An extension of Poincar\'e group based on generalized Fermi-Walker coordinates

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by  Josep Llosa

An extension of Poincaré group based on generalized
Fermi-Walker coordinates
Josep Llosa
Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB),
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Universitat de Barcelona, Martí Franquès 1, E08028 Barcelona, Spain
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2 February 6, 2017

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Abstract
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Theclassofacceleratedandrotatingreferenceframeshasbeenstudiedonthebasisofgeneralized

Fermi-Walker coordinates. We obtain the infinitesimal transformations connecting any two
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c of these frames and also their commutation relations. We thus have an infinite dimensional
q
extensionofthe Poincaréalgebraand,althoughitturnsoutto beAbelianextension,andhence
-
r trivial, it is noteworthy that, contrarily to Lorentz boosts, acceleration and rotational boost
g
generators commute with each other and with the generators of Poincaré group as well.
[

1 PACS number: 02.40.Ky,02.20.Tw, 02.20.Sv, 04:20.Cv,
v
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7 1 Introduction
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0 The laws of Newtonian mechanics hold in all inertial reference frames, which are in uniform rec-
. tilinear motion with respect to each other. This is known as the principle of relativity of Galilei.
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0 However, thisprincipleofrelativitycanbeextendedtoarbitrary rigidframes,whichareinarbitrary
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translational and rotational motion with respect to each other, i. e. they are mutually related by
1
coordinate transformations like
:
v
Xi x′i = Ri(t)xj +si(t), t′ = t+t ,
j 0
r
a where Ri(t) is an orthogonal matrix and si(t) arbitrary functions of time. The laws of Newtonian
j
mechanics have the same form in any of these coordinate systems, provided that the necessary
inertial force fields —dragging, Coriolis, centrifugal, ...— are included.
According to the special theory of relativity the laws of physics hold in all Lorentzian reference
frames, the relative motion of any couple of these frames is rectilinear and uniform, and the coor-
dinates in any pair of these frames are connected by a Poincaré transformation. Endeavouring to
set up a theory of gravity consistent with his theory of relativity, Einstein initially aimed to gener-
alize the theory of relativity to accelerated motions [1], but he soon abandoned this idea in favour
1
ExtendedPrinciple6 v0.tex 2
of the principle of general covariance. Its invariance group, namely spacetime diffeomorphisms, is
much wider than Poincaré group but, as soon Kretschmann pointed out [2], [3] “since any theory,
whatever its physical content, can be rewritten in a generally covariant form, the group of general
coordinate transformations is physically irrelevant” [4]. Moreover, in Kretschmann’s view, special
relativity is the one with the relativity postulate of largest content; indeed, its isometry group is a
ten-parameter group, which is the largest isometry group in four dimensions, whereas for generic
spacetimes in general relativity the isometry group reduces to the identity.
Morerecently otherauthors haveinsistedintheconvenience ofrestricting generalcovariance [5]and
evenaPrincipleofrestrictedcovariancehasbeenexplicitlystated[6]. Thereisalsointheliteraturea
renewed interestinaccelerated referenceframes. Inanon-relativistic contextweshouldmention the
extensionsofGalileialgebratoencompassacceleration[7],[8],[9]and,fromarelativisticstandpoint,
the approach by Mashhoon [10]in viewof its possibleapplication tonon-local gravitational theories
[11].
Our aim is to find an extension of the principle of relativity, meaning that there is a class of
coordinate systems —larger that the Lorentzian class but more restrictive than general curvilinear
coordinates– such that the laws of physics have the same form, including inertial force fields if
necessary; in as much the same way as the extended principle of relativity for rigid reference frames
holds in Newtonian mechanics, as commented at the start.
Fermi-Walker (FW) coordinates [12], [13] are characterized by an arbitrary origin worldline, whose
proper velocity stands for the time axis, whereas the triad of space axis evolve without rotating in a
specific way, namely Fermi-Walker transport [14], just to stay orthogonal to proper velocity. Often
these coordinates have been seen as the relativistic generalization of the coordinates associated to
an accelerated, non-rotating observer [12], [15], [16].
Thisclassofobserversandassociatedcoordinatescanbeextendedbyallowingthespaceaxestohave
an arbitrary rotational motion [12], [17], [18]. We shall refer to these as generalized Fermi-Walker
1
coordinates (GFW) .
In section 2 we outline the main features of FW and GFW coordinate systems in Minkowski space-
time and see the specific form of Minkowski interval when written in these coordinates. Then we
prove that the latter specific form is exclusive of GFW coordinates and state a kind of uniqueness
result associated to them.
The Minkowski metric components in these coordinates are ten well defined functions, namely
g (Xj,T;f (T)) that involve six functions of time f (T), J = 1...6. These functions are con-
µν J J
nected with the origin proper acceleration and the angular velocity of the spatial triad of axes.
In passing from one GFW system of coordinates to another the g (Xj,T;f ) are form invariant,
µν J
i.e. considered as functions of the ten variables Xj, T and f they do not change. However they
J
are not isometries in the proper sense because, if we consider the components as functions of the
four coordinates, namely g (Xj,T) := g (Xj,T;f (T)), their values do change because the six
µν µν J
functions f (T) differ from one GFW system to the other. This dual feature incline us to use the
J
term generalized isometries [19].
We then study the infinitesimal transformations connecting two GFW coordinate systems, as the
1
Gourgoulhon [18]uses the term “coordinates with respect to a generic observer”
ExtendedPrinciple6 v0.tex 3
solutions of a generalized Killing equation, and derive the infinitesimal generators . As these trans-
formationsimplynotonlyachangeinthespacetimecoordinates(Xj,T)butalsointhesixfunctions
f (T) characterizing the GFW system, the infinitesimal generators act on a manifold that is much
J
larger than merely Minkowski spacetime (an infinite dimensional one, actually). These generators
span an infinite dimensional extension of Poincaré algebra which includes acceleration and rotation,
which may be taken as the mathematical embodying of an extension of the principle of special
relativity abiding arbitrary translational and rotational motions.
2 Generalized Fermi-Walker coordinates
Let zµ(τ) be a timelike worldline in ordinary Minkowski spacetime, and uµ = z˙µ(τ) and and
aµ = z¨(τ) the proper velocity and acceleration 4-vectors. (We take c= 1, Greek indices run from 1
to4andLatinindicesfrom1to3; xµ refertoLorentziancoordinatesandthesummationconvention
is always understood unless the contrary is explicitely said.)
A 4-vector wµ(τ) is Fermi-Walker transported [13] along zµ(τ) if
dwµ
= (uµa −u aµ) wν (1)
ν ν
dτ
Let us now consider an orthonormal tetrad, eˆµ (τ), which is FW transported along zµ(τ) and such
(α)
that eˆµ = uµ. For a given point in spacetime, xµ, the Fermi-Walker coordinates [13], [18] with
(4)
space origin on zµ(τ) are:
The time T(xν), given as an implicit function by
[xµ−zµ(T)] u (T)= 0 (2)
µ
The space coordinates Xi, defined by
Xˆi = [x −z (T(x))] eˆµ (T(x)) (3)
µ µ (i)
FW coordinates are the local coordinates of a non-rotating accelerated observer [12]. A natural
generalization, that also includes arbitrary rotational motion, is based on the notion of generalized
Fermi-Walker (GFW) transport [12], [17] of a vector wµ along the worldline zµ(τ):
dwµ
= Ωµ(τ)wν (4)
dτ ν
where
Ωµ = uµa −aµu −ǫµ ωαuβ (5)
ν ν ν ναβ
is the (spacetime) angular velocity and ωα(τ) is an arbitrary vector that is orthogonal to uµ that
we shall call proper angular velocity 4-vector.
ExtendedPrinciple6 v0.tex 4
Consider now a new tetrad {eν (τ)} , with eν (τ) = uν(τ), that is GFW transported along
(α) α=1...4 (4)
zµ(τ). We shall write:
e = η eν , eν(α) = eν ηβα
ν(α) νµ (α) (β)
and the components of the angular velocity in this comoving base are:
Ωˆα = Ωµe(α)eν , Ωˆ4 = Ωˆi = aˆi, Ωˆi = ǫi ωˆk (6)
β ν µ (β) i 4 j jk
where ωˆl = ω eν .
ν (l)
On the basis of the origin worldline zµ(τ) and the space axes eµ we can introduce the generalized
(i)
Fermi-Walker coordinates of a point xν: (a) the time T(x) is defined as in equation (2) and (b) the
space coordinates are
Xj =[x −z (T(x))] eµ (T(x)) (7)
µ µ (i)
The inverse coordinate transformation — from GFW to Lorentzian coordinates— is
T, Xl −→ xµ, xµ = zµ(T)+Xleµ (T) (8)
(l)
(cid:16) (cid:17)
whence it easily follows that
dxµ = 1+X~ ·~a uµ+ǫl ωˆkXieµ dT +eµ dXl (9)
ik (l) (l)
h(cid:16) (cid:17) i
and the Minkowski metric in GFW coordinates is
2 2
ds2 = dX~2+2dT dX~ · X~ ×~ω −dT2 1+X~ ·~a − X~ ×~ω (10)
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where the usual standard 3-vector notation has been introduced for the sake of brevity, with ~ω =
(ωˆ1,ωˆ2,ωˆ3) and~a = (aˆ1,aˆ2,aˆ3).
Had we to compare these GFW coordinates with the ordinary FW coordinates based on the same
world line, we should obtain Xj = Rj(T)Xˆi, where Rj(τ) is a rotation matrix satisfying
i i
R˙j(τ) = ǫl ωˆk(τ)Rj (τ)
i ik l
(ǫl = ǫ isthethreedimensionalLevi-Civitasymbol,regardlessthepositionoftheLatinindices).
jk ljk
It is worth to remark that, whereas aµ, ων and Ω are the components of respectively proper ac-
µν
celeration, proper angular velocity vector and spacetime angular velocity on an external Lorentzian
coordinate base, aˆj(τ), ωˆj(τ) and Ωˆ (τ) are the components of these objects with respect to the
αβ
GFW transported base. This is why we shall refer to them as intrinsic proper acceleration and so
on.
ExtendedPrinciple6 v0.tex 5
2.1 The GFW reference frame with origin zµ(τ) and angular velocity Ωµ (τ)
ν
µ
Any GFW transported tetrad with angular velocity Ω (τ) is a solution of the linear ordinary
ν
differential system (4-5). Its general solution is
eµ (τ) = Λµ (τ)eν (0) (11)
(α) ν (α)
µ
where Λ (τ) is a solution of the differential system
ν
Λ˙µ = Ωµ(τ)Λρ , with Λµ (0) = δµ (12)
ν ρ ν ν ν
Therefore two tetrads, eµ = e′µ = uµ, that are GFW transported along the same worldline with
(α) (α)
the same angular velocity will only differ in their initial values and as, besides eµ = e′µ = uµ,
(4) (4)
these initial values are connected by a space rotation
4
e′µ (0) = eµ (0)Rβ , with Rβ = R4 = δβ,
(α) (β) α 4 β 4
β=1
X
Ri being a constant orthogonal matrix. Combining the latter with (11) we have that
j
i,j=1...3
(cid:16) (cid:17)
e′µ (τ) = eµ (τ)Rβ (13)
(α) (β) α
Hence all GFW transported tetrads along a given worldline with the same angular velocity are the
same apart from an initial space rotation and, according to the definition the GFW coordinates
based on any of these tetrads will differ at most in a constant rotation:
τ′ = τ , Xi = Ri X′j
j
Given a GFW coordinate system with origin zµ(τ) and angular velocity Ωµ (τ), the 3-parameter
α
congruence of worldlines Xi =constant, τ ∈ R are the “history” of a place in the reference space
associated to the GFW coordinates and, as commented above, in any other GFW coordinated
system based on zµ(τ) and Ωµ (τ) we shall still have that X′i = R−1 i Xj = constant.
α j
The equation defining this 3-parameter congruence is
(cid:0) (cid:1)
ϕµ(T, X~) = zµ(T)+Xieµ (T), τ ∈R (14)
(i)
and the proper time rate at the place X~ is
2 2
dτ = dT 1+X~ ·~a(τ) − X~ ×~ω ;
r
h i h i
This is the time ticked by a stationary atomic clock at X~ and it coincides with T at the origin. In
general, τ 6= T and usually the readings of proper time τ by two stationary clocks at two different
ExtendedPrinciple6 v0.tex 6
places willnotkeepsynchronized. Itisthus more convenient tousethesynchronous time T instead
of local proper time.
The proper velocity vector of the worldline X~ =constant at the synchronous time T is
1+X~ ·~a uµ+ǫl Xiωˆkeµ
Uµ(T,X~) = ik (l) (15)
(cid:16) (cid:17)
2 2
1+X~ ·~a − X~ ×~ω
r
h i h i
The first term corresponds to the origin translational velocity, whereas the second term reflects the
rotational motion.
Due to the presence of a square root in the denominator, the domain of validity of the GFW
coordinates is restricted to the region 1+X~ ·~a > X~ ×~ω and the equality defines the horizon of
the GFW coordinate system. (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
3 Uniqueness
We now prove that the expression(10) of the metric is exclusive forGFW coordinates in Minkowski
spacetime.
Proposition 1 If in some coordinate system Xi,T the spacetime metric has the form (10), then
there is a worldline zµ(τ) and an orthonormal tetrad eµ (τ), α = 1...4, such that Xi,T are
(cid:0) (cid:1) (α)
the GFW coordinates based on that worldline and tetrad in a locally Minkowskian spacetime.
(cid:0) (cid:1)
Proof: It is straightforward to check that the Riemann tensor for the metric (10) vanishes; there-
fore the spacetime is locally Minkowskian.
Then consider the matrix Ωˆα(τ) defined by
β
Ωˆi (τ) = ǫi ωˆk(τ), Ωˆ4(τ) = Ωˆi (τ) = aˆi(τ) (16)
j jk i 4
where the functions aˆi and ωˆk are obtained from the coefficients in the metric (10). Then take
µ
e (τ) as the solution of the ordinary differential system
(α)
µ
de
(α) = eµ Ωˆβ (τ) (17)
dτ (β) α
µ µ
for some initial data {e (0)} , that form an orthonormal tetrad, with e (0) timelike. Due
(α) α=1...4 (4)
to the particular form of the matrix Ωˆβ , it is obvious that eµ (τ) is an orthonormal tetrad for all
α (α)
τ as well.
Being eµ a timelike vector, consider a worldline zµ(τ) such that z˙µ(τ) = eµ (τ) and the matrix
(4) (4)
Ωµ = Ωˆβ eµ e(α)
ν α (β) ν
ExtendedPrinciple6 v0.tex 7
(indices are raised and lowered with η = diag[+1,+1,+1,−1]). It is straightforward to see that
αβ
the tetrad eµ (τ) is GFW transported along zµ(τ) with an angular velocity Ωµ(τ).
(α) ν
Consider finally the coordinate transformation
Xi,T −→ xµ = zµ(T)+Xjeµ (T)
(j)
(cid:0) (cid:1)
whose Jacobian and inverse Jacobian are respectively given by
dxµ = z˙µ(T)+eµ (T)Ωˆα(T)Xj dT +eµ (T)dXj (18)
(α) j (j)
(cid:16) (cid:17)
e (T)dxµ e (T)Ωˆi (T)Xj
dT = − µ(4) , dXi = e + µ(4) j dxµ (19)
1+X~ ·~a(T) µ(i) 1+X~ ·~a(T) !
Substituting the latter in the expression (10) we obtain that, in the coordinates xµ, the invari-
ant interval is ds2 = η dxµdxν; hence xµ are Lorentzian coordinates and Xi, T are the GFW
µν
coordinates for the worldline zµ(τ) and the tetrad eµ . ✷
(α)
3.1 A geometric characterization of GFW coordinate systems
We shall see now that, if the hypersurfaces T =constant are hyperplanes in Minkowski spacetime
and Xj are Cartesian coordinates on these hyperplanes, then (Xj,T) are GFW coordinates apart
from a shift in the origin.
Proposition 2 Let (Xj,T) be a system of coordinates such that the Minkowski metric spatial com-
ponents are g = δ , then: (a) the hypersurfaces T =constant are flat, (b) the rank of their
ij ij
extrinsic curvature K is at most one, and (c) it satisfies that ∂ K = 0.
ij [i j]l
Proof: The metric restricted to the hypersurfaces T =constant, g = δ , is flat and the ambient
ij ij
metric is Minkowski metric. Then, if K is the second fundamental form, Gauss equation [20]
ij
implies that K K = 0.
i[j l]k
The latter equation having the same symmetries as a Riemann tensor and being three the number
of effective dimensions, it is equivalent to its trace (il), that is:
KjKi −KiKj = 0
i k i k
whichimpliesthatKj hastwoeigenvalues, namely Ki (simple)and0(double), hencerankKj ≤ 1.
i i i
The relation ∂ K = 0 is a consequence of the Codazzi-Mainardi equation [20] and the fact that
[i j]l
Xj are Cartesian coordinates for the first fundamental form on the hypersurfaces T =constant. ✷
Proposition 3 If the metric spatial components are g = δ and the second fundamental form on
ij ij
T =constant vanishes, then (Xj,T) are GFW coordinates apart from a shift in the origin.
ExtendedPrinciple6 v0.tex 8
Proof: Let us define v = g and N2 =~v2−g , then the inverse spacetime metric components
i 4i 44
are:
gij = δ −N−2v v , g4i = N−2v , g44 = −N−2 = det(gµν)6= 0
ij i j i
and the connexion symbols are:
{ij|k} = 0, {ij|4} = ∂ v , {4i|j} = ∂ v
(i j), [i j]
(20)

{44|i} = ∂ v +N ∂ N +2~v·∂~v , {44|4} = N ∂ N +2~v ·∂ ~v
T i i i T T 
The unit covector normal to the hypersurfaces T =constant and the second fundamental form are
respectively
n = −N δ4 and K = −N Γ4 =N−1∂ v
a a ij ij (i j)
ThevanishingofK thenimpliesthat ∂ v = W isskewsymmetric, whoseintegrability conditions,
ij i j ij
∂ W = ∂ W , combined with by the Jacobi identity, imply that ∂ W = 0, that is
l ij i lj j il
W = ǫ ωˆk(T) and therefore v = ǫ Xjωˆk(T)+V (T) (21)
ij ijk i ijk i
So far we have used the restrictions imposed by Gauss and Codazzi-Mainardi equations. We have
still to exploit the vanishing of the components R . Including that ∂ v =0 and the connexion
4i4j (i j)
symbols (20), we arrive at
R = 0 ⇔ ∂ N = 0,
4i4j ij
that is
∃ ~a(T) and B(T) such that N = B +X~ ·~a
and, provided that B 6= 0, the time coordinate T can be redefined so that B = 1.
Therefore, in these coordinates the Minkowski metric reads
2 2
ds2 = dX~2+2dT dX~ · X~ ×~ω+V~ +dT2 − 1+X~ ·~a + X~ ×~ω+V~ (22)
(cid:18) (cid:19)
(cid:16) (cid:17) h i h i
Finally,ifweshifttheoriginas: X˜j = Xj+Mj(T),whereM~ (T)isasolutionof∂ M~ +ω~×M~ = V~ ,
T
the Minkowski spacetime interval in the new coordinates has the form (10), i. e. (X˜j,T) are GFW
coordinates. ✷
4 Generalized isometries
Deriving a closed expression for the transformation relating two different GFW coordinate systems
would imply to invert the transformation law (14), which is not feasible in general. However we
can obtain expressions for infinitesimal transformations with the help of the notion of generalized
isometry [19].
In a GFW coordinate system the invariant interval has a very specific shape (10) that involves six
arbitrary functions aˆi(T) and ωˆj(T). The transformation formulae relating two GFW coordinate
ExtendedPrinciple6 v0.tex 9
systems, Xµ = (X~,T) −→ X′µ = (X~′,T′), must preserve this overall shape but, perhaps, with
a different sextuple of functions, aˆ′i(t),ωˆ′j(t) . We shall call this transformation a generalized
isometry [19] because the interval is:
(cid:0) (cid:1)
ds2 = g (Xν,f (X))dXαdXβ = g (X′ν,f′(X′))dX′αdX′β (23)
αβ I αβ I
Although the functions g (X,f ) are the same, the metric coefficients, g (X) = g (X,f (X)),
αβ I αβ αβ I
′ ′
are different because the values f (X) change to f (X ) in passing from one system to the other.
I I
This is the reason why the transformation is not actually an isometry and we need to introduce the
notion of generalized isometry.
In the present case f are six function that only depend on the coordinate T and can be arranged
I
as the skewsymmetric matrix
ǫ ωˆk(T) aˆi(T)
Ωˆ = ijk (24)
αβ −aˆj(T) 0 !
Consider now the infinitesimal transformation
X′α = Xα+εξα(X), f′(X) =f (X)+εΦ (X) (25)
I I I
and therefore f′(X′)= f (X)+ε [Φ (X)+ξαD f (X)], where X4 = T and D = ∂ .
I I I α I α ∂Xα
Substituting this in equation (23) and keeping only first order terms we obtain
ξαD g +2D ξαg +G (X) = 0 (26)
α µν (ν µ)α µν
where
∂g
µν
g (X) = g (X,f (X)) and G (X) = Φ
µν µν I µν I ∂f
I (cid:18) I (cid:19)(X,fI(X))
X
We have as many functions Φ as f and they can be also arranged in the skewsymmetric matrix
I I
Fˆ as
αβ
Fˆ = ǫ αˆk(T), Fˆ = −Fˆ = Aˆi(T) (27)
ij ijk i4 4i
in much the same way as we did for the f in the matrix (24).
I
Equation (26) can also be written as the generalized Killing equation
∇ ξ +∇ ξ +G = 0, (28)
µ ν ν µ µν
where ∇ is the Levi-Civita connexion for g and ξ = g ξν.
µν µ µν
Notice that the infinitesimal transformation (25) acts on Xα and also on the functions Ωˆ ; hence
αβ
the arena to represent their action is not Minkowski spacetime but rather the larger (infinite di-
mensional) manifold
M ⊂ X~,T,Ωˆ (t) ∈ R4×C0 R,R6 ,such that 1+X~ ·~a(T) > X~ ×~ω(T)
αβ
n(cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12)o
(cid:0) (cid:1) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
ExtendedPrinciple6 v0.tex 10
(the inequality is to ensure that the metric (10) is non-degenerate). The infinitesimal generator
then looks like:
δ
ξαD + dtFˆ (t) (29)
α ZR αβ δΩˆαβ(t)
where ξα depends on Xj, T and Ωˆ (t).
µν
The need of the additional information contained in the six functions Ωˆ (t) comes from the
αβ
fact that the coordinates (X~,T) are not enough to determine an event in spacetime unless we
further indicate the family of GFW observers to which these coordinates belong. To fix Ωˆ (t)
αβ
means choosing a subclass of GFW coordinate systems, that corresponding to GFW observers with
the same intrinsic spacetime angular velocity. Each one is characterized by the initial values of
the tetrad, eµ (0), and the initial point in the origin worldline, zµ(0), (the tetrad eµ (τ) and the
(α (α
worldline zµ(τ) are then obtained as in the proof of Proposition 1 in section 3). By the way, it is
the same kind and number of parameters as for Poincaré group, although they are not Lorentzian
observers because Ωˆ (t) 6= 0.
αβ
The 4-dimensional submanifolds M = X~,T,Ωˆ (t) ∈ M for a given Ωˆ (t) corresponds to
Ωˆ αβ αβ
the part of Minkowski spacetime which isn(cid:16)accessible to t(cid:17)he GFWo observers of the class determined
by this intrinsic spacetime angular velocity.
Fromaphysicalviewpointitisworthtoremarkherethatthemagnitudes Ωˆ aremeasurablebythe
αβ
GFW observer by means of accelerometers (aˆ = Ωˆ ) and gyrometers (Ωˆ ) along the observer’s
j 4j jk
spatial axes. As their measures do not need resorting to anything external, these magnitudes are
“absolute”, hence there is no sense in referring to them as “relative acceleration” or “relative angular
velocity”.
From the interval expression (10) we have that
2 2
g = 0, g = ǫ Xjωˆk, g = X~ ×ω~ − 1+X~ ·~a (30)
ij 4i ijk 44
(cid:16) (cid:17) (cid:16) (cid:17)
and, as it follows from (26),
1
G = 0, G = ǫ Xjαˆk, G = − 1+X~ ·~a X~ ·A~ + X~ ×~ω · X~ ×α~ (31)
ij 4i ijk 44
2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Using this and after a little algebra, the generalized Killing equation splits in several blocks as
space: D ξ = 0 (32)
(i j)
2
cross: − 1+X~ ·~a D ξ4+D ξ +ǫj XlωˆkD ξ −ξ ǫj ωˆk +ǫ Xlαˆk = 0 (33)
i 4 i lk i j j ik ilk
(cid:16) (cid:17)
time: − 1+X~ ·~a D 1+X~ ·~a ξ4 +ǫj XlωˆkD ξ +
4 lk 4 j
(cid:16) (cid:17) h(cid:16) (cid:17) i
1 1
D g ξ −ξ4ǫ Xlωˆk + G = 0 (34)
j 44 j jlk 44
2 2
(cid:16) (cid:17)