inside a narrow
0556-2821/2003/67共2兲/024025共5兲/$20.00
world tube surrounding the particle’s worldline to derive
equations of motion with radiation reaction effects included.
DeWitt and Brehme 关3兴 extended this approach to allow for
curvature in spacetime. Mino, Sasaki and Tanaka 关4兴 generalized it to include the gravitational self-force as well. Quinn
and Wald 关5兴 and Quinn 关6兴 have obtained similar results by
pursuing an axiomatic approach for the gravitational, electromagnetic and scalar field self-forces.
On a formal level, these analyses provide a coherent and
consistent treatment of the self-force. Analysis begins with a
given worldline ⌫, described by z a ( ␶ ), which obeys the Lorentz force law through the fixed background gravitational
and electromagnetic fields. The self-force gives the actual
worldline an acceleration away from ⌫, and it is the mass
times this acceleration that we wish to determine.
Towards this end, first the retarded field is obtained in
terms of the corresponding Green’s function. Traditionally
the Green’s function has been decomposed into ‘‘direct’’ and
‘‘tail’’ parts. The resulting ‘‘direct’’ part of the field at a point
x is determined completely by sources on the past null cone
of x; in flat spacetime this is the Liénard-Wiechert potential
for electromagnetism. The curvature of spacetime allows for
an additional contribution from sources within the past null
cone, and this is the ‘‘tail’’ part. The self-force on a particle is
then composed of two pieces: The first piece comes from the
direct part of the field and the acceleration of ⌫ in the background geometry; in flat spacetime this is the AbrahamLorentz-Dirac 共ALD兲 force. The second piece comes from
the tail part of the field and is present in curved space even if
⌫ is a geodesic. While the decomposition of the field into the
direct and tail parts has been useful for describing the selfforce, neither of these parts individually is a solution of the
field equation.
In flat spacetime Dirac 关2兴 decomposed the retarded electromagnetic field into two parts: 共i兲 the ‘‘mean of the advanced and retarded fields’’ which is an inhomogeneous field
resembling the Coulomb q/r piece of the scalar potential
near the particle, and 共ii兲 a ‘‘radiation’’ field, his Eqs. 共11兲
and 共13兲, which is a homogeneous solution of Maxwell’s
equations. He described the self-force as the interaction of
the particle with the radiation field, a well-defined vacuum
Maxwell field.
67 024025-1
©2003 The American Physical Society
PHYSICAL REVIEW D 67, 024025 共2003兲
STEVEN DETWEILER AND BERNARD F. WHITING
Previous descriptions of the self-force in curved spacetime 关3– 6兴 reduce to Dirac’s result in the flat spacetime limit.
And they provide clear, adequate expressions for the selfforce. However, they do not share the physical simplicity of
Dirac’s analysis where the force is described entirely in
terms of an identifiable, vacuum solution of the field equations. Indeed, for an electromagnetic field, the vector potential A self
a 关cf. Eq. 共14兲 for the corresponding scalar field expression兴 might be said to be responsible for the self-force.
But generally A self
a does not satisfy Eq. 共23兲 below, and the
current density J a , which would result from the application
of the operator on the left hand side of Eq. 共23兲 to A self
a ,
would have a nonvanishing charge distribution in the vicinity
of the particle. Furthermore, A self
a is nondifferentiable at the
location of the particle with four-velocity u a if (R ab
⫺ 16 g ab R)u b ⫽0 there. Similar statements hold for scalar and
gravitational fields.
In this paper we present a curved space generalization of
Dirac’s scheme. We find that, near the worldline ⌫, even in a
curved geometry, the retarded field can be decomposed into
two parts such that the first, A Sa , is an inhomogeneous solution to the field equation with a point source, similar to the
‘‘mean of the advanced and retarded fields,’’ while the second, A Ra , is a homogeneous solution which yields a complete
description of the self-force. These parts are related to but
distinct from the usual direct and tail parts. Scalar and gravitational fields are analyzed similarly.
Below, we first focus on the scalar field case, and then
briefly describe our results for the electromagnetic and gravitational fields.
␳ 共 x 兲 ⫽q
␺ 共 x 兲 ⫽4 ␲ q
where the derivative of the field is to be evaluated at the
location of the particle. Usually it is implicit that ␺ does not
include the field from the particle itself but is composed only
of the ‘‘external’’ field. The description of the self-force is
not so straightforward. However, one can introduce a quantity ␺ self, as in our Eq. 共14兲, which when substituted into the
right hand side of Eq. 共1兲 formally yields the scalar-field
self-force as described by Quinn 关6兴. Similar expressions for
the force in terms of the derivatives of the field are given in
Eq. 共28兲 and Eq. 共34兲 for electromagnetic and gravitational
fields.
The scalar field equation
ⵜ 2 ␺ ⬅ⵜ a ⵜa ␺ ⫽⫺4 ␲␳
共2兲
is formally solved in terms of a Green’s function,
ⵜ 2 G 共 x,z 兲 ⫽⫺ 共 ⫺g 兲 ⫺1/2␦ 4 共 x⫺z 兲 .
共3兲
The source function for a point charge moving along a
worldline ⌫, described by coordinates z a ( ␶ ), is
共4兲
冕
G 关 x,z 共 ␶ 兲兴 d␶ .
共5兲
A symmetric scalar field Green’s function is derived from
the Hadamard form to be
G sym共 x,z 兲 ⫽
1
关 u 共 x,z 兲 ␦ 共 ␴ 兲 ⫺ v共 x,z 兲 ⌰ 共 ⫺ ␴ 兲兴 ,
8␲
共6兲
where u(x,z) and v (x,z) are biscalars, the properties of
which are described by DeWitt and Brehme 关3兴. They are
determined by a local expansion in the vicinity of ⌫, and are
symmetric under interchange of x and z. ␴ is half of the
square of the distance measured along the geodesic from x to
z with ␴ ⬍0 for a timelike geodesic, and ␴ ⫽0 on the past
and future null cones of x. The expansions for the biscalars
u(x,z) and v (x,z) are known to be convergent within a finite
neighborhood of ⌫ if the geometry is analytic 关7兴. The ⌰
(⫺ ␴ ) guarantees that only when x and z are timelike related
is there a contribution from v (x,z). The terms in any Green’s
function containing u and v are frequently referred to as the
‘‘direct’’ and ‘‘tail’’ parts, respectively.
The direct part of G sym(x,z) has support only on the null
cone of x, and the resultant direct part of the field for a
particle moving along ⌫ is
sym
␺ dir
共 x 兲⫽
The force on a point particle with charge q moving in a
scalar field may be deduced from 共see discussion in 关6兴兲
共1兲
共 ⫺g 兲 ⫺1/2␦ 4 „x⫺z 共 ␶ 兲 … d␶ ,
where ␶ is the proper time along the worldline of the particle
with scalar charge q. The scalar field of this particle is
SCALAR FIELD
F a ⫽qⵜ a ␺ ,
冕
冋
qu 共 x,z 兲
2 ␴˙
册 冋
⫹
␶ ret
qu 共 x,z 兲
2 ␴˙
册
,
共7兲
␶ adv
where ␴˙ ⫽d␴ „x,z( ␶ )…/d␶ , and ␶ ret and ␶ adv refer to the
proper time of the intersection of ⌫ with the past and future
null cones of x, respectively.
The tail part of G sym(x,z) has support within both the past
and future null cones of x, and the resultant tail part of the
field for a particle moving along ⌫ is
sym
␺ tail
共 x 兲 ⫽⫺
q
2
冉冕 冕 冊
␶ ret
⫺⬁
⫹
⬁
␶ adv
v共 x,z 兲 d␶ .
共8兲
DeWitt and Brehme 关3兴 use local expansions in the vicinity of ⌫ to show that 关9兴
u 共 x,z 兲 ⫽1⫹
1
R ⵜ a ␴ ⵜ b ␴ ⫹O 共 r 3 兲 ,
12 ab
x→⌫;
共9兲
r is the proper distance from x to ⌫ measured along the
spatial geodesic which is orthogonal to ⌫. They also show
that the symmetric biscalar v (x,z) is a solution of the homogeneous wave equation,
ⵜ 2 v共 x,z 兲 ⫽0,
and that for x close to ⌫,
024025-2
共10兲
PHYSICAL REVIEW D 67, 024025 共2003兲
SELF-FORCE VIA A GREEN’S FUNCTION DECOMPOSITION
v共 x,z 兲 ⫽⫺
1
R 共 z 兲 ⫹O 共 r 兲 , x→⌫.
12
共11兲
⫺q v关 x,z 共 ␶ ret兲兴 ⵜa ␶ ret⫽q 关v ␴˙ ⫺1 ⵜa ␴ 兴 ␶ ret
⫽⫺
The retarded and advanced Green’s functions are derived
from G sym(x,z),
G ret共 x,z 兲 ⫽2⌰ 关 ⌺ 共 x 兲 ,z 兴 G sym共 x,z 兲
G adv共 x,z 兲 ⫽2⌰ 关 z,⌺ 共 x 兲兴 G sym共 x,z 兲 ,
共12兲
where ⌰ 关 ⌺(x),z 兴 ⫽1⫺⌰ 关 z,⌺(x) 兴 equals 1 if z is in the past
of a spacelike hypersurface ⌺(x) that intersects x, and is zero
otherwise. The G rad(x,z) implicitly used by Dirac in flat
spacetime is
G rad共 x,z 兲 ⫽G ret共 x,z 兲 ⫺G sym共 x,z 兲 .
共13兲
Note that the fields resulting from two different Green’s
functions that each obey Eq. 共3兲 necessarily differ by a homogeneous solution of Eq. 共2兲.
G ret(x,z) has reasonable causal structure, and we assume
for simplicity that ␺ ret is, in fact, the actual field resulting
from the source particle.
qR 共 x 兲
共 x a ⫺z a 兲 ⫹O 共 r 兲 , x→⌫,
12r
共15兲
the spatial part of which is not defined when x is on ⌫. In the
usual self-force analysis, one first averages ⵜa ␺ self over a
small, spatial two-sphere surrounding the particle, thus removing the spatial part of Eq. 共15兲. Then one takes the limit
of this average as the radius of the two-sphere goes to zero,
thereby obtaining a well defined contribution to the selfforce 关3– 6兴.
HOMOGENEOUS FIELD
We now provide an alternative expression for the field
responsible for the self-force.
Given one Green’s function which solves Eq. 共3兲, a second may be generated by adding to the first any biscalar
which is a homogeneous solution of Eq. 共3兲. v (x,z) is just
such a biscalar and is also symmetric, v (x,z)⫽ v (z,x) 关3兴.
Thus, a new symmetric Green’s function is
G S共 x,z 兲 ⬅G sym共 x,z 兲 ⫹
1
1
v共 x,z 兲 ⫽
关 u 共 x,z 兲 ␦ 共 ␴ 兲
8␲
8␲
⫹ v共 x,z 兲 ⌰ 共 ␴ 兲兴 ,
SELF-FORCE
Careful analyses 关3– 6兴 show that contributions to the selfforce result from both the source’s acceleration, if ⌫ is not a
geodesic, and from the curvature of spacetime. For these two
distinct possibilities, the self-force is a consequence of the
particle interacting with either the direct or the tail part of its
field, respectively. By following the detailed derivations in
关3– 6兴, we see that the self-force may be considered to result,
via Eq. 共1兲, from the interaction of the particle with the quantity
␺ self⫽⫺
冋
qu 共 x,z 兲
2 ␴˙
册
which has no support within the null cone. We use G S(x,z)
only in a local neighborhood of the particle, and do not depend upon any knowledge of its global existence. G S(x,z)
does have support on the null cone of x, just as G sym(x,z)
does, and also outside the null cone, at spacelike separated
points. The use of G S(x,z) is thus not complicated by the
need for knowledge of the entire past history of the source
and is amenable to local analysis. The corresponding field
␺ S共 x 兲 ⫽
␶ adv
⫺q
␶ ret
冕
␶ ret
⫺⬁
v关 x,z 共 ␶ 兲兴 d␶
共16兲
冋
qu 共 x,z 兲
2 ␴˙
册 冋
⫹
␶ ret
qu 共 x,z 兲
2 ␴˙
册
⫹
␶ adv
q
2
冕
␶ adv
␶ ret
v共 x,z 兲 d␶
共17兲
共14兲
is an inhomogeneous solution of Eq. 共2兲 just as ␺ ret is. In the
pioneering spirit of Dirac, it is natural to define
where, unlike Dirac’s radiation field, ␺ self is not a homogeneous solution of the field equation.
The first expression in Eq. 共14兲 is finite and differentiable
in the coincidence limit, x→⌫. When substituted into the
right hand side of Eq. 共1兲 this expression provides the curved
space generalization of the ALD force, and local expansions
of u(x,z) and ␴˙ (x,z) in 关3– 6兴 give the resultant force in
terms of the acceleration of ⌫ and components of the Riemann tensor.
The integral in Eq. 共14兲 comes from the tail part of the
Green’s function. Its derivative results, in part, from the implicit dependence of ␶ ret upon x. Quinn 关6兴 computes this
contribution to the derivative to be
G R共 x,z 兲 ⬅G ret共 x,z 兲 ⫺G S共 x,z 兲
共18兲
关cf. Eq. 共13兲兴. Remarkably, like G ret(x,z), G R(x,z) has no
support inside the future null cone. Corresponding to
G R(x,z), we construct
024025-3
␺ R⫽ ␺ ret⫺ ␺ S⫽⫺
⫻
冋
qu 共 x,z 兲
2 ␴˙
冉冕 冕 冊
␶ ret
⫺⬁
⫹
1
2
␶ adv
␶ ret
册
␶ adv
⫺q
␶ ret
v共 x,z 兲 d␶
共19兲
PHYSICAL REVIEW D 67, 024025 共2003兲
STEVEN DETWEILER AND BERNARD F. WHITING
analogous to Dirac’s radiation field. By construction ␺ R is a
homogeneous solution of Eq. 共2兲 and has the consequent
property that it is smooth in the coincidence limit, x→⌫. We
note its relation to ␺ self
1
␺ R⫽ ␺ self⫺ q
2
冕
␶ adv
␶ ret
v共 x,z 兲 d␶ .
共20兲
If (R ab ⫺ 16 g ab R)u b ⫽0 at the particle, then A self
a is nondifferentiable there.
Similar to the scalar field, A Sa is an inhomogeneous solution of Eq. 共23兲. That it also satisfies the gauge condition
共22兲 follows from an argument similar to that after Eq. 共3.37兲
in 关3兴. We have
ⵜ a A Sa ⫽
冕
ⵜa ⬘ 共 G SJ a ⬘ 兲 冑⫺g d4 x ⬘ ,
共26兲
Our most significant technical result is, in fact, that the
self-force is determined by the particle’s interaction with ␺ R,
since the integral term in Eq. 共20兲 gives no contribution to a
self-force. For a field point x which is near ⌫, the integrand
may be expanded using Eq. 共11兲. The dominant part of the
integral from ␶ ret to ␶ adv then brings in a factor of ␶ adv⫺ ␶ ret
⫽2r⫹O(r 2 ) times v (x,x), and the integral term of Eq. 共20兲
is
assuming that J a is conserved, where G S is the scalar
Green’s function of Eq. 共16兲. That G S has no support within
the past or future null cone implies that the integral is zero
when written as a boundary integral.
With the definition
1
⫺qr v共 x,x 兲 ⫹O 共 r 2 兲 ⫽ qrR 共 x 兲 ⫹O 共 r 2 兲 , x→⌫.
12
A Ra gives a homogeneous electromagnetic field. The electromagnetic self-force becomes
共21兲
The derivative of 121 qrR(x) gives an outward pointing, spatial unit vector near ⌫; this exactly cancels the troublesome
part of ⵜa ␺ self in Eq. 共15兲 which is thus absent from ⵜa ␺ R.
The derivative of the remainder term O(r 2 ) is zero in the
limit that x approaches ⌫ and gives no contribution to the
self-force. Thus, the self-force may be seen to be due exclusively to the interaction of the particle with ␺ R via Eq. 共1兲.
We regard this approach as preferable, because ␺ R is differentiable at the location of the particle, so that averaging is no
longer required in computing the self-force. Even more importantly, ␺ R is a homogeneous solution of Eq. 共2兲.
ELECTROMAGNETIC AND GRAVITATIONAL FIELDS
The analysis for the scalar field is easily generalizable to
both electromagnetic and gravitational fields by the addition
of extra indices to u(x,z) and v (x,z) to create corresponding
bivectors and bitensors, and by the introduction of ḡ ab ⬘ (x,z),
which is the bivector of parallel displacement 关3兴. The
primed indices below refer to the source point z, the
unprimed indices to the field point x as above. The definitions and relationships for the various Green’s functions follow the same pattern as above and are not repeated below.
For the electromagnetic field, the Lorentz gauge requires
ⵜa A a ⫽0,
共22兲
S
A Ra ⬅A ret
a ⫺A a ,
F a ⫽qg ac 共 ⵜc A Rb ⫺ⵜb A Rc 兲 ż b .
共23兲
DeWitt and Brehme 关3兴 show that
u ab ⬘ 共 x,z 兲 ⫽ḡ ab ⬘ 共 x,z 兲 u 共 x,z 兲 ,
and that 共as x→⌫)
冉
冊
1
1
v ab ⬘ 共 x,z 兲 ⫽ ḡ a c ⬘ R b ⬘ c ⬘ ⫺ g b ⬘ c ⬘ R ⫹O 共 r 兲 .
2
6
ⵜa h̄ ab ⫽0,
共29兲
and, with R ab ⫽0, the perturbed Einstein equations are
ⵜ 2 h̄ ab ⫹2R a c b d h̄ cd ⫽⫺16␲ T ab ,
共30兲
where h̄ ab ⫽h ab ⫺ 21 g ab h c c is the trace reversed version of the
metric perturbation h ab . Mino, Sasaki and Tanaka 关4兴 show
that
u abc ⬘ d ⬘ 共 x,z 兲 ⫽2ḡ ac ⬘ 共 x,z 兲 ḡ bd ⬘ 共 x,z 兲 u 共 x,z 兲 ,
共31兲
and that 共as x→⌫)
v abc ⬘ d ⬘ 共 x,z 兲 ⫽⫺ḡ a e ⬘ ḡ b f ⬘ R c ⬘ e ⬘ d ⬘ f ⬘ 共 z 兲 ⫹O 共 r 兲 .
共32兲
self
If R cadb u c u d ⫽0 at the particle, then h̄ ab
is nondifferentiable
there.
S
Similar to the electromagnetic field case, h̄ ab
is an inhomogeneous solution of Eq. 共30兲 satisfying the gauge condition 共29兲, if T ab is conserved. With the definition
R
ret
S
⬅h̄ ab
⫺h̄ ab
,
h̄ ab
R
h̄ ab
is differentiable on ⌫, and
冉
共33兲
冊
1
R
R
F a ⫽⫺ ␮ 共 g ab ⫹ż a ż b 兲 ż c ż d ⵜc h db
⫺ ⵜb h cd
.
2
共24兲
共25兲
共28兲
This combines with the Lorentz force law from the background to determine the actual worldline of the particle.
For the gravitational field, the harmonic gauge requires
so that Maxwell’s equations become
ⵜ 2 A a ⫺R a b A b ⫽⫺4 ␲ J a .
共27兲
共34兲
Subject to this force, the particle moves along a worldline
which is actually a geodesic for a metric composed of the
R
. While geodebackground geometry complemented by h ab
sic motion has been demonstrated in the past 关4,5兴, only in
024025-4
SELF-FORCE VIA A GREEN’S FUNCTION DECOMPOSITION
PHYSICAL REVIEW D 67, 024025 共2003兲
R
our case is the reference metric, g ab ⫹h ab
, a vacuum solution of the Einstein equations through O( ␮ ).
the metric with its tidal distortion from the ‘‘external,’’ homogeneous field 关8兴. As a consequence of Eq. 共34兲, the local
observer naturally sees that the worldline of the particle is a
geodesic in the combined, ‘‘external’’ homogeneous field
which he measures. Making only local measurements near
the worldline, the observer sees no radiation, no local source
for the ‘‘external’’ field and also no effect which he would be
compelled to describe as radiation reaction.
DISCUSSION
For the clearest demonstration of the impact of our analysis, we consider the free motion of a particle of small mass ␮
in the purely gravitational case. With no a priori knowledge
of the background geometry, an observer makes local measurements of the metric within a neighborhood of the worldline of the particle. That field has two distinct contributions.
R
—this
The first is the background metric combined with h ab
combination appears as an ‘‘external,’’ homogeneous field;
R
no local measurement distinguishes h ab
from the backS
; for free motion the
ground. The second comes from h ab
observer knows this contribution to be simply the ␮ /r part of
关1兴 C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation
共Freeman, San Francisco, 1973兲, p. 211.
关2兴 P.A.M. Dirac, Proc. R. Soc. London A167, 148 共1938兲.
关3兴 B.S. DeWitt and R.W. Brehme, Ann. Phys. 共N.Y.兲 9, 220
共1960兲.
关4兴 Y. Mino, M. Sasaki, and T. Tanaka, Phys. Rev. D 55, 3457
共1997兲.
关5兴 T.C. Quinn and R.M. Wald, Phys. Rev. D 56, 3381 共1997兲.
ACKNOWLEDGMENTS
We are grateful to Amos Ori and Robert Wald for comments on an earlier version of this paper. This research has
been supported in part by NSF Grant No. PHY-9800977
共B.F.W.兲 and NASA Grant No. NAGW-4864 共S.D.兲 with the
University of Florida.
关6兴 T.C. Quinn, Phys. Rev. D 62, 064029 共2000兲.
关7兴 J. Hadamard, Lectures on Cauchy’s Problem in Linear Differential Equations 共Yale University Press, New Haven, CT,
1923兲.
关8兴 S. Detweiler, Phys. Rev. Lett. 86, 1931 共2001兲.
关9兴 Our convention 2ⵜ[a ⵜb] ␰ c ⫽R abc d ␰ d agrees with that in Refs.
关5,6兴 and is the opposite of that used in 关3,4兴.
024025-5