Anti_de_Sitter4.ipynb

Kernel: SageMath 8.0

4-dimensional anti-de Sitter spacetime

This Jupyter/SageMath worksheet presents the computation of the curvature tensor and the Ricci tensor of the 4-dimensional anti-de Sitter spacetime $\mathrm{AdS}_4$. See the SageManifolds home page for more details about tensor calculus with the free computer algebra system SageMath.

%display latex

We declare first the spacetime manifold $M$ and the chart $X$ of Poincaré coordinates:

M = Manifold(4, 'M')
X.<t,x,y,z> = M.chart('t x y z:(0,+oo)')
X

X.coord_range()

The AdS metric is then defined as

g = M.lorentzian_metric('g')
L = var('L', domain='real')
g[0,0] = -L^2/z^2
g[1,1], g[2,2], g[3,3] = L^2/z^2, L^2/z^2, L^2/z^2
g.display()

The Christoffel symbols are (the vanishing ones and those that can be deduced by symmetry on the last two indices are not displayed)

g.christoffel_symbols_display()

The components of the Riemann curvature tensor are

g.riemann().display_comp(only_nonzero=True, only_nonredundant=True)

while those of the Ricci tensor are

g.ricci().display_comp(only_nonzero=True, only_nonredundant=True)

Another view of the same thing:

g.ricci().display()

The Ricci scalar turns out to be constant (as on any maximally symmetric spacetime):

g.ricci_scalar().display()

Finally, we check that the Einstein equation with the negative cosmological constant $\Lambda=-\frac{3}{L^2}$ is satisfied:

Lambda = -3/L^2
g.ricci() - 1/2*g.ricci_scalar()*g + Lambda*g == 0