PART III: EFFECTIVE GRAVITY FROM YANG-MILLS THEORIES
XI. CLASSICAL ASPECTS Let us now study some classical aspects of the theory
DRAFT
FIG. 3: The ratio κ2/γ2 as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of
48π2 55
√
5 e56
Λ−22.
Following the second approach, the scalesµand Λ can be estimated by fixing the current value of Newton’s constant toG−1≈1.491×1032T eV2. Thus, combining (165) and (171), one easily achieves
7.323×1031T eV2= Λ2 µ
Λ −70/9
ln µ2
Λ2
. (172)
We have little freedom to choose µas long as N κ2/16π2 <1. Let us work atµ2 = 2Λ2. Thus, N κ2/16π2≈0.393.
This provides ln(µ2/Λ2)≈0.693. Accepting these as reasonable values, we achieve for the renormalization group scale
Λ2≈2.122×1033T eV2, (173)
which is quite close (for a 1-loop semi-perturbative estimate) to Planck energyEp2= 1.491×1032T eV2. Although some approximations and extrapolations have been considered, we can interpret these values as a good result, indicating that the geometric phase of gravity appeared right before Planck scale. This means that right above Planck scale, where quantum mechanics starts to make sense, gravity is already in its geometric phase.
We can also estimate the value of the cosmological constant for the chosen scales. From (165) and (170), it provides
Λ2≈1.106×1032 T eV2. (174)
This is a huge amount of energy and is three orders of magnitude greater than quantum field predictions [121] of Λ2qf t ∼ −3.71×1028T eV2and would not cancel it to provide the observational data value Λ2obs∼1.686×10−68eV2. However, for 1-loop approximation at zero temperature, it is quite remarkable that a solution accommodating Newton constant and Planck energy scale could be found.
XI. CLASSICAL ASPECTS
DRAFT
for the vierbein, and (Exercise 3.17) 3
Λ2D ? Rab+ 2eb? Ta−εabcdTced= 0, (176) for the spin-connection. The last term in (175) is recognized as the Einstein term with cosmological constant.
Nevertheless, as discussed in Sect. X B, the cosmological constant term in (175) will sum with all vacuum terms of the matter fields action. Therefore, by this phenomenological argument, we should replace Λ7−→ Λ, only at this term.e Thus,
3
2Λ2Rbc?(Rbcea) +Tb?(Tbea) +D ? Ta−εabcd Rbced−Λe2 3 ebeced
!
= 0, (177)
whereΛ is the observational cosmological constant, which is extremely small. Equations (175) and (176) are the fulle field equations, valid at all sectors of the geometric phase. On the other hand, for a sufficient low energy scale where all other theories achieve their vacuum states, equations (177) and (176) must be employed.
The Einstein limit can be obtained with a very subtle argument: Since Λ2 is a very large quantity, all terms proportional to Λ−2 are small perturbations as long as the curvatures are not too strong. Hence, at zeroth order approximation, equations (177) and (176) reduce to
Tb?(Tbea) +D ? Ta−εabcd Rbced−Λe2 3 ebeced
!
= 0.
2eb? Ta−εabcdTced = 0. (178) The second of (178) is algebraic in the torsion and, hence, a solutionT = 0 is allowed49. Thus, (178) reduce to
εabcd Rbced−Λe2 3 ebeced
!
= 0. (179)
which is the well know Einstein equations, i.e. the first of (139) in vacuum.
B. Spherical symmetry static solutions
Let us consider the field equations (176) and (177) for vanishing torsion and very high values of Λ2, 3
2Λ2Rbc?(Rbcea)−εabcd Rbced−Λ˜2 3 ebeced
!
= 0. (180)
We remark here that the first term in (180) will be considered as a very small perturbation. In, (176), for vanishing torsion, the remaing term is virtually zero and can be safelly neglected since it does not affect any other dynamical term.
We take the usualansatz [10–13] for static and spherically symmetric metric, namely
e0=−eα(r)dt , e1=eβ(r)dr , e2=rdθ , e3=rsinθdφ . (181)
49It is important to understand that this limit can only be taken outside a matter distribution of for sufficient small spin densities, otherwise,T6= 0 in general.
DRAFT
Hence, equation (180) decomposes as σ
"
2
e−2β∂rβ r
2 +
1−e−2β r2
2# + 2
e−2β∂rβ r
+1−e−2β
r2 + 3λ = 0, σ
"
2
e−2β∂rα r
2 +
1−e−2β r2
2#
−2
e−2β∂rα r
+1−e−2β
r2 + 3λ = 0, σ
( h
e−(α+β)∂r e−β∂reαi2 +
e−2β∂rα r
2 +
e−2β∂rβ r
2) +
−e−(α+β)∂r e−β∂reα
−e−2β∂rα
r +e−2β∂rβ
r + 3λ = 0,
(182) fora= 0 ,a= 1 and a= 2, respectively. The constants in (182) are
σ = − 3 2Λ2 , λ = −Λ˜2
3 , (183)
We notice that the differential equations fora= 2 anda= 3 are identical. The remaining equations can be solve by perturbation theory around the well-known de Sitter Schwarzchild solution []. A long computation yields, at fourth order in perturbation theory,
e−2β = 1 +a1
r +a2r2−η b1
r +b2r2+b3
r4
−η2c1
r +c2r2+c3
r4 +c4
r7
+
− η3 d1
r +d2r2+d3
r4 +d4
r7 + d5
r10
−η4e1
r +e2r2+e3
r4 +e4
r7 + e5
r10+ e6
r13
,
(184) withη =σλ≡ 12ΛΛ˜22 being the small perturbation parameter. At zeroth order we havea1=−2GM, a2=λ. On the other hand, b1, c1, d1 and e1 are constants of integration while the rest of the constants have dependence in these first ones.
Clearly, forr2GM, the asymptotic solution is the de Sitter spacetime, as expected, ds2=
1 + ˜Λpr2
dt2+ 1
1 + ˜Λpr2+r2dΩ2 , (185)
with ˜Λp=a2−b2η−c2η2−d2η3−e2η4.
C. Cosmology
The details of this section can be found in [135]. Again, let us start by considering only the subtle vanishing torsion case. Considering a homogeneous and isotropic metric means that there exists a special spacetime foliation where each spatial section is maximally symmetric50. Therefore, the metric must be of a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) type [134]
ds2=−dt2+a2(t) dr2
1−kr2+r2dΩ2
, (186)
50A maximally symmetric space is a space that has the same number of symmetries as the Euclidean space. In other words, a space is maximally symmetric if it hasn(n+ 1)/2 linearly independent Killing vectors for ann-dimensional space
DRAFT
wherea(t) is the scale factor, dΩ2is the solid angle and the constantk∈ {−1,0,1}defines the curvature of the spatial sections. As usually used in the literature, it is convenient the following variables
l≡ ¨a a, h≡
a˙ a
2 + k
a2. (187)
We also define,ρΛ≡χ−1Λ2 andρΛ˜ ≡χ−1Λ˜2. Hence, substituting the metric (186) in the field equations51(176) and (177), it is found
3 2χρΛ
h2−h+χ
3(ρ+ρΛ˜) = 0, 3
2χρΛ
l2−l−χ
6(ρ+ 3p−2ρΛ˜) = 0, 1
(χρΛ)2∂t(l+h) = 0. (188)
These are the complete equations governing the evolution of the cosmological scenario. We will explore them in different energy scale sectors. We notice that the third equation in (188) is a novelty information. In typical theories this equation is not present. It arises from the fact that we have a curvature squared term and the fact that we are considering the first order formalism.
1. The Λ-CDM model
The infrared sector is characterized by a low energy regime for gravity, i.e. weak curvature regime. The characteristic energy scale of the model is given by Λ2 or, equivalently, throughρΛ. Thus, we have to compare the linear terms h andl with ρΛ. Ifhand lρΛ, then the quadratic terms in (188) become negligible. Effectively, one can reach the IR regime by taking the limit Λ2→+∞in (176) and (177). Hence52,
h=χ
3(ρ+ρΛ˜), (189)
l=−χ
6(ρ+ 3p−2ρΛ˜), (190)
which are exactly the Einstein field equations with a cosmological term. Therefore, the present stage of the Universe is correctly described by the model.
2. High curvature regime
Let us now consider a strong curvature regime in which the curvature square term is comparable with the other terms. In contrast to the IR regime, now the quadratic curvature becomes comparable to ρΛ and can no longer be neglected in (188). In what follows we shall outline separately the evolution of a vacuum from a matter filled spacetime.
• Vacuum model
The dynamical equations for a vacuum universe in the UV sector follow from (188) by setting ρ=p= 0, 3
2χρΛh2−h+χ
3 ρΛ˜ = 0, 3
2χρΛ
l2−l+χ
3 ρΛ˜ = 0,
∂t(l+h) = 0. (191)
51Perfect fluit energy-momentum tensor has to be added in (177).
52See Sect. XI A for the argumentation to drop the third equation (188).
DRAFT
The first two equations above are algebraic relations for handl respectively. In fact, both equations have the same structure showing thathandl share the same spectrum. It is straightforward to identify the roots as
(h, l) = Λ2dS±= Λ2 3
1±
r
1−2ρΛ˜
ρΛ
, (192)
Since both handl are constant, the third equation in (191) is automatically satisfied. The Ricci scalar reads R= 6(l+h) which for l =hgives R= 12 Λ2dS± and for l 6=hwe haveR= 4Λ2. The evolution of the scale factor can automatically be integrated from the equationh= Λ2dS±. The possible solutions read
a(t) = 1 Λ2dS±
cosh (ΛdS±t) ; k=−1, exp (ΛdS±t) ; k= 0, sinh (ΛdS±t) ; k= 1,
(193)
where ΛdS± ≡q
Λ2dS±. One can immediately recognize these solutions being three different foliations of a de Sitter universe with a cosmological constant given by Λ2dS±.
The deceleration parameter is defined as q =−¨aaa˙2 =−l(h−k/a2)−1 =−1/ 1−kΛ−2dS±a−2
, which is always negative. Hence, an accelerated expanding phase is manifest.
The effective cosmological constant depends both onρΛ˜ andρΛ. In the limit ˜Λ2/Λ21, we can expand up to first order and obtain
Λ2dS±≈ Λ2 3
"
1± 1−Λ˜2 Λ2
!#
. (194)
The root Λ2dS+ is approximately proportional to Λ2 while the root Λ2dS− is approximately proportional to Λ˜2. Given the enormous value of Λ2, the first root represents an universe with a violent de Sitter phase.
Therefore, this phase can be associated with an inflationary expansion. On the other hand, the second root would correspond to a smooth accelerating phase similar to the late time expansion in the ΛCDM model. This result may be a Hint that a general solution connecting the inflationary era with the ΛCDM model could be developed.
• Perfect fluid model
In the second order formalism one has to provide an equation of state that in this case reduces to give a functional dependence of the pressure in terms of the energy density, i.e. p=p(ρ). In this case, the equation of state of the fluid is of major importance to completely specific the dynamic system. In contrast, the first order gravity analyzed in this paper has an extra set of dynamical field equations (188). They are a consequence of the dynamically independent character of the spin connection field. In principle we have three variables to determine the evolution, namely, the scale factor, the energy density and the pressure. However, with the thermodynamic equation of state we would have four equation for three variables and the system would be overdetermined.
Indeed, the set of equations (188) is sufficient to establish the time evolution of all the variables. A possible way to reconcile this situation with a thermodynamic description of matter is to interpretation the UV sector as a regime where the gravitational field does not distinguish the nature of the matter fields, i.e. all perfect fluids gravitate in the same manner in the UV regime. For the two thermodynamic quantities it is found (see fig. 4)
ρ=3ξ0(4χρΛ− R0) 4χ2ρΛ
a−4− 9ξ02 2χ2ρΛ
a−8−
ρΛ˜−R0
4χ + R20 32χ2ρΛ
, (195)
p=−ρ+
4χ− R0
χ2ρΛ
ξ0a−4, (196)
The particular caseξ0= 0 freezes the value of the energy density and the pressure becomesp=−ρ. Therefore, we have to assume the general case where ξ0 6= 0. Equations (195) and (196) show the universal behavior of the pressure and energy density independent of the equation of state of the fluid. Indeed, if one assumes a barotropic equation of statep=ωρwith constantω, equations (195) and (196) impliesω=−1. Thus, the only consistent barotropic equation of state in the UV sector isp=−ρ. It is clear then that the energy density and
DRAFT
FIG. 4: Behaviour of pressure and energy density according to scale factor.
pressure given by (195) and (196) must be associated with a non-conservation of the energy-momentum tensor.
Indeed, equations (188) are consistent with
˙ ρ+ 3a˙
a(ρ+p) + ρ˙2
8( ˙a/a)(ρ+ρΛ˜−ρΛ/2) = 0, (197) which verifies the non-conservation of the usual energy-momentum tensor. The first two terms are the usual components while the last non-linear term comes from the high curvature corrections.
The scale factor can also be determined as a(t) =
s
x0e±αt+9k2−3R0ξ0
R20x0
e∓αt+ 6k R0
, (198)
where α= p
R0/3 andx0 > 0 is a constant of integration. The above solution has non-trivial R0 = 0 and k= 0 limits.
Three different simple situations of (198) are displayed in Fig. 5. First, we take vanishing Ricci scalar but nonzero spatial section, providing
a(t) = q
ξ0k−k(t±p
|ξ0|)2. (199)
The±sign within the square term does not change qualitatively the evolution. Fork= 1 we have a big bang-big crunch solution with an initial and a final singularity. The scale factor reaches its maximal value attmax=√
ξ0
where a(tmax) =√
ξ0. The k=−1 has to disjoint branches. Hence, both branches have an initial singularity and expands or it is a collapsing universe with a future singularity. For the plus sign the initial singularity is located at t= 0 while the final singularity in the collapsing phase is att=−2√
ξ0. For flat spatial section the
FIG. 5: Time evolution of the scale factor.
constant ξ0 must be positive definite and we have a(t) =
q
±2p
ξ0t, (200)
DRAFT
where the constant of integration was chosen to locate the singularity att= 0. Once again we have a collapsing phase for t <0 that reaches the singularity and an expanding phase with initial singularity att= 0.
Finally, let us only comment that the characteristic of the general solution (198) depends on the interplay of the constantsk,R0 andξ0. In particular, the evolution can describe a bounce if the second term has the same sign as the first one, namely, ifR0<3/ξ0.
D. Dark stuff
Nowadays, dark matter and dark energy could be expected features from a good fundamental gravity theory. They can have a completely different origin. However, the fact that dark matter only interacts only with gravity could be a clue about its gravity origin.
Dark energy is not a problem ifΛ is at our disposal. Dark matter, on the other hand, is a bit more complicated.e There are two possible ways that the present approach could account for dark matter. The first one is to enlarge the gauge group to a bigger group encoding the SO(5). A known example is theSL(5,R) group, as in [36, 40]. In this case, the extra sector of the gauge group manifests itself in the geometric sector as a matter field which interacts only with gravity. Moreover, the mass of this field is very large since it is proportional to Λ2. This means that the range of this field is very very small. Hence, this field should very difficult to be detected, as expected from dark matter.
The other possibility is to remain with theSO(5) group and consider the contribution of the Gribov-Zwanziger fields.
This contribution is also small because these terms are proportional to~2, since it comes from 1-loop contributions.
Moreover, they would also interact only with gravity while a very small equivalence principle violation is expected due to the non-covariance of the Faddeev-Popov operator. this last property is not discarded from a possible dark matter behavior, see for instance [122–124].
Appendix A: Differential forms
This appendix is not a rigorous exposition of exterior differential forms. We provide here a brief survey about the main definitions and results only for operational matters. In other words, we provide here a kind of practical tutorial about differential forms. For details we refer to the vast existing literature on the topic, see for instance [62, 125–128].
1. Exterior product andp-forms
For generality purposes, let us considerD-dimensional manifoldMDas the starting scenario. For each differentiable curve passing at a generic pointx∈MD, there is a tangent vector field. Thetangent space Tx(MD) is the collection of all tangent vectors at x. The differential operators {∂µ}, with µ ∈ {1,· · · , D}, are the components for which a basis in Tx(MD) is constructed. The dual space Tx∗(MD) is called cotangent space and its basis is defined by the objectsdxµ. Such duality is imposed through the inner product rule
(∂µ, dxν) =δµν . (A1)
Theexterior product is typically represented by the symbol ∧. For two components of the basis forTx∗(MD), the exterior product is defined as the anti-commuting product
dxµ∧dxν =dxµ⊗dxν−dxν⊗dxµ=−dxν∧dxµ. (A2) In the case ofµ=νwe have a null exterior product, obviously. It is easy to see that the exterior product is associative.
For the sake of simplicity, from now on, we omit the∧symbol from the notation.
An exterior differential formω, or simply a p-form, is defined by ω= 1
p!ωµ1µ2···µpdxµ1dxµ2· · ·dxµp, (A3) where ωµ1µ2···µp are covariant tensor components of order p 6 d. Due to (A2), these components are completely anti-symmetric on their indexes. Moreover, the restrictionp6dfollows from the obvious fac
The exterior product between a p-form α and a q-form β results in a r-form γ such that r = p+q, which is non-vanishing ifr6d, given by
γ=αβ= (−1)pqβα , (A4)
DRAFT
where the factor ±1 comes from the anti-commuting nature of the exterior product: two generic forms will anti- commute only if both forms are odd in their ranks.
2. Exterior derivative The differential operatord, which is called exterior derivative, is defined by
d=∂νdxν . (A5)
It acts onp-form as follows dω= 1
p!(∂ν)dxν ωµ1µ2···µpdxµ1dxµ2· · ·dxµp
= 1
p!∂νωµ1µ2···µpdxνdxµ1dxµ2· · ·dxµp. (A6) It is evident that the exterior derivative is a map from the space ofp-forms to the space of (p+ 1)-form.
The exterior derivative has two important properties. The first of them concerns its nilpotency,d2= 0. The second is about its application on a product of forms and the Leibniz rule. To illustrate this second point, let us take a p-formαand aq-formβ. The action of the exterior derivative on the product is explicitly shown below,
d(αβ) =dαβ+ (−1)pαdβ . (A7)
If the exterior derivative acts on a sum of forms, then
d(c1α+c2β) =c1dα+c2dβ , (A8) wherec1andc2 are 0-forms.
3. The Hodge dual operator
The Hodge duality consists in an operation that maps ap-form into a dual (D−p)-form. The Hodge dual operator is a map as follows,
∗dxµ1dxµ2· · ·dxµp =
√g
(D−p)!µ1···µpµp+1···µddxµp+1· · · dxµD , (A9) whereg=|detgµν|. Hence, the Hodge dual associated to ap-formω is
∗ω=
√g
p!(D−p)!ωµ1···µpµ1···µpµp+1···µddxµp+1· · ·dxµD . (A10) The Hodge duality is an isomorphism that maps the space of allp-forms Ep in the space of allD−p-formsED−pas
Ep7−−→? ED−p . (A11)
Since the dimension of the spacep-form is given by dimEp=
D p
≡ D!
p!(D−p)! , (A12)
it is straightforward to see that
dimED−p = D
D−p
, (A13)
i.e., the dimension of the dual space has the same dimension of the space ofp-forms. The Hodge dual operator satisfies the linear property,
∗(c1α+c2β) =c1∗α+c2∗β , (A14)
wherec1andc2 are 0-forms. When the Hodge dual operator is twice employed, we obtain
DRAFT
??=
((−1)p(D−p) if MD is Euclidean,
(−1)p(D−p)+1 if MD is Minkowskian. .
Hence, the double Hodge dual operation brings back the p-form to the original shape up to a signal. Another interesting mathematical relation is about Levi-Civita tensor. In aD-dimensional Euclidean space the volume form is related to that tensor as
dx1· · ·dxD= 1
D!µ1···µDdxµ1· · ·dxµD ⇒dxµ1· · ·dxµD =µ1···µDdx1· · ·dxD , (A15) while in aD-dimensional Minkowskian space we have
dx0· · ·dxD−1= 1
D!µ1···µDdxµ1· · ·dxµD ⇒dxµ1· · ·dxµD =−µ1···µDdx0· · ·dxD−1 , (A16) since the full contraction between the Levi-Civita tensor is given as follow,
µ1···µDµ1···µD =
(D! for Euclidean space
−D! for Minkowskian space .
It is also possible to employ the vielbein to project the Levi-Civita tensor indexes, eaµ1
1eaµ2
2· · ·eaµk
ka1a2···akµk+1µk+2···µk+` = µ1µ2···µkµk+1µk+2···µk+` ,
eaµ11eaµ22· · ·eaµkka1a2···akµk+1µk+2···µk+` = µ1µ2···µkµk+1µk+2···µk+` . (A17) The inverse relations are easily found
eaµ1
1eaµ2
2· · ·eaµk
kµ1µ2···µkak+1ak+2···ak+` = a1a2···akak+1ak+2···ak+` ,
eaµ11eaµ22· · ·eaµkkµ1µ2···µkak+1ak+2···ak+` = a1a2···akak+1ak+2···ak+` . (A18)
4. Integration of differential forms The integral of ap-formω over an oriented region U ∈Rp is defined as
Z
U
ω= Z
U
1
p!ωµ1···µpdxµ1· · ·dxµp =o(x) Z
U
1
p!ωµ1···µpdx1· · ·dxp , (A19) whereo(x) =±1 depending on the orientation of the coordinate basis. The result is, obviously, a (p−1)-form.
An important property that uses the integration ofp-forms is contained in the definition of the inner product. Let αandβ be genericp-forms. Ifα∗β is anr-form, then we define the inner product on ap-dimensional manifoldMp as
(α, β) = Z
Md
α∗β = Z
U
β∗α . (A20)
Finally, the Gauss-Stokes’ theorem in this formalism can be enunciated:
Z
U
dω= I
∂U
ω . (A21)
Appendix B: Geometrical aspects of gauge theories
In this appendix we summarize some formal concepts about the geometrical structure of gauge theories. A parallel with the gravity geometrical structure is also presented. Notions about topological spaces, maps and manifolds may be required. Proofs are omitted. Standard references are [2, 62, 125]. See also [63] for a short discussion and [129]
for an advanced approach. For simplicity we employ differential form notation and setκ= 1.
DRAFT
1. Principal bundles and connections
We have learned about Lee groups at Sec. I C 1. Together with a differential manifold (spacetime, in this case), A Lie group form the basic structure for gauge theories. Such structure is a sophisticated product of topological spaces called principal bundles, which is a particular case of the more general concept of fibre bundles. Formally, a Fibre bundle is a spaceB(G, M, F), called total space, obeying the following conditions [62]
• B is a differential manifold;
• M, called the base space, is a differential manifold with an open covering{Ui};
• F, called fibre, is a differential manifold;
• Gis a Lie group, called structure group, with left action onF;
• ∃a surjective map53 π:B 7−→M, called projection map.
• ∃ a diffeomorphism54 φi : {Ui} ×F 7−→ π−1(Ui) such that π◦φi(p, f) = p where55 p ∈ M and f ∈ F. Equivalently, φi(p, f) :F 7−→Fpis a diffeomorphism.
• Whenever Ui∩Uj6=∅we have tij(p) =φ−1i (p, f)◦φj(p, f) :F 7−→F ∈G. Moreover,tij(p) must respect –Forp∈Ui,tii(p) = id;
–Forp∈Ui∩Uj,tij(p) =t−1ji (p);
–Forp∈Ui∩Uj∩Uk,tij(p)tjk(p) =tik(p);
The inverse of the projection defines the fiber at p ∈ M, π−1 : (p) = Fp, which is a “copy” of F at p. The diffeomorphism φi is called the local trivialization and is analogous to the concept of local Cartesian product. The mapstij is a smooth map betweenφi andφj and are called transition functions.
Further important definitions are:
• A section onB is a smooth maps :M 7−→B
π◦s=id. Thus, s(p) =π−1(p)∈Fp. A global section is a section defined independently ofUi while a local section is only defined inUi.
• A fibre bundle is a trivial bundle if all transition functions can be taken as identity maps, i.e. a trivial bundle is a direct (Cartesian) product M ×F. Equivalently, a trivial bundle is a fibre bundle which admits a global section. Nevertheless, a fibre bundle is also trivial if the base spaceM is contractible to a point.
A principal bundle is simply a fibre bundle where the fibre coincides with the structure group, F = G, and is denoted by P(G, M). Besides the left action of G onF =G, the right action ofGon the fibre can be defined by means of: Letu∈Gandq= (p, fi)∈P thenφ−1i (qu) = (p, fiu). Applyingφi(q) from the left in this last expression, we findqu=φi(p, fiu). The right action is mapFp7−→Fp and is possible because of the closure property of groups.
A connection on a principal bundle is the fibre bundle analogue of affine connections in Riemannian manifolds. It allows to compare vector in the tangent space ofP(G, M), providing the notion of parallelism in the total spaceP. To do this, we split the tangent space ofP,Tq(P), at a pointq= (p, g)∈P into its vertical and horizontal subspaces, namely Vq and Hq, respectively. The vertical space is tangent to a fiber Gp. As a consequence, as discussed in Sec. I C 1, it is equivalent to the algebra of G. The horizontal space is a complement of Vq, which is not unique.
Hence, aconnection is a unique split of Tq(P) obeying
• Tq(P) =Hq⊕Vq
• A vector field X ∈P split asX =XH+XQ, where XH ∈Hq andXV ∈Vq;
• At the same fibre, horizontal spaces are related by the right action of G, i.e. Rg:Hq 7−→Hqg.
53A surjective map between two topological spaces is a map whose image is entirely covered by the map.
54A diffeomorphism between two topological spaces is a smooth and continuous map whose inverse is also continuous and smooth, i.e. a diffeomorphism is a smooth homeomorphism.
55Obviously,q= (p, f) is a point of the total spaceB.