46 research outputs found
Spectral Evolution of the Universe
We derive the evolution equations for the spectra of the Universe.
Here "spectra" means the eigenvalues of the Laplacian defined on a space,
which contain the geometrical information on the space.
These equations are expected to be useful to analyze the evolution of the
geometrical structures of the Universe.
As an application, we investigate the time evolution of the spectral distance
between two Universes that are very close to each other; it is the first
necessary step for the detailed analysis of the model-fitting problem in
cosmology with the spectral scheme.
We find out a universal formula for the spectral distance between two very
close Universes, which turns out to be independent of the detailed form of the
distance nor the gravity theory. Then we investigate its time evolution with
the help of the evolution equations we derive.
We also formulate the criteria for a good cosmological model in terms of the
spectral distance.Comment: To appear in Phys. Rev.
Partition Function for (2+1)-Dimensional Einstein Gravity
Taking (2+1)-dimensional pure Einstein gravity for arbitrary genus as a
model, we investigate the relation between the partition function formally
defined on the entire phase space and the one written in terms of the reduced
phase space. In particular the case of is analyzed in detail.
By a suitable gauge-fixing, the partition function basically reduces to
the partition function defined for the reduced system, whose dynamical
variables are . [The 's are the Teichm\"uller
parameters, and the 's are their conjugate momenta.]
As for the case of , we find out that is also related with another
reduced form, whose dynamical variables are and .
[Here is a conjugate momentum to 2-volume .] A nontrivial factor
appears in the measure in terms of this type of reduced form. The factor turns
out to be a Faddeev-Popov determinant coming from the time-reparameterization
invariance inherent in this type of formulation. Thus the relation between two
reduced forms becomes transparent even in the context of quantum theory.
Furthermore for , a factor coming from the zero-modes of a differential
operator can appear in the path-integral measure in the reduced
representation of . It depends on the path-integral domain for the shift
vector in : If it is defined to include , the nontrivial factor
does not appear. On the other hand, if the integral domain is defined to
exclude , the factor appears in the measure. This factor can depend
on the dynamical variables, typically as a function of , and can influence
the semiclassical dynamics of the (2+1)-dimensional spacetime.
These results shall be significant from the viewpoint of quantum gravity.Comment: 21 pages. To appear in Physical Review D. The discussion on the
path-integral domain for the shift vector has been adde
Evolution of the discrepancy between a universe and its model
We study a fundamental issue in cosmology: Whether we can rely on a
cosmological model to understand the real history of the Universe. This
fundamental, still unresolved issue is often called the ``model-fitting problem
(or averaging problem) in cosmology''. Here we analyze this issue with the help
of the spectral scheme prepared in the preceding studies.
Choosing two specific spatial geometries that are very close to each other,
we investigate explicitly the time evolution of the spectral distance between
them; as two spatial geometries, we choose a flat 3-torus and a perturbed
geometry around it, mimicking the relation of a ``model universe'' and the
``real Universe''. Then we estimate the spectral distance between them and
investigate its time evolution explicitly. This analysis is done efficiently by
making use of the basic results of the standard linear structure-formation
theory.
We observe that, as far as the linear perturbation of geometry is valid, the
spectral distance does not increase with time prominently,rather it shows the
tendency to decrease. This result is compatible with the general belief in the
reliability of describing the Universe by means of a model, and calls for more
detailed studies along the same line including the investigation of wider class
of spacetimes and the analysis beyond the linear regime.Comment: To be published in Classical and Quantum Gravit
General Relativity in terms of Dirac Eigenvalues
The eigenvalues of the Dirac operator on a curved spacetime are
diffeomorphism-invariant functions of the geometry. They form an infinite set
of ``observables'' for general relativity. Recent work of Chamseddine and
Connes suggests that they can be taken as variables for an invariant
description of the gravitational field's dynamics. We compute the Poisson
brackets of these eigenvalues and find them in terms of the energy-momentum of
the eigenspinors and the propagator of the linearized Einstein equations. We
show that the eigenspinors' energy-momentum is the Jacobian matrix of the
change of coordinates from metric to eigenvalues. We also consider a minor
modification of the spectral action, which eliminates the disturbing huge
cosmological term and derive its equations of motion. These are satisfied if
the energy momentum of the trans Planckian eigenspinors scale linearly with the
eigenvalue; we argue that this requirement approximates the Einstein equations.Comment: 6 pages, RevTe
Relativistic dynamics of cylindrical shells of counter-rotating particles
Although infinite cylinders are not astrophysical entities, it is possible to
learn a great deal about the basic qualitative features of generation of
gravitational waves and the behavior of the matter conforming such shells in
the limits of very small radius. We describe the analytical model using kinetic
theory for the matter and the junction conditions through the shell to obtain
its equation of motion. The nature of the static solutions are analyzed, both
for a single shell as well as for two concentric shells. In this second case,
for a time dependent external shell, we integrate numerically the equation of
motion for several values of the constants of the system. Also, a brief
description in terms of the Komar mass is given to account for the
gravitational wave energy emitted by the system.Comment: 19 pages, 8 figure
The spectral representation of the spacetime structure: The `distance' between universes with different topologies
We investigate the representation of the geometrical information of the
universe in terms of the eigenvalues of the Laplacian defined on the universe.
We concentrate only on one specific problem along this line: To introduce a
concept of distance between universes in terms of the difference in the
spectra.
We can find out such a measure of closeness from a general discussion. The
basic properties of this `spectral distance' are then investigated. It can be
related to a reduced density matrix element in quantum cosmology. Thus,
calculating the spectral distance gives us an insight for the quantum
theoretical decoherence between two universes. The spectral distance does not
in general satisfy the triangular inequality, illustrating that it is not
equivalent to the distance defined by the DeWitt metric on the superspace.
We then pose a question: Whether two universes with different topologies
interfere with each other quantum mechanically? We concentrate on the
difference in the orientabilities. Several concrete models in 2-dimension are
set up, and the spectral distances between them are investigated: Tori and
Klein's bottles, spheres and real projective spaces. Quite surprisingly, we
find many cases of spaces with different orientabilities in which the spectral
distance turns out to be very short. It may suggest that, without any other
special mechanism, two such universes interfere with each other quite strongly.Comment: 47 page
Observables of the Euclidean Supergravity
The set of constraints under which the eigenvalues of the Dirac operator can
play the role of the dynamical variables for Euclidean supergravity is derived.
These constraints arise when the gauge invariance of the eigenvalues of the
Dirac operator is imposed. They impose conditions which restrict the
eigenspinors of the Dirac operator.Comment: Revised version, some misprints in the ecuations (11), (13) and (17)
corrected. The errors in the published version will appear cortected in a
future erratu
Can the entanglement entropy be the origin of black-hole entropy ?
Entanglement entropy is often speculated as a strong candidate for the origin
of the black-hole entropy. To judge whether this speculation is true or not, it
is effective to investigate the whole structure of thermodynamics obtained from
the entanglement entropy, rather than just to examine the apparent structure of
the entropy alone or to compare it with that of the black hole entropy. It is
because entropy acquires a physical significance only when it is related to the
energy and the temperature of a system. From this point of view, we construct a
`thermodynamics of entanglement' by introducing an entanglement energy and
compare it with the black-hole thermodynamics. We consider two possible
definitions of entanglement energy. Then we construct two different kinds of
thermodynamics by combining each of these different definitions of entanglement
energy with the entanglement entropy. We find that both of these two kinds of
thermodynamics show significant differences from the black-hole thermodynamics
if no gravitational effects are taken into account. These differences are in
particular highlighted in the context of the third law of thermodynamics.
Finally we see how inclusion of gravity alter the thermodynamics of the
entanglement. We give a suggestive argument that the thermodynamics of the
entanglement behaves like the black-hole thermodynamics if the gravitational
effects are included properly. Thus the entanglement entropy passes a
non-trivial check to be the origin of the black-hole entropy.Comment: 40 pages, Latex file, one figur
Euclidean Supergravity in Terms of Dirac Eigenvalues
It has been recently shown that the eigenvalues of the Dirac operator can be
considered as dynamical variables of Euclidean gravity. The purpose of this
paper is to explore the possiblity that the eigenvalues of the Dirac operator
might play the same role in the case of supergravity. It is shown that for this
purpose some primary constraints on covariant phase space as well as secondary
constraints on the eigenspinors must be imposed. The validity of primary
constraints under covariant transport is further analyzed. It is show that in
the this case restrictions on the tanget bundle and on the spinor bundle of
spacetime arise. The form of these restrictions is determined under some
simplifying assumptions. It is also shown that manifolds with flat curvature of
tangent bundle and spinor bundle and spinor bundle satisfy these restrictons
and thus they support the Dirac eigenvalues as global observables.Comment: Misprints and formulae corrected; to appear in Phys. Rev.
Thermodynamics of entanglement in Schwarzschild spacetime
Extending the analysis in our previous paper, we construct the entanglement
thermodynamics for a massless scalar field on the Schwarzschild spacetime.
Contrary to the flat case, the entanglement energy turns out to be
proportional to area radius of the boundary if it is near the horizon. This
peculiar behavior of can be understood by the red-shift effect caused
by the curved background. Combined with the behavior of the entanglement
entropy, this result yields, quite surprisingly, the entanglement
thermodynamics of the same structure as the black hole thermodynamics. On the
basis of these results, we discuss the relevance of the concept of entanglement
as the microscopic origin of the black hole thermodynamics.Comment: 27 pages, Latex file, 7 figures; revised to clarify our choice of the
state and to add references. Accepted for publication in Physical Review