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Actual state of discussion: 11th of March 2011 

Space-Time Quadrants of Dark Energy and how Einstein missed those

© 2011, Dr. Henryk Frystacki

Cosmology assumes dark energy to explain the expansion of space with increasing speed, and dark matter to understand the stability of galaxies. The current standard model of cosmology, the Lambda-CDM model, includes the cosmological constant “Lambda” in Einstein’s formulas of general relativity to consider the expansion of space. The observations of distance-red-shift relations of supernovae gave new significance to this constant [1, 2, 3, 4]. The following discussion of diversified space-time energy describes three space-time quadrants below the event horizon of length and time that complete the present picture of space-time. One quadrant of these three energy reservoirs appears with static energy, the second one with reversed energy, and the third with dynamic energy. All four quadrants expand space with a roll-out across the entire set-up. The reversed energy field tank below the event horizon of length and time causes additionally gravity in the length-time-grid of space-time, holding the galaxies together. This roll-out process adjusts the cosmological constant and could be a natural way to derive the cosmological constant used in cosmology from quantum physics. Measurements have shown that it is in the order of about 10-120 reduced Planck units which is by factor 10-120 smaller than assumed by quantum physics on Planck scale with the completely wrong assumption of scientists that quantum vacuum is equivalent to the cosmological constant [5, 6]. This assumption turned out to be the greatest error ever in physics that has been scientifically postulated and published.

The Planck length is related to Planck energy by the uncertainty principle. At Planck scale the concept of space-time collapses as quantum indeterminacy becomes absolute. The Compton wavelength gets into the range of the Schwarzschild radius of a black hole. The framework of the quantum field theory describes fundamental interactions and forces on quantum levels but cannot yet integrate gravity in a way that all fundamental forces are unified at this Planck scale. String theories, loop quantum gravity, and non-commutative geometry are attempts to integrate gravity into the quantum field theory. This discussion of diversified energies of an extended space-time model focuses on energy transformation aspects of space-time across event horizons.

A diversified energy reservoir below the event horizon for length and time and its interactions can be already studied in a reduced two-dimensional space-time grid. A reduced two-dimensional space-time grid of the special theory of relativity uses an x-axis for length and a y-axis for time. The general theory of relativity introduces mass caused curvatures into this picture that distort this two-dimensional grid into one direction of a third z-dimension. Staying in this grid of finished length and passed time above event horizons with x and y leads to the well-known formulas of Planck length and Planck time, using the reduced Planck constant, the gravitational constant, and the speed of light.

The extension of this two-dimensional grid of positive units for finished lengths and for passed time with opposing negative values that mark energy density levels below the event horizons of length and time leads to a space-time grid with four space-time quadrants instead of one space-time quadrant with length, time, and vacuum energy density. Staying below a Planck length and a Planck time makes these negative values possible without any inversion of time or turning space inside out because they affect only the vacuum energy density, but now within a new picture with three different energy tanks. Entering completely another space-time quadrant, it shows its full dimensions and the original length-time quadrant disappears below the changed new event horizons.

The chosen geometry for these four quadrants is an extension of the space-time diagrams of Robert W. Brehme [7, 8, 9]. These diagrams have been published in the American Journal of Physics and have been only used for educational purposes. An extended view with four quadrants of space-time brings scientific research aspects into this teaching method: Three additional space-time quadrants below and at the event horizons of length and time deliver 75% of the energy contributions for the space expansion by interaction with the first space-time quadrant and baryonic masses. The (-x, -y)-space-time quadrant opposes the (+x, +y)-quadrant which generates a 25% energy effect of cold dark matter by the grid distortions. A condensation of about 4.6% (to within 0.1%) of baryonic matter and their kinetic energies reduce the share of dark energy from 75% to about 72.1% (to within 1.5%) and of dark matter from 25% to about 23.3% (to within 1.3%) if these shares are taken from the estimates of cosmology and the WMAP seven years analysis [10, 11]. A model of three space-time quadrants of dark energy below the event horizons of length and time will represent these shares accurately.


Length contraction, time dilation, and the relativity of simultaneity of events of the special theory of relativity can be derived from the diagram in figure 1. A similar type of diagram has been introduced by Robert W. Brehme. In contrary to Minkowski’s well-known two-dimensional space-time diagram, Brehme’s diagram maintains the linear unchanged scales for all observers as it actually happens. Figure 1 combines a time dilation diagram on the upper left side with and a length contraction diagram on the upper right side by the use of a common y-axis. The relativity of the simultaneity of events makes this coincidence of a length axis of an inertial frame of reference at the speed of light and the time axis of a resting observer feasible, as the further discussion will show. 


                      Figure 1: Length contraction and time dilation with four space-time quadrants  

The (+x)-axis describes a length of an intended motion direction within a mass-free inertial frame of reference of a resting observer, subdivided into Planck length distances. The (+y)-axis shows the time on a clock of this resting observer, running at the same steady pace in each position on the x-axis. The right side of figure 1 describes the length contraction by projections of rotating lengths onto the x-axis and the caused change of simultaneity of events by the projection of this rotating length onto the y-axis. A contraction will be read differently by the resting observer or by the moving observer: The resting observer notices a reduction of the size of the moving inertial system, and the moving observer notices a reduction of the original space distance to the target. The rotation angle has to be matched with the relative speed increase of the moving observer. For this purpose any freely chosen length is rotated from the x-axis into the y-axis and then calibrated with a linear superimposed speed scale from 0 up to the speed of light, as demonstrated in figure 1. The Pythagoras formula leads to the correct length contraction values as x-projections on the x-axis and to the correct readings of the change of simultaneous events on any rotated x-length into serial events for the resting observer by projections of the rotating length onto the y-time axis.

Using the Pythagoras formula

                                                       results in the length contraction:


The length l is any unaltered length in the moving system, lv/c the projection value on y if we rotate l fully into y and stay with its length calibration. x reflects the reading of the length l by the resting observer on x by comparison with the unchanged x-scale, and at the same time the reading of the moving observer of the shrinking distance towards the target in space. Note that we have neither shifted the coordinates of the moving object along x due to the speed, nor the present time of the resting observer along the y-axis, indicated by the zero point: The superposition of two diagrams in the starting origin of a measurement simplifies the discussion of the energetic impacts. The zero point of each new measurement leaps for the resting observer along the y-axis due to the progress of time if the initial frame of reference of the resting observer serves as a frozen reference. A monitored length of the moving frame of reference leaps additionally Planck length by Planck length along the x-axis. However, this is not necessarily increasing the distance to the resting observer as it is possible to change the heading after each Planck time into any direction, even into an orbit around the resting observer. In case of translation along x of a Cartesian system, the zero-x-y-coordinates of the moving frame of reference shift in the view of the resting observer between the y-axis (zero Planck length per Planck time) and the bisector of x- and y-axis (speed of light with one Planck length per Planck time).

The Pythagoras formula can be applied in an identical way for the left side of figure 1. This way, the projection of an original time interval on the y-axis that rotates now from y to (-x) shows the reading of a decreasing time interval for a moving observer in relation to the resting observer: The time of the moving observer has been dilated in relation to the resting observer, causing the well-known fact that the moving observer reads a shorter time span between events in the original environment than the resting observer does. The stretched time scale of the moving observer is merely characterized by the reciprocal factor:


Speed stays constant for both observers within their inertial frame of reference because length and time change by the same ratio. Speed vectors can be easily added in the rotational construction across several inertial frames of references at different relative speed. The angle between the own time axis and the own length axis of any observer maintains always 90º. The energy tanks below event horizons for length and time make this possible. The (-x)-axis reflects the strength of time dilation that accommodates all events of any original time interval below the event horizon of time, i.e. without progressing y-time of the resting observer. The change of simultaneity for a resting observer by observation and rotation of a constant length l from the x-axis to the y-axis can be described by the formula part of relativistic mechanics for time intervals in moving systems [12]:

     The term    compensates the relativistic

length contraction:


that a resting observer will monitor, in other words it keeps the value of the contracting length l constant and equal to the initial x-value. One speed of light factor c changes this length value into a corresponding time difference, or, more precisely, into a corresponding time delay Δt of an event. The remaining quotient v/c reflects the projection value onto the y-axis. This projection is linked to the rotation angle that is ruled by this quotient v/c. 

This rotational picture is completed by the fourth axis (-y) that opposes the y-axis in the same way that (-x) is opposing x. This (-y)-axis can be understood as a clockwise rotated axis in relation to the x-axis, or anticlockwise rotated axis in relation to the (-x)-axis. Its value stays below a Planck time in relation to the y-axis, just like (-x) stays below a Planck length in relation to the x-axis. The (-y)-axis reflects the strength of a length contraction that accommodates all events on any original length below the event horizon, i.e. actually not being a length with noticeable events for the resting observer. (-x) and (-y) are neither a length nor a time in the environment of any observer with x-length and y-time but summarize all energy processes of the space-time grid below the event horizons for length and time on base of action and reaction and rolled up or out via the rotational construction. The (-y)-axis shows maximum accelerated processes below the event horizon for any observer with an existing x-length in the same way that x shows maximum acceleration of processes below the event horizon for any observer with an inertial frame of reference at the speed of light in relation to its origin with x-length and y-time. The rotational construction shows that space-time frames of any size and without any rest mass can only reach a maximum of relative speed of light.

The three space-time quadrants below the event horizon of length and time deliver with their energy tanks 75% of the energy contributions for the roll out of space on top of all interactions of the (+x, +y)-quadrant with the baryonic masses. The (-x, -y)-space-time quadrant opposes this (+x, +y)-quadrant and generates a 25%-energy effect of cold dark matter by the grid distortions. A condensation of about 5% of baryonic matter energy and their kinetic energies reduce this share of dark energy from 75% to about 72% and of dark matter from 25% to about 23%. This could be explained by an energy origin of baryonic matter of about 1% contributed by the (-x, -y)-quadrant, 1% by the (-x, +y)-quadrant, 1% by the (+x, -y)-quadrant, and 1% by the (+x, +y)-quadrant. This precipitated baryonic mass is entirely attributed to the (+x, +y)-length-time-quadrant. The fifth seeming percentage point of baryonic matter can be explained by a 2% reduction of the dark matter share if the (+x, +y)-share and the (-x, -y)-share of precipitated masses are both considered in the space-time curvatures of the (+x, +y)-quadrant that are caused by the presence of these masses. The mentioned rounded off mass share corresponds to the current WMAP estimates [11]. The measurement of 4.6% for masses leads with this distribution key to 72.24% dark energy and 23.16% dark matter. The latest WMAP-results show the difference of only 0.14% shifted from dark energy to dark matter. The total rest mass rotates summarizing length-time axes of coordinates anticlockwise in figure 1 just like kinetic energy and any other form of energy according to the principle of equivalence. Einstein’s four-dimensional space-time requires a causally related distribution. The general theory of relativity captures only this anticlockwise rotation together with the local contractions by Ricci-tensors and relativistic field equations. It does not consider a basic set-up of space-time alone as a vacuum energy construction with an overall distortion pattern that is caused by its four quadrants. Figure 1 demonstrates that space-time alone, i.e. without precipitation of masses, is already rolling out space and embeds the precipitated galaxies of masses with a dark cold matter effect of the (-x, -y)-quadrant.

All four quadrants expand space with a roll-out across the four axes. As opposing axes are completely below the event horizon of each other and successive axes of the rotational construction exactly at the event horizon of each other with quantized release of energy, the whole potential energy of each axis and the opposition of quadrants have to be considered for a balanced roll out of length and time. Each roll out of a Planck length affects the whole construction, adjusting the cosmological constant far below the value of models where quantum vacuum is equivalent to the cosmological constant. The measured cosmological constant is by a factor of 10-120 smaller, indicating distributed energies and the slowdown of space inflation by a rollout across interacting space-time quadrants.


Length appearance, length contraction, time appearance and time dilation can be connected with energy tanks below the event horizons for length and time. These energy tanks seem to cause dark energy and dark matter of the current models of cosmology. Planck length and Planck time are the observations of a finished process above the event horizon. They form a quantum grid on base of Planck energy and Heisenberg’s uncertainty principle. The (+x, +y)-field reacts to the presence of masses with (-x, -y)-caused distortions as described by the general theory of relativity.

The rotational construction of an extended space-time makes it possible to talk about an energy grid that accommodates the effects of the general theory of relativity and quantum mechanics, but also the phenomena of dark energy, dark matter, and baryonic matter. The mass of baryonic matter can be interpreted as a quantized interaction with the (-x, -y)-tank, making the (-x, -y)-area a candidate for a mass background field [13, 14]. There is no need for a Higgs Boson, as masses are based in this model on an interaction of partially rotated space-time zones with their reverse tank, being subject to grid distortions. The static sector delivers the electric charge of an elementary particle or of a nucleus formation, the dynamic sector the magnetic spin. The time dilation has been covered by the special and general theories of relativity as only one function of the (+y,-x)-tank. The split into static (+y,-x)-tank, reverse (-x,-y)-tank, and dynamic (+x, -y)-tank are new aspects for the discussions about space, time, matter, and vacuum energy density on Planck scale. The cosmological constant reflects in this model the constant negative pressure of the adiabatic roll out of space acting as repulsive force to gravity.



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