Anti-de Sitter spacetime

This worksheet demonstrates a few capabilities of SageManifolds (version 1.0, as included in SageMath 7.5) in computations regarding the 4-dimensional anti-de Sitter spacetime.

Click here to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command sage -n jupyter

NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:

In [1]:
version()
Out[1]:
'SageMath version 8.0.beta10, Release Date: 2017-06-11'

First we set up the notebook to display mathematical objects using LaTeX rendering:

In [2]:
%display latex

We also define a viewer for 3D plots (use 'threejs' or 'jmol' for interactive 3D graphics):

In [3]:
viewer3D = 'threejs' # must be 'threejs', 'jmol', 'tachyon' or None (default)

Spacetime manifold

We declare the anti-de Sitter spacetime as a 4-dimensional differentiable manifold:

In [4]:
M = Manifold(4, 'M', r'\mathcal{M}')
print(M); M
4-dimensional differentiable manifold M
Out[4]:

We consider hyperbolic coordinates $(\tau,\rho,\theta,\phi)$ on $\mathcal{M}$. Allowing for the standard coordinate singularities at $\rho=0$, $\theta=0$ or $\theta=\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\mathbb{R}^4$). If we restrict ourselves to regular coordinates (i.e. to considering only mathematically well defined charts), the hyperbolic coordinates cover only an open part of $\mathcal{M}$, which we call $\mathcal{M}_0$, on which $\rho$ spans the open interval $(0,+\infty)$, $\theta$ the open interval $(0,\pi)$ and $\phi$ the open interval $(0,2\pi)$. Therefore, we declare:

In [5]:
M0 = M.open_subset('M_0', r'\mathcal{M}_0' )
X_hyp.<ta,rh,th,ph> = M0.chart(r'ta:\tau rh:(0,+oo):\rho th:(0,pi):\theta ph:(0,2*pi):\phi')
print(X_hyp) ; X_hyp
Chart (M_0, (ta, rh, th, ph))
Out[5]:
In [6]:
X_hyp.coord_range()
Out[6]:

$\mathbb{R}^{2,3}$ as an ambient space

The AdS metric can be defined as that induced by the immersion of $\mathcal{M}$ in $\mathbb{R}^{2,3}$, the latter being nothing but $\mathbb{R}^5$ equipped with a flat pseudo-Riemannian metric of signature $(-,-,+,+,+)$. Let us construct $\mathbb{R}^{2,3}$ as a 5-dimensional manifold covered by canonical coordinates:

In [7]:
R23 = Manifold(5, 'R23', r'\mathbb{R}^{2,3}')
X23.<U,V,X,Y,Z> = R23.chart()
print(X23); X23
Chart (R23, (U, V, X, Y, Z))
Out[7]:

We introduce on $\mathbb{R}^{2,3}$ the flat pseudo-Riemannian metric $h$ of signature $(-,-,+,+,+)$

In [8]:
h = R23.metric('h', signature=1)
h[0,0], h[1,1], h[2,2], h[3,3], h[4,4] = -1, -1, 1, 1, 1
h.display()
Out[8]:

The AdS immersion into $\mathbb{R}^{2,3}$ is defined as a differential map $\Phi$ from $\mathcal{M}$ to $\mathbb{R}^{2,3}$, by providing its expression in terms of $\mathcal{M}$'s default chart (which is X_hyp = $(\mathcal{M}_0,(\tau,\rho,\theta,\phi))$ ) and $\mathbb{R}^{2,3}$'s default chart (which is X23 = $(\mathbb{R}^{2,3},(U,V,X,Y,Z))$ ):

In [9]:
var('l', latex_name=r'\ell', domain='real')
assume(l>0)
Phi = M.diff_map(R23, [l*cosh(rh)*cos(ta/l),
                      l*cosh(rh)*sin(ta/l),
                      l*sinh(rh)*sin(th)*cos(ph),
                      l*sinh(rh)*sin(th)*sin(ph),
                      l*sinh(rh)*cos(th)],
                 name='Phi', latex_name=r'\Phi')
print(Phi); Phi.display()
Differentiable map Phi from the 4-dimensional differentiable manifold M to the 5-dimensional differentiable manifold R23
Out[9]:

The constant $\ell$ is the AdS length parameter. Considering AdS metric as a solution of vacuum Einstein equation with negative cosmological constant $\Lambda$, one has $\ell = \sqrt{-3/\Lambda}$.

Let us evaluate the image of a point via the map $\Phi$:

In [10]:
p = M((ta, rh, th, ph), name='p'); print(p)
Point p on the 4-dimensional differentiable manifold M

The coordinates of $p$ in the chart X_hyp:

In [11]:
X_hyp(p)
Out[11]:
In [12]:
q = Phi(p); print(q)
Point Phi(p) on the 5-dimensional differentiable manifold R23
In [13]:
X23(q)
Out[13]:

The image of $\mathcal{M}$ by the immersion $\Phi$ is a hyperboloid of one sheet, of equation $$-U^2-V^2+X^2+Y^2+Z^2=-\ell^2.$$ Indeed:

In [14]:
(Uq,Vq,Xq,Yq,Zq) = X23(q)
s = - Uq^2 - Vq^2 + Xq^2 + Yq^2 + Zq^2
s.simplify_full()
Out[14]:

We may use the immersion $\Phi$ to draw the coordinate grid $(\tau,\rho)$ in terms of the coordinates $(U,V,X)$ for $\theta=\pi/2$ and $\phi=0$ ($X\geq 0$ part) or $\phi=\pi$ ($X\leq 0$ part). The red (rep. grey) curves are those for which $\rho={\rm const}$ (resp. $\tau={\rm const}$):

In [15]:
graph_hyp = X_hyp.plot(X23, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:0}, 
                    ranges={ta:(0,2*pi), rh:(0,2)}, number_values=9, 
                    color={ta:'red', rh:'grey'}, thickness=2, parameters={l:1}, 
                    label_axes=False)  # phi = 0 => X > 0 part
graph_hyp += X_hyp.plot(X23, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:pi},
                    ranges={ta:(0,2*pi), rh:(0,2)}, number_values=9, 
                    color={ta:'red', rh:'grey'}, thickness=2, parameters={l:1}, 
                    label_axes=False)  # phi = pi => X < 0 part
show(graph_hyp, aspect_ratio=1, viewer=viewer3D, online=True,
     axes_labels=['V','X','U'])

To have a nicer picture, we add the plot of the hyperboloid obtained by parametric_plot with $(\tau,\rho)$ as parameters and the expressions of $(U,V,X)$ in terms of $(\tau,\rho)$ deduced from the coordinate representation of $\Phi$:

In [16]:
Phi.coord_functions() # the default pair of charts (X_hyp, X23) is assumed
Out[16]:
In [17]:
Ug = Phi.coord_functions()[0](ta,rh,pi/2,0).subs({l:1})  # l=1 substituted to have numerical values
Vg = Phi.coord_functions()[1](ta,rh,pi/2,0).subs({l:1})
Xg = Phi.coord_functions()[2](ta,rh,pi/2,0).subs({l:1})
Ug, Vg, Xg
Out[17]:
In [18]:
hyperboloid = parametric_plot3d([Vg, Xg, Ug], (ta,0,2*pi), (rh,-2,2), color=(1.,1.,0.9))
graph_hyp += hyperboloid
show(graph_hyp, aspect_ratio=1, viewer=viewer3D, online=True,
     axes_labels=['V','X','U'])

Spacetime metric

As mentionned above, the AdS metric $g$ on $\mathcal{M}$ is that induced by the flat metric $h$ on $\mathbb{R}^{2,3}$, i.e.$g$ is the pullback of $h$ by the differentiable map $\Phi$:

In [19]:
g = M.lorentzian_metric('g')
g.set( Phi.pullback(h) )

The expression of $g$ in terms of $\mathcal{M}$'s default frame is found to be

In [20]:
g.display()
Out[20]:
In [21]:
g[:]
Out[21]:

Curvature

The Riemann tensor of $g$ is

In [22]:
Riem = g.riemann()
print(Riem)
Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M
In [23]:
Riem.display_comp(only_nonredundant=True)
Out[23]:

The Ricci tensor:

In [24]:
Ric = g.ricci()
print(Ric)
Ric.display()
Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M
Out[24]:
In [25]:
Ric[:]
Out[25]:

The Ricci scalar:

In [26]:
R = g.ricci_scalar()
print(R)
R.display()
Scalar field r(g) on the 4-dimensional differentiable manifold M
Out[26]:

We recover the fact that AdS spacetime has a constant curvature. It is indeed a maximally symmetric space. In particular, the Riemann tensor is expressible as $$ R^i_{\ \, jlk} = \frac{R}{n(n-1)} \left( \delta^i_{\ \, k} g_{jl} - \delta^i_{\ \, l} g_{jk} \right), $$ where $n$ is the dimension of $\mathcal{M}$: $n=4$ in the present case. Let us check this formula here, under the form $R^i_{\ \, jlk} = -\frac{R}{6} g_{j[k} \delta^i_{\ \, l]}$:

In [27]:
delta = M.tangent_identity_field() 
Riem == - (R/6)*(g*delta).antisymmetrize(2,3)  # 2,3 = last positions of the type-(1,3) tensor g*delta
Out[27]:

We may also check that AdS metric is a solution of the vacuum Einstein equation with (negative) cosmological constant $\Lambda = - 3/\ell^2$:

In [28]:
Lambda = -3/l^2
Ric - 1/2*R*g + Lambda*g == 0
Out[28]:

Radial null geodesics

The radial null geodesics of AdS spacetime are given by $$ \tau = \pm 2 \ell \left( \mathrm{atan} \left(\mathrm{e}^\rho\right) - \frac{\pi}{4} \right) + \tau_0,$$ where $\tau_0$ is a constant (the value of $\tau$ at $\rho=0$).

Let us consider a finite family of these geodesics for $\theta=\pi/2$ and $\phi=0$ or $\pi$:

In [29]:
null_geod_out = [M.curve({X_hyp: [2*l*(atan(exp(rh)) - pi/4) + 2*pi*i/8, rh, pi/2, 0]}, 
                          (rh, 0, +oo)) for i in range(9)]
null_geod_in = [M.curve({X_hyp: [-2*l*(atan(exp(rh)) - pi/4) + 2*pi*i/8, rh, pi/2, pi]}, 
                         (rh, 0, +oo)) for i in range(9)]
In [30]:
print(null_geod_out[0])
Curve in the 4-dimensional differentiable manifold M
In [31]:
null_geod_out[0].display()
Out[31]:
In [32]:
null_geod_in[0].display()
Out[32]:

Let us plot the null geodesics in terms of the coordinates $(\tau,\rho)$:

In [33]:
graph = Graphics()
for geod in null_geod_out:
    graph += geod.plot(X_hyp, ambient_coords=(rh,ta), prange=(0,4),
                       parameters={l:1}, color='green', thickness=2)
for geod in null_geod_in:
    graph += geod.plot(X_hyp, ambient_coords=(rh,ta), prange=(0,4),
                       parameters={l:1}, color='green', thickness=2)
graph += X_hyp.plot(X_hyp, ambient_coords=(rh,ta), fixed_coords={th:0, ph:pi}, 
                    ranges={ta:(-pi/2,5*pi/2), rh:(0,4)}, 
                    number_values={ta: 13, rh: 9},
                    color={ta:'red', rh:'grey'}, parameters={l:1})
show(graph, aspect_ratio=1/2)

We can get a 3D view of the radial null geodesics via the isometric immersion $\Phi$:

In [34]:
graph = Graphics()
for geod in null_geod_out:
    graph += geod.plot(X23, mapping=Phi, ambient_coords=(V,X,U), prange=(0,2),
                       parameters={l:1}, color='green', thickness=2, 
                       plot_points=20, label_axes=False)
for geod in null_geod_in:
    graph += geod.plot(X23, mapping=Phi, ambient_coords=(V,X,U), prange=(0,2),
                       parameters={l:1}, color='green', thickness=2, 
                       plot_points=20, label_axes=False)
show(graph+graph_hyp, aspect_ratio=1, viewer=viewer3D, online=True,
     axes_labels=['V','X','U'])

Hence the image by $\Phi$ of the radial null geodesics are straight lines of $\mathbb{R}^{2,3}$. This is not surprising since $\Phi$ is an isometric immersion and the null geodesics of $\mathbb{R}^{2,3}$ are straight lines. Note that the null geodesics of $\mathrm{AdS}_4$ rule the hyperboloid (remember that a hyperboloid of one sheet is a ruled surface).

"Static" coordinates

Let us introduce spherical coordinates $(\tau,R,\theta,\phi)$ on the AdS spacetime via the coordinate change $$R = \ell \sinh(\rho) $$ Despiste the $(\tau,\rho,\theta,\phi)$ coordinates are as adapted to the spacetime staticity as the $(\tau,R,\theta,\phi)$ coordinates, the latter ones are usually called "static" coordinates.

In [35]:
X_stat.<ta,R,th,ph> = M0.chart(r'ta:\tau R:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
print(X_stat); X_stat
Chart (M_0, (ta, R, th, ph))
Out[35]:
In [36]:
X_stat.coord_range()
Out[36]:
In [37]:
hyp_to_stat = X_hyp.transition_map(X_stat, [ta, l*sinh(rh), th, ph])
hyp_to_stat.display()
Out[37]:
In [38]:
hyp_to_stat.set_inverse(ta, asinh(R/l), th, ph, verbose=True)
stat_to_hyp = hyp_to_stat.inverse()
stat_to_hyp.display()
Check of the inverse coordinate transformation:
  ta == ta
  rh == arcsinh(sinh(rh))
  th == th
  ph == ph
  ta == ta
  R == R
  th == th
  ph == ph
Out[38]:

The expression of the metric tensor in the new coordinates is

In [39]:
g.display(X_stat.frame(), X_stat)
Out[39]:

Similarly, the expression of the Riemann tensor is

In [40]:
Riem.display_comp(X_stat