A couple of metrics in generalrepytivity

A couple of metrics in generalrepytivity

January 18, 2018

A couple of metrics in generalrepytivity #

As a small tutorial on generalrepytivity1, this blogpost explains how to create a metric tensor in generalrepytivity and, with it, how to get all the usual geometric invariants one is interested in (such as the Riemann tensor, the Ricci tensor and the scalar curvature). You can find all the code in this article in this jupyter notebook.

Installing the library #

generalrepytivity is now on pypi, so installing it is easy using pip:

pip install generalrepytivity

Importing the library #

We heavily recommend importing it along with sympy (the symbolic computation library of python):

import generalrepytivity as gr
import sympy

Tensors come with a fancy LaTeX printing function, so if you’re working on a jupyter notebook I recommend setting the printing function of sympy to use latex:

sympy.init_printing(use_latex=True)

Creating a tensor in generalrepytivity #

To create a Tensor object, one must specify three things:

  • coordinates, which are a list of sympy symbols.
  • _type, a tuple of integers (p,q).
  • values, a dictionary of the non-zero values (whose keys are tuples of multiindices).

For example, say you want to create the following tensor in \(\mathbb{R}^4\) with coordinates \(t, x, y, z\) :

\[ \Gamma = (x^2 + y^2)\frac{\partial}{\partial t}\otimes \frac{\partial}{\partial x}\otimes dx\otimes dz + \sin(t)\frac{\partial}{\partial x}\otimes \frac{\partial}{\partial y}\otimes dt \otimes dt \]

in this case, we would need the sympy symbols t, x, y, z for the coordinates, the _type would be (2,2) (because \(\Gamma\) is a (2,2)-tensor) and for the values note that the components of \(\Gamma\) in these coordinates are the following:

\[\Gamma^{0, 1}_{1, 3} = x^2 + y^2\] \[\Gamma^{1, 2}_{0, 0} = \sin(t)\]

so the dictionary of values should look like this:

values = {
    ((0, 1), (1, 3)): x**2 + y**2,
    ((1, 2), (0, 0)): sympy.sin(t)
}

In summary:

t, x, y, z = sympy.symbols('t x y z')
values = {
    ((0,1), (1, 3)): x**2 + y**2,
    ((1,2), (0, 0)): sympy.sin(t)
}

Gamma = gr.Tensor([t, x, y, z], (2,2), values)

and you can access the components of the tensor by indexing:

Gamma[(0,1), (1,3)] == x**2 + y**2
True

Metrics #

A metric \(g\) is just a (0,2)-tensor, so we could create it using a values dict just like we did above but, because (0,2)-tensors are so important (and because they can be represented by matrices), generalrepytivity has a simpler way of creating them. The function to use is gr.get_tensor_from_matrix, which has the following syntax:

gr.get_tensor_from_matrix(matrix, coordinates)

where matrix is a square sympy matrix and the coordinates are just like above (i.e. a list of sympy symbols).

Examples of metrics #

In this blogpost we will deal with three different solutions to Einstein’s equations: Gödel’s metric, Schwarzschild’s metric and FRLW.

Gödel’s metric #

In 1949, Gödel proposed a solution to Einstein’s Field Equations which described a rotating universe in which closed world-lines are possible (that is, you could in some way influence the past). Because of this violation of causality, Gödel’s metric is just interesting from a theoretical perspective, not from a modeling one.

Gödel’s metric is

\[g = a^2(dx_0\otimes dx_0 - dx_1\otimes dx_1 + (e^{2x_1}/2)dx_2\otimes dx_2 - dx_3\otimes dx_3 + e^{x_1}dx_0\otimes dx_2 + e^{x_1}dx_2 \otimes dx_0) \]

To enter it into generalrepytivity, we could use the matrix representation:

x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')
a = sympy.Symbol('a')
A = a**2 * sympy.Matrix([
    [1, 0, sympy.exp(x1), 0],
    [0, -1, 0, 0],
    [sympy.exp(x1), 0, sympy.exp(2*x1)/2, 0],
    [0,0,0,-1]])

g_godel = gr.get_tensor_from_matrix(A, [x0, x1, x2, x3])

Schwarzschild metric #

Schwarzschild metric is generally used when it comes to modeling spherical objects and their influence in the geometry of spacetime. It comes after assuming a static and spherically symmetrical spacetime.

Schwarzschild metric is (in units such that \(c=1\) and \(G_N = 1\) )

\[g = -\left(1 - \frac{2 m}{r}\right)dt \otimes dt + \left(1 - \frac{2 m}{r}\right)^{-1}dr \otimes dr + (r^{2} \sin^{2}{\left (\theta \right )})d\phi \otimes d\phi + r^{2}d\theta \otimes d\theta \]

where \(m\) is the mass of the spherical object. To review the canon way of creating tensors, lets use a values dictionary:

t, r, theta, phi = sympy.symbols('t r \\theta \\phi')
m = sympy.Symbol('m')
values = {
    ((), (0,0)): -(1-2*m/r),
    ((), (1,1)): 1/(1-2*m/r),
    ((), (2,2)): r**2,
    ((), (3,3)): r**2*sympy.sin(theta)**2
}
g_sch = gr.Tensor([t, r, theta, phi], (0, 2), values)

FRLW #

The FRLW metric (which stands for Friedmann-Lemaître-Robertson-Walker) models a homogenous, isotropic and either expanding or contracting (depending on a constant) universe.

Its tensor representation in coordinates is

\[g = -dx_0\otimes dx_0 + A(x_0)^2(dx_1\otimes dx_1 + dx_2\otimes dx_2 + dx_3\otimes dx_3)\]

where \(A(x_0) = \epsilon x_0^q\) with \(0<q <1\) and \(\epsilon\in\mathbb{R}\) . Using matrices:

x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')
epsilon, q = sympy.symbols('\\epsilon q')
A = epsilon * (x0 ** q)
g_matrix = sympy.diag(-1, A**2, A**2, A**2)
g_FRLW = gr.get_tensor_from_matrix(g_matrix, [x0, x1, x2, x3])

The Spacetime object #

A good way of making a summary of the usual geometric invariants one computes using a metric is by using the Spacetime object in generalrepytivity. It takes just one argument: the metric.

Godel_spacetime = gr.Spacetime(g_godel)
Sch_spacetime = gr.Spacetime(g_sch)
FRLW_spacetime = gr.Spacetime(g_FRLW)

Each object now holds:

  • The Christoffel symbols christoffel_symbols.
  • The Riemann tensor Riem.
  • The Ricci tensor Ric.
  • The scalar curvature R.

For example, Schwarzschild’s metric was created in order to be Ricci flat (i.e. \(\text{Ric} = 0\) ). Let’s verify this:

Sch_spacetime.Ric == 0
True

Printing a summary in LaTeX #

Lastly, the Spacetime object comes with a print-to-file function which accepts two format options: tex or txt. Running the command

Godel_spacetime.print_summary('godel.tex', _format='tex')

generates a .tex file with a summary of all the values of the christoffel_symbols, Riem, Ric and R.

  1. This is the first of a series of posts about generalrepytivity, a python toolbox I made for computations related to tensors and general relativity. ↩︎