1. Introduction

2. Mapping between Spaces with Mixed Covariant and Contravariant Bases

2.3. Proposition and Proof

Proof. 

Let ${\mathbb{R}}^4$ be an Euclidean space with canonical basis $\left\{{\mathrm{e}}_{\mathsf{\mu }}\right\}$ and dual basis $\left\{{\mathrm{e}}^{\mathsf{\mu }}\right\}$; $\mathbb{R}(3,1)$ a Minkowski space with orthonormalized basis $\left\{{\mathrm{f}}_{\mathsf{\mu }}\right\}$. It is to be shown that, upon a change of basis, the element ${\mathrm{e}}^0$ of the dual Euclidean basis transforms like the element ${\mathrm{f}}_0$ of the Minkowski basis.

The orientation preserving changes of basis in Euclidean space $SO\left(4\right)=\{\mathrm{R}\in GL\left(4\right)\mid {\mathrm{R}}^{\mathrm{T}}\mathrm{R}=\mathbb{1},det\mathrm{R}=1\}$ can be expressed as an exponential series:

$$\mathrm{R}=e^{t\mathrm{A}}{\mathrm{R}}^{-1}={\mathrm{R}}^{\mathrm{T}}=e^{-t\mathrm{A}}.$$

(11)

Where t is the transformation parameter and $\mathrm{A}$ is a skew symmetric matrix (${\mathrm{A}}^{\mathrm{T}}=-\mathrm{A}$). The skew symmetry of these matrices is what leads to the orthogonality of the finite transformations and the coordinate vectors to change with the transposed transformation rule (Eq. 8).

The matrices $\mathrm{A}$ build, together with the commutator $[\mathrm{A},\mathrm{B}]=\mathrm{AB}-\mathrm{BA}$, the Lie-algebra $so\left(4\right)$ of the Lie-group $SO\left(4\right)$: the algebra that generates the infinitesimal orientation preserving coordinate transformations in ${\mathbb{R}}^4$, which through exponential mapping span the whole Lie-group $SO\left(4\right)$ (for introductions see e.g. [10,11]).

In the case of infinitesimal transformations ($t\ll 1$) it is sufficient to only consider the first terms of the exponential series. The special role of the Lie-algebra is seen here:

$$\mathrm{R}=e^{t\mathrm{A}}\approx (1+t\mathrm{A}){\mathrm{R}}^{-1}=e^{-t\mathrm{A}}\approx (1-t\mathrm{A}).$$

(12)

The elements $\mathrm{A}$ of the Lie-algebra can once more be expressed as a linear combination of a basis, which in the case of $so\left(4\right)$ consists of six skew symmetrical matrices, e.g.:

$${\mathrm{L}}_1=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right){\mathrm{L}}_2=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right){\mathrm{L}}_3=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$

(13)

$${\mathrm{K}}_1=\left(\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right){\mathrm{K}}_2=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right){\mathrm{K}}_3=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right)$$

(14)

$$\begin{array}{cc}\hfill \mathrm{with}& [{\mathrm{L}}_{\mathrm{i}},{\mathrm{L}}_{\mathrm{j}}]={\epsilon }_{ijk}{\mathrm{L}}_{\mathrm{k}}\hfill \\ \hfill & [{\mathrm{K}}_{\mathrm{i}},{\mathrm{K}}_{\mathrm{j}}]={\epsilon }_{ijk}{\mathrm{L}}_{\mathrm{k}}\hfill \\ \hfill & [{\mathrm{L}}_{\mathrm{i}},{\mathrm{K}}_{\mathrm{j}}]={\epsilon }_{ijk}{\mathrm{K}}_{\mathrm{k}},\hfill \end{array}$$

(15)

and therefore

$$t\mathrm{A}=\sum _{i=1,2,3}\left(t^i{\mathrm{L}}_{\mathrm{i}}+s^i{\mathrm{K}}_{\mathrm{i}}\right).$$

(16)

To apply a change of the mixed basis, a transformation rule has to be constructed which transforms the element ${\mathrm{e}}_0$ of the basis with the same coefficients that a normalized coordinate vector ${\mathrm{v}}^0={(1,0,0,0)}^{\mathbf{T}}$ is transformed with. Using transformation rule Eq. 7 and definition Eq. 14, the infinitesimal change of coordinate vector ${\mathrm{v}}^0$ can be written as follows:

$$\begin{array}{cc}\hfill {\mathrm{v}}^{\prime 0}& =\left(1-t\mathrm{A}\right){\mathrm{v}}^0\hfill \\ \hfill & =\left(1-\sum _{i=1,2,3}{s}^i{\mathrm{K}}_{\mathrm{i}}\right){\mathrm{v}}^0\hfill \\ \hfill & =\left(1+\sum _{i=1,2,3}{s}^i{\mathrm{K}}_{\mathrm{i}}^{-1}\right){\mathrm{v}}^0.\hfill \end{array}$$

(17)

The (pseudo-)inverse ${\mathrm{K}}_{\mathrm{i}}^{-1}=-{\mathrm{K}}_{\mathrm{i}}$ denotes the inverse of ${\mathrm{K}}_{\mathrm{i}}$ on the subspace spanned by ${\mathrm{e}}_{\mathrm{i}}$ and ${\mathrm{e}}_0$. In componentwise notation this reads:

$$\begin{array}{cc}\hfill {\left(v^{\prime 0}\right)}^{\nu }& =\left({\delta }_0^{\nu }-t{A}_0^{\nu }\right){\left(v^0\right)}^0\hfill \\ \hfill & =\left({\delta }_0^{\nu }-\sum _{i=1,2,3}{s}^i{\left(K_i\right)}_0^{\nu }\right){\left(v^0\right)}^0\hfill \\ \hfill & =\left({\delta }_0^{\nu }+\sum _{i=1,2,3}{s}^i{\left(K_i^{-1}\right)}_0^{\nu }\right){\left(v^0\right)}^0.\hfill \end{array}$$

(18)

Eq. 18 is rearranged to show the change $\mathrm{dv}=({\mathrm{v}}^{\prime 0}-{\mathrm{v}}^0)$ caused by the transformation. It holds for the individual components:

$$\begin{array}{cc}\hfill d{v}^{\nu }={\left(v^{\prime 0}\right)}^{\nu }-{\left(v^0\right)}^{\nu }& =\sum _{i=1,2,3}{s}^i{\left(K_i^{-1}\right)}_0^{\nu }{\left(v^0\right)}^0\hfill \\ \hfill & =\sum _{i=1,2,3}{s}^i{\left(K_i^{-1}\right)}_0^{\nu }.\hfill \end{array}$$

(19)

On the other hand, the transformation rule to obtain ${\mathrm{e}}_0^{\prime }$ is given by (from Eq. 6):

$$\begin{array}{cc}\hfill {\mathrm{e}}_0^{\prime }& =\sum _{\nu }{\mathrm{e}}_{\mathbf{\nu }}\left({\delta }_0^{\nu }+t{A}_0^{\nu }\right)\hfill \\ \hfill & =\sum _{\nu }{\mathrm{e}}_{\mathbf{\nu }}\left({\delta }_0^{\nu }+\sum _{i=1,2,3}{s}^i{\left(K_i\right)}_0^{\nu }\right).\hfill \end{array}$$

(20)

Rearrangement of Eq. 20 provides the change $\mathrm{de}={\mathrm{e}}_0^{\prime }-{\mathrm{e}}_0$ induced by the transformation:

$$\mathrm{de}={\mathrm{e}}_0^{\prime }-{\mathrm{e}}_0=\sum _{\nu ,i}{\mathrm{e}}_{\mathbf{\nu }}{s}^i{\left(K_i\right)}_0^{\nu },$$

(21)

which results in component notation:

$$\begin{array}{cc}\hfill d{e}_{\nu }={\left(e_0^{\prime }-e_0\right)}_{\nu }& =\sum _{i=1,2,3}{\left(e_{\nu }\right)}_{\nu }{s}^i{\left(K_i\right)}_0^{\nu }.\hfill \end{array}$$

(22)

Sought is an infinitesimal basis transformation with which the individual entries of change $d{v}^{\nu }$ of the vector (Eq. 19) coincide with the entries of change $d{e}_{\nu }$ of the basis vector (Eq. 22). In order for this to be true for all transformations, the comparison of these two equations yields that the entries of the first row ${\left(K_i\right)}_0^{\nu }$ of matrices ${\mathrm{K}}_{\mathrm{i}}$ must be equal to the corresponding entries ${\left(K_i^{-1}\right)}_0^{\nu }$ of their inverted matrices ${\mathrm{K}}_{\mathrm{i}}^{-1}$.

As expected, this condition cannot be fulfilled by regular (non-trivial) Euclidean transformations, since in this case the ${\mathrm{K}}_{\mathrm{i}}$ are skew symmetric and orthogonal, and thus ${\mathrm{K}}_{\mathrm{i}}^{-1}=-{\mathrm{K}}_{\mathrm{i}}$. To nonetheless find a change of basis which has the same effect on ${\mathrm{e}}_0$ as it has on ${\mathrm{v}}^0$, one must depart from changes of basis in the ordinary Euclidean sense and search for a new set of transformations.

In order for the new base transformations to be useful, they must again function as a Lie algebra, such that finite transformations can be generated from the infinitesimal transformations using the exponential map. The starting point is thus given by a new set of infinitesimal generators $\overline{\mathrm{A}}$ (marked with a bar) of the same form as in Eq. 16:

$$t\overline{\mathrm{A}}=\sum _{i=1,2,3}\left(t^i{\mathrm{L}}_{\mathrm{i}}+{\overline{s}}^i{\overline{\mathrm{K}}}_{\mathrm{i}}\right).$$

(23)

According to the previous considerations, different elements ${\overline{\mathrm{K}}}_{\mathrm{i}}$ of the basis are sought, which are their own inverses in their spanned subspace:

$${\left(\overline{\phantom{K}}{K}_i\right)}_0^{\nu }\stackrel{!}{=}{\left(\overline{\phantom{K}}{K}_i^{-1}\right)}_0^{\nu }.$$

(24)

Since this condition alone does not give a unique form for the matrices ${\overline{\mathrm{K}}}_{\mathrm{i}}$, some additional requirements can be applied. The transformation of the other basis vectors should remain the same as before, thus the coefficients in columns $j\ne 0$ stay unchanged with respect to Eq. 14:

$${\left(\overline{\phantom{K}}{K}_i\right)}_j^{\nu }\stackrel{!}{=}{\left(K_i\right)}_j^{\nu }j=1,2,3.$$

(25)

Furthermore, volume conservation can be preserved:

$${\left(\overline{\phantom{K}}{K}_i\right)}_0^0\stackrel{!}{=}{\left(K_i\right)}_0^0=0.$$

(26)

Considering conditions Eqs. 24, 25 and 26, the following new transformation matrices can be constructed:

$${\overline{\mathrm{K}}}_1=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right){\overline{\mathrm{K}}}_2=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right){\overline{\mathrm{K}}}_3={\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right)}_{.}$$

(27)

The new elements retain the matrix form of the transformations, yet without orthogonality, since the skew symmetry of the Lie-algebra’s elements had to be abandoned. The generators Eq. 27 are inserted back into Eq. 23, which now describes the infinitesimal basis transformations between mixed bases according to Definition 2.

On the other hand, the unchanged matrices $\left\{{\mathrm{L}}_{\mathrm{i}}\right\}$ (Eq. 13) build, together with the new matrices $\left\{{\overline{\mathrm{K}}}_{\mathrm{i}}\right\}$ (Eq. 27) and the commutator as a Lie bracket, the Lie-algebra $so(3,1)$ with elements $\overline{\mathrm{A}}\in so(3,1)$. By means of the exponential mapping this Lie-Algebra translates into the proper Lorentz group $SO(3,1)$, where:

$$\Lambda =e^{t\overline{A}}\Lambda \in SO(3,1).$$

(28)

Yet the elements of the proper Lorentz group $\Lambda$ are defined as those orientation preserving changes of basis taking place in the Minkowski space $\mathbb{R}(3,1)$, with metric tensor $g_{\mu \nu }^M=\mathrm{diag}\left(-1,+1,+1,+1\right)$ (see e.g. [12]).

The determined infinitesimal transformation rule according to Eq. 23 for mixed bases in Euclidean space as specified in Definition 1 thus corresponds exactly to the infinitesimal transformation rule for a canonical base $\left\{{\mathrm{f}}_{\mathsf{\mu }}\right\}$ of Minkowski space. Moreover, since only the basis vector ${\mathrm{e}}_0$ has been changed, the Euclidean space with mixed basis can be directly substituted by a Minkowski space with canonical basis:

$$\begin{array}{cc}\hfill {\mathrm{e}}^0& ⟼{\mathrm{f}}_0\hfill \\ \hfill {\mathrm{e}}_{\mathrm{i}}& ⟼{\mathrm{f}}_{\mathrm{i}}i=1,2,3.\hfill \end{array}$$

(29)

${\mathrm{e}}^0$ transforms like the element ${\mathrm{f}}_0$ of the Minkowski basis and vice versa, and the discovered transformation is equivalent to a regular change of basis in Minkowski space.

3. Interpretation as Distance Measurement by a Flat Observer

3.2. Proposition and Proof

By Definition 4, a flat observer cannot directly take distance measurements along the direction normal to its submanifold parallel to ${\mathrm{e}}_0$. Under certain conditions, however, indirect distance measurements in this direction may become possible:

Lemma 1. 

A flat observer according to Definition 4 can make distance measurements in the ${\mathrm{e}}_0$ direction if:

A) 

A measurable object $\mathrm{O}$ according to Definition 5 with spatial extent in the ${\mathrm{e}}_0$ direction is present, it intersects the measurement neighborhood ${\epsilon }_M$ of the flat observer $(\mathrm{O}\cap {\epsilon }_M\ne \left\{\right\})$, and

B) 

there is relative displacement between ${\epsilon }_M$ and the object in the ${\mathrm{e}}_0$ direction.

Proof. 

A)

As per Definition 3, to perform any measurement, the metric in Eq. 30 must be used. Measurable objects are defined as the input quantities for this metric. They are also the only way the flat observer can measure distances in the ${\mathrm{e}}_0$ direction, since a basis vector is not available. Hence, for the measurement in this direction, a measurable object $\mathrm{O}$ must be present.

To measure a distance other than zero in some direction, the two vectors inserted into the metric must differ in the coordinate of interest. A length $d(\mathrm{dv},\mathrm{dw})$ normal to M can only be defined by two vectors $\mathrm{dv}$, $\mathrm{dw}$ with different values in the ${\mathrm{e}}_0$ component. This difference between the two vectors $\mathrm{dv}-\mathrm{dw}$ can just as well be defined as a measurable object $\mathrm{O}$ with a non zero spatial extent in the ${\mathrm{e}}_0$ direction. Finally, a flat observer can only perform measurements in its neighborhood ${\epsilon }_M$, thus the measurable object must intersect that neighborhood.

B)

Since the flat observer only has the overview of one single coordinate in the ${\mathrm{e}}_0$ direction, a relative shift between $\mathrm{O}$ and ${\epsilon }_M$ is necessary to observe at least two different coordinate values.

For further illustration of this concept, see the three-dimensional schematic example in Figure 2.

Lemma 2.

The measurement of a distance normal to the submanifold according to Lemma 1 by a flat observer according to Definition 4 produces a contravariantly transforming scale.

Proof. 

For a flat observer, the only measurable relation to the outside of its submanifold M is given by the measurable object $\mathrm{O}$. The basis vector ${\mathrm{e}}_0$ is by Definition 4 not available for measurement.

Since there are no further references, a standard length in the ${\mathrm{e}}_0$ direction can only be defined using object $\mathrm{O}$, and any additional distance measurement is subsequently quantified as multiples of this standard length. However, the measured standard length used as the basis element and the resulting scale transform contravariantly, since a measurable object $\mathrm{O}$ is a vectorial quantity, as stated in Corollary 2.

The effect of a contravariant measurement is illustrated in Figure 3 with a three-dimensional example.

These considerations lead to the following proposition:

Proof. 

The proof is divided into two parts. The first part is to show that the Minkowski metric can be understood as a measurement rule in an Euclidean space if a mixed basis $\left\{{\mathrm{e}}^0,{\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right\}$ is used when carrying out the measurement. This part was proven in Section 2.3.

The second part is covered by Lemma 2, which shows that a flat observer uses such a mixed basis, where the scale for distance measurement is covariant within its submanifold and contravariant when normal to its submanifold. Thus, the argumentation is consistent and Proposition 2 is proven.

4. Conclusions

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