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first published 24. May 2002 revised upload |
| Spaces of Endomorphisms on Pseudo-Orthogonal 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
There is a direct and <,>-orthogonal decomposition ⊞ of this Jordan algebra, given by
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 here in ipmWeylP. The pseudo-orthogonal vector space (𝕍,<,>) decomposes into the disjoint union of the two subsets
| ℝ are the real numbers  these are the classical categories on pseudo-orthogonal vector spaces throughout this article <,>-skew-adjoint means <Ex,y> = − <x,Ey> for any vectors x, y , endomorphisms A, E and <,>-adjoint means <Ax,y> = <x,Ay> | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Pseudo- Orthogonal Involutions and Reflections | For a number ઠ from the ground-field and a given vector t outside the null-cone define the linear transformation ઠᦔt on this vector space by
Structure Theorem of Involutions and Reflections
As an involution ᦔt has the associate involution − ᦔt and hence the projector ⼫t
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 pseudo-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 ⸻mmmx,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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Algebra and Geometry of Pseudo- Orthogonal Vector Spaces | A simple and fundamental algebra structure on such a pseudo-orthogonal vector space is the Jordan algebra composition
Contrary to Lie algebras the left multiplications L▣(x) are not derivations ( or anti-derivations ). But sums of their commutators L▣(x)L▣(y) − L▣(y)L▣(x)
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
which shows that the multiplication ₪ is not commutative. Unlike in associative categories 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 ※.
The left multiplication Sx in a symmetric space is given by
 ( especially Se(y) = y−1 ) .
Compatability of <,> and ₪ reads
Loos has shown in his second chapter, that defining for any Jordan algebra Since the Sx are homogeneous of degree −1 in the 2nd variable these Ƥ(x)-transformations turn out to be linear with
The 2nd symmetric space axiom identifies the neutral displacement by
More general powers are given in a Jordan algebra with unity element, which is ( not associative ) but power-associative, in the ordinary way: Defining
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 In this example it becomes after a verification
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) 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 ? | typically s,t,u,v,w,x,y,z 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 for non-linear transformations we use script letters those are non-classical categories
For symmetric spaces the quadratic representation corresponds to the adjoint representation of associative and Lie structures ! i.e. to a map Ad : ※ → 𝔸ut (※,₪) subject to Ad(x ₪ y) = Ad(x) ₪ Ad(y) = Ad(x) ○ Ad(y)−1○ Ad(x) aredisplacements more useful in relativity and quantum mechanics than translations ? are theresimilar constructions for < t , t > = 0 ? giving rise to a structure theory of massless particles like photons, gravitons, ... | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Null- or Light-Cones | Both 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
But note – For Ĭ on the null-cone we get a non-linear mapthis does not make the null-cone itself a symmetric space, which remains an open question. If we could define a symmetric space multiplication thereon, a representation of it should lead to a classification of interactions ❗
| how to find a <>-compatible topoloy ? reflections belong to the category of symmetric spaces | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Curvature on Space-Time | The real high-dimensional vector space curv(𝕍,<,>) of pseudo-Riemannian curvature structures is defined by bilinear mapsR : 𝕍 × 𝕍 → 𝕖nd 𝕍
| 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: The
Ш ○ Ω = 0 , Singer & Thorpe's Structure Theorem of Curvature Spaces
The 1st class of more general examples is given for any t ∈ 𝕍 by
◆ ρRt,s(x,y) = − (n−2) ( <t,s><x,y> − <t,x><s,y> − <t,y><s,x> ) ,
In [T84] there is a more detailed well-known construction of curvature structures from semi-simple Lie- and Jordan algebras and Lie triples.◆ LRt,s = − (n−2) ( <t,s> id𝕍 − o+t,s) ,
◆ Sc(Rt,s) = − (n−1)(n−2) <t,s> .
The 2nd class is given for any pair A,B of <,>-self-adjoint endomorphims on 𝕍 by
◆ ρRA,B(x,y) = ... ,
( reproducing the canonical bilinear form ≪,≫ of endomorphisms ) and especially◆ LRoA,B = −½ ( AB+BA−trace(A) B−trace(B) A ) ,
◆ Sc(RoA,B) = − ( trace (AB) − trace(A) trace(B) )
◆ ρΩ(A)(x,y) = ... , ... ,
As only chance to look inside 𝕚mage Ш remains Nomizu's construction of
◆ Ω(A) = − ( A−trace(A) idVV), Ω(idVV) = (n−1) idVV ,
◆ Sc(Ω(A)) = (n−1) trace(A) , Sc(Ro) = n(n−1) .
the 3rd class of curvature structures ИE,D , defined as
◆ ρИE,D(x,y) = 3/2 ( <Ex,Dy>+ <Dx,Ey>) ,
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◆ LИE,D= 3/4 ( ED+DE ) ,◆ Sc(ИE,D) = − 3 trace(ED) .
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
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 curvature structure theory - a great step for the mankind up from linear to multilinear algebra
great mathematical achievement, well-known as quantum mechanics' statistical operators but never used in gravity and the symplectic phase-space formulation of classical mechanics why? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| In- and Covariance Groups | Chasing the pseudo-orthogonal groups and Lie algebras on (𝕍,<,>) around the diagrams was done in [T78], where even dilatations
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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Einstein's Gravitational Field Equations | Einsteins gravitational field equations without cosmological term were given in our notation in [ T78 p 1123 ] and [T84] as
| n = 4 and the signature +,−,−,− is the special case of physical relativity | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Gravitational and Electromagnetic Radiation | Gravitational waves, first postulated by Henri Poincaré 1905 in the context of special relativity [ Po i p 1507 ], and described mathematically by André Lichnérowics [ Li c 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
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' 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 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 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 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 statistics. | to get rid of Äther, reformulate Maxwell's equations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Petrov's Classification | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 <,>-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 [ Li c ] 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
| must have a better solution el.mag. field ⊑ grav.field ⊔ ⊔ el.mag. wave ⊑ grav.wave | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Post's Gravity- Contribution to Electromagnetic Intensities | [ 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 coadjoint ad-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 drastic physical effects in regions of strong gravitation | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Cosmological Models | 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 quantization | 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. |