II. Event connection in multi-dimensional complex Minkowski models

One major feature of the Einstein equation, E = mc², is the "connecting quantity," the velocity of light squared or c². Einstein, A. Lorentz and H. Minkowski developed the description of the relationship of space and time. This relation is given as S² = x² - c² t² in which X = (x,y,z) represents the three components of space and the component of time, t, is related again by the velocity of light square, c². This equation is the four-dimensional equivalent of the Pythagorean theorem (the squares of the sums of the sides of a right triangle are equal to the square of the hypotenuse), except in this general form, time as the fourth dimension, is included. The quantity, S. is denoted as the spacetime "distance" in four "space", consisting of three spatial dimensions and one temporal dimension. This distance is an invariant or constant.

We can use this relationship to clearly define causal events in spacetime, in which we can use the Minkowski diagram or "light-cone" to define this relationship. The limitations on the cause-effect relationships is defined by one of Einstein's axioms, that is, that no material object can exceed transmission at greater than the velocity of light. The light cone then demonstrates which events can have a connection to what causes.

Although one of Einstein's axioms in special relativity denies physical matter access to the space-like region, or v > c, the structure of his theory does not preclude such a connection, as pointed out by Gary Feinberg at Columbia University in 1967⁽⁷⁾ Only a specific type of massive particle or signal termed a "tachyon" (which means swift in Greek) can exist for v > c signals. This type of particle has an imaginary mass, that is, a mass multiplied by

. Although no physical tachyon has been discovered, the use of imaginary numbers and quantities is very useful and is prevalent in physics as long as the final physical result is real. We will make use of imaginary dimensions in our model, as is done in analyticity or causality conditions in elementary particle physics.

In order to modify the causal connections of the spacetime metric, S² = x² - c²t², we need to introduce a set of complex dimensions. What we propose to do is introduce four real dimensions and four imaginary dimensions in such a way that locations or events, which are distant in four "space", appear Juxtaposed in our multi-dimensional space. The eight space model can be utilized in such a way as to make two events (which we can denote Event A and Event B) appear simultaneous, or continuous, ant can be described in eight space.

We represent the possible ways we may "mix" time and space dimensions to make this possible in Figure 1.

In relativity causal connection of events are described by the invariance of the Minkowski line element which, in special relativity is given as

S² = /x/² - c²t² (1)

where c is the velocity of light, where the exchange of information between an event cause and event effect are limited by the velocity of light.

It is proposed that the ordinary Minkowski four-space (three spatial, one temporal coordinates) might simply be the real part of an eight dimensional complex spacetime or that real space is a projective slice through a complex eight space! For this generalized coordinate model we let the spatial coordinates x¹ ® x + iX, and similarly for time, t¹ ® t + ii, where i =

and x¹ and t¹ are real numbers. Analogous to the expression for the square of the distance between two points in Minkowski 4-space, we thus transform from real (x,t) space to complex (x¹,t¹) space.

We then introduce complex spacetime-like coordinates as a space-like part X and time-like part T as imaginary parts of x ant t. Now we have the 8-space Minkowski invariant line elements⁽⁷⁾ as,

S² = /x¹/² - c²/t¹/² (2)

for

x¹ = x + iX (3a)

and

t¹ = t + ii (3b)

Figure 1.

In the complex space multi-dimensional model, we introduce, in addition to the usual orthogonal four space, four imaginary components, three spatial and one temporal. This is necessary in order to model remote event connection, and to retain the causality and symmetry conditions in physics. as our complex dimensional components.
Then,

x'² - /x'/² = x² = x² (4a)

and

t'² /t'/² = t² + r² (4b)

Recalling that the square of a complex number is given as

/x'/² = x'x'* = (x + ix)(x - ix) = x² + x² (5)

for x and X real. Therefore, we have for the 8-space line element, in which the metrical line element S² is real,

S² = x² c²t² + X² - c²i² (6a)

Causality is defined by remaining on the light cone, in real spacetime, as

s² = x² - c²t² = x² t² (6b)

We can use atomic units so that we have the condition c =1 for simplicity, then generalized causality in complex spacetime is defined by

s² = X² = t² + X²

in the x, t, X, i generalized light cone 8-dimensional "space".

Let us calculate the interval spatial separation between two event occurrences (x₁,X₁) ant (x₂,X₂) with real spatial separation Dx = x₂ - x₁ and imaginary spatial separation DX = X₂ -X₁ and real temporal separation Dt = t₂ - t₁ to and imaginary temporal separation Di = i₂-i₁. Then the distance along the line element is Ds² = (x² + x² + t² -i2) and it must be true that the line interval is a real separation. Then for the choice of units in which c = 1,

Ds² = (x₂ -x₁)² + (X₂ - X₁)²- (t₂ -t₁)² - (i₂ - i₁)² (8)

Ds² = (x₂ -x₁)² + (x₂ - x₁)² for the space part (9)
- (t₂ - t₁)² - (i₂ - i₁)² for the time part

Because of the relative signs of the real and imaginary space and time components and in order to achieve the causality connectedness condition between the two events, of Ds² = 0, we must "mix" space and time. That is, we use the imaginary time component to effect a zero spatial separation and we use the imaginary space component to effect a zero time separation.^(7,11) We identify (x₁,t₁) with a receiver event (Event B) remotely perceiving information from an event target (x₂,t₂) (Event A).

Figure 2 represents the case where there is no time element, or Dt = 0, and in which Dx

0. Then in the simplest causal connection in which DX = 0,

Ds² = 0 = (x₂ - x₁)² - (x₂ - x₁)² (10)

By introducing the complex time component, one can achieve a condition in which the apparent separation in the real physical plane defined by x, t is zero, given access to the imaginary time in the X, i plane. Note that the x,t plane defines the usual light cone relations.

The causality condition in 8-space is then the full 8- space metric

Ds² = 0 = (x₂ - x₁)² - (i₂ - i₁)² - (t₂ - t₁)² - (X₂ - X₁)² (11)

We can no longer represent this as a three-dimensional diagram (in a two-dimensional figure). We can rewrite the above expression as where the left side relates to that aspect of remote connection in space ant the right site refers to the time event correlation. The complex time and space in the 8-space representations make appear to that modality that one can access distance information as though it is not distant ant access information about future events as though it were being accessed in current time in the present.

(X₂ - x₁) - (i₂ -i₁)² = (t₂ -t₁)² - (X₂ -X₁) (12)

Figure 2.

Complex time model of remote connectedness. We have the usual physical spatial separation of events on the x axis in the xt plane which appears separated by a zero separation by "moving" in the Xi plane.

In fact, we cannot define the velocity of light or any physical velocity as unity, c = 1, which is often done for convenience of calculation. We have worked out the details of the connection velocities for both the real time and "precognitive" or future time remote connection cases (see Figure 3) for the real time velocities. Since velocity is a derivative quantity of space with respect to time (such as miles per hour), we must calculate, in general complex derivative forms for V_ti and V_xi which vectorially add to the real velocity V_xt.⁽⁷⁾ We can experimentally examine the time delay in spatially separated events between the Target Event B and Receiver Event B (see Figure 2).

Possible correlation of events may be possible over planetary distances, using space exploration vehicles. One possible remote phenomena to examine, in this regard, is Mach's principle (see Ref. 16). Also we have examined in detail a "holographic" motel of perception.⁽¹⁷⁾

We will now turn our attension to the remote connectedness properties which may occur in the quantum domain. We will examine the properties of solutions to the Schrödinger equation in the complex eight-dimensional space.

Figure 3.

For real time "remote connection", we represent the relationship of velocities of signal propagation 'or the case of an event A "observing" an event B. accessing information about in the complex eight-space representation. From the prospective of event C, event A ant B are contiguous; that is, they appear as if they are not spatially separated. For real time remote connections, a five-dimensional space representation is adequate. In compact notation, where each set of three spatial dimensions is taken as a single spatial dimension, this representation becomes a three-space.⁽¹⁰⁾ (XBL 783-2441)