Towards Fractal Gravity

In an extension of speculations that physical space–time is a fractal which might itself be embedded in a high-dimensional continuum, it is hypothesized to “compensate” for local variations of the fractal dimension by instead varying the metric in such as way that the intrinsic (as seen from an embedded observer) dimensionality remains an integer. Thereby, an extrinsic fractal continuum is intrinsically perceived as a classical continuum. Conversely, it is suggested that any variation of the metric from its Euclidean (or Minkowskian) form can be “shifted” to nontrivial fractal topology. Thereby “holes” or “gaps” in spacetime could give rise to (increased) curvature.

Embedded observers [1-3] and agents are operationally bound by self-reflexive, intrinsic methods and means available from within the very system they exist.Such observers have no access to extrinsic, Platonistic entities which are beyond their operational physical capacities.(They may, nonetheless, have inspirational "afflatus" or ideas about some external truth; but would not be able to prove this in any effable way [4] beyond zero-knowledge proof methods.) In particular, based on Hausdorff measures and dimension theory [5][6][7][8][9][10][11] of fractals [12], it has been suggested that, while (i) extrinsically and ontologically, space-time might be a fractal set with possibly non-integer dimension [13,14], (ii) intrinsically and epistemically, that is, from an operational point of view, it might appear as if observers embedded in such fractals would experience not much phenomenological differences as compared to "inhabiting" standard continua such as R n [15][16][17][18][19].In other words, the fractal space-time concept can be put to some extreme by speculating that, for all practical purposes, intrinsically embedded observers cannot differentiate between, say, three-dimensional continua R 3 and some continuous fractal which is a (possibly stochastic) generalization of the Cantor set of Hausdorff dimension three [18], and which is embedded in a larger-dimensional continuum, say, R d , with d > 3.
I suggest here to take a further speculative step by shifting the nontrivial topological structure of such fractals to the metric of the (embedding) space.That is, even for non-integer dimensions, intrinsic observers could, for all practical purposes, not differentiate whether they either exist in a space with standard (Euclidean, Minkowski) metric whose support is a fractal; or whether their support is a classical, integer dimensional continuum (say, R 3 ), but the Riemannian metric of the space is somehow non-standard and, in particular, non-Euclidean or non-Minkowskian.
For the sake of an intuitive, informal example of why "cutting out holes" in a given set and "gluing together" the remaining pieces might affect geometric properties of the object, consider a situation depicted in Fig. 1, in which segments of a unit circle are eliminated, and the remaining pieces form a new circle of smaller radius.
Another fractal example is (as often) of the Cantor set type.Suppose from a unit circle the middle third segment 2π 3 , 4π 3 is cut out, such that the two pieces 0, 2π 3 and 4π 3 , 2π remain, as is depicted in Fig. 2(a).From these remaining pieces, the respective middle third segments are cut out again, as is depicted in Fig. 2(b)-(e); and so on ad infinitum.Thereby a continuum of measure zero is obtained: at the n'th construction stage, encode each first remaining third by 0 and each third remaining third by 1, and associate these respective bits with the n'th digits of a binary number.In the limit this construction creates the binary unit continuum [0, 1].However, at each con- struction stage, the set "loses" one third of its length, so that, in the limit this length converges to zero; that is, lim n→∞ So, effectively, the "price" of scale independence of the measure is the non-intuitive fact that the dimension of this set is not a natural number.In an ad hoc attempt to maintain some positive integer dimensionality of the set one may go one step further and turn to changing the metric; thereby forcing the dimensional parameter to be a natural number again.
For the sake of an example, note that the volume of a ball of radius r in d-dimensional Euclidean space is V (d, r) = √ πr d /Γ(d/2 + 1).By "shifting" the dimensionality d to the "curvature" r; that is, by one obtains a "radius" r associated with the Cantor set by inserting d = log(2)/log(3); that is, By abduction one may infer the following general desiderandum for the parametrization of "volume" as it relates to fractal dimensionality and curvature: Thereby the terms (i) fractal dimension d on the left hand side of (3) refers to the Hausdorff dimension of the fractal object, as seen extrinsically, whereby the object is embedded in a space of extrinsic, higher dimensionality n; (ii) outer, extrinsic curvature, parametrized by the radius R on the left hand side of (3), stands for the curvature of the fractal object within an embedding space; (iii) target dimension m on the right hand side of (3), refers to the intrinsic, dimension of the object "forced" to be a natural number; thereby the fractal set will, operationally and intrinsically, not be perceived as fractal but rather as a conventional continuum R m of smaller or equal dimensionality than the embedding space, but of higher or equal dimensionality than the fractal; that is, (iv) intrinsic curvature, parametrized by the radius r on the lright hand side of (3), refers to the curvature experienced intrinsically upon pretension of the target dimensionionality.
Corresponding to (4), as compared to the extrinsic radius, one obtains a smaller or equal intrinsic radius; that is Of course, these considerations are tentative, highly speculative and need further scrutiny.
Many issues and questions remain, among them how to conceptualize the shift (back & forth) from the "fractality of the continuum" to the metric; and vice versa.Also, it needs to be seen how to obtain curvature from an originally flat (zero curvature) spacetime.In the end, there might appear a possibility to extend the formalism of general relativity by "punching" scale invariant "holes" or "gaps" into space-time; thereby creating a theory of gravity which generalizes relativity theory by fractal geometric support with non-curved standard metrics.

ACKNOWLEDGMENTS
The idea to fractal gravity emerged from a skype conversation of the author with Hui Deng, Tyler Hill and Barry Sanders on July 5th, 2017 on their beautiful paper [20] on light scattering in any dimension.Further formalizations were discussed and investigated with Ludwig Staiger and Cristian S. Calude.I am grateful to Thomas Sommer for discussions and a critical reading of the manuscript.Nevertheless, I am to blame for all misconceptions and omissions.
[1] Tommaso Toffoli, "The role of the observer in uniform systems," in Applied General Systems Research: Recent Developments and Trends, edited by George J. Klir (Plenum Press, Springer US, New York, London, and Boston, MA, 1978) pp.395-400.
FIG. 1. Intuitive informal example why "cutting out holes" in a continuum might yield different radii if one "glues" together the remaining pieces.(a) consider an original circle with radius 1; (b) pieces of 30 degrees are cut out of (a), thereby effectively reducing the length of the set by a factor of two; (c) those pieces are "glued together" to yield a half-circle; (d) alternatively one can draw a full circle with a reduced radius of half the original radius.

FIG. 2 .
FIG. 2. Fractal example why "cutting out holes" or "creating gaps" in a continuum in a scale invariant way might yield different radii if the remaining pieces are scaled by the Hausdorff dimension and subsequently "pasted" together.(a) consider an original circle with radius 1; (b) the middle third segment 2π 3 , 4π 3 is cut out of (a), thereby effectively reducing the length of the set by a factor of 1 3 ; (c) the middle third segments 2π 9 , 4π 9 and 14π 9 , 16π 9 are cut out of the remaining segments in (b), thereby effectively reducing the length of the set by a factor of 1 3 ; (d)-(e) shows the iteration of this construction; (f) alternatively one can draw a full circle with a pasting of the upscaled segments and a reduced radius r ≈ 0.8 from Eq. (2).