Faddeev gravity action on the piecewise constant fundamental vector fields

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1 Faddeev gravity action on the piecewise constant fundamental vector fields arxiv:2.703v [gr-qc] 23 Dec 20 V.M. Khatsymovsky Budker Institute of Nuclear Physics of Siberian Branch Russian Academy of Sciences Novosibirsk, , Russia address: Abstract In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of d = 0 -vector fields. We derive the minisuperspace (discrete) Faddeev action by evaluating the Faddeev action on the spacetime composed of the (flat) -simplices with constant -vector fields. This is an analog of the Regge action obtained by evaluating the Hilbert-Einstein action on the spacetime composed of the flat -simplices. One of the new features of this formulation is that the simplices are not required to coincide on their common faces. Also an analog of the Barbero-Immirzi parameter γ can be introduced in this formalism. keywords: Einstein gravity; Regge calculus; composite metric; Faddeev gravity; discrete gravity Introduction Regge calculus [] is a minisuperspace formulation of gravity, that is, exact general relativity (GR) on a family of the metric fields (piecewise flat ones) sufficiently large to approximate any metric with any accuracy. Spacetime is taken as the set of the flat d tetrahedra. Let σ n be n-dimensional tetrahedron or n-simplex. Then the Einstein

2 2 action is a sum over triangles σ 2, 2 R gd x = σ 2 A σ 2α σ 2. () Here, A σ 2 is area of the triangle σ 2, α σ 2 is the defect angle on the triangle σ 2. Regge calculus is expected to be the baze for quantizing GR [2, 3, ] difficult to implement by the other methods because of the formal nonrenormalizability of GR. Let us consider some another minisuperspace theory based on the Faddeev formulation of gravity [5]. In the Faddeev gravity, the metric is composed of d = 0 -vector fields, g λµ = fλ A f µa. (2) Here, λ,µ,...=, 2, 3, ; A,B,...=,..., d. Simple example: locally, d Riemannian space can be considered as a hyper-surface in the 0d Euclidean space. If f A (x) were its coordinates, then we would have fλ A = fa / x λ. (3) IntheFaddeevgravity, however, fλ A (x)areindependent variables. Wecanintroduce an alternative connection Γ λµν = fλ A f µa,ν (f µa,ν ν f µa ), Γλ µν = g λρ Γρµν, () its curvature is K λ µνρ = Γ λ µρ,ν Γ λ µν,ρ + Γ λ σν Γ σ µρ Γ λ σρ Γ σ µν = Π AB (f λ A,ν f µb,ρ f λ A,ρ f µb,ν). (5) Here, Π AB = δ AB f λ A f λb (6) is a projector. Note that it makes usual and covariant derivatives equivalent, Π AB f λb,µ = Π AB µ f λb. (7) Then we can write out the Faddeev action, S = g λν g µρ K λµνρ gd x + ǫ λµνρ K λµνρ d x 2γ = Π [ AB (fa,λf λ µ B,µ fa,µf λ µ B,λ ) g γ ǫλµνρ f λa,µ f νb,ρ ]d x. (8)

3 Here we have generalized it by adding the P-odd /γ-term where γ is an analog [6] of the Barbero-Immirzi parameter [7]-[0] in the usual Einstein gravity (in the connection representation). The variation of the action reads ( δs 2 = f A gδfa λ µ K µ λ ) 2 δµ λ K ǫµνρσ 2γ g K λνρσ +Π [ AB fb,ν ν Tµ λµ +fν B,µ Tµ νλ +fν B,λ Tµ µν ] + ǫτµνρ 2γ g (g λσg κτ g λτ g κσ )fb,ρ κ Tσ µν. (9) Here, the Π AB -part of the field eqs δs = 0 turns out to be a linear homogeneous system for torsion Tµν λ = fλ A (fa µ,ν fa ν,µ ) with a nondegenerate 2 2 matrix leading to Tµν λ = 0. This gives Γ λ µν = Γλ µν, the Cristoffel. Then Kλ µνρ = Rλ µνρ, the Riemann for which ǫ µνρσ R λνρσ 0, and the fµ A -part of the field eqs turns out to be the Einstein eqs. 3 2 The piecewise constant fields In GR, taking the metric as ds 2 = g nn (dx n ) 2 +g αβ dx α dx β (0) where n is any of, 2, 3,, and α,β,γ,... n, we can separate out the derivatives over x n squared in the action as follows, R gd x = (g αγ g βδ g αβ g γδ )g αβ,n g γδ,n g nn gd x+... () If the metric istaken to bepiecewise constant (withg λµ = constinside the-simplices), this implies continuity condition for the induced on the faces metric, g αβ x n =x n 0 +0 g αβ x n =x n 0 0 = 0 x n 0, (2) if some 3-face is described locally by x n = x n 0. That is, the simplices should coincide on their common faces. action In the Faddeev gravity, the fields f A λ S = Π [ AB (fa,λf λ µ B,µ fa,µf λ µ B,λ ) g can be taken to be piecewise constant. The γ ǫλµνρ f λa,µ f νb,ρ ]d x (3)

4 does not contain the square of any derivative and does not require continuity condition on fλ A (x) and thus on the metric g λµ = fλ A f µa. The simplices are not required to coincide on their common faces, and we can take fλ A (x) = const independently in these -simplices. 3 The discrete Faddeev action To evaluatethe discrete action[], we divide IR asthe set ofpoints x = (x,x 2,x 3,x ) by the hypersurfaces a λ x λ + b = 0 (mathematical hyperplanes) into polytopes, in particular, -simplices. f A λ Then we note that the expression are independent in these -simplices. Q AB = f λ A,λ fµ B,µ fµ A,λ fλ B,µ () )iszeroiffa λ dependsonlyononecoordinate: contributionofthe3-simplicesvanishes, and Q AB = const δ(x )δ(x 2 ) in a neighborhood of the 2-simplex σ 2 at x = 0,x 2 = 0 (triangle in x 3,x -plane); 2) is a full derivative so that Q AB = λ (f λ A µ f µ B f µ A µ f λ B) (5) Q AB dx dx 2 = where the contour C encircles the triangle σ 2. This gives C (f Adf 2 B f 2 Adf B) (6) n 2Q AB = δ(x )δ(x 2 ) [fa(σ i)f B(σ 2 i+) i= fa (σ i+ )f2 B (σ i )]+(A B) (7) where the -simplices σ i,i = n surround the triangle σ2. The action is γ 2 ΠAB (σ 2 ) n i= { [f A(σ i)f 2 B(σ i+) f A (σ i+ )f2 B (σ i )] g(σ 2 ) [ fa (σ i )f 3B(σ i+ ) f A(σ i+ )f 3B(σ i )]} ( x 3 σ x 3 σ x 3 σ x σ (8) 3) where σ 2 is formed by the edges σ3,σ with the -vectors xλ σ, x λ 3 σ.

5 5 An invariant form of the action We can rewrite this action in some invariant variables. This invariance is that one w. r. t. the choice of the coordinates x λ assigned to the vertices. Choose some two edges σ,σ 2 in addition to the above σ 3,σ spanning σ 2 so that σ,σ 2,σ 3,σ span some σi 0 σ 2. The difference between x λ of the endpoints of σµ is xλ σ. Invariant edge µ variables f σ λ A or f A σ λ are defined by f λ A = µ f σ µ A x λ σ µ, fa σ λ = f A µ xµ σ λ. (9) Then we can write the contribution of σ 2 to the action, { n [ 2 ΠAB (σ 2 ) f σ A (σi)f σ 2 B (σi+) i= f σ A (σ i+ )fσ 2 B (σ i ) ] det gσ λ σ µ γ [ fσ A (σ i)f σ 3 B(σ i+) f σ A(σ i+ )f σ 3 B (σ i )]} (20) where Π A B=δ A B λf A σ λf σ λ B, g σ λ σ µ =fa σ λf σ µ A (metric edge components). 5 Restoring the continuum action from the discrete minisuperspace one We can obtain the continuum Faddeev action from the found discrete minisuperspace one. This reverse transition is intended to show that the information encoded in the minisuperspace formulation is sufficient to reproduce the essential degrees of freedom of the continuum theory. This procedure is analogous to the work [2] for the usual GR where the Regge action has been shown to tend to the Hilbert-Einstein one if simplicial decomposition of the given smooth manifold is made finer and finer. To do this, we choose the fields to approximate some fixed smooth fλ A (x) arbitrarily closely by making decomposition of spacetime into the -simplices finer and finer (the coordinate steps along the edges x λ σ 0 ). Then we choose the discrete variable f λ A (σ ) to be fa λ(x) at x = x σ, some central point in σ. The procedure is most easily illustrated by the case n = : [ f A (σi)f B(σ 2 i+) f A(σ i+)f B(σ 2 i) ] i=

6 = u f A vf 2 B vf A uf 2 B (2) 6 where u f f(σ3 ) f(σ ) and vf f(σ ) f(σ 2 ) in the continuum limit become derivatives. This gives us the basic structure (some two-dimensional Jacobian) which being summed over the different contravariant components f λ... and variations between the different pairs of simplices f = f(σ ) f(σ ) just leads to f λ A,λ fµ B,µ f µ A,λ fλ B,µ and thus to the Faddeev action. The /γ-part of the action is considered completely analogously. 6 Possibility of the cubic decomposition of spacetime A new interesting property of the Faddeev formulation as compared to the Regge calculus is the possibility to decompose the curved spacetime into the (flat) cubes. Consider decomposition of IR into the rectangular parallelepipeds or cuboids {x x λ 0 +nλ ε λ x λ x λ 0 +(nλ +)ε λ,λ =,2,3,}. (22) Then f λ A(x) is assumed to be constant in each of these cuboids. Above derivation of the minisuperspace Faddeev action remains the same if σ are not -simplices but cuboids. In Regge calculus, the continuity of metric on the faces leads to that the spacetime composed of flat cuboids is flat as well. In the Faddeev gravity, continuity of fa λ (x) is not required, and the flat cuboids can be used for modeling the curved spacetime. The cuboid action looks just as the naively discretized action or the continuum one (8) in which the derivatives are substituted by their finite difference counterparts. It is much simpler than the simplicial one and, at the same time, it is a minisuperspace action. 7 Conclusion To summarize, we have constructed some discrete formalism analogous to the Regge calculus, but based not on the usual GR, but on its Faddeev form. Some new features of this formalism are the following. )Theinvariantphysical variablesareedge0-vectorsfσ A (σ )(orf A σ (σ )),independent

7 7 for the different -simplices σ containing the edge σ. 2) The -simplices do not necessarily coincide on their common faces. 3) /γ-term is present which is an analogue of the so called Barbero-Immirzi parity odd term in the connection representation of the GR action. An advantage of the discrete Faddeev formalism is the possibility to use cubic decomposition instead of the simplicial one for modeling the curved spacetime due to possibility of the field fa λ discontinuities in the Faddeev gravity. The cuboid action looks simple as the naive discretization of the continuum one. Also this possibility of the field fa λ discontinuities allows one to evaluate the area spectrum of any surface as the sum of spectra of independent triangles. In blackhole physics, reasonable physical arguments require a discrete area spectrum, but up to now, only one gravity theory (LQG) allows to obtain it [3] - [5]. We hope that the discrete Faddeev gravity approach is also able to handle this task. The present work was supported by the Ministry of Education and Science of the Russian Federation. References [] Regge T. 96 Nuovo Cimento [2] Regge T. and Williams R. M Journ. Math. Phys. 396 [arxiv: [gr-qc]]. [3] Hamber H. W Gen. Rel. Grav. 87 [arxiv: [gr-qc]]. [] Ambjorn J., Goerlich A., Jurkiewicz J., and Loll R. 202 Physics Reports [arxiv: [hep-th]]. [5] Faddeev L. D. 20 Theor. Math. Phys [6] Khatsymovsky V. M. 203 Class.Quant.Grav [arXiv: [gr-qc]]. [7] Barbero J. F. 995 Phys. Rev. D [arxiv:900 [gr-qc]]. [8] Holst S. 996 Phys. Rev. D [arxiv:95026 [gr-qc]]. [9] Immirzi G. 997 Nucl. Phys. Proc. Suppl [arxiv: [gr-qc]].

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