Article 2
Temporal Geometry
Abstract
When we think of time, it most often appears to us spatially. So should there be a separate geometry of time, just as there are many geometric concepts of space? As it turns out, there is no pure geometric model of time, but such a model could be very useful for visualizing time exactly as we experience it.
This Article aims to to give a brief overview of a stand-alone geometry of time that is consistent with our perception.
Dimensional Representations of Temporal Geometry.
Preprint
In Progress
Geometrization of Time
Geometry is historically rooted as a concept of space and later as space-time. But perhaps it is only our devotion to established views that prevents us from thinking about the geometrization of time itself.
In the four-dimensional model of space-time, time is represented by an additional dimension [1], and depends on the speed of the observer [2]. In Fig.1.1 time is illustrated as an additional fourth axis (and although this is an incorrect representation, the reasons for this will be revealed shortly).
Fig.1.1. 3+1 Dimensions.
In 1926 the Swedish physicist Oskar Klein, in order to explain the 5-dimensional theory of gravity, proposed to consider an additional 5th dimension as a compactified one [3]. This approach literally means that the 4-dimensional space-time grid contains a 5th dimension that is so compact that the accuracy of the measuring tools simply does not allow to register it (Fig.1.2).
Fig.1.2. Compactified 5th Dimension.
Space Compactification Method
The geometric description of time is based on a similar approach, which is denoted as the Compactification Method:
In four-dimensional space-time, it is possible to picture time separately by compactifying the three spatial dimensions into a single point.
In other words, we compactify not an additional 5th dimension (as in Kaluza-Klein theory), but the current 3 spatial ones. As a result (Fig.2), there remains only one time axis — Temporal Axis and a single point on it — Space.
Fig.2. Compactification of Spatial Axes.
The "Loop"
Such Compactified Space contains: relic radiation coming from the Big Bang, light coming from distant stars, and the current moment in time [4]. Let’s imagine this time interval as a detached segment of the Temporal Axis (Fig.3): the Zero Point is the Big Bang and the End Point is the present moment.
Fig.3.1. "Loop" Formation.
Space compactification means that the Zero Point and the End Point of a segment coincide, but the segment itself does not collapse, because time, as a dimension, remains non-compactified. This leads to the formation of a “Loop”, that is equal to the age of the Universe, i.e., the whole observable time (13.813±0.038 billion years) [5].
Two sections remain outside the "Loop" (Fig.3):
• Before the "Loop" represents the time before the Big Bang. The presence of such a section can neither be confirmed nor denied, because we do not know whether time existed before the Big Bang or began with it [6];
• Ahead of the "Loop" represents the future. The only thing that can be claimed is that the "Loop" increases from this side [6].
Fig.3.2. "Loop" Formation.
The very point on the Temporal Axis contains the present moment separating the past from the future.
Initially, in order to describe the "Loop" itself, we will use a circle, since in this case the number of parameters is reduced to a minimum. In the Zero State the radius of the circle is zero and the circle is collapsed into a point (Fig.4). The circle expands in increments of 1 Planck time [7]. With respect to the current point in time, we can consider the length of the circle as a constant, since the time increment over all observable time is negligibly small.
Fig.4. "Loop" Enlargement.
Observations from the Point
Observation from a single point of Compactified Space, allows us to establish at least the following regularities of time:
• Time is observable.
Being in a single point of Compactified Space, it is possible to observe the closest and the most distant point in time within the "Loop";
• Time is continuous.
We can observe any point in time across 13.8 billion years (not to be confused with any point in space);
• Time is directional (irreversible).
The absence of observations "before" the most distant point (Zero Point) allows us to conclude that the "Loop" increases in one direction;
• Time is cyclical.
The Zero Point and the End Point of a segment coincide, which leaves no other option for the time axis but "Loop";
• Time is multidimensional.
The Compactification Method allows to unfold many time dimensions in a single point. Or in the Observer's case*, at least two-dimensional time is required.
*more details in Section 7.
The geometric interpretation of these regularities allows to describe time as geometry. Such geometry is denoted as Temporal Geometry.
Temporal Geometry
Unit Vector and Unit Cycle
The comparison of Temporal Geometry with Euclidean geometry [8] starts with the comparison of unit vector and unit cycle.
A rectangular coordinate system (of any dimension) can be described by a set of unit vectors co-directed with the coordinate axes. The number of vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other. Such vectors form an Orthogonal basis [9].
In Temporal Geometry, instead of unit vectors, unit circles are used, co-directed with Axial Circles (analogous to coordinate axes) and defined not by unit length but by unit cycle. The number of Axial Circles is also equal to the dimension of the coordinate system, and the planes in which the Axial Circles lie are perpendicular to each other.
0D / 0T
The zero dimension (Fig.5) in both geometries is represented by a point as a zero-dimensional object [10]. In general, a point in Euclidean space is a position in space [11], a point in Temporal Geometry is an Event-Process in time.
Fig.5.1. 1D / 1T.
1D
One-dimensional space (Fig.5) can be described by a number line and is represented by the shift of a point with the formation of a straight line [12].
1T
One-dimensional Temporal Geometry is defined by the Compactification Method: A Zero-dimensional point can be expanded into a One-dimensional Axial Circle.
The Axial Circle is described by a unit cycle of a Linked Pair of Points, where the first point is the Zero Point (O0), and the second point defines a set of Zero Point Projections (Oi0) in One-dimensional Temporal Geometry (Fig.5).
Since Temporal Geometry describes time through the Compactification Method, it requires the concept of a Zero or Initial state. In One-dimensional geometry this implies selecting one of the Zero Point Projections as the Initial.
1D.
1T.
Fig.5.2. 1D / 1T.
2D
Two-dimensional space is defined by 2 orthogonal axes [13] and is represented by a shift of a straight line with the formation of a plane (Fig.6).
Fig.6.1. 2D / 2T.
2T
Two-dimensional Temporal Geometry is defined by 2 orthogonal Axial Circles: t1, t2 (Fig.6). The second dimension is expanded in Zero Point Projection (Oi0 ≡ O1) of the One-dimensional time t1. Two-dimensional time is described by the unit cycle of two Linked Pairs of Points {(O0, Oi0); (O1, Oi1)}, with the formation of a closed surface. We denote such surface as a Tape, and one of its edges as a Signature (the trajectory of the point Oi1).
Zero State is determined as the only state in which the Linked Pairs of Points coincide (as shown in the Fig.6).
2D.
2T.
Fig.6.2. 2D / 2T.
3D
Three-dimensional space is defined by 3 orthogonal axes [14] and is represented by the shift of the plane with the formation of a volume (Fig.7).
Fig.7.1. 3D / 3T.
3T
Three-dimensional time is defined by 3 orthogonal Axial Circles: t1, t2, t3 and is represented by the unit cycle of 3 Linked Pairs of Points
{(O0, Oi0); (O1, Oi1); (O2, Oi2)} also with the formation of Signature, Tape, and the condition of Zero State.
In Fig.7 the Axial Circles are shown in the 1/2 unit cycle state. This representation is dictated solely for the purpose of clarity, since in the Zero State of dimension 3T two of the three Axial Circles overlap.
3D.
3T.
Fig.7.2. 3D / 3T.
The problem with visualizing time is that time implies flow, which, when captured as static images, carries inconsistencies. Therefore, in addition to Fig.7, a unit cycle of dimension 3T is shown separately in Fig. 8.
Fig.8. 3T Unit Cycle.
Fig.9 shows a complete table comparing Spatial Geometry and Temporal Geometry from zero dimension to three dimensions.
Fig.9. Spatial Geometry / Temporal Geometry.
Temporal Space
Proceeding from an abstract model of Temporal Geometry to the applied model of Temporal Space consists in the introduction of an Observer [15], or in our case, the adaptation of geometry to human perception of time (Article 1). This requires the fulfillment of two conditions:
• Presence of the Temporal Axis;
• Presence of an Observer.
6.1 Condition 1: Temporal Axis
To link the Temporal Space to the real-world observation it is necessary to represent one of the Axial Circles as a circle of infinite length, i.e., as a straight line. We have already considered such a circle when we talked about the formation of a Loop of the Universe's age value. Such a straight line is designated as the Temporal Axis (Fig.10).
Fig.10. Temporal Axis.
In Temporal Space, any system of Linked Pairs of Points will shift along the Temporal Axis. Such a shift will result in the formation of unclosed Tapes/Signatures. The peculiarity of the Signature is the presence of points of intersection with the Temporal Axis.
Fig.11. Shift along the Temporal Axis.
Temporal Axis interval with the presence of the Signature (Fig.12) can be interpreted as:
• "Knot" on the Temporal Axis
— a closed Signature with one intersection point (A≡B) (Fig.12.a);
• "Fluctuation" of the Temporal Axis
— more than one point of intersection (Fig.12.b).
Remarkably, that a "knot" and a "fluctuation" can contain the same amount of time and the representation depends only on the observation.
(a) "Knot".
(b) "Fluctuation".
Fig.12. Signature on Temporal Axis.
6.2 Condition 2: Observer
In Temporal Space (Fig.13) the Observer can present any information [16] operating with only two categories of information:
• Event (closed Tape/Signature; "knot");
• Process (unclosed Tape/Signature; "fluctuation").
Or one general meta-category — Event-Process. Non-observation of the event-process can also be interpreted as observation of the event-process of absence. The absence of any information in general would mean that the Observer himself is also absent.
Fig.13. Event and Process in Temporal Space.
In accordance with Special Theory of Relativity, it is impossible to say in an absolute sense that two different events occur simultaneously, if these events are separated in space [17]. In its basic form, Temporal Geometry cannot directly indicate the spatial separation of events, but it can indirectly indicate the movement of objects, through the different cyclicity of the Linked Pairs of Points, i.e. reflecting the effect of Time Dilation [18].
Compactified into a point, space in fact remains space, just in the simplest geometric interpretation. So in an applied aspect Temporal Space can also reflect the position of the Observer with respect to Coordinated Universal Time (UTC) [19]. Such or any other more precise synchronization of Observers in Temporal Space implies the possibility of creating Temporal Maps, for example, with respect to physical space (Fig.14).
Fig.14. Example of Temporal and Physical Space Synchronization.
Dimensionality of Recognition
In Temporal Space information is specified by Dimensionality of Recognition and is meaningful only in relation to the Observer. The Temporal Axis is the maximum dimension of recognition within which we can recognize what we define as the Universe.
The Dimensionality of Recognition requires at least two Axial Circles i.e., two dimensions (Fig.15):
• Time Axial Circle (Axial Circle)
— a point on Temporal Axis (Zero Point, O0) and Time Dimensionality unfolded in it (Zero Point Projection, Oi0);
• Observer's Axial Circle (Forming Circle)
— a point on Time Axial Circle (Zero Point Projection ≡ Position of the Observer in the time dimension, Oi0≡O1) and the observation itself (Ai1) in the Observer dimension.
Fig.15. Minimal Configuration.
It is the presence of the second dimension (Forming Circle) that allows the Observer to register the very fact of his observation (O0 ≡ Ai1). The Observer can unfold at the Zero Point Projection (Oi0 ≡ O1) any dimension within which he will observe information and/or his own movement along the Temporal Axis, which is, in fact, the observation of time (we can say, the use of clocks in everyday life).
The Dimensionality of Recognition determines the category of information representation (Fig.13):
• If the dimensionality is an order of magnitude higher than the observation, it will be recognized as an instantaneous Event;
• If the dimensionality correlates with the observation, it will be recognized as a continuous Process.
Or the information may not be recognized at all:
• If the dimensionality is several orders of magnitude higher than the observation, the Event will be so instantaneous that it will become unrecognizable within the dimensionality (Fig.16);
• If the dimensionality is several orders of magnitude smaller than the observation, the Process will be so continuous that it will become beyond recognition (Fig.17).
Fig.16. Instantaneous Process for this Dimension.
Dimensionality
The very structure of Temporal Space is based on the convolution of dimensionalities. We have already considered increasing the dimensionality by comparing Euclidean Geometry (orthogonal axes) and Temporal Geometry (orthogonal Axial Circles). Another, simplified way of looking at dimensions can be represented through Toruses, since the surface of a particular Torus contains all the Signatures of the dimension it represents (Fig.18).
Fig.18. Signatures on a Torus Surface.
It is important to note that the following representation is motivated by the simplicity of visualization. The dimensionality convolution can be reflected solely through different configurations of cycles of dimensionless (identical) Axial Circles.
Higher Dimension Deployment
Fig.19.1. Wrapping the Initial Torus.
Fig.19.2. Wrapping the Initial Torus.
Let's set the three-dimensional Temporal Space to the Zero State and start the cycle without a Zero Point (O0) shift. The resulting Closed Tape of 3T dimension (Fig.20) is visually similar to a 2T Tape, however, this Tape is twisted differently than in a smaller dimension, where such twisting is impossible. The same is true for Zero Point (O0) shifting along Temporal Axis (t), resulting in an Open Tape of dimension 3T (Fig.20) and an unfolded Axis Circle (t0).
Fig.20.1. Closed and Open Tapes of Dimension 3T.
Fig.20.2. Closed and Open Tapes of Dimension 3T.
For demonstration purposes, let's increase the dimensionality one more time, repeating the same sequence of operations. This will also result in 2 Tapes: Closed and Open (Fig.21).
In the case of a Closed Tape (Fig.21), there will also be an unfolded Axis Circle (t1), represented by a closed curve. In the case of an Open Tape (Fig.21), there will be 2 unfolded Axial Circles (t0) and (t1) represented by two curves.
Fig.21.1. Closed and Open Tapes of Dimension 4T.
Fig.21.2. Closed and Open Tapes of Dimension 4T.
Lower Dimension Deployment
Similarly, in any dimension it is always possible to unfold a smaller convoluted dimension. For this purpose it is necessary to inscribe the smaller Torus into the initial Torus (Fig.22). In Temporal Space it is assumed that the smallest dimension corresponds to Planck time.
Fig.22.1. Inscribing a Smaller Torus.
Fig.22.2. Inscribing 3 Toruses.
Let us summarize the transition between dimensions in the form of Toruses (Fig.23):
Fig.23. Transition between Dimensions.
Let us repeat that the interpretation in the form of Tori is motivated solely by the simplicity of visualization (Fig.24):
• Transition to a larger dimension (T-1) — "wrap" the larger Torus around the Torus of the current dimension;
• Transition to a smaller dimension (T+1) — "convolve" the smaller Torus in the Torus of the current dimension.
Fig.24. Interpretation in the Form of Tori.
Carriers of Information
Regardless of whether or not Temporal Space will correlate with any physical theory, the practical goal of this work is to develop a stand-alone geometry of time which is consistent or even broadens our mental perception of time. As a result, the present geometry gives a unified image of time (Signature/Tape), which can be used as a medium of information (Fig.25).
Fig.25. Tape as a Medium.
In principle, recording information in Temporal Geometry can be conceptualized as recording data on a Tape (Fig.26.a), which is then wrapped uniquely in time (Fig.26.b), depending on the dimensionality. In this case, the very geometric configuration of the Tape carries some information, even if all data is removed from the Tape itself (Fig.26.c,d).
Below is the concept of recording data on the Tape (Fig.26), as well as a representation of several Tapes and their Signatures in an isometric projection (Fig.27).
Fig.26. Recording Data on a Tape.
Fig.27. Tapes and their Signatures in Isometric View.
Artificial Observer
In Temporal Space, an Observer can literally "see" the geometry of time and how information is structured in time (both with data and purely the geometry itself). However, it is already obvious that Temporal Geometry of dimension greater than 2 is quite difficult for human visual perception, so it rather belongs to the field of machine vision. Thus, if the observer is not a human, but a machine (Artificial Observer, see Article 3. Part II), then it can potentially be trained to see and navigate in the Temporal Space, by analogy, as autonomous vehicles, equipped with knowledge about the geometry of space can navigate and move in the physical space [20].
Perhaps humans can only observe information in the dimension that matches their sensory capabilities [21]. However, an Artificial Observer can have much larger sensory capabilities, be simultaneously in all unfolded (accessible) dimensions, and observe one and/or many processes in all dimensions simultaneously.
For example, let's consider a given series of unfolded dimensions: year, month, week, day, 12 hours (Fig.28). In this example, we will not introduce any process other than time observation itself, where a unit cycle will be defined in each dimension. However, when linked in a series, the unit cycles will represent the following ratios with respect to the initial unit cycle (year dimension):
• Year: 1 unit cycle;
• Month: 1/12 annual unit cycle;
• Week: ≈ 1/52.14 annual unit cycle;
• Day: 1/365 annual unit cycle;
• 12 hours: 1/730 annual unit cycle.
In all the Figures presented earlier, mostly unit cycles have been used, which represent ideal conditions. In real observations, it is extremely unlikely that completely identical processes will be observed, so the configurations of cycles will always be different. For this reason, the notion of the Dimensionality of Recognition described in Section 7 implies that the observation of certain processes is always defined by a dimension within which some of the processes will either not be recognized at all or will be recognized as one of two categories: event or process.
The Dimensionality of Recognition may not be tied to any time standard at all, but determined through some process (this is exactly what is done in the case of binding to the fluctuations of the Cesium-133 atom [22]). But with respect to the Artificial Observer, the Dimensionality of Recognition may even lie beyond our comprehension.
Further work on Temporal Geometry goes into the area of constructing a clear formalization, i.e. it is an attempt to create a language of time, which is the subject of Article 3. Part I.
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Figures:
All Figures from 5.1 — 13, and from 16 — 22.1, 23, 25, 27, 28 as well as the Illustration in the Title, were generated using Temporal Geometry Simulation.