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 complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

/pdf/computer-algebra-in-gravity-research-12vc2z8g6k.pdf

Computer algebra in gravity research

A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of the Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton–York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Godel-type spacetimes in three-dimensional general relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Godel-type spacetimes is studied and the results compared with previous work on four-dimensional Godel-type spacetimes.

Equivalence of three-dimensional spacetimes

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

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 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.

D Pollney

Papers

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

Classifying geometries in general relativity: III. Classification in practice

Classifying geometries in general relativity: II. Spinor tools