**Escape of Time**

**©** 2010, Dr. Henryk Frystacki

**1. ****Introduction**

Two equations of relativistic mechanics describe time dilation and length contraction of the special theory of relativity STR with the coordinates x’, y’, z’ and t’ of a moving system S’ at the speed v, and with the coordinates x, y, z and t of the initial starting system S. Δx and Δx’ are lengths, Δt and Δt’ are time intervals read on clocks and compared with each other. System S’ moves along the x-axis. c stands for the speed of light in a vacuum.

Important note: Time in non-coinciding x’-coordinates in S’ differs, if evaluated in S. This describes a relativity of simultaneity of events. For two events in the same space location in S’ (x_{2}’= x_{1}’) but at a different time t_{2}’≠ t_{1}’** **time dilation of the special theory of relativity is described with x_{2}’- x_{1}’= 0 by:

The time period of a moving observer is shorter in comparison with the time period of a remaining observer in the starting location: Any moving observer stays in the present of the remaining one, but ages slower. The corresponding length contraction is based on proven constancy of speed of light, having the impact that the original distance Δx to the destination is getting shorter for the moving observer, expressed by Δx’:

Limiting time and length now by Planck-time and Planck-length and transforming the translations in three-dimensional space in an avant-garde way into rotary processes in space-time results in complementary views on time, length, speed, and space-time curvatures of the general theory of relativity GTR. Quantum mechanics gets its feasible “space-time view” on Heisenberg’s uncertainty principle. A rotary set-up of space-time reveals interchangeable space-time dimensions and an overall super symmetry of space-time. It does allow the discussion of the special theory of relativity and the general theory of relativity in combination with all quantum mechanical aspects, mass generation, and dark energy sources of an expanding universe.

**2. ****Complementary view on space-time**

For the following discussions, subjective simultaneity of events defines the perceived space and is defined by all events throughout space that fall into a single Planck-time of an observer’s individual subjective time. Three-dimensional space of this observer is constructed by simultaneity of events across distances. Original x-coordinates of relativistic mechanics with a Cartesian system describe in the following discussion the distance between any two points in three-dimensional space that appear with simultaneity of events. x’-coordinates stay on a chosen x-line. Let us postulate that there cannot be any coincidence of two or more events within one Planck-time in the same observer’s subjective space location within one single Planck length and combine this postulate with the proven concept of the constancy of speed of light, implying the dimensional stability and process continuity within a moving inertial system. This complementary view causes some remarkable changes.

Postulating that there cannot be a coincidence of simultaneous events in the same space location that is defined by a subjective single Planck length during a subjective Planck-time of an observer, any expansion or any shrinking of a homogeneous and isotropic space can only be explained by the corresponding change of simultaneity of events, if space is defined by all observable areas with a simultaneity of events. An event is defined by one single quantum leap of time or length. According to relativistic mechanics, any remaining observer in a starting location reads a different time along x’ of any moving object.

The term

of the formulas of relativistic mechanics describes different time in different locations along x’ of the moving object. Neglecting the non-linear relativistic effect, a time span is given by the reduced formula Δt = (x_{2}’/c – x_{1}’/c)v/c: The observer in the object will further on register simultaneity of events across the entire moving inertial system at any relative speed. The remaining observer in the starting system, however, will read sequential events instead, spread over the period Δt. This period increases linearly with linear increase of relative speed. This picture changes, however, if subjective space is defined by simultaneity of events that construct space. In this case, Δx’ will be successively replaced by time intervals, with the leaping function on Planck length and Planck time level, if length and time are limited at these values. Δx’ of the moving object is transformed into a time interval, and this interval is dilated just like any other observed time interval at relative speed.

In the reduced formula, this replacement quantity of Planck lengths by Planck time intervals depends on the ratio v/c of relative speed and maximum possible speed of light. Reaching theoretically speed of light, simultaneous events in the moving system are sequential events in the starting system. This coverage and the non-simultaneity replacement of length by time can be visualized with the rotation picture of figure 1, looking in there only on Planck length x-axis, on Planck time y-axis, on the broken line, and on the linearly increasing speed ratio on the y-axis from v/c = 0 up to v/c = 1. Considering no motion below the level of one Planck length and one Planck time because of being the minimum levels for any process, any speed increase shows always leaps, and the maximum acceleration a_{max} equals c/t_{p}. Planck length and Planck time cause the hatched areas of figure 1 below Planck levels, allowing deviating positions of zero points of superimposed diagrams within the limits of one Planck length and one Planck time.

** Figure 1: Replacement of distance by time caused by change of simultaneity**

An observer that remains in the starting environment would read for the doted length arrow of a moving object 3 Planck lengths by projection on the original length scale. This picture, however, cannot be correct, because it would imply a reduction of the moving system from originally 6 Planck lengths down to the length of the rotated doted arrow between the zero point and the broken line calculable by (3^{2}+4^{2})^{1/2 }= 5. This means that the step by step replacement of one Planck length by Planck time and treating them in the diagram equally by scaling each Planck time with a speed of light factor or scaling length by a speed of light divisor would lead to the subjective length reduction of the moving object for an observer in this moving inertial system and in the demonstrated case by exactly one Planck length, having the impact that light would disperse across this reduced length faster. This is in contradiction with all discoveries of physics. 6 Planck lengths will stay constant in the moving system, being the basic prerequisite for the constancy of speed of light that is observable from any observation post. This proven fact can be easily integrated into figure 1, but keeping for the moment a linear y-axis unchanged, and according to a speed caused length-time replacement of by the reduced formula Δt = (x_{2}’/c – x_{1}’/c)v/c, considering a superimposed relativistic factor (1-v^{2}/c^{2})^{-1/2} for S’-distance (x_{2}’ – x_{1}’) that keeps an original length of the x-axis constant during the rotation process in figure 1, because of keeping for example the 6 Planck lengths on the 6-unit-bow from zero speed up to speed of light, not considering the leap functions and successive replacement of length by time for evaluations in the starting base. Figure 1 shows the replacement of length by time, but with a constant Δx’: the increase of Δt is slowed down during the rotation of Δx’ in space-time by the relativistic factor.

Keeping the dimensions of the moving inertial system and a starting system constant develops the circular bow in figure 1, to identify exchange leaps and to capture non-linear developments with stable linear x-axis and y-axis of a starting inertial system: In case that an original length of 6 l_{P} of the length-time exchange on the y-axis is re-calibrated linearly from 0 speed up to speed of light c, figure 1 shows relative length contraction xl_{P }= integer 6l_{P}(1-v^{2}/c^{2})^{1/2} for x=1,2,3,4,5,6 according to the Pythagoras formula 6^{2}l_{P}^{2}(v^{2}/c^{2}) + x^{2}l_{P}^{2} = 6^{2}l_{P}^{2}, consistent with STR but with the quantum leaps that are necessary to separate simultaneous events in the subjective space of simultaneity. The length contraction is derived by projection on the x-axis.

Any observer of the inertial starting environment cannot shift a subjective space-time position within this diagram but stays in a subjective central zero position for the start of consecutive events and measurement. This leads to an uncertainty in evaluations if cause and effect are not distinguishable, for example if the observer has left the starting position with relative speed, or if the starting position moves in relation to the observer. Figure 1 shows that any perception of simultaneity of events is subjective and that distances are generated by separation, in case that two or more simultaneous events cannot coincide in the space frame of a subjective Planck length within the interval of one subjective Planck time. Space may be interpreted as individual three-dimensional separation result for simultaneous events and any kind of space expansion would require a process that increases areas of simultaneous events at the expense of serial events, considering dilation and contraction effects, and other equivalent processes.

The important finding is the fact that simultaneity of events or sequential events could be generated by an exchange of time and length and that this process can be mathematically and physically described by a rotary process in space-time, defining space as the subjective area of simultaneity.

Figure 1 keeps the dimensions of the inertial starting system stable by the perpendicular and linear construction of the length-time exchange frame. However, the y-axis of a moving inertial system can be rotated together with its x-axis to show the correct developments and directions of relative time dilations of STR. This is demonstrated in the rotation picture 2. Note that the t_{R0D}-axis describes a continuous reference time of any starting system. The measurement of relative time developments starts with setting t=0 on a clock in the moving object and on another clock remaining in the starting position.

Figure 2 shows the geometrical derivation of the relativistic time dilation: t_{R} describes the time of an object, moving with the speed v_{rel} = v relatively to the initial system with time t_{R0D}.

** ****Figure 2: Relative time dilation in a moving inertial system**

Anticlockwise t_{R}-rotation causes relative t_{R}-dilation. A moving observer does not notice the dilation within the moving inertial system, but a faster pace of time in the original starting environment with its relatively contracted length frame. Figure 2 shows an example with the ratio 5/6 between the pace of t_{R} and the pace of t_{R0D}.

The twin paradox describes the relative slow down of time for the twin that initiated the motion, resulting in relative time acceleration for the twin that remains in the starting inertial system, staying in each other’s present by comparing continuously differently running clocks.

The minimum of subjective time is one Planck time t_{P} = 5.931·10^{-44} s, causing leaps in this space-time-speed diagram. t_{R1D} is a theoretical support axis, showing a relative standstill of time and an infinite dilation in relation to t_{R0D}. Linear speed calibration corresponds to the subjective possibility to linearly increase relative speed against any starting system. Not considering quantum leaps, this calibration is defined by

The interval on t_{R1D} develops with the linear increasing speed relation factor v/c and Δt_{R1D} is defined by Δt_{R1D} = xt_{P1D}, with x = 1,2,3… and Planck-time t_{P1D} of t_{R1D}. Δt_{R0D }is defined by xt_{P0D} with x =1,2,3… and Δt_{R} is defined by Δt_{R} = xt_{PR} with x = 1,2,3…, introducing a relative Planck time t_{P0D} for t_{R0D} and a relative Planck time t_{PR} for t_{R}. Δ is henceforth indicating leaping functions.

Superposition of speed in a moving system upon the speed of the moving system itself leads in such a diagram to the non-linear speed addition of the STR and to the absolute speed of light barrier. Using the equation of Pythagoras for

Δt_{R0D}^{2} = Δt_{R}^{2} - Δt_{R1D}^{2}

results in Δt_{R0D}^{2} / Δt_{R}^{2} = 1 – v^{2}/c^{2}

The inverse equation describes an energetic prolongation of the time t_{R} to express for example the slower energy consumption by the relativistic factor in comparison with the original time frame, if the calibration of time is not adapted according to the dilation:

Δt_{R}^{2} / Δt_{R0D}^{2} = 1/(1 – v^{2}/c^{2})

Considering the necessary re-calibration because of time dilation with the stretching of Δt_{R} results consequently in the inverse ratio for an observer on t_{R0D}:

Δt_{R0D}^{2} / Δt_{R}^{2} = 1/(1 – v^{2}/c^{2})

An observed time t_{R} is dilated in comparison to t_{R0D}. t_{R} runs in fact relatively slower than t_{R0D}, though the moving object with time t_{R} does neither leap into the past nor into the future of t_{R0D }but stays in the present, which has been proved in many experiments in circular accelerators.

It is possible to combine figure 1 and 2. The turn angles for an object have the same value. A straight movement in three-dimensional space has been transformed into a rotary process in space-time, if monitored from the central position of each event within a homogeneous, isotropic space. Although t_{R1D} is maximum dilated in relation to t_{R0D}, the rectangular construction of these two time axes seems to define the total segment of t ≥ 0, in case that any negative time development into the past of events is forbidden, i.e. any development beyond the hatched areas. The relative shifting of the zero point of events within the Planck frames can generate immense effects, because of the involved relative time dilation or time contraction. Static pressure and dynamic acceleration of processes against t_{R0D} are capable to generate simultaneity and three-dimensional space. Figure 1 shows that the length of the moving object would fully coincide with the time axis t_{R0D} in case of reaching speed of light. Defining additionally space by observable simultaneity of events transforms this length of the object into an interval on t_{R0D} that is dilated in correspondence to the relativistic length contraction. If the moving observer at the speed of light reads simultaneity across the moving vehicle, but the remaining observer of the starting environment reads sequential events within the vehicle, the picture suggests itself that we may read only simultaneity of events throughout space because of being ourselves on another track in space-time at the speed of light. This would mean a maximum dilation of our time t_{R0D} against another perpendicular time line t_{R3D} and passive speed of light on a fourth time axis t_{R2D} that was originally a length but contracted into a Planck length because of the speed of light, and simultaneously rotated into t_{R3D}. If this assumption is valid for baryonic masses and environment, rotation process will generate space distances with simultaneity of events, and t_{R3D} changes into distances with the minimum of a Planck length. t_{R0D} is maximum dilated in relation to t_{R3D} to ensure this simultaneity. This way, all t_{R2D}- and t_{R3D}-processes can be completely described within the hatched areas of simultaneity in figure 1 and 2, without backwards running time in a picture with four axes t_{R0D},t_{R1D},t_{R2D},t_{R3D}, as long as t_{R0D} runs continuously forward. t_{R2D} opposes t_{R0D} and t_{R1D} opposes t_{R3D}. Four axes that are rotated by 90 degrees are just the result of chaining consecutive and simultaneous events with individual relative zero points within the hatched areas against all other events, maintaining always t_{R0D} > 0.

This fourth axis t_{R2D} is simultaneously the 90 degrees turned speed calibrated time axis for all superimposed relative motions at active speed of light with t_{R1D} time, like photons. Gravitation, inertia, and also three-dimensional space are in this model only possible because of the hatched tidal tension areas with the tension function of t_{R2D} and the maximum acceleration of t_{R3D}-processes against t_{R0D} that generates Planck length tension and space with simultaneity of events. Axis t_{R2D} may be calibrated for the observation from t_{R0D} with a/a_{max}, introducing acceleration a, and the maximum possible acceleration a_{max} = c/t_{P} that describes acceleration from zero to speed of light during a Planck-time. t_{R0D}, t_{R3D}, and t_{R1D} are capable to describe the special theory of relativity. t_{R2D} allows the realization of space-time curvatures of the general theory of relativity. Defining t_{R2D} as an equal parameter to t_{R0D}, t_{R1D}, t_{R3D} leads to quantity replacement and dilations between t_{R1D}, t_{R2D}, and t_{R3D}.

**3. ****Conclusions**** **

** ****Figure 3: Space-time-speed-acceleration leap frame**

Space-time-speed-acceleration leap frames as shown in figure 3 should have the following features in order to entirely conform to relativistic mechanics and quantum mechanics:

The frame that is defined by subjective Planck time and Planck length has a central location for any event within space-time, in case of the homogeneous and isotropic development of space-time. The leap frame has a fixed structure and dimension for events, but differs from other frames in location and relative tension and acceleration parameters, generating simultaneous or serial events, depending on the nature and location of an observer. It is the physical basis to build up three-dimensional space by spatial separation of all simultaneous events, and of four-dimensional space-time with serial events on a subjective time line. It could also explain the generation and precipitation of matter.

The leap-frames gets conclusive evidence, considering time, length, speed, and acceleration to be combined rotated manifestations of identical types of energy components, developing in clusters either propulsion or tension energy that can be cut down to an energy quantum of action but not any further.

Evaluation of gravitational features of ± t_{R2D}-acceleration without capturing additional exchangeable features of ± t_{R0D}-, ± t_{R1D}-, and ± t_{R3D}-components misses 75% of total energy contributions that can increase the space area of simultaneity. Speed processes captured by v/c on t_{R1D} reduce this value. Cosmological analyses derive comparable 73% of dark energy.

The rotation concept implies that the observer with time t_{R0D} moves unnoticeably at the speed of light on the length l_{R2D}, causing the simultaneity of events on t_{R3D} and the three-dimensional l_{R3D}-space instead of sequential events of t_{R3D}. The idea requires two different kinds of speed of light, an active speed of light, and a passive speed of light. Any initiated process with an active speed of light leads to static processes relatively to the observation base, like in electrostatics. Passive speed of light of an observer is system immanent and can be identified by the simultaneity of extremely fast processes in subjective space, like in magnetism. The dualism of wave and particles seems to be based on active and passive speed of light processes. System immanent passive speed of light on a length l_{R2D} does not conflict with the fact that baryonic masses cannot reach active speed of light in the l_{R3D}-space, as these two axes are perpendicular to each other: They are interacting just like reactive power, for example electric tension and electric current that are 90 degrees out of phase. t_{R2D} opposes to time development by time dilation and t_{R1D} opposes to t_{R3D} by length contraction.

The transformation of translation processes in space-time into rotation processes of combined space-time-speed-acceleration views intends to stimulate the discussions about the nature of space-time and its quantum physical aspects in a complementary direction. Describing space-time with combinations of vacuum energy elements could explain the quantum mechanical effects on top of the special theory of relativity and general theory of relativity. This approach indicates hidden energy components and leap processes that may form a new basis for combining quantum physics with the theory of relativity and with cosmology. The relative change of space-time quantities by rotary symmetry supports the maintenance of a space-time continuum.