Alternative form of Minkowski metric?

[stanleyfapps]

I've tried to find this addressed in other threads without success, so I apologize if it has already been addressed.

In Coleman and De Luccia (Gravitational effects on and of vacuum decay), they suggest that by analytic continuation ($$\[\xi = i\tau \]$$):


$$\[ds^2 = -d\xi ^2 - \rho (\xi )^2(d\Omega_S )^2\]$$

becomes

$$\[ds^2 = d\tau ^2 - \rho (i\tau )^2(d\Omega_T )^2\]$$

and if $$\[\rho (\xi ) = \xi \]$$, this is "ordinary Minkowski space." However, it doesn't look like Minkowski space to me. Does anyone know what transformation makes this look like ordinary Minkowski space?

How does this transform to look like Minkowski space?

$$\[ds^2 = d\tau ^2 + \tau^2(d\Omega_T )^2\]$$

 

[bcrowell]

How does this transform to look like Minkowski space?

$$\[ds^2 = d\tau ^2 + \tau^2(d\Omega_T )^2\]$$

Is $\Omega_T$ a solid angle in two dimensions (theta and phi), or is it a solid angle in three dimensions (theta, phi, and some other angle like eta)? In other words, do you even have 4 dimensions total, or just 3?

What are the definitions of $\Omega_S$ and $\Omega_T$, and how do they differ from one another?

 

[stanleyfapps]

$$\[d\Omega_T \]$$ is the element of length on a unit (3-dimensional) hyperboloid with spacelike normal vector in Minkowski space.

$$\[d\Omega_S\]$$ is the element of length for a unit hyperboloid with timelike normal vector in Minkowski space.