Curvature Structures and Gravitational Waves ϟ Incepta Physica Mathematica

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

Classify Lorentz × dilatation- (i.e. Weyl-) group orbits in the real high-di­men­sio­nal vec­tor spa­ce curv(VV,<,>)of  pseu­do-Rie­man­nian curva­ture struc­tures, i.e. a bilinear map
    R : VV ×VV→ end VV
subject for any  x,y,v,z ∈ VVto Singer & Thorpe's three axioms [ST]
skew symmetryR(y,x) = − R(x,y) ,
 pseudo-orthogonalderivations <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 ( VV,<,>),es­pe­ci­al­ly, for the phys­ical case of Min­kow­ski space, those of di­men­sion 4 [Nom], [T84] ( and many more publications ). There always is the
trivial curvature structureo Ro(y,x) z = <y,z> x − <x,z> y ,
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
RA,B(x,y) = ½ ( Ro(Ax,By) + Ro(Bx,Ay) )
define elements RA,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.
Singer & Thorpe
theory -
a great step
for the
from linear to
multilinear algebra
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
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. 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'. To ex­pe­ri­men­tal­ly veri­fy the existence of gra­vi­ta­tion­al waves it is necessary to get an idea of the freq­uen­cy for every such pure ( i.e. not given by an elec­tro­mag­netic wave ) gra­vi­ta­tion­al wa­ve orbit ( but how? ).
 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 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
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 sta­ 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
curvature space
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 structure theory of cur­va­ture:
NomE,B(x,y)  =  RE,B(x,y) − <Ex,y>B − <Bx,y>E
NomE,B(x,y)  =  ½ ( Ro(Ex,By) + Ro(Bx,Ey) ) − <Ex,y>B − <Bx,y>E
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.
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 ( with res­pect to the Min­kow­ski form ) skew‌-‌ad­joint intensity B. In [T78] and [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 ad­jointad‌-‌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.
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)

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 Pura & Appl. 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.
[ST]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 Pseudo-Rie­man­nian Cur­va­ture Ten­sors – Elec­tro­mag­ne­tic Implies Gra­vi­ta­tio­nal Ra­dia­tion  Springer Lecture Notes in Mathematics 1156 [1984] p 317-339 connects electromagnetic and gravitational radiation.
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