Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type
Albert Much, Marcos Rosenbaum, José David Vergara, Diego Vidal-Cruzprieto
aa r X i v : . [ m a t h - ph ] M a y Quantum-Corrected Einstein Equations for aNoncommutative Spacetime of Lie-Algebraic Type
Albert Much ∗ 1 , Marcos Rosenbaum †2 , José David Vergara ‡3 , and DiegoVidal-Cruzprieto §41 Centro de Ciencias Matemáticas, UNAM, Morelia, Michoacán, Mexico
Instituto de Ciencias Nucleares, UNAM, D.F., MéxicoJuly 11, 2018
Abstract
A general formula for the curvature of a central metric, w.r.t a noncommutativespacetime of general Lie-algebraic type is calculated by using the generalized braidingformalism. Furthermore, we calculate geometric quantities such as the Riemann tensorand the Ricci tensor and scalar in order to produce quantum corrections to the Einsteinfield equations.
Contents
The complete theory of quantum gravity is an open problem and is considered tobe the most defying problem of our time. Several approaches have been thoroughlyanalyzed, in order to solve this long standing problem. A particular interesting andphysically well-motivated approach is the program initiated by noncommutativegeometry.Non-commutative geometry is, in addition, to being a generalized formalism of ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] the commutative framework, a novel approach that arises from a deep and meaningfulconceptual pondering between general relativity and quantum mechanics. Theprocedure of going into smaller scales leads us to the point where our descriptionof spacetime as a continuum is neither physically nor mathematically well-defined.This is the subject of investigation and the result of the so called "geometrical"measurement problem, [DFR95]. The argument combines the uncertainty principleand the Schwarzschild radius which in turn imply that the measurement of aspacetime point with arbitrary precision is not possible since the energy involved inthis process creates a black hole and thus spacetime, around the Planck length, doesnot have a continuous structure.Hence, the introduction of a noncommutative structure, i.e. a quantized spacetime, isfrom a physical point of view well motivated, it also has been noted in [RVJ07] thatit is an emergent phenomenon arising from reparametrizing a field theory upon anextended phase space. In particular, the geometrical measurement problem and itssolution are ontologically equivalent to solving the motion of the electron in the atomby using a quantized version of the phase space. The quantization process, concerningthe quantum mechanical objects is straight forward from an algebraic point of view.In particular, the observables namely the position and the momentum are promotedto operators (noncommuting) on a Hilbert space. An equivalent path is to deformthe product of the algebra of functions on phase space, [Lan14]. However, in ourapproach to a noncommutative spacetime we take the algebraic (operator-valued)approach.Moreover, in the commutative geometrical case the Einstein field equationsconnect spacetime and energy in a well-known manner. In particular, the curvatureof the spacetime is a main ingredient to solve the field equations. Since a non-commutative algebra, i.e. a noncommutative geometry, is the generalization of thecommutative geometrical case the motivation of this work is to investigate the possiblegeneralization of the Einstein field equations. Therefore, we want to investigatethe possibility of writing down an explicit equation that is the generalization of theEinstein field equations in noncommutative geometry that additionally displays theconnection between spacetime and matter, while taking the noncommutativity intoaccount. The intended generalization should come in terms of quantum correctionsto the usual Einstein field tensor.The main difference in this letter with regards to works that use the quantiza-tion deformation procedure, i.e. deforming the commutative algebra of functions (seefor example [HR06], [AC10], [ABD + A whose commutator relations are of the form (see [Wes03,Equation 2]) [ x µ , x ν ] = C µνλ x λ . The choice to study this case is based on the richness of examples given in [BM14],where quantum corrections up to first order (of the Planck length) are produced inaddition to the "classical" Einstein tensor. Hence, this letter intends to generalizethis approach to general noncommutative spacetimes of a Lie-algebraic type withoutthe reference to a specific model.In particular, as in the standard approach to noncommutative geometry, we define the connection as a linear map that acts on one-forms in the following fashion,[BM14], [Lan14]: ∇ : Ω → Ω ⊗ A Ω ∇ ( aω ) = da ⊗ A ω + a ∇ ( ω ) ∇ ( ωa ) = ( ∇ ω ) a + σ ( ω ⊗ A da ) , where a ∈ A , ω ∈ E is the module and σ is a bimodular map known as the generalizedbraiding, where σ is obtained by σ ( ω ⊗ A da ) = ∇ ( ωa ) − ( ∇ ω ) a = ∇ ([ ω, a ]) − [( ∇ ω ) , a ] − ∇ ( aω ) − a ∇ ( ω )= da ⊗ A ω + [ a, ( ∇ ω )] + ∇ ([ ω, a ]) By using this formula the authors in [BM14] were able to generate corrections of firstorder in their deformation parameter and finally calculate the corresponding Riccitensor and Ricci scalar, from where quantum corrected Einstein field equations wereobtained. In this work, as already mentioned, this is done for a general Lie-algebraicspacetime.Our main motivation is to obtain a general formula for the quantum-correctedconnection regardless of the algebra under consideration. This is achieved by makingextensive use of a requirement that arises in all the before-mentioned examples: theline element has to be in the center of the algebra. Up to date, central bi-modules(see [Lan14, Chapter 6-9], [MP96], [MT88] and references therein) play a fundamentalrole in formulating geometric quantities in the noncommutative geometry (NCG)approach. Hence, the condition of centrality of the metric tensor is essential in allour considerations. Moreover, the centrality of the metric tensor is closely relatedto keeping some of its classical tensorial features, and in particular it allows usto invert it without any ambiguity, see [BM14]. By having an expression for theconnection in a noncommutative Lie-algebraic spacetime we are able to render theRiemann, Ricci and Einstein tensors; these derive ntities are defined analogously totheir classical counterparts and allow us to derive their classical counterparts plusquantum corrections up to first order in the parameter related to Planck’s length.Similar questions have been posed for quantum cosmology seen as a minisuperspaceof quantum gravity, by making use of a noncommutative algebra in the context ofthe Bianchi-I model [RVJM14].Our framework goes as follows: first we need to choose a noncommutative al-gebra for our spacetime, for now we shall restrict ourselves to work with a Lie-algebratype. After this we define a differential calculus where an arbitrary quantity arises,this is due to the fact that a symmetric factor may be added if we consider the sum oftwo commutators; however, we deal with this ambiguity by the end of our procedure.The next step is to demand that our line element is a central element of the algebra.Next, we follow the algebraic formulation of connections made by Koszul [Kos86]where we take advantage of the generalized braiding to obtain a quantum correctionfor the connection. This renders one of our main results, which is a formula for thequantum-corrected Christoffel symbols, up to first order for any algebra, its associateddifferential calculus and any metric. From this point we follow the definitions thatare analogues of their counterparts in classical Riemannian geometry; we obtain theRiemann, Ricci and Einstein tensors, with quantum corrections up to first order.The organization of this paper is as follows; The second section sets the pre-cise mathematical framework in order. The third section derives the most generalformula for equivalents of the geometric entities that are needed for the EinsteinField equations w.r.t. a noncommutative spacetime of Lie-algebraic type.
Conventions . We use d = n + 1 , for n ∈ N and the Greek letters are split into µ, ν = 0 , . . . , n and we choose the following convention for the Minkowski scalarproduct of d -dimensional vectors, a · b = − a b + ~a · ~b . In this section, we first define the algebra that is an essential element in this work.The calculations in the next sections rely on the fact that we have an associative andunital algebra and therefore we are able to define the universal differential calculuscorresponding to this algebra. The universal differential calculus allows to define analgebra of forms, similar to the differential forms, but rather than using functionsand the exterior derivative the idea in NCG is to use the algebra of those functions.Hence, a more general and sophisticated framework is formulated that extends thecommutative case. In particular, the forms are generated by a tensor product w.r.t.the algebra and the derivatives are divided into two cases. The cases are split into thoseof exterior derivative and those of interior derivatives: Outer and Inner (or Interior)derivatives. These derivatives are the generators of the algebra of derivations.
Definition (Spacetime) . Consider a Lie-algebraic spacetime, i.e. the coordi-nates are generators of a noncommutative, associative and unital algebra that fulfillthe following commutation relations, (see [Wes03, Equation 2]) [ x µ , x ν ] = C µνλ x λ . (2.1)with structure constants C µνλ ∈ C for each µ, ν and λ .This definition serves as a starting point for defining a differential calculus, but itneeds to be complemented with the notion of a universal differential algebra. Definition (Universal differential algebra) . Consider an associative algebra A with unit over C , we define the universal differential algebra of forms (c.f. [Lan14,Chapter 7, Section 1] and [Con95, Chapter 3, Section 1]) which is denoted by Ω( A ) = L p Ω p ( A ) as:For p = 0 it is the algebra itself, i.e. Ω ( A ) = A . The space Ω ( A ) of one-formsis generated, as a left A -module by a C -linear operator d : A → Ω ( A ) , called theuniversal differential, which satisfies the relations, d = 0 , d ( ab ) = ( da ) b + adb, ∀ a, b ∈ A . (2.2)If Ω ( A ) is a left (right) A -module we can induce a right (left) A -module structurevia the universal differential given in Equation (2.2), which makes Ω ( A ) a bi-module.With this notion we are ready to build the Ω p ( A ) -space as Ω p ( A ) = Ω ( A ) ⊗ A · · · ⊗ A Ω ( A ) | {z } p − times . An immediate consequence of our definition is that the differential algebra of forms isgraded.
Proposition (Deformation of the differential structure) . Let the noncommu-tative, associative and unital algebra A be defined by the Relations (2.1). Then, theapplication of the universal differential (2.2) on the algebra has the following solution, [ dx µ , x ν ] = (cid:18) C µνλ + S µνλ (cid:19) dx λ =: D µνλ dx λ , (2.3) where the constant tensor components S µνλ ∈ C are symmetric in µ, ν . .2 Centrality condition 5 Proof.
We act with the universal differential on the commutator, the left hand siderenders commutators each of them contain a one-form basis and a generator, the righthand side is just the structure constants of the Lie-algebra contracted with a oneform-basis, d [ x µ , x ν ] = [ dx µ , x ν ] + [ x µ , dx ν ]= (cid:18) C µνλ + S µνλ (cid:19) dx λ − (cid:18) C νµλ + S νµλ (cid:19) dx λ = C µνλ dx λ . Precisely, the notion of a universal differential algebra, allows us to regard dx µ asa one-form basis for the space Ω ( A ) . In the introduction we stated and explained that in addition to the differential calculus,the centrality (w.r.t. the algebra) of the metric is an important requirement. In thissection the implications of the centrality requirement are investigated. We begin bydefining the line element as a tensor product of two one-forms.
Definition (Center of two-tensors) . Let us define the center of two-tensor,which will be denoted by Z (Ω ( A ) ⊗ A Ω ( A )) as the following set Z (Ω ( A ) ⊗ A Ω ( A )) := (cid:8) z ∈ Ω ( A ) ⊗ A Ω ( A ) | [ z, a ] = 0 , ∀ a ∈ A (cid:9) . Definition (Line element) . Let the metric g be element of the center of two-forms, i.e. g ∈ Z (Ω ( A ) ⊗ A Ω ( A )) . The expression for the metric in terms of thebasis, that is a tensor product of two one forms, is given by g = g µν dx µ ⊗ A dx ν , where we assume symmetry for the metric components g µν = g νµ . Proposition (Centrality condition) . Let g ∈ Ω ( A ) ⊗ Ω ( A ) be the line element.Then, the requirement of centrality for the metric tensor, i.e. g ∈ Z (Ω ( A ) ⊗ Ω ( A )) ,has the following solution, [ x λ , g µν ] = D αλµ g αν + D αλν g αµ (2.4) Proof.
Demanding centrality means that [ x λ , g ] = 0 , thus ! = [ x λ , g ] = [ x λ , g µν dx µ ⊗ dx ν ]= [ x λ , g µν ] dx µ ⊗ dx ν + g µν [ x λ , dx µ ⊗ dx ν ]= [ x λ , g µν ] dx µ ⊗ dx ν − D µλα g µν dx α ⊗ dx ν − D νλα g µν dx µ ⊗ dx α where in the last lines we used the Leibniz rule and the solution of the commutatorrelation between the algebra and the differentials given in Equation (2.3). Remark . If we choose the metric and the algebra (choose structure constants),then the symmetric term can be automatically found. Although it is also possible tochoose the structure constants and the symmetric term in order to find the metric,this is a rather unusual path. For the rest of this text we omit the subscript on the tensor product.
We now introduce the concept of the connection for noncommutative algebras anduse it to obtain first order corrections to the covariant derivative, the Christoffelsymbols and the curvature quantities. At the end of this section we shall write down,as well, the Einstein tensor plus the quantum corrections that we obtain by using theconcept of a bi-modular map.As for the concept of the connection which has to be understood as the gener-alization of the Koszul formula [Kos86], see [MT88] and [MP96] we give the followingdefinition.
Definition (Connection) . The connection ∇ is a linear map that acts on one-forms in the following fashion ∇ : Ω ( A ) → Ω ( A ) ⊗ A Ω ( A ) ∇ ( aω ) = da ⊗ A ω + a ∇ ( ω ) ∇ ( ωa ) = ( ∇ ω ) a + σ ( ω ⊗ A da ) where a ∈ A , ω ∈ E is the module and the symbol σ is a bi-modular map known asgeneralized braiding.Next, in order to induce quantum corrections, i.e. corrections in orders of mag-nitude of the structure constant parameters we use the definition of the covariantderivatives and the bi-modular map. First, we give the definition of the bi-modularmap. Definition (Generalized braiding) . The bi-modular map σ is obtained by usingboth expressions of covariant derivatives w.r.t. a left or right module (see Definition3.1). Thus, σ ( ω ⊗ A da ) = ∇ ( ωa ) − ( ∇ ω ) a = ∇ ([ ω, a ]) − [( ∇ ω ) , a ] − ∇ ( aω ) − a ∇ ( ω )= da ⊗ A ω + [ a, ( ∇ ω )] + ∇ ([ ω, a ]) (3.1)In [BM14, Equation 5.6] the covariant derivative was defined in such manner thatquantum corrections can be produced by using the covariant derivative to zeroth orderand the bi-modular map, and it is given by the following definition. Definition (Covariant derivative) . The covariant derivative up to first orderin the structure constants is given by ∇ ( dx µ ) = 12 ( I + σ ) ◦ ∇ ( dx µ ) , (3.2)where the zero order (in the structure constants) of the covariant derivative has beendenoted by ∇ .It should be noted that in the commutative case the generalized braiding becomesjust a flipping. This recovers the usual definition for the classical covariant derivative. Remark . In the subsequent discussion all equalities have to be understood to holdup to first order in the structure constants D . Having explicated the above mathematical structures, we can now derive whatwe consider the most basic formula for our present work, which is the covariantderivative up to first order in the deformation parameters D . This formula will .1 General formula 7 then allows us to extend the Einstein Relativity to the noncommutative frame-work in the further sections. By using the covariant derivative defining Equation(3.2) we calculate the explicit outcome of the covariant derivative for our algebra (2.1).First let us introduce the new symbols ˜Γ , d ˜Γ , d q Γ and Σ , where the first is adecomposition of the Christoffel symbol into its purely classical and purely quantumparts. The second is this symbol differentiated and expanded into a one-form basiswhile the third is the commutator of the Christoffel symbol with the one-form basis.That is ˜Γ µρσ := Γ µρσ + q Γ µρσ , d ˜Γ µρσ =: ˜Γ µρσλ dx λ , Σ σµρσλ dx λ := [ dx σ , Γ µρσ ] d q Γ µρσ =: q Γ µρσλ dx λ . (3.3) Theorem [General formula] The covariant derivative (see Equation (3.2))for the most general Lie-algebraic type of noncommutative spacetime, up to first-orderin the structure constants, is given in terms of the zero-order connection as follows, ∇ ( dx µ ) = − ˜Γ µρσ dx ρ ⊗ dx σ (3.4) where ˜Γ is explicitly given by ˜Γ µρσ = Γ µρσ + 12 Γ µαβ ( D λβρ Γ αλσ + D λβσ Γ αρλ − D αβλ Γ λρσ ) −
12 Γ µαβ [ x β , Γ αρσ ] (3.5) where here Γ denotes the connection of zero-order (in the structure constants).Proof. The Koszul formula for the connection involves the classical connection and ageneralized braiding acting upon it ∇ ( dx µ ) = 12 ∇ ( dx µ ) + 12 σ ( ∇ ( dx µ )) = −
12 Γ µρσ dx ρ ⊗ dx σ − σ (Γ µρσ dx ρ ⊗ dx σ ) , while for the sake of simplicity we are going to focus on the second term, which isprecisely the generalized braiding given in Equation (3.1). Thus, σ (Γ µρσ dx ρ ⊗ dx σ ) = dx σ Γ µρσ ⊗ dx ρ + [ x σ , ∇ (Γ µρσ dx ρ )] + ∇ [Γ µρσ dx ρ , x σ ] . Next, we make use of the third expression in Equations (3.3) to pull Γ µρσ through dx σ ,then the generalized braiding becomes Γ µρσ dx σ ⊗ dx ρ + Σ σµρσλ dx λ ⊗ dx ρ + [ x σ , d (Γ µρσ ) ⊗ dx ρ − Γ µρσ Γ ραβ dx α ⊗ dx β ]+ ∇ (Γ µρσ D ρσλ dx λ + [Γ µρσ , x σ ] dx ρ )=(Γ µσρ + Σ λµσλρ ) dx ρ ⊗ dx σ + [ x σ , d (Γ µρσ ) ⊗ dx ρ ] − [ x σ , Γ µρσ Γ ραβ dx α ⊗ dx β ]+ D ρσλ ( d (Γ µρσ ) ⊗ dx λ − Γ µρσ Γ λαβ dx α ⊗ dx β ) + d ([Γ µρσ , x σ ]) ⊗ dx ρ − [Γ µρσ , x σ ]Γ ραβ dx α ⊗ dx β , where the last four terms come from applying the covariant derivative keeping in mindthe Leibniz rule and the difference between operating it on a zero-form and a one-form.Our calculation of the generalized braiding results in σ (Γ µρσ dx ρ ⊗ dx σ ) =(Γ µσρ + Σ λµσλρ ) dx ρ ⊗ dx σ − D ρσλ d (Γ µρσ ) ⊗ dx λ + [ x σ , d (Γ µρσ )] ⊗ dx ρ − [ x σ , Γ µρσ ]Γ ραβ dx α ⊗ dx β − Γ µρσ [ x σ , Γ ραβ dx α ⊗ dx β ]+ D ρσλ ( d (Γ µρσ ) ⊗ dx λ − Γ µρσ Γ λαβ dx α ⊗ dx β ) + [ d (Γ µρσ ) , x σ ] ⊗ dx ρ + [Γ µρσ , dx σ ] | {z } = − Σ σµρσλ dx λ ⊗ dx ρ − [Γ µρσ , x σ ]Γ ραβ dx α ⊗ dx β , .2 Geometrical Quantities and the Einstein Tensor 8 and, after some fairly straight further cancellations, we arrive at Γ µσρ dx ρ ⊗ dx σ − Γ µαβ [ x β , Γ αρσ dx ρ ⊗ dx σ ] − D αβλ Γ µαβ Γ λρσ dx ρ ⊗ dx σ . Furthermore, note that the remaining commutator results in two terms, one of whichyields [ x β , dx ρ ⊗ dx σ ] = − ( D ρβλ dx λ ⊗ dx σ + D σβλ dx ρ ⊗ dx λ ) whereby, rearranging indicesin such a way that the two-form basis is a common factor for the whole expression,we arrive to the following final result for the generalized braiding σ (Γ µρσ dx ρ ⊗ dx σ ) = (cid:16) Γ µσρ + Γ µαβ ( D λβρ Γ αλσ + D λβσ Γ αρλ − D αβλ Γ λρσ ) − Γ µαβ [ x β , Γ αρσ ] (cid:17) dx ρ ⊗ dx σ . Remark . In the rest of the paper we refer to Equation (3.4) as the general for-mula for the covariant derivative of the most general Lie-algebraic type of a noncom-mutative spacetime.The general formula is the formula for the covariant derivative of the most generalLie-algebraic type of a noncommutative spacetime. If one sets the deformation con-stants, i.e. the structure constants D , equal to zero one obtains the classical case. Forthe quantum terms being unequal to zero there is an interesting term that remains.It is the last term, that depends on the commutator of the connection Γ with thealgebra. In the following we give expressions for that term for specific cases. Proposition
The commutator of the generators of the algebra x µ ∈ A andthe covariant derivative of the differential of the algebra ∇ ( dx ν ) ∈ Ω ( A ) ⊗ A Ω ( A ) isgiven by [ x µ , ∇ ( dx ν )] =( D λµρ Γ νλσ + D λµσ Γ νρλ − [ x µ , Γ νρσ ]) dx ρ ⊗ dx σ , (3.6) and it holds to all orders in the structure constants. Therefore if [ x µ , ∇ ( dx ν )] = 0 wehave [ x µ , Γ νρσ ] = D λµρ Γ νλσ + D λµσ Γ νρλ . (3.7) Moreover, let the connection be a central element, i.e. ∇ ( dx ν ) ∈ Z (Ω ( A ) ⊗ A Ω ( A )) ,then the general formula reduces to ∇ ( dx µ ) = − (cid:18) Γ µρσ − D αβλ Γ µαβ Γ λρσ (cid:19) dx ρ ⊗ dx σ Proof.
The calculation is straight-forward and uses the specific form of the covariantderivative and the solution (2.3) of the commutator. Thus [ x µ , ∇ ( dx ν )] = − [ x µ , Γ νρσ dx ρ ⊗ dx σ ]= − [ x µ , Γ νρσ ] dx ρ ⊗ dx σ − Γ νρσ [ x µ , dx ρ ⊗ dx σ ]= ( D λµρ Γ νλσ + D λµσ Γ νρλ − [ x µ , Γ νρσ ]) dx ρ ⊗ dx σ , where in the commutator we omitted terms that are of order one since the commutatorwould generate a higher order in the structure constants. The second part followsimmediately from the former proposition. Next, we give the definition of the Riemann tensor in the context of NCG. It is givenas a combination of the exterior derivative d , the wedge product ∧ , which maps thetensor product of two elements of the algebra of forms to the skew-symmetric product,and finally the covariant derivative. .2 Geometrical Quantities and the Einstein Tensor 9 Definition
Let ω µ ∈ Ω ( A ) and ∇ be the connection, then the Riemanntensor is given as R ( ω µ ) :=( d ⊗ I − ( ∧ ⊗ I ) ◦ ( I ⊗ ∇ )) ∇ ( ω µ ) . (3.8)By using the former definition and the explicit formula for the covariant derivativefor a general Lie-algebraic noncommutative spacetime we calculate the Riemann tensorexplicitly. Note that this formula, as well, holds in general. Proposition
The
Riemann tensor (see Equation (3.8)) for the most gen-eral Lie-algebraic type of noncommutative spacetime, up to first-order in the structureconstants, is given in terms of the formerly defined symbols as follows, ˜ R µσαρ = ˜Γ µρσα − ˜Γ µασρ + ˜Γ µαλ ˜Γ λρσ − ˜Γ µρλ ˜Γ λασ + Γ µλβ (Σ λβρσα − Σ λβασρ ) . (3.9) In terms of the classical Riemann tensor plus corrections from the quantum part ofthe Christoffel symbol, the new Riemann tensor reads, ˜ R µσαρ = R µσαρ + q Γ µρσα − q Γ µασρ + Γ µαλ q Γ λρσ + q Γ µαλ Γ λρσ − Γ µρλ q Γ λασ − q Γ µρλ Γ λασ + Γ µλβ (Σ λβρσα − Σ λβασρ ) , where q Γ is explicitly given by, q ˜Γ µρσ = 12 Γ µαβ ( D λβρ Γ αλσ + D λβσ Γ αρλ − D αβλ Γ λρσ ) −
12 Γ µαβ [ x β , Γ αρσ ] . Proof.
It should be noted that since our calculation is up to first order, taking theclassical Christoffel symbol suffices. The action of the curvature upon the coordinatedbasis of one-forms is given in terms of its defining Equation (3.8) R ( dx µ ) :=( d ⊗ I − ( ∧ ⊗ I ) ◦ ( I ⊗ ∇ )) ∇ ( dx µ ) , by inserting the equation ∇ ( dx µ ) = − ˜Γ µρσ dx ρ ⊗ dx σ into the definition of the Riemanntensor one has R ( dx µ ) = − ( d ⊗ I − ( ∧ ⊗ I ) ◦ ( I ⊗ ∇ ))(˜Γ µρσ dx ρ ⊗ dx σ )= − d (˜Γ µρσ dx ρ ) ⊗ dx σ + (˜Γ µρσ dx ρ ∧ ∇ ( dx σ ))= − d (˜Γ µρσ ) ∧ dx ρ ⊗ dx σ − ˜Γ µρσ dx ρ ∧ ˜Γ σαβ dx α ⊗ dx β . Since we should obtain a three-form, we have to take all elements of the algebra tothe left, c.f. [Lan14, Chapter 7, Equation (7.12)]. In order to proceed we need tocommute a Christoffel symbol with an element of the basis of one-forms. Given thatour calculation is up to first order, we only consider the classical part of the Christoffelsymbol multiplying the commutator. R ( dx µ ) = − d ˜Γ µρσ ∧ dx ρ ⊗ dx σ − ˜Γ µρσ ˜Γ σαβ dx ρ ∧ dx α ⊗ dx β − Γ µρσ [ dx ρ , Γ σαβ ] ∧ dx α ⊗ dx β . Next by rearranging the indices of the former expression and using the definition ofthe commutator of a one-form with a classical Christoffel symbol given in the thirdexpression of Equations (3.3) we get R ( dx µ ) = − (˜Γ µαβλ + ˜Γ µλσ ˜Γ σαβ + Γ µρσ Σ ρσαβλ ) dx λ ∧ dx α ⊗ dx β = − (˜Γ µαβλ + ˜Γ µλσ ˜Γ σαβ + Γ µρσ Σ ρσαβλ − ˜Γ µλβα − ˜Γ µασ ˜Γ σλβ − Γ µρσ Σ ρσλβα ) dx λ ⊗ dx α ⊗ dx β . Therefore, the components of the Riemann tensor are ˜ R µσαρ = ˜Γ µρσα − ˜Γ µασρ + ˜Γ µαλ ˜Γ λρσ − ˜Γ µρλ ˜Γ λασ + Γ µλβ (Σ λβρσα − Σ λβασρ ) . .2 Geometrical Quantities and the Einstein Tensor 10 Writing down the quantum corrected Christoffel symbol as its classical plus quantumparts we arrive to our main result ˜ R µσαρ = R µσαρ + q Γ µρσα − q Γ µασρ + Γ µαλ q Γ λρσ + q Γ µαλ Γ λρσ − Γ µρλ q Γ λασ − q Γ µρλ Γ λασ + Γ µλβ (Σ λβρσα − Σ λβασρ ) . In order to calculate the Einstein tensor we need the noncommutative analoguesof the Ricci tensor and scalar. In order to avoid certain ambiguities encountered in[BM14], we define these quantities as one does in usual geometry. The motivationtherein is two-fold. The first reason is, as already pointed out, to avoid certain am-biguities. The second reason comes from the core of the deformation quantizationargument. Namely, up to zero order in the deformation parameters, which in our caseare represented by the structure constants, the observables or quantities at hand arethe classical ones. Hence, since we consider the obtained Christoffel symbols as theclassical ones plus quantum corrections, the Ricci tensor and scalar that are obtainedclassically by the trace have to be in the quantum case up to first order obtainable inthe same manner. Moreover, the metric itself has no deformation parameter explicitly.Hence, the (left) inverse metric should obey the same property. Therefore, we givein the following the definition of the Ricci tensor and scalar explicitely by using thegeneral formula for the covariant derivative and in particular by using the result ofthe Riemann-tensor (see Proposition 3.2).
Definition (Ricci tensor and scalar) . The Ricci tensor is the trace of theRiemann tensor over the first and third indices and the Ricci scalar is the trace overthe two indices of the Ricci tensor, i.e. ˜ R βλ := ˜ R µβµλ , ˜ R := g µν ˜ R µν . Next, give the explicit formulas for the Ricci tensor and scalar by using the gen-eral formula for the covariant derivative and in particular by using the result of theRiemann-tensor (see Proposition 3.2).
Proposition [Ricci tensor and Ricci scalar] The Ricci tensor for the most gen-eral Lie-algebraic type of noncommutative spacetime, up to first-order in the structureconstants, is given by ˜ R σρ = ˜Γ µρσµ − ˜Γ µµσρ + ˜Γ µµλ ˜Γ λρσ − ˜Γ µρλ ˜Γ λµσ + Γ µλβ (Σ λβρσµ − Σ λβµσρ ) (3.10) and by rewriting the quantum corrected Riemann symbol as a classical part plus apurely quantum one, we obtain ˜ R σρ = R σρ + q Γ µρσµ − q Γ µµσρ + Γ µµλ q Γ λρσ + q Γ µµλ Γ λρσ − Γ µρλ q Γ λµσ − q Γ µρλ Γ λµσ + Γ µλβ (Σ λβρσµ − Σ λβµσρ ) . Thus taking the trace, leads us to the Ricci scalar that reads ˜ R = R + g ρσ (cid:16) q Γ µρσµ − q Γ µµσρ + Γ µµλ q Γ λρσ + q Γ µµλ Γ λρσ − Γ µρλ q Γ λµσ − q Γ µρλ Γ λµσ +Γ µλβ (Σ λβρσµ − Σ λβµσρ ) (cid:17) . (3.11)In addition, using the former results of the Ricci tensor and its scalar, we finallyturn our attention to the observable that is of real physical importance, i.e. theEinstein tensor. Since we do not have a general definition of covariance, we impose thatthe Einstein tensor has the same form as the original. Of course, this is an assumptionbut we consider that it is consistent in the sense of having an appropriate classical limit.Hence, the quantum corrected Einstein tensor for a general Lie-algebraic spacetime isexplicitly expressed by the following theorem. .3 Bicrossproduct model 11 Theorem
The Einstein tensor is defined analogously to the classical by takingthe classical and quantum parts of the Ricci tensor and scalar (see former Proposition),i.e. ˜ G σρ := ˜ R σρ −
12 ˜ Rg σρ . Hence, the explicit Einstein tensor reads ˜ G σρ = G σρ + q Γ µρσµ − q Γ µµσρ + Γ µµλ q Γ λρσ + q Γ µµλ Γ λρσ − Γ µρλ q Γ λµσ − q Γ µρλ Γ λµσ + Γ µλβ (Σ λβρσµ − Σ λβµσρ ) − g αβ (cid:16) q Γ µαβµ − q Γ µµβα + Γ µµλ q Γ λαβ + q Γ µµλ Γ λαβ − Γ µαλ q Γ λµβ − q Γ µαλ Γ λµβ + Γ µλτ (Σ λταβµ − Σ λτµβα ) (cid:1) g σρ . (3.12) Proof.
It is immediate from Proposition 3.3.
Since our result is a generalization of the results obtained in [BM14] and thereforean important consistency check is to see if the general formula reproduces the resultsobtained by the original authors. The algebra of the so called Bicrossproduct modelis defined in the following:
Definition (Bicrossproduct model algebra) . The Lie-algebra for the bi-crossproduct model of spacetime is [ x, t ] = λx [ x, dt ] = λdx [ t, dt ] = λdt, λ ∈ C / R . The former definition implies on the constants that we denoted by D , which are thesum of the structure constants C and the symmetric constants S (see Equation (2.3)),the following equalities D = − λ D = − λ. Moreover, since the general formula makes extensive use of the zeroth-orderChristoffel symbols, and in order to compare our findings with the bicrossproductmodel we state their exact form in the following, [BM14, Appendix A, A.1] Γ µν = (cid:18) − bt x − (1 + 2 bt ) x − (1 + 2 bt ) − tx − (1 + bt ) (cid:19) , Γ µν = (cid:18) − bx bt bt − bx − t (cid:19) . In order to simplify the calculation further we introduce the central variable v = xdt − tdx , as the original authors did, in order to reduce the expressions for theclassical connections which are given as ∇ ( dx ) = 2 bx − v ⊗ v ∇ ( v ) = − x − v ⊗ dx, We shall as well make extensive use of the following relation ∇ ( v ) = ∇ ( xdt − tdx ) = dx ⊗ dt − dt ⊗ dx + x ∇ ( dt ) − t ∇ ( dx ) , (3.13)which although it is an identity, we want to bring attention to it. The main cause isthat all the calculations in [BM14, Section 5] are expressed in terms of v and dx whileours are in dt and dx . Lemma (Classical connection on dt ) . The classical connection acting on dt is anelement of the two forms of the bicrossproduct model and it is given in terms of thecentral element v as ∇ ( dt ) = x − ( − v ⊗ dx + 2 btv ⊗ v − dx ⊗ v ) . .3 Bicrossproduct model 12 Proof.
The use of Equation (3.13) allows us to rewrite the connection ∇ ( dt ) in termsof the covariant derivative on the central variable v and the covariant derivative onthe differential of x , i.e. ∇ ( dt ) = x − ( ∇ ( v ) + t ∇ ( dx ) − dx ⊗ dt + dt ⊗ dx ) . = x − ( − x − v ⊗ dx + x − btv ⊗ v − dx ⊗ dt + dt ⊗ dx )= x − ( − v ⊗ dx + 2 btv ⊗ v − dx ⊗ v ) . where in the last lines we used the explicit form of the central element v = xdt − tdx and solved for dt , i.e. dt = v + tdx .By using the former results we are ready to calculate the Christoffel symbol whichis induced from our general formula. Proposition
The quantum corrected connection for dx in our formalism is ˜Γ σα =2 b (cid:18) − x t + λ/ t + λ/ − x − t ( t + λ ) (cid:19) . This result matches the authors findings in [BM14, Proposition 5.1].Proof.
We proceed by a direct computation of the commutators between the genera-tors and the covariant derivatives, the first quantity we consider is [ t, ∇ ( dt )] =[ t, x − ( − v ⊗ dx + 2 btv ⊗ v − dx ⊗ v )]=2 λx − ( − v ⊗ dx + 2 btv ⊗ v − dx ⊗ v ) = 2 λ ∇ ( dt )=2 λx − (cid:0) − x (1 + 2 bt ) dt ⊗ dx + 2 t (1 + bt ) dx ⊗ dx + 2 btx dt ⊗ dt − x (1 + 2 bt ) dx ⊗ dt (cid:1) + O ( λ ) , while for the second we find [ x, ∇ ( dt )] =[ x, x − ( − v ⊗ dx + 2 btv ⊗ v − dx ⊗ v )]= 2 bx − [ x, t ] v ⊗ v = 2 bλx − v ⊗ v = λ ∇ ( dx ) . For the remaining covariant derivatives we have the following commutator relations [ t, ∇ ( dx )] =2 b [ t, x − ] v ⊗ v = 2 bλx − v ⊗ v = λ ∇ ( dx ) , [ x, ∇ ( dx )] =0 . Next, for the purpose of calculating the general connection, i.e. the zero order Christof-fel plus the quantum corrected one we use Proposition (3.6) which followed from thegeneral formula. Hence, the result is written in the following form ˜Γ τσα =Γ τσα + 12 Γ τνµ ([ x µ , ∇ ( dx ν )] σα − D νµλ Γ λσα ) , where the common notation [ x µ , ∇ ( dx ν )] σα dx σ ⊗ dx α := [ x µ , ∇ ( dx ν )] has beenused. Further, by substituting the value for the classical Christoffel symbols andall the commutators and by setting τ = 1 the quantum corrected Christoffel-symbolreads, ˜Γ σα =Γ σα + 12 Γ ([ t, ∇ ( dt )] σα − D Γ σα ) + 12 Γ ([ x, ∇ ( dt )] σα − D Γ σα )+ 12 Γ ([ t, ∇ ( dx )] σα )=Γ σα − bx (2 λ ∇ ( dt ) σα + λ Γ σα ) + bt ( λ ∇ ( dx ) σα + λ Γ σα ) + bt ( λ ∇ ( dx ) σα )=Γ σα − bλx ( − σα + Γ σα ) + bt ( − λ Γ σα + λ Γ σα ) − bλt Γ σα =(1 − bλt )Γ σα + bλx Γ σα and by summarizing the result in matrix notation the proof is completed. Note thatthis result indeed matches the authors findings in [BM14, Proposition 5.1]. By using the works of ([Con95], [MP96], [MT88] and [BM14]) we calculated aformula for the covariant derivative for a general noncommutative spacetime ofthe Lie-algebraic type, see Theorem 3.1. The main ingredient that entered thiswork was to demand the centrality of the metric tensor (or the line-element asusually Physicists refer to) and the exterior derivative d that can be defined forany associative unital algebra. This is done by using the universal differential calculus.The general formula allowed us to calculate the corresponding geometrical en-tities such as the the Riemann tensor and the Ricci scalar. This furthermore allowedus to write the Einstein-tensor in terms of the "classical" one, where here classicalrefers to the zero order in the deformation parameter, plus quantum correctionsstemming from the noncommutativity of the assumed algebra (see Equation (3.11)for the explicit formula).The next line of work in this context is investigating physical consequences ofthis formula and examining the physical reality related to those models. Furthermore,with all the geometry issues settled the next steps in progress are to explore thedynamics of matter in a noncommutative spacetime. The interest w.r.t. matter comesfrom the argument that the noncommutative geometry may give origin to matter, see[CC96] and the review [Ros01]. By applying our general formulas to specific modelswe intend to see if this statement holds in the context of noncommutative spacetimesof the Lie-algebraic type. This however is current work in progress.Another interesting aspect is the question if other types of noncommutativealgebras produce similar results, i.e. are our corrections unique w.r.t. arbitrary non-commutative spacetimes or are they a specific result of noncommutative spacetimesof the Lie-algebraic type? References [ABD +
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