Erratum: “A characterization of causal automorphisms by wave equations” [J. Math. Phys. 53, 032507 (2012)]

The second author, recently, has found that there are some errors in the proof of the Theorem 2.3 in Ref. 1. However, none of the results or conclusions is affected by the error and it seems to be better to show that Theorem 2.3 can be directly obtained from Theorem 2.1 in Ref. 1 by use of known results in conformal geometry. In this process, we can see that the characteristic differences between

$R12$ and

$\mathbb {R}^n_1$

$R1n$ with n ⩾ 3 come from the conformal geometry.

The following is Theorem 2.1 in Ref. 1.

Theorem 2.1. Let (x₁, ⋯, x_{n + 1}) be the standard coordinate system on 

$R1n+1$ and (y₁, ⋯, y_n+1) be another coordinate system on 

$\mathbb {R}^{n+1}_1$

$R1n+1$ with n ⩾ 2. Assume that, for any smooth function φ, 

$\sum \limits _{i=1}^n \frac{\partial ^2 \varphi }{\partial x_i^2} - \frac{\partial ^2 \varphi }{\partial x_{n+1}^2} = 0$

$∑i=1n∂2φ∂xi2−∂2φ∂xn+12=0$ if and only if 

$\sum \limits _{i=1}^n \frac{\partial ^2 \varphi }{\partial y_i^2} - \frac{\partial ^2 \varphi }{\partial y_{n+1}^2} = 0$

$∑i=1n∂2φ∂yi2−∂2φ∂yn+12=0$⁠. Then, all rows of Jacobian matrix 

$\big ( \frac{\partial y_i}{\partial x_j} \big )$

$∂yi∂xj$ are mutually orthogonal and have the same length. Furthermore, the first n rows are spacelike vectors and the last row is a timelike vector.

Let us denote the operator

$∑i=1n∂2∂xi2−∂2∂xn+12$ by □ and

$\sum \limits _{i=1}^n \frac{\partial ^2 }{\partial y_i^2} - \frac{\partial ^2 }{\partial y_{n+1}^2}$

$∑i=1n∂2∂yi2−∂2∂yn+12$ by □′. Then Theorem 2.1 states that the condition, □φ = 0 ⇔ □′φ = 0 for any C² function φ, implies that the Jacobian of F is Lorentz matrix multiplied by a positive function, where

$F : \mathbb {R}^{n+1}_1 \rightarrow \mathbb {R}^{n+1}_1$

$F:R1n+1→R1n+1$ given by (y₁, ⋯, y_n+1) = F(x₁, ⋯, x_n+1) is a diffeomorphism. Essentially, this implies that the diffeomorphism F is a conformal transformation if □φ = 0 ⇔ □′φ = 0 for any C² function φ.

Then, we can use the following known result, called Liouville's theorem.

Theorem 2.2. Every C⁴ conformal transformation of a region of pseudo-Euclidean space of dimension ⩾3 is a composite of isometries, dilations, and inversions.

Proof. This is Theorem 15.2 in Ref. 2.

It is a well-known fact that any causal automorphisms on

$R1n+1$ with n ⩾ 2 is C^∞ and so we can apply the above theorem to Theorem 2.1. Also, since inversion has a singularity, it cannot be defined on the whole of

$\mathbb {R}^{n+1}_1$

$R1n+1$ and thus any conformal transformation is a composite of isometries and dilations. Therefore, we have the following, which is Theorem 2.3 in Ref. 1.

Theorem 2.3. Let 

$F:R1n+1→R1n+1$ given by (y₁, ⋯, y_n+1) = F(x₁, ⋯, x_n+1) be a diffeomorphism with n ⩾ 2. Then the necessary and sufficient condition for F to be a causal automorphism on 

$\mathbb {R}^{n+1}_1$

$R1n+1$ is that 

$\frac{\partial y_{n+1}}{\partial x_{n+1}} \ge 0$

$∂yn+1∂xn+1≥0$⁠, and for any smooth function φ, 

$\sum \limits _{i=1}^n \frac{\partial ^2 \varphi }{\partial x_i^2} - \frac{\partial ^2 \varphi }{\partial x_{n+1}^2} = 0$

$∑i=1n∂2φ∂xi2−∂2φ∂xn+12=0$ if and only if 

$\sum \limits _{i=1}^n \frac{\partial ^2 \varphi }{\partial y_i^2} - \frac{\partial ^2 \varphi }{\partial y_{n+1}^2} = 0$

$∑i=1n∂2φ∂yi2−∂2φ∂yn+12=0$⁠.

Recently, it is shown that causal automorphisms on

$R12$ can also be characterized by invariance of wave equations. Therefore, if Liouvilles's theorem holds for two-dimensional space, causal automorphism on

$\mathbb {R}^2_1$

$R12$ must have the same form as in Theorem 2.3, which is not true. In other words, the characteristic difference between

$\mathbb {R}^2_1$

$R12$ and

$\mathbb {R}^n_1$

$R1n$ with n ⩾ 3 comes from the truth of Liouville's theorem.