On noncommutative spherically symmetric spaces

Two families of noncommutative extensions are given of a general space-time metric with spherical symmetry, both based on the matrix truncation of the functions on the sphere of symmetry. The first family uses the truncation to foliate space as an infinite set of spheres, is of dimension four and necessarily time-dependent; the second can be time-dependent or static, is of dimension five and uses the truncation to foliate the internal space.


Introduction
Finding an operator description of realistic gravitational configurations is an important task of any theory aiming to describe quantum gravity. Such description, here referred to as 'quantization', would incorporate at least effectively properties of spacetime beyond the classical regime; it could as well be fundamental. The basic aspect of quantization is representation through operators: concrete representation yields the eigenvalues and the eigenstates of the operators of position, metric and curvature. Therefore, possible outcomes of quantization can be discretization of spacetime, if the spectra of coordinates are discrete, or resolution of singularities, if the classically singular values do not belong to the spectrum of curvature invariants. Another important question, of interest in noncommutative geometry, is whether the algebra of operators A together with the calculus which supports the given gravitational configuration can be in some sense regarded as a 'noncommutative space', that is, whether the notions as connection and metric can be introduced geometrically or are just quantities defined 'externally', as fields with given properties.
In spite of the progress which has been made in the recent years in noncommutative geometry and in quantum gravity in general, some important results are still lacking. For physics the most interesting spacetimes like the Schwarzschild black hole or the Friedmann-Robertson-Walker (FRW) cosmology have not been 'quantized' yet, at least not to a common consent.
Relevant or 'realistic' models have been obtained only in two dimensions and reduce essentially to the quantum line and the fuzzy sphere. A generalization of the quantum line has been proposed in every dimension [1], and in every even dimension there is a simple smooth noncommutative space with planar symmetry defined by the constant commutators J µν 0 between coordinates, [x µ , x ν ] = ikJ µν = ikJ µν 0 . (1.1) Usually J µν 0 is taken to be nonsingular; constantk defines the length scale at which the effects of noncommutativity become significant. However, these spaces are not isotropic except in two spatial dimensions while the important classical configurations have spherical symmetry.
Therefore, if we aim to find 'realistic' quantum spacetimes we need to construct four-dimensional spaces with exact or slightly deformed spherical symmetry, and this is the main objective of our paper. The approach we are using is that of noncommutative differential geometry because we believe that one should be able to describe quantum or quantized gravity as the classical: through geometry. We shall see that, although noncommutative spaces which we obtain here do not have exactly the desired classical limits, the formalism which we use seems to be an appropriate tool. We hope that it will be possible to find solutions yet closer to the classical ones by dimensional extension of the algebras here analyzed.
In ordinary differential geometry a metric is introduced to measure the distance between points. Since noncommutative geometry is essentially without points the concept might seem in this case rather pointless. It is possible however to carry over one of the definitions of a metric which is used in commutative geometry and to give it a meaning in certain noncommutative cases as a measure of distance. In ordinary differential geometry if ξ and η are 1-forms then the value of the inner product g(ξ ⊗ η) at a given point does not depend on the values of ξ and η at any other point. This condition can be expressed as the bilinearity condition, that for any function f We assume that this condition remains valid in the noncommutative case. Without the bilinearity condition it is not possible to distinguish for example in ordinary spacetime a metric which assigns a function to a vector field in such a way that the value at a given point depend only on the vector at that point from one which is some sort of convolution over the entire manifold.

General formalism
A noncommutative geometry as we define it [2] consists of an associative algebra A (here referred to as 'noncommutative space') over which sits a differential calculus Ω * (A). Within this structure we single out a set of n 1-forms θ α , a noncommutative extension of the moving frame or n-bein of classical differential geometry. Conversely one can define the differential calculus to be such that any chosen special set of 1-forms commute with the elements of the algebra [x µ , θ α ] = 0. (2.1) We refer to it as a frame since it seems to be the most natural generalization of the moving frame of Cartan. In fact, the bimodule structure of the 1-forms uniquely determines a differential calculus over the algebra, With the frame it is relatively easy at least to first order to calculate the objects of interest in differential geometry: beyond first order more imagination will certainly be necessary.

Kinematics
If instead of the 1-form θ α we had an element f of the algebra, then condition (2.1) could be written to first order as and states that the center of the algebra is trivial. With the frame element however we can only state that the condition is first order in the derivatives. One can think of it then as a constraint. We shall find a second condition e α C α βγ = 0 (2.4) in terms of the Ricci rotation coefficients which one can think of as a gauge condition.
The derivations e β dual to θ α , θ α (e β ) = δ α β , are usually assumed to be inner, This is the simplest and in some important cases [2,3] like matrix spaces the only choice to define vector fields. Sometimes however the condition (2.5) cannot be imposed: in the case of quantized phase space of quantum mechanics for example, momenta p α = −i δ µ α ∂ µ do not belong to the algebra of coordinates. When coordinates commute the dimension of the phase space is necessarily twice the dimension of the configuration space. In noncommutative geometry it need not be so: the phase space can be identical to the configuration space, that is, 'half' of its classical analogon. The answer to the question: how many operators generate A? defines as we shall see the initial conditions of the problem which we are trying to solve.
The momenta necessarily satisfy a quadratic relation of the form This 'structure equation' follows from the conditions which we impose on the differential; it gives the same information about geometry of the space as the set of derivations e α or the set of frame elements θ α . The differential df of a function f ∈ A is given by when the module of 1-forms is free and the frame is used as basis. We can write to first order The iǫ here is the product of ik and the square µ 2 of a macroscopic mass scale parameter µ, which we can relate for example with the Schwarzschild mass m or with the cosmological constant Λ. As beforek defines the scale of noncommutativity through the commutation and if we take for example thatk ∼ l 2 P l and µ 2 ∼ Λ we obtain ǫ ∼ 10 −122 . The coefficients Q αβ γδ , F β γδ , K βγ are antisymmetric in the lower pair of indices while Q αβ γδ is symmetric in the upper pair; they are hermitian elements of the center of the algebra A. Formula (2.6) is equivalent to writing the Ricci rotation coefficients C γ αβ , [e α , e β ] = C γ αβ e γ (2.10) as linear expressions in the momenta, It can also be considered as the definition of a Poisson structure, to which it is known [4,5,6] one can associate a curvature.
All derivations here are taken in the semi-classical approximation that is to leading order in ǫ.
This is in fact the only possibility to do concrete calculations when one is solving a problem like ours: to find an a priori unknown algebra, that is, commutators J µν (x). Then we have for example It is important to note and easy to see that in this approximation the formalism has diffeomorphism invariance, that is the commutators transform as tensors. Indeed if we make the change of variables x ′µ = x ′µ (x ρ ), using (2.12) we obtain This means that the diffeomorphism symmetry is present not only in the classical limit but also in the linear order ink.
To define and study a noncommutative space A there are two equivalent paths (the 2-fold way). Starting with a set of vector fields e α on a smooth manifold which satisfy (2.10) one can define the momenta as the solutions to the equations [p α , x µ ] = e µ α , (2.14) which follow from the expression dx µ = e µ α θ α .
(2.15) Relation (2.6) then leads to a set of important consistency conditions which one has to solve.
Alternatively, one can start from a moving frame θ α on a classical manifold satisfying and search for an analogous noncommutative frame, extending thus Ω * (C) to Ω * (A). In both cases one obtains the essential ingredients to construct a noncommutative geometry, an algebra and a compatible differential calculus. We find however that there are restrictions on the ingredients which arise from the Jacobi identities.

Dynamics
The commutator (2.9) must satisfy Jacobi identities which can be written in the form of vanishing of an anomaly. We shall accept the assumption that this condition is the only obstruction to the associativity of the product. There are also Leibniz rules, obtained by replacing one position generator x λ by a momentum p α . As we are restricting our attention to the case when momenta depend on coordinates, the additional Jacobi identities follow from (2.17). The Leibniz rules can be written in the form It is easily seen that the analogous compatibility conditions for equation (2.9), are equivalent in fact to the quadratic relation (2.6) for the momenta.

Real-space foliation
The simplest spherically symmetric noncommutative space is the fuzzy sphere [7,8]. Anticipating that isotropic spaces in higher dimensions contain it as a subspace, we will review its basic properties. Two angular variables of the polar coordinate system on the fuzzy sphere are described by three Cartesian coordinates which satisfy commutation relations of the algebra of rotations. The Lie algebra so 3 has irreducible representations by three n × n complex matrices J a ∈ M n which we normalize such that Let x a , a = 1, 2, 3 , be the quantized coordinates. Quantization is defined by relation which is an analog of the Bohr quantization and postulates that the area of the quantum sphere contains an integer number of elementary cells of areak. It defines also the radius r since from the Casimir relation of so 3 for large n we obtain The commutator of the coordinates is then given by Relation (3.3) suggests that the three-dimensional space in polar coordinates is, or can be represented as a direct sum of all irreducible representations of so 3 : this model is called the 'onion model'. As we need a four-dimensional manifold we shall not use it; simple extensions by one algebra generator do not have the desired properties and limits too [9,10]. Therefore we add the radius and the time as coordinates independent on x a . That is, we consider space-time A = A ′ ⊗ A ′′ to be the tensor product of the fuzzy-sphere algebra A ′ and an algebra A ′′ generated by r and t.
The momenta on the fuzzy sphere can be identified with the coordinates, We have therefore and if we let π a b = δ a b − 1 r 2 x a x b be the radial projection onto the sphere, we obtain from definition (2.7) the expression for the differentials dx a , and inversely, In the leading order ink we can write (3.10)

The momentum algebra
Since the information about symmetries is contained in the frame derivatives e α that is in p α , the momentum algebra * is the best starting point to search for A. It has in addition a fixed quadratic form which reduces the number of possible Ansätze. We investigate then the extensions of the so 3 algebra by two operators p 0 and p 4 which we introduce in place of r and t as generators of A ′′ . We start with an algebra of a general form 15) * We refer to the algebras generated by the coordinates and the momenta respectively as the position algebra and the momentum algebra although we assume in this section they are one and the same algebra.
which manifestly respects rotational symmetry. In accordance with the requirement that the algebra be quadratic we write The a k , b k and c k are constants while Ξ is a general quadratic polynomial in p 0 and p 4 .
When we impose Jacobi identities on (3.11-3.15) we obtain various restrictions. For example we find and from these relations we conclude that π 0 and π 4 are mutually proportional, The constants are also not independent: we have The most important implication of the Jacobi constraints is that there exists always a linear combination of p 0 and p 4 which commutes with p a : it is in fact equal to Π. Using Π, the momentum algebra simplifies to We adopt here the convention that the momenta are antihermitian operators.
It is easy to rewrite algebra (3.22) in the tensor-product form by introducing the hermitian frame components ζ a of a vector (which we will later use as coordinate), (3.25) We see that A indeed contains an so 3 as a subalgebra; A is a tensor product of the so 3 generated by three ζ a and a quadratic A ′′ defined by relation However, differential calculus defined by (p a , Π, π 4 ) is not a tensor-product calculus because the momenta do not belong to the factor algebras. This property is desired as otherwise for the metric we would obtain a simple product-metric.
We mentioned already that within the frame formalism the momentum algebra determines space-time geometry. Of course there is a freedom in the choice of the linear conection, which then uniquely gives the torsion and the curvature; these expressions one can show are generically quadratic in the momenta, [2,11]. But momenta are a priori unknown functions of coordinates. Therefore although our model is essentially fixed, to understand it in more details and to find its classical limit we need to determine coordinates.

The position algebra
The functional dependence of p α on x µ is not unique, but as everything else it is constrained by the algebraic structure of A and the required properties of the commutative limit. In the present case there is only one spatial 3-vector in the algebra so it is natural to assume that coordinates x a are proportional to it. In fact we already introduced the hermitian ζ a , p a = Πζ a , as generators which provide the tensor-product form of the algebra (3.25). We can thus choose ζ a as coordinates, x a = ζ a ; one can see easily from (3.43) that the choice of the proportionality factor does not influence the form of the angular part of the line element.
Of course, only two of the ζ a are independent because ζ 2 = ζ a ζ a is the Casimir of A. The operators Π and π 4 have to be mutually independent functions of r and t. An almost obvious Ansatz for these functions is Coordinate components of the (inverse) metric can be found from relation where the frame components e µ α are given by e µ α (x) = [p α , x µ ]. One can also use the inverse θ α µ of e µ α , g µν = θ α µ θ β ν η αβ .
(3.30) Two given expressions are equivalent within the precision we are working with, that is to the operator-ordering ambiguities. We can also write the line element as and then express the frame 1-forms through the differentials, θ α = θ α µ (x) dx µ , to obtain g µν . The constant frame metric g αβ = η αβ here has the signature (− + + + +) .
The commutator of r and t, can be obtained from compatibility of (3.22) with (3.32). Using it we find easily For differentials of the coordinates we have we introduced ζ i = δ i a ζ a . From these expressions we obtain the frame components We can calculate the metric by applying (3.30); the nonzero elements are We see immediately that in the classical limitk → 0 off-diagonal terms g 0i vanish. Further, since we can assume that the Casimir ζ 2 = 1 4 ǫ 2 (n 2 − 1) is small, we have 1 − ζ 2 > 0. The angular part of the metric is, as expected, proportional to the projector on the sphere π ab , The prefactor in (3.38) is also correct: the inverse of g ij is on the sphere given by and therefore the angular part of the line element is In the classical limit we obtain We have a time-dependent metric. Namely, we cannot assume that G(t) = G 0 , a constant, because momenta Π and π 4 have to be functionally independent. Introducing instead of t a new variable τĠ we can simplify the line element to with the curvature scalar R (3.50) Since the Ansatz (3.28) which we used does not give a static metric which could be interpreted as an extension of the Schwarzschild black hole, perhaps it is possible to modify it to obtain the extension of the FRW cosmology? Then instead of (3.44) we would need the angular line element ds 2 Ω = a 2 (t)r 2 dΩ . This corresponds presumably to a more general dependence, We have where constants α, β, γ and λ satisfy βγ − αλ = cµ 2 = 0. (3.54) The change the variables to   Using (3.30) we find: Again the off-diagonal components g 0i and g 4i vanish in the commutative limit and the angular part is proportional to the projector on the sphere: the corresponding classical metric is spherically symmetric, non-static and block-diagonal. Inverting the angular part we get that is, The (r, t) part on the other hand gives (3.71) One can further simplify using variables ρ and τ and obtain (3.72) Having the line element written as this it is easy to recognize that the change of coordinates LG → e −λτ (3.73) gives back the metric (3.47). This is a manifestation of diffeomorphism invariance of the formalism, and shows that the noncommutative space which we obtained is in fact unique.

A representation
The coordinates of the solution (3.28) satisfy the commutation relation e λτ r = q −1 re λτ .
where τ is the dilatator, or by We have here an indication of the importance of the calculi in the description of the geometries: the differential calculus usually introduced on a space described by the Heisenberg algebra is flat, while here the subspace (r, τ ) has a constant negative curvature, [12]. One can find even an example of an algebra over which there are two different calculi with geometries having as commutative limit two different topologies.

Internal-space foliation
We have seen that in the previous section geometry of the space-time was fully determined by momenta and by their commutators. Such situation is typical when we apply the noncommutative frame formalism in its 'minimal' version, that is when all p α ∈ A. A close relation between the algebraic and the geometric structures is however a general feature of any noncommutative geometry, also in cases when we need to extend the space-time in order to obtain the appropriate phase space or the prescribed symmetries.
We shall attempt now to find static spherically symmetric metric starting from the algebra of coordinates. This means that we will first make an Ansatz for the position commutators and for the frame, and then analyse the implied consistency relations. This approach is perhaps more intuitive and seems easier because the position algebra is not restricted in its form like the momentum algebra. However, the overall number of equations remains the same if we constrain the momenta to belong to A: a real advance comes if we allow for some of the derivations to be external. As a result, we shall find a way to extend A and to obtain static solutions.

The position algebra
Assume that A is generated by operators x µ = (ξ a , ρ, r, t) with for some large n ξ a = 2 n J a . (4.1) Variable ρ is a fifth generator which we can consider as a Kaluza-Klein extension. We expect a general spherically symmetric solution to depend on ρ, r and t but we shall restrict our attention to the static case.
The multiplication table is and it gives to first order the equation (4.6) or if we introduce γ 0 and γ 4 as J 0 = Jγ 0 , J 4 = Jγ 4 , As a convenient solution we can choose The last equation is identically fulfilled in the static case.
For comparison with the previous section we introduce x a = ρξ a , x a x b δ ab = ρ 2 and the

The frame
As the angular part of the frame we choose the Mauer-Cartan frame of the group multiplied by a function h to account for the change in volume of the 3-sphere as we move along the radial; the radial and the time-like components of the frame are, we suppose, diagonal. We obtain for the frame in five dimensions (4.11) The f , g and h are functions of ρ, r and t. As on the fuzzy sphere this Ansatz gives ρ dρ + dρ ρ = 0 (4.12) which is unusual because it implies dρ = 0 in linear order even though ρ is not in the center.
In particular, dρ = 0 in the classical limit. This is a specific feature of our calculus (4.11) and is a consequence of the fact that the algebra of coordinates, we shall see, is different from the algebra of momenta.
To be consistent, differential calculus (4.11) has to be compatible with algebra (4.2-4.3) and this compatibility is expressed by equations (2.18). To leading order the right-hand side of these equations is given by while to calculate the left-hand side we use the property that the elements of the frame basis commute with A, (4.14) An analysis of all constraints which follow from these requirement yields: We must impose additional relations which assure that the exterior multiplication of two frame 1-forms is well defined. These relations follow from the definition of the wedge product [2] θ α θ β = P αβ γδ θ γ θ δ . (4.16) In lowest order we obtain where Q αβ γδ are constants introduced in (2.8). In our case, (4.17) become the following equations: [θ a , θ b ] = 0 (4.18) There is one remaining relation for [θ a , θ 0 ] which can be obtained from The first set of equations (4.18-4.22) gives the following constraints: and from the second set we obtain: the C k are constants.

Solutions
We obtained a relatively complicated set of equations, but as we are looking for static solutions we can assume that no function depends on time. In fact, the proposed choice J 4 = 0, L = 2ρ 2 /nk satisfies almost all of the equations. The first nontrivial constraint is J 0′ = 0 and it implies J 0 = J 0 (ρ). The remaining are: whereas (4.25) is satisfied identically with C 4 = 0. We have three equations for three frame functions f , g and h.
When we are solving (4.28-4.30) we should not forget the diffeomorphism invariance. There are essentially two different cases. In the simplest case the frame does not depend on ρ, and equations reduce to ordinary differential equations. We can define the radial coordinate arbitrarily, for example by fixing the frame function as g = 1; or h = r; or f g = 1. Let us briefly review these choices and the corresponding solutions.
For g = 1 the radius r is the geodesic normal coordinate and (4.28-4.30) become: The solution is h = h 0 e −γr , f = f 0 (cosh 2βr + 1), (4.32) where β, h 0 and f 0 are the integration constants. The corresponding classical limit is easily derived, and it has the scalar curvature If radius r is defined such that the area of the sphere be 4πr 2 , that is h = r, we obtain We have where f obeys 1 2 We have therefore in case when the metric does not depend on ρ three families of solutions: esentially they represent one noncommutative space, except perhaps for singular values of the integration constants when the spaces might differ. This space is static and curved and, since everything depends on just one variable r, has a straightforward classical limit.
What happens when we include the dependence on ρ in the frame functions? We have seen that ρ is essentially quantum variable so perhaps we can expect solutions which are physically more interesting. The set of equations (4.28-4.30) is in this case a set of partial equations in r and ρ. † This implies that we have many more solutions, and we shall here restrict to the case when both conditions h = r and f g = 1 are fulfilled. The equation for h then gives where J 0 is an arbitrary function J 0 = J 0 (ρ) of ρ. Imposing f g = 1 we obtain that the remaining two equations are consistent for C = 0 and (r∂ r − ρ∂ ρ ) log f = 1. with the corresponding metric Although solution (4.47) is restricted in its form, in the classical limit it gives practically all static metrics because clasically, ρ as a constant. For example, taking and assuming that ρ = γ we obtain the Schwarzschild metric.

A representation
Noncommutative space A is a tensor product A = A ′ ⊗ A ′′ of a first factor generated by the ξ a and a second factor generated by ρ, r and t. In this particular case the coordinates satisfy a Lie bracket relation and the corresponding Lie algebra is a singular contraction of sl 2 (R): the five position generators are closely related † Note that derivatives ∂ρ in these equations only come from evaluating commutators with ρ.
to the six generators of the Lie algebra of the Lorentz group. The A ′′ can be represented for example on the Hilbert space H ′′ of square integrable functions of two variables (s, u), in analogy with the representation discussed in the previous section, (4.51)

Conclusions
We found in this paper two families of noncommutative spherically symmetric geometries which can be considered as extensions of static and cosmological solutions to the Einstein equations. The problem of extending of a classical spherically symmetric geometry to a noncommutative space (associative algebra A) might seem at the first sight easy. However, the simplest way of extension, the tensor product, is not satisfactory because no classically relevant metric is a product metric. Therefore one has to extend both the algebra and the associated differential calculus nontrivially, and that gives nontrivial set of constraints.
Following the intuition that gravity is related to geometry we used a noncommutative version of the Cartan frame formalism [2]. The main input in this formalism is the set of tetrads or the frame. In the basis of the frame 1-forms the metric components are constants and the differential is defined naturally, with respect to the given or required gravitational configuration. However the algebraic structures are rigid and there are constraints: first, the Jacobi constraints in the algebra and second, the consistency constraints necessary for the compatibility of the algebra with the calculus. One of the significant features of the formalism is the possibility to generalize the diffeomorphism invariance. To first order in noncommutativity this invariance is exactly the same as in the Einstein gravity and one is allowed to choose the most convenient set of coordinates.
Our concrete problem was formulated as: how to extend the fuzzy sphere to a four-dimensional static or cosmological space? We first started by extending the symmetry, that is the momentum algebra, assuming at the same time that the frame derivations are inner. This assumption gave an essentially unique solution for A: an algebra generated by five coordinates (ζ a , r, t) or by five momenta (p a , Π, π 4 ) constrained by one relation that is, possessing one Casimir operator, ζ 2 . This algebra with the associated calculus describes a spherically symmetric nonstatic space, which however is not spatially homogeneous that is, not isotropic. Nonetheless the space has some interesting properties for example the Hubble parameter is given by noncommutativity. Everything in the model was essentially fixed: the Jacobi constraints behaved as field equations.
In the second part of the paper we extended the original algebra by adding one variable, so A was generated by six coordinates (ξ a , ρ, r, t) constrained by one constraint, or by five unconstrained coordinates (x a , r, t). In this setup, as we have seen, one could obtain practically all static spherically symmetric configurations as classical limit. The price to pay was an additional variable ρ with somewhat unusual properties: the differential calcus implied that dρ = 0 in the classical limit while ρ was not a constant (we have for example [ρ, t] = 0). In principle one would consider the existence of an element of the algebra which is not a constant and nevertheless has vanishing differential an undesirable feature. However such elements are not uncommon, we mention as example the dilatator Λ of the quantum line. The exact role of ρ in this specific case remains to be understood better, along with the question of the appropriate representation; in any case the classical relation dρ = 0 suggests that a natural way to interpret ρ is as a Kaluza-Klein parameter which measures the internal space. In addition, one can rather easily see that in this model the momenta cannot belong to A. Namely, solving equations (2.14) for p α we obtain the solution ikp a = δ ab ξ b , p 0 = p 0 (ρ, r), but the remaining p 4 = p 4 (ρ, r, t) is inconsistent with the assumption e a 4 = 0 which gives diagonal and static metric.
Momentum operators are, in classical gravity, always external, p α = e µ α ∂ µ . In addition when we deal with commutative space, the Jacobi identities and the de Rham consistency conditions are trivially satisfied. In the opposite, maximally constrained noncommutative case when the phase and the configuration spaces are identical, the Jacobi identities and the compatibility conditions practically fix the dynamics and the geometry. Here we observe that, with increase of the phase space, constraints become less restrictive and we have more freedom to choose solutions: more noncommutative geometries can be defined consistently. It is certainly possible to continue along these lines and find other interesting quantum spaces with nondegenerate noncommutativity, correct symmetries and desired commutative limits, and that is what we plan to investigate in our future work.