2026-08-30
Geometrical Coordinate Splitting between Observers
A New Look at the Lorentz Transformations
The same event can be observed from different reference frames. For one observer, it has coordinates \(ct,x\), for another: \(ct',x'\). The event remains one, but its coordinate descriptions are different. Typically, these two sets of coordinates are written separately and then related by Lorentz transformations.
In this article, we will look at the same problem differently. Is it possible to place two descriptions of a single event in a single mathematical object without mixing them and without declaring one reference frame dominant? It turns out that an idempotent splitting of unity provides a suitable language for this.
The basic idea is as follows: it is not the event itself that is split between observers, but its coordinate description. The coordinates become two-component, while the space-time interval remains unitary.
1. Unity as a Hidden Structure
The number \(1\) is usually perceived as the simplest indivisible object. However, in an idempotent basis, it admits an internal decomposition.
\[\tag{1} \boxed{1=\ep+\em},\] where the two components have the properties
\[\tag{2} \ep^2=\ep,\qquad \em^{\,2}=\em,\qquad \ep\em=0.\] The origin and geometric meaning of this decomposition are discussed in more detail in the article "From Euler's Formula to Split Geometry". Here, we only need the main property of idempotents: they allow two independent components to be stored within a single object.
Let two arbitrary values \(A\) and \(B\) be combined in an expression.
\[\tag{3} \mathcal A=\ep A+\em B.\] Thanks to the condition \(\ep\em=0\), the components are not multiplied or mixed. But if their values are the same, \(A=B\), the splitting collapses:
\[\tag{4} \mathcal A=\ep A+\em A=(\ep+\em)A=A.\] Splitting only occurs when the values of the two idempotent components differ. When they coincide, the structured expression again looks like a regular scalar.
2. Two Projectors — Two Observers
Consider two inertial reference frames \(S\) and \(S'\). Their axes are oriented in the same direction, and their coordinate origins coincide at \(t=t'=0\). System \(S'\) moves relative to \(S\) along the \(x\) axis with constant velocity \(v\).
One event receives two coordinate descriptions in these systems:
\[\tag{5} X_S=(ct,x),\qquad X_{S'}=(ct',x').\] Assign each description to its own idempotent component:
\[\tag{6} \boxed{\ep\longleftrightarrow S',\qquad \em\longleftrightarrow S.}\] Then, for any coordinate of one event, we can introduce a joint representation.
\[\tag{7} \boxed{\mathcal A_{S'S}=\ep A'+\em A}.\] In this interpretation, the equalities \(\ep^2=\ep\) and \(\em^{\,2}=\em\) mean that re-selecting the description of the same observer changes nothing. The condition \(\ep\em=0\) preserves the two descriptions independently, and the sum \(\ep+\em=1\) combines them into a single complete relative object.
The idempotents here do not assert that a physical event has split into two parts. They serve as two independent geometric labels, assigning coordinates to different observers.
3. Geometrically split coordinates
For the time and space coordinates, we define
\[\tag{8} \boxed{\begin{aligned} \mathcal T_{S'S}&=\ep\,ct'+\em\,ct,\\ \mathcal X_{S'S}&=\ep\,x'+\em\,x. \end{aligned}}\] The quantity \(\mathcal T_{S'S}\) contains two time readings, and \(\mathcal X_{S'S}\) contains two spatial coordinate values. In this case, both pairs refer to the same event.
If the coordinate descriptions match, then \(ct'=ct\) and \(x'=x\). Idempotents immediately add up to one:
\[\tag{9} \begin{aligned} \mathcal T_{SS}&=(\ep+\em)ct=ct,\\ \mathcal X_{SS}&=(\ep+\em)x=x. \end{aligned}\] Thus, a regular coordinate turns out to be a special diagonal case of a split coordinate:
\[\tag{10} \boxed{A'=A\quad\Longrightarrow\quad \ep A'+\em A=A.}\] Even different coordinate systems can assign identical coordinates to a single event—for example, to a common origin. Therefore, splitting reveals not simply the presence of two observers, but the difference in their coordinate descriptions of a given event.
4. The hyperbolic part as a measure of difference
The difference between two idempotents forms a hyperbolic unit
\[\tag{11} \j=\ep-\em,\qquad \j^2=1.\] From here
\[\tag{12} \ep=\frac{1+\j}{2},\qquad \em=\frac{1-\j}{2}.\] ByLet's represent these expressions in formula (7):
\[\tag{13} \boxed{\mathcal A_{S'S}=\frac{A'+A}{2}+\j\frac{A'-A}{2}.}\] The first part contains the average of the two coordinate descriptions, and the coefficient of \(\j\) is half their difference. For time and space, this yields
\[\tag{14} \begin{aligned} \mathcal T_{S'S}&=\frac{ct'+ct}{2}+\j\frac{ct'-ct}{2},\\ \mathcal X_{S'S}&=\frac{x'+x}{2}+\j\frac{x'-x}{2}. \end{aligned}\] If both observers obtain the same value, the coefficient of \(\j\) vanishes. Therefore, in this construction, the hyperbolic component has a clear meaning:
\[\tag{15} \boxed{\j\text{-component}\quad\longleftrightarrow\quad \text{difference in coordinate descriptions}.}\] 5. Lorentz Transformations
The coordinates of two inertial observers are related by the usual Lorentz transformations:
\[\tag{16} \boxed{\begin{aligned} ct'&=\gamma(ct-\beta x),\\ x'&=\gamma(x-\beta ct), \end{aligned}}\] where
\[\tag{17} \beta=\frac{v}{c},\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}.\] In matrix form, these expressions are written as
\[\tag{18} \begin{pmatrix}ct'\\ x'\end{pmatrix}=\Lambda(\beta)\begin{pmatrix}ct\\ x\end{pmatrix},\qquad \Lambda(\beta)=\begin{pmatrix}\gamma&-\gamma\beta\\ -\gamma\beta&\gamma\end{pmatrix}.\] The matrix \(\Lambda(\beta)\) transforms the event description from system \(S\) to system \(S'\). Now we combine the transformed and original descriptions in a single idempotent entry.
6. Idempotent notation of a transformation
The joint coordinates of two observers can be represented by the formula
\[\tag{19} \boxed{\begin{pmatrix}\mathcal T_{S'S}\\ \mathcal X_{S'S}\end{pmatrix}=\left[\ep\Lambda(\beta)+\em I\right]\begin{pmatrix}ct\\ x\end{pmatrix},}\] where \(I\) is the identity matrix. Expanding (19), we obtain
\[\tag{20} \boxed{\begin{aligned} \mathcal T_{S'S}&=\ep\,\gamma(ct-\beta x)+\em\,ct,\\ \mathcal X_{S'S}&=\ep\,\gamma(x-\beta ct)+\em\,x. \end{aligned}}\] The component \(\ep\Lambda(\beta)\) contains the coordinates obtained by observer \(S'\). The component \(\em I\) preserves the original coordinates of observer \(S\). The Lorentz transformation turns out to be selective: it acts on one idempotent branch, while the other serves as the starting point of comparison.
In the absence of relative motion
\[\tag{21} \beta=0,\qquad \gamma=1,\qquad \Lambda(0)=I.\] Therefore, the joint description operator collapses:
\[\tag{22} \ep\Lambda(0)+\em I=(\ep+\em)I=I.\] Relative motion does not create idempotents: they are already present in the hidden decomposition of unity. The motion makes the difference between the two components observable, since the coordinates of the \(S\) and \(S'\) systems no longer coincide.
7. The coordinates are split, the interval is unitary.
Despite the difference in coordinates, both observers receive the same space-time interval:
\[\tag{23} s^2=c^2t^2-x^2,\qquad s'^2=c^2t'^2-x'^2,\qquad s'^2=s^2.\] We define a joint representation of the interval:
\[\tag{24} \mathcal S^2_{S'S}=\ep s'^2+\em s^2.\] Since \(s'^2=s^2\), the two idempotent components again coincide:
\[\tag{25} \boxed{\mathcal S^2_{S'S}=\ep s^2+\em s^2=(\ep+\em)s^2=s^2.}\] The same result follows directly from split coordinates. Due to the orthogonality of \(\ep\em=0\), the mixed products vanish:
\[\tag{26} \begin{aligned} \mathcal T_{S'S}^{\,2}-\mathcal X_{S'S}^{\,2} &={}\ep(c^2t'^2-x'^2)+\em(c^2t^2-x^2)\\ &=\ep s'^2+\em s^2=s^2. \end{aligned}\] The coordinates of a single event are split between observers, but the space-time interval remains unsplit. Idempotents preserve the difference in relative descriptions and disappear from the quantity common to both systems.
8. A Numerical Example
Consider system \(S'\) moving with velocity \(v=0.6c\). Then
\[\tag{27} \beta=0.6,\qquad \gamma=\frac{1}{\sqrt{1-0.6^2}}=1.25.\] Let the event in the system \(S\) have coordinates \(ct=10\) and \(x=0\), measured in the same units of length. According to Lorentz's formulas:
\[\tag{28} ct'=1{}25(10-0{}6\cdot0)=12{}5,\qquad x'=1{}25(0-0{}6\cdot10)=-7{}5.\] The joint coordinates are as follows
\[\tag{29} \mathcal T=12{}5\ep+10\em,\qquad \mathcal X=-7{}5\ep+0\em.\] The coordinate values differ, but the intervals are the same:
\[\tag{30} 12.5^2-(-7.5)^2=156.25-56.25=100.\] \[\tag{31} 10^2-0^2=100.\] Therefore, the idempotent interval collapses into an ordinary number:
\[\tag{32} \mathcal T^2-\mathcal X^2=100\ep+100\em=100.\] 9. Permutation of Observers
Neither of the two observers need be physically allocated. Let's introduce an operation that permutes idempotents:
\[\tag{33} \ep^*=\em,\qquad \em^*=\ep.\] It changes the order of the two coordinate descriptions:
\[\tag{34} \left(\mathcal A_{S'S}\right)^*=\ep A+\em A'=\mathcal A_{SS'}.\] At the same time, the relative velocity changes sign:
\[\tag{35} S\leftrightarrow S'\qquad\Longrightarrow\qquad \beta\leftrightarrow-\beta.\] Thus, the choice of component \(\ep\) for \(S'\), and \(\em\) for \(S\) is only the order of notation. Rearranging the observers swaps the components but does not change the physical interval.
10. Generalization to 3+1 Spacetime
So far, we have considered motion along a single spatial axis. In four-dimensional spacetime, an event has coordinates. \[\tag{36} X_S^\mu=(ct,x,y,z),\qquad X_{S'}^\mu=(ct',x',y',z').\]
The idempotent union of two descriptions retains the same form:
\[\tag{37} \boxed{\mathcal X_{S'S}^{\mu}=\ep X_{S'}^{\mu}+\em X_S^{\mu}.}\] If the coordinates are related by a common Lorentz matrix. \[\tag{38} X_{S'}^{\mu}=\Lambda^{\mu}{}_{\nu}X_S^{\nu},\]
then the shared object is written as
\[\tag{39} \boxed{\mathcal X_{S'S}^{\mu}=\left(\ep\Lambda^{\mu}{}_{\nu}+\em\delta^{\mu}{}_{\nu}\right)X_S^{\nu}.}\] Its interval remains an ordinary scalar:
\[\tag{40} \boxed{\eta_{\mu\nu}\mathcal X_{S'S}^{\mu}\mathcal X_{S'S}^{\nu}=c^2t^2-x^2-y^2-z^2=s^2.}\] Thus, the principle is i ndependent of the number of spatial coordinates. Idempotents separate the observers' descriptions, and the Minkowski metric identifies the quantity they share.
Conclusion
We started with a simple factorization of unity \(1=\ep+\em\) and assigned two idempotents the meaning of independent projectors onto the coordinate descriptions of two observers. This allowed us to combine the coordinates of a single event in a single geometric object, storing each description in its own component.
When coordinates coincide, the idempotents add up to unity, and the split object becomes an ordinary coordinate. With relative motion, the values diverge, and the hyperbolic component expresses the difference between them. The Lorentz transformation acts selectively: one component contains the transformed coordinates, while the other preserves the original reference frame.
The main result can be written in compact form:
\[\tag{41} \boxed{ \begin{gathered} \mathcal T_{S'S}=\ep\,ct'+\em\,ct, \qquad \mathcal X_{S'S}=\ep\,x'+\em\,x, \\ \mathcal T_{S'S}^{\,2}-\mathcal X_{S'S}^{\,2}=s^2. \end{gathered} } \] The coordinates of a single event are geometrically split between different observers, while the space-time interval belongs to their common, unsplit part.
The proposed notation does not replace special relativity and does not alter the Lorentz transformations. It provides a different mathematical view of their structure: relative coordinates are stored in independent idempotent components, and the invariant arises where these components reunite into unity.