Notation regarding generalized Minkowski space

Question

In section 12 of the book Surfaces in classical geometries: A treatment by moving frames by Gary R. Jensen, Emilio Musso and Lorenzo Nicolodi (see preview here), Möbius geometry is described.

They introduce the generalized Minkowski space of signature $(4,1)$ as follows:

Let $R^{4,1}$ denote $R^5$ with a Lorentzian inner product. Let $\epsilon_0,\dots,\epsilon_4$ denote the standard orthonormal basis of $R^{4,1}$ given by the standard orthonormal basis $\epsilon_0,\dots,\epsilon_3$ of the Euclidean space $R^4$ and with $\langle \epsilon_4, \epsilon_4\rangle=-1$. The Lorentzian inner products $\langle \epsilon_a, \epsilon_b\rangle$, for $a,b=0,\dots,4$, are the entries of the matrix $$\begin{pmatrix} I_4 & 0 \\ 0 & -1 \end{pmatrix}.$$ Write elements of $R^{4,1}=R^4\oplus R\epsilon_4$ as $x + t\epsilon_4$, where $x\in R^4$ and $t\in R$. The Lorentzian inner product is then $$\langle x+s\epsilon_4,y+t\epsilon_4\rangle=x\cdot y-st.$$

My problem is that I would expect vectors in $R^{4,1}$ to have dimension $5$, just like vectors in the Minkowski space have dimension $4$. They actually start stating that $R^{4,1}$ is $R^5$ with a Lorentzian inner product. This could suggest that it is just a typo (two actually), but later on in the chapter they are consistent on this matter: vectors in $R^{4,1}$ have $4$ components. For instance, they consider a mapping which is the sum of a vector in $\mathbb{S}^3$ with the vector $\epsilon_4$. Therefore I assume there's no typo and it's just my poor understanding...

Can anyone help understanding this construction and how it works?

Answer 1

So when they write

Write elements of $R^{4,1}=R^4\oplus Re_4$ as $x+te_4$, where $x\in R^4$ and $t\in R$.

what they actually mean is that $x$ is in the $4$-dimensional subspace of $R^{4,1}$ spanned by $\{e_1,e_2,e_3,e_4\}$. It doesn't mean that $x\in R^4$ literally, $x$ is still an element of a $5$-dimensional space but one of the coordinates is zerod.

For example if we take the standard basis $e_i=(0,\ldots,0,1,0,\ldots, 0)$ with $1$ at $i$-th posisition then the statement can be rewritten as

Write elements of $R^{4,1}$ as $(x_0,x_1,x_2,x_3,0)+t(0,0,0,0,1)$.

More generally the precise statement should be

Write elements of $R^{4,1}$ as $x+te_4$ where $x=x_0e_0+x_1e_1+x_2e_2+x_3e_3$ for $x_0,x_1,x_2,x_3,t\in R$.

Here $R^4$ means $span(e_0,e_1,e_2,e_3)$. It is a $4$-dimensional subspace of $R^5$.