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xAct - Example Problems

Jeremy Sakstein
University of Pennsylvania
sakstein@physics.upenn.edu

These are a short set of example problems designed to help you familiarize yourself with xAct and its extensions. They are designed to be executed during my tutorial but can be completed independently. My own solutions are available upon request.

Part of the fun of xAct is figuring out new ways of accomplishing what you need and learning new functions. For this reason, I have included a few components that can be done far quicker with functions not covered in the main tutorial, and, in some cases, new objects that you will need to figure out how to call (for example the Levi-Civita tensor). I encourage you to look at the documentations, especially xTras: http://www.xact.es/xtras/documentation/guide/xTras.html.

We begin by loading all the packages required (with the exception of xPand)

Block[{Print},
<<xAct`xTensor`;
<<xAct`xPert`;
<<xAct`xTras`;
<<xAct`xCoba`;
];

The warm-up exercise just gets you to load the packages, you should quit the kernel and reload this every time.

1) Warm up

Define a manifold M with (xTras) dimension dim endowed with a metric g and covariant derivative CD. The free indices should be b-p. You should quit the kernel and re-run your code every time you start a new problem.

2) Scalar-Gauss-Bonnet Theory

The Lagrangian for Einstein-dilaton-Gauss-Bonnet gravity is

(R/2) - (1/4) (((∇ϕ))^2) + (η/8) f(ϕ)𝒢,

where ϕ is dimensionless, we are working in units where 8πG=c=1. The equation of motion for the metric is (see e.g. hep-th/9511071 equation 5 with appropriate convention changes)

(Subscript[G, ab])= (1/2) (Subscript[∇, a])ϕ (Subscript[∇, b])ϕ - (1/4)  (Subscript[g, ab]) ((((Subscript[∇, c])ϕ))^2) - (η/8) (Subscript[𝒦, ab])

where

(Subscript[𝒦, ab]) = ((Subscript[g, ac]) (Subscript[g, bd]) + (Subscript[g, ad]) (Subscript[g, bc])) (η^edfh) (Subscript[∇, j]) ( (Subscript[((OverTilde[R])^cj), fh]) (Subscript[∇, e]) f(ϕ))

with (Subscript[((OverTilde[R])^cj), fh]) = (η^cjab) (Subscript[R, abfh]) and (η^cjab) = (Sqrt[-g]) (ϵ^cjab) being the Levi-Civita tensor.

Use xAct to derive the tensor equation of motion for this theory and verify that the above equations are indeed correct.

3) Slowly rotating neutron stars

In this exercise we will extend the formalism for relativistic stars to the slowly rotating case. A slowly rotating star (to first-order in rotation) has the metric:

(ds^2) = -(e^(-ν(r))) (dt^2) + (e^(λ(r)))(dr^2) + (r^2)((dθ)^2) + (r^2)(Sin^2)(θ) ((dϕ)^2) - 2[Ω - ω(r)](r^2)(Sin^2)(θ) dtdϕ

where Ω is the star’s angular velocity and ω(r) is the fluid’s angular velocity as measured by a freely-falling observer. 

Using xCoba, derive the equation of motion for ω(r) for both general relativity and the John theory (with ϕ = qt asymptotically) studied in the tutorial. In the case of GR, you should match to Hartle’s formular given in 1967Ap J. . .150.1005H, equation 46, and in the case of John you should match to 1603.04876 equation 30.