
Algebra and Geometry
of
Pseudo Orthogonal Vector Spaces  Given a real pseudoorthogonal vector space (VV,<,>)of dimension n as in [T78] and [T84], there are two subsetsin 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 positivedefinite case. The more interesting and underinvestigated first one is a cone of dimension n−1, but not convex, say nullcone, the second one is an open subset, hence of dimension n. If <,> is positivedefinite, the nullcone becomes the zero vector 0 only, and the second one a punctured plane. For indefinite <,> both are noncompact and may have different connectivity components.
A simple algebra structure on such a pseudoorthogonal vector space is the Jordan algebra compositionfor 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 nontrivial ( if so nilpotent ) Lie algebra? And what for symplectic vector spaces? For t ∈ ※(VV,<,>)there is the neutral elementof this Jordan algebra (VV, ▣t )with inverseSince Jordan algebras are ( not associative but ) powerassociative 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 ₪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 byBecause of the 3rd symmetric space axiom it is an ( socalled 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 nullcone such that its inner automorphism is the identity map on ※. Compatability of <,> and ₪ reads
  x   y 
<Su(x) , Su(y) > = <u,u>²  < 
 , 
 >  , 
  <x,x>   <y,y> 
i.e. we get a nonlinear transformation of conformal type outside the nullcone and for l inside the nullcone a nonlinear map
 <l,y> 
Sl : y ι➥ 2 
 l  ,  Sl : ※(VV,<,>) → ╳(VV,<,>) 
 <y,y> 
from outside into the nullcone, which is not defined only on the nullcone. 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( 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 nullcone ) 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
 <u,u> 
<Ƥ(u) x , Ƥ(u) y> = ( 
 )² <x,y> 
 <e,e> 
describing the compatibility of <,> and displacements as a conformal property. The 2nd symmetric space axiom identifies the neutral displacement by Does for the first subset, the nullcone ╳(VV,<,>)exist a similiar symmetric space multiplication, perhaps choosing the given vector t inside the nullcone?  typically s,t,u,v,w,x,y,z are vectors of VV
l even in the nullcone ╳( , )
wherein we frequently drop arguments
capital letters A,B, ... are reserved for linear transformations on VV
for nonlinear transformations we use script letters
those are nonclassical categories
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)
displacements more useful in
relativity and quantum mechanics
than translations? 
Spaces of Endomorphisms on PseudoOrthogonal Vector Spaces  Up to now all categories were subsets of the pseudoorthogonal vector space. Now we turn to spaces of endomorphisms on VV. The automorphism or pseudoorthogonal group of the pseudoothogonal vector space isits derivation or pseudoorthogonal Lie algebra ( with respect to the commutator ) isThey 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 isfor u ∈ VV,giving the typical commutation relations in a basisfree form  note that these also can be found for 2nd power polynomials in its Clifford algebra. Its antiderivation or pseudoorthogonal Jordan algebra ( with respect to the anticommutator ) iswith the typical, generating linearly elementAgain, 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 aswhich is the special case s= ± t of the more general <,>selfadjoint, but in general not pseudoorthogonal endomorphism
 2 
Jt,s =^{+} idVV− 
 o^{+}t,s . 
 <t,s> 
There is a direct and <,>orthogonal decomposition ⊞ of this Jordan algebra, given by
 1   1 
A = 
 trace(A) id_{VV} ⊞ ( A − 
 trace(A) id_{VV}) , 
 n   n 
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 thereupon
throughout this article <,>skewadjoint means
<Ex,y> = − <x,Ey>
for any vectors x, y and endomorphisms A, E and
<,>adjoint
means
<Ax,y> = <x,Ay> 
Curvature on SpaceTime  The real highdimensional vector space curv(VV,<,>)of pseudoRiemannian curvature structures is defined by bilinear mapssubject for any x,y,v,z ∈ VVto Singer & Thorpe's three axioms [ST]on a real pseudoorthogonal 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 thewhich defines a 1dimensional subspace in the curvature space, called the space of scalar curvature. It will be used below to give three classes of examples. Note that R_{o}(x,y) = 2 o_{x,y} , but the Bianchi identity does not reduce to the Jacobi identity for R_{o}.
 this axiomatic has been overlooked in physics 
The Structure Theory of Curvature  Mathematical data, in terms of which the structure theory of curvature spaces can be formulated, are constructed in terms of basic linear algebraic structures. Theis a symmetric bilinear form, which may be used to define R semisimple if it is nondegenerate and compact if it is positive or negativedefinite. Since <,> is nondegenerate Witt's theorem gives a unique ( necessarily <>selfadjoint ) endomorphism on VV ,theand a linear form in the dual space VV*,theIn addition we need the endomorphismwith an injective linearespecially Ω(id_{VV})= R_{o} , and the nonlinearMoreover there are two projectors on curv(VV,<,>),thewhere idempotence and moreover commutatability are easily verifiedsee [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 wherefrom we get the direct vector space decomposition of type kernШ ⊞ imageШ ofuniquely 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,<,>)bywhich, as was shown in [T84], comes from a Jordan algebra, in fact that on a real pseudoorthogonal vector space, which was studied elsewhere on this webpage and is basic for the quantization of curved spacetimes. It is constructed for the Jordan composition ▣t asSince it is quadratic in t polarization t ι➥t+s gives even more general curvature structuresNote that since the last term herein is a curvature structure the ½( )sum also is one. ObviouslyBecause the last equation is square in t it can be polarized t ι➥t+s towhich 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,smodifications of R_{o} areIn [T84] there is a more detailed wellknown construction of curvature structures from semisimple Lie and Jordan algebras and Lie triples. The 2nd class is given for any pair A,B of <,>selfadjoint endomorphims on VVbydefining elements R_{o}^{A,B} in the curvature space. The special case B = id_{VV}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,and for C = L_{RoA,B}with a sorrow consequence: No insight into imageШ by this 2nd class of examples. In fact Ω is a linear bijectionwhich carries over any structure from the ( Jordan algebra of ) selfadjoint transformations into the curvature space. The full diagram of short exact sequences was given in [T78 p 1120]. The data of this A,Bmodification of R_{o} are( reproducing the canonical bilinear form ≪,≫ of endomorphisms ) and especiallyAs only chance to look inside imageШ remains Nomizu's construction of
the 3rd class of curvature structures И_{E,D} , defined asin terms of two <,>skewadjoint endomorphisms E and D, similar but not identical to the 2nd class construction of curvature in terms of two <,>selfadjoint endomorphisms. We call curvature structures of this type em. The data of this E,Dmodification of R_{o} areThere 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 energymomentum 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,Dwhere the anticommutator of two <,>skewadjoint endomorphisms is <,>selfadjoint.
A 4th ,mixed' class of curvature structures is given by one <,>selfadjoint A and one <,>−skewadjoint E. If they are supposed to be linear in their two parameters, then [T84 p 14] there are only the two caseswhere • is the coadjoint action by commutators, and they lie in the first two subspaces of the above curvature space decomposition.
Note that these curvature structures must be linearcombinations of elements of the pseudoorthogonal 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 curvature structure theory  a great step for the mankind up from linear to multilinear algebraprojectors are a great mathematical achievement,
wellknown as quantum mechanics' statistical operators
but never used in gravity and the symplectic phasespace formulation of classical mechanics
why? 
In and Covariance Groups  Chasing the pseudoorthogonal groups and Lie algebras on (VV,<,>)around was done in [T78], where even dilatationsare included to a larger linear transformation group GGof dimension 1+½ n(n−1), called the linear conformal or sometimes Weyl group of the pseudoorthogonal 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 wellknown Ad, ad way. Blowing up the pseudoorthogonal 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 nonlinear transformations, in this special symmetric space in square ones in the left variable. The direct decomposition of the curvature space commutes with these Ad, adactions, 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 GGorbits 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 
Einstein's Gravitational Field Equations  Einsteins gravitational field equations without cosmological term were given in our notation in [T78 p 1123] and [T84] as
 Sc(n) 
L_{R} − 2 
 id_{VV}  =  ⓖ T 
 n 
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 formone, which corresponds to lowring or highring indices in the index notation. Also from Sc(R) = − ⓖ trace T_{R} which in some cases is easier to handle.  n = 4 and the the signature +,−,−,− is the special case of general relativity 
Gravitational and Electromagnetic Radiation  Gravitational waves [Lic p 45] are defined for any x,y,z ∈ VVin terms of a (necessarily) lightlike ( for indefinite <,> this means on the nullspace ) vector l ∈ VVwithIf this is the case R is called a gravitational wave and l its wave vector. This is an orbit property with respect to the pseudoorthogonal group, even with respect to its Weyl group. For R a gravitational wave with respect to the wave vector l, the main theorem
 ρ_{R}(x,y) = τ <x,l><y,l> for some real τ, i.e. ρ_{R} is degenerate and R not semisimple,
 L_{R} = τ o^{+}(l,l),
 Sc(R) = 0 ( i.e. R ∈ image(ℇ−Ш) ⊞ imageШ ), trivial curvatures of ℝR_{o} cannot be waves,
 l is an eigenvector of eigenvalue 0 for L_{R}
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 <,>selfadjoint endomorphims, gravitational waves therein are parametrized by lightlike 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. wavelength, 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' means curvature structure
in the following 
Classification Still Unsolved  But which Weyl curvatures are determined by wave vectors? Since the (full) Weyl group acts transitively on the nullspace, 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)(n3)/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 ) <,>selfadjoint 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 timelike or spacelike 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 a further decomposition of the space of Weyl curvatures ? 
Categories Matter!  Electromagnetic orbits are given by two physical tensor fields, the field strengths E ( a 3vector E together with a skew 3×3 matrix H ) and the intensities D ( a 3vector 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 positivedefinite metric in spacetime 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 <,>skewadjoint! 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, wellknown in physics from Fermi statistics.  to get rid of Äther, reformulate Maxwell's equations 
Petrov's Classification  Petrov's classification by bivectors is one in the n(n+1)/2dimensional Jordan algebra of <,>selfadjoint 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 eigenvaluebased curvature classification is one by orbits of the Weyl group.  has to be generalized to the whole curvature space 
Embedding Electromagnetic Fields 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 <,>skewadjoint 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) lightlike vector l such that for any x,y,z ∈ VVand the same equations for the intensity D. Clearly symmetrization is equivalent toCertainly for most physical media D = ε E and especially for the vacuum physisists write ε_{o} for this constant. That it is not an universal constant, but depends on the curvature at any position in spacetime follows from  must have a better solution
el.mag. field ⊑ grav.field ⊔ ⊔ el.mag. wave ⊑ grav.wave 
Post's Gravity Contribution to Electromagnetic Intensities  Post [Pos] remarks that if E and D are linearly connected ( like in most media on earth ) the 4tensor 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 <,>selfadjoint A ( Post has the identity ) and a <,>skewadjoint electromagnetic field E, which, when inserted into the given curvature, delivers the <,>skewadjoint intensity D. In [T84] it was shown that for this ,mixed' case, there are only solutions in the space of nonWeyl 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 coadjoint adaction by inner derivations [T84]. So the only nontrivial embedding of electromagnetics into gravity remains Nomizu's! But this only is the case because all three constructions, that by two <,>selfadjoint or that by two −skewadjoint endomorphisms or the mixed one, start from the trivial 1dimensional 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 <,>skewadjoint B(x,y) can't be ) and the Bianchiidentity ( 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 pseudoorthogonal 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 spacetime 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 curvaturemodified 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 spacetime huge ) number, the curvature radius of that position.  B(x,y) = R(x,Ey)
for any vectors x, y of Minkowski space can have drastic physical effects in regions of strong gravitation 
Cosmological Models  Physical spacetime is not a vector space as in this study sofar, but a pseudoRiemannian manifold of dimension 4 and signature +,−,−,−. <,> becomes the eigentime, 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 spacetime, giving rise to a LeviCivita connection, which in turn gives rise to a curvature structure. GGis the structure group of these bundles.
A cosmological model is a spacetime, 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 spacetime is not a cosmological model, but in each point of spacetime there is a cosmological model, which approximates actual spacetime 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.  
