Flat deformation theorem and symmetries in spacetime

0.35 MB·English

by  Josep Llosa

Flat deformation theorem and symmetries in spa
etime.
1 2
Josep Llosa , Jaume Carot
1
Departament de Físi
a Fonamental, Universitat de Bar
elona, Spain
9
0 2
0 Departament de Físi
a, Universitat de les Illes Balears, Spain
2

n
a January 9, 2009
J

9
Abstra
t

]
c g
q The (cid:29)at deformation theorem states that given a semi-Riemannian analyti
metri
on
- F c
r a manifold, lo
ally there always exists a two-form , a s
alar fun
tion , and an arbitrarily
g x F c
[ pres
ribed s
alar
onstraint depending on the point of the manifold and on and , say
Ψ(c,F,x) = 0 η = cg ǫF2
3 , su
h that the deformed metri
− is semi-Riemannian and (cid:29)at. In
v this paper we (cid:28)rst show that the above result implies that every (Lorentzian analyti
) metri
0 g ηab := agab 2bk(alb) η
3 may be written in the extended Kerr-S
hild form, namely − where is
a
0 ka, la kal = 1
(cid:29)at and are two null
ove
tors su
h that − ; next we show how the symmetries
1 g η g
. of are
onne
ted to those of , more pre
isely; we show that if the original metri
admits
9
0 a Conformal Killing ve
tor (in
luding Killing ve
tors and homotheties), then the deformation
8 η
may be
arried out in a way su
h that the (cid:29)at deformed metri
`inherits' that symmetry.
0
:
v
i
X
1 Introdu
tion
r
a
g
ab
It has been re
ently proved [1℄ that, given a semi-Riemannian analyti
metri
on a manifold M,
F c
ab
there exists a 2-form and a s
alar fun
tion su
h that:
Ψ(c,F ,x) = 0 x
ab
1. An arbitrary s
alar
onstraint , ∈ M, is ful(cid:28)lled and
2. The so-
alled `deformed metri
'
η = cg ǫF2 where ǫ = 1 and F2 := F gcdF
ab ab − ab ± ab ac db (1)
1
is semi-Riemannian and (cid:29)at
This result was
alled Flat Deformation Theorem. For the purposes of the present paper, we shall
only
onsider the four-dimensional Lorentzian
ase.
The proof of the above theorem was based on the existen
e of solutions for a partial di(cid:27)erential
η
ab
system that is derived from the
ondition that is (cid:29)at. As a
onsequen
e of the arbitrariness in
η
ab
the
hoi
e of the Cau
hy hypersurfa
e and Cau
hy data, the deformation (1) leading to a (cid:29)at is
by no means unique. Furthermore, asthe Cau
hy-Kovalewski theorem isa
ornerstone in the proof,
the validity of the theorem is limited to the analyti

ategory.
The purpose of the present paper is to deal with the question of how the symmetries of the metri
g η g
ab ab ab
are re(cid:29)e
ted upon the deformed metri
, more pre
isely: assuming that admits a Killing
Xa F c Xa
ab
ve
tor (cid:28)eld , we ask whether it is possible to
hoose and in (1) su
h that is also a
η
ab
Killing ve
tor (cid:28)eld for . We shall prove that the answer is in the a(cid:30)rmative in the
ase of
non-null Killing ve
tors and that the symmetry is thus somehow `inherited' along the deformation.
The paper is stru
tured as follows: se
tion 2
ontains some algebrai
developments on the
on-
sequen
es of the deformation law (1) for a 4-dimensional spa
etime whi
h will allow us to state
it in a number of alternative ways, thus illustrating di(cid:27)erent features of the deformation law. In
1
se
tion 3 we present the formalism and prove some intermediate results in order to demonstrate
the theorem alluded to in the previous paragraph. It is worth noti
ing that in order to prove it,
the problem is reformulated on the 3-dimensional quotient manifold (see se
tion 4.2), so that a
dimensional redu
tion o

urs. Se
tion 5
ontains a generalization of the above result to the
ase
of (non-null) Conformal Killing Ve
tors. Finally, in se
tion 6, we present some examples whi
h we
believe may be of interest due to their physi
al relevan
e. We put some te
hni
al developments in
the appendi
es in order to make the paper more readable. Also for this reason, we do not insist at
every intermediate step on the lo
al
hara
ter of the results presented here, but the reader should
bear this in mind.
1
This formalism was developed in a numberof referen
es, notably[4℄and [5℄ whi
h will be used in se
tion 4. We
presentit herein a way well suited to ourpurposes.
2
2 Algebrai

onsequen
es of the deformation law
F
ab
Consider now the 2-form whose existen
e is granted by the deformation theorem [1℄; there are
two possibilities, either it is
x , y , k , l g = x x +y y 2k l
a a a a ab a b a b (a b)
(a) singular (or null), then, a tetrad { } exists su
h that −
and
F = 2k x and then F2 = k k
ab [a b] ab − a b (2)
or else it is
(b) non-singular (or non-null), in whi
h
ase a tetrad su
h as the one above exists in terms of
F
ab
whi
h reads
F = 2Bx y +2Ek l and then F2 = B2 (x x +y y ) 2E2k l
ab − [a b] [a b] ab − a b a b − (a b) (3)
E B Fa := gacF
where and are fun
tions related to the algebrai
invariants of b cb. If either
B E
or is zero, the resulting 2-form is timelike or spa
elike respe
tively. If neither of them
vanishes, the 2-form is said to be non-simple.
η = cg +ǫk k
ab ab a b
In the singular
ase, the deformation law (1) reads or, equivalently,
1 ǫ
g = η k k
ab ab a b
c − c (4)
k ka = 0 η g
a ab ab
with and (cid:29)at. That is, is a
onformal Kerr-S
hild metri
[2℄. The singular
ase is
therefore non-generi
and en
ompasses a rather restri
ted
lass of metri
s.
In the non-singular
ase, from equations (1) and (3) we have that:
η = ag +bS
ab ab ab
(5)
a = c + ǫB2 b = ǫ(B2 + E2) S = 2k l
ab (a b)
with , − and − . As it was shown in [1℄, this is the
generi

ase in the sense that the (cid:29)at deformation (5)
an always be a
hieved for any analyti
semi-Riemannian metri
.
Ψ(c,F ,x) = 0 a
ab
Noti
e that the arbitrary s
alar
onstraint has no
onsequen
es on the fa
tors
b f(c,E,B) = 0
and in (5). Indeed, in
luding (3) the s
alar
onstraint may be written as or,
3
f˜(c,a,b) = 0 c = c(a,b)
equivalently, as a relation whi
h, at most,
an be used to determine to
F
ab

hoose one amongst the many 2-forms
ompatible with (5).
We have hitherto proved that:
g
ab
Proposition 1 Let be a Lorentzian analyti
metri
on a spa
etime M. Lo
ally there exist two
a b k l k la = 1
a a a
s
alars, and , and two null ve
tors, and , su
h that − and the metri
η := ag 2bk l
ab ab (a b)
− (6)
is Lorentzian and (cid:29)at.
The above expression vaguely reminds a
onformal Kerr-S
hild transformation, but in the present
k l
a a

asetwonon-parallelnullve
tors, and ,o

ur. Weshallhen
eforth
allthisexpressionextended
Kerr-S
hild form and proposition 1
an be restated as:
Any Lorentzian analyti
metri

an be written in extended Kerr-S
hild form.
An equivalent statement is
g
ab
Proposition 2 Let be a Lorentzian analyti
metri
on a spa
etime M. Lo
ally there exist two
a b S
ab
s
alars, and , and a hyperboli
2-plane su
h that the metri
η := ag +bS
ab ab ab
(7)
is Lorentzian and (cid:29)at.
Sa
Noti
e that b is a 2-dimensional proje
tor:
SaSd = Sa , Sa = 2
d b b a
(8)
ka, la H :=
ab
whi
h proje
ts ve
tors onto the hyperboli
plane spanned by { }. If we now denote
g S
ab ab
− , i.e. the
omplementary proje
tor, then:
HaHd = Ha, Ha = 2, and SaHd = HaSd = 0
d b b a d b d b
(9)
4
H S
ab ab
isthen theellipti
2-plane spanned by any two spa
elike ve
tors orthogonal to , inparti
ular
xa,ya H = 2x y
ab (a b)
, the spa
elike ve
tors in the
hosen tetrad, i.e. , and it is then possible to write
H
ab
the deformation (1) in a way similar to that given by (7) but in terms of the (ellipti
) proje
tor
S
ab
instead of the , namely:
η := a¯g +¯bH
ab ab ab
(10)
a¯ ¯b
where and are s
alars.
From the
omments and developments above and taking (7) into a

ount, we
an write
g := H +S and η := (a+b)S +aH ,
ab ab ab ab ab ab
(11)
S g
ab ab
that is, the almost-produ
t stru
ture [3℄ de(cid:28)ned by is
ompatible with both metri
s, and
η
ab
, and therefore we
an state
g
ab
Proposition 3 Let be a Lorentzian analyti
metri
on a spa
etime M. Lo
ally it exists a
η g
ab ab
Lorentzian (cid:29)at metri
that shares with an almost-produ
t stru
ture.
3 Spa
etimes admitting a (non-null) Killing ve
tor
In this se
tion we are going to set up and develop the formalism and basi
results whi
h will be
later used in order to prove the result stated in the introdu
tion; namely: that if the metri
admits
an isometry, it is always possible to preserve it in the (cid:29)at deformed metri
.
η Xa ξ := η Xb
ab2 a ab
Let Mbe a spa
etime with an arbitrary metri
admitting aKilling ve
tor . Let
l := ξ Xa l = 0
a
and . Assume that the Killing is non-null, that is: 6 , and denote by S the set of all
Xa
3
orbits of , whi
h we assume to be a 3-manifold (the quotient manifold) .
π π : π(x) = O
x
Weshall designate by the
anoni
al proje
tion M −→ S where istheorbit through
x Xa
the point ∈ M of the 1-parameter group generated by .
The proje
tor: 1
ha := δa Xaξ
b b − l b (12)
2 ηab
Note: does notdesignate the(cid:29)atmetri
at this point. We usethiXsanotation here for later
onvenien
e.
3
It
an be shown that lo
ally this is always the
ase if (cid:28)xedpoints of are ex
luded.
5
T Xa
proje
ts ve
tors in M onto ve
tors that are transverse (orthogonal) to . There is a bije
tion
T′a... Ta...
[4℄ between tensor (cid:28)elds b... on S and the tensor (cid:28)elds b... on M that ful(cid:28)ll:
XbTa... = 0, ξ Ta... = 0 and Ta... = 0
b... a b... LX b... (13)
Xa ξ Xa
a
that is, those whi
h are transverse to and and Lie invariant along . Following Gero
h [4℄
(cid:19)While it isuseful
on
eptually to have thethree-dimensional manifold S, itplays nofurther logi
al
role inthe formalism. We shallhereafter drop theprimes: we shall
ontinue to speakof tensor (cid:28)elds
being on S, merely as a shorthand way of saying that the (cid:28)eld (formally, on M) satis(cid:28)es (13)(cid:20)
l = 0
As 6 the proje
ted metri
1
h := η ξ ξ
ab ab a b
− l (14)
indu
es a semi-Riemannian metri
on the quotient manifold S, the so1-
alled `quotient metri
'. Its
+1 + 1 sign(l) hab := ηab XaXb
signature is − . We shall designate by − l the inverse quotient
habh = ha
bc c
metri
, that is: .
3.1 The Killing equation
η = 0 ξ ξ + ξ = 0
X ab a b b a a b
From L it follows that ∇ is skew-symmetri
, that is: ∇ ∇ where ∇ stands
η
for the
ovariant derivative asso
iated to .
ξ = 0 Xal = 0 l := l Xa ξ
X a a a a a b
We also have that L and , where ∇ . Sin
e is non-null, ∇
an be
de
omposed as:
2 ξ := 2f ξ +Θ with f := log l and Θ Xb = 0
a b [a b] ab ab
∇ | | (15)
Θ = Θ
ab ba
− is related with the vorti
ity of the Killing (cid:29)ow. We shall use the above form for the
Killing equation in the sequel.
3.2 The Levi-Civita
onne
tion on S
Ta...
Let b... be a tensor (cid:28)eld on S and de(cid:28)ne:
D Ta... := ha hnhk Tm...
c b... m b c∇k n... (16)
6
Ta... ha Xa
Clearly, it is a tensor (cid:28)eld on S, sin
e b... and b both satisfy (13), and, sin
e is a KV, the Lie
D
a
derivative with respe
t to it
ommutes with ∇; further it
an be easily proved that is a linear
f D f
a

onne
tion: indeed, it is linear, it satis(cid:28)es the Leibniz rule and for any s
alar fun
tion on S,
f D h = 0
c ab
is the gradient of . Moreover, it
an be also shown that it is torsion-free and that (this
D
last result holds trivially); therefore, is the Levi-Civita
onne
tion on S (see [4℄).
va wb
Let now , be two ve
tor (cid:28)elds on S, then taking into a

ount (13), (15) and (16) one easily
gets 1
D wa = wa+ XaΘ vbwc
v v bc
∇ 2l (17)
D wa := vbD wa
v b
where . Noti
e the formal similarity between this formula and Gauss equation for
Θ
bc
hypersurfa
es, even though S is not a submanifold and we have the skew-symmetri
instead of
the se
ond fundamental form.
3.3 The Riemann tensor on S
va h
ab
Consider next a ve
tor (cid:28)eld on S endowed with the quotient metri
and its asso
iated Levi-
D c
Civita
onne
tion a as de(cid:28)ned above in (16). We aim at
al
ulating the Riemann tensor Rdab for
this
onne
tion.
[D ,D ]vc = vd c
From the Ri

i identities, a b Rdab, we have that
1
= R⊥ + Θ Θ +Θ Θ ,
Rabcd abcd 2l ab cd [ac b]d
(cid:0) (cid:1)
R⊥ := hmhnhphqR Θ Θ +Θ Θ +Θ Θ = 0
where abcd a b c d mnpq. Using the identity ab cd ac db ad bc that follows
dim = 3
from the fa
t that S , we then arrive at
3
= R⊥ + Θ Θ
Rabcd abcd 4l ab cd (18)
R ξ =
abcd a b c
The remaining
omponents of follow from the se
ond order Killing equation [6℄, ∇ ∇
R Xd := R
dabc Xabc
whi
h, taking into a

ount (15), leads to:
1 1
R⊥ = D Θ + f Θ
Xabc 2 a bc 2 [b ac] (19)
1 1 1
R = D l Θ bΘ + l l
XaXc −2 a c− 4 a bc 4l a c (20)
7
We have thus shown that the entire Riemann tensor on M may be expressed in terms of the
ξ
a
kinemati
invariants of and the Riemann tensor on S asso
iated to the Levi-Civita
onne
tion
D h
a ab
of the proje
ted (quotient) metri
.
3.4 Lift of a metri
from S to M
We have hitherto shown how a semi-Riemannian metri

an be proje
ted from M to S. We shall
Xa
now
onsider the
onverse
ase. As before, let be a ve
tor (cid:28)eld on M and let S be the set of its
orbits, whi
h we take to be a manifold a

ording to the reasoning at the beginning of the present
π :
se
tion. Further, let M → S be the
anoni
al proje
tion.
h (+ + σ) σ = 1
ab
Let now be a semi-Riemannian metri
on S having
onstant signature , ± . We
π∗h = h
ab ab
shalldenotebythesamesymbolthepulledba
kmetri
onM,i.e.: ,whi
hisdegenerate
h Xb = 0 h = 0 η
ab X ab ab
be
ause , moreover, L . The point now is: does it exist a metri
on M su
h
Xa h
ab
that: (a) admits as a Killing ve
tor and (b) has as the quotient metri
?
ξ := η Xb l = ξ Xa
a ab a
Ifitexists, a relation similar to (14) must hold, with and . Hen
e, the solution
ξ ξ = 0 l := ξ Xa
a X a a
is not unique, be
ause we may
hoose any
ove
tor su
h that L and that has
4

onstant sign ; then taking 1
η := h + ξ ξ
ab ab a b
l (21)
Xa η h
ab ab
as the lifted metri
, all the required
onditions are satis(cid:28)ed (namely: is a KV of and
is its quotient metri
). Then, if no further
ondition is demanded, equations (18), (19) and (20)
η h
ab ab
merely relate the Riemann tensors for both metri
s, and . However, if we require the lifted
η
ab
metri
to ful(cid:28)ll some supplementary
ondition, e.g. to be (cid:29)at, then these be
ome equations on
ξ h
a ab
the
hosen and the given , mu
h in the same way as the Gauss
urvature equation and the
Codazzi-Mainardi equations set up
onditions on the way that a submanifold
an be immersed in
an ambient spa
e.
ξ ξ = 0 α
a X a a
The
hoi
e of is restri
ted by the
ondition L . Assume that a 1-form on M su
h that
α Xa = 1 α = 0 ξ ξ = l(α +µ )
a X a a a a a
and L is given. Then, the sought
an be written as , with
l := ξ Xa µ Xa = 0
a a
and . It
an be easily proved that:
ξ = 0 Xl = 0 and µ = 0
X a X a
L ⇔ L
4 (+ + +−)
The sign is to be
hosen so thatthelifted metri
has therequiredsignature
8
α α Xa = 1 α = 0 ξ
a a X a a
Hen
e, given a 1-form on M su
h that and L ,
hoosing is equivalent to
l = 0 µ ξ = l(α +µ )
a a a a

hoosing a fun
tion 6 on S, a 1-form on S and taking .
The exterior derivative of this expression yields
2
(dξ) = l ξ +l(dµ) and Θ = l(dµ) +l(dα)
ab l [a b] ab ab ab ab (22)
where (15) has been taken into a

ount.
l µ
a
In terms of and , taking (22) into a

ount, the equations (18), (19) and (20) read:
3l
R⊥ = (dµ) (dµ)
abcd Rabcd− 4 ab cd (23)
1 1
R⊥ = D [l(dµ) ]+ l (dµ)
Xabc 2 a bc 2 [b ac] (24)
1 l2
R = D l (dµ) (dµ) hbd
XaXc a c ad bc
−2 − 4 (25)
l µ h
a ab
that are equations for , and to be solved on S.
3.5 Hypersurfa
es and Killing ve
tors
Σ π−1Σ Xa
Let be a surfa
e in S, then is a hypersurfa
e in M and the Killing ve
tor is tangent to
it. The following diagram is
ommutative:
π
(η, ,R) // (h,D, )
∇ MOO SOO R
J j
(η′, ′,R′) π−1Σ π // Σ (h′,D′, ′)
∇ R
J j
where and are the respe
tive embeddings.
η′ Φ ′ R′
Werespe
tivelydenoteby ab, ab,∇ and abcd the(cid:28)rstandse
ondfundamentalforms,theindu
ed
π−1Σ

onne
tion and the intrinsi

urvature on as a hypersurfa
e of the Riemannian manifold
( ,η ) h′ φ D′ ′ Σ
M ab . Similarly, we denote by ab, ab, and Rabcd the
orresponding obje
ts on regarded
( ,h )
ab
as a hypersurfa
e in S .
na η π−1Σ Xa π−1Σ ξ na = 0
a
Let be the unit ve
tor -normal to . Sin
e is tangent to , then . Further-
na = 0 Va π−1Σ Va π−1Σ
X X
more, L . Indeed, for any tangent to we have that L is also tangent to
9
Xa η naVb = 0
ab X
and, using that is a Killing ve
tor (cid:28)eld, we easily arrive at L , whi
h implies that
na na na η nanb = 0 na = 0
X ab X X
L ∝ . On the other hand, as is unit, L , when
e it follows that L .
na h Σ
Therefore, is also a ve
tor in S and is the unit ve
tor -normal to .
π−1Σ Σ φ = Φ⊥
It
an be easily proved that the se
ond fundamental forms for and satisfy that: ab ab.
Vb π−1Σ
On the other hand, for any ve
tor (cid:28)eld tangent to , we have that
1
Φ XaVb = n Vb = ξ nb = (dξ) Vanb
ab X b V b ab
∇ −∇ −2
Van = 0 Vbn = 0
X a b
where in the se
ond equality we have used that L and that . The above
(dξ) nb := (dξ) f nb := f
ab an b n
equation implies, putting and ,
1 1 1
Φ Xa = (dξ) = f ξ + Θ
ab nb n b nb
2 2 2 (26)
where (15) has been taken into a

ount. Therefore,
1 1
Φ = φ + f ξ ξ + Θ ξ
ab ab n a b n(a b)
2l l (27)
4 Flat deformation
The aim of this se
tion is to prove the main result in this paper, namely,
( ,g ) g Xa
ab ab
Theorem 1 Let M be a spa
etime with a metri
admitting a non-null Killing ve
tor .
Lo
ally there exists a deformation law
η = ag +bH
ab ab ab
(28)
a b H g η
ab ab
where and are two s
alars, is a 2-dimensional proje
tor on a -ellipti
plane and is (cid:29)at
Xa
and also admits as a Killing ve
tor.
It will be
onvenient for our purposes to prove the following result previously:
Xa g η b = 0
ab ab
Proposition 4 Let be a Killing ve
tor for and let be de(cid:28)ned by (28) with 6 , then:
η = 0 a = b = 0 and H = 0
X ab X X X ab
L ⇔ L L L (29)
10