We compare three different models of rotating neutron star spacetimes: i) the Hartle-Thorne (1968) slow-rotation approximation, keeping terms up to second order in the stellar angular velocity; ii) the exact analytic vacuum solution of Manko et al. (2000a, 2000b); and iii) a numerical solution of the full Einstein equations. In the first part of the paper we estimate the limits of validity of the slow-rotation expansion by computing relative errors in the spacetime’s quadrupole moment Q and in the corotating and counterrotating radii of Innermost Stable Circular Orbits (ISCOs) R±. We integrate the Hartle-Thorne structure equations for five representative equations of state. Then we match these models to numerical solutions of the Einstein equations, imposing the condition that the gravitational mass and angular momentum of the models be the same. We find that the Hartle-Thorne approximation gives very good predictions for the ISCO radii, with R± accurate to better than 1% even for the fastest millisecond pulsars. At these rotational rates the accuracy on Q is � 20%, and better for longer periods. In the second part of the paper we focus on the exterior vacuum spacetimes, comparing the Hartle-Thorne approximation and the Manko analytic solution to the numerical models using Newman-Penrose (NP) coordinateindependent quantities. For all three spacetimes we introduce a physically motivated ‘quasi-Kinnersley’ NP frame. In this frame we evaluate a quantity, the speciality index S, measuring the deviation of each stellar model from Petrov Type D. Deviations from speciality on the equatorial plane are smaller than 5% at star radii for the faster rotating models, and rapidly decrease for slower rotation rates and with distance. We find that, at leading order, the deviation from Type D is proportional to (Q QKerr). Our main conclusion is that the Hartle-Thorne approximation is very reliable for most astrophysical applications.

https://researchportal.port.ac.uk/portal/files/124574/0405146.pdf

Rotating neutron stars: an invariant comparison of approximate and numerical space-time models

The Invar tensor package

The Teukolsky master equation is the basic tool for the study of perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky radial equation and the Teukolsky angular equation. We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions. Large classes of new exact solutions are found and classified with respect to their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and introduce collimated one-way running waves. It is shown that a proper linear combination of such solutions can present bounded one-way running waves. This type of waves may be suitable as models of the observed astrophysical jets.

https://iopscience.iop.org/article/10.1088/0264-9381/27/13/135001/pdf

Classes of exact solutions to the Teukolsky master equation

The Teukolsky Master Equation is the basic tool for study of perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky Radial Equation and the Teukolsky Angular Equation. We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions. Large classes of new exact solutions are found and classified with respect to their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and introduce collimated one-way-running waves. It is shown that a proper linear combination of such solutions can present bounded one-way-running waves. This type of waves may be suitable as models of the observed astrophysical jets.

Classes of Exact Solutions to the Teukolsky Master Equation

A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson and type D (anti-)Nariai or Plebanski–Hacyan spaces, and their generalizations. All possibilities are discussed and interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type D pure radiation or explicit type II electrovacuum waves in the (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein–Maxwell fields with a non-vanishing cosmological constant.

Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes

We study conformally flat pure radiation spacetimes by means of their invariant classifications and show that no such spacetime requires higher than the fourth covariant derivative of the Riemann tensor in its invariant classification. Additional side results that we obtain are as follows. The Edgar - Ludwig metric for conformally flat pure radiation is shown to be a true generalization of the Wils metric; the subclass of the Edgar - Ludwig spacetimes which admit exactly one Killing vector is identified, generalizing Koutras's G1 subclass of the Wils metric; it is shown that the Edgar - Ludwig spacetimes with no Killing vectors can further be split into three discrete subsets, depending on the Cartan invariant where the fourth essential coordinate is found.

The invariant classification of conformally flat pure radiation spacetimes

This is the first in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we briefly discuss the Cartan-Karlhede invariant classification of geometries and the significance of the standard form of a spinor. We then present algorithms for putting the Weyl spinor, Ricci spinor and general spinors into standard form.

https://iopscience.iop.org/article/10.1088/0264-9381/17/3/306/pdf

Classifying geometries in general relativity: I. Standard forms for symmetric spinors

This is the third in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we describe how the Cartan-Karlhede method for classifying a geometry is accomplished in practice. We give as an example the classification of the Edgar-Lugwig conformally flat pure radiation metrics.

Classifying geometries in general relativity: III. Classification in practice

We present the first example of a metric whose invariant classification requires the fifth covariant derivative of the Riemann tensor. The existence of this metric raises the known practical bound for invariant classifications, reconciles the theoretical and practical upper limits for invariant classifications of Petrov type N spacetimes, and brings the practical limits for invariant classifications to within one of the theoretical limits for a spacetime in general.

A spacetime whose invariant classification requires the fifth covariant derivative of the Riemann tensor

This is the second in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we discuss the choice of computer algebra platform and the general relativity application package. The derivative operators needed in the Cartan-Karlhede classification algorithm and the behaviour of symmetric spinors under frame rotations are then presented.

Jim E F Skea

Papers

The invariant classification of conformally flat pure radiation spacetimes

Classifying geometries in general relativity: I. Standard forms for symmetric spinors

Classifying geometries in general relativity: III. Classification in practice

A spacetime whose invariant classification requires the fifth covariant derivative of the Riemann tensor

Classifying geometries in general relativity: II. Spinor tools