504
R. Plamondon
sum of independent or quasi-independent random variables. The key feature of this
theorem, which makes it practical for the study of complex systems, is that although
the details of the individual sub-processes are unknown, the behavior of the whole
system can be predicted, under some non-restrictive conditions, to converge toward
multivariate Gaussian functions.
In this paper, we use this modeling approach to conjecture about emergent
gravity, speculating from a possible connection between the Bayes’s law and the
Einstein’s general relativity equation. The idea of modifying gravity to come up
with new relativistic field descriptions has been proposed time and again in the
last decades to provide among other things alternative explanations to some open
problems in astronomy and astrophysics [4, 5, 6]. These extensions aimed at
correcting and enlarging Einstein’s theory to encompass several shortcomings when
cosmological, astrophysical, mathematical, and quantum mechanical observations
and objections are taken into account [7, 8, 9]. In this mindset, in a recent paper
[10], the static non-empty symmetric geometry described by a metric based on an
erfc gravitational potential has been proposed and studied in detail. This new metric
provides a consistent set of predictions and interpretations regarding some open
problems in the solar system, like the fly-by anomalies, the secular increase of the
astronomical unit, the residual Pioneers’ delays [11].
In the present manuscript, a fundamental question is addressed: can we lay the
foundations for an emergent model that predicts the existence of an erfc potential
using the central limit theorem? In the next section, starting from a complementarity
that has been pointed out by Wheeler, we propose a comparison between Einstein’s
equation and Bayes’s law of conditional probabilities and use it to support our
analogical and speculative argumentation. The whole framework relies on a global
probabilistic description of a star formation from which a fundamental law of
gravitation comes out as a consequence of an asymptotic convergence predicted by
the Central Limit Theorem. In Sect. 2, we put the general relativity in a probabilistic
context and in Sect. 3, we present the conditions under which a weighted Newton’s
law automatically emerges from this new scheme and then conclude.
1 Introducing a Probabilistic Context in General Relativity
Einstein’s gravitation equation, which links the space-time curvature tensor G to the
energy-momentum tensor T,
G = KT
(1)
has been encapsulated by Wheeler as, “Space-time tells matter how to move; matter
tells space-time how to curve.” [12]. This points out an interesting interdependence
that can be used to put general relativity into a probabilistic context if this assertion
is converted into a general and fundamental premise:
R. Plamondon
sum of independent or quasi-independent random variables. The key feature of this
theorem, which makes it practical for the study of complex systems, is that although
the details of the individual sub-processes are unknown, the behavior of the whole
system can be predicted, under some non-restrictive conditions, to converge toward
multivariate Gaussian functions.
In this paper, we use this modeling approach to conjecture about emergent
gravity, speculating from a possible connection between the Bayes’s law and the
Einstein’s general relativity equation. The idea of modifying gravity to come up
with new relativistic field descriptions has been proposed time and again in the
last decades to provide among other things alternative explanations to some open
problems in astronomy and astrophysics [4, 5, 6]. These extensions aimed at
correcting and enlarging Einstein’s theory to encompass several shortcomings when
cosmological, astrophysical, mathematical, and quantum mechanical observations
and objections are taken into account [7, 8, 9]. In this mindset, in a recent paper
[10], the static non-empty symmetric geometry described by a metric based on an
erfc gravitational potential has been proposed and studied in detail. This new metric
provides a consistent set of predictions and interpretations regarding some open
problems in the solar system, like the fly-by anomalies, the secular increase of the
astronomical unit, the residual Pioneers’ delays [11].
In the present manuscript, a fundamental question is addressed: can we lay the
foundations for an emergent model that predicts the existence of an erfc potential
using the central limit theorem? In the next section, starting from a complementarity
that has been pointed out by Wheeler, we propose a comparison between Einstein’s
equation and Bayes’s law of conditional probabilities and use it to support our
analogical and speculative argumentation. The whole framework relies on a global
probabilistic description of a star formation from which a fundamental law of
gravitation comes out as a consequence of an asymptotic convergence predicted by
the Central Limit Theorem. In Sect. 2, we put the general relativity in a probabilistic
context and in Sect. 3, we present the conditions under which a weighted Newton’s
law automatically emerges from this new scheme and then conclude.
1 Introducing a Probabilistic Context in General Relativity
Einstein’s gravitation equation, which links the space-time curvature tensor G to the
energy-momentum tensor T,
G = KT
(1)
has been encapsulated by Wheeler as, “Space-time tells matter how to move; matter
tells space-time how to curve.” [12]. This points out an interesting interdependence
that can be used to put general relativity into a probabilistic context if this assertion
is converted into a general and fundamental premise: