Although Schwarzschild black holes are the simplest textbook example (Hawking
and Ellis 1973), a simplified or toy model rather than a physically realistic one, the
Schwarzschild radius is profoundly important nonetheless. This feature may be the
key to understanding the quantum nature of gravity, because the surface area at that
radius is quantized, and this sets its information-carrying capacity. If we treat gravity
as a process of graviton condensation, information storage and processing at the
event horizon can be explained as behavior at the quantum critical point of a BEC—
according to recent work by Dvali and Gomez (2014)—and this is worthy to explore
further. It opens up the possibility that laboratory studies of various types of quantum
condensation can be used to probe the nature of BH event horizons (Steinhauer
2016) (Herdman et al. 2017), and thereby give us a better understanding of gravity
itself.
Sakharov (1967) first proposed analogies, which ascribe to the action of gravity
the same nature as BEC formation—but at that time Bose–Einstein condensation
was seen as only a curious theoretical possibility. In the current era, we know that
BECs will reliably form if we cool an appropriate sample far enough (Anderson et al.
1995). This is now within the reach of almost any University Physics department, as
well, because the process has been perfected and miniaturized (Hänsel et al. 2001).
But the analogy of gravity with Bose–Einstein condensation offers a window on how
gravity’s quantum mechanical action gives rise to well-known classical properties.
The same analogy is seen in features of the Mandelbrot Set (M) at the Misiurewicz
point near (À1.543689, 0i), and in associated and related figures of M, which the
author began studying more than 30 years ago (Dickau 2016a). This point is a
solution to ((c
2 + c)
2 + c)
2 + c ¼ (c
2 + c)
2 + c, and it is one of the few places in M
where an exact analytical solution (its precise location) is possible to obtain. When
points are highlighted, where the function’s iterand diminishes monotonically over
3 iterations (i.e., the Mandelbrot Butterfly), the special significance of this spot is
easy to discern. But examined in the standard rendering; it is an unassuming spot
where a collection of telephone poles diminishes in size to extinction, then grows on
the other side in opposite phase—a feature that is easily missed. Nonetheless, since
this location in M is an analogy for both black hole event horizons and Bose–
Einstein condensation, it is relevant to unifying the quantum and relativistic views of
Physics. The remainder of this paper discusses what we can learn from these
overlapping analogies, where multiple descriptions afford a single congruent result.
7.2 Entropic Gravity, Condensation, and Dimensional
Reduction
When people ask “what is gravity?” the standard answers do not mention entropy or
thermodynamics, and describe it as an attractive force between massive objects, state
that it is due to the curvature of space, or say it is one of the four fundamental forces;
but there are other answers. It has been suggested by Jacobson (1995), Verlinde
(2011), Padmanabhan (2010a, b), and others, that instead of a fundamental force of
68
J. J. Dickau
and Ellis 1973), a simplified or toy model rather than a physically realistic one, the
Schwarzschild radius is profoundly important nonetheless. This feature may be the
key to understanding the quantum nature of gravity, because the surface area at that
radius is quantized, and this sets its information-carrying capacity. If we treat gravity
as a process of graviton condensation, information storage and processing at the
event horizon can be explained as behavior at the quantum critical point of a BEC—
according to recent work by Dvali and Gomez (2014)—and this is worthy to explore
further. It opens up the possibility that laboratory studies of various types of quantum
condensation can be used to probe the nature of BH event horizons (Steinhauer
2016) (Herdman et al. 2017), and thereby give us a better understanding of gravity
itself.
Sakharov (1967) first proposed analogies, which ascribe to the action of gravity
the same nature as BEC formation—but at that time Bose–Einstein condensation
was seen as only a curious theoretical possibility. In the current era, we know that
BECs will reliably form if we cool an appropriate sample far enough (Anderson et al.
1995). This is now within the reach of almost any University Physics department, as
well, because the process has been perfected and miniaturized (Hänsel et al. 2001).
But the analogy of gravity with Bose–Einstein condensation offers a window on how
gravity’s quantum mechanical action gives rise to well-known classical properties.
The same analogy is seen in features of the Mandelbrot Set (M) at the Misiurewicz
point near (À1.543689, 0i), and in associated and related figures of M, which the
author began studying more than 30 years ago (Dickau 2016a). This point is a
solution to ((c
2 + c)
2 + c)
2 + c ¼ (c
2 + c)
2 + c, and it is one of the few places in M
where an exact analytical solution (its precise location) is possible to obtain. When
points are highlighted, where the function’s iterand diminishes monotonically over
3 iterations (i.e., the Mandelbrot Butterfly), the special significance of this spot is
easy to discern. But examined in the standard rendering; it is an unassuming spot
where a collection of telephone poles diminishes in size to extinction, then grows on
the other side in opposite phase—a feature that is easily missed. Nonetheless, since
this location in M is an analogy for both black hole event horizons and Bose–
Einstein condensation, it is relevant to unifying the quantum and relativistic views of
Physics. The remainder of this paper discusses what we can learn from these
overlapping analogies, where multiple descriptions afford a single congruent result.
7.2 Entropic Gravity, Condensation, and Dimensional
Reduction
When people ask “what is gravity?” the standard answers do not mention entropy or
thermodynamics, and describe it as an attractive force between massive objects, state
that it is due to the curvature of space, or say it is one of the four fundamental forces;
but there are other answers. It has been suggested by Jacobson (1995), Verlinde
(2011), Padmanabhan (2010a, b), and others, that instead of a fundamental force of
68
J. J. Dickau