Specialized Orthonormal Frames and Embedding
We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-form...
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irk-123456789-1492152019-02-20T01:29:08Z Specialized Orthonormal Frames and Embedding Estabrook, F.B. We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-forms of the framing. The embeddings can be isometric, as in minimal surfaces or Regge-Teitelboim gravity, or torsion-free, as in Einstein vacuum gravity. Involutive exterior differential systems are given, and their Cartan character tables calculated to express the well-posedness of the underlying partial differential embedding and specialization equations. 2013 Article Specialized Orthonormal Frames and Embedding / F.B. Estabrook // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 6 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83C20; 57R40; 58A15 DOI: http://dx.doi.org/10.3842/SIGMA.2013.012 http://dspace.nbuv.gov.ua/handle/123456789/149215 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-forms of the framing. The embeddings can be isometric, as in minimal surfaces or Regge-Teitelboim gravity, or torsion-free, as in Einstein vacuum gravity. Involutive exterior differential systems are given, and their Cartan character tables calculated to express the well-posedness of the underlying partial differential embedding and specialization equations. |
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Estabrook, F.B. |
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Estabrook, F.B. Specialized Orthonormal Frames and Embedding Symmetry, Integrability and Geometry: Methods and Applications |
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Estabrook, F.B. |
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Estabrook, F.B. |
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Specialized Orthonormal Frames and Embedding |
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Specialized Orthonormal Frames and Embedding |
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Specialized Orthonormal Frames and Embedding |
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Specialized Orthonormal Frames and Embedding |
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Specialized Orthonormal Frames and Embedding |
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specialized orthonormal frames and embedding |
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Інститут математики НАН України |
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Specialized Orthonormal Frames and Embedding / F.B. Estabrook // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 6 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT estabrookfb specializedorthonormalframesandembedding |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 012, 5 pages
Specialized Orthonormal Frames and Embedding?
Frank B. ESTABROOK
Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109 USA
E-mail: frank.b.estabrook@jpl.nasa.gov
Received October 09, 2012, in final form February 12, 2013; Published online February 15, 2013
http://dx.doi.org/10.3842/SIGMA.2013.012
Abstract. We discuss some specializations of the frames of flat orthonormal frame bundles
over geometries of indefinite signature, and the resulting symmetries of families of embedded
Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear
constraints on the connection 1-forms of the framing. The embeddings can be isometric,
as in minimal surfaces or Regge–Teitelboim gravity, or torsion-free, as in Einstein vacuum
gravity. Involutive exterior differential systems are given, and their Cartan character tables
calculated to express the well-posedness of the underlying partial differential embedding
and specialization equations.
Key words: embedding; orthonormal frames; Cartan theory
2010 Mathematics Subject Classification: 83C20; 57R40; 58A15
1 Introduction
The class C of a Riemannian space is the minimum number of extra dimensions required for it
to be locally embeddable in a flat space (various signatures possible, which we ignore for the
moment; often we write repeated indices as “covariant”, but proper signs must be inserted in such
summations). Also our discussions are all local, with no global existence implied for solutions
of the equivalent partial differential equation. So class C 4-metrics are those embeddable in
4+C flat dimensions, C ≤ 6, and the embedding of their associated orthonormal frame bundles
(Cartan moving frame formalism, basis forms θi, connection forms ωij = −ωji, i = 1, . . . , 4)
brings in C + 4C + C(C − 1)/2 additional orthonormal basis forms θA and connection forms
ωiA = −ωAi and ωAB = −ωBA, A = 1, . . . , C. Taking µ, ν, etc. to span the combined range
of indices 1, . . . , 4 + C of the embedding space, the 1-forms θµ and ωµν = −ωνµ satisfy the
Cartan–Maurer structure equations of a flat orthonormal frame bundle ASO(4 + C).1
Isometric embedding requires the θA to vanish upon pullback, and together with their exterior
derivatives the 2-forms ωAi ∧ θi, and dynamic 4-forms derived from the Hilbert Lagrangian,
ωiA ∧ Rjk ∧ θlεijkl, they generate well posed exterior differential systems (EDS) (Rij are the
induced Riemann 2-forms dωij + ωik ∧ ωkj ) [4]. These isometric EDS’s for solutions of Regge–
Teitelboim (RT) gravity, of various classes, have been expounded in a recent paper [2].
?This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants
and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html
1Thus Einstein vacuum general relativity can be set as an exterior differential system on the group space
ASO(10), of dimension 55. This is arguably an elegant improvement over the more customary setting of this
field theory in terms of coordinate components, which means for a first order set of PDE’s setting the EDS
on the second frame bundle over a four dimensional base, of dimensionality 4 + 16 + 40 = 60. The first EDS
has 21 Cauchy characteristic vectors so in principle 21 variables can be eliminated, leaving a system of partial
differential equations in 34 variables, 4 in involution, independent, and 30 dependent. The second setting, when
using orthonormal tetrad frames, as in the dyadic and Newman–Penrose formalisms, only reduces to 44 variables
but has 10 degrees of gauge freedom together with 10 conservation laws [1].
mailto:frank.b.estabrook@jpl.nasa.gov
http://dx.doi.org/10.3842/SIGMA.2013.012
http://www.emis.de/journals/SIGMA/SDE2012.html
2 F.B. Estabrook
Among the solutions of RT theory are those that additionally annul the compatible Ricci
3-forms ωiA ∧ ωAj ∧ θkεijkl. This more restricted vacuum Einstein theory can conveniently be
directly set as a (not strictly isometric) embedding EDS on the orthonormal frame bundle over
10 space, requiring only the four torsion 2-forms ωiA ∧ θA to pull back to vanish, together with
their exterior derivatives and the dynamic Ricci 3-forms. A number of vacuum solutions, as well
as other physically interesting non-vacuum metrics, have been classified both by embedding class
and motion group, or metric symmetry [6]. Here we treat both EDS’s for embedded families
of solutions of RT theory with special symmetry and class ≤6, and new EDS for families of
solutions of Einstein theory with special symmetry in class 6.
In [2] an explicit six dimensional framing for embedding static spherically symmetric (SSS)
metrics was found, and solutions of Regge–Teitelboim theory reduced to quadrature, using a co-
ordinate imbedding due to Ikeda (other types of class 2 SSS coordinate embeddings exist; with
two new ones, all possible have since been given by Paston and Sheykin [5]). The Schwarzschild
vacuum solution of orthodox general relativity was a particular one of these. The connection
forms there derived were not explicitly given; we write them below in Section 2. We extract
the algebraic specialization of the O(6) frame fibers they imply. The specialization is a set of
self closed linear relations between the ωµν . A closely related specialization is also presented
and Cartan characters of embedding RT EDS’s using these specialized frames are calculated,
showing only the second of them to be well posed. We take it to be the prototype for a new
method of frame specialization that can be used for formulating EDS’s for families of solutions
having specialized metric symmetries.
In Section 3 we outline this technique for specializing the fibers of ASO(4+C) frame bundles,
and present calculated Cartan character tables of EDS’s for isometric embedding of RT theory
in them, showing them to be well posed and involutory. Section 4 briefly expounds an illus-
trative toy problem, isometrically embedding a two dimensional variational problem in a frame
specialized flat 3-space. In Section 5 we give results for four specializations of the fibers of
ASO(6, 4), the orthonormal frame bundle over flat ten-dimensional space. With torsion 2-forms
and Ricci 3-forms, we find EDS’s for families of Einstein vacuum solutions that are well posed
and involutory.
To summarize, we have calculated the Cartan characters, and well posedness, of a number
of EDS for embedded 4-spaces in specialized higher dimension flat orthonormal frame bundles,
and we conjecture that their solutions have intrinsic symmetry properties such as Riemannian
submersions, isometries and Killing tensors. The specializations we consider are sets of self
closed linear relations between the ωµν . Families of metrics with symmetries have heretofore
been difficult to treat with Cartan moving frame or tetrad formalism [3].
2 A static spherically symmetric embedding
In [2], treating class 2 static spherically symmetrical spacetimes, which all have four isome-
tries, they were imbedded in a flat 6-space coordinatized by r, θ, φ, t, z5, z6, signature
(−1,−1,−1, 1, 1,−1); The 6-space was framed with a family of orthonormal hexad bases adapted
to the (already known) metrics of the embedded space-times: This framing could have been even
more general, and its connection forms were not there written out, but they allow us to induce
the specialization of connection forms that succinctly expresses the desired symmetry; the fra-
ming is
θ1 =
(
dr +
(
dz5 + dz6
)
c′[r]− 2c′[r]t(2c′[r]tdr + 2c[r]dt)
−
(
dz5 − dz6
)(
g′[r] + c′[r]t2
))/√
1 + 4c′[r]g′[r],
θ2 = rdθ, θ3 = r sin[θ]dφ, θ4 = tdz5 − tdz6 + 2c[r]dt+ 2c′[r]tdr,
Specialized Orthonormal Frames and Embedding 3
θ5 =
(
dr + dz5
(
g′[r]− c′[r]t2 + c′[r] + 1/(2c′[r])
)
− 2tc′[r](2c′[r]tdr + 2c[r]dt)
−dz6
(
g′[r]− c′[r]t2 − c′[r] + 1/(2c′[r])
))/√
1 + 4c′[r]g′[r],
θ6 = dr + dz5/(2c′[r])− dz6/(2c′[r]),
c[r] and g′[r] are arbitrary functions. The flat metric is
−θ1θ1 − θ2θ2 − θ3θ3 + θ4θ4 + θ5θ5 − θ6θ6,
and the embedding is isometric, θ5 = 0, θ6 = 0.
The exterior derivatives of the above, expanded on them, give an explicit set of specialized
the 15 connection 1-forms ωµν
ω12 =
θ2
r
√
1 + 4c′[r]g′[r]
, ω13 =
θ3
r
√
1 + 4c′[r]g′[r]
, ω14 =
c′[r] (θ4 − 2tθ6c
′[r])
c[r]
√
1 + 4c′[r]g′[r]
,
ω15 =
(
θ1 − θ5 + θ6
√
1 + 4c′[r]g′[r]
)(
−
(
1 + 2c′[r]g′[r]
)
c′′[r]
+ 2c′[r]2g′′[r]
)/(
c′[r]
(
1 + 4c′[r]g′[r]
)3/2)
,
ω16 = −
((
θ1 − θ5
)√
1 + 4c′[r]g′[r] + θ6
(
1 + 4c′[r]g′[r]
))
c′′[r]
/(
c′[r]
(
1 + 4c′[r]g′[r]
)3/2)
,
ω23 =
Cot[θ]θ3
r
, ω24 = 0, ω25 = − θ2
r
√
1 + 4c′[r]g′[r]
, ω26 = −θ2
r
, ω34 = 0,
ω35 = − θ3
r
√
1 + 4c′[r]g′[r]
, ω36 = −θ3
r
, ω45 =
c′[r] (−θ4 + 2tθ6c
′[r])
c[r]
√
1 + 4c′[r]g′[r]
, ω46 = 0,
ω56 = −
((
θ1 − θ5
)√
1 + 4c′[r]g′[r] + θ6
(
1 + 4c′[r]g′[r]
))
c′′[r]
/(
c′[r]
(
1 + 4c′[r]g′[r]
)3/2)
.
To generalize from this coordinate presentation, we note from it that the algebraic frame
fiber specialization for this SSS embedding is apparently generated by the annulling of seven
1-forms ω12 +ω25, ω13 +ω35, ω14 +ω45, ω16 +ω65, ω24, ω34, ω64. It can be checked that: exterior
derivatives of these connection 1-forms all vanish by virtue of the 1-forms themselves. Otherwise
said, they are self-closed, or a completely integral Pfaffian system (c.i.p.s.). Let us call them
a set of symmetry forms.
We might consider an isometric embedding EDS with this frame specialization, set on the 21
dimensional orthonormal frame bundle over six dimensions θi, i = 1, . . . , 4 and θA, A = 5, 6. The
EDS is generated by the nine 1-forms (θA, symmetry forms), together with their closure 2-forms.
Taking the signature −1,−1,−1, 1, 1,−1 we calculate the EDS to have Cartan character table
21(9, 2, 2, 2)4 and to be in involution with respect to the θi. Since ω23 does not appear in the
generators there is one Cauchy Characteristic vector. There is one degree of gauge freedom:
s4 = 1. If we further add to this EDS the two dynamic 4-forms of Regge–Teitelboim gravity,
however, we no longer have a well posed system: all four θi are then not in involution. So this
isometric embedding EDS is apparently not exactly equivalent to the coordinate presentation
of [2]. The coordinate presentation has of course further specialized the framing (only two
arbitrary functions of one metric variable come in), and we are in this present work concerned
with finding well posed involutory families of symmetric solutions.
A possible resolution discussed in Section 3 is to consider an EDS generated by the closely
related c.i.p.s. ω12 + ω25, ω13 + ω35, ω14 + ω45, ω16 + ω65, ω42 + ω26, ω43 + ω36. This set of
six symmetry forms leads to a well posed class 2 EDS for RT theory, with character table
21(8, 2, 2, 4)4, in involution with respect to the θi and has one degree of gauge freedom s4 = 1.
4 F.B. Estabrook
3 A systematic specialization of frames in embedding spaces
The first four of the symmetry forms for Schwarzschild embedding found above are a c.i.p.s. in
their own right: ω12 +ω25, ω13 +ω35, ω14 +ω45, ω16 +ω65. The exterior derivative of any of these
pulls back to vanish modulo the set. This is also true if all four are written with minus signs
between the two terms. The index 1, or in general i, is from the set of imbedded forms, and the
index 5 or in general A is from the coset of embedding dimensions. The pulled back form ω15, or
in general ωiA, is exact modulo the set. Remarkably, this construction of a c.i.p.s. works for all di-
mensions and signatures, so long as the signatures of i and A are of opposite sign! We remember
the Codazzi equation dωiA+ωiµ∧ωµA = 0 and notice that all the quadratic terms vanish modulo
these c.i.p.s. This is the most basic construction of specializing symmetry forms we have found.
Further systematic addition of these specializing sets of symmetry forms can go on as long as
more pairs, say j, B, etc., of opposite signature are available (omitting previous i, A, etc. pairs
from each successive summation), and a number of Cartan characteristic table calculations have
shown that well posed RT (isometric) EDS’s are always found. Moreover, we find this again in
the class 6 case for Einstein vacuum (torsion-free embedding). In Section 5 we explicitly write
these four c.i.p.s., having numbers of generators 8, 8+6 = 14, 8+6+4 = 18 and 8+6+4+2 = 20
respectively.
Our systematic construction of these c.i.p.s. for specializing frame curvature forms, in
ASO(n,m), appears to be new, but a number of other c.i.p.s. for specialized frames are of
course well known and important. When a c.i.p.s. is imposed on the fiber algebra O(n,m) the
remaining moving frame forms are those of a subbundle, which can be labeled by its fiber algebra
and base dimension. For example the case of a subbundle ASU(m) of ASO(m,m), and further
isometric embedding of ASO(m) in that, is explicitly treated as an EDS for special Lagrangian
calibration in [4, p. 201] (the Cartan character table given there should read s2 = n − 1).
Various other interesting manifolds discussed there similarly can be understood as discovery of
c.i.p.s. for subbundles of orthonormal frame bundles ASO(m,n); taken together with isometric
embedding derived from minimizing calibration forms, EDS are found for associative, coasso-
ciative and Cayley geometries. We should also stress that, in distinction to previous such work
(e.g., Calabi–Yau manifolds), our new specializations have so far only been been treated locally,
through calculation of Cartan characters of the various embedding EDS.
4 The simplest case: a specialized embedding
of minimal 2-spaces in flat 2+1-space
With moving frame bases θi, i = 1, 2, and θA, A = 3, signature (1, 1,−1) we begin with an EDS
generated by θ3, its exterior derivative ω13 ∧ θ1 + ω23 ∧ θ2, and the new specialization which in
three dimensions is just the single form ω12−ω13. The character table is 6(2, 1)2 with one degree
of gauge freedom, well posed and involutory on θ1 and θ2. As RT theory does not exist in this
low dimension we then consider extremal surfaces, with Lagrangian Λ = θ1∧θ2, whose variation
generates the EDS to be the isometric embedding form θ3 together with the dynamic 2-form
ω13 ∧ θ2 − ω23 ∧ θ1. This EDS for specialized minimal surfaces remains well posed, involutory
on θ1 and θ2, with table 6(2, 2)2.
Integration, introduction of scalar variables to explicitly find the PDE’s, begins with the
closed 1-form ω23, leading to a coordinate we call u. Sequentially from the structure (moving
frame) equations and 2-forms in the EDS we find further closed 1-forms, and can introduce
variables v, x and y. The result is
ω23 = −u−1du, ω13 = u−1dv, θ2 = udx, θ1 = udy,
and the two 2-forms require ux − vy = uy + vx = 0.
Specialized Orthonormal Frames and Embedding 5
The induced intrinsic 2-metric on any of the family of solutions is then seen to be
ds2 = θ1θ1 + θ2θ2 = u2
(
dx2 + dy2
)
,
where u(x, y) satisfies Laplace’s equation. The Gaussian curvature is (u2x+u2y)/u
4. We have not
yet been able to characterize a common geometric property of this family of embedded minimal
surfaces, resulting from various solutions of Laplace’s equation; but it may be relevant that at
least the plane, point particle and dipole solutions all lead to metrics that support a Killing
vector field.
5 Application to Einstein gravity
Not surprisingly the symmetry specializations we have found to work with Regge–Teitelboim
gravity of various classes appear to work also with Einstein gravity, but only in the class 6
case, embedding in flat 10-dimensional space with the required signature. Then indeed we find
specialized frame bundles for setting EDS’s for families of solutions of vacuum general relativity.
This means we drop the strict isometric embedding 1-forms and return to the EDS (equally
generated from the Hilbert Lagrangian) of four torsion 2-forms and four Ricci 3-forms.
Let us number bases i = 1, 2, 3, 4 and A = 5, 6, 7, 8, 9, 0, use signature (−1,−1,−1, 1, 1, 1, 1,
−1, 1, 1) and base our first specialization on indices 1 and 5. Take the symmetry forms to be the
c.i.p.s. (ω12 +ω25, ω13 +ω35, ω14 +ω45, ω16−ω56, ω17−ω57, ω18−ω58, ω19−ω59, ω10−ω50). The
EDS is generated by these eight 1-forms, four torsion 2-forms ωiA∧θA, their exterior derivatives,
and the dynamic Ricci 3-forms ωiA ∧ωAj ∧ θkεijkl. Explicit calculation of the Cartan characters
of this EDS shows it to be well posed and involutory on the θi: 55(8, 4, 12, 14)4 with three
degrees of gauge freedom. There are 10 Cauchy characteristic vectors (ω67, ω68, ω69, ω60, ω78,
ω79, ω70, ω89, ω80, ω90 do not appear in the generators of the EDS).
The second specialization adds the additional six symmetry forms based on indices 2 and 6:
(ω23+ω36, ω24+ω46, ω27−ω67, ω28−ω68, ω29−ω69, ω20−ω60). Again well posed; 55(14, 4, 12, 14)4,
one degrees of gauge freedom and six CC vectors (ω78, ω79, ω70, ω89, ω80, ω90 do not appear).
The third specialization adds four symmetry forms based on 3 and 7, 55(18, 4, 12, 14) with
2 CC’s. The fourth specialization adds yet two more, based on 4 and 8, so 20 in all. This results
in 55(20, 4, 12, 14)4, with one remaining gauge freedom.
Acknowledgements. I thank the JPL Office of the Chief Scientist for a visiting appoint-
ment during which this research was carried out, and the Science Division for hospitality. My
colleagues John W. Armstrong, Curt Cutler, Massimo Tinto and Michele Vallisneri gave constant
stimulus and support.
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http://arxiv.org/abs/1202.1204
http://dx.doi.org/10.1017/CBO9780511535185
1 Introduction
2 A static spherically symmetric embedding
3 A systematic specialization of frames in embedding spaces
4 The simplest case: a specialized embedding of minimal 2-spaces in flat 2+1-space
5 Application to Einstein gravity
References
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