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Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes

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Abstract

We study the fractional gravity for spacetimes with non-integer fractional derivatives. Our constructions are based on a formalism with the fractional Caputo derivative and integral calculus adapted to nonholonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange–Finsler and Hamilton–Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, ellipsoid gravitational configurations and imbedding of such objects in solitonic backgrounds.

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Notes

  1. We follow the system of notations with left abstract labels explained in Refs. [45, 52], see also next section.

  2. In “integer” differential geometry, there are used also equivalent terms like anholonomic and non-integrable manifolds. A nonholonomic manifold of integer dimension is defined by a pair \((\mathbf{V},\mathcal{N})\), where V is a manifold and \(\mathcal{N}\) is a nonintegrable distribution on V, see the next section for a generalization to non-integer dimensions.

  3. The left Riemann–Liouville (RL) derivative is

    $${}_{_{1}x}\overset{\alpha}{\partial}_{x}f(x):=\frac{1}{\Gamma (s-\alpha)}\biggl( \frac{\partial}{\partial x}\biggr) ^{s}\int_{_{1}x}^{x}(x-x^{\prime})^{s-\alpha-1}f(x^{\prime})\,dx^{\prime},$$

    where Γ is the Euler’s gamma function. The left fractional Liouville derivative of order α, where s−1<α<s, with respect to coordinate x is \(\overset{\alpha }{\partial }_{x}f(x):=\lim_{_{1}x\rightarrow -\infty }{}_{\ _{1}x}\overset{\alpha }{\partial }_{x}f(x)\).

    The right RL derivative is \({}_{x}\overset{\alpha }{\partial }_{_{2}x}f(x):=\frac{1}{\Gamma (s-\alpha )}( -\frac{\partial }{\partial x}) ^{s}\int_{x}^{_{2}x}(x^{\prime }-x)^{s-\alpha -1}f(x^{\prime })\,dx^{\prime }\). The right fractional Liouville derivative is \({}_{x}\overset{\alpha }{\partial }f(x^{k}):=\lim_{_{2}x\rightarrow \infty }{}_{\ x}\overset{\alpha }{\partial }_{_{2}x}f(x)\).

    The fractional RL derivative of a constant C is not zero but, for instance, \({}_{_{1}x}\overset{\alpha}{\partial}_{x}C=C\frac{(x-_{1}x)^{-\alpha}}{\Gamma(1-\alpha)}\). Integro-differential constructions based only on such derivatives seem to be very cumbersome and has a number of properties which are very different from similar ones for the integer calculus.

  4. This follows from the fundamental theorem of fractional calculus [35]

  5. For the “integer” calculus, we use as local coordinate co-bases/-frames the differentials dx j=(dx j)α=1. For 0<α<1 we have dx=(dx)1−α(dx)α. The “fractional” symbol (dx j)α, related to \(\overset{\alpha}{d}x^{j}\), is used instead of dx i for elaborating a co-vector/differential form calculus, see below the formula (6).

  6. We to not follow alternative constructions with RL fractional derivative [1, 17] because in that direction it is not clear how to prove integrability of the RL-fractional Einstein equations.

  7. The N-adapted coefficients are explicitly determined by the (13),

    We can verify that introducing above formulas into (15) we obtain that \(\widehat{T}_{\ jk}^{i}=0\) and \(\widehat{T}_{\ bc}^{a}=0\), but \(\widehat{T}_{\ ja}^{i},\widehat{T}_{\ ji}^{a}\) and \(\widehat{T}_{\ bi}^{a}\) are not zero, and that the metricity conditions are satisfied in component form.

  8. The N-adapted coefficients of distortion tensor \(Z_{\ \alpha\beta }^{\gamma}\) are computed

  9. Such transforms of geometric objects (with deformations of the frame, metric, connections and other fundamental geometric structures) are more general than those considered for the Cartan’s moving frame method, when the geometric objects are re-defined equivalently with respect to necessary systems of reference.

  10. ε is an eccentricity because, for integer configurations, the coefficient \(h_{4}=b^{2}=\eta_{4}(\xi,\vartheta ,\varphi,\bar{\theta})\varpi^{2}(\xi)\) becomes zero for data (53) if \(r_{+}\simeq{2\mu_{0}}/[{1+\varepsilon\frac {q_{0}(r)}{4\mu ^{2}}\sin(\omega_{0}\varphi+\varphi_{0})}]\); this is the “parametric” equation for an ellipse r +(φ) for any fixed values \(\frac{q_{0}(r)}{4\mu^{2}},\omega_{0},\varphi_{0}\) and μ 0.

  11. A function η can be a solution of any three dimensional solitonic and/or other nonlinear wave equations

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  1. Science Department, University “Al. I. Cuza” Iaşi, 54 Lascar Catargi Street, Iaşi, 700107, Romania

    Sergiu I. Vacaru

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Vacaru, S.I. Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes. Int J Theor Phys 51, 1338–1359 (2012). https://doi.org/10.1007/s10773-011-1010-9

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