On algebraically special vacuum spacetimes in five dimensions

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by  Harvey S. Reall

On algebraically special vacuum spacetimes in five
dimensions
Harvey S. Reall, Alexander A.H. Graham and Carl P. Turner
∗ † ‡
3
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
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Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
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January 28, 2013
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] Abstract
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Vacuum solutions admitting ahypersurface-orthogonalrepeated principalnulldirec-
-
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g tion are an important class of 4d algebraically special spacetimes. We investigate the
[ 5d analogues of suchsolutions: vacuumspacetimes admittingahypersurface-orthogonal

2 multiple Weyl aligned nulldirection (WAND). Such spacetimes fall into 4families deter-
v minedbytherankofthe3 3matrixthatdefinestheexpansionandshearofthemultiple
7 ×
WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the
5
9 2 remaining families. We show how to define coordinates which lead to a considerable
5 simplificationoftheEinsteinequationwithcosmological constant. Therank2casegives
.
1 warped product and Kaluza-Klein versions of the 4d Robinson-Trautman solutions as
1
well as some new solutions. The rank 1 case gives product, or analytically continued
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1 Schwarzschild, spacetimes.
:
v
i
X 1 Introduction
r
a
The technique for solving the 4d Einstein equation that has received most attention is search-
ing for solutions with algebraically special Weyl tensor [1]. This is how the Kerr solution
was discovered. It is natural to exploit the same technique to solve the Einstein equation for
d > 4 spacetime dimensions, especially as this is one of the few techniques which works with
a cosmological constant. This paper will consider only Einstein spacetimes, i.e. solutions
of the vacuum Einstein equation with a cosmological constant. We will look for solutions
which are algebraically special in the classification of Coley, Milson, Pravda and Pravdova
[2]. Such a solution admits a multiple Weyl aligned null direction (multiple WAND), the
higher-dimensional generalization of a repeated principal null direction.
∗[email protected]
†[email protected]
‡[email protected]
1
In 4d, the Einstein equation simplifies considerably when the repeated principal null di-
rection is hypersurface-orthogonal. There are two families of such solutions: the Robinson-
Trautman (RT) family and the Kundt family [1]. RT spacetimes are defined by the property
of admitting a null geodesic congruence with vanishing shear and rotation, but non-vanishing
expansion. Thisfamilyincludes theSchwarzschild solution, theC-metric, andtime-dependent
spacetimes which approach Schwarzschild asymptotically. Kundt spacetimes are defined by
the property of admitting a null geodesic congruence with vanishing expansion, rotation and
shear. This family includes pp-waves, and the near-horizon geometry of the extreme Kerr
black hole.
Higher-dimensional RT spacetimes were investigated in Ref. [3]. It was found that these
are significantly less rich than in 4d, containing certain simple time-independent generaliza-
tionsoftheSchwarzschild solutionbutnothingthatcouldbeidentifiedasahigher-dimensional
analogue of the C-metric, and no time-dependent generalizations of Schwarzschild. Higher-
dimensional Kundt spacetimes were discussed in Refs. [4, 5].
A possible reason why the higher-dimensional RT solutions are less interesting than in 4d
is that a multiple WAND in d > 4 dimensions need not be shear-free. Hence, unlike in 4d, the
RTandKundtfamiliesarenotthemostgeneralsolutionsadmittingahypersurface orthogonal
multiple WAND. For example, a Schwarzschild black string belongs to neither family. So it
seems worthwhile investigating more general solutions admitting a hypersurface-orthogonal
multiple WAND. Ref. [6] studied some general properties of such solutions. Coordinates were
introduced analogous to those used for RT or Kundt solutions. The dependence of the metric
on one of these coordinates was fully determined for the more special algebraic types.
Recently, Ref. [7] investigated algebraically special solutions in 5d. In the case of a
hypersurface-orthogonal multiple WAND, it was found that there exist two classes of space-
time that are distinct from the RT and Kundt classes. To describe these classes, we need
to introduce some notation. Recall that the expansion, rotation and shear of a null vector
field ℓa are defined as follows. Introduce a set of d 2 orthonormal spacelike vectors ma,
− i
orthogonal to ℓa. Define the (d 2) (d 2) optical matrix of ℓa
− × −
ρ = mamb ℓ (1)
ij i j∇b a
The expansion, rotation and shear are the trace, antisymmetric part, and traceless symmetric
part of ρ . We are interested in the case of a hypersurface orthogonal multiple WAND ℓa, for
ij
which ρ is symmetric. In this case, in 5d, Ref. [7] found that the eigenvalues of the 3 3
ij
×
matrix ρ are either a,a,a , a,a,0 , a,0,0 or 0,0,0 , where a = 0. The first of these
ij
{ } { } { } { } 6
is the RT family and the last is Kundt family. We will investigate the two other possibilities.
Both classes are non-empty e.g. they contain the Schwarzschild black string solution, and the
product dS S2 respectively.
3
×
Weproceedasfollows. Thefirststepistointroducecoordinatesadaptedtothegeometrical
properties of these spacetimes. For RT or Kundt spacetimes, the procedure for doing this is
well-known. However, in the cases of interest to us we have to work harder. In section 2 we
show that these spacetimes admit foliations by null submanifolds which enable one to define
a natural set of coordinates adapted to the structure of ρ . We then investigate the vacuum
ij
Einstein equation, allowing for a cosmological constant.
2
In section 3 we consider the case for which ρ has rank 2. The general solution involves
ij
a function m similar to the ”mass” parameter of the 4d RT solutions. For m = 0 we find
6
that the solution must belong to one of the following classes (i) for any cosmological constant
Λ, a warped product involving a 4d vacuum RT solution (equation (84)); (ii) for Λ = 0, the
Kaluza-Klein oxidation of a 4d RT solution with a null Maxwell field (equation (89)); (iii)
for Λ = 0, a new class with metric given in equation (94), for which the Einstein equation
reduces to a pair of elliptic PDEs in 2 dimensions; (iv) for Λ < 0 a similar class, with metric
given in (100), involving the solutions of PDEs in 3 dimensions. For m = 0 the Einstein
equation reduces to a pair of equations identical to those governing type III (or more special)
RT solutions of 4d Einstein-Maxwell theory (section 3.3).
When ρ has rank 1, we determine all solutions explicitly in section 4. They coincide with
ij
the solutions obtained in Ref. [8]. They are either product spacetimes (dS S2 or adS H2)
3 3
× ×
or analytically continued versions of a generalized Schwarzschild solution (equation (134)).
Examples of the latter are the Kaluza-Klein bubble of Ref. [9] and the anti-de Sitter soliton
of Ref. [10].
Finally, we discuss a 6d case. Ref. [11] obtained necessary conditions on the eigenvalues of
the optical matrixof a hypersurface-orthogonal multiple WAND inany number of dimensions.
In 6d, there is the interesting possibility of eigenvalues a,a,b,b with a,b = 0, a = b. The
{ } 6 6
coordinates of section 2 can be introduced for this case. However, the only solutions of the
Einstein equation (with cosmological constant) in this class are conformally flat.
2 Canonical form for metric
2.1 Introduction of coordinates
We will make use of the higher-dimensional generalization of the Geroch-Held-Penrose (GHP)
formalism introduced in Ref. [12]. Our notation will follow the notation of that reference.
Introduce a null basis e ℓ,e n,e m , i = 2,...,d 1 satisfying
0 1 i i
{ ≡ ≡ ≡ } −
ℓ2 = n2 = ℓ m = n m = 0 ℓ n = 1 m m = δ (2)
i i i j ij
· · · ·
Greek indices µ,ν,...refer to thisbasis. Latinindices a,b,c...areabstract indices. Introduce
the following notation for the connection components [13]
i
L = ℓ , N = n , M = (m ) (3)
µν ν µ µν ν µ µν ν i µ
∇ ∇ ∇
Consider an Einstein spacetime, in any number of dimensions, admitting a multiple WAND.
Ref. [8] proved that such a spacetime must admit a geodesic multiple WAND so there is
no loss of generality in restricting attention to geodesic multiple WANDs. We start with
a couple of useful Lemmas. Note that we do not yet assume our multiple WAND to be
hypersurface-orthogonal.
Lemma 1. Let ℓa be a geodesic multiple WAND (not necessarily affinely parameterized)
such that (i) ρ (defined in (1)) has a block diagonal structure with blocks ρ and ρ , i.e.,
ij IJ αβ
3
ρ = ρ = 0; (ii) ρ = λδ for some function λ; (iii) λ is not an eigenvalue of ρ . Then
Iα αI IJ IJ αβ
the distribution span ℓ,m ( I) is integrable.
I
D ≡ { ∀ }
Proof. Using (i), equation NP1 of Ref. [12] with i = α, j = I gives
β α α
ρ M = ρ M = λ M (4)
αβ I0 JI J0 I0
α
where we used (ii) in the final equality. Now (iii) implies that M = 0, i.e.,
J0
m (ℓ m ) = 0 (5)
J α
· ·∇
Consider[ℓ,m ]. Firstwehaveℓ [ℓ,m ] = ℓaℓb m ℓamb ℓ = m ℓb ℓa (1/2)mb ℓ2 =
i · i ∇b ia− i∇b a − ia ∇b − i∇b
0 using the geodesicequation forℓa. Wealsohave m [ℓ,m ] = m (ℓ m ) m (m ℓ) =
α I α I α I
· · ·∇ − · ·∇
m (ℓ m ) ρ = 0. These results imply [ℓ,m ] .
I α αI I
− · ·∇ − ∈ D
Using (i) and (ii), equation NP3 of Ref. [12] with i = α, j = J, k = K reduces to
β α
ρ M = λ M (6)
αβ [JK] [JK]
α
and hence (iii) implies M = 0, i.e.,
[JK]
m m m = 0 (7)
[J K] α
· ·∇
(cid:16) (cid:17)
Finally, consider [m ,m ]. We have ℓ [m ,m ] = 2m (m ℓ) = 2ρ = 0 using
J K J K [K J] KJ
· − · ·∇ −
(ii). We also have mα [m ,m ] = 2m (m m ) = 0 using the result just obtained.
J K [K J] α
· − · ·∇
These results imply [m ,m ] . This concludes the proof.
J K
∈ D
Lemma 2. The basis vectors m can be chosen to be parallelly transported along geodesics
i
with tangent ℓa whilst preserving the properties (i), (ii), (iii) of Lemma 1.
Proof. LetVa beparallellytransportedℓa Vb = 0withV1 = V ℓ = 0(aconditionpreserved
a
∇ ·
by parallel transport). Then
i i i i
ℓa∂ V = ℓa (V m ) = Vµ M = V0 M +V M = V M (8)
a i a i µ0 00 j j0 j j0
∇ ·
i
where, in the final equality, we used M = ℓ ℓb ma = maℓb ℓ = 0 (from the geodesic
α 00 a ∇b i − i ∇b a
equation). Using (5) (i.e. M = 0) we have
J0
I α
ℓ ∂V = V M , ℓ ∂V = V M (9)
I J J0 α β β0
· ·
Hence if all V (all V ) initially are zero then all V (all V ) remain zero under parallel
I α I α
transport. Now define m′ to be a set of basis vectors which coincide initially with m and
i i
are parallelly transported along a geodesic with tangent ℓ. Using the result just derived, m′
I
remains orthogonal to m and m′ remains orthogonal to m (and both remain orthogonal to
α α I
ℓ). It follows that m′ is a linear combination of ℓ and the m , and m′ is a linear combination
I J α
of ℓ and the m . Hence the new basis is related to the old one by a null rotation [12] about
β
4
ℓ (which does not change ρ ) and a spin [12] which does not mix m with m and hence
ij I α
preserves the structure of ρ .
ij
In the rest of this paper, we will impose the extra condition that the geodesic multiple
WAND ℓa is hypersurface-orthogonal, which is equivalent to vanishing rotation:
ρ = 0. (10)
[ij]
Let ρ , ι = 1,...,N be the distinct eigenvalues of ρ = ρ . Choose the spatial basis
ι ij (ij)
vectors m = m ,...,m so that m spans the ιth eigenspace of ρ . Then ρ
{ i} { (I1) (IN)} { (Iι)} ij (ij)
is diagonal. Lemma 1 implies that the distribution ℓ,m is integrable for all ι.
Dι ≡ { Iι}
Therefore each eigenspace of ρ determines a foliation by null submanifolds of spacetime.
ij ι
N
Let us focus on the case in which N = 2, i.e., ρ has two distinct eigenvalues. For d = 5
ij
this is the only possibility other than Robinson-Trautman or Kundt [7] (we also know that
ρ = 0 in this case although we shall not assume this yet). We can introduce coordinates
2
adapted to these null foliations as follows.
ℓ is hypersurface orthogonal. Introduce a function u with du = 0 so that ℓ is orthogonal to
6
surfaces ofconstant u. Using a boost [12] we canarrangethatℓ = (du) , which implies that
a a
−
ℓ is tangent to the affinely parameterized null geodesic generators of the surfaces of constant
u.
Let the surfaces have dimension n+1 and introduce a chart
1
N
(X˜0,...,X˜n,Y˜n+1,...,Y˜d−1) (11)
so that the surfaces are surfaces of constant Y˜. These lie within the surfaces of constant
1
N
u therefore u is constant over each surface and hence cannot depend on X˜. Therefore
1
u = u(Y˜). Hence we can define new coordNinates (X˜,yn+1,...,yd−2,u) where each y is a
function of the Y˜s and the surfaces are surfaces of constant yn+1,...,yd−2,u. Next pick
1
N
some spacelike hypersurface Σ transverse to ℓ and complete (yn+1,...,yd−2,u) to a chart
(y1,...yd−2,u) on this surface. Finally, assign coordinates (u,r,y1,...,yd−2) to the point
affine parameter distance r from this surface along the integral curve of ℓ (a geodesic) starting
from (y1,...yd−2,u). Since we already know that u and yn+1,...yd−2 are constant along
such geodesics, this agrees with our previous definitions of these quantities. We now have a
chart (u,r,y1,...yd−2) for which ℓa = ∂/∂r, ℓ = (du) , and the surfaces are surfaces
a a 1
− N
of constant u,yn+1,...,yd−2. On any such surface, we have a coordinate chart defined by
(r,y1,...,yn).
Repeat this procedure for the foliationdefined by , using the same surface Σ. This gives
2
N
a coordinate chart (u,r,x1,...,xd−2) with the property that the surfaces are surfaces of
2
N
constant u,x1,...,xn (after reordering the x-coordinates). The coordinates u,r are common
to both charts.
Consider η = dx1 ... dxn dyn+1 ... dyd−2. If η = 0 then some linear combination of
∧ ∧ ∧ ∧ ∧
dx1,...,dxn equals a linear combination of dyn+1,...,dyd−2. But then there is some 1-form
other than (du) = ℓ that is normal to both and , which is a contradiction. Hence
a a 1 2
− N N
η = 0.
6
Now ℓ = ∂/∂r (in either chart) so ℓ dyµ = ℓ dxµ = 0 hence ℓ η = 0. Therefore
· · ·
ℓ (dr η) = η = 0 and hence dr η = 0. It follows that (r,x1,...,xn,yn+1,...,yd−2) is
· ∧ 6 ∧ 6
5
a good coordinate chart on surfaces of constant u and then (u,r,x1,...,xn,yn+1,...,yd−2)
is a good coordinate chart on spacetime with the property that surfaces are surfaces
1
N
of constant u,yn+1,...,yd−2 and are surfaces of constant u,x1,...,xn. The distribu-
2
N
tion spanned by ∂/∂r,∂/∂x1,...,∂/∂xn is tangent to and the distribution spanned by
1
{ } N
∂/∂r,∂/∂yn+1,...,∂/∂yd−2 is tangent to . Hence these distributions are orthogonal.
2
{ } N
The metric now must take the form
ds2 = grrdu2 2dudr+g dxM +grMdu dxN +grNdu
MN
− −
+ g dyA +grAdu dyB(cid:16)+grBdu (cid:17)(cid:16) (cid:17) (12)
AB
(cid:16) (cid:17)(cid:16) (cid:17)
where indices M,N run from 1 to n and A,B run from n+1 to d 2. This is a special case
−
of the metric discussed in Ref. [6]. Henceforth we will use (as in the above Lemmas) indices
I,J,... instead of I ,J ,... and α,β,... instead of I ,J ,.... We will use m,n,... to denote
1 1 2 2
either M or A. Note that the basis vectors m obey mu = mu = mA = mM = 0.
i I α I α
2.2 Further simplifications
We will show how the above metric can be simplified further by imposing the multiple WAND
condition and some components of the Einstein equation. We will do this by solving some of
the ”Newman-Penrose” (NP) equations of Ref. [12].
We have chosen ℓa to be tangent to affinely parameterized geodesics, and will take ma
i
to be parallely transported along these geodesics. It follows that na also must be parallely
transported, i.e., all basis vectors are parallely transported.
In our parallely transported basis, equation NP1 of Ref. [12] (with the WAND condition)
after shifting r appropriately reduces to
1 1
ρ = , ρ = (13)
1 2
r r +a¯(u,x,y)
where a¯(u,x,y) = 0 and we will allow for the possibility a¯ = corresponding to ρ = 0 (we
2
6 ∞
know a¯ = when d = 5). All equations below are valid in the limit a¯ unless otherwise
∞ → ∞
noted. Here, and henceforth, an overbar denotes a quantity independent of r.
Calculating ρ gives
ij
1
ρ = mmmn∂ g (14)
ij 2 i j r mn
and hence
1
∂ g = ρ m m (15)
r mn ij im jn
2
Taking µ = m in ℓ mµ = 0 and using (15) gives an equation which can be solved to give
·∇ i
1 1
mm(u,r,x,y) = m¯m(u,x,y), mm(u,r,x,y) = m¯m(u,x,y) (16)
I r I α 1+r/a¯ α
Taking µ = r gives
∂ grm = 2(∂ mr)mm (17)
r r i i
6
We now have
1 1
gMN = m¯Mm¯N, gAB = m¯Am¯B (18)
r2 I I (1+r/a¯)2 α α
and hence
g = r2g¯ (u,x,y), g = (1+r/a¯)2g¯ (u,x,y) (19)
MN MN AB AB
Since dℓ = 0 we have
L = L τ (20)
1j j1 j
≡
Equation NP3 of Ref. [12] is
k ρ = 0. (21)
[j |i|k]
Setting j = J,i = I,k = K gives
1
mr +τ = 0 if n > 1 (22)
r J J
(recall that n is the number of x-coordinates, the restriction n > 1 arises because (21) is
antisymmetric on jk). Setting j = J,i = I,k = γ gives
γ 1
M = (1+r/a¯) mr +τ δ (23)
IJ r γ γ IJ
(cid:18) (cid:19)
Setting j = I,i = α,k = β gives
I r 1
M = mr +mM∂ a¯ +τ δ (24)
αβ −a¯ r+a¯ I I M I αβ
(cid:20) (cid:16) (cid:17) (cid:21)
j = β,I = α,K = γ gives
1
mr +mA∂ a¯ +τ = 0 if d 2 n > 1, a¯ = (25)
(r+a¯) α α A α − − 6 ∞
(cid:16) (cid:17)
The remaining components of (21) are trivial.
Since the basis is parallelly propagated, τ′ N = 0. Equation NP2 of Ref. [12] (with
i ≡ i0
vanishing curvature terms) reduces to
1 1
τ = τ¯ (u,x,y), τ = τ¯ (u,x,y) (26)
I I α α
r 1+r/a¯
The orthogonality relations of the basis vectors give
n = 1, n = mrm (27)
r m − i im
Using this we can calulate τ = L directly to obtain
i i1
1
τ = ∂ mr + m mrm ∂ gmn (28)
i − r i 2 im j jn r
Since we know the r-dependence of the LHS we now can obtain (using (18))
1
mr = τ¯ + m¯r (29)
I − I r I
7
r 1
mr = τ¯ + m¯r (30)
α −1+r/a¯ α 1+r/a¯ α
Now using (22) gives
m¯r = 0 if n > 1 (31)
I
and (25) gives
m¯r = a¯τ¯ m¯A∂ a¯ if d 2 n > 1, a¯ = (32)
α − α − α A − − 6 ∞
A calculation from the definition of the LHS gives
α 1 1 1
M = τ¯ + m¯r δ m m mA∂ gMN (33)
IJ 1+r/a¯ − α r α IJ − 2(1+r/a¯) IM JN α A
(cid:18) (cid:19)
Comparing with (23) gives
1
∂ g¯ = 2g¯ m¯B τ¯ + m¯r g¯ (34)
A MN AB α α a¯ α MN
(cid:18) (cid:19)
and hence
g¯ (u,x,y) = P¯(u,x,y)−2h (u,x) (35)
MN MN
where
∂ P¯ 1
A = g¯ m¯B τ¯ + m¯r (36)
P¯ − AB α α a¯ α
(cid:18) (cid:19)
Combining with (32) gives (absorbing a function of u,x into h )
MN
P¯ = a¯ if d 2 n > 1, a¯ = (37)
− − 6 ∞
Now we calculate from the definition of the LHS
I 1 1 1
M = τ¯ + m¯r δ m m m¯M∂ gAB (38)
αβ r +a¯ − I r I αβ − 2r αA βB I M
(cid:18) (cid:19)
Comparing with (24) gives
2
∂ g¯ = g¯ g¯ m¯Nm¯r (39)
M AB −a¯ AB MN I I
(cid:16) (cid:17)
and hence
g¯ (u,x,y) = Q¯(u,x,y)−2h (u,y) (40)
AB AB
where
∂ Q¯ 1
M = g¯ m¯Nm¯r (41)
Q¯ a¯ MN I I
Combining with (31) gives (absorbing a function of u,y into h )
AB
Q¯(u,x,y) = 1 if a¯ = or n > 1 (42)
∞
Now we return to (17). This gives
1
grM = m¯rm¯M +g¯rM (43)
r2 I I
8
so, from (31),
grM = g¯rM if n > 1 (44)
and, from (41),
a¯ ∂ Q¯
grM = g¯MN N +g¯rM if a¯ = (45)
r2 Q¯ 6 ∞
From (17) we also obtain
a¯ 1
grA = τ¯ + m¯r m¯A +g¯rA if a¯ = (46)
(1+r/a¯)2 α a¯ α α 6 ∞
(cid:18) (cid:19)
grA = 2rτ¯ m¯A +g¯rA if a¯ = (47)
− α α ∞
Note that the coefficient in the r-dependent piece involves (in both cases)
1
τ¯ + m¯r m¯A = g¯AB∂ P¯/P¯ (48)
α a¯ α α − B
(cid:18) (cid:19)
In summary, we have shown that the metric components in (12) must take the following form:
g (u,r,x,y) = r2P¯(u,x,y)−2h (u,x) (49)
MN MN
where P¯ = a¯ if d 2 n > 1 and a¯ =
− − 6 ∞
g (u,r,x,y) = (1+r/a¯(u,x,y))2Q¯(u,x,y)−2h (u,y) (50)
AB AB
where Q¯ = 1 if a¯ = or n > 1,
∞
1
grM(u,r,x,y) = f¯rM(u,x,y)+g¯rM(u,x,y), (51)
r2
where f¯rM = 0 if n > 1 and
a¯ ∂ Q¯
f¯rM = P¯2hMN N if a¯ = (52)
r2 Q¯ 6 ∞
(f¯rM is unconstrained if n = 1, a¯ = .)
∞
a¯ ∂ P¯
grA = Q¯2hAB B +g¯rA(u,x,y) if a¯ = (53)
−(1+r/a¯)2 P¯ 6 ∞
∂ P¯
grA = 2rQ¯2hAB B +g¯rA(u,x,y) if a¯ = (54)
P¯ ∞
where hMN, hAB are the inverses of h , h .
MN AB
In deriving these results, we used only equations NP1, NP2 and NP3 with vanishing (Ricci
and Weyl) curvature components (as is the case for an Einstein spacetime with ℓa a multiple
WAND). The metric deduced above is the most general metric admitting a hypersurface-
orthogonal geodesic multiple WAND, for which ρ has precisely two distinct eigenvalues, and
ij
satisfies R = R = R = 0.
rr rM rA
9
2.3 The 5d case
In 5d, Ref. [7] proved that a¯ = so there are just two cases to consider, n = 2 and n = 1.
∞
For n = 2 we can perform a coordinate transformation x x′(x,u) to arrange that
h (u,x) = δ (absorbing a conformal factor into P¯). h has→a single component h(u,y)
MN MN AB
and y y′(y,u) can be used to set h = 1. The metric is
→
r2 ∂ P 2
ds2 = Udu2 2dudr+ (dx+Adu)2 + dy + C +2r y du (55)
− − P2 " P ! #
where we have dropped the bar on P. Here bold font denotes objects transforming as vectors
in 2d Euclidean space with coordinates x. The functional dependence is U = U(u,r,x,y),
P = P(u,x,y), A = A(u,x,y), C = C(u,x,y).
For n = 1, h (u,x) has a single component which we can set equal to 1 using x
MN
x′(x,u). Although we have Q¯ = 1 in this case, a transformation y y′(y,u) can be used→to
→
achieve h (u,y) = Q(u,y)−2δ for some function Q. The metric is
AB AB
r2 A 2 1 P 2
ds2 = Udu2 2dudr+ dx+ +B du + dy+ C+2rQ2∇ du (56)
− − P2 r2 Q2 P
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:18) (cid:19) (cid:21)
Again we have dropped the bar on P. Here bold font denotes objects transforming as vectors
in 2d Euclidean space with coordinates y and is the gradient operator in this space. The
∇
functional dependence is U = U(u,r,x,y), P = P(u,x,y), A = A(u,x,y), B = B(u,x,y)
C = C(u,x,y), Q = Q(u,y).
As mentioned above, these metrics satisfy R = R = R = 0. We will determine the
rr rx ry
conditions arising from the remaining components of the Einstein equation with cosmological
constant:
R = Λg (57)
ab ab
The calculations were performed using Mathematica.
3 5d metric with rank 2 optical matrix
3.1 The Einstein equation
For the metric (55), the components of the Einstein equation in the xM,y directions give
R = 0 ∂ AM = 0 (58)
My y
⇔
so A = A(u,x)
R = 0 ∂ A2 +∂ A1 = 0 (59)
12 1 2
⇔
R R = 0 ∂ A1 ∂ A2 = 0 (60)
11 22 1 2
− ⇔ −
These equations are equivalent to (lowering indices with δ )
MN
1
∂ A = Aδ (61)
(M N) MN
2∇·
10