On the Discrete Version of the Kerr Geometry

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keywords: Einstein theory of gravity; minisuperspace model; piecewise ﬂat spacetime; Regge calculus; Kerr black hole

1 Introduction

General relativity (GR), formulated on a subclass of the Riemannian manifolds, the piecewise flat manifolds or simplicial complexes composed of flat 4-dimensional tetrahedra or 4-simplices, was proposed by Regge in 1961 [1] as a discrete approximation to ordinary GR on smooth manifolds. The conviction that in this case we do not lose essential degrees of freedom is based on the possibility of using a piecewise flat manifold to arbitrarily accurately approximate an arbitrary Riemannian manifold [2, 3]. This formulation fits well into the discrete approach to defining the functional integral in quantum gravity [4], a proven tool for extracting physical quantities [5, 6]. Regge’s discrete coordinateless formulation serves here as a regularization tool that is not tied to any coordinate system. The functional integral can be regularized in a similar, but more computable way, by limiting the set of 4-simplices to several of their types in the Causal Dynamical Triangulations approach [7]. It can also be assumed that the structure of spacetime at small distances is given by a piecewise flat manifold [8]. The application of the Regge calculus to cosmology was considered in the works [9, 10, 11, 12]. There we are talking about the numerical analysis of cosmological models; their emergence in the originally quantum Causal Dynamical Triangulations approach was considered in [13]. Approximation by a simplicial complex was probably first applied to the geometries of Schwarzschild and Reissner-Nordström in [14], where it was used for numerical estimates in GR. Subsequently, in [15], for such systems, more efficient computational lattice methods in numerical GR were proposed. Quantum analysis in the framework of Loop Quantum Gravity was applied to the Schwarzschild black hole in [16, 17]. In this system, the singularity was resolved due to a nonzero area quantum (of the order of the Plank scale), which plays here the same regularizing role as the elementary area in lattice theory. We have analyzed the Schwarzschild black hole [18], including the case of slow rotation (Lense-Thirring metric) [19] in simplicial gravity. The approach of these papers, which minimally involves the concept of a path integral, is reduced to solving classical discrete equations in which the elementary length, which plays the role of the 3 lattice spacing, is of quantum origin. We now aim to analyze a discrete version of the Kerr black hole. We first briefly describe the approach in Section 2. Section 3 gives a metric ansatz for which discrete equations are to be solved. The solution to these discrete equations can be obtained from a similar solution to the discrete Schwarzschild problem using analytic continuation, and continuation issues are discussed in Section 4. The discrete metric in a neighborhood of the singular points (more exactly, those that were singular in the continuum) is found in Section 5. The Riemann tensor when approaching the singular points from the vacuum side (that is, from the points where the Ricci tensor is equal to zero) is found in Section 6. The distribution of gravity sources (that is, of nonzero Ricci or Einstein tensors), mainly in the vicinity of the singular points, is found in Section 7.

2 The method

We can think of triangulation as a measurement process that creates the corresponding simplicial complex and piecewise flat spacetime as the simplest structure that describes this triangulation (as opposed to the case where 4-simplices have intrinsic curvature). Then it is important that there is a mechanism that fixes the scale of edge lengths (or elementary lengths) at some nonzero level [20]. This is due to a specific form of the functional integral measure, which has a maximum at certain values of the edge lengths. Ideally, the Regge calculus strategy implies averaging over possible simplicial structures; consider here the case of some fixed structure. To get a well-defined functional integral in Regge calculus, an appropriate choice of variables is important. In this respect, the connection variables introduced by Fröhlich [21] are promising. We take Regge action in terms of both tetrad variables (edge vectors) and independent on them connection matrices or SO(3,1) rotations living on the tetrahedra or 3-simplices, in particular, set the action to be a combination of contributions from self-dual and anti-self-dual connection matrices separately [22]. The use of connection variables is motivated by difficulties of passing in a non-singular way to the continuous time limit and to the canonical Hamiltonian form of the action in terms of purely metric (length or vector) variables. Excluding the connection classically gives the original Regge action. With such a form of the action in the larger configuration superspace of length/vec- 4 tor and connection variables, the Jacobian of the Poisson brackets of constraints, which determines the canonical path integral measure, still has singularities, which is typical for discrete theories, but we can define such a measure in a non-singular way in a more extended configuration superspace in which the area tensors living on the triangles or 2- simplices are independent. Then we can restore the full discrete measure which reduces to the found canonical one in the continuous time limit, no matter what coordinate is chosen as time. This full measure can be projected onto the physical hypersurface of unambiguously defined edge lengths in the extended configuration superspace by inserting the corresponding δ-function factor. In the resulting functional integral, we can perform integration over the connection. After functional integration over the connection, we have a functional integral in terms of edge vectors only,

exp[iS(ℓ, Ω)]( )dµ(ℓ) Ω= exp[iS˜(ℓ)]( )F (ℓ)Dℓ, (1) · D · Z Z as a functional on the functions of the set of the edge lengths ℓ =(l1,...,ln). Here we can use different methods to compute the modulus F (ℓ) (measure) and the argument S˜(ℓ) (phase) of the result, which lead to expansions over different parameters. Namely, parameterizing metric by the ADM lapse-shift functions[23], we extract F (ℓ) from the expansion of the functional integral over the scale N of the discrete lapse-shifts (N, N) (certain edge vectors). S˜(ℓ) is extracted from the stationary phase expansion of the functional integral. The parameters of these expansions both can be small if the length scale b (5) below is formally large in the Planck scale √G units. Using namely such expansions for F (ℓ) and S˜(ℓ) is justified by that nonzero contributions into F (ℓ) and S˜(ℓ) arise just in the leading orders of the corresponding expansions of the functional integral. In particular, S˜(ℓ) in the leading order of the stationary phase expansion is just the Regge action S(ℓ), since it follows from S(ℓ, Ω) by excluding Ω classically by definition of S˜(ℓ) and construction of this expansion. The measure in the path integral is such that the perturbative expansion generated by this integral provides a leading order contribution that has a maximum at a certain optimal initial point ℓ0 =(l01,...,l0n) (background metric). Roughly speaking, this is the maximum point of the measure. In somewhat more detail, we pass from ℓ to some new variable u = (u1,...,un) that reduces the measure F (ℓ)Dℓ to the Lebesgue one 5

Du, 2 1 ∂ S(ℓ0) ∂lj (u0) ∂ll(u0) S(ℓ)= ∆uk∆um + ..., (2) 2 ∂lj ∂ll ∂uk ∂um j,k,l,mX ∆u = u u0, at − ∂S(l ) 0 =0, (3) ∂lj and the determinant of the second order form in the exponent should have a minimum at this point, or 1 ∂2S(ℓ ) − F (ℓ )2 det 0 (4) 0 ∂l ∂l i k should have a maximum. This, in particular, sets the scale of the edge lengths (of the spatial and diagonal edges, but not of the temporal ones, whose vectors, the discrete analogs of the lapse-shift functions, are ﬁxed by hand). The equations of motion (3) themselves do not ﬁx such a variable. The function (4) has a bell-shaped form and a pronounced maximum when the typical area of the spatial and diagonal triangles is b2/2, where the typical length scale is

b = 32G(η 5)/3. (5) − p Here, η is a fundamental parameter that characterizes the quantum extension of the η 4 theory and parameterizes the factors Vσ4 in the path-integral measure, where Vσ is the 4-volume of the 4-simplex σ4. Now the task is to find a Kerr-like solution as an optimal background metric for the perturbative expansion in the framework of the functional integral discussed above. That is, we should solve the Regge skeleton equations (3) together with the requirement to maximize the measure (4), which fixes the length scale b (5). We have considered [24] a simplicial complex, in which certain coordinates are given to the vertices, so that the geometry can be described by a piecewise constant metric. The Regge action can be calculated in terms of this metric through intermediate definition of discrete Christoffel symbols and written in the form of an expansion over metric variations from 4-simplex to 4-simplex. In particular, we can take the simplest periodic simplicial complex, the cell of which is topologically a cube divided into 4! = 24 4-simplices with the help of the diagonals drawn from one of its vertices [25]. Then the leading order term in this expansion is a finite-difference form of the Hilbert-Einstein 6 action,

λµ √g, λ =∆ M λ ∆ M λ +M λ M σ M λ M σ , K λµ K µνρ ν ρµ − ρ νµ νσ ρµ − ρσ νµ 4 cubes −X 1 M λ = gλρ(∆ g + ∆ g ∆ g ), ∆ =1 T . (6) µν 2 ν µρ µ ρν − ρ µν λ − λ

λ Tλ is the shift operator along the coordinate x by 1. When calculating the functional measure, we use the expansion over the discrete lapse-shift vectors as constant parameters. That is, we can introduce coordinates of the vertices so that the metric included in (6) has (N, N) = const. A particular case is the metric in a synchronous frame, (N, N) = (1, 0). In the leading order over metric variations, our task is reduced to solving the ﬁnite-diﬀerence form of the Einstein equations in a synchronous frame. Kerr geometry can be described by convenient metrics in one case or another [26], we start with the Boyer-Lindquist metric [27],

r r ρ2 r r ds2 = 1+ g dt2 + dr2 + ρ2dθ2 + r2 + a2 + a2 g sin2 θ sin2 θdϕ2 − ρ2 ρ2 r r △ 2a g sin2 θdϕdt, (7) − ρ2 where ρ2 = r2 + a2 cos2 θ, = r2 r r + a2. △ − g

For this geometry, we can introduce a synchronous frame similar to that in the case of the Lemaitre metric for the Schwarzschild geometry. Such a metric for the Kerr geometry under consideration can be obtained in a similar way, by binding the coordinates to a set of freely moving particles, and we can write for it an expression of the form[28],

2 2 2 rr1 2 2 r sin(2θ) r sin θ ds = dτ + dr 2a I√r1dr1dθ +2a√rg √r1dr1dϕ1 − ρ2 1 − ρ2 ρ2 2 2 2 4 r sin (2θ) 2 2 3 r sin(2θ) sin θ + ρ + a I dθ 2a √rg Idθdϕ1 ρ2 − ρ2 r r ∞ dr + r2 + a2 + a2 g sin2 θ sin2 θdϕ2, where I = . (8) 2 1 2 2 ρ r r (r + a ) Z

Here r is a function of τ, r1, θ by means of p

3/2 3/2 2 2 2 2 r r ∞ r + a cos θ dr τ = 1 − + √r . (9) 2 2 3 √rg r r (r + a ) − √rg Z " # p By excluding r1 in favor of r, we obtain an analogue of the Painlev´e-Gullstrand metric 7 for the Schwarzschild geometry, the Doran metric[29],

r r ρ2 r r ds2 = 1+ g dτ 2 + dr2 + ρ2dθ2 +2 g dτdr − ρ2 r2 + a2 r2 + a2 r r r r r +2a g sin2 θdrdϕ +2a g sin2 θdτdϕ r2 + a2 1 ρ2 1 r r r + r2 + a2 + a2 g sin2 θ sin2 θdϕ2. (10) ρ2 1 As a typical example, consider a triangulation of the system or a possible form of the simplicial complex, if possible, almost everywhere of the simplest periodic structure, in a section along r1, τ passing through the singularity r = 0, θ = π/2. This section consists of two regions (ϕ = const in them): θ = π/2 and r varies from to 0 (the 1 ∞ equatorial plane outside the singularity ring); r = 0 and θ changes from π/2 to 0 (the equatorial plane inside the singularity ring).

If τ = const, ϕ1 = const, θ = π/2 and r is changing, then the invariant length along the coordinate line dr1 or dr is determined as ρ2 r2 rdr r2 ds2 = dr2 = dr2, ds (at r a) , s (11) r2 + a2 r2 + a2 ≈ a ≪ ≈ 2a

2 (according to ds in terms of τ, r, θ, ϕ1 (10)), and if s covers n typical spacings, s = nb, then the n-th vertex has r √2abn. If the singularity corresponds to r = r at ≈ 1 1(0) τ = τ0, then (9) gives that the n-th vertex from it at the same τ corresponds to r1 = r1(n), and the world line r1 = r1(n) intersects r =0 at τ = τ0 + ∆τn such that

r 3/2 3 3/2 3/2 3 r 3 5/2 3 5/4 √rg∆τn = r r = dr r (2abn) . (12) 2 1(n) − 1(0) 2 √r2 + a2 ≈ 5a ≈ 5a Z0

If τ = const, ϕ1 = const, r = 0 and θ is changing, the invariant length along the coordinate line dr1 or dθ is determined as

a π 2 ds2 = ρ2dθ2 = a2 cos2 θdθ2,s = a(1 sin θ) θ (at 2 π θ 1) (13) − ≈ 2 2 − π 2 − ≪ 2 (according to ds in terms of τ, r, θ, ϕ1 (10)), and if s covers n typical spacings, s = nb, then the ( n)-th vertex from θ = π/2 has (π/2 θ)2 2bn/a. Then (9) gives that − − ≈ the ( n)-th vertex from the singularity at τ = τ0 corresponds to r1 = r1( n), and the − − world line r1 = r1( n) intersects r =0 at τ = τ0 + ∆τ n such that − − 2 2 3 3/2 3/2 3 ∞ a cos θ 1/2 √rg∆τ n = r r = dr 6I0a bn, 1( n) 1(0) 2 2 2 − − − −2 0 r (r + a ) ≈ − Z ∞ dy 2 I0 = = [Γ(1/4)]p/(4√π)=1.854.... (14) 4 0 1+ y Z p 8

r=0, π/2 θ= − r = √2ab 3 (τ τ ) r 2 0 √ g √6b/a √4b/a √2b/a 0✭✟✟ r = √4ab − ✭✟  z }| ❜ ✟{ ✟ r = √6ab  ❜ ✟ ✟   π  θ = 2  PP ❛❛ ✱✱ r = √8ab  P  ✱

✱ ✱  ❤❤❤ ❤❤❤ ✱ ✱ r = √10ab  r = 0 ❉  ✱ ✱ ★  θ = 0 ❉ ✱ ✱ ★  ❉ ✱ ✱ ★ ✚✚  ★ ★ ✚ ❅ ❇ ★ ★ ✚ ❅ ❇ ★ ★ ✚ ✑✑ ✑ ✧ ✧ ✟ 3/2 3/2 ✑ ✑ ✧ ✧ ✟ ✟ r r ✑ ✑ ✧ ✧ ✟ ✟ 1 − 1(0) 3I a3/2 6I b√a 2 1 0 15/4 25/4 35/4 45/4 55/4 6(2a)1/4b5/4 − 0 0 ×− − ×

Figure 1: A triangulation using, if possible, almost everywhere the simplest periodic structure obtained from the cubic subdivision in the Lemaitre type coordinates in a section along r1, τ passing through the singularity.

This is illustrated by Fig. 1. The difference in r1 between two neighboring world lines (12) or (14), which in a sense characterizes the density of world lines, is different on both sides of the singularity, (12) and (14). This leads to irregularities in the nearest neighborhood of the singularity in the simplest used periodic simplicial structure. Of course, there is also a violation of the triangulation regularity connected with that we aim to choose ∆τ b for the difference in τ between the leaves τ = const to ≃ deal with 4-simplices that are not too shrunk in any direction. Normally the leaves go through the vertices at the intersections of the world lines r1 = const with the singularity. But if the difference in τ between neighboring such vertices (defined by the difference in r1), ∆τn+1 ∆τn (12) or ∆τ n ∆τ n 1 (14) is significantly less or − − − − − greater than b, then either only some of the vertices at the intersections of the lines r1 = const with the singularity belong to a leaf τ = const (the rest of these vertices are connected by edges with the nearest vertices of the nearest leaves) or additional leaves between the two under consideration are introduced (and the vertices at their intersections with the singularity are connected by edges with the nearest vertices in the τ = const leaves). In both cases, there is again a violation of the triangulation regularity in the nearest neighborhood of the singularity. In our analysis, we should consider the general simplicial metric, which is close at large distances in a certain topology to the Kerr one, here in a synchronous frame 9 of reference. We use the simplest periodic (almost everywhere) simplicial complex obtained by a triangulation like the above one, and the action in the leading order over metric variations from simplex to simplex is a finite-difference form of the Hilbert-

Einstein action (6), where the interpolating smooth metric gλµ is close in a certain topology to the Kerr one. In principle, subsequent orders over metric variations could be taken into account; the corresponding terms should be written out in (6); in these orders, we should distinguish between the metrics in different 4-simplices inside the 4-cube. In the leading order over metric variations, just considered in the present paper, the situation is simplified. Finite differences obey the same rules as the corresponding derivatives; in particular, in this order there is ”diffeomorphism invariance”: in a finite-difference expression, we can go from a metric close to the Kerr metric in a synchronous frame of reference to a metric close to the Kerr metric in some other frame of reference. As the latter reference system, we will choose the Kerr-Schild coordinate system [30, 31]. Now we do not need to know the details of triangulation in the intermediate synchronous frame; only its existence is important in principle.

3 The metric ansatz

So, consider the original continuum metric in the Kerr-Schild coordinates,

ds2 = dτ 2 + dx2 + dx2 + dx2 − 1 2 3 r r3 rx ax rx + ax x 2 + g dτ 1 − 2 dx 2 1 dx 3 dx . (15) r4 + a2x2 − r2 + a2 1 − r2 + a2 2 − r 3 3 Here, for convenience, the coordinates xk (whose diﬀerentials are in fact, of course, contravariant vectors) will be denoted by subscripts. The quantity r obeys the equation connecting the radial oblate spheroidal coordinate r with the Cartesian coordinates x1, x2, x3, r4 r2(x2 + x2 + x2 a2) a2x2 =0. (16) − 1 2 3 − − 3 This metric is a particular case of the algebraically special metric [30, 31, 32]

g = η + l l , lλl =0, lλ = ηλµl , η = diag( 1, 1, 1, 1). (17) λµ λµ λ µ λ µ λµ − It is convenient to denote

l = l n , n =( 1, n )=( 1, n , n , n ), so that l l = l2n n , (18) λ 0 λ λ − λ − 1 2 3 λ µ 0 λ µ 10 and (15) corresponds to r r3 rx ax rx + ax x l l = g 1, 1 − 2 , 2 1 , 3 . (19) 0 λ r4 + a2x2 − r2 + a2 r2 + a2 r 3 The metric functions (15) have the only singularity on the ring

2 2 2 x1 + x2 = a , x3 =0. (20)

Also note the following behaviour of r and these metric functions as x3 tends to 0, a x x2 | 3| 1+ O 3 at x2 + x2 < a2 a2 x2 x2 x2 ξ2 1 2  − 1 − 2 − 3  x3 2 2 2 r =  pa x3 1+ O | a | at x1 + x2 = a , (21)  | |  2 p 2 2 2 2 x3 2 2 2 x1 + x2 + x3 a 1+ O 2 at x1 + x2 > a  − ξ   p r a x x2  g | 3| 1+ O 3 at x2 + x2 < a2 (a2 x2 x2 x2)3/2 ξ2 1 2  − 1 − 2 − 3 rg x l2 =  1+ O | 3| at x2 + x2 = a2 , (22) 0  2 a x a 1 2  3  | | r x2 p g 1+ O 3 at x2 + x2 > a2 x2 + x2 + x2 a2 ξ2 1 2  1 2 3  −  x2 + x2 + x2 x2  p1 1 2 3 sgn x 1+ O 3 at x2 + x2 < a2 − a2 3 ξ2 1 2  r  x3 x3 2 2 2 n3 =  | a | sgn x3 1+ O | a | at x1 + x2 = a , (23)   q 2 x3 x3 2 2 2 1+ O 2 at x1 + x2 > a  x2 + x2 + x2 a2 ξ  1 2 3 −  where a2ξ2 (x2+px2+x2 a2)2. We observe the presence of x and sgn x here, as well ≡ 1 2 3− | 3| 3 as in n , n , which include x linearly through r. This leads to the appearance of δ- 1 2 | 3| function source terms in the Einstein equations. The support for the source distribution contains the singularity ring, and it is of interest to analyze the system within the discrete approach, paying attention to the neighborhood of the singularity (which was such in the continuum), including the source terms. In the most general case, we can consider an (overdetermined) decomposition of any metric gλµ into some g(0)λµ and some lλlµ,

λ λ λµ gλµ = g(0)λµ + lλlµ, l lλ =0, l = g(0)lµ. (24)

λ λ The Riemann tensors Rµνρ and R(0)µνρ for the metrics gλµ and g(0)λµ, respectively, are related as [32]

Rλ = Rλ + Cλ Cλ + Cλ Cσ Cλ Cσ , (25) µνρ (0)µνρ ∇ν µρ −∇ρ µν σν µρ − σρ µν 11

is the covariant diﬀerentiation operator with respect to g , ∇λ (0)λµ 1 Cλ = (l lλ)+ (l lλ) λ(l l )+ lλlσ (l l ) (Cλ = 0). (26) µν 2 ∇µ ν ∇ν µ −∇ µ ν ∇σ µ ν µλ This gives the Ricci tensor components,

R = R + Cν Cν Cρ , (27) λµ (0)λµ ∇ν λµ − ρµ λν and thus the Einstein equations. If a ﬁnite diﬀerence form of the Einstein equations is considered, we have the Kerr solution in the leading order over metric variations (in the region where such variations are small, that is, in a certain sense far from singularities), in particular, g(0)λµ = ηλµ. Then the vacuum equations take the form

Cν Cν Cρ =0. (28) ∇ν λµ − ρµ λν

It is an overdetermined system for lλ. Moreover, starting from the answer, note that rg as an arbitrary parameter of the solution enters lλ through the coeﬃcient √rg. λ 2 n Then Cµν is a combination of rg- and rg-terms and (28) is a combination of rg -terms, n =1, 2, 3, 4. Due to the arbitrariness of rg, such equations should be fulﬁlled in each order of rg. In the order rg, (28) reads

(l lν )+ (l lν)Cν ν(l l )=0. (29) ∇ν∇λ µ ∇ν∇µ λ λµ −∇ν∇ λ µ In particular, the 00- and 0k-components, taking into account stationarity, l = ∇0 λ 0, are ∇2 2 (l0)=0 (30) and (l lm) ∇2(l l )=0, (31) ∇m∇k 0 − 0 k respectively, outside the source region. The divergence k( ) of the left-hand side ∇ · of (31) is identically zero, and there are two independent equations among (31). In principle, taking into account the condition

lλl l2 + l2 + l2 + l2 =0, (32) λ ≡ − 0 1 2 3

(30), (31) and (32) present four equations for four values lλ. A solution to (31) can be given as l l = η + a ∇χ, (33) 0 k ∇k × 12 where a = (0, 0, a) and χ obeys ∇2χ =0 (34)

2 (the metric (15) indeed has such a form). Then, provided l0 satisﬁes (30), an equation for χ follows by substituting this l into (32),

(∇η + a ∇χ)2 = l4. (35) × 0 But for our purposes (analytic continuation) we need an explicit closed expression for the solution, which we have not for the nonlinear partial diﬀerential equation (35).

2 Therefore, we consider the simple ansatz for the metric in which l0 is subject to (30) and the other components are related to it in the same way as in the exact solution.

2 The fact that l0 from (15) obeys (30) can be evident if we represent this function as a superposition of two Newton potentials with the centers shifted along x3 by the imaginary values +ia and ia, − r3 1 1 2 = + , (36) r4 + a2x2 r + ia cos θ r ia cos θ 3 − r ia cos θ = x2 + x2 +(x ia)2, (37) ± 1 2 3 ± r3 1 q 1 2 = + 4 2 2 2 2 2 2 2 2 r + a x3 x + x +(x + ia) x + x +(x ia) 1 2 3 1 2 3 − 1 1 p + p, y = (0, 0,y3), y3 = a sgn y3, (38) ≡ (x + iy)2 (x iy)2 − (x (x , x , x )). Thisp is a truncatedp form of the Newman-Janis trick [33] (or, in ≡ 1 2 3 the Kerr-Schild coordinates, [34]), with which one obtains the Kerr geometry from the Schwarzschild geometry. This prompts us to formulate the discretization problem as that of the analytic continuation of the discrete Schwarzschild problem (i. e., the discrete Newton potential) to complex coordinates, namely, as the problem of shifting by ia along some axis, which thus becomes the rotational axis. ± Important is the behaviour of the metric near the singularity. In the considered coordinates, the only singularity in the continuum theory is that in the overall scale of lλ, i. e., of l0 if we take l0, n as independent metric functions. This singularity is achieved at r 0, cos θ = x /r 0, i. e., at (x + iy)2 0 (or at (x iy)2 0). → 3 → → − → In the discrete case, the complex Newton potential 1/ (x + iy)2 in (38) is re- placed by an unknown function φ(x) (close to 1/ (x + iyp)2 at large distances from the singularity ring). This φ(x) is not singular atpr 0, cos θ 0 (it is cut oﬀ at → → a suﬃciently small length). This can be expressed by saying that φ(x) is Newton’s 13

(complex) potential, but corresponding to some effective reff and cos θeff that do not vanish at r = 0, cos θ = 0, 1 1 1 1 φ(x)= , reff = Re , cos θeff = Im . (39) reff + iy3 cos θeff φ y3 φ In the continuum theory, or, approximately, at large distances from the singularity,

2 reﬀ = r, θeﬀ = θ. In our simple ansatz for the metric, l0 obeys (30), and n is expressed in terms of the complex potential φ and the polar angle ϕ, as in the continuum solution.

This is equivalent to taking as r, θ their effective values reff , θeff (39); ϕ is defined by x2/x1 = tg ϕ. Thus, we have

2 l0 = rg Re φ (40) 1 2 1 2 rx1 ax2 x1 Re φ− ax2 a (Im φ− ) n1 = − − − (41) 2 2 2 2 2 1 2 r + a ⇒ a x1 + x2 sa + (Re φ− ) 1 2 1 2 rx2 + ax1 x2 Repφ− + ax1 a (Im φ− ) n2 = − (42) 2 2 2 2 2 1 2 r + a ⇒ a x1 + x2 sa + (Re φ− ) x3 1 1 n3 = Impφ− , (43) r ⇒ y3 where φ is obtained by continuation to complex coordinates,

φ(x)= φ0(x + iy), y = (0, 0, a sgn y3), (44) of the solution of the ﬁnite-diﬀerence Poisson equation with a point source,

3 0 at x =0 ∆ ∆ φ (x)= 6 (45) j j 0  j=1 C at x =0, X  such that at large distances it approximates Newton’s continuum potential, 1 φ0(x) , x . (46) → √x2 →∞ 1 A priori, we still do not exclude the presence of vertices at which Im(φ− ) > a | | ( cos θ > 1, which would lead to imaginary n , n ), which would require modifying | eff | 1 2 our ansatz in the corresponding region. If the vertices are located near the singularity (this is just the case that interests us here), φ is large and cos θ 1. At large | | | eff | ≪ distances, the discrete solution begins to approach the continuum one, and cos θeff approaches cos θ. The most dangerous region seems to be where θ is close to 0, in which cos θ values greater than 1 are not excluded due to lattice artifacts. In this | eff | case, it is natural to put n = (0, 0, 1) at the corresponding vertices. Note that the ± details of our ansatz for n do not affect the results obtained (in the leading order over metric variations). 14 4 Analytic continuation

The discrete analogue of Newton’s potential has the form 1 π/b π/b π/b d3p πb2 exp(ipx) x φ0( )= 3 3 2 . (47) r ⇒ π/b π/b π/b (2π) sin (pjb/2) Z− Z− Z− j=1 Evidently, passing to complex coordinates under theP integral sign cannot be made because of arising an unlimitedly growing exponent of p. Let us ﬁrst integrate over, say, dp3. The corresponding integral is known to be

π/b π dp3 exp(ip3x3) dq3 exp(iq3ν) 3 2 = π/b 2π sin (pjb/2) π πb ch κ cos q3 Z− j=1 Z− − P 2 sin(πν) 1 ∞ ( 1)n exp( κn) = +2ν − − , (48) πb sh κ ν ν2 n2 " n=1 # X − q p b, ν x /b, ch κ =3 cos q cos q . j ≡ j ≡ 3 − 1 − 2 Now we can pass to complex x , so Im ν = a/b. But this leads to a non-physically 3 ± exponentially large right-hand side of (48) at a small length scale b due to the presence of sin(πν) there. On the other hand, it is admissible to add to (48) a function of ν, which vanishes for integer ν. We add to the sum in (48) an analytic function f(ν) − that has no poles for integer ν. We denote the (modiﬁed) sum in (48) multiplied by sin(πν)/π as

1 ∞ ( 1)n exp( κn) sin(πν) sin(πν) g(ν) +2ν − − f(ν) . (49) ≡ ν ν2 n2 π − π " n=1 # X − For an integer ν, we have

g(ν) = exp( κ ν ), ν =0, 1, 2,.... (50) − | | | | Continuation to complex ν gives

g(ν) = exp( κ√ν2), √ν2 = ν sgn Re ν, (51) − where √ν2 is deﬁned by √1 = 1 on the complex plane with the cut at ν2 ( , 0], ∈ −∞ that is, along the imaginary axis Re ν = 0. This means

π/b dp3 exp(ip3x3) 2 x3 + iy3 3 2 exp κ sgn x3 at x3 x3 + iy3 (52) π/b 2π sin (pjb/2) ⇒ b sh κ − b ⇒ Z− j=1 for the analyticP continuation of (48) and implies

1 g(ν) ∞ ( 1)n[exp( κn) g(ν)] f(ν)= − +2ν − − − , (53) ν ν2 n2 n=1 X − 15 so f(ν) indeed has no poles for integer ν. This uses the partial fraction expansion for sine, π 1 ∞ ( 1)n = +2ν − . (54) sin(πν) ν ν2 n2 n=1 X − In fact, this corresponds to calculating the integral (48) by taking only the residue at the pole inside the contour formed by the segments [ π, +π], [ π, π + iL], [ π + − − − − iL, +π + iL], [+π, +π + iL], L sgn ν, in the complex q plane without taking into →∞· account the contributions from the last three segments, [ π, π+iL], [ π+iL, +π+iL], − − − [+π, +π + iL]. This does not change the integral at physical (integer) points, but removes the aforementioned non-physical (inﬁnite, as b tends to zero) terms. The uniqueness of such a continuation from the discrete set of points (of course, on the class of analytic functions) is based on some discussion of the fact that any nontrivial addition to g(ν), which vanishes for integer ν, must contain a factor that is the product of the factors (1 ν/n), n = 1, 2,... , and ν (Weierstrass factorization − ± ± theorem), which leads to sin(πν),

∞ ν2 sin(πν) ν 1 = , (55) − n2 π n=1 Y exponentially diverging at Im ν = a/b at b 0. This uses the product | | → ∞ → expansion for sine. Now we can pass to complex coordinates under the integral sign. So far, we can do this without restricting the range of admissible values of the real parts of the coordinates

3 3 (IR ) only for x : if we pass to complex x1, x2,

x x + iy , x + iy , (56) 1 ⇒ 1 1 2 2 2 2 2 then for x3 < y1 + y2 the integrand grows exponentially in a certain region. That is, starting with the integration over dp3, we can relatively easily analyze the case of rotation around x3. Thus, we have

φ(x) = φ0(x1, x2, x3 + iy3) π π dq1dq2 x1 x2 x3 + iy3 = exp iq1 + iq2 κ sgn x3 , (57) π π 2πb sh κ b b − b Z− Z− y = a = a sgn y . By redeﬁning q q sgn(x y ), we rewrite this as 3 ± 3 j ⇒ j 3 3 π π dq1dq2 s x3 φ(x)= exp i [q1x1 + q2x2 aκ] | |κ , (58) π π 2πb sh κ b − − b Z− Z− s sgn(x y ). ≡ 3 3 16 5 The metric in the discrete case

In the continuum limit, b 0, we return from q to p = q /b and note that generally → j j j (not at singular points) the cut-oﬀ eﬀect of the exponent exp(ipx) leads to limiting p , not q (say, by 1/r if r a). Then, in this limit, κ = b p2 + p2, and after passing j j ≫ 1 2 to polar variables, p1 = ρ cos ϕ, p2 = ρ sin ϕ, the integralp (58) gives the continuum expression for the analytically continued Newton potential,

2π dϕ ∞ φ = dρ exp ρ [is (x cos ϕ + x sin ϕ a) x ] 2π { 1 2 − −| 3| } Z0 Z0 2π dϕ 1 1 = = 0 2π 2 2 2 2 2 Z x3 is x1 + x2 cos φ a ( x3 + isa) + x + x | | − − | | 1 2 1 p q = . (59) (x + iy)2

Here, the cut-oﬀp eﬀect of the exponent and, therefore, the applicability of the continuum expression is determined, roughly, by max x2 + x2 a , x , that is, by 1 2 − | 3| the distance from the singularity ring. In relation top what should this distance be large? The answer to this question, as a by-product, will follow from the analysis in the neighborhood of the singularity ring, to which we now turn.

To this end, we pass to the variables λj = sin(qj/2), j =1, 2, then the integral (58) takes the form

+1 +1 dλ1dλ2 1/2 2 2 2 2 2 2 − φ = 1 λ1 1 λ2 1+ λ1 + λ2 λ1 + λ2 1 1 πb − − Z− Z− 1 exp 2is (x arcsin λ + x arcsin λ ) · b 1 1 2 2 (isa + x ) arsh 2 (1 + λ2 + λ2)(λ2 + λ2) . (60) − | 3| 1 2 1 2 q

We pass to polar coordinates for the integration variables λ1, λ2 and for the point x1, x2,

λ1 = Λ cos ϕ 1 Λ cos ϕ 1 x1 = r0 cos ϕ0 , − ≤ ≤ , , (61) λ = Λ sin ϕ  1 Λ sin ϕ 1  D x = r sin ϕ  2  − ≤ ≤  2 0 0  so that   

dΛdϕ 1/2 1 φ = 1 Λ4 + Λ4 +Λ6 cos2 ϕ sin2 ϕ − exp 2isr [cos ϕ πb − b 0 0 ZZD h arcsin (Λ cos ϕ) + sin ϕ arcsin (Λ sin ϕ)] (isa + x ) arsh 2Λ√1+Λ2 . (62) · 0 − | 3| i 17

For small b, a contribution to this integral dominates from the region in which the functions arcsin and arsh in the exponent are small, that is, the values of Λ are small enough to at least lie in the convergence domain of the Taylor series for the exponent in the integration region (this refers to the function arsh, for which the Taylor series D converges at 2Λ√1+Λ2 1, that is, at Λ 1/ 2 √2+1 ). Making such a Taylor ≤ ≤ series expansion, we obtain q

dΛdϕ 1/2 1 φ = 1 Λ4 + Λ4 +Λ6 cos2 ϕ sin2 ϕ − exp 2is πb − b ZZD 1 3 3 3 Λ[r cos(ϕ ϕ) a]+3− isΛ r cos ϕ cos ϕ + sin ϕ sin ϕ + a x (2Λ · 0 − 0 − 0 0 0 −| 3| 1 3 5 5 5 5 5 3− Λ + isr cos ϕ O Λ cos ϕ +isr sin ϕ O Λ sin ϕ + x O Λ . (63) − 0 0 0 0 | 3| It is seen that φ can sharply increase at r0 = a, x3 = 0 due to a qualitative decrease in the suppression effect by the exponential at these points. Thus, this is a refined position of the (regularized) singularity. At these points, the linear term in Λ in the expansion of the exponent over Λ, ϕ ϕ vanishes, and the contribution of large Λ, − 0 ϕ ϕ is suppressed by the next cubic terms Λ3 and Λ(ϕ ϕ )2. As a result, the − 0 − 0 dominant contribution comes from the typical values of Λ, ϕ ϕ defined by aΛ3 b, − 0 ∼ aΛ(ϕ ϕ )2 b, that is, Λ (b/a)1/3, ϕ ϕ (b/a)1/3. − 0 ∼ ∼ | − 0| ∼ Thus, the typical values of Λ, ϕ ϕ are proportional to √3 b. Therefore, √3 b is a − 0 natural small parameter. Then we consider a neighborhood of the established analogue of the singularity ring and write the coordinates in the form

x1 = a cos ϕ0 + k1b  x2 = a sin ϕ0 + k2b , (64)   x3 = k3b   where k1, k2, k3 are integers of order 1, and expand in (fractional) powers of b, at- tributing b1/3 to Λ and ϕ ϕ . The expression for φ reads − 0 dΛdϕ is 1 φ = exp F , F = aΛ(ϕ ϕ )2 + a 1 + cos4 ϕ + sin4 ϕ Λ3 πb b − − 0 3 0 0 ZZD 1 a sin(4ϕ )Λ3 (ϕ ϕ )+ O Λ(ϕ ϕ )4, Λ3(ϕ ϕ )2, Λ5 −4 0 − 0 − 0 − 0 +2bΛ(k cos ϕ + k sin ϕ + is k )+ O bΛ(ϕ ϕ ), bΛ4 . (65) 1 0 2 0 | 3| − 0 Shown are F terms of order b (Λ(ϕ ϕ )2 and Λ3) and b4/3 (Λ3(ϕ ϕ ) and bΛ). − 0 − 0 The subsequent terms O(... ) are b5/3 and less. In particular, the dependence of the ∼ 18 denominator on Λ in (63), when rewritten in the exponential form, adds O (bΛ4) = O b7/3 to F . After integration over ϕ, we get

∞ isπ 2 6 3b 6 3 3b 2 φ = exp dw exp iw +2 (isk12 k3 ) w , (66) − 4 √πab saµ " saµ −| | # Z0 4 4 1/3 2 µ = 1 + cos ϕ0 + sin ϕ0,k12 = k1 cos ϕ0 + k2 sin ϕ0, Λ=(3b/(aµ)) w .

With the help of contour integration in the complex plane of w, we pass from the oscillating exponent to a monotonic one (mainly) by a complex rotation of the variable, w = u exp(isπ/12),

∞ isπ 2 6 3b 6 3 3b isπ 2 φ = exp du exp u +2 exp (isk12 k3 ) u . − 6 √πab saµ "− saµ 6 −| | # Z0 (67) First consider the case (b/a)1/3 ik k 1. (68) | 12 −| 3|| ≫ Then we can neglect the u6 term in the exponent and ﬁnd

1 φ = , x12 bk12 = x1 cos ϕ0 + x2 sin ϕ0. (69) 2a [x + is x ] ≡ 12 | 3| p If we optimally choose a point on the ring (a cos ϕ0, a sin ϕ0, 0), relative to which the coordinates (bk1, bk2, bk3) of the current vertex are determined, so that ϕ0 is the polar angle of x, then

x =(x a cos ϕ ) cos ϕ +(x a sin ϕ ) sin ϕ = x2 + x2 a. (70) 12 1 − 0 0 2 − 0 0 1 2 − q At max x2 + x2 a , x a, (71) 1 2 − | 3| ≪ q we have

2a [x + is x ] x2 + x2 +(isa + x )2 =(x + iy)2 (72) 12 | 3| ≈ 1 2 | 3| and thus (69) reproduces the continuum function (59) near the ring. In principle, the exact continuum answer for all points would follow if the above integration over ϕ would be exact (giving some Bessel function): at sufficiently large distances from the ring, the dominating contribution to the integral is provided by smaller values of Λ, the latter enters the coefficient at 1 cos(ϕ ϕ ), therefore, considerable values of − − 0 19 the angle ϕ ϕ can give a significant contribution, so the quadratic approximation of − 0 1 cos(ϕ ϕ ) used above is not suitable if (71) is not satisfied. − − 0 Thus, the answer becomes the continuum one at the distances satisfying (68) or

3 max x2 + x2 a , x √ab2. (73) 1 2 − | 3| ≫ q

This estimate is stronger than the simplest one available, such as in the Schwarzschild case, which states that the distances should be much larger than b itself. In the vicinity of the ring, we expand over isk k , 12 −| 3| 1 1 aµ isπ φ = Γ 3 exp +2√π (isk k ) a√3πµ 6 3b − 6 12 −| 3| r 4π 3b isπ + 3 exp (isk k )2 Γ(1/6) aµ 6 12 −| 3| s 2 3b 2/3 isπ + Γ(1/6) exp (isk k )3 + ... , (74) 9 aµ 3 12 −| 3| # 1 1 3b isπ = a 3πµ 3 exp φ Γ(1/6) aµ 6 ( s p 2/3 2√π 3b isπ 2 exp (isk12 k3 )+0 (isk12 k3 ) −(Γ(1/6))2 aµ 3 −| | · −| | 8π3/2 2 3b 4/3 2isπ + exp (isk k )3 + ... , (75) 4 − 9Γ(1/6) aµ 3 12 −| 3| (Γ(1/6)) ) so the real and imaginary parts in the orders of interest are

1 1 aµ Re φ = √3Γ 3 4√π k 2a√3πµ 6 3b − | 3| r

4π 3 3b 2 2 + √3k12 + √3k3 +2k12 k3 + ... , (76) Γ(1/6)saµ − | | # 2/3 1 1 √3πµ 3 3b 2√π 3b Re = √3+ k12√3 k3 a φ 2Γ(1/6) "saµ Γ(1/6) aµ −| | 2 2 +0 k12 +0 k12 k3 +0 k3 + ... , (77) · · | | · # 2/3 1 1 √3πµ 3 3b 2√π 3b Im = sgn k3 k12 sgn k3 + k3√3 y3 φ 2Γ(1/6) "saµ − Γ(1/6) aµ 2 2 +0 k12 +0 k12 k3 +0 k3 + ... . (78) · · | | · # 20 6 Riemann tensor

The metric functions obtained have kinks and breaks at x3 = 0 (the extension of the functions from integer to arbitrary coordinates is implied). Consider ﬁrst the case x = 0. We substitute the metric (40) - (43) with φ found, (76) - (78), into (the 3 6 discrete form of) the fully covariant Riemann tensor,

1 R = [∆ ∆ (l l ) + ∆ ∆ (l l ) ∆ ∆ (l l ) ∆ ∆ (l l )] λµνρ 2b2 µ ν λ ρ λ ρ µ ν − µ ρ λ ν − λ ν µ ρ +(ηστ lσlτ )(C C C C ) , (79) − σ,µν τ,λρ − σ,µρ τ,λν 1 C = [∆ (l l ) + ∆ (l l ) ∆ (l l )] . (80) λ,µν 2b µ ν λ ν λ µ − λ µ ν

Acting by the finite-difference operators ∆λ, ∆λ∆µ on the metric in the neighborhood of the singularity ring, we consider the minimal order in the expansion of the metric over k, which does not lead to zero, linear for ∆λ and bilinear for ∆λ∆µ, since the expansion proceeds effectively over √3 bk, and higher orders are suppressed by powers of √3 b. And on polynomials of such minimal orders, the finite differences act as derivatives. For the found expansion of the metric for orders in b for finite differences, we have

∆ Re φ = O(1) (at λ =3), ∆ ∆ Re φ b1/3, ∆ n b2/3, λ λ µ ∼ λ µ ∼ ∆ ∆ n b5/3. (81) λ µ ν ∼

2/3 The last formula takes into account the dependence of n1, n2 (41), (42) both on b k1, 2/3 b k2 due to our expansion of φ, and on bk1, bk2 through the already present dependence on x1, x2 in addition to our expansion. Note also that the metric eﬀectively

3 3 depends on k1, k2 via √bk12 = √b (k1 cos ϕ0 + k2 sin ϕ0) in our expansion; the already existing dependence of n1, n2 on x1, x2 leads to a weaker (by a power of b) dependence on k1, k2 through bk1, bk2. Therefore,

n(0)λ∆ (l l ) b, where n(0)λ = (1, sin ϕ , cos ϕ , 0) (82) λ µ ν ∼ − 0 0 is the value of nλ on the singularity ring (k = 0) in the absence of discreteness correc- tions (b 0). nλ itself on the ring diﬀers from n(0)λ by O b1/3 , so → nλ∆ Re φ b1/3, nλ∆ n b, nλ∆ (l l ) b1/3. (83) λ ∼ λ µ ∼ λ µ ν ∼

With such estimates, the leading contribution to R can be distinguished at b 0. λµνρ → 21

The general form of the term with the second ﬁnite diﬀerence is as follows:

∆µ∆ν [(Re φ)nλnρ]=(∆µ∆ν Re φ) nλnρ

+ [(∆µ Re φ) ∆ν (nλnρ)+(∆ν Re φ) ∆µ (nλnρ)]

1/3 2/3 +(Re φ)∆µ∆ν (nλnρ)= O b + O b + O (b) , (84) with orders for the resulting three terms, respectively, of which the ﬁrst is the largest,

(∆ ∆ Re φ) n n b1/3. (85) µ ν λ ρ ∼

All other terms in Rλµνρ, including those that are bilinear in Cλ,µν, are smaller (have higher orders in b). Roughly speaking, ﬁnite diﬀerences should be concentrated on Re φ. As a result, the main contribution to R at x = 0, b 0 takes the form λµνρ 3 6 → r R = g (n n ∆ ∆ Re φ + n n ∆ ∆ Re φ λµνρ 2b2 λ ρ µ ν µ ν λ ρ 5/3 n n ∆ ∆ Re φ n n ∆ ∆ Re φ) b− . (86) − µ ρ λ ν − λ ν µ ρ ∼

In the neighborhood of x3 = 0, it is appropriate to use the more accurate Hermitian deﬁnition 1 ∆ ∆ + ∆ ∆ ∆ ∆ (87) −2 λ µ µ λ ≡ − (λ µ) instead of ∆λ∆µ for the discrete version of ∂λ∂µ, similar to the deﬁnition used in the discrete Laplacian (45). Otherwise, calculating ∆3∆3f(k3) for some metric function f(k3), ∆ ∆ f(k )= f(k ) 2f(k 1) + f(k 2), (88) 3 3 3 3 − 3 − 3 − at k = 1, will require knowledge of f( 1), that is, passing through the non-analyticity 3 − point k3 = 0.

In the context of our calculation, we need to contract two vectors of the type ∆λA, σ λµνρ ∆µB, say, when ﬁnding Cσ,µν Cλρ above or RλµνρR below (if instead of ∆λA and/or

∆µB we have a value proportional to lλ and/or lµ, the contraction procedure is trivial and is done using the ηλµ metric). We have

gλµ(∆ A)∆ B = ηλµ(∆ A)∆ B lλ(∆ A)lµ∆ B = ηλµ(∆ A)∆ B λ µ λ µ − λ µ λ µ (Re φ)(nλ∆ A)nµ∆ B = ηλµ(∆ A)(∆ B) 1+ O b1/3 (89) − λ µ λ µ λ according to the above estimates for n ∆λ, whose action introduces an additional 1/3 smallness b in comparison with one ∆λ (acting on the functions of k3 and mainly 22

1/3 k12) and for Re φ = O b− . Thus, in the leading order, the contraction is performed by ηλµ.

The found leading contribution to Rλµνρ in the neighborhood of the singularity ring at x = 0 (86) can be written in the form of three matrices A, B, C: 3 6

R0k0l = Akl, Rklmn = ǫklpǫmnrCpr, R0klm = ǫlmnBnk, (90) of which A, C are symmetric. These matrices are

√3 cos2 ϕ √3 cos ϕ sin ϕ sgn x cos ϕ 0 0 0 − 3 0 A = C =  √3 cos ϕ sin ϕ √3 sin2 ϕ sgn x sin ϕ  , (91) − 0 0 0 − 3 0    sgn x3 cos ϕ0 sgn x3 sin ϕ0 √3   − − −   2  sgn x3 cos ϕ0 sgn x3 cos ϕ0 sin ϕ0 √3 cos ϕ0  2  B = sgn x3 cos ϕ0 sin ϕ0 sgn x3 sin ϕ0 √3 sin ϕ0 . (92)    √3 cos ϕ0 √3 sin ϕ0 sgn x3   −    They have the properties

A = C , A =0, B = B , (93) kl − kl kk kl lk which are equivalent to the vacuum Einstein equations Rλµ = 0, and Bkk = 0, which is equivalent to the identity R0123 + R0231 + R0312 = 0. A and B can be combined into one complex D,

2 cos ϕ0 cos ϕ0 sin ϕ0 i cos ϕ0 √  2  D = A + iB = 3+ i sgn x3 cos ϕ0 sin ϕ0 sin ϕ0 i sin ϕ0 . (94)    i cos ϕ0 i sin ϕ0 1   −    Two complex eigenvalues of D (the third eigenvalue is equal to minus the sum of these two due to vanishing of the trace of D) provide information about four independent curvature invariants. So, all eigenvalues are equal to zero. This means that in the leading order in b at the points under consideration (on the singularity ring) there are no invariants measuring the curvature. Meanwhile, the vanishing of the Kretschmann scalar follows immediately upon the direct use of the found general form of Rλµνρ (86),

2 r 2 R Rλµνρ = g nλnµ∆ ∆ Re φ =0, (95) λµνρ b4 λ µ λ since n ∆λ gives zero when acting on the metric in the leading order in b, as mentioned above. 23

Some non-invariant characteristic of the Riemann tensor can be given, such as a

λµνρ mean value of its component. The positive contribution to RλµνρR is due to Rklmn and R , 72 components in total (say R and R are considered diﬀerent 0k0l klmn − lkmn components), the negative contribution is due to R0klm, 72 components in total. These λµνρ contributions are equal in absolute value to get zero RλµνρR ,

λµνρ klmn 0klm 0k0l RλµνρR = RklmnR +4R0klmR +4R0k0lR , (96)

so the root mean square value of the physical component of Rλµνρ is

1 klmn 0k0l 1 0klm Rλˆµˆνˆρˆ = (RklmnR +4R0k0lR )= 4R0klmR | | r72 r−72 · 4√2π r = g . (97) 37/6Γ(1/6)µ5/6 a4/3b5/3

In the continuum theory, the Kretschmann scalar is equal to [35]

6 2 4 2 4 2 4 6 6 λµνρ 2 r 15a r cos θ + 15a r cos θ a cos θ RλµνρR = 12r − − (98) g (r2 + a2 cos2 θ)6 12r2 = Re g . [(x + iy)2]3

We can see that the space of (x1, x2, x3) is divided by three spheres into six regions, λµνρ in each of which the RλµνρR sign is constant. Each of these surfaces contains the singularity ring, and at any point of the ring, the tangent planes to them divide the space into six equal sectors of 60 degrees each, in which the signs of this value alternate,

λµνρ see Fig. 2. Therefore, it is not surprising that the regularized value of RλµνρR on the ring turns out to be zero.

7 Source distribution

The curvature includes a discrete analogue of the δ-function with support at x3 = 0 2 due to terms in b Rλµνρ of the type

n n ∆ ∆ Re φ δ , (99) λ µ 3 3 ∼ k30

δ is the Kronecker symbol arising from the double diﬀerentiation of the modulus k k30 | 3| in the expansion of Re φ. This is also singled out by a larger value O (1) compared to the above O b1/3 in the analyticity region x = 0. Then we will show that this 3 6 contribution just leads to Rλµ = 0, that is, to the gravity sources. 6 24

λµνρ RλµνρR < 0

λµνρ RλµνρR > 0

600 600 λµνρ 0 0 λµνρ singularity ring RλµνρR < 0 60 60 RλµνρR > 0 0 60 600

λµνρ RλµνρR > 0

λµνρ RλµνρR < 0

λµνρ Figure 2: The sign of the continuum RλµνρR in the space.

In (99), we use a more precise Hermitian definition ∆ ∆ for the discrete version − 3 3 2 of ∂3 , similar to that used in the discrete Laplacian (45), which can be important at the nonanalyticity points (in this case, using ∆2 instead of ∆ ∆ results in δ instead 3 − 3 3 k31 of δk30). The dependence on k is also present in n , λ =1, 2, but with a smaller coefficient, | 3| λ b2/3, and the appropriate terms in b2R are suppressed by a power of b, despite ∼ λµνρ 1/3 the large value b− of Re φ, ∼ (Re φ)∆ ∆ (n n ) b1/3δ , λ,µ =3. (100) 3 3 λ µ ∼ k30 6 The symbol δ can also arise from the (single) differentiation of n sgn k over k30 3 ∼ 3 k3, but the corresponding terms in the ∆∆g part of Rλµνρ are suppressed by a power of b, ∆ (∆ n )n Re φ = (∆ ∆ n )n Re φ + b1/3δ . (101) λ 3 3 µ λ 3 3 µ ···∼ k30 2 In the (∆g) part of Rλµνρ, factors such as

∆ (l l ) = ∆ (n n Re φ) δ (102) 3 λ 3 3 λ 3 ∼ k30 could presumably contain the contribution of interest to us, since their product could 2 give (δk30) = δk30. However, such a product [∆3 (lλl3)] [∆3 (lµl3)] can only appear in 25

Rλµνρ from C3,λ3C3,µ3, but C3,λ3 does not contain ∆3 (lλl3). The latter is contained in C , and the potentially interesting term is C C , λ,µ,ν = 3. This includes λ,33 λ,33 λ,µν 6 1/3 terms with δk30, but they are suppressed by b compared to the leading one, (99),

[∆ (n n Re φ)] [∆ (n n Re φ)] b1/3δ , λ,µ,ν,ρ =3. (103) 3 λ 3 µ ν ρ ∼ k30 6

As a result, the leading discrete δ-function-like (Kronecker symbol) contribution to

Rλµνρ in the neighborhood of the singularity ring takes the form

r 2 r R = g n n ∆ ∆ Re φ = g n n δ . (104) 3λ3µ 2b2 λ µ 3 3 √3µ ab2 λ µ k30

The Riemann tensor at x = 0 (86) and in the vicinity of x = 0 (104) diﬀer in the 3 6 3 5/3 2 values and spatial distributions, b− and b− δk30. Note that the contribution (86) also occurs at x3 = 0 in addition to (104), since these contributions come from diﬀerent terms in Re φ, bilinear in k for (86) and proportional to k for (104) ((86) is simply | 3| 1/3 suppressed at x3 = 0 by b in comparison with (104)). Therefore, it is appropriate to use the same expression for Rλµνρ for all k3,

r R = g n n ∆ ∆ Re φ n n ∆ ∆ Re φ λµνρ 2b2 − λ ρ (µ ν) − µ ν (λ ρ)

+nµnρ∆(λ∆ν) Re φ + nλnν ∆(µ∆ρ) Re φ . (105) In addition, (105) through (104) is an exhaustive δ-function-like contribution to

Rλµνρ in the leading order over metric variations also at points located at a sufficiently large distance (73) from the ring. Indeed, this contribution follows when twice dif- ferentiating Re φ in the terms ∆3∆3(nλnµ Re φ). Here we can also differentiate twice 2 2 2 nλ, λ = 1, 2 (or nµ). This operation gives δk30 at x1 + x2 < a , that is, x3 = 0 and Re φ = 0 in factors. So, for our purposes, only Re φ needs to be differentiated twice in ∆3∆3(nλnµ Re φ). δk30 also appears in ∆3n3 or ∆3n3, that is, in the terms 2 2 2 ∆3∆λ(n3nµ Re φ) or ∆3∆λ(n3nµ Re φ). But δk30 provides Re φ = 0 for x1 + x2 < a and, therefore, also ∆λ Re φ = 0, λ = 1, 2. So, such terms do not contribute to δk30. 2 In the (∆g) -part of Rλµνρ, as in the above consideration of the same question in the vicinity of the ring, δk30 could arise due to the terms (103) from ∆3n3 or ∆3n3. Again, this provides Re φ = 0 among the factors, and this contribution disappears. Thus, in the leading order over metric variations, either in the vicinity of the ring, or at macroscopic distances (73) from the ring, the source contribution in the curvature 26 is

r δ part of R = δ part of g ∆ ∆ Re φ n n δ3δ3 + n n δ3δ3 k30 λµνρ k30 2b2 3 3 λ ν µ ρ µ ρ λ ν n n δ3δ3 n n δ3δ3 . (106) − λ ρ µ ν − µ ν λ ρ Then we have for the Einstein tensor G = R R(η + l l )/2, λµ λµ − λµ λ µ r δ part of G = δ part of g ∆ ∆ Re φ m , k30 λµ k30 2b2 3 3 λµ 1 n2 n n 0 − 3 1 2  2  2 3 3 n1 1 n2 n1n2 0 mλµ = nλnµ + n3ηλµ n3nλδµ n3nµδλ = − . (107) − −  2   n2 n1n2 1 n1 0   −     0 0 00      On the left-hand side, we know a priori that the Kerr metric (reproduced at macroscopic distances (73) from the ring with our discrete φ (59) in the leading order over metric variations) is a vacuum solution outside sources, and our discrete solution in the vicinity of the ring obeys R =0 at x = 0 (93), that is, λµ 3 6

δk30 part of Gλµ = Gλµ. (108)

On the right-hand side, φ is an analytic continuation of the solution to the Poisson equation with a point source, and we can check that it is itself a vacuum solution outside δ-function-like sources. Indeed, a simple calculation for the exact (within the framework of our ansatz, of course) φ (58) gives

3 π π dq1dq2 s x3 ∆ ∆ φ(x)= δ 3 exp i [q x + q x aκ] | |κ . (109) j j k 0 πb b 1 1 2 2 − − b j=1 π π X Z− Z− In passing, we note the following relation useful for estimating if φ is known:

3 ∂ ∆ ∆ φ(x)= 2δ sh b φ. (110) j j − k30 ∂ x j=1 3 X | | Thus, 3 3

δk30 part of ∆j∆j Re φ = ∆j∆j Re φ. (111) j=1 j=1 X X Here, in turn,

δk30 part of ∆j∆j Re φ = δk30 part of ∆3∆3 Re φ, (112) j=1 X 27 since the diﬀerentiations over k1, k2 do not lead to δ-function analogs. Thus,

r 3 G = g ∆ ∆ Re φ m . (113) λµ 2b2 j j λµ j=1 ! X

The matrix structure of Gλµ is such that

µ µ Gλµn =0, that is, Gλµl =0. (114)

Due to this, the indices of Gλµ can be raised/lowered using ηλµ.

As a consistency check, consider the vanishing of the divergence of Gλµ. For consistency, consider this at different distances, although this may be of particular interest for the vicinity of the singularity ring, while at sufficiently large distances from the ring, the formalism is close to continual, where such an identity is fulfilled in a routine way. Denoting the covariant finite difference by Dλ, we have

1 1 1 D Gλ = ∆ √ gGλ Gλν∆ g = ∆ Gλ Gλν∆ (l l ) λ µ √ g λ − µ − 2 µ λν λ µ − 2 µ λ µ − λ = ∆λGµ. (115)

The neglect of the second term in the leading order over metric variations in the last equality is possible due to (114).

2 At macroscopic distances from the ring, taking our φ (59) or the known l0/rg (22), we ﬁnd

3 2ab ∆j∆j Re φ = δk30, (116) − 2 2 2 3/2 j=1 (a x1 x2) X − − rg a Gλµ = δk30mλµ, (117) − b (a2 x2 x2)3/2 − 1 − 2 λ and the direct calculation gives the vanishing DλGµ in the leading order over metric variations,

λ λν DλGµ = η ∆ν Gλµ x2 + x2 ax ax 0 1 2 − 2 1  2 2  rg ax2 a x1 x1x2 0 = δk30 ∆0, ∆1, ∆2, ∆3 − − − −ab −  2 2   ax1 x1x2 a x2 0   − −     0 0 00    1   3/2 = 0, 0, 0, 0 . (118) ·(a2 x2 x2) # − 1 − 2 28

1/3 In the vicinity of the singularity ring, n3 has smallness O b , and in the leading order over metric variations mλµ = nλnµ (for λ,µ = 3; otherwise 0) and 6 3 4 1 ∆ ∆ Re φ = δ , (119) j j √3µ a k30 j=1 X 2 r G = g δ n n ,λ,µ =3, otherwise G =0, (120) λµ √3µ ab2 k30 λ µ 6 λµ r 3 r 3 D Gλ = g ∆ mλ ∆ ∆ Re φ = g n nα∆ ∆ ∆ Re φ =0, (121) λ µ 2b2 λ µ j j 2b2 µ α j j j=1 j=1 X X where α = 1, 2, µ = 3 in (121), but for µ = 3 we get the same zero D Gλ identi- 6 λ µ cally. Here, since our expansion of metric functions goes in b1/3k, the normal order

1/3 of magnitude associated with ∆λ is O b (this is obtained by the action of ∆λ on 3 2/3 j=1 ∆j∆j Re φ; O b is obtained when ∆λ acts on nµ, µ = 1, 2). Now, in (121), 2/3 Pa combination n1∆1+ n2∆2 is formed, with which the order of magnitude O b is associated with the additional smallness O b1/3 , as we considered above, (81- 83). (In the leading order, n1∆1 + n2∆2 = sin ϕ0∆ 1 + cos ϕ0∆2 can be considered as a ﬁnite- − 3 diﬀerence version of the derivative over ϕ, so that the independence of j=1 ∆j∆j Re φ λ on ϕ is essential here.) This just gives the vanishing DλGµ in the leadingP order over metric variations or, here, over b. (109) implicitly uses a >> b (via order of magnitude estimates, (99-103)), but for comparison it is useful to formally consider its right-hand side at a = 0,

3 δ π x π x ∆ ∆ φ(x) = k30 exp iq 1 dq exp iq 2 dq j j πb 1 b 1 2 b 2 j=1 π π X Z− Z− 4π = δ δ δ , x = m b, j =1, 2. (122) b m10 m20 k30 j j This is an analogue of the continuum ∇2φ =4πδ3(x). − At a = 0, consider an analogue of the continuum ∇2φ d3x, that is, the sum 6 − 3 1 N N π Rπ d2q b3 ∆ ∆ φ = exp is q m + q m b2 j j π 1 1 2 2 m1m2k3 j=1 m1= N m2= N π π − − Z− Z− n h X X π π X2 X a d q sin[q1(N +1/2)] sin[q2(N +1/2)] κ (q1, q2) = − b π π π sin(q1/2) sin(q2/2) a io Z− Z− exp i κ (q , q ) . (123) · − b 1 2 h i Here N >> 1. At N >> a/b, the sines effectively behave as δ-functions here, sin[q (N +1/2)] 1 2πδ(q), (124) sin(q1/2) ⇒ 29 and we have 3 1 b3 ∆ ∆ φ =4π, (125) b2 j j 1 2 3 j=1 mXm k X the same, as in the case a = 0. But at a = 0, the source is not pointlike. The contribution of the inner vertices of 6 2 2 2 the disk x1 + x2 < a , located at macroscopic distances (73) from the ring, to the sum (125) can be estimated by passing to an integral of the discrete Laplacian of φ (116) with the lower cutoff a1/3b2/3 for these distances, ∼ 3 3 1 2aδ(x3) 3 3 a b ∆j∆jφ = d x . (126) b2 − 2 2 2 3/2 ∼ − b disk j=1 (a x1 x2) r X X Z − − This contribution is infinite at b 0 and has the sign opposite to that required. → The contribution to this sum from the neighborhood of the ring should also be taken into account. To estimate, we take the value of the discrete Laplacian of φ to be (119) in a band along the ring with width a1/3b2/3. The number of vertices in this ∼ band is the area of the band divided by b2, and it is (a/b)4/3. The corresponding ∼ contribution to the sum (125) is

3 1 b a 4/3 a b3 ∆ ∆ φ 3 . (127) b2 j j ∼ a b ∼ b ring j=1 r X X This value has the same degree of divergence at b 0 as (126), but has the required → sign. The sum (125) is the result of the mutual cancellation of large numbers. In the continuum GR, we have d2p ∇2φ = δ(x ) exp i p x + p x (a i0) p2 + p2 − 3 π 1 1 2 2 − − 1 2 ZZ q (a iε)δ(x ) = 2 lim − 3 . (128) − ε 0 [(a iε)2 x2 x2]3/2 → − − 1 − 2 For a small but nonzero ε, the multiplier at δ(x3) here depends on x1, x2 (i.e., on x2 + x2), as shown in Fig. 3. Trying to write the limiting ε 0 distribution in 1 2 → pclosed form as an ordinary function, we get something like 2aθ (a2 x2 x2) 1 ∇2 1 2 2 2 Re φ = − − δ(x3)+ δ a x1 + x2 δ(x3). (129) − − (a2 x2 x2)3/2 0 − − 1 − 2 q The δ-function term with support on the ring turns out to have a formally infinite coefficient. In the above discrete approach, such a term is finite and phenomenologically in the continuum notations looks as

2 2/3 1/3 2 2 ∇ Re φ + a− b− δ a x + x δ(x ). (130) − ∼··· − 1 2 3 q 30 × ) 3 x ( δ = φ Re 2 0 −∇

0 a 2 2 x1 + x2 p Figure 3: The source distribution, regularized in the continuum (128) (or qualitative in the discrete approach; the factor at δ(x3), as written here, or at δk30, respectively).

8 Conclusion

In the functional integral framework, the problem of the discrete Kerr geometry (like the problem of any other discrete classical solution) arises as the problem of finding the optimal background metric for the perturbative expansion within this framework. In addition to the classical equations of motion (3), we also have the condition of maximizing the functional measure (4) (once the measure turns out to have maxima), which fixes the elementary length scale b. In the leading order over metric variations from simplex to simplex, the equations of motion are reduced to the finite-difference form of Einstein’s equations. To solve analytically in general form the full Einstein system, especially discrete, seems to be an unattainable task, but we can use an ansatz with one unknown complex function φ which gives the Kerr metric in the continuum case. In the discrete case, we obtain a solution close to the continuum Kerr one at large distances, having a finite metric and effective curvature on the ring that was singular in the continuum. A distinctive feature of this system is that the typical distance from the singularity

3 at which the metric approaches the continuum one is √ab2 which is much larger than such a distance b in the discrete Schwarzschild case (we assume a >> b, since this case is physically more signiﬁcant). That is, the lattice eﬀects in the vicinity of the singularity are enhanced here in comparison with the Schwarzschild case. The Riemann tensor components are found in the vicinity of the ring. Those with

3 sources R = 0 have support in the plane x = 0 on a disk which is a O(√ab2)- λµ 6 3 31 neighborhood of the disk x2 + x2 a2 (formed by the singularity ring). The vacuum 1 2 ≤ 5/3 R (i.e., at x = 0) in the vicinity of the ring is O(b− ) in the leading order at λµνρ 3 6 λµνρ small b. It has RλµνρR = 0, and all four curvature invariants vanish on the ring in this order. In the neighborhood of the singularity ring at x = 0, R has R = 0 3 λµνρ λµ 6 2 and is O(b− ).

The source distribution Gλµ in the continuum can be written as a limit of a regularized expression (or a derivative of a generalized function). Heuristically, it can be represented as the sum of the δ-function on the ring with an infinite coefficient and a smooth function of the opposite sign on the disk formed by the ring, diverging when approaching the ring. In the discrete version, there are no infinities; phenomenologically, Gλµ can be written as the sum of the δ-function on the ring with a coefficient 2/3 1/3 O(a− b− ) and a smooth function of the opposite sign on the disk, cut off at a

3 distance O(√ab2) from the ring. Integrally, the large (at small b) contribution of the ring is partially cancelled by the inner part of the disk, and this corresponds to rg, as in the Schwarzschild problem. Physically, we have a rather exotic required energy-momentum tensor, such that the total energy on the ring is many times (for small b) greater than the mass that determines rg, and this excess is in total compensated by the negative contribution of the inner part of the disk.

λ The fulﬁllment of the discrete version of the conservation law DλGµ = 0 in the leading order over metric variations is also veriﬁed, including, which is especially interesting, on the singularity ring.

Acknowledgments

The present work was supported by the Ministry of Education and Science of the Russian Federation.

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