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

Spaces of
Vector Spaces
Given a real pseudo-orthogonal vector space (𝕍,<,>) of dimension n, i.e. <,> a non-de­ge­ne­rate sym­me­tric bi­li­near form, as in [ T78] and [T84], the  automorphism-  or  pseudo-orthogonal group of the pseu­do‌-‌or­tho­go­nal vec­tor space  is
    𝕆(𝕍,<,>)   = { G ∈ 𝕖nd(𝕍,ℝ / <Gx,Gy> = <x,y>  ∀ x,y ∈ 𝕍 } ,
its derivation- or pseudo-orthogonal Lie algebra ( with respect to the commutator ) is
    𝕠(𝕍,<,>)   = { E ∈ 𝕖nd(𝕍,ℝ / <Ex,y>+<x,Ey> = 0  ∀ x,y ∈ 𝕍 } .
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 ( <x,u>y − <y,u>x )  with   ox,y = − ox,y = − oy,xand trace ox,y = 0
for  u ∈ 𝕍, giving the typical commutation relations basis-free
    [ ox,y , oz,w ]  =  <x,z> oy,w − <x,w> oy,z − <y,z> ox,w + <y,w> ox,z
with respect to the commutator [ , ] - 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 gi­ven by the vec­tor space of the <,>-self‌-‌ad­joint en­do­mor­phisms
    𝕠+(𝕍,<,>)  = { A ∈ 𝕖nd(𝕍,ℝ / <Ax,y> = <x,Ay>  ∀ x,y ∈ 𝕍 }
with the generating linearly standard element
       o+x,y : u  ι½ ( <x,u>y + <y,u>x )with o+x,y =  o+x,y =  o+y,xand trace o+x,y = <x,y> ,
and the defining, rather simple anti-commutation relations
    { o+x,x, o+y,y}+  :=  ½ ( o+x,xo+y,y+ o+y,yo+x,x)   =   <x,y> o+x,y ,
which by polarizing twice ( or directly ) give the more general form
    { o+x,y, o+z,w}+  = ¼ ( <x,z> o+y,w+ <x,w> o+y,z+ <y,z> o+x,w+ <y,w> o+x,z) .
They give rise to the hope to find a corresponding composition law for the symmetric space on the set of in­ver­tib­le self-adjoint endomophisms. However, in general not all elements in the space of self-adjoint en­do­mor­phims are of this form, they are only finite sums ( the number of terms of which being between 1 and the di­men­sion n ) of these typical ge­ne­ra­ting ele­ments o+x,y.
There is a direct and <,>-orthogonal decomposition  ⊞  of this Jordan algebra, given by
A   =   trace(A) id𝕍 ⊞   ( A − trace(A) id𝕍) ,
in a direct sum of scalar multiples of the identity and the 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 write the letter s or S in front.
Exponentiation leads to this fourth category of symmetric spaces in the sense of Ottmar Loos, the lo­cal (tan­gent) struc­ture of which is exactly the real part of the Jordan algebra, stu­died for Hil­bert spaces in ipmWeylP.
 The pseudo-orthogonal vector space  (𝕍,<,>)  decomposes into the disjoint union of the two subsets
(𝕍,<,>)= { t ∈ 𝕍 / < t, t>0 } of  time- or space-like vectors t  outside the null-cone,
(𝕍,<,>)= { Ĭ ∈ 𝕍 / < Ĭ , Ĭ >= 0 } of  light-like  vectors Ĭon the null-cone,
which are not subspaces. The first is a flat open subspace of dimension n, the second a connected curved cone of dimension n-1. This decomposition is left invariant under the action of the pseudo-orthogonal group.  

are the
real numbers

these are the
classical categories
on pseudo-orthogonal
vector spaces

throughout this article

<Ex,y> = − <x,Ey>

for any vectors  x, y ,
endomorphisms  A, E



<Ax,y> = <x,Ay>

For a number from the ground-field and a given vector  t  outside the null-cone de­fine the linear trans­for­ma­tion t on this vector space by
t u = u + ઠ
twhich is( id𝕍 +
o+t,t) u
in obvious notations: With in place of the running vector from the underlying vector space, clearly
t ○t   =   id𝕍 + ઠ (ઠ+2)
tandλt  =  t
for a non-vanishing λ from the ground-field, i.e. the 's are not linear but homogenous of degree 0 in  t . This means that the vectors of a whole ray t  map onto the same reflection. For any such vec­tor t we get the

Structure Theorem of Involutions and Reflections

t  is invertible<=>ઠ ≠ − 1,   in which case the inverse is  εt  with  ε = −ઠ / (ઠ+1) .
t  is non-invertible<=>ઠ = − 1,   in which case it is a projector, sayt .
t  is involutive<=>ઠ = − 2,   in which case it is pseudo-orthogonal, i.e. leaves <,>
,    invariant, and is called a reflection.
Such an endomorphism t  is <>-self-adjoint, i.e. in this Jordan algebra of endomorphisms and there­fore ne­ver in the pseu­do-or­tho­go­nal Lie algebra. In the case of involutions we drop the , writing
t u = u − 2
t = ( id𝕍
o+t,t) u  ,
which reflects any element of the one-dimensional span of t at its <>-perpendicular n-1‌-‌di­men­sio­nal com­ple­men­ta­ry subspace, i.e. for any λ of the ground-field
t u = −u  <=>  u = λt   butt u = u  <=>  <t,u>=0  
holds. It is the special case   s = ± t  of the more general <,>-self-adjoint, but in general not pseu­do‌-‌or­tho­go­nal en­do­mor­phism
t,s  =  id𝕍
o+t,s   i.e. simply   t,su  =  u  − 
( <t,u>s + <s,u>t )
if t and s are not <,>-perpendicular. Sometimes the group generated by reflections is called the  Weyl group  of the pseu­do‌-‌or­tho­go­nal vec­tor space [ L&N p 362], not to be con­fused with the Weyl group whose Lie al­ge­bra is the Hei­sen­berg Lie al­ge­bra or else the group of pseu­do-or­tho­go­nal trans­for­ma­tions plus di­la­ta­tions. For re­flec­tions we get the fundamental formula
    (ff) x  ᦔy  ᦔx  =  ᦔxy
[L&N p 351]. Reflections are not a group, but (ff) means, that they are a symmetric subspace with res­pect to the Loos-multiplication on the group ( and hence symmetric space ) of all invertible linear transforma­tions and es­pe­cial­ly the pseudo-or­thogo­nal group, seen as a symmetric space.
 As an involution t has the associate involution  − ᦔt and hence the projector t
− ᦔtid𝕍 − ⼫t= id𝕍
t  .
Obviously t projects onto the one-dimensional span of t and its complementary projector onto the <,>-per­pen­di­cular n−1-dimensional subspace.
Since so far all these transformations are linear, they are (outer) automorphisms of the addi­tive abe­lian group  ( 𝕍, + )  on the underlying vector space. To get Loos' formulation of a symmetric space we there­fore can take  æ = ᦔt for involution. Then the fixed point group or genera­lized uni­ta­ry group be­comes the set of vec­tors x with  <t,x> = 0, which is the orthogonal com­ple­ment of the li­near span of t, and the set of fixed points be­comes this linear span it­self. In this flat case both sub­sets are sym­me­tric spaces with respect to the Loos mul­ti­pli­ca­tion, which be­comes ac­cor­ding to (g₪) in the general theory thereon
x ₪ y = x y–1 x   which here is equal to   x − y + x  i.e.  2x − y x,y ∈ 𝕍æ ,
since we are in the additive abelian group of a vector space.
Problem: Find involutions and projectors, which are not of this reflection type, even not re­flec­tions at a more than one‌-‌di­men­sio­nal sub­space. And re­mem­ber: This on­ly works since t is not light‌-‌like, i.‌e. not on the light‌– or more ge­ne­ral not on the null‌–cone.
play an
essential role
in the general theory of
root systems


Vector Spaces
A simple and fundamental 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> twriting this as= L(x) y
for any  t ∈ 𝕍. a special case of those described on this website on Hilbert spaces for use in quan­ti­za­tion. This left multiplication is the sum of a <,>-symmetric and a <,>-skew endomorphism
    L(x) =  <t,x> id𝕍 − 2 ot,x   with    trace L(x) = n <t,x> ,
which facilitates many calculations. The two symmetric bilinear forms of this Jor­dan al­ge­bra are the ca­no­ni­cal
trace L(xt y) − trace L(x) trace L(y) =  n ((2−n)<x,t><y,t> − <t,t><x,y> )
one, which is non-degenerate for  n > 1, i.e. a non-trivial case, and the always non-degenerate
trace L(xt y) − trace L(x) trace L(y) =  n (   2   <x,t><y,t> − <t,t><x,y> )
one, which is used in the structure theory of Jordan algebras.
  Contrary to Lie algebras the left mul­ti­pli­ca­tions L(x) are not derivations ( or anti-de­ri­va­tions ). But sums of their commutators
are, the socalled inner derivations of the Jordan algebra. In our special Jordan algebra they have the ty­pi­cal com­mutation relations of the pseudo-orthogonal Lie algebta, which is tedious to verify.

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 ∈ (𝕍,<,>)  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 (𝕍, ▣t) and the 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 ( clearly(x−1)−1 = x) .
<x,x> <x,x>
The (open) subset  𝕀nv(𝕍, ▣)  of invertible elements of a Jordan algebra, which here is the  outside of the null‌-‌cone (𝕍,<,>) , has a sym­me­tric space com­po­si­tion , which here is
(o₪)x ₪ y  =  2 
x  −
y( especially  x−1  =  e ₪ x ) ,
which even is defined for a vector  x  on the null-cone  (𝕍,<,>) . 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 ( see be­low – eve­ry­thing taken from the books of O. Loos ). Ob­vious­ly
 e ₪ e = e  ,  e−1 = e  but  x−1 = e ₪ x  ≠ x ₪ e =  2
x −
e  ,
which shows that the multiplication is not commutative. Unlike in associative categories
( x ₪ y )−1  = e ₪ ( x ₪ y ) = ( e ₪ x ) ₪ ( e ₪ y ) =  x−1 ₪ y−1 ,
i.e. inversion is an automorphism of the symmetric space like all left-multiplications with an ele­ment of the vec­tor space.
 The left multiplication  Sx in a symmetric space is given by
    Sx(y)  =  x ₪ y( especially Se(y) = y−1 ) .
Because of the 3rd symmetric space axiom it is an ( so-called inner ) automorphism of the sym­me­tric spa­ce, here (𝕍,<,>) , because of the 2nd it even is involutive. Hence the 1st three sym­me­tric space axi­oms state that  every point is a fixed point of an involutive automorphism  and the 4‌th axi­om states that it is isolated.
Application to 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. The symmetric spa­ce  (,₪)  ad­mits the Jor­dan al­ge­bra  (𝕍, ▣t)  as its (local) tangent structure, cor­res­pon­ding to the Lie al­ge­bra as the (lo­cal) tan­gent struc­ture for a Lie group.
Loos has shown in his second chapter, that defining for any Jordan algebra
Ƥ(x) y  =   Sx( Se(y) )=  x ₪ (e ₪ y)
( where we drop the dependency on the neutral element e on the left hand side ) there is no dan­ger of con­fu­sion with the quadratic representation
    Ƥ(x) y=  2 L(x)L(x) − L(x ▣ x )
of the Jordan algebra since they coincide outside the null-cone. The Ƥ(x) generate a nor­mal sub­group of the automorphism group of the sym­me­tric space 𝕀nv(𝕍, ▣)  of invertible elements for any Jor­dan alge­bra with a neu­tral ele­ment. It is cal­led the  group of dis­pla­ce­ments  be­cause it re­du­ces to translations in the flat case [ Loo p 66]. Its di­men­sion is n, that of the vec­tor space  𝕍  of the Jor­dan al­ge­bra.
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 ,
the inverse in the displacement group for  x ∈ 𝕍 being
    Ƥ(x)−1 = Ƥ(x−1)( however  Sx−1 = Sx ) .
This easily is proved [Loo p 72] by applying  Ƥ  to  y = Ƥ(y) y−1  in the  fundamental formula
    Ƥ(Ƥ(x)y)  = Ƥ(x) Ƥ(y) Ƥ(x)
for Jordan algebras. Substituting herein y−1 for y and remembering that  Ƥ(x) y−1 = x ₪ y
    Ƥ(x ₪ y)  = Ƥ(x) Ƥ(y)−1Ƥ(x)  = Ƥ(x) ₪ Ƥ(y)
results, where the symmetric multiplication on the right hand side is the usual  f ₪ g = f g−1f  one in as­so­cia­tive structures. This means that the fundamental formula of the Jordan algebra gives
    Ƥ : x  ιƤ(x)  ,  Ƥ  : 𝕀nv(𝕍, ▣)  →  𝔸ut( 𝕀nv(𝕍, ▣) , ₪)
as the adjoint - or better self re­pre­sen­ta­tion of the symme­tric space: The set 𝕀nv(𝕍, ▣)  of in­ver­tib­le ele­ments, here the out­side of the null-cone, thus turns out to be a re­pre­sentation space of the sym­me­tric mul­ti­pli­ca­tion. How­ever, it is not defined on the null‌-‌cone! Another version of the the self-representation is
    S : x  ιSx  ,   S  : 𝕀nv(𝕍, ▣)  →  𝔸ut(𝕀nv(𝕍, ▣) , ₪)
if we insert the symmetric space definition of Ƥ into the fundamental formula to get
    Sx ₪ y   =   Sx○ Sy−1○ Sx   =   Sx ₪ Sy ,
dividing by the involutive inversion Se.
More general powers are given in a Jordan algebra with unity element, which is ( not as­so­cia­tive ) but po­wer-as­sociative, in the ordinary way: De­fi­ning
    x0 = e , x1 = x , xn+2 = Ƥ(x) xn , x−n = e ₪ xn
they coincide with -powers: Using induction
    Ƥ(xn) = Ƥ(x)n , xm ₪ xn = x2m−n , (xm)n = xm n ,
where 2m−n is the symmetric multiplication of an additive abelian group. Herein the first equa­tion gives the pos­si­bility to define an exponential series like in associative algebras with
Ƥ( ex )  =  exp ( Ƥ(x) ) .
This exponential series maps vectors into vectors in the set of invertible elements. One expects that ex­po­nen­tials ‌-‌ge­ne­rate the con­nec­tivity component of e. The exponential series allows the de­ter­mination of an in­fi­ni­tesimal transformtion
    𝕝ieS : x  ι𝕝ieSx  ,    𝕝ieS  : 𝕍  →  𝕞ap(𝕍, ▣ )
of the -left multiplication Sx if we expand this nonlinear transformation as
    Se+τ x+⋅⋅⋅(u)   =   Se(u)  +  τ 𝕝ieSx(u)  + ⋅⋅⋅
with  eτ x = e + τ x + terms with higher powers of τ and collect the terms which are linear in τ to get
u  ι𝕝ieSx(u)  =  ( <e,u>x + <x,u>e − <e,x>u )
with the bracket the linear transformation  2 o+x,e<x,e> id  on  𝕍  with trace  (1−n)<x,e>  and a di­la­ta­tion.
The in­fi­ni­te­si­mal transformation of the quadratic representation follows by right-composing with Se, i.e. by applying the infinitesimal left multiplication to the inverse, to give
u  ι( 𝕝ieƤ(x) ) u  =  (𝕝ieSx○ Se) u  =  2 L(x) u ,
i.e. the infinitesimal quadratic representation of a Jordan algebra is the left multiplication.
In this example it becomes after a verification
u  ι( < t ,u>x + <x, t >u − <u,x> t )  =  2 L(x) u .
Thus commutators of infinitesimal left multiplications generate the Lie algebra of the in­fi­ni­te­si­mal dis­place­ments, but themselves they admit no simple composition law in the general case.
Expanding in (o₪) both va­ri­ab­les x,y as exponential series [ Loo p 95] and collecting the terms linear in the sca­lar pa­ra­me­ter of this se­ries, we get the ad­di­tive abe­lian sym­me­tric space composition a

(₪a)u ₪a v  = 2 u − v  ,

which is abelian in the sense of symmetric spaces [Loo p 134]. These straight­for­ward cal­cu­la­tions al­so work in the more general complex unitary case.
 Is there on the null- or light-cone, taken as underlying manifold of dimension n−1, in­stead of the n-di­men­sio­nal space of invertible elements, a si­mi­lar sym­me­tric space composition ? 


are vectors of  𝕍 

l even in the
null-cone  ⛌( , ) ,

wherein we frequently
drop arguments,

capital letters  A,B,E,D ...
are reserved for
linear transformations
on  𝕍  and

non-linear transformations
we use
script letters

those are

Jordan algebras
in the box

symmetric spaces
quadratic representation
to the
adjoint representation
associative and Lie
i.e. to 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

are there
similar constructions
 < t , t >  =  0

giving rise to
structure theory
massless particles
photons, gravitons, ...
Null- or Light-ConesBoth subsets and inherit their topological structure from that one of 𝕍, but this must be made com­pa­tib­le with <,>, which is standard only in the positive-definite case. The more interesting and un­der­in­ve­sti­ga­ted second subset is a cone of di­men­sion n−1, but not convex. The first one is an open sub­set, hence of di­men­sion n. If <,> is positive‌-‌definite, the null-cone becomes the zero vec­tor 0 only, and the first one a punc­tur­ed plane. For indefinite <,> both are non‌-‌com­pact. The null-cone has one connectivity com­po­nent, its com­ple­men­tary set  outside the null-cone  has two, three or four con­nec­ti­vi­ty com­po­nents, de­pen­ding on di­men­sion and signature.
 The invertibility of reflections results in the

 Geometry of Null-Cones   

Two non-vanishing vectors Ĭ and Ĭ' in the null-cone
can be reflected into each other by the reflection    J Ĭ + Ĭ'
if they are not <,>-perpendicular.
Reflections hence act transitively on null-cones and
     ᦔ : t    ι ᦔ t      ,       ᦔ  : (VV,<,>) ᦔ
defines a bijection of the symmetric space of the vec-
tors outside the null-cone into the symmetric space
of reflections on .
For the proof note that null-cones neither are subspaces nor convex This makes both disjoint sets re­pre­sen­tation spaces of the symmetric space of all invertible elements of our Jordan algebra, both not vector spaces.
But note – this does not make the null-cone itself a symmetric space, which remains an open question.
 For Ĭ on the null‌-‌cone we get a non-linear map
SĬ  : y  ι 2
Ĭ ,SĬ  : (𝕍,<,>)  →  (𝕍,<,>)
from outside onto the null-cone, which is not defined only on the null-cone itself. 
how to find
<>-compatible topoloy

belong to the
symmetric spaces
The real high-dimensional vector space  curv(𝕍,<,>)  of  pseu­do-Rie­man­nian cur­va­ture struc­tur­es is defined by bi­li­near maps
    R : 𝕍 × 𝕍𝕖nd 𝕍
subject for any  x,y,v,z ∈ 𝕍 to Singer & Thorpe's three axioms [ S&T]
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  (𝕍,<,>), 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 ½ Ro(x,y) = oy,x, 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
Ricci form ρR(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 𝕍, the
Ricci transformation LR  from R<LRx,y> = ρR(x,y)
and a linear form in the dual space 𝕍*, the
curvature scalarSc(R) = trace LR  .
In addition we need an endomorphism, the
Ricci mapRic : R  ι RoLR,id𝕍 = Ω(LR)  ,   Ric : curv(𝕍,<,>)  → curv(𝕍,<,>)
with an injective linear
Ricci Jordan mapΩ : A ι Ω(A) = RoA,id𝕍 ,   Ω : 𝕠+(𝕍,<,>)  → curv(𝕍,<,>) ,
especially  Ω(id𝕍) = Ro , and the non-linear
and the<R(x,y) y,x>
sectional curvature sec(R)  = 
   ,    sec : curv(𝕍,<,>)  →   .
<Ro(x,y) y,x>
Moreover there are two projectors on  curv(𝕍,<,>) , 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 [ S&T]
Singer & Thorpe's Structure Theorem of Curvature Spaces 
R ∈ 𝕚mage Ш <=> the Ricciform ρR vanishes,
 i.e. ρШR = 0 , LШR= 0 , Sc(ШR) = 0
R ∈  Ro <=> the sectional curvature is constant
R ∈  Ro ⊞   𝕚mage Ш <=> LR is a multiple of the identity)
R ∈ 𝕚mage(ℇ−Ш)  ⊞   𝕚mage Ш <=> the scalar curvature of R vanishes ,
wherefrom we get the direct vector space decomposition of type  𝕜ern Ш ⊞ 𝕚mage Ш  of
    curv(VV,<,>)  =    Ro   ⊞   𝕚mage(ℇ−Ш)   ⊞   𝕚mage Ш  ,
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 ∈ 𝕍 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(ᦔt) .
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> id𝕍 − 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  𝕍  by
RoA,B(x,y)  = ½ ( Ro(Ax,By) + Ro(Bx,Ay) ) ,
defining elements RoA,B in the curvature space. The special case  B = id𝕍  of 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 𝕚mage Ш by this 2nd class of examples. In fact  Ω  is a li­near bi­jec­tion
Ω  :  𝕠+(VV,<,>)→ 𝕜ern Ш  =   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 𝕚mage Ш  remains Nomizu's construction of
 the 3rd class of curvature structures ИE,D, defined as
ИE,D(x,y)= Ro E,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  (𝕍,<,>)  around the diagrams was done in [T78], where even dilatations
    x  ι➥  λ x  ,  𝕍𝕍
are included to a larger linear transformation group  𝔾  of 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  𝔾  is 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  𝕜ern Ω  it is traced back to the clas­si­fi­ca­tion of  𝔾-or­bits in  𝕠+(,), one of the clas­si­cal real sim­ple Jor­dan al­ge­bras of en­do­mor­phisms, but in­side  𝕚mage Ш  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 ,
chasing 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
id𝕍 =  ⓖ TR
with the gravitational constant and  T ∈ 𝕠+(𝕍,<,>) 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 move­ment 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
id𝕍 =  ⓖ 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 
id𝕍 ) ,
which in some cases is easier to handle.

n = 4
the signature
is the special case
physical relativity
Gravitational waves, first postulated by Henri Poincaré 1905 in the context of special relativity [ Poi p 1507], and described mathematically by André Lichnérowics [ Lic p 45], are defined for any  x,y,z ∈ 𝕍  in terms of a (necessarily) light‌-‌li­ke ( for in­de­fi­nite <,> this means on the null‌-‌space ) vec­tor  l ∈ 𝕍 with
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&thinsnp;, 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) 𝕚mage Ш ) , the 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 ½ n(n+1)-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 ∈ 𝕍
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 p 26] 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 p 26] 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 ). How­ever, 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  𝕍  above, 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. 𝔾  is the struc­ture group of these bund­les.
A cosmological model is a space-time, whose curvature sections lie in one  𝔾-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 bet­ter than the flat 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.
can actual space-time
even be
better approximated
by using
third order derivatives
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 Background 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  Above only the first volume is used. We generalize his symme­tric mul­ti­pli­ca­tion from hyperboloids (spheres) to arbitrary elements outside null-cones, this being well-known to the Artin school, and this in turn to the complex case of pseudo-Hilbert spaces. One of the main points of Loos is that instead of studying  CC*-algebras in quan­tum me­cha­nics, one should study al­ge­bras with an in­volu­tion ( in­stead of the *-anti-involution).
[L&N]O Loos, E Neher  Reflection Systems and Partial Root Systems  Forum Math 23 [2011] p 349-411  see also
O Loos, E Neher  Locally Finite Root Sys­tems   Mem Am Math Soc 171 n°811 [2004] p 1-214
[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
[Poi]H Poincaré  Sur la Dynamique de l'Électron  Comptes Rendus de l'Académie de France  [1905] p 1504-1508
[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.
[S&T]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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