Tomimatsu-Sato Metric Research Articles

The Tomimatsu–Sato Metric Reloaded

In this work, we derive exact analytic formulae for the inner and outer surfaces representing the boundary of the ergoregion appearing in the Tomimatsu–Sato (TS) metric. Exact expressions for the radii of the ergoregion in prolate spheroidal coordinates and in Boyer-Lindquist coordinates are obtained. We also found that in addition to the ring-shaped naked singularity, there is an event horizon placed in the inner region inside the aforementioned curvature singularity. In comparing our results with previous studies, we also uncovered and corrected several errors in the literature. Finally, we provide tables of numerical values for the inner and outer boundaries of the ergoregion for different values of the rotational parameter. We hope this study will be a useful resource for all researchers interested in the Tomimatsu–Sato metric.

A Note on the Factor Structure of Some Non-Rational Vacuum Metrics

In this note we point out that a large class of stationary, axisymmetric, vacuum solutions of the Einstein equations which are not representable as rational functions of some coordinates has a factor structure similar to that of Tomimatsu-Sato metrics.

Remarks on the Charged, Magnetized Tomimatsu-Sato delta = 2 Solution

The full metric describing a charged, magnetized generalization of the Tomimatsu-Sato (TS) $\delta=2$ solution is presented in a concise explicit form. We use it to investigate some physical properties of the solution; in particular, we point out the existence of naked ring singularities in the hyperextreme TS metrics, the fact previously overlooked by the researchers, and we also demonstrate that the ring singularities can be eliminated by sufficiently strong magnetic fields in the subextreme case, while in the hyperextreme case the magnetic field can move singularities to the equatorial plane.

A Solution of the Einstein Equation in a Stationary Gravitational Field Due to a Rotating Source

The stationary and axially symmetric gravitational field due to a rotating deformed source can be treated by the Ernst equation. We prove analytically the validity of the solution of the Ernst equation that was presented as a possible solution in a previous paper for arbitrary real values of the deformation parameter δ. By means of analytic continuations of the solution we obtain a new solution in oblate spheroidal coordinates. This leads to a metric different from the Tomimatsu-Sato metric. From these solutions for real δ we determine the metric functions both in prolate and oblate spheroidal coordinates.

Applications of maple to general relativity

The present paper describes the Riemann package written in the Maple programming language. The package has been developed for tensor component calculations in General Relativity and for some tensor abstract manipulations. It allows the user to perform tensor algebra operations, such as addition, multiplication and contraction of tensors. It is also possible to create new tensors with defined symmetries and to apply Maple functions to the tensor components. We give an example of an application of the Riemman package using the Tomimatsu-Sato metric, and also compare it to other Maple tensor packages.

Generalizations of the Kerr-Newman and charged Tomimatsu-Sato metrics in the Jordan-Brans-Dicke theory

A procedure to construct stationary axisymmetric solutions of the Jordan-Brans-Dicke field equations with electromagnetic sources is obtained solutions, since they are “compositions” of Weyl static solutions with the given stationary ones, are equipped with several parameters, as many as one wishes.

Factor structure of the Tomimatsu–Sato metrics

It is shown that the factoring property of the Tomimatsu–Sato metrics follows from the structure of special Hankel determinants. A set of linear algebraic equations determining the factors is found. The factors of the first five Tomimatsu–Sato metrics are tabulated.

The Application of Tomimatsu-Sato Metric to the Gravitational Field of Ordinary Spinning Object

An expression of the Tomimatsu-Sato metric in the Schwarzschild coordinates is found. In it those small terms of are omitted. The resulting expression is applied to the description of gravitational field of ordinary spinning objects such as the sun. The Landau-Lifshitz method used to find the deflection of light and precession of perihelia in the Schwarzschild field is generalized to the gravitational field of a spinning object. It is shown that the expressions of the deflection of light and precession of perihelia in the Tomimatsu-Sato field are the same as those in the Kerr field. Tomimatsu and Sato have conjectured that their metric is more appropriate for describing the gravitational field of ordinary spinning object than Kerr metric. In order to check this conjecture one must consider the terms smaller than .

An exact rotating dyon solution with the Tomimatsu-Sato metric

We present an exact rotating dyon solution for which the space-time metric takes the Tomimatsu-Sato form with an arbitrary integer distortion parameter δ. Our solution reduces to the rotating monopole solution for vanishing electric charge.

Multi-Soliton Solutions of the Einstein Equation and the Tomimatsu-Sato Metric

Multi-Soliton Solutions of the Einstein Equation and the Tomimatsu-Sato Metric

Axisymmetric gravitational fields: a nonlinear differential equation that admits a series of exact eigenfunction solutions

A nonlinear ordinary differential equation is presented. This equation is shown to occur as a particular integral of Einstein’s equations of gravitation, and may be used to generate a class of solutions which contains the Kerr, and the Tomimatsu–Sato metrics as particular cases.

A note on the first Tomimatsu-Sato metric

A note on the first Tomimatsu-Sato metric

Time-like geodesics in the first Tomimatsu-Sato metric

A numerical study of the time-like geodesics in the first axially symmetric solution of the Einstein vacuum field equations given by Tomimatsu and Sato is made. The study is made on the equatorial plane, following the method employed by Bardeen, Press and Teukolsky (1972) in the Kerr field.

Charged Tomimatsu-Sato metrics

We show that the solutions of Einstein-Maxwell field equations can be constructed from the Tomimatsu-Sato solutions of Einstein's field equations for empty space simply by inspection when δ has integral values.

Limits of the Tomimatsu-Sato gravitational field

The Tomimatsu-Sato (TS) solutions of the Einstein field equations are studied in several limiting cases. In the weak-field limit we construct two Newtonian models for the source, one consisting of a rotating disc of radius a/n, the other made up of n complex point multipoles. The ``extreme'' limit q = 1 is also examined in detail, and we find there are many distinct ways of taking this limit. We are thereby led to a new two-parameter family of exact solutions which, unlike the TS metrics, are not asymptotically flat.

Event horizon of the Tomimatsu-Sato metrics

Recen t ly GIBBOI~'S and RUSSEL-CLAltK (1) have s ta ted t h a t our new solution (3) to E ins t e in ' s v a c u u m field equat ion does n o t conta in an even t horizon. Af te r the publ ica t ion of this new solution, which is a s ta t ionary , ax i symmet r ic , a sympto t i ca l ly flat, exac t solution, we have fu r ther found an in te res t ing series of exac t solutions (a) including the Ker r solution and the above new solut ion as special members . This new series of solutions, which we call the T-S solutions hereaf ter , e~)ntain the three parameters , i.e. the mass m, t he angular m o m e n t u m J ( ~ m2q) and the dis tor t ion pa rame te r 6, which specifics t he W e y l solutions in the s ta t ic case of q = 0. Fo r example, the quadrupole momen t s of the g rav i t a t iona l fields are given as

Geodesic Motions in the Tomimatsu-Sato Metric

Geodesic Motions in the Tomimatsu-Sato Metric