CoCalc -- Lifshitz_coordinates.ipynb

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Jupyter notebook Lifshitz_coordinates.ipynb

Project: Lifshitz BH

Path: Lifshitz_coordinates.ipynb

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Kernel: SageMath 6.10

Various coordinate systems in anisotropic 5d Lifshitz spacetimes

When $\nu=1$ , all these coordinate systems cover the Poincaré patch of ${\rm AdS}_5$ spacetime.

In [1]:

%display latex

In [2]:

M = Manifold(5, 'M')
print M

5-dimensional differentiable manifold M

For $\nu=1$ , $M$ is nothing but the Poincaré patch of ${\rm AdS}_5$ .

Coordinates $(t,x,y_1,y_2,r)$

In [3]:

X1.<t,x,y1,y2,r> = M.chart(r't x y1:y_1 y2:y_2 r')
X1

In [4]:

var('nu', latex_name=r'\nu', domain='real')
g = M.lorentzian_metric('g')
g[0,0] = -exp(2*nu*r)
g[1,1] = exp(2*nu*r)
g[2,2] = exp(2*r)
g[3,3] = g[2,2]
g[4,4] = 1
g.display()

Coordinate $\tilde r = e^r$

In [5]:

X2.<t,x,y1,y2,R> = M.chart(r't x y1:y_1 y2:y_2 R:\tilde{r}:(0,+oo)')
X2

In [6]:

X2_to_X1 = X2.transition_map(X1, [t, x, y1, y2, ln(R)])
X2_to_X1.display()

In [7]:

X2_to_X1.inverse().display()

In [8]:

g.display(X2.frame(), X2)

Coordinate $\rho = {\tilde r}^\nu$

In [9]:

X3.<t,x,y1,y2,rho> = M.chart(r't x y1:y_1 y2:y_2 rho:\rho:(0,+oo)')
X3

In [10]:

X2_to_X3 = X2.transition_map(X3, [t, x, y1, y2, R^nu])
X2_to_X3.display()

In [11]:

X2_to_X3.set_inverse(t, x, y1, y2, rho^(1/nu), verbose=True)

Check of the inverse coordinate transformation:
   t == t
   x == x
   y1 == y1
   y2 == y2
   R == R
   t == t
   x == x
   y1 == y1
   y2 == y2
   rho == rho

In [12]:

X2_to_X3.inverse().display()

In [13]:

g.display(X3.frame(), X3)

Coordinate $z = 1/\rho = {\tilde r}^{-\nu}$

In [14]:

X4.<t,x,y1,y2,z> = M.chart(r't x y1:y_1 y2:y_2 z:(0,+oo)')
X4

In [15]:

X3_to_X4 = X3.transition_map(X4, [t, x, y1, y2, 1/rho])
X3_to_X4.display()

In [16]:

X3_to_X4.inverse().display()

In [17]:

g.display(X4.frame(), X4)

When $\nu=1$ , $(t,x,y_1,y_2,z)$ are the so-called Poincaré coordinates, i.e. the standard coordinates on the Poincaré patch of ${\rm AdS}_5$ spacetime.

In [ ]:

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Various coordinate systems in anisotropic 5d Lifshitz spacetimes

Coordinates $(t,x,y_1,y_2,r)$

Coordinate $\tilde r = e^r$

Coordinate $\rho = {\tilde r}^\nu$

Coordinate $z = 1/\rho = {\tilde r}^{-\nu}$

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Various coordinate systems in anisotropic 5d Lifshitz spacetimes

Coordinates (t,x,y1,y2,r)(t,x,y_1,y_2,r)(t,x,y1​,y2​,r)

Coordinate r~=er\tilde r = e^rr~=er

Coordinate ρ=r~ν\rho = {\tilde r}^\nuρ=r~ν

Coordinate z=1/ρ=r~−νz = 1/\rho = {\tilde r}^{-\nu}z=1/ρ=r~−ν

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Coordinates $(t,x,y_1,y_2,r)$

Coordinate $\tilde r = e^r$

Coordinate $\rho = {\tilde r}^\nu$

Coordinate $z = 1/\rho = {\tilde r}^{-\nu}$