On conformally flat and type N pure radiation metrics

0.17 MB·English

by  Jiri Podolsky

Noname manuscript No.
(will be inserted by the editor)
Jiˇr´ı Podolsky´ · Ondˇrej Prikryl
On conformally flat and type N pure
radiation metrics
9
0
0
2

n
a Received:22 August 2008 / Accepted:date
J

9

Abstract We study pure radiation spacetimes of algebraic types O and N
]
c withapossiblecosmologicalconstant.Inparticular,wepresentexplicittrans-
q formations which put these metrics, that were recently re-derivedby Edgar,
-
VickersandMachadoRamos,intoageneralOzsv´ath–Robinson–R´ozgaform.
r
g By putting all such metrics into the unified coordinate system we confirm
[ thattheirderivationbasedontheGIFformalismiscorrect.Weidentifyonly

2 few trivial differences.
v
Keywords Kundt spacetimes conformally flat solutions pure radiation
4
· ·
4
PACS 04.20.Jb 04.30.Nk
9 ·
4
.
2
1 1 Introduction
8
0
In recent papers [1,2,3,4], Edgar, Vickers and Machado Ramos systemati-
:
v cally re-derived a general family of pure radiation spacetimes which are of
i algebraic types N and O. Using the advantages of the generalized invariant
X
formalism (GIF) [5], which combines the features of standardGeroch–Held–
r Penrose(GHP)andnullrotationinvariantformalisms,theyobtainedacom-
a
plete class of conformally flat and type N solutions of the field equations,
possibly admitting a cosmologicalconstant Λ.
Thesemetricswerepresentedin[1,2,3,4]inthecontravariantformsofgij
using various different coordinates. Our main aim here is to compare these
(apparentlydistinct)metricsbyputtingallofthemintoacommoncoordinate
system which is more suitable for physical and geometrical interpretation.
This will also elucidate relations of these solutions to previous works, in
Instituteof Theoretical Physics, Faculty of Mathematics and Physics,
CharlesUniversityinPrague,VHoleˇsoviˇck´ach2,18000Prague8,CzechRepublic
Tel.: [phone], E-mail: podolsky‘AT’mbox.troja.mff.cuni.cz
2
particular those presented in [6,7,8,9,10]. We will explicitly demonstrate
that all such metrics can be conveniently written in the form
2 Q2 Q2 (Q2) Q
ds2 = dζdζ¯ 2 dudv+ κ v2 ,u v H du2, (1)
P2 − P2 P2 − P2 − P
(cid:18) (cid:19)
where
P =1+ 1Λζζ¯, (2)
6
Q =(1 1Λζζ¯)α+β¯ζ+βζ¯, (3)
− 6
and
κ= 1Λα2+2ββ¯, (4)
3
where α(u), β(u) are functions of u, and H(ζ,ζ¯,u) is a function of ζ,ζ¯,u.
The general metric (1)–(4) was first presented by Ozsv´ath, Robinson and
Ro´zga in [6]. It represents all type N or conformally flat Kundt spacetimes
witha cosmologicalconstantwhichareeither vacuumorcontainpure radia-
tion. Indeed, with the null tetrad k=∂ , l=(P2/Q2)∂ +(P4/2Q4)F ∂ ,
v u r
m=P∂ζ¯ where F =guu, the only non-zero curvature tensor components
are given by
P4
Ψ = 1(P H) , (5)
4 2 ,ζζ Q3
P3
Φ22 = 12(P2H,ζζ¯+ 31ΛH)Q3 . (6)
The complete family of conformally flat Kundt spacetimes with pure
radiationareobtainedbysetting Ψ =0,inwhichcasethe functionH takes
4
the particular form
(u)+ ¯(u)ζ+ (u)ζ¯+ (u)ζζ¯
H = A B B C , (7)
1+ 1Λζζ¯
6
where (u), (u) and (u) are arbitrary functions of u, with and real.
A B C A C
The pure radiation component is then
P3
Φ = 1 + 1Λ . (8)
22 2 C 6 A Q3
(cid:0) (cid:1)
Thespacetimesarevacuum when Φ22 =0,i.e. P2H,ζζ¯+ 13ΛH =0.This
equation has a general solution H =(f,ζ +f¯,ζ¯)− 31ΛP−1(ζ¯f +ζf¯), where
f(ζ,u) is an arbitrary function of ζ and u, holomorphic in ζ. In all other
cases, the spacetimes contain pure radiation and are of type N or O.
In particular, for conformally flat and vacuum spacetimes, = 1Λ .
C −6 A
SuchfunctionsH oftheform(7)correspondtof =c (u)+c (u)ζ+c (u)ζ2,
0 1 2
where c (u) are complex functions of u related to and . These solu-
i
A B
tions, with f quadraticin ζ, are isometric to Minkowski(if Λ=0), de Sitter
(ifΛ>0)andanti-deSitterspacetime(ifΛ<0),see[9].Forallotherchoices
of H, the Kundt spacetimes (1)–(4) describe exact non-expanding pure ra-
diation and/or gravitationalwaves.
3
2 Type N spacetimes with pure radiation
The class of type N pure radiation solutions of Einstein’s field equations
with Λ=0 within the Kundt family was obtained in [2]. In the coordinates
(t,n,a,b) of equation (66) therein, the contravariantmetric tensor reads1
0 1/a 0 0
1/a L/an/a 0
gij = 0 −n/a 1 0, (9)
0 0 0 1
 
 
wherethearbitraryfunctionL(t,ξ,ξ¯)dependsonthreerealcoordinatest,a,b
via the complex variable ξ =a+ib. The inverse matrix to (9) is
(n2+aL)a n 0
−
a 0 0 0
g = , (10)
ij  n 0 1 0
−
0 0 0 1
 
 
so that the line element can be written as
ds2 =da2+db2 2ndtda+2adtdn+(n2+aL)dt2. (11)
−
Now, performing the transformation and re-labelling
n
x=a, y =b, v = , u=t, (12)
−2a
the metric becomes
ds2 =dx2+dy2 4x2dudv+(4x2v2+xL)du2, (13)
−
whichisthestandardexplicitformoftheKundttypeNmetrics,seee.g.[11].
Introducing the complex spatial parameter ζ = 1 (x+iy), we obtain
√2
ds2 =2dζdζ¯ 2(ζ+ζ¯)2dudv+ 2(ζ+ζ¯)2v2 (ζ+ζ¯)H du2. (14)
− −
(cid:2) (cid:3)
This is the particular case P =1, Q=ζ+ζ¯, H = 1 L of the general
−√2
metric (1) of Kundt type N spacetimes given in [6,9], which corresponds to
the choice of parameters Λ=0, α=0, β =1, i.e. κ=2.
1 Hereand in thefollowing we are changing thesignature to(−+++).
4
3 Type O spacetimes with pure radiation
Next, we will examine the family of solutions obtained in [1]. In the coordi-
nates (t,n,a,b), these conformally flat spacetimes are given by
0 1/a 0 0
1/a (2S 2Ma a2 b2)/a n/a E/a
gij = 0 − n/−a − 1 0 , (15)
0 E/a 0 1
 
 
where E,M,S are arbitrary functions of t (see equation (68) in [1]). The
correspondingcovariantmetric tensorcomponents are obtainedby inverting
this matrix, i.e.
(n2+E2+aL) a n E
− −
a 0 0 0
g = , (16)
ij  n 0 1 0 
−
E 0 0 1
 − 
 
where
L a2+b2+2Ma 2S. (17)
≡ −
The metric thus takes the explicit form
ds2 =da2+db2 2ndtda 2Edtdb+2adtdn+(n2+E2+aL)dt2, (18)
− −
which, for E =0, obviously reduces to (11). With the transformation and
re-labelling
n
x=a, y =b Edt, v = , u=t, (19)
− −2a
Z
the metric becomes
ds2 =dx2+dy2 4x2dudv+(4x2v2+xL)du2, (20)
−
where the function L is
L=x2+y2+2Mx+ 2 Edu y 2S+ Edu 2. (21)
−
This is againthe standard Kundt m(cid:0) eRtric (1(cid:1)3), but the(cid:0)fRunction(cid:1) L now has a
special form (21). It is fully consistent with the assumed conformal flatness
which requires L to be at most quadratic in the spatialcoordinates x and y,
with the coefficients being arbitraryfunctions of u. Introducing the complex
coordinate ζ = 1 (x+iy), we obtain the metric (14) in which L is
√2
L(ζ,ζ,u)=2ζζ¯+f(u)ζ+g(u)ζ¯+h(u), (22)
wheref,g,hareanyfunctions ofu.Thisexactlycorrespondstothe function
H given by expression (7) in the case when Λ=0 and (u) is constant.
C
5
4 Type O pure radiation spacetimes with a negative cosmological
constant
Now, we will analyse the solutions described by equation (103) in [3] which
represent a class of conformally flat pure radiation metrics with Λ<0 such
that ττ¯+ 1Λ=0, where τ is the corresponding spin coefficient. The con-
6
travariant metric tensor components in coordinates (r,n,m,b) are
0 1 0 0
−
1 V mν (r) bν (r)
gij =m2 − − − 4 − 4 , (23)
 0 mν (r) 2λ2 0 
4
−
0 bν (r) 0 2λ2
 − 4 
 
where
m
V =3nν (r) ν (r)(b2+m2) ν (r)b +ν (r), (24)
4 − 5 − 6 − λ2 3
in which ν (r), ν (r), ν (r), ν (r) are arbitrary functions of r. The inverse
3 4 5 6
matrix is
V˜ 2λ2 mν (r) bν (r)
4 4
1 −2λ2 −0 − 0 − 0
gij = 2λ2m2  m−ν4(r) 0 1 0 , (25)
−
 bν4(r) 0 0 1 
 − 
 
with
V˜ 2λ2V ν2(r)(m2+b2), (26)
≡− − 4
so that the metric reads
1
ds2 = dm2+db2 2mν (r)drdm 2bν (r)drdb
2λ2m2 − 4 − 4
(cid:16)
4λ2drdn V˜ dr2 . (27)
− −
(cid:17)
Applying the transformation
x=mR2(r), y =bR2(r), v =nR3(r), (28)
where
1
R(r)=exp ν (r)dr , (29)
4
− 2
(cid:16) Z (cid:17)
we obtain
1
ds2 = dx2+dy2 4λ2R(r)drdv+2λ2H˜ dr2 , (30)
2λ2x2 −
(cid:16) (cid:17)
in which H˜ has the form
R2(r)
H˜ ν (r)(x2+y2) ν (r)R2(r)y x+ν (r)R4(r). (31)
≡− 5 − 6 − λ2 3
Finally, the transformation from r to the new coordinate u,
u=2λ2 R(r)dr, (32)
Z
6
puts the above metric to
1
ds2 = dx2+dy2 2dudv+Hdu2 , (33)
2λ2x2 −
(cid:0) (cid:1)
where the function H is obtained from H˜ by substitution for r,
H(x,y,u)=A(u)(x2+y2)+B(u)x+C(u)y+D(u), (34)
in which A,B,C,D are arbitrary2 functions of u. With the identification
Λ
λ2 , (35)
≡−6
where Λ<0 is a negative cosmological constant, this is exactly the confor-
mally flatsubfamily ofthe Siklos solutionspresentedfor the firsttime in [7].
Using the transformation ζ = 6 (x+ 1 +iy)/(x 1 +iy), v = 12r,
− −Λ 2 − 2 Λ
see[8],themetric(33)isputintoqtheOzsv´ath–Robinson–R´ozgaform(1)–(7)
with α=1, β = 1Λ, i.e. Q= 1+ 1Λζ 1+ 1Λζ¯ and κ=0.
−6 −6 −6
q (cid:16) q (cid:17)(cid:16) q (cid:17)
5 Other type O pure radiation spacetimes with a cosmological
constant
In recent work[4], Edgarand Machado Ramos extended their investigations
to a complete family of conformally flat pure radiation metrics with Λ=0
for which ττ¯+ 1Λ=0. The correspondingcontravariantmetric tensor co6 m-
6 6
ponents given by equation (86) therein, using the coordinates (t˜,c,a,x˜), are
0 k1/4 0 0
−a(32−k)
 
k1/4 8 Z 2k(356Λ2a4+1)c k1/4γ3(t˜)
gij =−a(23−k) ak1/2(32−k)2 a(23−k) − a(32−k) , (36)
 
 0 2k(3a56(Λ232−ak4)+1)c 4k2 0 
 
 0 −ka1(/324γ−3k(t)˜) 0 4k 
where γ (t˜) is an arbitrary function of t˜, and k is given by3
3
k = 1Λa2+ 1. (37)
6 2
2 NoticethatduringthederivationwehaveobtainedB(u)=−1/(2λ4).However,
an arbitrary function B(u) can be set to any constant value by the coordinate
transformation x=efx′, y=efy′, u= e2fdu′, v=v′+ 1f˙(x′2+y′2), using a
2
′
suitable function f(u), see [7].
3 Wehavere-labeled thecosmological pRarameter Λusedin[4]to 1Λ,whereΛis
6
now thestandard cosmological constant. See also relation (35).
7
The function Z may have three distinct forms, namely
(i) for Λ>0: Z =γ (t˜) cos 1Λx˜ +2√2aγ (t˜)+z(a,c), (38)
1 6 2
(cid:16)q (cid:17)
(ii) for Λ<0: Z =γ (t˜) cosh 1Λx˜ +2√2aγ (t˜)+z(a,c), (39)
1 −6 2
1(cid:16)q (cid:17)
(iii) for Λ<0: Z =± −241Λ −2exp ∓ −16Λx˜ +2√2aγ2(t˜)+z(a,c).(40)
(cid:0) (cid:1) (cid:16) q (cid:17)
Here γ (t˜) and γ (t˜) are arbitrary functions, and
1 2
12k1/2
z(a,c) + 1 Λ2k1/2a3c2(25Λ2a4 1Λa2+13). (41)
≡− Λ 288 36 − 3
The corresponding inverse matrix to (36) is
−8kaZ+ γ342(kt˜) + (356Λ2ka14/+21)2c2 a(kk1−/423) (356Λ2k2a5/44+1)c −γ34(kt˜)
 a(kk1−/423) 0 0 0 
g = ,
ij  
 (356Λ2a4+1)c 0 1 0 
 2k5/4 4k2 
 
 
 γ3(t˜) 0 0 1 
 − 4k 4k 
 (42)
so that the metric can be expressed as
1 1 ( 5 Λ2a4+1)c γ (t˜)
ds2 = da2+ dx˜2+ 36 dt˜da 3 dt˜dx˜ (43)
4k2 4k k5/4 − 2k
+a(2k−3)dt˜dc+ 8aZ + γ32(t˜) + (356Λ2a4+1)2c2 dt˜2.
k1/4 − k 4k k1/2
(cid:18) (cid:19)
It is possible to apply the transformation
1
x= x˜ γ (t˜)dt˜ , (44)
3
√2 −
(cid:18) Z (cid:19)
1 1Λa2 1
v = k3/4 6 − c, (45)
−2κ a
u=4˜t, (46)
where κ is a suitable non-vanishing constant. This puts the metric (43) to a
more compact form
1 1 κa2 κa2 a
ds2 = da2+ dx2 2 dudv+ κ v2 H˜ du2, (47)
4k2 2k − 2k 2k − 2k
(cid:18) (cid:19)
8
in which the function H˜(a,x,u) reads
12k1/2
(i) H˜ =γ (u)cos 1Λ(√2x+ γ (u)du) +2√2aγ (u) ,
1 6 3 2 − Λ
(cid:16)q R (cid:17) 12k1/2
(ii) H˜ =γ (u)cosh 1Λ(√2x+ γ (u)du) +2√2aγ (u) ,
1 −6 3 2 − Λ
(iii) H˜ =± −241Λ −21(cid:16)exqp ∓ −31Λ x Rexp ∓ −(cid:17)61Λ γ3(u)du
(cid:0) (cid:1) (cid:16) q (cid:17) (cid:16) q R 12k(cid:17)1/2
+2√2aγ (u) . (48)
2
− Λ
Now, by comparing (47) with (1) we immediately conclude that the two
metrics are identical provided
κa2 Q2
= , (49)
2k P2
1 1 2
da2+ dx2 = dζdζ¯, (50)
4k2 2k P2
a Q
H˜ = H, (51)
2k P
with constant α, β. In view of (37), the condition (49) leads to the relation
Q
a= , (52)
κP2 1ΛQ2
− 3
q
whereP andQaregivenby (2),(3).The condition(50)canthenbe usedto
findtherelationbetweentherealcoordinatexandthecomplexcoordinateζ.
Infact,itcanthusbe shownthatbothrelations(49)and(50)aresatisfiedif
1 (1 1Λζζ¯)α+β¯ζ+βζ¯
a = − 6 , (53)
√2 (1Λβ¯ζ2+ 1Λαζ β)(1Λβζ¯2+ 1Λαζ¯ β¯)
6 3 − 6 3 −
√κq dζ
x= i F(ζ) F¯(ζ¯) , where F(ζ)= , (54)
2 − 1Λβ¯ζ2+ 1Λαζ β
Z 6 3 −
(cid:0) (cid:1)
with κ= 1Λα2+2ββ¯=0, cf. (4). Of course, the function F can be inte-
3 6
grated as
3
F(ζ) = lnζ for β =0, (55)
Λα
3 Λ
F(ζ) = 2 arctanh (β¯ζ+α) for β =0. (56)
− Λκ 3κ 6
r r
(cid:16) (cid:17)
The transformation (53), (54) puts the line element (47) explicitly to the
Ozsv´ath–Robinson–R´ozgametric form (1), namely
2 Q2 Q2 Q
ds2 = dζdζ¯ 2 dudv+ κ v2 H du2, (57)
P2 − P2 P2 − P
(cid:18) (cid:19)
9
where P =1+ 1Λζζ¯ and Q=(1 1Λζζ¯)α+β¯ζ+βζ¯,withconstantα,β.
6 − 6
Moreover,using (51), the function H can be expressed as
√2
H = (1Λβ¯ζ2+ 1Λαζ β)(1Λβζ¯2+ 1Λαζ¯ β¯)H˜ , (58)
κP 6 3 − 6 3 −
q
In view of the above three possible forms (i)–(iii) of H˜, and considering
relations (52)–(56), it can be argued that the function H takes the form
(7) for the most generalconformally flat solution of the Ozsv´ath–Robinson–
Ro´zga family of spacetimes.
Asshownin[6,9],therearevariousgeometricallydistinctsubclassesofso-
lutionsrepresentedbythemetric(57).Thesesubclassescanbedistinguished
by the cosmological constant Λ and sign of the function κ defined in (4).
Eachis representedby the correspondingcanonicalchoice of the parameters
α and β. The case κ=0, which corresponds to the Siklos spacetimes with
ττ¯+ 1Λ=0 and Λ<0, was described in previous section 4. The solutions
6
inthe presentsection5arecharacterisedbyττ¯+ 1Λ=0,andcorrespondto
6 6
the cases κ=0, Λ=0.
6 6
For Λ>0, there is only the subclass κ>0 of possible solutions, see (4).
Itscanonicalrepresentationisgivenbyα=0,β =1,sothat P =1+ 1Λζζ¯,
6
Q=ζ+ζ¯,andκ=2.Inthiscasethetransformation(53),(54)simplifiesto
1 (ζ+ζ¯)
a = √2 , (59)
(1 1Λζ2)(1 1Λζ¯2)
− 6 − 6
q
3
x= i arctanh 1Λζ arctanh 1Λζ¯ , (60)
− Λ 6 − 6
r (cid:18) (cid:16)q (cid:17) (cid:16)q (cid:17)(cid:19)
1+ 1Λζ 1 1Λζ¯
i 3 6 − 6
= ln .
−2rΛ (cid:16)1+q1Λζ¯(cid:17)(cid:16)1 q1Λζ(cid:17)
6 − 6
(cid:16) q (cid:17)(cid:16) q (cid:17)
A straightforward calculation now shows that the function H given by (58)
takes exactly the form (7) where
1 6
(u)= γ (u) cos 1Λ γ (u)du √2 , (61)
A √2 1 6 3 − Λ
1 (cid:18) (cid:16)q R (cid:17) (cid:19)
(u)= 2γ (u) i 1Λγ (u) sin 1Λ γ (u)du , (62)
B √2 2 − 6 1 6 3
(u)= 1Λ(cid:18) (u) 2,q (cid:16)q R (cid:17)(cid:19) (63)
C −6 A −
which follows from the case (i) of H˜.
If Λ<0, there are two subclasses of possible solutions for κ=0. When
6
κ>0, the canonical representation is again given by α=0, β =1, and the
transformation has the form (59), (60), with the replacement Λ Λ and
→−
arctanh arctan in(60).Similarly,thetrigonometricfunctionsin(61)–(63)
→
arereplacedby the correspondinghyperbolic functions,whichcorrespondto
the case (ii) of H˜ in (48). Similar results can be obtained for the case (iii).
10
When κ<0, the canonical representation of these spacetimes is α=1,
β =0, so that P =1+ 1Λζζ¯, Q=1 1Λζζ¯ and κ= 1Λ<0. In this case
6 − 6 3
the transformation (53), (54) is simply
1 3 1 1Λζζ¯
a = − 6 , (64)
√2 Λ ζζ¯
| |
1 3 pζ
x= ln . (65)
2s Λ ζ¯
| | (cid:18) (cid:19)
Notice that x is purely imaginary in this case. Indeed, introducing the polar
parametrisation ζ =ρ exp(iϕ), the transformation becomes
1 3 1 1Λρ2 3
a= − 6 , x=i ϕ. (66)
√2 Λ ρ s Λ
| | | |
However, this is consistent with the form of the metric (47). It follows from
therelation(49)thatthesignoftheparameterκisthesameasthesignofthe
function k, so that k <0 in this case, and the coordinate x must (formally)
be purely imaginary. In fact, 2k = 1Λa2+1= 1(3/Λ)((1+ 1Λρ2)2ρ 2 <0,
3 2 6 −
and the expression (50) gives
1 1 2
da2+ dx2 = (dρ2+ρ2dϕ2). (67)
4k2 2k (1+ 1Λρ2)2
6
Again, the function H for the case (ii) takes the form (7) in which
6√2
(u)= γ (u) 3/Λ ,
2
A Λ − | |
1 (cid:16) p (cid:17)
(u)= − γ (u) cosh 1 Λ γ (u)du i sinh 1 Λ γ (u)du ,
B √2 1 6| | 3 − 6| | 3
(u)= 1Λ (u)(cid:18) 2 (cid:16)6q/Λ . R (cid:17) (cid:16)q R (cid:17)(cid:19)(68)
C −6 A − | |
Analogousresultsarealspoobtainedforthecase(iii)of(48),providedalinear
combination of the terms with different signs (that is with their upper and
lower choices) is considered.
6 Summary and conclusions
Inthiscontributionwehaveanalyzedpureradiationspacetimeswhichareof
algebraictypes N and O. In particular,we haveconcentratedon the metrics
presented in recent works [1,2,3,4]. We have found explicit transformations
which put all these solutions to the Ozsv´ath–Robinson–R´ozgaform (1)–(4),
first given in [6]. This conveniently represents an entire family of type N or
conformallyflatKundtspacetimeswithanarbitrarycosmologicalconstantΛ,
which are either vacuum or contain pure radiation.
By putting the above metrics into the unified coordinate system of (1)
we have confirmed that their derivation based on the GIF formalism, as