7.3 The Part Played by the Mandelbrot Set
As it turns out, the scenario I spelled out above—where a higher-dimensional
volume is everted to create the 4-d spacetime bubble we now inhabit, and where
the action of gravity at a Schwarzschild horizon is like Bose–Einstein condensation—is encoded in the shape of the Mandelbrot Set—and in its family of related and
associated figures. This is something the author has struggled to understand for more
than 30 years (Dickau 1987), wondering if it would turn out to be of use to Physics.
A sketch of these ideas was presented on a poster at GR21 in 2016 (Dickau 2016b),
but my talk at FFP15 is the first attempt to spell out to the Physics community how
one can connect the dots to join Mandelbrot gravity with a larger body of theoretical
Physics relating to gravity and cosmology. The truly remarkable thing is that M
displays or reproduces features of 5-d ! 4-d DGP gravity and the correspondence of
Schwarzschild event horizons with Bose–Einstein condensation, with no adjusting
factors or constants put in by hand. This is mainly due to the fact that M is maximally
asymmetric, but contains many symmetries, and the ubiquitous role played by
symmetry-breaking and asymmetry shaping the laws of Physics (Dickau 2018).
The Mandelbrot Set is symmetrical across the real axis, but grossly asymmetrical
along it, having a positive maximum of (0.25, 0i) and a much greater extent in the
negative, with a negative extremum at (À2, 0i). We know that M displays the
progression to chaos, such that iterating the core equation z ! z
2 + c in the reals
yields a bifurcation diagram exactly mimicking the logistic map—where at each split
point the boundary of M folds back on itself. The first bifurcation point is at
(À0.75, 0i) where the maximum of the cardioid opens into the circular disc centered
at (À1, 0i). The feature at (À0.75, 0i) resembles the folding of space in a braneworld
model like DGP gravity, or the 5-d black hole ! 4-d spacetime bubble cited above.
That space can fold back on itself is the crucial generalization here, but dimensionality can be higher before and lower after such a transition. Though the Mandelbrot
Set is seen as a two-dimensional object, living in the complex numbers, it is defined
up to the octonions. The familiar representation of M is a projection of the set onto
any one imaginary dimension, but the hypercomplex numbers (H and ), with 3 and
7 imaginaries, respectively, afford M with more range for variations. What makes
the transition at (À0.75, 0i) more than a curious visual metaphor, however, is that M
encodes Cartan’s rolling ball analogy for G 2 symmetries (Fig. 7.1), where G 2 is the
smallest exceptional Lie group and the exceptional Lie groups all come from the
octonions.
In regard to what M tells us about gravity though, the bifurcation map in the Reals
gives us the essential clue. While a bifurcation diagram mainly shows how the
trajectories diverge, we observe there is a spot where all of the divergent trajectories
appear to converge, which is a Misiurewicz point. The corresponding location in
M ~ (À1.543689012692, 0i) is relatively unassuming, but it has a special dual
significance. It represents both the quantum critical point of Bose–Einstein condensation and the event horizon of a Schwarzschild black hole. This is where gravitation
by condensation is depicted in M. We note that this value satisfies the equation
7 Gravitation by Condensation
71
As it turns out, the scenario I spelled out above—where a higher-dimensional
volume is everted to create the 4-d spacetime bubble we now inhabit, and where
the action of gravity at a Schwarzschild horizon is like Bose–Einstein condensation—is encoded in the shape of the Mandelbrot Set—and in its family of related and
associated figures. This is something the author has struggled to understand for more
than 30 years (Dickau 1987), wondering if it would turn out to be of use to Physics.
A sketch of these ideas was presented on a poster at GR21 in 2016 (Dickau 2016b),
but my talk at FFP15 is the first attempt to spell out to the Physics community how
one can connect the dots to join Mandelbrot gravity with a larger body of theoretical
Physics relating to gravity and cosmology. The truly remarkable thing is that M
displays or reproduces features of 5-d ! 4-d DGP gravity and the correspondence of
Schwarzschild event horizons with Bose–Einstein condensation, with no adjusting
factors or constants put in by hand. This is mainly due to the fact that M is maximally
asymmetric, but contains many symmetries, and the ubiquitous role played by
symmetry-breaking and asymmetry shaping the laws of Physics (Dickau 2018).
The Mandelbrot Set is symmetrical across the real axis, but grossly asymmetrical
along it, having a positive maximum of (0.25, 0i) and a much greater extent in the
negative, with a negative extremum at (À2, 0i). We know that M displays the
progression to chaos, such that iterating the core equation z ! z
2 + c in the reals
yields a bifurcation diagram exactly mimicking the logistic map—where at each split
point the boundary of M folds back on itself. The first bifurcation point is at
(À0.75, 0i) where the maximum of the cardioid opens into the circular disc centered
at (À1, 0i). The feature at (À0.75, 0i) resembles the folding of space in a braneworld
model like DGP gravity, or the 5-d black hole ! 4-d spacetime bubble cited above.
That space can fold back on itself is the crucial generalization here, but dimensionality can be higher before and lower after such a transition. Though the Mandelbrot
Set is seen as a two-dimensional object, living in the complex numbers, it is defined
up to the octonions. The familiar representation of M is a projection of the set onto
any one imaginary dimension, but the hypercomplex numbers (H and ), with 3 and
7 imaginaries, respectively, afford M with more range for variations. What makes
the transition at (À0.75, 0i) more than a curious visual metaphor, however, is that M
encodes Cartan’s rolling ball analogy for G 2 symmetries (Fig. 7.1), where G 2 is the
smallest exceptional Lie group and the exceptional Lie groups all come from the
octonions.
In regard to what M tells us about gravity though, the bifurcation map in the Reals
gives us the essential clue. While a bifurcation diagram mainly shows how the
trajectories diverge, we observe there is a spot where all of the divergent trajectories
appear to converge, which is a Misiurewicz point. The corresponding location in
M ~ (À1.543689012692, 0i) is relatively unassuming, but it has a special dual
significance. It represents both the quantum critical point of Bose–Einstein condensation and the event horizon of a Schwarzschild black hole. This is where gravitation
by condensation is depicted in M. We note that this value satisfies the equation
7 Gravitation by Condensation
71