Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e., Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (respectively, embed) the most general class of (torsional) Galilean (respectively, Carrollian) connections.

1.
E.
Cartan
, “
Sur les variétés à connexion affine et la théorie de la relativité généralisée
,”
Ann. Sci. Éc. Norm. Supér.
40
(
3
),
325
412
(
1923
);
E.
Cartan
,
Ann. Sci. Éc. Norm. Supér.
42
(
3
),
17
88
(
1925
) (in French);
E.
Cartan
,
On Manifolds With an Affine Connection and the Theory of General Relativity
, translated by
A.
Magnon
and
A.
Ashtekar
(
Bibliopolis
,
1986
) (in English).
2.
K.
Friedrichs
, “
Eine invariante formulierung des Newtonschen gravitationsgesetzes und des grenzüberganges vom Einsteinschen zum Newtonschen gesetz
,”
Math. Ann.
98
,
566
(
1928
) (in German).
3.
A. N.
Bernal
and
M.
Sánchez
, “
Leibnizian, Galilean and Newtonian structures of space-time
,”
J. Math. Phys.
44
,
1129
(
2003
); e-print arXiv:gr-qc/0211030.
4.
X.
Bekaert
and
K.
Morand
, “
Connections and dynamical trajectories in generalised Newton-Cartan gravity. I. An intrinsic view
,”
J. Math. Phys.
57
(
2
),
022507
(
2016
); e-print arXiv:1412.8212.
5.
M.
Crampin
, “
On differentiable manifolds with degenerate metrics
,”
Proc. Cambridge Philos. Soc.
64
,
307
(
1968
).
6.
H. P.
Künzle
, “
Galilei and Lorentz structures on space-time: Comparison of the corresponding geometry and physics
,”
Ann. Inst. Henri Poincaré A
17
,
337
(
1972
).
7.
C.
Duval
and
H. P.
Künzle
, “
Sur les connexions newtoniennes et l’extension non triviale du groupe de Galilée
,”
C. R. Acad. Sci. Paris
285
,
813
(
1977
) (in French).
8.
M.
Trümper
, “
Lagrangian mechanics and the geometry of configuration spacetime
,”
Ann. Phys.
149
,
203
(
1983
).
9.
O.
Andreev
,
M.
Haack
, and
S.
Hofmann
, “
On nonrelativistic diffeomorphism invariance
,”
Phys. Rev. D
89
,
064012
(
2014
); e-print arXiv:1309.7231;
M.
Geracie
,
D. T.
Son
,
C.
Wu
, and
S. F.
Wu
, “
Spacetime symmetries of the quantum Hall effect
,”
Phys. Rev. D
91
:
045030
(
2015
); e-print arXiv:1407.1252;
A.
Gromov
and
A. G.
Abanov
, “
Thermal Hall effect and geometry with torsion
,”
Phys. Rev. Lett.
114
,
016802
(
2014
); e-print arXiv:1407.2908;
B.
Bradlyn
and
N.
Read
, “
Low-energy effective theory in the bulk for transport in a topological phase
,”
Phys. Rev. B
91
,
125303
(
2015
); e-print arXiv:1407.2911;
T.
Brauner
,
S.
Endlich
,
A.
Monin
, and
R.
Penco
, “
General coordinate invariance in quantum many-body systems
,”
Phys. Rev. D
90
,
105016
(
2014
); e-print arXiv:1407.7730;
S.
Moroz
and
C.
Hoyos
, “
Effective theory of two-dimensional chiral superfluids: Gauge duality and Newton-Cartan formulation
,”
Phys. Rev. B
91
,
064508
(
2015
); e-print arXiv:1408.5911;
M.
Geracie
and
D. T.
Son
, “
Hydrodynamics on the lowest Landau level
,”
J. High Energy Phys.
1506
,
044
(
2015
); e-print arXiv:1408.6843;
M.
Geracie
,
S.
Golkar
and
M. M.
Roberts
, “
Hall viscosity, spin density, and torsion
,” e-print arXiv:1410.2574 (
2014
).
10.
K.
Jensen
, “
On the coupling of Galilean-invariant field theories to curved spacetime
,” e-print arXiv:1408.6855 (
2014
).
11.
D. T.
Son
, “
Newton-cartan geometry and the quantum Hall effect
,” e-print arXiv:1306.0638 (
2013
).
12.
B.
Carter
and
I. M.
Khalatnikov
, “
Canonically covariant formulation of Landau’s Newtonian superfluid dynamics
,”
Rev. Math. Phys.
6
,
277
304
(
1994
).
13.
M. H.
Christensen
,
J.
Hartong
,
N. A.
Obers
, and
B.
Rollier
, “
Torsional Newton-Cartan geometry and Lifshitz holography
,”
Phys. Rev. D
89
(
6
),
061901
(
2014
); e-print arXiv:1311.4794;
J.
Hartong
,
E.
Kiritsis
, and
N. A.
Obers
, “
Lifshitz space-times for schrödinger holography
,”
Phys. Lett. B
746
,
318
(
2015
); e-print arXiv:1409.1519;
J.
Hartong
,
E.
Kiritsis
, and
N. A.
Obers
, “
Schrödinger invariance from Lifshitz isometries in holography and field theory
,”
Phys. Rev. D
92
,
066003
(
2015
); e-print arXiv:1409.1522;
E. A.
Bergshoeff
,
J.
Hartong
, and
J.
Rosseel
, “
Torsional Newton-Cartan geometry and the schrödinger algebra
,”
Classical Quantum Gravity
32
(
13
),
135017
(
2015
); e-print arXiv:1409.5555;
J.
Hartong
,
E.
Kiritsis
, and
N. A.
Obers
, “
Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum
,”
J. High Energy Phys.
1508
,
006
(
2015
); e-print arXiv:1502.00228.
14.
M. H.
Christensen
,
J.
Hartong
,
N. A.
Obers
, and
B.
Rollier
, “
Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography
,”
J. High Energy Phys.
1401
,
057
(
2014
); e-print arXiv:1311.6471.
15.
R.
Banerjee
and
P.
Mukherjee
, “
Dynamical construction of Hořava-Lifshitz geometry
,” e-print arXiv:1502.02880;
J.
Hartong
and
N. A.
Obers
, “
Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry
,”
J. High Energy Phys.
1507
,
155
(
2015
); e-print arXiv:1504.07461.
16.
M.
Geracie
,
K.
Prabhu
, and
M. M.
Roberts
, “
Curved non-relativistic spacetimes, Newtonian gravitation and massive matter
,”
J. Math. Phys.
56
(
10
),
103505
(
2015
); e-print arXiv:1503.02682.
17.
L. P.
Eisenhart
, “
Dynamical trajectories and geodesics
,”
Ann. Math.
30
,
591
(
1929
).
18.
M. W.
Brinkmann
, “
On Riemann spaces conformal to Euclidean space
,”
Proc. Natl. Acad. Sci. U. S. A.
9
,
1
(
1923
);
[PubMed]
M. W.
Brinkmann
, “
Einstein spaces which are mapped conformally on each other
,”
Math. Ann.
94
,
119
(
1925
).
19.
C.
Duval
,
G. W.
Gibbons
, and
P.
Horvathy
, “
Celestial mechanics, conformal structures and gravitational waves
,”
Phys. Rev. D
43
,
3907
(
1991
); e-print arXiv:hep-th/0512188.
20.
X.
Bekaert
and
K.
Morand
, “
Embedding nonrelativistic physics inside a gravitational wave
,”
Phys. Rev. D
88
(
6
),
063008
(
2013
); e-print arXiv:1307.6263.
21.
C.
Duval
,
G.
Burdet
,
H. P.
Kunzle
, and
M.
Perrin
, “
Bargmann structures and Newton-Cartan theory
,”
Phys. Rev. D
31
,
1841
(
1985
).
22.

PLATO “The Republic” Book VII.

23.
E.
Minguzzi
, “
Classical aspects of lightlike dimensional reduction
,”
Classical Quantum Gravity
23
,
7085
(
2006
); e-print arXiv:gr-qc/0610011.
24.
C.
Duval
,
P. A.
Horváthy
, and
L.
Palla
, “
Conformal symmetry of the coupled Chern-Simons and gauged nonlinear Schrodinger equations
,”
Phys. Lett. B
325
,
39
(
1994
); e-print arXiv:hep-th/9401065;
C.
Duval
,
P. A.
Horváthy
, and
L.
Palla
, “
Conformal properties of Chern-Simons vortices in external fields
,”
Phys. Rev. D
50
,
6658
(
1994
); e-print arXiv:hep-th/9404047;
C.
Duval
,
P. A.
Horváthy
and
L.
Palla
, “
Spinor vortices in nonrelativistic Chern-Simons theory
,”
Phys. Rev. D
52
,
4700
(
1995
); e-print arXiv:hep-th/9503061;
C.
Duval
,
P. A.
Horváthy
, and
L.
Palla
, “
Spinors in non-relativistic Chern-Simons electrodynamics
,”
Ann. Phys.
249
,
265
(
1996
); e-print arXiv:hep-th/9510114.
25.
M.
Hassaine
and
P. A.
Horváthy
, “
Field dependent symmetries of a nonrelativistic fluid model
,”
Ann. Phys.
282
,
218
(
2000
); e-print arXiv:math-ph/9904022;
M.
Hassaine
and
P. A.
Horváthy
, “
Symmetries of fluid dynamics with polytropic exponent
,”
Phys. Lett. A
279
,
215
(
2001
); e-print arXiv:hep-th/0009092;
M. D.
Montigny
,
F. C.
Khanna
, and
A. E.
Santana
, “
Lorentz-like covariant equations of non-relativistic fluids
,”
J. Phys. A: Math. Gen.
36
,
2009
(
2003
).
26.
G. W.
Gibbons
and
C. E.
Patricot
, “
Newton-Hooke space-times, Hpp waves and the cosmological constant
,”
Classical Quantum Gravity
20
,
5225
(
2003
); e-print arXiv:hep-th/0308200.
27.
C.
Duval
,
M.
Hassaine
, and
P. A.
Horvathy
, “
The Geometry of Schrodinger symmetry in gravity background/non-relativistic CFT
,”
Ann. Phys.
324
,
1158
(
2009
); e-print arXiv:0809.3128;
C.
Duval
and
P. A.
Horváthy
, “
Non-relativistic conformal symmetries and Newton-Cartan structures
,”
J. Phys. A: Math. Gen.
42
,
465206
(
2009
); e-print arXiv:0904.0531.
28.
G. W.
Gibbons
and
C. N.
Pope
, “
Kohn’s theorem, Larmor’s equivalence principle and the Newton-Hooke group
,”
Ann. Phys.
326
,
1760
(
2011
); e-print arXiv:1010.2455;
P. M.
Zhang
and
P. A.
Horváthy
, “
Kohn’s theorem and Galilean symmetry
,”
Phys. Lett. B
702
,
177
(
2011
); e-print arXiv:1105.4401;
K.
Andrzejewski
,
J.
Gonera
, and
P.
Kosinski
, “
Nonlinear realizations, the orbit method and Kohn’s theorem
,”
Phys. Lett. B
711
,
439
(
2012
); e-print arXiv:1203.3311.
29.
M.
Cariglia
, “
Hidden symmetries of Eisenhart lift metrics and the Dirac equation with flux
,”
Phys. Rev. D
86
,
084050
(
2012
); e-print arXiv:1206.0022;
M.
Cariglia
and
G.
Gibbons
, “
Generalised Eisenhart lift of the Toda chain
,”
J. Math. Phys.
55
,
022701
(
2014
); e-print arXiv:1312.2019;
M.
Cariglia
,
G. W.
Gibbons
,
J. W.
van Holten
,
P. A.
Horváthy
, and
P.-M.
Zhang
, “
Conformal killing tensors and covariant Hamiltonian dynamics
,”
J. Math. Phys.
55
,
122702
(
2014
); e-print arXiv:1404.3422.
30.

By homogeneous Bargmann group, we mean the homogeneous Galilei group in a d + 2 dimensional representation inherited from that of the Bargmann group, cf. Ref. 22.

31.
V.
Bargmann
, “
On Unitary ray representations of continuous groups
,”
Ann. Math.
59
,
1
(
1954
).
32.
R.
Andringa
,
E.
Bergshoeff
,
S.
Panda
, and
M.
de Roo
Newtonian gravity and the Bargmann algebra
,”
Classical Quantum Gravity
28
,
105011
(
2011
); e-print arXiv:1011.1145.
33.
K.
Morand
, “
Nonrelativistic symmetries and Newton-Cartan gravity
,” Ph.D. thesis,
Université François Rabelais de Tours
,
2014
, available at http://inspirehep.net/record/1411872/files/kevin.morand_4532.pdf.
34.
U.
Niederer
, “
The maximal kinematical invariance group of the free Schrödinger equation
,”
Helv. Phys. Acta
45
,
802
(
1972
).
35.
J.
Gomis
and
J. M.
Pons
, “
The centralizer subalgebras of Poincare IO(3,1)
,”
Il Nuovo Cimento A
47
,
166
(
1978
).
36.
G.
Burdet
,
M.
Perrin
, and
P.
Sorba
, “
About the non-relativistic structure of the conformal algebra
,”
Commun. Math. Phys.
34
85
(
1973
).
37.
H.
Bacry
and
J. M.
Lévy-Leblond
, “
Possible kinematics
,”
J. Math. Phys.
9
,
1605
(
1968
).
38.
J. M.
Lévy-Leblond
, “
Une nouvelle limite non-relativiste du groupe de Poincaré
,”
Ann. Inst. Henri Poincaré A
3
1
(
1965
).
39.
C.
Duval
,
G. W.
Gibbons
,
P. A.
Horvathy
, and
P. M.
Zhang
, “
Carroll versus Newton and Galilei: Two dual non-einsteinian concepts of time
,”
Classical Quantum Gravity
31
,
085016
(
2014
); e-print arXiv:1402.0657.
40.
C.
Duval
,
G. W.
Gibbons
, and
P. A.
Horvathy
, “
Conformal Carroll groups and BMS symmetry
,”
Classical Quantum Gravity
31
,
092001
(
2014
); e-print arXiv:1402.5894.
41.
C.
Duval
,
G. W.
Gibbons
, and
P. A.
Horvathy
, “
Conformal Carroll groups
,”
J. Phys. A: Math. Gen.
47
(
33
),
335204
(
2014
); e-print arXiv:1403.4213.
42.
E.
Bergshoeff
,
J.
Gomis
, and
G.
Longhi
, “
Dynamics of Carroll particles
,”
Classical Quantum Gravity
31
(
20
),
205009
(
2014
); e-print arXiv:1405.2264.
43.
J.
Kowalski-Glikman
and
T.
Trzesniewski
, “
Deformed Carroll particle from 2+1 gravity
,”
Phys. Lett. B
737
,
267
(
2014
); e-print arXiv:1408.0154;
E.
Bergshoeff
,
J.
Gomis
, and
L.
Parra
, “
The symmetries of the Carroll superparticle
,”
J. Phys. A: Math. Gen.
49
(
18
),
185402
(
2016
); e-print arXiv:1503.06083.
44.
C.
Barrabes
and
W.
Israel
, “
Thin shells in general relativity and cosmology: The Lightlike limit
,”
Phys. Rev. D
43
,
1129
(
1991
).
45.

The term (Galilean, Carrollian, etc.) “manifold” will stand for a smooth manifold endowed with a metric structure and a compatible connection.

46.
K.
Morand
, “
Embedding Galilean and Carrollian geometries II. Leibniz vs Bargmann
” (to appear).
47.
M.
Henneaux
, “
Zero Hamiltonian signature spacetimes
,”
Bull. Soc. Math. de Belgique
31
,
47
(
1979
).
48.
J. M.
Lee
,
Introduction to Smooth Manifolds
(
Springer
,
2003
).
49.
J.
Dieudonné
,
Éléments d’Analyse, Tome III
(
Gauthier-Villars
,
1970
).
50.

We denote igld+2=gld+2Rd+2 the affine Lie algebra in d + 2 dimensions.

51.

Let g be a Lie algebra with hg being a Lie subalgebra (i.e., h,hh) and ig an ideal (i.e., i,gi). The algebra g is the semidirect sum of i with h if it admits the decomposition g=hi as h-modules. The subalgebra h will be called the homogeneous part of g.

52.

The leaf t = 0 is special in that only the homogeneous Carroll algebra acts faithfully.

53.

Actually, when dealing with non-Riemannian geometries, the definition of the vector space W(M) will sometimes necessitate a piece of the metric structure (e.g., W(M,ψ) for Galilean connections and W(M,ξ) for Carrollian and ambient Galilean connections).

54.

The isomorphism is not necessarily canonical but may depend on the choice of an element of an affine space (cf., e.g., Lemma 4.2 of Ref. 5 or Subsection 4 of the  Appendix).

55.

Indeed, an important discrepancy between Riemannian and non-Riemannian geometries lies in the fact that, in the latter case, only a subset of “metric” structures admits torsionfree compatible connections (e.g., Leibnizian structures with closed absolute clock in the Galilean case or invariant Carroll structures in the Carrollian case).

56.

We pass on an important subtlety that arises in nonrelativistic cases (in contradistinction with the relativistic case), namely, that there may not be a canonical choice of a natural origin (and thus of affine map Θ) but rather a class of such origins (dubbed special connections in our terminology). The class of special connections forms an affine space which can be “resolved,” i.e., which is isomorphic to a “simpler” affine space [e.g., the affine space of fields of observers to define special Galilean connections, cf. Proposition (3.9) in Ref. 5, or the one of principal connections to define special Carrollian connections, cf. the  Appendix].

57.

In the relativistic case, the presence of a canonical origin (the Levi-Civita connection) ensures that the classification is a vector space. However, in the nonrelativistic cases, the lack of a canonical origin (cf. footnote 57) prevents the classification to possess a natural structure of vector space. In the latter case, a classification is thus an affine space, isomorphic to DM,m, obtained by taking the product of W(M) by the affine space used in the definition of Θ and then quotienting by the model vector space of the latter (cf. Proposition B.4 in Ref. 5 for precise statements).

58.
Note that our choice to restrict to invariant Leibnizian pairs in the definition of ΘN,A does not constrain the space of torsionfree ambient Galilean connections considered but only the representation we give of such connections. In other words, it only narrows the space of origins of DM,ξ,ψ,γ (i.e., the space of torsionfree special ambient connections, cf. below). If one chooses to relax this invariance condition, the map gets modified according to
ΘN,A:ΓFN,Aμν=2γNλμνNλ+2γNλμAνLξNλ,ΣAμν=μAν+AμLξAν.
59.
Relaxing the invariance condition on the Leibnizian pair (N, A), the torsionless special connection spanning the kernel of the map in the equation in Ref. 59 reads
ΓN,Aμνλ=ξλμAν+Nλμψν+12hAλρμγNρν+νγNρμργNμνξλAμLξAνhAλρψμAνLξγNρσNσ.
60.

Note that the second relation of Proposition 3.10 is invariant under the transformation VV + , for all fCM (and similarly for W) so that for a given field of observers N, the bilinear form defined by relation 2 really acts on ΓKerψ/Span ξ and then constrains dd+12 components.

61.

We pursue with the notation convention used in Ref. 5 and make use of the same symbols for the various spaces and maps encountered in the torsionfree and torsional cases in order to emphasise the similitude in the logic of the arguments.

62.
One can go one step further and package tensors Σ and U into a single object T possessing the symmetries of a torsion tensor, just like in the relativistic case. Explicitly, one constructs an isomorphism between WM,ξ and ΓT*M2T*M as
ϕA:WM,ξΓT*M2T*M:Σμν,Uλ|μνTAλ|μνUλ|μν+AλΣ[μν]+AμΣνλ
whose inverse isomorphism reads
ϕA1:ΓT*M2T*MWM,ξ:Tλ|μνΣAμνξλTμ|λν+Tν|λμ+Tλ|μν,UAλ|μνTλ|μνξρAλTρ|μν+AμTν|ρλ+AμTρ|νλAμTλ|νρ.
Composing φA and ϕA leads to a resolution of VM,ξ,ψ,γ. Explicitly, we define the isomorphism ΦAϕAφA reading as
ΦA:VM,ξ,ψ,γΓT*M2T*M:SμνλTAλ|μν(γNλρ+AλAρ)S[μν]ρ+AρAμSνλρ+ψλγNσμSνρσNρ,
where NFOM,ψ is an arbitrary field of observers. The inverse isomorphism takes the form
ΦA1:ΓT*M2T*MVM,ξ,ψ,γ:Tλ|μνSAμνλ=PAνσ(hAλρ+ξλξρ)Tμ|ρσ+Tσ|ρμ+Tρ|μσ.
63.

Note that the tensor Uλ|μν in (3.25) is independent of the choice of field of observers N, hence the absence of superscript on U.

64.
Similarly to the torsionfree case, one can relax the invariance condition on the Leibnizian pair and define the following affine map:
Θ:LPM,ξ,ψ×DM,ξ,ψ,γWM,ξ:N,A,ΓΣAμν=μAν+AλΓ[μν]λ+AμLξAν,UN,Aλ|μν=γNλρΓ[μν]ρ12AμPLνρPLλσLξγNρσ+ψλγNσμνNσγNρμAνLξNρ.
65.
Relaxing the invariance condition on the Leibnizian pair N,A, the torsional special connection spanning the kernel of the map in the equation in Ref. 65 reads
ΓN,Aμνλ=ξλμAν+Nλμψν+12hAλρμγNρν+νγNρμργNμνξλAμLξAν12hAλρPLμσAνLξγNρσ+2ψμAνLξγNρσNσ.
66.

Recall that in the Lorentzian case, the arbitrariness is encoded in the unconstrained torsion tensor, i.e., in a section of the vector bundle 2T*MTM whose fibers have dimension d+1d+222.

67.

Note that relations 1–2 of (*) together with UN,Aλ|μν=0 ensure that FNμνΓ2Annξ. Similarly, the fact that ΣA[μν]=0 guarantees that ΣA(μν)Γ2Annξ.

68.

Note that the invariance of γ and N ensure LξγN=0 so that the second component of ΘN,A does not depend on A and will then be denoted as UN.

69.
J.
Hartong
, “
Gauging the Carroll algebra and ultra-relativistic gravity
,”
J. High Energy Phys.
1508
,
069
(
2015
); e-print arXiv:1505.05011.
70.
K. L.
Duggal
and
A.
Bejancu
,
Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications
(
Kluwer Academic
,
1996
).
71.
If one considers a generic Ehresmann connection AECM,ξ, the map ΘA becomes ΘA:ΓΣAμν=μAν+AμLξAν while the generic torsionfree Carrollian connection reads
Γμνλ=ξλμAν+12hAλρμγρν+νγρμργμνξλAμLξAν+ξλΣAμν.
72.
T.
Dereli
,
S.
Kocak
, and
M.
Limoncu
, “
Newton-Cartan connections with torsion
,” e-print arXiv:gr-qc/0402116.
73.
One can go one step further and package tensors Σ and U in a single object T by constructing an isomorphism between WM,ξ and ΓT*M2T*M as
ϕA:WM,ξΓT*M2T*M:Σμν,Uλ|μνTAλ|μνUλ|μν+AλΣ[μν]+AμΣνλ
whose inverse isomorphism reads
ϕA1:ΓT*M2T*MWM,ξ:Tλ|μνΣAμνξλTμ|λν+Tν|λμ+Tλ|μνUAλ|μνTλ|μνξρAλTρ|μν+AμTν|ρλ+AμTρ|νλAμTλ|νρ.
Composing φA and ϕA leads to a resolution of VM,ξ,γ. Explicitly, we define the isomorphism ΦAϕAφA reading as
ΦA:VM,ξ,γΓT*M2T*M:SμνλTAλ|μν(γNλρ+AλAρ)S[μν]ρ+AρAμSνλρ
while the inverse isomorphism takes the form
ΦA1:ΓT*M2T*MVM,ξ,γ:Tλ|μνSAμνλ=PAνσ(hAλρ+ξλξρ)Tμ|ρσ+Tσ|ρμ+Tρ|μσ.
74.
If one insists in dealing with a generic (i.e., not necessarily invariant) Ehresmann connection, the previous map can be modified as
Θ:ECM,ξ×DM,ξ,γWM,ξ:A,ΓΣAμν=μAν+AλΓ[μν]λ+AμLξAν,UAλ|μν=γλρΓ[μν]ρ12AμLξγνλ.
75.
If one relaxes the invariance condition on the Ehresmann connection A, the most general torsional Carrollian connection compatible with CM,ξ,γ can be expressed as
Γμνλ=ξλμAν+12hAλρμγρν+νγρμργμνξλAμLξAν12hAλρLξγμρ+ξλΣAμν+hAλρUAμ|ρν+UAν|ρμ+UAρ|μν.
76.
J. A.
Wheeler
,
C. W.
Misner
, and
K. S.
Thorne
,
Gravitation
(
W. H. Freeman and Co.
,
1973
), Chap. 12;
D. B.
Malament
,
Topics in the Foundations of General Relativity and Newtonian Gravitation Theory
(
University of Chicago
,
2012
), Chap. 4.
You do not currently have access to this content.