arXiv subjects

Kouji Nakamura

Publications and source records attributed to Kouji Nakamura.

At least 55 records · Page 3Linked to original sources

arXiv2007

"Gauge" in General Relativity: Second-order general relativistic gauge-invariant perturbation theory

As in the case of the other gauge field theories, there is so called ``gauge'' also in general relativity. This ``gauge'' is unphysical degree of freedom. There are two kinds of ``gauges'' in general relativity. These are called the first- and the second-kind of gauges, respectively. The gauge of the first kind is just coordinate system on a single manifold. On the other hand, the gauge of the second kind arises in the general relativistic perturbations. Through the precise distinction of these two concepts of ``gauges'', we develop second-order gauge-invariant general relativistic perturbation theory.

gr-qc↗

arXiv2007

Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --

Along the general framework of the gauge invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we formulate the second order gauge invariant cosmological perturbation theory in a four dimensional homogeneous isotropic universe. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We derive the all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. All equations are derived in terms of gauge invariant variables without any gauge fixing. These equations imply that the second order vector and tensor modes may be generated due to the mode-mode coupling of the linear order scalar perturbations. We also briefly discuss the main progress of this work by the comparison with some literatures.

gr-qc↗

arXiv2006

Short note on second-order gauge-invariant cosmological perturbation theory

The second order perturbations in Friedmann-Robertson-Walker universe filled with a perfect fluid are completely formulated in the gauge invariant manner without any gauge fixing. All components of the Einstein equations are derived neglecting the first order vector and tensor modes. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the non-linear effects of the Einstein equations from the first order density perturbations.

gr-qc↗

arXiv2006

Gauge-invariant Formulation of the Second-order Cosmological Perturbations

Gauge invariant treatments of the second order cosmological perturbation in a four dimensional homogeneous isotropic universe filled with the perfect fluid are completely formulated without any gauge fixing. We derive all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the scalar-scalar mode coupling of the linear order perturbations as the result of the non-linear effects of the Einstein equations.

gr-qc↗

arXiv2006

Second Order Gauge Invariant Perturbation Theory -- Perturbative curvatures in the two-parameter case --

Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied.

gr-qc↗

arXiv2004

Causal Structure and Gravitational Waves in Brane World Cosmology

The causal structure of the flat brane universe of RSII type is re-investigated to clarify the boundary conditions for stochastic gravitational waves. In terms of the Gaussian normal coordinate of the brane, a singularity of the equation for gravitational waves appears in the bulk. We show that this singularity corresponds to the ``seam singularity'' which is a singular subspace on the brane universe. Based upon the causal structure, we discuss the boundary conditions for gravitational waves in the bulk. Introducing a null coordinate, we propose a numerical procedure to solve gravitational waves with appropriate boundary conditions and show some examples of our numerical results. This procedure could be also applied in scalar type perturbations. The problem in the choice of the initial condition for gravitational waves is briefly discussed.

hep-th↗

arXiv2004

Continuous Limit of Multiple Lens Effect and the Optical Scalar Equation

We study the continuous limit of the multiple gravitational lensing theory based on the thin lens approximation. Under the approximation, we define a new, light-path dependent angular diameter distance $\tilde{d}$ and show that it satisfies the optical scalar equation. The distance provides relations between quantities used in the gravitational lensing theory (the convergence-, the shear- and the twist-term) and those used in the scalar optics theory (the rates of expansion, shear and rotation).

astro-ph↗

arXiv2004

Stochastic Gravitational Wave Background in Brane World Cosmology

We investigate the cosmological evolution of gravitational waves in Friedman-Robertson-Walker brane world models embedded in a five dimensional anti de-Sitter spacetime. To predict the spectrum of stochastic gravitational background at present generated in the inflationary phase, we numerically calculate the evolution of gravitational waves according to the method developed in our previous paper [K. Ichiki and K. Nakamura, arXiv:hep-th/0310282]. The resulting logarithmic energy spectrum is significantly altered from that of the standard four dimensional inflationary models in the frequency range f > f_{AdS} ~ 10^{-4} Hz, which corresponds to the curvature scale we set of anti de-Sitter spacetime, $l ~ 1$mm.

astro-ph↗

arXiv2004

General framework of higher order gauge invariant perturbation theory

Based on the gauge invariant variables proposed in [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime is considered. We derived formulae of the perturbative Einstein tensor of each order, which have the similar form to the definitions of gauge invariant variables for arbitrary perturbative fields. As a result, each order Einstein equation is necessarily given in terms of gauge invariant variables.

gr-qc↗

arXiv2003

Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations

The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. For each order metric perturbation, we define the variable which is defined by the appropriate combination with lower order metric perturbations. Under the gauge transformation, this variable is transformed in the manner similar to the gauge transformation of the linear order metric perturbation. We confirm this up to third order. This implies that gauge invariant variables for higher order metric perturbations can be found by using a procedure similar to that for linear order metric perturbations. We also derive gauge invariant combinations for the perturbation of an arbitrary physical variable, other than the spacetime metric, up to third order.

gr-qc↗

arXiv2003

Comparison of the oscillatory behaviors of a gravitating Nambu-Goto string with a test string

Comparison of the oscillatory behavior of a gravitating infinite Nambu-Goto string and a test string is investigated using the general relativistic gauge invariant perturbation technique with two infinitesimal parameters on a flat spacetime background. Due to the existence of the pp-wave exact solution, we see that the conclusion that the dynamical degree of freedom of an infinite Nambu-Goto string is completely determined by that of gravitational waves, which was reached in our previous works [K. Nakamura, A. Ishibashi and H. Ishihara, Phys. Rev. D{\bf 62} (2002), 101502(R); K. Nakamura and H. Ishihara, Phys. Rev. D{\bf 63} (2001), 127501.], do not contradict to the dynamics of a test string. We also briefly discuss the implication of this result.

gr-qc↗

arXiv2003

Hoop Conjecture and Cosmic Censorship in the Brane-World

The initial data of gravity for a cylindrical matter distribution confined on the brane is studied in the framework of the single brane Randall-Sundrum scenario. In this scenario, 5-dimensional aspect of gravity appears in the short range gravitational interaction. We found that the sufficiently thin configuration of matter leads to the formation of the marginal surface even if the configuration is infinitely long. This means that the hoop conjecture proposed by Thorne does not hold in the Randall-Sundrum scenario; Even if a mass $M$ does not get compacted into a region whose circumference in every direction is ${\cal C}\le 4πGM$, black holes with horizons can form in the Randall-Sundrum scenario.

gr-qc↗

arXiv2003

Continuous Limit of Multiple Gravitational Lens Effect and Average Magnification Factor

We show that the gravitational magnification factor averaged over all configurations of lenses in a locally inhomogeneous universe satisfy a second order differential equation with redshift $z$ by taking the continuous limit of multi-plane gravitational lens equation (the number $N$ of lenses $\to\infty$) and that the gravitationally magnified Dyer-Roeder distance in a clumpy universe becomes to that of the Friedmann-Lema\^ıtre universe for arbitrary values of the density parameter $Ω_{0}$ and of a mass fraction $\barα$ (smoothness parameter).

astro-ph↗

arXiv2003

Hoop Conjecture and Black Holes on a Brane

The initial data of gravity for a cylindrical matter distribution confined to a brane are studied in the framework of the single-brane Randall-Sundrum scenario. In this scenario, the 5-dimensional nature of gravity appears in the short-range gravitational interaction. We find that a sufficiently thin configuration of matter leads to the formation of a marginal surface, even if the configuration is infinitely long. This implies that the hoop conjecture proposed by Thorne does not hold on the brane: Even if a mass $M$ does not become compacted into a region whose circumference ${\cal C}$ in every direction satisfies ${\cal C}> 4πGM$, black holes with horizons can form in the Randall-Sundrum scenario.

gr-qc↗

arXiv2002

Does a Nambu-Goto wall emit gravitational waves? -- Cylindrical Nambu-Goto wall as an example of gravitating non-spherical walls --

Gravitational field of a cylindrical Nambu-Goto wall in the vacuum spacetime is considered in order to clarify the interaction between Nambu-Goto membranes and gravitational waves. If one neglects the emission of gravitational waves by the wall motion, the spacetime becomes singular. It is also shown that the emission of gravitational waves does occur by the motion of the cylindrical wall if the initial data is singularity free. The energy loss rate due to radiation of gravitational waves agrees with that estimated from the test wall motion and the quadrupole formula for the gravitational wave emission. This is quite different from the oscillatory behavior of gravitating Nambu-Goto membranes: the presence of gravity induces the wall to lose its dynamical degree of freedom.

gr-qc↗

arXiv2002

Initial condition of a thick loop cosmic string and linear perturbations

The initial data of the gravitational field produced by a loop thick string is considered. We show that a thick loop is not a geodesic on the initial hypersurface, while a loop conical singularity is. This suggests that there is the ``{\it critical thickness}'' of a string, at which the linear perturbation theory with a flat space background fails to describe the gravity of a loop cosmic string. Using the above initial data, we also show that the linear perturbation around flat space is plausible if the string thickness is larger than $\sim5\times10^{-3}a$, where $a$ is the curvature radius of the loop.

gr-qc↗

arXiv2001

Dynamics of a string coupled to gravitational waves II - Perturbations propagate along an infinite Nambu-Goto string

The perturbative modes propagating along an infinite string are investigated within the framework of the gauge invariant perturbation formalism on a spacetime containing a self-gravitating straight string with a finite thickness. These modes are not included in our previous analysis. We reconstruct the perturbation formalism to discuss these modes and solve the linearized Einstein equation within the first order with respect to the string oscillation amplitude. In the thin string case, we show that the oscillations of an infinite string must involve the propagation of cosmic string traveling wave.

gr-qc↗

arXiv2000

Dynamics of a string coupled to gravitational waves - Gravitational wave scattering by a Nambu-Goto straight string

We study the perturbative dynamics of an infinite gravitating Nambu-Goto string within the general-relativistic perturbation framework. We develop the gauge invariant metric perturbation on a spacetime containing a self-gravitating straight string with a finite thickness and solve the linearized Einstein equation. In the thin string case, we show that the string does not emit gravitational waves by its free oscillation in the first order with respect to its oscillation amplitude, nevertheless the string actually bends when the incidental gravitational waves go through it.

gr-qc↗