Journal Club – ARTIGO Identification of a Gravitational

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Identification of a Gravitational Arrow of Time
Julian Barbour, Tim Koslowski e Flavio Mercati
21 de novembro de 2014,
apresentando: Boselli
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Artigo
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Artigo
Artigos
I
Identification of a Gravitational Arrow of Time, J. Barbour, T.
Koslowski, and F. Mercati, PHYSICAL REVIEW LETTERS
113, 181101 (2014)
I
Identification of a Gravitational Arrows of Time, J. Barbour,
T. Koslowski, and F. Mercati, arXiv:1310.5167.
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Proposta
Our fundamental assumptions are:
1. The Universe is a closed dynamical system. In Newtonian
gravity (NG), this means an ‘island universe’ of N point
particles. In general relativity (GR), the Universe must be
spatially closed. This is not in conflict with current
observations.
2. A notion of universal simultaneity exists. This is built into NG.
3. Since all measurements are relational, only shapes are
physical. We will define the complexity, denoted CS , of any
complete shape of the Universe. It is a pure number.
arXiv:1310.5167
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Outline
Outline
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O Modelo
I
Eliminando a Escala
I
Complexidade
I
A Flexa do Tempo
I
Conclusões
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O Modelo
O Modelo
O Universo é um sistema com:
Etot = 0,
Jtot = 0,
Ptot = 0
Solução para o sistema de N-corpos (t → ±∞):
t
Os centros de massa se separam linearmente com o tempo.
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Eliminando a Escala
Eliminando a escala
Partı́cula a com: posição ra , momento pa e massa ma .
Problema Newtoniano com Etot = 0
Etot =
N
X
pa · pa
a=1
2ma
+ VNew ,
VNew =
X ma mb
a<b
rab
Função homogênea de grau k
f (αx1 , . . . , αxn ) = αk f (x1 , . . . , xn )
satisfaz a similaridade dinâmica (similar ao teorema do virial);
reescala:
ra → αra , t → α1−k/2 t
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Eliminando a Escala
Escala global: Momento de Inércia do centro de massa.
Icm :=
N
X
ma ||ra − rcm ||2
a=1
Resultado da teoria de Lagrange-Jacobi
Icm := 4Ecm − 2(2 + k)Vk
Para Newton com k = −1 e Ecm ≥ 0 segue que → Icm > 0.
Momento dilatacional (1/2 da derivada de Icm ):
D :=
N
X
a=1
a
rcm
a · pcm ,
pacm = pa −
N
1 X b
r
N
b=1
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Eliminando a Escala
Consequências: Icm é uma função côncava e D é monótona
CS
Icm
D0
D
arXiv:1310.5167v1
t
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Eliminando a Escala
Como D é monótono, pode ser usado como escala de tempo τ
A evolução temporal será dada pelo Hamiltoniano H:
!
N
X
1/2
a
a
2
H(τ ) = ln
π ·π +τ
− ln Icm |VNew |
a=1
onde π a é o momento de forma (shape)
r
Icm a
a
π =
p − Dσa
ma
Onde σa é a coordenada de forma
r
ma cm
σa =
r
Icm a
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Eliminando a Escala
Each point of S (a shape) is an objective state of the
system, freed of unphysical properties like the overall orientation
in absolute space, the position of the center of mass of the
Universe, and the total scale.
The Hamiltonian H(τ ) is a function of time τ , the shape
degrees of freedom and their conjugate momenta. This
allows us to eliminate the evolution of scale from the
problem and express the dynamics purely on shape space.
The usual description with scale can be reconstructed from the
solution on shape space, for which it is necessary to specify a
nominal initial value of Icm (due to dynamical similarity, this value
is completely conventional and unmeasurable).
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Complexidade
Complexidade
“If we ignore the nominal scale of the system, what characterizes
best its overall state?”
There are two simple lengths that characterize the system. One is
the root-mean-square length `rms :
sX
1
2 = 1 I 1/2 ,
`rms :=
ma mb rab
cm
mtot
mtot
a<b
Measures the overall size of the system.
The other is the mean harmonic length `mhl :
1
`mhl
:=
1 X ma mb
1
= 2 |VNew | .
2
mtot r <a rab
mtot
Measures how close to each other are the tightest particle pairs
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Complexidade
It is some dimensionless measure of inhomogeneity that
distinguishes the central ‘turning point’ shown in Figure from the
states either side.
CS
t
CS :=
`rms
`mhl
(Complexidade)
It is a good measure of non-uniformity or clustering
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Complexidade
t
CS
t
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Escala agindo como atrito
A hamiltoniana dependente de τ
H(τ ) = ln
N
X
!
πa · πa
− ln (Icm |VNew |)
a=1
pode ser colocada em uma forma independente de τ
ω a = π a /D;
λ = log τ
H0 = log
N
X
!
a
a
ω ·ω +1
− log CS
a=1
Com as equações de movimento
dσa
∂H0
=
,
dλ
∂ω a
→ Momento não canônico −ω a
dω a
∂H0
=−
− ωa
dλ
∂σa
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Escala agindo como atrito
Notice that the change of variables that we performed allows us to
describe only half of each solution: the half before or after D = 0.
Each typical solution of the N-body problem maps into two
solutions of the equations with friction. This and the fact that the
potential − ln CS has no local minima (only infinitely deep
potential wells) explain why CS grows secularly either side of a
unique minimum.
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A Flexa do Tempo
The Arrow of Time
In the above, we have reformulated the dynamics of the N-body
problem as a dynamical system on shape space whose motion is
controlled by a potential − ln CS and is subject to linear friction.
This explains intuitively why CS and therefore ln CS grows
secularly: it is essentially minus the potential energy of a system
with friction. Moreover, one can use the results of to show that
typically there is a lower bound on CS that grows without bound
for large |τ |. This result holds also for the atypical solutions that
reach Icm = 0. However, we have to assume that at least one
bound system forms, and no particle escapes in finite times like in
Xia’s solution [Ann. Math. 135 (1992) 411–468]
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A Flexa do Tempo
Conclusão (1)
In the light of this circumstance, it is very natural to identify an
arrow of time with the direction in which structure, measured in
our case by CS , grows. We then have a dynamically- enforced
scenario with one past (the minimum of CS , which occurs near
τ = 0) and two futures. The growth-of-complexity arrow always
points away from the unique past. In the atypical solutions that
terminate with Icm = 0 there is one past and only one future.
CS
t
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Resumo
Resumo
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O Universo é um sistema
com:
I
Dinâmica de N-copros é
reformulada como um
sistema dinâmico no
espaço de forma.
I
O movimento é
controlado por um
potencial − ln CS sujeito a
atrito linear
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Conseqüência: CS ) cresce
(H0 é mantido constante)
Etot = 0, Jtot = 0, Ptot = 0
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Mudança de escala com
Icm e D (Lagrange-Jacobi)
H → H(τ ) (espaço de
forma)
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Definição de CS
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Redefinição de H com
λ = log(τ )
H → H0 (ω a , CS )
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Conclusões
Conclusões
Our results are a ‘proof of principle’: all the solutions of a
time-symmetric dynamical law suited to approximate our Universe
have a strongly time-asymmetric behavior for internal observers.
So far as we know, this conclusion is new. It follows from the exact
Lagrange–Jacobi relation and special properties of the model’s
potential. Of course, it has long been known that gravity causes
clustering of an initially uniform matter distribution. In our
Universe, this is reflected above all in the formation of galaxies and
is the most striking macroscopic arrow. Our novelty is that in all
solutions of the N-body problem the dynamical law guarantees,
without any past hypothesis, an epoch of relative uniformity out of
which growth of structure must be observed. We conclude that
the origin of time’s arrow is not necessarily to be sought in
initial conditions but rather in the structure of the law which
governs the Universe.