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 $g$ 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 $g=1$ is analyzed in detail. By a suitable gauge-fixing, the partition function $Z$ basically reduces to the partition function defined for the reduced system, whose dynamical variables are $(\tau^A, p_A)$. [The $\tau^A$'s are the Teichm\"uller parameters, and the $p_A$'s are their conjugate momenta.] As for the case of $g=1$, we find out that $Z$ is also related with another reduced form, whose dynamical variables are $(\tau^A, p_A)$ and $(V, \sigma)$. [Here $\sigma$ is a conjugate momentum to 2-volume $V$.] 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 $g=1$, a factor coming from the zero-modes of a differential operator $P_1$ can appear in the path-integral measure in the reduced representation of $Z$. It depends on the path-integral domain for the shift vector in $Z$: If it is defined to include $\ker P_1$, the nontrivial factor does not appear. On the other hand, if the integral domain is defined to exclude $\ker P_1$, the factor appears in the measure. This factor can depend on the dynamical variables, typically as a function of $V$, 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

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

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

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

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

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 $E_{ent}$ turns out to be proportional to area radius of the boundary if it is near the horizon. This peculiar behavior of $E_{ent}$ 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

Back-Reaction on the Topological Degrees of Freedom in (2+1)-Dimensional Spacetime

We investigate the back-reaction effect of the quantum field on the topological degrees of freedom in (2+1)-dimensional toroidal universe, ${\cal M} \simeq T^2\times {\bf R}$. Constructing a homogeneous model of the toroidal universe, we examine explicitly the back-reaction effect of the Casimir energy of a massless, conformally coupled scalar field, with a conformal vacuum. The back-reaction causes an instability of the universe: The torus becomes thinner and thinner as it evolves, while its total 2-volume (area) becomes smaller and smaller. The back-reaction caused by the Casimir energy can be compared with the influence of the negative cosmological constant: Both of them make the system unstable and the torus becomes thinner and thinner in shape. On the other hand, the Casimir energy is a complicated function of the Teichm\"uller parameters $(\tau^1, \tau^2)$ causing highly non-trivial dynamical evolutions, while the cosmological constant is simply a constant. Since the spatial section is a 2-torus, we shall write down the partition function of this system, fixing the path-integral measure for gravity modes, with the help of the techniques developed in string theories. We show explicitly that the partition function expressed in terms of the canonical variables corresponding to the (redundantly large) original phase space, is reduced to the partition function defined in terms of the physical-phase-space variables with a standard Liouville measure. This result is compatible with the general theory of the path integral for the 1st-class constrained systems.Comment: 42 pages, phyzzx.tex, Figures will be sent on reques