Various coordinate systems in anisotropic 5d Lifshitz spacetimes

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

%display latex

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

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, x, {y_1}, {y_2}, r)\right)$$

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()

$$\newcommand{\Bold}[1]{\mathbf{#1}}g = -e^{\left(2 \, {\nu} r\right)} \mathrm{d} t\otimes \mathrm{d} t + e^{\left(2 \, {\nu} r\right)} \mathrm{d} x\otimes \mathrm{d} x + e^{\left(2 \, r\right)} \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + e^{\left(2 \, r\right)} \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} +\mathrm{d} r\otimes \mathrm{d} r$$

Coordinate $\tilde r = e^r$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, x, {y_1}, {y_2}, {\tilde{r}})\right)$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ r & = & \log\left({\tilde{r}}\right) \end{array}\right.$$

X2_to_X1.inverse().display()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ {\tilde{r}} & = & e^{r} \end{array}\right.$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}g = -{\tilde{r}}^{2 \, {\nu}} \mathrm{d} t\otimes \mathrm{d} t + {\tilde{r}}^{2 \, {\nu}} \mathrm{d} x\otimes \mathrm{d} x + {\tilde{r}}^{2} \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + {\tilde{r}}^{2} \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + \frac{1}{{\tilde{r}}^{2}} \mathrm{d} {\tilde{r}}\otimes \mathrm{d} {\tilde{r}}$$

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

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, x, {y_1}, {y_2}, {\rho})\right)$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ {\rho} & = & {\tilde{r}}^{{\nu}} \end{array}\right.$$

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

X2_to_X3.inverse().display()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ {\tilde{r}} & = & {\rho}^{\left(\frac{1}{{\nu}}\right)} \end{array}\right.$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}g = -{\rho}^{2} \mathrm{d} t\otimes \mathrm{d} t + {\rho}^{2} \mathrm{d} x\otimes \mathrm{d} x + {\rho}^{\frac{2}{{\nu}}} \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + {\rho}^{\frac{2}{{\nu}}} \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + \frac{1}{{\nu}^{2} {\rho}^{2}} \mathrm{d} {\rho}\otimes \mathrm{d} {\rho}$$

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

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, x, {y_1}, {y_2}, z)\right)$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ z & = & \frac{1}{{\rho}} \end{array}\right.$$

X3_to_X4.inverse().display()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} t & = & t \\ x & = & x \\ {y_1} & = & {y_1} \\ {y_2} & = & {y_2} \\ {\rho} & = & \frac{1}{z} \end{array}\right.$$

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

$$\newcommand{\Bold}[1]{\mathbf{#1}}g = -\frac{1}{z^{2}} \mathrm{d} t\otimes \mathrm{d} t + \frac{1}{z^{2}} \mathrm{d} x\otimes \mathrm{d} x + \left( z^{-\frac{2}{{\nu}}} \right) \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + \left( z^{-\frac{2}{{\nu}}} \right) \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + \frac{1}{{\nu}^{2} z^{2}} \mathrm{d} z\otimes \mathrm{d} z$$

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.