Curvature Structures and Gravitational Waves ϟ Incepta Physica Mathematica

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
Classify Lorentz × dilatation- (i.e. Weyl-) group orbits in the high-di­men­sio­nal vec­tor spa­ce of Pseu­do-Rie­man­nian curva­ture struc­tures on a pseu­do-ortho­gonal vec­tor space, es­pe­ci­al­ly ( for the phys­ical case of 4-di­men­sional Min­kow­ski space ) those of dimension 4. In
  • H Tilgner   The Group Structure of Pseudo-Riemannian Curvature Struc­tur­es  J. Math. Phys. 19 [1978] p 1118-1125
  • H Tilgner   Conformal Orbits of Electromagnetic Pseudo-Rie­man­nian Cur­va­ture Ten­sors – Elec­tro­mag­ne­tic Implies Gra­vi­ta­tion­al Rad­ia­tion   Lec­ture Notes in Mathe­ma-
    tics 1156 [1984] p 317-339
Weyl group actions were chased around decomposition and dia­grams of Singer & Thor­pe's (linear) space of all curvature structures.
Singer & Thorpe
theory -
a great step
for the
from linear to
multilinear algebra
Gravitational waves are defined in terms of a light-like vec­tor [Lic], an­ni­hi­lated by the wave curvature structure ( lets drop the ter­minus struc­ture in the fol­low­ing ). This is an orbit pro­per­ty with res­pect to the pseu­do-or­tho­gonal group, even its Weyl group. Can­di­dates of gra­vi­ta­tion­al wave or­bits are those special wave or­bits which are not of elec­tro­mag­ne­tic ori­gin – these are not strong enough to be measured and they can't be ,seen'. It should be possib­le to derive the freq­uen­cy for every such pure gravita­tion­al wave orbit, in order to get an ex­pe­ri­men­tal veri­fi­ca­tion of gra­vi­ta­tion­al waves.
 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!
curvature structure

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 ones. 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
great mathematical

well-known as
quantum mechanics'
statistical operators

but never used

in gravity

and the symplectic
formulation of
classical mechanics

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 with respect to the Min­kowski form! 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 get rid of
Petrov's classification by bi-vec­tors is one in the n(n-1)/2-dimensional Lie alge­bra of 2nd po­wer ele­ments in this Clif­ford algebra, which if for­mula­ted basis-free gives a very ele­gant for­mu­la­tion of the com­muta­tion re­lations of pseu­do-ortho­gonal Lie al­ge­bras. There­fore his eigen­val­ue-based cur­va­ture clas­si­fi­ca­tion is one by or­bits of the above Weyl group in this Lie al­gebra. Which of Petrov's or­bits lie in the space of Weyl cur­va­ture and which not? 
into Gravity
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 original structure theory of cur­va­ture. There remains to show that Nomizu's con­struction leads to a phy­si­cal elec­tro­mag­ne­tic field, sol­ving Ein­steins field equa­tions in terms of a (which?) elec­tro­mag­ne­tic ener­gy-mo­men­tum ten­sor.
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 elec­tro­magnetic field E, which, when in­sert­ed in­to the given cur­va­ture, de­livers the skew-adjoint ( with res­pect to the Min­kow­ski form ) intensity B. In [♦ p 14] 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 adjointad-action by inner deri­va­tions [♦]. 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 scalar 1-di­men­sional curva­ture.
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)

for all
vectors x,y
Minkowski space
Literature with comments
[Lic]A Lichnérowics  Ondes et Radiations Électromagnétique et Gravitationelles en Relativité Générale  Annali di Mate­ma­ti­ca Pur. & App. 4 [1960] p 1-95 is a very elegant representation of the concept, although written in the old-fashioned index notation.
[Nom]K Nomizu  The Decomposition of Generalized Curvature Tensor Fields  p 335-345 in  Differential Geometry, Papers in Ho­nor of Ken­ta­ro Yano  Kinukuniya, Tokyo [1972] no ISBN
[Pos]E J Post  Formal Structure of Electromagnetics  North Holland, Amsterdam [1962] no ISBN is an inspiring book, though written in an old-fashi­oned way with indices: The electromagnetic constituency 4-tensor between the 2-tensors (4×4 matrices) of field-strengths and -in­ten­si­ties has the same symmetries as a curvature tensor.
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