| Classify Lorentz × dilatation- (i.e. Weyl-) group orbits in the real high-dimensional vector space curv(VV,<,>)of pseudo-Riemannian curvature structures, i.e. of bilinear mapssubject for any x,y,v,z ∈ VVto Singer & Thorpe's three axioms [ST]|
on a real pseudo-orthogonal vector space ( VV,<,>),especially, for the physical case of Minkowski space, those of dimension 4 [Nom], [T84] ( and many more publications ). There always is the
|skew symmetry||R(y,x) = − R(x,y) ,|
| pseudo-orthogonal||derivations|| <||R(x,y) u , v > + < u , R(x,y) v >|| = 0 ,|
|Bianchi-identity||R(x,y) z + R(z,x) y + R(y,z) x|| = 0|
which defines a 1-dimensional subspace in the curvature space, called the space of scalar curvatures, and gives rise to a huge amount of more examples: Indeed for any pair A,B of <,>-self-adjoint endomorphims of VV
|trivial curvature structureo|| Ro(y,x) z = <y,z> x − <x,z> y ,|
define elements RoA,B in the curvature space. In [T78] and [T84] Weyl group actions were chased around decomposition and diagrams of Singer & Thorpe's space of these curvature structures.
|RoA,B(x,y) = ½ ( Ro(Ax,By) + Ro(Bx,Ay) )||Singer & Thorpe|
a great step
from linear to
| Gravitational waves [Lic p 45] are defined for any x,y,z ∈ VVin terms of a (necessarily) light-like ( for indefinite <,> this means on the null-space ) vector l ∈ VVwith |
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. Candidates of 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? ).
|annihilation||R(x,y) l = 0 ,|
|symmetrization||<l,x> R(y,z) + <l,y> R(z,x) + <l,z> R(x,y) = 0 .|
Mathematically on a Pseudo-Riemannian manifold there is a Levi-Civita connection which gives rise to a curvature. It remains to show, that a wave curvature lies the the subspace of Weyl curvatures only(?). The other curvatures, are parametrized by self-adjoint ( with respect to the Minkowski form ) transformations. Since their Jordan algebra is simple, there can be no such light-like vectors at all!
in the following
| But which Weyl curvatures are determined by wave vectors? Since the (full) Weyl group acts transitively on the null-cone, 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, which then would exhaust the space of Weyl curvatures, which 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.|
A structure theory usually means that there is a decomposition into ( the direct sum of ) subspaces in terms of projectors, like 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.
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 endomorphisms below. There is such a construction by using tensor products of such a vector, to get an endomorphism and insert it into the three constructions below. In the 'mixed' case - the electromagnetic one - we cannot arrive by this construction in the space of Weyl curvatures. In the other two cases we have to find 'odd' constructions in order to arrive only in the space of Weyl curvatures.
|projectors are a|
but never used
and the symplectic
| 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 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|
| Petrov's classification by bi-vectors is one in the n(n+1)/2-dimensional Jordan algebra of <,>-self-adjoint endomorphisms, since those are generated linearly by pairs of vectors of VV.It is mapped [T84] by the linear map Ω onto the cosmological subspace of the curvature space. Since this Jordan algebra is one of the classical real simple ones, this classification is given by Dynkin diagrams. However, the complementary subspace of the Weyl curvature structures is not arrived at by such an easy map.|
Clearly Petrov's eigenvalue-based curvature classification is one by orbits of the Weyl group.
|has to be generalized|
to the whole
| Implementing electromagnetics into general relativity usually is done by inserting the 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.|
Nomizu [Nom] has given a direct method to construct a curvature from two <,>-skew-adjoint endomorphisms ( take electromagnetic E and B ) similar to the construction of curvature in terms of two <,>-self-adjoint endomorphisms in Singer & Thorpe's structure theory of curvature:
defines elements in the curvature space for any two <,>-skew endomorphisms E,B. There remains to show that Nomizu's construction leads from a physical electromagnetic field, given by field strength E and intensity B to a gravitational field, solving Einsteins field equations in terms of a (which?) electromagnetic energy-momentum tensor.
|RoE,B(x,y) = RoE,B(x,y) − <Ex,y>B − <Bx,y>E|
|ЍE,B(x,y) = ½ ( Ro(Ex,By) + Ro(Bx,Ey) ) − <Ex,y>B − <Bx,y>E|
Post [Pos] remarks that if E and B are linearly connected ( like in most media ) 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 B. In [T84] it was shown that for this ,mixed' case, there are only solutions in the space of non-Weyl curvatures, none in the space of Weyl ones, and if A especially is the identity, even these two vanish. In fact, the only possible curvatures constructed from such A's and E's are of the form E•A and the trace thereof, where the • is the natural ( Lie- on Jordan ) algebraical 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.
|throughout this article|
<Ex,y> = − <x,Ey>
for any vectors x, y
and endomorphisms A, E
<Ax,y> = <x,Ay>
| 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 two mathematical approaches from the trivial curvature, but this time more general from the given 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 ) and the Bianchi-identity ( that's why they are not interesting for mathematics ). But it is clear from the construction that the third still holds, i.e. the outcome B is in the pseudo-orthogonal Lie algebra and hence may be interpreted as an electromagnetic field intensity. This has a physical consequence: The full gravitational field ( not only its scalar part, i.e. the trivial curvature ) contributes to electromagnetic field intensities. Since there is no point in space-time without gravitation this contribution never vanishes. Half way between two adjacent galaxies, this contribution is neglectible, but near the center of a galaxy, near the event horizon of a black hole, or for a strong gravitational wave, 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 the usual one.||B(x,y) = R(x,Ey)|
vectors x, y of
| || |