Sep 18, 2015 09:51 PM
Identification of a Gravitational Arrow of Time
Julian Barbour,1 Tim Koslowski,2 and Flavio Mercati3,*
1College Farm, South Newington, Banbury, Oxon OX15 4JG, United Kingdom
2University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada
3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
(Received 8 May 2014; published 29 October 2014)
It is widely believed that special initial conditions must be imposed on any time-symmetric law if its
solutions are to exhibit behavior of any kind that defines an “arrow of time.”We show that this is not so. The
simplest nontrivial time-symmetric law that can be used to model a dynamically closed universe is the
Newtonian N-body problem with vanishing total energy and angular momentum. Because of special
properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a
uniquely defined point into two halves. In each, a well-defined measure of shape complexity fluctuates but
grows irreversibly between rising bounds from that point. Structures that store dynamical information
are created as the complexity grows and act as “records.” Each solution can be viewed as having a single
past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and
will only be aware of the records of one branch and deduce a unique past and future direction from
inspection of the available records.
DOI: 10.1103/PhysRevLett.113.181101 PACS numbers: 04.40.-b, 04.20.-q
Many different phenomena in the Universe are time
asymmetric and define an arrow of time that points in the
same direction everywhere at all times [1]. Attempts to
explain how this arrow could arise from time-symmetric
laws often invoke a “past hypothesis”: the initial condition
with which the Universe came into existence must have
been very special. This is based on thermodynamic
reasoning, which seems to make a spontaneous emergence
of an arrow of time very unlikely. Although thermodynamics
works very well for subsystems, provided gravity is
not a dominant force, self-gravitating systems exhibit
“antithermodynamic” behavior that is not fully understood.
Since the Universe is the ultimate self-gravitating system
and since it cannot be treated as any subsystem, its behavior
may well confound thermodynamic expectations.
In this Letter, we present a gravitational model in which
this is the case. In all of its typical solutions, internal
observers will find a manifest arrow of time, the nature of
which we are able to precisely characterize. We emphasize
that in this Letter we make no claim to explain all the
various arrows of time. We are making just one point: an
arrow of time does arise in at least one case without any
special initial condition, which may therefore be dispensable
for all the arrows. In this connection, we mention that
in Ref. [2] (Sec. II), Carroll and Chen conjectured that the
thermodynamic arrow of time might have a time-symmetric
explanation through entropy arrows much like the complexity
and information arrows we find.
The model.—The Newtonian N-body problem with
vanishing total energy Etot ¼ 0, momentum Ptot ¼ 0,
and angular momentum Jtot ¼ 0 is a useful model of
the Universe in many respects [3]. As we show below,
these conditions match the intuition that only relational
degrees of freedom of the Universe should have physical
significance [4–6]. A total angular momentum Jtot and a
total energy Etot would define, respectively, an external
frame in which the Universe is rotating and an absolute
unit of time. Moreover the conditions Jtot ¼ Ptot ¼ 0 and
Etot ¼ 0 ensure scale invariance and are close analogues
of the Arnowitt-Deser-Misner constraints of Hamiltonian
general relativity (in the spatially closed case) [7]. These
properties, along with the attractivity of gravity, are
architectonic and likely to be shared by any fundamental
law of the Universe.
First support for our claim follows from the rigorous
results of Ref. [8] that asymptotically (as the Newtonian
time t → ∞) N-body solutions with Etot ¼ 0 and Jtot ¼0
typically “evaporate” into subsystems [sets of particles
with separations bounded by Oðt2=3Þ] whose centers of
mass separate linearly with t as t → ∞. (The separation
linear in t is, of course, a manifestation of Newton’s first
law in the asymptotic regime. Our result arises from the
combination of this behavior with gravity’s role in creating
subsystems that mostly then become stably bound.) Each
subsystem consists of individual particles and/or clusters
whose constituents remain close to each other; i.e., the
distances between constituent particles of a cluster are
bounded by a constant for all times. One finds in numerical
simulations that the bulk of the clusters are two-body
Kepler pairs whose motion asymptotes into elliptical
Keplerian motion. For reasons we next discuss, this
t → ∞ behavior occurs in all typical solutions on either
side of a uniquely defined point of minimum expansion, as
illustrated in Fig. 1.
PRL 113, 181101 (2014)
Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS week ending
31 OCTOBER 2014
0031-9007=14=113(18)=181101(5) 181101-1 © 2014 American Physical Society
(...)
Complexity is a prerequisite for storage of information in
local subsystems and therefore the formation of records.We
nownote that a notion of local records canbefound in amodel
as simple as our Etot ¼ 0 and Jtot ¼ 0 N-body problem.
Recall that the system, for large jtj, breaks up typically
into disjoint subsystems drifting apart linearly in
Newtonian time t. These subsystems get more and more
isolated, and one can associate dynamically generated local
information with them. For this, we use the result of
Marchal and Saari [8] that as t → ∞ each subsystem J
develops asymptotically conserved quantities:
EJ(t)=EJ(∞) + O(t^−5/3)
JJ(t)=JJ (∞) + O(t−2/3);
XJ(t/t) =VJ (∞)+O(t^−1/3)
For more, see: https://physics.aps.org/featured-article...113.181101
Julian Barbour,1 Tim Koslowski,2 and Flavio Mercati3,*
1College Farm, South Newington, Banbury, Oxon OX15 4JG, United Kingdom
2University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada
3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
(Received 8 May 2014; published 29 October 2014)
It is widely believed that special initial conditions must be imposed on any time-symmetric law if its
solutions are to exhibit behavior of any kind that defines an “arrow of time.”We show that this is not so. The
simplest nontrivial time-symmetric law that can be used to model a dynamically closed universe is the
Newtonian N-body problem with vanishing total energy and angular momentum. Because of special
properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a
uniquely defined point into two halves. In each, a well-defined measure of shape complexity fluctuates but
grows irreversibly between rising bounds from that point. Structures that store dynamical information
are created as the complexity grows and act as “records.” Each solution can be viewed as having a single
past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and
will only be aware of the records of one branch and deduce a unique past and future direction from
inspection of the available records.
DOI: 10.1103/PhysRevLett.113.181101 PACS numbers: 04.40.-b, 04.20.-q
Many different phenomena in the Universe are time
asymmetric and define an arrow of time that points in the
same direction everywhere at all times [1]. Attempts to
explain how this arrow could arise from time-symmetric
laws often invoke a “past hypothesis”: the initial condition
with which the Universe came into existence must have
been very special. This is based on thermodynamic
reasoning, which seems to make a spontaneous emergence
of an arrow of time very unlikely. Although thermodynamics
works very well for subsystems, provided gravity is
not a dominant force, self-gravitating systems exhibit
“antithermodynamic” behavior that is not fully understood.
Since the Universe is the ultimate self-gravitating system
and since it cannot be treated as any subsystem, its behavior
may well confound thermodynamic expectations.
In this Letter, we present a gravitational model in which
this is the case. In all of its typical solutions, internal
observers will find a manifest arrow of time, the nature of
which we are able to precisely characterize. We emphasize
that in this Letter we make no claim to explain all the
various arrows of time. We are making just one point: an
arrow of time does arise in at least one case without any
special initial condition, which may therefore be dispensable
for all the arrows. In this connection, we mention that
in Ref. [2] (Sec. II), Carroll and Chen conjectured that the
thermodynamic arrow of time might have a time-symmetric
explanation through entropy arrows much like the complexity
and information arrows we find.
The model.—The Newtonian N-body problem with
vanishing total energy Etot ¼ 0, momentum Ptot ¼ 0,
and angular momentum Jtot ¼ 0 is a useful model of
the Universe in many respects [3]. As we show below,
these conditions match the intuition that only relational
degrees of freedom of the Universe should have physical
significance [4–6]. A total angular momentum Jtot and a
total energy Etot would define, respectively, an external
frame in which the Universe is rotating and an absolute
unit of time. Moreover the conditions Jtot ¼ Ptot ¼ 0 and
Etot ¼ 0 ensure scale invariance and are close analogues
of the Arnowitt-Deser-Misner constraints of Hamiltonian
general relativity (in the spatially closed case) [7]. These
properties, along with the attractivity of gravity, are
architectonic and likely to be shared by any fundamental
law of the Universe.
First support for our claim follows from the rigorous
results of Ref. [8] that asymptotically (as the Newtonian
time t → ∞) N-body solutions with Etot ¼ 0 and Jtot ¼0
typically “evaporate” into subsystems [sets of particles
with separations bounded by Oðt2=3Þ] whose centers of
mass separate linearly with t as t → ∞. (The separation
linear in t is, of course, a manifestation of Newton’s first
law in the asymptotic regime. Our result arises from the
combination of this behavior with gravity’s role in creating
subsystems that mostly then become stably bound.) Each
subsystem consists of individual particles and/or clusters
whose constituents remain close to each other; i.e., the
distances between constituent particles of a cluster are
bounded by a constant for all times. One finds in numerical
simulations that the bulk of the clusters are two-body
Kepler pairs whose motion asymptotes into elliptical
Keplerian motion. For reasons we next discuss, this
t → ∞ behavior occurs in all typical solutions on either
side of a uniquely defined point of minimum expansion, as
illustrated in Fig. 1.
PRL 113, 181101 (2014)
Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS week ending
31 OCTOBER 2014
0031-9007=14=113(18)=181101(5) 181101-1 © 2014 American Physical Society
(...)
Complexity is a prerequisite for storage of information in
local subsystems and therefore the formation of records.We
nownote that a notion of local records canbefound in amodel
as simple as our Etot ¼ 0 and Jtot ¼ 0 N-body problem.
Recall that the system, for large jtj, breaks up typically
into disjoint subsystems drifting apart linearly in
Newtonian time t. These subsystems get more and more
isolated, and one can associate dynamically generated local
information with them. For this, we use the result of
Marchal and Saari [8] that as t → ∞ each subsystem J
develops asymptotically conserved quantities:
EJ(t)=EJ(∞) + O(t^−5/3)
JJ(t)=JJ (∞) + O(t−2/3);
XJ(t/t) =VJ (∞)+O(t^−1/3)
For more, see: https://physics.aps.org/featured-article...113.181101