Curvature Structures and Gravitational Waves ϟ Incepta Physica Mathematica literature

Curvature Structures
Gravitational Waves
Hans Tilgner


 Lichnérowics has gi­ven an ele­gant ma­the­ma­ti­cal pre­sen­ta­tion of gra­vi­ta­tio­nal wa­ves, which fits in­to the frame­work of Sin­ger & Thor­pe's- axi­oma­ti­sa­tion and de­com­po­si­tion of cur­va­ture struc­tures on pseu­do-Rie­man­nian ma­ni­folds, na­tu­ral­ly co­var­iant to the pseu­do‌-‌or­tho­go­nal group plus di­la­ta­tions. — Post has no­tic­ed that the elec­tro­mag­ne­tic con­sis­tu­en­cy 4×4 ten­sor has the same sym­me­tries as a cur­va­ture ten­sor! — Com­bi­ning the­se re­sults with cur­va­ture con­struc­tions by No­mi­zu for elec­tro­mag­ne­tic field strengths and –in­ten­si­ties, one can de­com­pose the high‌-‌di­men­sio­nal space of all cur­va­ture struc­tures in­to group or­bits, which com­mute with the Sin­ger & Thor­pe di­rect vec­tor space sum of Weyl- and Ein­stein- sub­spaces.

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first published
24. May 2002

revised upload



Vector Spaces
Given a real pseudo-orthogonal vector space (VV,<,>)of dimension n as in [T78] and [T84], there are two subsets
 ╳(VV,<,>){  l ∈ VV/  <l,l>= 0 } of  light-like  vectors and 
(VV,<,>){ s ∈ VV/ <s,s>0 } of  time-  or  space-like  vectors
in VV,which decompose this vector space into a disjoint union.
Both inherit their topological structure from that one of VV,but this must be made compatible with <,>, which is standard only in the positive-definite case. The more interesting and under­in­ve­sti­ga­ted first one is a cone of dimension n−1, but not convex, say null-cone, the se­cond one is an open subset, hence of dimension n. If <,> is positive‌-‌definite, the null-cone becomes the zero vec­tor 0 only, and the second one a punc­tur­ed plane. For indefinite <,> both are non‌-‌com­pact and may have different con­nec­ti­vi­ty com­po­nents.
A simple algebra structure on such a pseudo-orthogonal vector space is the Jordan algebra com­po­si­tion
x  ▣t  y   =   <y,t> x + <x,t> y − <x,y> t
for any  t ∈ VV,special case of those described on this website on Hilbert spaces for use in quan­ti­za­tion.
Exercise: Dropping the last term in this composition and changing the plus to a minus sign, do we get a non-trivial ( if so nilpotent ) Lie algebra? And what for symplectic vector spaces?
For  t ∈ (VV,<,>)there is the neutral element
e =
 with <e,e> =
 ( normalizing  ê :=
  to get  <ê,ê> = 1 )
<t,t><t,t> <t,t> 
of this Jordan algebra (VV, ▣t )with inverse
x−1  =  2
t  − 
x  especially  t−1 = <e,e>e
<x,x><t,t>² <x,x><t,t>
x−1  =  2
e  − 
x .
<x,x> <x,x>
Since Jordan algebras are ( not associative but ) power-associative there is an exponential se­ri­es, which maps vectors into vectors in the set of invertible elements, given in terms of this Jor­dan com­po­si­tion  ▣t or equivalently by that following one of the symmetric space [Loo]: The spa­ce of in­ver­tib­le elements of this Jordan algebra, which is (VV,<,>) ,has a sym­me­tric space com­po­si­tion
(o₪) x ₪ y  =  2 
x  −
especially  x−1  =  e ₪ x ,
which even is defined for a vector  x ∈ ╳(VV,<,>).Note how the symmetric composition o₪ ge­ne­ra­li­zes the inverse, which is a special case of more general negative powers in the symmetric space and in the Jordan al­ge­bra together, since powers in both structures coincide if suitably de­fined. In the symmetric space let the left multiplication  Sx be given by
    Sx(y)  =  x ₪ y .
Because of the 3rd symmetric space axiom it is an ( so-called inner ) automorphism of the sym­me­tric spa­ce (VV,<,>),because of the 2nd it even is involutive. Application on x resp. −x shows, that there is no vector x outside the null‌-‌co­ne such that its in­ner au­to­morphism is the identity map on .
 Compatability of <,> and reads
<Su(x) , Su(y) >  =  <u,u>² <
> ,
i.e. we get a non-linear transformation of conformal type outside the null-cone and for l inside the null‌-‌cone a non-linear map
Sl  : y  ι 2
l ,Sl  : (VV,<,>) →   ╳(VV,<,>)
from outside into the null-cone, which is not defined only on the null-cone. The symmetric spa­ce (,₪) admits the Jordan al­ge­bra (VV, ▣t )as its (local) tangent structure, cor­res­pon­ding to the Lie al­ge­bra as the (local) tangent struc­ture for a Lie group. One expects that ex­po­nen­tials ‌-‌ge­ne­rate the con­nec­tivity component of e.
Loos has shown in his second chapter, that defining for any Jordan algebra
Ƥ(x) (y)  =   Sx( Se(y) )=  x ₪ (e ₪ y)
( where on the left hand side the dependency on the neutral element e is dropped for a moment, and there is no danger of confusion with the quadratic representation on a Jordan alge­bra - they co­in­ci­de outside the null-cone ) the Ƥ(x) generate a nor­mal sub­group of the automorphism group of the symmetric space of invertible elements for any Jordan algebra with an identity element. It is cal­led  group of dis­pla­ce­ments  be­cause it re­du­ces to translations in the flat case [Loo]. Its di­men­sion is n, that of the vector space VV.
Since the Sx are homogeneous of degree −1 in the 2nd variable these Ƥ(x)‌-‌transforma­tions turn out to be li­near with
<Ƥ(u) x , Ƥ(u) y>  =  (
)²  <x,y>
describing the compatibility of <,> and displacements as a conformal property. The 2nd sym­me­tric spa­ce axiom identifies the neutral displacement by
     Ƥ(e) y = e ₪ (e ₪ y) = id y ,  id =  Se○ Se ,  here also  = Sê○ Sê .
Does for the first subset, the null-cone  ╳(VV,<,>)exist a similiar symmetric space mul­ti­pli­ca­tion, per­haps choosing the given vector t inside the null-cone?
are vectors of  VV

l even in the
null-cone  ╳( , )

wherein we frequently
drop arguments

capital letters  A,B, ...
are reserved for
linear transformations
on  VV

non-linear transformations
we use
script letters

those are

is there for
symmetric spaces
something like an
adjoint representation
for groups
i.e. a map

Ad : ※ → Aut (※,₪)

subject to

Ad(x ₪ y) = Ad(x) ₪ Ad(y)

= Ad(x) ○ Ad(y)−1○ Ad(x)

more useful in

quantum mechanics

Spaces of
Vector Spaces
Up to now all categories were subsets of the pseudo-orthogonal vector space. Now we turn to spa­ces of endomorphisms on VV.
The automorphism- or pseudo-orthogonal group of the pseudo-othogonal vector space is
    OO (VV,<,>)  ={ G ∈ end(VV,ℝ / <Gx,Gy> = <x,y>  ∀ x,y ∈ VV} ,
its derivation- or pseudo-orthogonal Lie algebra ( with respect to the commutator ) is
     oo (VV,<,>)  ={ E ∈ end(VV,ℝ / <Ex,y>+<x,Ey> = 0  ∀ x,y ∈ VV} .
They are related via the standard exponential series for endomorphisms, exponentiating ele­ments of the Lie algebra to elements of the group. A ty­pi­cal, ge­ne­rating linearly element is
     ox,y : u  ι½ ( <y,u>x − <x,u>y ) with  ox,y = − ox,y = − oy,xand trace ox,y= 0
for  u ∈ VV,giving the typical commutation relations in a basis-free form - note that these also can be found for 2nd power polynomials in its Clifford algebra. Its anti-derivation- or pseu­do‌-‌or­tho­go­nal Jor­dan algebra ( with respect to the an­ti‌-‌com­mu­ta­tor ) is
    oo+(VV,<,>)  ={ A ∈ end(VV, / <Ax,y> = <x,Ay>  ∀ x,y ∈ VV}
with the typical, generating linearly element
     o+x,y : u  ι½ ( <y,u>x + <x,u>y )with o+x,y =  o+x,y =  o+y,xand trace (o+x,y) = <x,y> .
Again, the elements of this Jordan algebra are finite sums of these typical generating elements. Ex­po­nen­tiation leads to a fourth category in the set of invertible endomorphisms, which has a sym­me­tric space composition.
Another standard element of this Jordan algebra is the  reflection at  t ∈ ※( , ), defined as
Jt u=u − 2
t=( idVV
o+t,t) u with Jt t = − t and Jt○Jt = idVV ,
which is the special case   s= ± t  of the more general <,>-self-adjoint, but in general not pseu­do‌-‌or­tho­gonal endomorphism
Jt,s  =+  idVV
o+t,s .
There is a direct and <,>-orthogonal decomposition  ⊞  of this Jordan algebra, given by
A   =  
 trace(A) idVV ⊞   ( A −
 trace(A) idVV) ,
into multiples of the identity and traceless endomorphisms, which are embedded as the first two sub­spaces in the decomposition of the space of curvature structures below.
In case of trace­less en­do­mor­phisms, or in the group case of elements with determinant 1, we put the letter s or S in front.
and those the
classical categories

throughout this article

<Ex,y> = − <x,Ey>

for any vectors  x, y
and endomorphisms  A, E



<Ax,y> = <x,Ay>
The real high-dimensional vector space curv(VV,<,>)of  pseu­do-Rie­man­nian cur­va­ture struc­tur­es is defined by bilinear maps
    R : VV ×VV→ end VV
subject for any  x,y,v,z ∈ VVto Singer & Thorpe's three axioms [ST]
skew symmetryR(y,x) = − R(x,y) ,
pseudo-orthogonal derivations<R(x,y) u , v > + < u , R(x,y) v > = 0 ,
Bianchi identityR(x,y) z + R(z,x) y + R(y,z) x = 0
on a real pseudo-orthogonal vector space ( VV,<,>),especially for physical Minkowski space, tho­se of di­men­sion 4 and Minkowski signature [Nom], [T84] ( and many more publications ). Note that the second axiom some­times is substituted for another equivalent one. There al­ways is the
    trivial curvature structure   Ro(x,y) z = <y,z> x − <x,z> y ,
which defines a 1-dimensional subspace in the curvature space, called the space of scalar cur­va­ture. It will be used below to give three classes of examples. Note that  Ro(x,y) = 2 ox,y, but the Bi­anchi iden­ti­ty does not reduce to the Jacobi identity for Ro.
this axiomatic
has been overlooked
in physics
Structure Theory
 Mathematical data, in terms of which the structure theory of curvature spaces can be for­mu­la­ted, are constructed in terms of basic linear al­ge­bra­ic struc­tures. The
RRicci formand theρR(x,y) = trace ( u  ι R(u,x) y )
is a symmetric bilinear form, which may be used to define R semi‌-‌sim­ple if it is non‌-‌de­ge­ne­rate and compact if it is positive‌- or negative-definite. Since <,> is non‌-de‌ge­ne­rate Witt's theorem gi­ves a unique ( necessarily <>-self-adjoint ) endomorphism on VV ,the
Ricci transformation  LR  from R<LRx,y> = ρR(x,y)
and a linear form in the dual space VV*,the
Rcurvature scalarand the Sc(R) = trace LR .
In addition we need the endomorphism
Ricci mapRanRic : R ι RoLR,idVV= Ω(LR) ,  Ric : curv(VV,<,>)  →curv(VV,<,>)
with an injective linear
Ricci Jordan mapand the Ω : A ι Ω(A) = RoA,idVV,   Ω : oo+(VV,<,>)  →curv(VV,<,>) ,
especially  Ω(idVV)= Ro , and the non-linear
and the<R(x,y) y,x>
sectional curvature sec(R)  = 
  ,   sec : curv(VV,<,>)  →   .
<Ro(x,y) y,x>
Moreover there are two projectors on curv(VV,<,>),the
Einstein projectorR(x,y)  =  R −
n (n −1)
and the
Weyl projectorШR(x,y)  =  R −
 Ω(LR) + 
 Ro  ,
n −2(n −1) (n −2)
where idempotence and moreover commutatability are easily verified
    ℇ = ℇ , Ш ○ Ш = Ш , ℇ ○ Ш = Ш ○ ℇ = Ш,(ℇШ) ○ (ℇШ) = (ℇШ) ,
    Ш ○ Ω = 0 ,
see [T78 sec 3] for a more complete list of formulas and a commutative diagram of short exact se­quen­ces, visualizing the following structure decomposition. We get [ST]

Singer & Thorpe's Structure Theorem of Curvature Spaces 
R ∈ imageШRthe Ricciform ρR vanishes,
 i.e. ρШR = 0 , LШR= 0 , Sc(ШR) = 0
R ∈  Rothe sectional curvature is constanto
R ∈  Ro ⊞   imageШRLR is a multiple of the identity)
R ∈ image(ℇ−Ш)  ⊞   imageШRthe scalar curvature of R vanishes(
wherefrom we get the direct vector space decomposition of type  kernШ ⊞ imageШ of
    curv(VV,<,>)  =    Ro   ⊞   image(ℇ−Ш)   ⊞   imageШ  ,
uniquely decomposing every curvature structure into a first cos­mo­lo­gi­cal part  kern(ℇ−Ш) and a se­cond Weyl part. This should be compared to the decomposition of every electro­mag­ne­tic field in­to a sta­tio­na­ry and a wave part.
 The 1st class of more general examples is given for any  t ∈ ( VV,<,>)by
    Rt(x,y)  =  <x,t> Ro(t,y) + <t,y> Ro(x,t) − <t,t> Ro(x,y)  ,
which, as was shown in [T84], comes from a Jordan algebra, in fact that on a real pseu­do‌-‌or­tho­go­nal vector space, which was stu­died else­where on this web­page and is basic for the quan­ti­zation of curved space‌-‌times. It is con­struc­ted for the Jordan composition  ▣t as
     x ▣t ( y ▣t z) − y ▣t ( x ▣t z )  =  Rt(x,y) .
Since it is qua­dra­tic in t polarization  t ιt+s  gives even more ge­ne­ral curvature structures
  Rt,s(x,y) = ½ ( Rt+s(x,y) − Rt(x,y) − Rs(x,y) )
 = ½ ( <x,t>Ro(s,y) + <x,s>Ro(t,y) + <y,t>Ro(x,s) + <y,s>Ro(x,t) ) − <t,s>Ro(x,y) .
Note that since the last term herein is a curvature structure the ½( )-sum also is one. Obviously
    Rs,t = Rt,s , Rt,t = Rt  but  Rt = − <t,t> Ω(Jt) .
Because the last equation is square in t it can be polarized  t ιt+s  to
    Rt,s  =  − Ω ( <t,s> idVV− 2 o+t,s ) ,
which can be easily verified directly. This sorrow result shows that we do not get access to the space of Weyl curvatures by the two vec­tor pa­ra­me­ters only. We call curvature structures of this type weak, they take place in  kernШ only. The sec­tio­nal cur­vature of these curvature structures turn out not to be constant, i.e. they de­pend on x,y. The other da­ta of these t,s‌-‌mo­di­fi­ca­tions of Ro are
◆ ρRt,s(x,y)  = − (n−2) ( <t,s><x,y> − <t,x><s,y> − <t,y><s,x> ) ,

◆ LRt,s = − (n−2) ( <t,s> idVV − o+t,s),

◆ Sc(Rt,s) = − (n−1)(n−2) <t,s> .
In [T84] there is a more detailed well-known construction of curvature structures from se­mi‌-‌sim­ple Lie‌- and Jor­dan al­ge­bras and Lie triples.
 The 2nd class is given for any pair  A,B  of <,>‌-‌self‌-‌ad­joint en­do­morphims on VVby
RoA,B(x,y)  = ½ ( Ro(Ax,By) + Ro(Bx,Ay) ) ,
defining elements RoA,B in the curvature space. The special case  B = idVVof which was used abo­ve to con­struct the map Ω. We call curvature structures of this type strong. There is linearity in the variables A,B,
RoB,A = RoA,B and RoA,B = ½ ( RoA+B,A+B − RoA,A − RoB,B ) ,
and for  C = LRoA,B
Ric(RoA,B)  =  RoC,idVV  =  Ω(C)    especially   Ric(Ro)  =  (n−1) Ro ,
with a sorrow consequence: No insight into  imageШ by this 2nd class of examples. In fact  Ω  is a li­near bijection
Ω  :  oo+(VV,<,>)→ kernШ  =   Ro  ⊞  image(ℇ−Ш)  ,
which carries over any structure from the ( Jordan algebra of ) self-adjoint transformations into the cur­va­ture space. The full diagram of short exact sequences was given in [T78 p 1120]. The da­ta of this A,B-mo­di­fi­ca­tion of Ro are
◆ ρRA,B(x,y)  = ...  ,
◆ LRoA,B = −½ ( AB+BA−trace(A) B−trace(B) A ) ,

◆ Sc(RoA,B) = − ( trace (AB) − trace(A) trace(B) )
( reproducing the canonical bilinear form ≪,≫ of endomorphisms ) and especially
◆ ρΩ(A)(x,y) = ... , ... ,
◆ Ω(A) = − ( A−trace(A) idVV), Ω(idVV) = (n−1) idVV ,

◆ Sc(Ω(A)) = (n−1) trace(A) , Sc(Ro) = n(n−1) .
As only chance to look inside  imageШ  remains Nomizu's construction of
 the 3rd class of curvature structures ИE,D, defined as
ИE,D(x,y)=RoE,D(x,y) − <Ex,y>D − <Dx,y>E
=½ ( Ro(Ex,Dy) + Ro(Dx,Ey) ) − <Ex,y>D − <Dx,y>E
in terms of two <,>-skew-adjoint endomorphisms E and D, similar but not identical to the 2nd class con­struction of curvature in terms of two <,>‌-‌self‌-‌ad­joint en­do­mor­phis­ms. We call cur­va­ture struc­tures of this type em. The data of this E,D-modification of Ro are
◆ ρИE,D(x,y) = 3/2 ( <Ex,Dy>+ <Dx,Ey>) ,
◆ LИE,D=  3/4 ( ED+DE ) ,
◆ Sc(ИE,D) = − 3 trace(ED) .
There remains to show that and how Nomi­zu's con­struc­tion leads from a phy­si­cal elec­tro­mag­ne­tic field, given by elec­tro­mag­ne­tic field strength E and -‌in­ten­sity D, to a gra­vi­ta­tional field, sol­ving Ein­steins field equa­tions in terms of a ( but which? ) elec­tro­mag­ne­tic ener­gy‌-‌mo­men­tum ten­sor. Here E com­bi­nes the elec­tric field strength E and the mag­ne­tic field strength H, where­as D is the elec­tric in­ten­si­ty D com­bi­ned with the mag­ne­tic in­ten­si­ty B. Looking for more curvature struc­tures there are besides И only two more solutions, if one supposes bi­linearity in the two pa­ra­me­ters E‌,‌D
 trace(ED) Ro  and  Ω(ED+DE) ,
where the anti-commutator of two <,>-‌skew‌-‌ad­joint endomorphisms is <,>-‌self‌-‌ad­joint.
 A 4th ,mixed' class of curvature structures is given by one <,>-self-adjoint A and one <,>‌−‌skew‌-‌ad­joint E. If they are supposed to be linear in their two parameters, then [T84 p 14] there are on­ly the two cases
 trace(EA) Ro  and  Ω(E•A) ,
where is the co-adjoint action by commutators, and they lie in the first two subspaces of the above curvature space decomposition.
Note that these curvature structures must be linear-combinations of ele­ments of the pseu­do‌-‌or­tho­gonal Lie algebra. In [T78] and [T84] Lie group and -‌al­ge­bra ac­tions on these clas­ses of ex­am­ples of cur­va­ture struc­tures were given, to­gether with their data.
Singer & Thorpe
theory -
a great step
for the
from linear to
multilinear algebra

projectors are a
great mathematical

well-known as
quantum mechanics'
statistical operators

but never used
in gravity
and the symplectic
formulation of
classical mechanics

In- and
Chasing the pseudo-orthogonal groups and Lie algebras on (VV,<,>)around was done in [T78], where even dilatations
    x  ι➥  λ x  ,  VVVV
are included to a larger linear transformation group  GGof dimension 1+½ n(n−1), called the li­ne­ar con­for­mal or some­times Weyl group of the pseu­do‌-‌or­tho­go­nal vec­tor space ( not to be con­fu­sed with the Weyl group in the clas­si­fi­ca­tion of real sim­ple Lie al­ge­bras ). It is straight­fore­ward to chase the group‌- and Lie al­ge­bra • ac­tions around the dia­grams and de­com­po­si­tions once they are de­fined in the well‌-‌known Ad, ad- way. Blo­wing up the pseu­do‌-‌or­tho­go­nal group is not ne­ces­sa­ry, but it col­lects or­bits of dif­feo­mor­phic shape in­to one. More­over it is easy to check that GGis a sub­group of linear transformations in the au­to­mor­phism group of the sym­me­tric space (,₪). Since o₪ is not linear in its two variables, infact it is homogeneous of degree 2 in x and of de­gree −1 in y, this auto­mor­phism group need not be a li­near group. Loos de­fines the  group of dis­place­ments  for any Jor­dan al­gebra as a normal sub­group in the auto­mor­phism group by left mul­ti­pli­cations, which re­sults in non‌-‌li­near trans­formations, in this special symmetric space in square ones in the left variable.
The direct decomposition of the curvature space commutes with these Ad, ad-ac­tions, i.‌e. is in­va­ri­ant, hence the no­ta­tion . There­fore the clas­si­fi­cation of group or­bits takes place in­side the three vec­tor space com­po­nents of the cur­va­ture space se­pe­rate­ly. In­side kernΩ it is traced back to the clas­si­fi­ca­tion of  GG‌-‌or­bits inoo+‍( , ) ,one of the clas­si­cal real sim­ple Jor­dan al­ge­bras of en­do­mor­phisms, but in­side imageШ it is com­ple­tely un­known and pro­bab­ly more in­vol­ved as that un­sol­ved one of nil­po­tent and sol­vab­le Lie al­ge­bras.
 A structure theory usually means that there is a decomposition into ( the di­rect sum of ) sub­spa­ces in terms of pro­jec­tors, like the above ‌-‌Sin­ger & Thor­pe's one of all cur­va­tures, such that the­re is a struc­ture theo­rem. A clas­si­fi­ca­tion by or­bits is a re­fine­ment of such a struc­ture theo­ry with the help of a trans­for­ma­tion group, if such a group exists and if it com­mu­tes with the pro­jec­tors on­to the sub­spaces.
including dilatations
by a real factor  λ
chases this around
formulas and diagrams
Field Equations
Einsteins gravitational field equations without cosmological term were given in our notation in [T78 p 1123] and [T84] as
LR − 2
idVV =  ⓖ T
with the gravitational constant and  T ∈ o+(<,>) the energy momentum trans­for­ma­tion. is an uni­ver­sal constant - attempts by Jordan and Dirac in the early days of general relativity to as­sume a timely decrease in order to explain the evolution of con­tinents on earth and their drift could not be veri­fied experimen­tally so far. Physically T is given by the distribution of sources of the gra­vi­ta­tio­nal field, mathematically it should be supplied with an index R. There also is the shor­ter form
idVV =  ⓖ TR
and the equivalent ρR(x,y)
<x,y> =  ⓖ <TRx,y>
one, which corresponds to lowring or highring indices in the index notation. Also from  Sc(R) = − ⓖ trace TR
trace TR
there is the equivalent form  LR =   ⓖ (  TR − 2 
idVV) ,
which in some cases is easier to handle.

n = 4
and the
the signature
is the special case
general relativity
Gravitational waves [Lic p 45] are defined for any  x,y,z ∈ VVin terms of a (necessarily) light‌-‌li­ke ( for in­de­fi­nite <,> this means on the null‌-‌space ) vec­tor  l ∈ VVwith
annihilationR(x,y) l = 0 ,
symmetrization<l,x> R(y,z) + <l,y> R(z,x) + <l,z> R(x,y) = 0 .
If this is the case R is called a gravitational wave and l its wave vector. This is an or­bit pro­per­ty with res­pect to the pseu­do-or­tho­gonal group, even with respect to its Weyl group. For R a gra­vi­ta­tio­nal wave with respect to the wave vector l, the main theorem
  • ρR(x,y) = τ <x,l><y,l>   for some real τ, i.e. ρR is degenerate and R not semi-simple,
  • LR = τ o+(l,l),
  • Sc(R) = 0  ( i.e. R ∈ image(ℇ−Ш) imageШ ), trivial curvatures of Ro cannot be waves,
  • l is an eigenvector of eigenvalue 0 for LR
on gravitational waves was proven in [T84]. Hence Ω(o+(l,l)) are examples of gravitational waves in image(ℇ−Ш) and even, and as it was shown there al­so, the only ones in this sub­space of the cur­va­ture space. Like the cur­va­ture struc­tures in this sub­space are para­me­trized com­ple­te­ly by <,>‌-‌self‌-‌ad­joint en­do­mor­phims, gra­vi­ta­tional waves there­in are para­me­trized by light‌-‌like vec­tors, both via the li­near map Ω.
Can­di­dates of phy­si­cal gra­vi­ta­tion­al wave or­bits are those special wave or­bits which are not of elec­tro­mag­ne­tic ori­gin, be­cause the lat­ter are not strong enough to be mea­sured and they can be ,seen'. To ex­pe­ri­men­tal­ly veri­fy the exi­stence of gra­vi­ta­tion­al waves it is necessary to get an idea of the freq­uen­cy resp. wave‌-‌length, both de­fined in terms of this wave vec­tor, for eve­ry such pure ( i.e. not given by an elec­tro­mag­netic wave ) gra­vi­ta­tion­al wa­ve or­bit - but how?
curvature structure

in the following
But which Weyl curvatures are determined by wave vectors? Since the (full) Weyl group acts tran­si­ti­ve­ly on the null-space, and group act­ions per­mute with Weyl‌- and Ein­stein‌- pro­jec­tors in Sin­ger & Thorpe's cur­va­ture space, one per­haps can show, that there is only one wave orbit in the sub­space of Weyl curva­tures. But it is unlikely that one such orbit ex­hausts the space of Weyl cur­va­tures, since this is a ( aw­ful­ly high ) n(n+1)(n+2)(n-3)/12 - dimen­sio­nal sub­space in the whole cur­va­ture spa­ce, for the dimension of which one has to add n(n+1)/2, the di­men­sion of the ( Jor­dan al­ge­bra of ) <,>‌-‌self‌-‌ad­joint en­do­mor­phisms.
 If there is only one (gravitational) wave orbit there arises the quest­ion, what are the other or­bits? Are one or two ( the num­ber depends on the sig­nature of the un­der­ly­ing bi­li­ne­ar form ) gi­ven by time‌-‌like or space‌-‌like vec­tors of the un­der­ly­ing vec­tor space? For this we need an ex­pli­ci­te con­struc­tion of a cur­va­ture in terms of these vec­tors, like we have one in the three ca­ses by vec­tors and endo­mor­phisms above. There is such a con­struc­tion by using ten­sor pro­ducts of such a vec­tor, to get an en­do­mor­phism and insert it into these three con­structions. In the 'mi­xed' case - the elec­tro­mag­netic one - we can­not arrive by this con­struction in the space of Weyl cur­va­tures. In the other cases we have to find 'odd' con­struc­tions in or­der to ar­rive only in the space of Weyl cur­va­tures.
is there
further decomposition
the space of
Weyl curvatures
Electromagnetic orbits are given by two physical tensor fields, the field strengths E ( a 3-vec­tor E together with a skew 3×3 matrix H ) and the intensities D ( a 3-vec­tor D together with a skew 3×3 matrix B ) in media without struc­ture and spe­cial re­la­ti­vity's flat Min­kow­ski space with a li­near de­pen­dence in be­tween. Phy­sicists take them as skew 4×4 matrices – which is er­ron­ous! Be­cause this means the in­tro­duc­tion of a po­si­tive‌-‌de­fi­nite me­tric in space‌-‌time and an Äther‌(-‌ca­te­go­ry) which – doesn't exist! We have to for­mu­late elec­tro­dy­na­mics en­tire­ly in the Min­kow­ski ca­te­go­ry, which means that E and B have to be <,>‌-‌skew‌-‌ad­joint! Even ex­te­rior al­ge­bra for­mu­la­tions by ex­te­rior differential forms lead into a dead end – we have to take a Clif­ford al­ge­bra ( over the Min­kow­ski space ) in­stead, well‌-‌known in phy­sics from Fer­mi sta­tis­ get rid of
Petrov's classification by bi-vec­tors is one in the n(n+1)/2-dimensional Jordan alge­bra of <,>‌-‌self‌-‌ad­joint endomorphisms, since those are generated linearly by pairs of vectors of VV.It is map­ped [T84] by the linear map Ω on­to the cosmological sub­space of the cur­va­ture space. Since this Jor­dan al­ge­bra is one of the clas­si­cal real simple ones, this clas­si­fi­ca­tion is given by Dyn­kin dia­grams. How­ever, the comp­le­men­ta­ry sub­space of the Weyl curvature struc­tures is not ar­rived at by such an easy map.
Clearly Petrov's eigenvalue‌-‌based curvature classification is one by orbits of the Weyl group.
has to be generalized
to the whole
curvature space
into Gravity
Implementing an electromagnetic field into general relativity usually is done by inserting its ener­gy mo­men­tum tensor into Ein­steins gra­vita­tional field equa­tions as a source ( i.e. on the right hand side ) and sol­ving for metric, connection and curvature. But this on­ly is an in­di­rect me­thod. Hence there is the question whether there exists a direct me­thod, starting from a <,>‌-‌skew‌-‌ad­joint elec­tro­mag­ne­tic field strength E and an -‌in­ten­si­ty D? So far we don't know any such di­rect con­struc­tion.
 If this ener­gy mo­men­tum ten­sor is that one of an electro­magnetic field only, like in most re­gions of the universe, we have the electrovac special case
Lichnérowics [Lic] has given an axiomatic characterization of electromagnetic waves, like that one above of graviational waves: The pair (E, D) is said to be an electromagnetic wave if there exists a (necessarily) light‌-‌like vector l such that for any x,y,z ∈ VV
annihilationE l = 0 ,
symmetrization<l,x><y,Ez> + <l,y><z,Ex> + <l,z><x,Ey> = 0
and the same equations for the intensity D. Clearly symmetrization is equivalent to
symmetrization <l,x>Ey = <l,y>Ex + <x,Ey>l ,
 <l,x>Dy = <l,y>Dx + <x,Dy>l .
Certainly for most physical media  D = ε E  and especially for the vacuum physisists write εo for this constant. That it is not an universal constant, but depends on the curvature at any position in space‌-‌time follows from
must have a
better solution

el.mag. field  ⊑  grav.field
⊔      ⊔
el.mag. wave  ⊑  grav.wave
Post [Pos] remarks that if E and D are linearly connected ( like in most media on earth ) the 4‌-‌ten­sor re­la­ting them has the same sym­metries as a cur­va­ture ten­sor, i.e. ful­fills Sin­ger & Thor­pe's cur­va­ture axioms. Since he starts from only one field strength E, his method leads to a first quest­ion: Start in be­tween Sin­ger & Thor­pe's and No­mi­zu's cur­va­ture con­struc­tions by one <,>‌-‌self‌-‌ad­joint A ( Post has the iden­ti­ty ) and a <,>‌-‌skew‌-‌ad­joint elec­tro­mag­ne­tic field E, which, when in­sert­ed in­to the given cur­va­ture, de­li­vers the <,>‌-‌skew‌-‌ad­joint in­tensity D. In [T84] it was shown that for this ,mixed' case, there are on­ly so­lu­tions in the space of non‌-‌Weyl cur­va­tures, none in the spa­ce of Weyl ones, and if A especial­ly is the iden­tity, even these two vanish. In fact, the only pos­sib­le cur­va­tures con­struc­ted from such A's and E's are of the form E•A and the trace there­of, where the is the natural ( Lie‌- on Jor­dan ) al­ge­bra­ical co­ad­jointad‌-‌ac­tion by in­ner de­ri­va­tions [T84]. So the only non‌-‌tri­vial em­bed­ding of elec­tro­mag­ne­tics in­to gra­vi­ty re­mains No­mi­zu's! But this only is the case be­cause all three con­struc­tions, that by two <,>‌-‌self‌-‌ad­joint or that by two −­skew‌-‌ad­joint en­do­mor­phisms or the mixed one, start from the tri­vial 1‌-‌di­men­sio­nal sca­lar cur­va­ture - more ge­ne­ral ones do not work.
Post's construction inverted leads to more: Given an elec­tro­mag­ne­tic field strength E in a gra­vi­ta­tional field, given by its cur­vature R, con­struct the elec­tro­mag­netic field in­ten­sity B in the same way as the three mathe­mati­cal ap­proaches from the tri­vial cur­va­ture, but this time more ge­ne­ral from the given curvature R. Clear­ly this mo­di­fi­ca­tion of R by E no lon­ger is a cur­va­ture: Two of Sin­ger & Thor­pe's cur­va­ture axioms are not ful­fil­led – <,>‌-‌skew sym­me­try ( be­cause E is <,>‌-‌skew‌-‌ad­joint B(x,y) can't be ) and the Bi­an­chi‌-‌iden­ti­ty ( that's why they are not in­te­rest­ing for ma­the­ma­tics - they con­tri­bute no­thing to the struc­ture theo­ry ). But it is clear from the con­struc­tion that the third still holds, i.e. the out­come B is in the pseu­do‌-‌or­tho­go­nal Lie al­ge­bra and hen­ce may be in­ter­pre­ted as an el­ec­tro­mag­ne­tic field in­ten­si­ty. This has a phy­si­cal con­se­quen­ce: The full gra­vi­ta­tio­nal field ( not on­ly its sca­lar part, i.e. the tri­vi­al cur­va­ture ) con­tri­butes to el­ec­tro­mag­ne­tic field  in­ten­si­ties. Since there is no point in spa­ce‌-‌time with­out gra­vi­ta­tion this con­tri­bu­tion ne­ver vani­shes. Half way be­tween two ad­ja­cent ga­lax­ies, this con­tri­bu­tion is ne­glec­tib­le, but near the cen­ter of a ga­la­xy, near the event ho­ri­zon of a black hole, for a strong gra­vi­ta­tio­nal wave or in the first moments of a big bang model these cur­va­ture‌-‌mo­di­fied in­ten­si­ties B can have mea­su­rab­le phy­si­cal ef­fects. On­ly in a uni­ver­se with con­stant cur­va­ture, actu­al­ly that one by Ir­ving Se­gal, stu­died on this web­page, this con­tri­bu­tion re­du­ces to a ( in some  po­si­tions of space‌-‌time huge ) number, the cur­va­ture radius of that po­si­tion.
B(x,y) = R(x,Ey)

for any
vectors x, y of
Minkowski space
can have
physical effects
regions of strong
Physical space-time is not a vector space as in this study sofar, but a pseudo-Rieman­nian ma­ni­fold of dimension 4 and signature +,−,−,−. <,> becomes the eigen‌-‌time, vector fields, i.e. sec­tions of the tangent bundle take the role of the ele­ments of the vector space VVabove, over which there are bundles whose fibres are curvature spaces. Curvature structures are sections in such a cur­va­ture bund­le. A dynamical sys­tem is a geodesic ( with respect to <,> ) in space‌-‌time, giving rise to a Levi‌-‌Civi­ta con­nec­tion, which in turn gives rise to a curvature struc­ture.  GGis the structure group of these bundles.
A cosmological model is a space-time, whose curvature sections lie in one GG - orbit.Hence the clas­si­fi­cation of these orbits in the curvature space is exactly the classification of cos­mo­lo­gi­cal mo­dels. Needless to say, actual space‌-‌time is not a cosmological model, but in each point of space-time there is a cosmological model, which approximates actual space‌-‌time best, es­pe­cial­ly better than the tangent Minkowski space. Equivalently one can develop the concept of lo­cal cur­va­ture structure, as Loos has given the concept of locally symmetric spaces in his books.
Also Possible
Assuming a big bang cosmology, in the first moments of the universe
strong electromagnetic fields alone
can have led to riddles in the cosmic back ground radiation.
Literature with comments
[Lic]A Lichnérowics  Ondes et Radiations Électromagnétique et Gravitationelles en Relativité Générale  Annali di Mate­ma­ti­ca Pura & Appl. 4 [1960] p 1-95  is a very elegant representation of the concept, although written in the old‌-­fashioned index notation.
[Loo]O Loos  Symmetric Spaces I + II  Benjamin N.Y. [1969] no ISBN
[Nom]K Nomizu  The Decomposition of Generalized Curvature Tensor Fields  p 335-345 in  Differential Geometry, Papers in Ho­nor of Ken­ta­ro Yano  Kinukuniya, Tokyo [1972] no ISBN
[Pos]E J Post  Formal Structure of Electromagnetics  North Holland, Amsterdam [1962] no ISBN is an inspiring book, though writ­ten in an old‌-‌fashio­ned way with indices: The electromagnetic constituency 4-tensor between the 2-tensors (4×4 matrices) of field-strengths and -in­ten­si­ties has the same symmetries as a curvature tensor.
[ST]M Singer, J A Thorpe  The Curvature of 4-Dimensional Einstein Spaces  in  Global Analysis, Papers in Ho­nor of K Kodaira  Prince­ton University Press, Princeton N.J. [1968] no ISBN
[T78]H Tilgner  The Group Structure of Pseudo-Riemannian Curvature Spaces  J.Math.Phys. 19 [1978] p 1118-1125  shows how ele­gant­ly in­variance groups can be chased around Singer & Thorpe's curvature diagrams.
[T84]H Tilgner  Conformal Orbits of Electromagnetic Rie­man­nian Cur­va­ture Ten­sors – Elec­tro­mag­ne­tic Implies Gra­vi­ta­tio­nal Ra­dia­tion  p 317-339 in Springer Lecture Notes in Mathematics 1156 [1984]  ISBN 0 387 15994 0  has more details and references on the ma­the­ma­ti­cal description of electromagnetic and gravitational radiation.
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