The Seven Classes of the Einstein Equations

0.11 MB·English

by  Sergey E. Stepanov

The Seven Classes of the Einstein Equations
Sergey E. Stepanov
Professor, Chair of Mathematics, Finance Academy under the Government of the
Russian Federation, 49, Leningradsky Prospect, Moscow, 125993 Russia.
e-mail: [email protected]
Irina I. Tsyganok
Professor, Chair of General Scientific Disciplines, Vladimir branch of Russian Uni-
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versity of Cooperation, 14, Vorovskogo str., Vladimir, 6000001 Russia.
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0 e-mail: [email protected]
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Abstract. In currentpaperwe referto thegeometrical classification oftheEinstein

n equationswhich has beendeveloped byoneof theauthorsof thispaper. Thisclassi-
a fication was based on the classical theory for decomposition of the tensor product of
J
representations into irreducible components, which is studied in the elementary rep-

6 resentation theory for orthogonal groups. We return to this result for more detailed
2 investigation of classes of theEinstein equations.

] Mathematics Subject Classification (2000): 53Z05; 83C22
G
Keywords: Einsteinequations,classification, invarianttensorandmetriccharacter-
D
istics.
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h
t
a
m
Introduction
[

1 Method of classification of Riemannian manifolds, endowed with tensor struc-
v ture, on the basis of decomposition of irreducible tensor product of represen-
3 tations (irreducible with respect to the action of orthogonal group) into irre-
7
ducible components has become traditional in differential geometry (see, e.g.,
6
[1] - [5]).
4
. Thismethodisusedintheoreticalphysics(see,forexample,[6];[7];[8]). For
1
example,inthepaper[7]Einstein-Cartanmanifoldswereclassifiedonthebasis
0
ofirreducibledecompositionofthetorsiontensorofanaffine-metricconnection
0
1 (irreducible with respect to the action of the Lorentz group). Current results
: in this modern today researchdirection in the General Relativity Theory were
v
i systemized in paper [7].
X
At current paper we return back to the work [8] of one of the authors and
r study in details proposed in this work classification of The Einstein equations
a
on the basis of method mentioned above.
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1 The Einstein equations
The space-time (M,g) in the General Relativity Theory is a smooth four-
dimensional manifold M, endowed with metrics g of the Lorentz signature
(- + + +).
Inanarbitrarymap(U,ϕ)withthe localcoordinatesystem{x0,x1,x2,x3}
the metric g of (M,g) is defined by its components g = g(∂ ,∂ ), where
ij i j
∂ =∂/∂xi for i,j,k,l,...=0,1,2,3. These components define the Christoffel
i
symbolsΓk =2−1gkl(∂ g +∂ g −∂ g )oftheLevi-Civitaconnection∇and
ij i lj j li l ij
the operator of the covariantdifferention defined by ∇ Xk =∂ Xk+ΓjXl for
i i il
X =Xk∂ .
k
In the General Relativity Theory a metric tensor g is interpreted as gravi-
tational potential and is related by the Einstein equations
1 1
Ric− sg =T : R − sg =T (1.1)
ij ij ij
2 2
to mass-energy distribution, which generates the gravitational field. Here Ric
is the Ricci tensorof metric g ofa space-time (M,g), which canbe found from
identity Ric(∂ ,∂ ) = R = Rk for components Rl of torsion tensor R,
i j ij ikj ikj
which,intheirturn,canbefoundfromequalityRk Xl =∇ ∇ Xk−∇ ∇ Xk,
lij i j j i
s := gijR – is scalar curvature of metrics g for (gij) = (g )−1 and, finally,
ij ij
T = T(∂ ,∂ ) – are components of known energy-momentum tensor T of
ij i j
matter.
The curvature tensor R satisfies well-known the Bianchi identities
dR=0: ∇ Rk +∇ Rk +∇ Rk =0. (1.2)
m lij i ljm j lmi
From these identities, in particular, following equalities can be found
d∗R=−dRic: −∇ Rk =∇ R −∇ R =0, (1.3)
k lij j li i lj
which lead to other the Bianchi identities
2d∗Ric=−ds: 2gkj∇ R =∇ s. (1.4)
j ki i
Equations (1.1) are complemented by ”conservation laws”
d∗T =0: −gik∇ T =0, (1.5)
i kj
which are derived from The Einstein equations on the basis of the Bianchi
identities (1.4).
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2 Invariantly defined seven classes
of the Einstein equations
In the paper [8] was proved that on a pseudo-Riemannian manifold (M,g)
the bundle Ω(M) ⊂ T∗M ⊗ S2M, which fiber in each point x ∈ M con-
sists of linear mappings Ω : T M → R such that Ω(X,Y,Z) = Ω(X,Z,Y)
x
n
and Ω(e ,e ,X) = 0 for arbitrary vectors X,Y and Z orthonormal basis
i i
iP=1
{e1,e2,...,en} of space TxM, has point-wise irreducible decomposition (ir-
reducible with respect to the action of pseudo-orthogonal group) Ω(M) =
Ω1(M)⊕Ω2(M)⊕Ω3(M).
If as(M,g) we take space-time, then ∇T ∈Ω(M). As the result six classes
of the Einstein equations can be determined via invariant approach Ω and
α
Ω ⊕Ω (α,β,γ = 1,2,3 β < γ), for each of them covariant derivatives ∇T
β γ
of energy-momentum tensors T of matter are cross-section of relevant invari-
antsubbundles Ω1(M),Ω2(M),Ω3(M)andtheir directsums Ω1(M)⊕Ω2(M),
Ω1(M)⊕Ω3(M) Ω2(M)⊕Ω3(M). Seventh classes determined in [8] satisfy
the condition ∇T =0.
3 The class Ω and integrals of
1
geodesic equations
Class Ω1 of the Einstein equations is selected via condition (see, [8])
∗
δ T =0: ∇ T +∇ T +∇ T =0. (3.1)
k ij i jk j ki
∗
As trace T = −s, then from (3.1) follows that d(trace T) = 2d T = 0, i.e.
g g
s=const. In this case equations (3.1) can be re-written in the following form
∗
δ Ric=0:∇ R +∇ R +∇ R =0. Reverseisevident,i.e. fromcondition
k ij i jk j ki
∗
δ Ric=0 can be deduced that s=const, and then equations (3.1).
If each solution xk = xk(s) of geodesic equations on pseudo-Riemannian
manifold (M,g) satisfies condition a dxidxj = const for symmetric tensor
ij ds ds
field a with local components a , then it’s said that such geodesic equations
ij
admit first quadratic integral (see, for example, [9]. In fact, following identity
∗
needs to be true δ a = 0. At the same time tensor field a, determined by
identityabove,iscalledtheKillingtensor(see[10]). Inourcase,firstquadratic
integral of geodesic equations will be R dxidxj = const and Ricci tensor will
ij ds ds
be consequently the Killing tensor.
Following theorem was proved:
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Theorem 1. The Einstein equations belong to the class Ω2 if and only if
in the space-time (M,g) geodesic lines equations admit following first quadratic
integral R dxidxj =const for Ricci tensor R =Ric(∂ ,∂ ) of metrics g.
ij ds ds ij i j
Remark. Letusunderlinethattheoryoffirstintegralsequationsofgeodesic
andsymmetrictensorKillingfieldshasmanyapplicationsinmechanics,general
relativity theory and other physics research(see, for example, [10] and [11]).
4 Class Ω and the Young-Mills equations
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Class Ω2 of The Einstein equations is selected via condition (see, [8])
dT =0: ∇ T =∇ T . (4.1)
k ij i kj
Considering equations (3.1) together with The Einstein equations we get that
s=const, and consequently we get following equations
dRic=0: ∇ R =∇ R . (4.2)
k ij i kj
Equations (4.2) are known as Codazzi equations (see, for details, [12]). Easy
to prove that equations (4.1) follow from Codazzi equations (4.2). In fact,
∗
from equations (4.2) follows that d Ric=−ds. Comparing this equation with
Bianchi identities (1.4) we conclude that s = const. As the result equations
(4.1) will be true. So, equations (4.1) and (4.2) are equivalent.
For constructing an example of space-time (M,g) with ∇T ∈ Ω2(M), we
will take n-dimensional (n ≥ 4) conformally flat pseudo-Riemannian manifold
(M,g), for which as known (see, for example, [9]) following equations are ful-
filled
1
d Ric− s·g =0.
(cid:20) 2(n−1) (cid:21)
If we suppose here that s = const we will get equations (4.2). Therefore, on
conformally flat space-time (M,g) with the constant scalar curvature s the
Einstein equations belong to the class Ω2.
On arbitrary n-dimensional pseudo-Riemannian manifold (M,g) the Weyl
projective curvature tensor P is defined by (see, for example, [9]) compo-
nents P =R − 1 (g R −g R ) in arbitrarymap(U,ϕ) with a local
lijk lijk n−1 lj ik lk ij
coordinate system {x0,x1,x2,x3}. For n > 2 conversion into zero of tensor
P characterises manifold of constant curvature (see [9]). Easy to show that
due to (1.3) covariant differential is d∗P = −n−2d Ric. We will say that
n−1
Weyl projective curvature tensor is a harmonic tensor if d∗P = 0. Definition
∗
is explained by the fact that from condition d P = 0 automatically follows
Bianchi identities dP = 0. Therefore if tensor P is considered as second form
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P :Λ2(TM)→Λ2(TM), then this form will be simultaneously closed and co-
closedandso a harmonicform(see, for example,[13]). Conditionfor the Weyl
projective curvature tensor P to be harmonic leads us to Codazzi equations
(4.2), which are equivalent as was proved to (4.1). Following theorem is true:
Theorem 2. The Einstein equations belong to the class Ω2 if and only if
the Weyl projective curvature tensor P of space-time (M,g) is harmonic.
It is well known (see, for example, [13]) that connection ∇ˆ in main bundle
π : E → M over pseudo-Riemannian manifold (M,g) with fiber metrics g is
E
called the Young-Mills field, if its curvature Rˆ together with Bianchiidentities
dRˆ =0 satisfies the Young-Mills equations d∗Rˆ =0.
IfweconsiderthatE =TM,g =g andwetakeLevi-Civitaconnection∇
E
of pseudo-Riemannian metrics g as connection ∇ˆ, then Young-Mills equations
∗
d R=0 due to equalities (1.3)willbecome Codazziequations(4.2), whichare
equivalent to equations (4.1). Following theorem is true:
Theorem 3. The Einstein equations belong to the class Ω2 if and only if
the Levi-Civita connection ∇ of metrics g of space-time (M,g) considered as
connection in bundle TM is the Young-Mills field.
Remark. Yang-Mills theory and the other variational theory as Seiberg-
Witten theory have been developed greatly and influenced to topology and
physics(see[14];[15];[16];[17]andetc),especiallyinthecaseof4-dimensional
manifolds. Yang-Millstheoryappearedindifferentialgeometry(see,for exam-
ple, [18], p. 443) as pseudo-Riemannian manifolds with harmonic curvature.
This means that pseudo-Riemannianmanifolds (M,g) of which curvature ten-
∗
sor R of the Levi-Civita connection ∇ satisfies d R=0, i.e. ∇ is a Yang-Mills
connection, taking E = TM, the tangent bundle of M, and g = g as in our
E
pieces 4.
5 The class Ω and geodesic mappings
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The class Ω2 of the Einstein equations is selected via condition (see, [8])
1
∇ R = (4(∂ s)g +(∂ s)g +(∂ s)g ). (5.1)
k ij k ij i kj j ik
18
Let us recall that diffeomorphism f : (M,g) → (M¯,g¯) of pseudo-Riemannian
manifolds of dimension n ≥ 2 is called geodesic mapping (see [19], p. 70), if
each geodesic curve of manifold (M,g) is transferred via this mapping onto
geodesic curve of manifold (M¯,g¯).
An arbitrary (n + 1)-dimensional (n ≥ 1) pseudo-Riemannian manifold
(M,g) which scalar curvature s 6= const and the Ricci tensor Ric satisfies the
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following equations:
n−2 2n
∇ R = (∂ s)g +(∂ s)g +(∂ s)g (5.2)
k ij k ij i kj j ik
2(n−1)(n+2)(cid:18)n−2 (cid:19)
foralocalcoordinatesystem{x0,x1,...,xn}iscalledamanifolds Ln+1. These
manifolds were introduced by N.S. Sinyukov (see [19], pp. 131-132). Manifold
Ln+1 are an example of pseudo-Riemannian manifolds with non-constant cur-
vaturewhichadmitnon-trivialgeodesicmappings. Easytocheckthatforn=3
equations(5.1)and(5.2)areequivalentandthatiswhyspaces-times(M,g)for
whichT ∈Ω3 areSinyukovspacesLn+1andduetothisreasonadmitnontrivial
geodesic mappings.
Following theorem is true:
Theorem 4. The Einstein equations belong to the class Ω2 if and only if
the space-time (M,g) is a Sinyukov space L4.
In addition, we call that if (M,g) is an (n+1)-dimensional Sinyukov man-
ifold then the metric form ds2 of the manifold (M,g) has the following form
(see [19], pp. 111-117and [20])
n
ds2 =g00(x0)(dx0)2+f(x0) gab(x1,...,xn)dxadxb (5.3)
aX,b=1
for special local coordinate system {x0,x1,...,xn}. The function f(x0), com-
ponents g00(x0) and gab(x1,...,xn) of the metric form (5.3) satisfy some de-
fined properties which can be found in the paper [20]. From this we conclude
thataspace-time(M,g)withtheEinsteinequationsoftheclassΩ2 isawarped
product space-time and the Robertson-Wolker space-time, in particular (see,
for example, [21]).
6 Three other classes
In brief we will describe remaining three classes of the Einstein equations. At
the beginning lets consider class of the Einstein equations Ω1 ⊕ Ω2 defined
by the following condition gij∇ T = 0 or equivalent condition s = const.
k ij
Following theorem is true:
Theorem 5. The Einstein equations belong to the class Ω1 ⊕Ω2 if and
only if the metric g of the space-time (M,g) has constant scalar curvature.
Next class of the Einstein equations Ω2 ⊕Ω3 is defined by the following
condition d T − 1(trace T)g = 0. To obtain its analytical characterisation
3 g
lets use the(cid:2)Weyl tensor W o(cid:3)f conformal curvature (see, for example, [9]). It
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is known that on arbitrary pseudo-Riemannian manifold (M,g) of dimension
n≥3 this tensor follows equation (see, for example, [9] and [18])
n−3 1
∗
d W =− d Ric− s·g . (6.1)
n−2 (cid:20) 2(n−1) (cid:21)
We’ll say that the Weyl tensor of conformal curvature is a harmonic tensor if
∗
d W =0(seealso[18]). Definitionisexplainedbythefactthatfromcondition
∗
d W = 0 (see [9]) automatically follows Bianchi identities dW = 0. Therefore
if the Weyl tensor W of conformal curvature is considered as second form
W : Λ2(TM) → Λ2(TM), then this form will be simultaneously closed and
coclosed and so harmonic (see [13]).
If the equation T − 1(trace T)g =Ric− 1s·g is true, then for space-time
3 g 6
(M,g)dueto(6.1)condition∇T ∈Ω2(M)⊕Ω3(M)isequivalenttoharmonicity
∗
requirement d W =0 for the Weyl tensor W of conformal curvature.
Following was proved:
Theorem 6. The Einstein equations belongtotheclass Ω2⊕Ω3 if andonly
if the tensor W of conformal curvature of the space-time (M,g) is harmonic.
The example of a space-time (M,g) with ∇T ∈ Ω2(M)⊕Ω3(M) is a con-
formally flat space-time (M,g), this means that W ≡ 0 (see [9], p 116) and,
∗
consequently, condition d W =0 is fulfilled automatically.
The last, third, class Ω1⊕Ω3 is defined by the following condition
δ∗ T − 1(trace T)g = 0. On the basis of the identity T − 1(trace T)g =
6 g 6 g
Ri(cid:2)c− 1s·g, our con(cid:3)dition turns into following differential equations
3
1
∗
δ (Ric− s·g)=0:
3
1
∇ R +∇ R +∇ R = ((∂ s)g +(∂ s)g +(∂ s)g ). (6.2)
k ij i jk j ki k ij i jk j ki
3
In this case R dxidxj = const will be first quadratic integral of light-like
ij ds ds
geodesic equations xk = xk(s) of the space-time (M,g). It is obvious that on
a Sinyukov manifold L4 equations (6.2) become identity.
Remark. The seventh class of the Einstein equations is defined by the
property ∇T = 0. Chaki and Ray have studied a space-time with covariant-
constant energy-momentum tensor T (see [22]). In addition, two particular
types of such space-times were considered and the nature of each was deter-
mined (see also [23]).
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