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Imaginary Relativity is a idea which completes Einstein's Theory of Relativity Theory. According to the recommended idea, the speed of light is not constant and absolute in an inertial frame. On the other hand, the speed of a light pulse in an inertial frame, moving with velocity u, equals to an imaginary value instead of the constant value c as it is considered in Einstein's Theory of Relativity Theory.

This idea first was published “Journal WSEAS TRANSACTIONS on COMMUNICATIONS, Volume 9 Issue 2, February 2010[1] by Mehran Rezaei.

## Introduction

Special Relativity theory is a way which helps observers in different reference frames to compare the results of their observations. This theory provides the possibility to express the laws of physics by using mathematical methods in different reference frames. The necessity for a new survey of fundamental quality of time and space is one of the results of this possibility. Using Lorentz Transformations and supposing a varying time as well as a constant speed for light, Einstein constructed equations leading to a series of wonderful results such as time expansion, length contraction, mass change due to speed change and the equivalency of mass and energy. These concepts have been experienced accurately in different ways so far and the special relativity theory has been approved in all of the experiments. Today, there are some observations which confirm speeds over the speed of light. Scientists in NEC Company succeeded to increase the speed of a single ray higher than the speed of ordinary light[2]. In this experiment, they pass a light ray through a cesium atomic chamber prepared specifically for the experiment. The light ray reached to the end of the chamber in 62 nanoseconds earlier than it does in regular conditions. According to a theory, in processes that messages moves with a speed higher than the speed of light, there is possibility for the particles called “Tachyon” to exist[3]. Beside these, there are other experiments and experimental samples provided by researchers from different parts of the world that can be easily searched and studied on the web. There is no doubt that accepting these velocities and theories are one of the strangest phenomena of modern physics, because according to the [relativity theory]] no physical process can be occurred in the velocities upper than the speed of light. Is it really possible to have velocities upper than the speed of light? Most scientists take a defensive reasoning when they face these real events. They say that at velocities upper than the speed of light, systems behaves erratically, that is why the speed of light can be good criteria to evaluate the rightfulness of other measurements. Therefore, if we observe that the velocity of a phenomenon is higher than the speed of light, we should look for the origin of error in our experiment. Imaginary Relativity idea which is discussed in this paper intends to indicate that the speed of light can be higher than 2.988×〖10〗^8 m/s, but as an imaginary number. On the other hand, it is supposed that in an inertial frame, the speed of light is not constant and absolute and it depends on velocity of the frame imaginarily. All relations and equations resulted from this theory is imaginary and as we will see in continuation, they have two main characteristics:

• The real part in imaginary relations is the same as classical value at low speed limit.
• The normalized value of imaginary relations is the same as Einstein’s Special Relativity relations at high speeds.

These results can be achieved using Lorentz new transformations which will be proven in this paper. These transformations open a new sight to a new definition of relativity which is called here Imaginary Relativity.

## The Imaginary Speed of Light

Consider inertial frame S' moving with velocity u and acceleration zero. In this frame, a light pulse is emitted in direction u with velocity c. What is the speed of light according to an observer located in frame S'? The answer, according to Einstein’s second postulate, is c. However, the answer according to imaginary relativity theory is different. It is assumed that velocity of frame S' affects imaginarily on the light pulse. It means that an imaginary frame is formed so that its imaginary part is u and its absolute (normalized) value is c. According to figure 1:

$C = \sqrt{c^2-u^2}\pm iu \,$



</p>

(1)

(The positive or negative signs represent the motion of light pulse in the opposite or the same direction of the frame respectively).

The absolute value of C (|C|) always equals c. This can be considered as a required condition for having a constant value for the speed of light in an inertial frame. It should be noticed that the speed of light has changed because of the imaginary effect of the frame velocity and includes a real part (√(c^2-u^2 )) and an imaginary part (u). Considering the relation between u and c in equation (1): If u=0, then C=c . It means that in a rest frame, the speed of light is always a constant value c. If u=c, then C=ic . It means that light has no physical effect in the real world as if it has been disappeared. If u>c, then C=i√(u^2-c^2 )±iu=iu(√(1-c^2⁄u^2 )±1). In this case, such as before case, C is entirely imaginary with an unknown effect in the real world. Using Euler formula, equation (1) can be written as follows:

\begin{align} C &= ce^i\theta \end{align} (2)

Where

θ=tan^(-1)⁡〖(u/c)/√(1-u^2/c^2 )〗   (3)


Variations of C versus u in figure 2 indicate that as much as the frame velocity increases, C approaches to behave more and more imaginary. On the contrary, as much as u decreases, C approaches to behave more and more real.

Therefore equation (1) indicates that velocity of a light pulse in a moving frame is C and not c. This is against the first postulate of Einstein. On the other hand, an observer who moves in a moving frame with a constant acceleration measures the velocity of the light pulse as imaginary. It is the same theory which defines and discusses as Imaginary Relativity in here. According to his postulates,Einstein claimed that an observer, who moves in a moving frame with a constant acceleration, never can set an experiment to realize whether the frame is moving or not. But, according to Imaginary Relativity theory, if observer can measure imaginary velocity of a light pulse, then he/she can realize if he/she is in move or rest using equation (1). Unfortunately, an observer never can feel an imaginary speed in his/her world just because of being located in real world. He/she always feels the effect of both real and imaginary speeds (in other words, the normal value of equations of imaginary relativity) simultaneously. Therefore, as it can be seen, absolute or normalized value of equation (1) equal c. It is the same as it is stated by Einstein’s special relativity postulates.

## Consequences of imaginary speed of light in special relativity

In this section, one of the important consequences of Imaginary Speed of Light in Special Relativity in space and time fields will be discussed and its results will be compared with special relativity equations. In the next sections, time and space fields will be investigated by observers located in different reference frames. It should be mentioned here that C stands for a complex number in equation (1) and c stands for absolute value c (c=2.988×〖10〗^8 m/s).

### Lorentz Transformations

Here, the aim is to see how Lorentz Transformations are changed when the speed of light is considered imaginary for a moving observer. These transformations help us to compare the measurements of space-time coordination faster for two observers who are moving relative to each other.

#### Space-time Transformations

Let us consider two observers A and B. Observer A sees an event in reference inertial frame S and determines its time-space coordination as x ,y ,z and t . In another inertial frame S', observer B who moves with velocity u, sees the same event with time-space coordination as x^',y^',z^' and t'.The most general figure for the transformation equations of the two observers , observing the same event at the same time is as follows:

x^'=a_11 x+a_12 y+a_13 z+a_14 t
y^'=a_21 x+a_22 y+a_23 z+a_24 t
z^'=a_31 x+a_32 y+a_33 z+a_34    (4)
t'=a_41 x+a_42 y+a_43 z+a_44 t


Now, the problem is to find the sixteen unknown coefficients in the above equations. Many books explain how the above equations can be solved[4]. Therefore, the complete solution is avoided here and just the simplified results of the solution as well as the important coefficients are considered as follow:

x^'=a_11 (x-ut)
y^'=y
z^'=z                (5)
t^'=a_41 x+a_44 t


Now, It is enough to find the coefficients a_41 ,a_11 and a_44 . To do this, it is supposed that at the time t=0 a spherical electromagnetic wave leaves the origin S which coincides origin ' . This wave is emitted in inertial frames S and S' with velocities c and C respectively. Therefore, development of the wave for each of the frames is characterized by the equation of a sphere which its radius is increased by increasing the rate of light speed versus time:

x^2+y^2+z^2=c^2 t^2 (6)


And

〖x'〗^2+〖y'〗^2+〖z'〗^2=C^2 〖t'〗^2   (7)


By replacing x', y^'and z' with their values from equation (5) in equation (7):

〖a_11〗^2 〖(x-ut)〗^2+y^2+z^2=C^2 〖(a_41 x+a_44 t)〗^2


And after rearrangement of the above equation:

(〖a_11〗^2-C^2 〖a_41〗^2 ) x^2+y^2+z^2-2(u〖a_11〗^2+C^2 a_41 a_44 )xt=(C^2 〖a_44〗^2-u^2 〖a_11〗^2)t^2


Whereas the above equation is equivalent with equation (6), their equivalent coefficients should be equaled to:

C^2 〖a_44〗^2-u^2 〖a_11〗^2=c^2
〖a_11〗^2-C^2 〖a_41〗^2=1
u〖a_11〗^2+C^2 a_41 a_44=0


Now, there are three equations with three unknowns. The result of solution of the above equations is as follows:

a_44=c/C/√(1-u^2/c^2 )
a_11=1⁄√(1-u^2⁄c^2 )              (8)
a_41=-u/Cc/√(1-u^2/c^2 )


By substituting the above results in equation (5), new equations for Lorentz Transformation under ‘’’Imaginary Relativity’’’ Theory will be formed as follow:

x^'=(x-ut)/√(1-u^2/c^2 )
y^'=y
z^'=z                              (9)
t^'=((c/C)t-(u/Cc)x)/√(1-u^2/c^2 )


As it can be seen, the equations which are related to place have not changed and just the equation related to time has changed. On the other hand, length Contraction is invariant under Imaginary Relativity Theory. But simply, it can be proved that Time Dilation is imaginary equation .So that:

t=Ct'/(c√(1-u^2⁄c^2 ))             (10)


This equation is the same known equation as Time Dilation except for the coefficient C/c . This coefficient indicates that Time Dilation is an imaginary phenomenon. To make it clear to understand, It is enough to substitute the value of C from equation (1) in equation (10)

t=t'±it'(u⁄c)/√(1-u^2⁄c^2 )        (11)


Equation (11) indicates that Time Dilation includes two values: an absolute real value which is the same measured value by observer S' and an imaginary value which expands with the coefficient( (u⁄c))⁄√(1-u^2/c^2 ). (Negative or positive sign depends on the direction of). The normalized value of t (distance of t from the origin of the coordination) is:

|t|=t'/√(1-u^2/c^2 )               (12)


This is the same equation as Einstein’s equation for Time Dilation.

#### Transformation of Velocity

For a special case, It is considered that all velocities are in the same direction (x-x') of two reference inertial frames S and S'. It is supposed that frame S is earth and frame S' is a train moving with velocity u relative to earth. The speed of a passenger located in train (frame S') is v' and its place in train versus time is calculated as x^'=v't' .The aim is to calculate the speed of the passenger relative to earth. Using Lorentz Transformation Equations (9) and replacing x' and t' with their values:

x-ut=v^' [(c/C)t-(u/Cc)x]


The above equation can be written as follows:

x=((c/C) v^'+u)/(1+v^' u/cC) t


If the velocity of the passenger relative to earth is shown with v, its place relative to earth during time t is earned from equation x=vt . Therefore, the above equation can be written as:

v=((c/C) v^'+u)/(1+v^' u/cC)       (13)


Whereas the velocity transformation is in the direction x-x' , an index x is added to v and v':

v_x=((c/C) 〖v^'〗_x+u)/(1+v^' u/cC)  (14)


The result will be more complicated if direction of the velocity is perpendicular to the length of relative movement. In a similar way [2], it can be proved for direction y-y' that:

v_y=(C/c)  (〖v^'〗_y √(1-u^2⁄c^2 ))/(1+〖v^'〗_x (u/c^2 ) )     (15)


And direction z-z' is

v_z=(C/c)  (〖v^'〗_z √(1-u^2⁄c^2 ))/(1+〖v^'〗_x (u/c^2 ) )    (16)


To get the parameters of velocity measured in frame S in terms of the parameters of velocity measured in frame S', It is enough to change the lower indices in equations (14), (15) and (16) and to replace –u with u. In Table 1, a comparison has been made between Lorentz transformation under c and Lorentz new transformation under C.

### Relativistic Mechanics

At this section, the dynamic concepts of energy and momentum from the perspective of Imaginary Relativity Theory will be discussed.

#### Relativistic Momentum

Let us start discussion in this section with classical equation of momentum P=mV. However, before doing that it is necessary to define proper time. Proper time τ is defined as the measured time by a fixed clock which shows the time of two events in a frame [3]. For instance, if a fixed clock in frame S records two events in the frame, then t'=τ and the equation (10) changes as follows:

t= Cτ/(c√(1-u^2/c^2 ))     (17)


It should be noted that proper time is always the minimum time difference measured between two events. Unfixed clocks always measure longer time difference. Therefore, if proper time τ is used instead of ordinary time t, a justified calculation for relativistic momentum can be made as follows:

P=m dx/dτ=mdx/dt  dt/dτ


As it is accepted in classical mechanics, u is considered as u=dx/dt. Although none of the observers comes to an agreement about dx/dt, they agree about dx/dτ, in which proper time dτ has been measured by a moving object. dt/dτ is obtained from equation (17). In this equation, u stands for the velocity of the moving reference frame relative to the fixed frame. Replacing dt/dτ with its value from equation (17):

P= muC/(c√(1-u^2⁄c^2 ))     (18)


Equation (18) presents a new definition of momentum as an imaginary phenomenon which can be called ‘’’Imaginary Momentum’’’. It should be noticed that the new equation for low values of u/c leads to the classical equation for momentum as P=mu . To keep the appearance of the new equation like the classical one, the mass appeared in equation (18) is defined as stationary mass m_°. Therefore, Imaginary relativistic mass can be defined as follows:

m= (Cm_°)/(c√(1-u^2⁄c^2 ))         (19)


Equation (19) is imaginary mass and indicates how relativistic mass of an object, moving with velocity u, changes according to variations of u. It can be resulted immediately that m=m_° (stationary mass) when u=0 (object is at rest). Generally speaking, m→m_° when u/c→0 . It indicates that Newtonian limit of relativistic mass m is m_°. Besides, statement C/c in the equation shows imaginary nature of m which can be written as follows (by replacing C with its value from equation (1)):

m=m_°+im_°  (u⁄c)/√(1-u^2⁄c^2 )    (20)


As a matter of fact, the real part of m always equals m_° which is a constant value. However the imaginary part of m increases with the rate of (u/c)/√(1-u^2/c^2 ) as u increases. Like prior equations, absolute value of m (|m|) is the same as mass in Einstein’s relativity theory:

|m|=m_°/√(1-u^2⁄c^2 )         (21)


Albert Einstein did not show much interest in equation (21). He was worried that accepting relativistic mass m invalidated the concept of mass as an inherent and invariance characteristic. Perhaps, equation (20) can heal this solicitous a little (at least because of existing m_° in the real part), and can give us a strong perspective of mass. However, it should be stated that today, “relativistic mass” and “rest mass” are old expressions.

#### Relativistic Energy

After recommending a new definition for linear momentum (equation (18)), it is time to concentrate on force and energy. According to definition of kinetic energy (which is defined as done work on a particle):

W_12=∫_1^2▒〖F.dr〗=K_2-K_1     (22)


Equation of second law of Newton can be corrected in order to include a new definition of linear momentum:

F=dP/dt=d/dt (mu)      (23)


In which, m is relativistic mass. If the movement starts from the rest situation (K_1=0), and velocity is on the same length as force is, F in equation (22) can be replaced with its value from equation (23):

W=K=∫▒〖d/dt(mu)×udt〗=∫_0^u▒ud(mu)   (24)


(It should be noticed that dr=udt). By integrating to equation (24):

K=mu^2-∫_0^u▒mudu    (25)


Replacing m with its value from equation (20) and integrating to the second part of the equation:

K=u^2 m_° (1+i (u⁄c)/√(1-u^2⁄c^2 ))-∫_0^u▒〖um_° (1+i (u/c)/√(1-u^2/c^2 ))du〗


After doing algebraic and simplification operations:

K=1/2 m_° u^2 [1+i((u/c+c/u)/√(1-u^2/c^2 )-c^2/u^2   sin^(-1)⁡〖u/c〗 ) ]  (26)


Equation (26) is named ‘’’Imaginary Relativistic kinetic Energy’’’. As it was expected, the real part of the equation is the same as equation of classical kinetic energy (1/2 m_° u^2). It is just the imaginary part which may seem a little complicated. For low speeds, u≪c, the value of imaginary part approaches to zero and K approaches to the same result as classical equation. Physicists believe that momentum is a more fundamental concept than kinetic energy (for instance, there is no law for conservation of kinetic energy). Therefore, the equation which relates mass and energy should include momentum instead of kinetic energy. Starting with equation (18) for momentum:

P=m_° u(1+i (u⁄c)/√(1-u^2⁄c^2 ))
P^2 c^2=〖m_°〗^2 u^2 c^2 (1+i (u⁄c)/√(1-u^2⁄c^2 ))^2
P^2 c^2=〖m_°〗^2 c^4 (u^2/c^2 ) (1+i (u⁄c)/√(1-u^2⁄c^2 ))^2    (27)


If it is supposed that γ=1+i (u⁄c)/√(1-u^2⁄c^2 ) , then:

u^2/c^2 =1-1/(γγ ̅ )       (28)


Knowing that γ ̅ is complex conjugate of γ , and replacing u^2⁄c^2 from equation (27) with its value from equation (28):

P^2 c^2=〖m_°〗^2 c^4 (1- 1/(γγ ̅ )  )γ^2


If Eand E_° stands for total energy (mc^2) and rest energy (m_° c^2) respectively, after algebraic operations and replacing equivalent values, the above equation can be written as follows:

E^2=(E_°  C/c)^2+P^2 c^2    (29)


Equation (29) is a very useful equation in dynamics, because it connects the total energy of a particle to its momentum and kinetic energy. For a Photon which has no rest mass, equation (29) is written as follows:

E=Pc   (30)


It means that the total energy of photon is just because of its motion. Besides, C/c in equation (29) indicates that the value of E is imaginary. In general, E_° C/c can be defined as energy E_C as follows:

E_C=E_°  C/c=m_° c^2  C/c
E_C=m_° cC   (31)


Equation (31) is called Imaginary Energy. Therefore, equation (29) can be written as follows:

E^2=〖E_C〗^2+P^2 c^2    (32)


Comparing this equation with general equation:

E^2=〖E_°〗^2+P^2 c^2     (33)


Indicates that in spite of E_C , E_° as rest energy is independent of the velocity of object. If replacing C in equation (31) with its value from equation (1):

E_C=m_° c^2 (√(1-u^2⁄c^2 )  +i u⁄c)    (34)


According to the author of this paper, E_C has a better and more general concept than E_° . In Einstein’s equations, each object has the rest energy m_° c^2 which is completely independent of its velocity. It means that if the speed of an object increases with a constant acceleration, there will be no change in the energy. The total energy should be calculated from equation (33). However, equation (34) contradicts this statement. According to this equation, if the speed of an object increases, the real value of the object’s energy (E_C) decreases, but its imaginary value increases. If the object moves with the speed of light, the real value of the energy reaches to its minimum (zero) and on the contrary, the imaginary value of energy increases to its maximum (one). In this case, the equation (33) changes as follows:

E_C=im_° c^2      (35)


That is completely imaginary. If the object is in absolute rest (u=0), then:

E_C=m_° c^2       (36)


Which is the samerest energy (E_°) appears in Einstein’s equations. It is necessary to be reminded that the normalized value of imaginary equation (34) leads to the rest energy of Einstein’s special relativity equation:

〖|E〗_C |=m_° c^2=E_°    (37)


The result was expectable such as prior imaginary equations. Besides, E to E_C ratio always equals:

E/E_C =√(1-u^2⁄c^2 )    (38)


The interesting point is that the above ratio is exactly the same as E to E_° ratio in special relativity equations.

## Consequences of imaginary speed of light in quantum physics

In this section, quantum concepts from the perspective of ‘’’imaginary relativity’’’ will be examined and it will be shown that these concepts are simpler and more convincing in confrontation new reasoning.

### Wave properties of particles

In classical physic, there are fundamental differences between the laws of wave and particle motions. Projectiles’ motion is based on the laws of particle dynamics and Newtonian mechanics, but wave’s motion is subjected to interference and diffraction and cannot be explained with [Newtonian mechanic]]s. Each particle carries energy which is limited in a small region of space. However, wave propagates its energy out to the space in different wave fronts. Considering these clear differences, Quantum Theory is a try to justify and to make a relationship between particle and wave natures. Although it seems to be incompatible logically, it should be confessed that it is practicable using ‘’’Imaginary Relativity’’’ ideay which provides us with a combination of particle and wave behaviors simultaneously. In 1924, Louis de Broglie in a boldly hypothesis presented in his PhD dissertation stated that not only light but also all matter has the duality nature of particle-wave [5]. Having no experimental background, de Broglie suggested that for every object moving with momentum P , there is a related wave with the length λ according to the following equation:

λ=h/P        (39)


Where h is Plank constant equals to 6.626×〖10〗^(-34) Js and λ is wave length of object or de Broglie wave length. By replacing P with mu :

λ=h/mu       (40)


Where m is imaginary mass. Substituting m for its value from equation (20):

λ=h/(m_° u(1+i (u⁄c)/√(1-u^2⁄c^2 )))     (41)


And after algebraic simplification of the above equation:

λ=h/(m_° u) (1-u^2/c^2 )(1-i (u⁄c)/√(1-u^2⁄c^2 ))   (42)


Equation (42) opens a new perspective of the ratio wave length to mass and velocity. As it can be seen from prior equations, the real part of the equations indicates the classic or low-velocity region and the imaginary part presents imaginary or high-velocity region. Therefore, it is acceptable to suppose that big-mass objects at low velocities pose wave length calculated as follows:

λ=h/(m_° u) (1-u^2/c^2 )   (43)


Louis de Broglie] had the same supposition, but his recommended formula was the equation (57). Whereas the statement 1- u^2⁄c^2 at low speeds nearly equals one (1), it is expected that the results of equations (43) and (40) are equal in the velocity confine. Considering equation (1), the equation (42) can be written as follows:

λ=hC/(m_° uc) √(1-u^2/c^2 )    (44)


Such as other equations and principles of Imaginary Relativity, it is expected that the normalized value of the above equation equals to Special Relativity equation. After algebraic operations and simplification:

|λ|=h/(m_° u) √(1-u^2/c^2 )    (45)


And by considering equation (21):

|λ|=h/|m|u                     (46)


Equation (46) is the same as De Broglie's formula with relativistic mass. The question which arises here is that if it is accepted that each moving particle has a wave with the length λ . How have wave and particle been put together? On the other hand, what is the nature of what waves? In continuation, it will be shown how ‘’’Imaginary Relativity’’’ idea will help our imagination and perception to answer this question. As a start point, imaginary relativistic mass in equation (20) will be discussed:

m=m_°+im_°  (u⁄c)/√(1-u^2⁄c^2 )          (47)


The real part of this equation has nothing to say, but there is a doubt about the imaginary part. By replacing m_°⁄√(1-u^2⁄c^2 ) with the normalized value of equation (21):

m=m_°+i (|m|u)/c       (48)


By substituting |m| for its value from equation (46):

m=m_°+i h/(|λ|c)    (49)


The above equation shows that every particle with the rest mass m_° carries a wave with the length λ . As a matter of fact, ‘’’this equation provides the possibility to unify the duality of wave and particle relations’’’, the statement which seemed impossible before [4]. In order to discover behavior of particles moving with or near the speed of light, scientist made different experiments to prove the above statement which apparently seems impossible to be unified. However, all the experiments indicated that wave and particle natures do exist simultaneously instead of being separately. Equation (49) proves the above statement. Besides, this equation is a convincing reason for existence of mass-less particles. Considering m_°=0 in equation (49):

m=i h/(|λ|c)      (50)


Or

|λ|=i h/mc        (51)


Comparing this equation with equation (46) demonstrates that equation (51) is the wave length of a particle moving with the speed of light (c). This particle, as it is known, is photon.

### Imaginary energy

To calculate the energy of a photon, it is enough to multiply both sides of equation (68) by c^2 and then replace mc^2 with E (Relativistic Energy):

E=i hc/(|λ|)              (52)


Whereas c/(|λ|) equals [[[frequency of photon]] (ν ):

E=ihν    (53)


The only difference between equation (53) and Einstein’s equation (E=hν) is just in existence of i in equation (53). The question which arises here is that what imaginary energy means in general. According to opinion of the author of this paper,’’’ imaginary energy is a kind of energy which is not distributed continuously in an environment, but it just can be delivered entirely to an object as a package’’’. An example makes the issue more clear. Let us consider a person who dives into water of a pool from a diving board. A part of energy of the person is changed to the waves which shake other swimmers into the water. If it was observed that diving a person into the water made another person jumped from the pool on the diving board, we had to accept that the given energy to the first diver (because of diving), had not been distributed to the expanding wave front, but it had been transferred to the jumper as a concentrated or imaginary energy[6]. The fact that energetic photoelectrons are released from the surface of metal immediately can be explained as follows: Whereas the energy of a falling photon is imaginary according to equation (53), it cannot be distributed continuously over the surface of metal. Therefore, the absorbed photon gives up its whole energy to an electron and the electron is delivered very fast. Although Einstein also used the above mentioned justification to explain photoelectric effect (by using the statement quantum energy instead of imaginary energy), his equation, E=hν indicates a real energy and cannot justify his reasoning. In a more general case, it can be said surely that quantum equations cannot explain the experimental nature of quantum mathematically and always need to be accompanied with justifications, reasoning or even philosophical statements. However, Imaginary Relativity Theory and the imaginary part of equation (53) justify the nature of quantum’s behavior.

### Wave description of particles

In physics, a wave packet is an envelope or packet containing a number of plane waves having different wave numbers or wavelengths, chosen such that their phases and amplitudes interfere constructively over a small region of space[7]. Consider a function such as ψ(x) having information about the situation of a particle in a specific time. If it is supposed that ψ(x) is a complex number, the square of ψ(x) tells us the probability of observing a particle in a specific place and time per length unit. In general:

dP/dx=〖|ψ(x)|〗^2    (54)


Since the probability of finding a particle in anywhere equals one (1), the normalization condition over ψ(x) can be written as follows:

∫_(-∞)^(+∞)▒〖|ψ(x)|〗^2  dx=1     (55)


If the wave number k is a continuous variable, the wave can be shown as a combination of terms e^ikx. The distribution of wave numbers is given by function g(k). According to integral transformation of Fourier, function ψ(x)can be written in terms of g(k) as follows:

ψ(x)=1/√2π ∫_(-∞)^(+∞)▒〖g(k) e^ikx 〗 dk      (56)


Function g(k) can be calculated based on (x) :

g(k)=1/√2π ∫_(-∞)^(+∞)▒〖ψ(x) e^(-ikx) 〗 dx        (57)


Let us suppose that the probability of finding a particle is the same within the confines of –L<x<L . The first thing to do is to make the distribution function of wave number (ψ(x)) for a square wave packet. Whereas the Probability of finding a particle is the same within the confines of –L<x<L , ψ(x) is a constant function. The best choice for ψ(x) is equation (49), because for a constant velocity lower than the speed of light (u≪c), mass is constant within the space confines. By replacing k in equation (49) from the below relation:

k=2π/λ     (58)


We will have:

m=ψ(x)=m_°+i (h|k|)/2πc       (59)


To simplify equations, h/2π is shown by ℏ in quantum physics. Therefore:

ψ(x)=m_°+i (ℏ|k|)/c      (60)


Now, k can be calculated from the above equation so that the normalization condition (equation 55) is satisfied:

∫_(-L)^(+L)▒〖|m_°+i (ℏ|k|)/c|〗^2  dx=1        (61)


And after solving of integral:

(〖m_°〗^2+(ℏ^2 k^2)/c^2 )2L=1
k=c/ℏ √(1/2L-〖m_°〗^2 )        (62)


Equation (61) determines the value of k which makes function ψ(x) normalized by replacing k from equation (62) within the confines L<x<-L , in equation (93):

ψ(x)=m_°+i√(1/2L-〖m_°〗^2 )  (63)


And for x<-L or x>L  :

ψ(x)=0    (64)


It should be noted that according to equation (61), the area under the curve is 1 and its absolute value is calculated from equation (64) as follows:

|ψ(x)|=√(1⁄2L)      (65)


Now, the function g(k) can be obtained from equation (57). Please note that ψ(x) is constant relative to integral’s variable: g(k)=1/√2π ∫_(-L)^(+L)▒〖(m_°+i√(1⁄2L-〖m_°〗^2 )) e^(-ikx) 〗 dx=((m_°+i√(1⁄2L-〖m_°〗^2 )))/√2π ∫_(-L)^(+L)▒e^(-ikx) dx After calculation of the integral and simplification:

g(k)=1/√π (√2 m_°+i√(1⁄L-2〖m_°〗^2 ))   sin⁡(kL)/k   (66)


g(k) is a sinc and imaginary function. Distribution of wave numbers (here, wave packet) is a function of ψ(x). To find the probability of the distribution in terms of, the square of absolute value of g(k) should be calculated:

|〖g(k)|〗^2=1/πL  sin⁡(kL)^2/k^2  (67)


Wave number is within the confines –π⁄L<k<π/L and the area under the curve is unit (1) in the space k.

## Space-time

In physics, space-time is any mathematical model that combines space and time into a single continuum. Einstein used a three-dimensional geometric figure termed the light cone to represent the usual four-space metric or Minkowski metric in a two-dimensional plane, based on the conic section diagrams. This geometric picture is formed from a figure with two axes, the ordinate is time, t and the abscissa is formed from the three dimensions of space as one axis X = x, y, z. The speed of light forms the sides of the two cones apex to apex with the t axis in the vertical direction.The purpose of this picture is to define the relationship between events in four spaces. For events connected by signals of u<c, where c is the velocity of light, events occur within the top of the light cone (forward time) or bottom (past time). These are termed time-like signal. Event connections outside the light cone surface, u=c, are connected by u>c and are called space-like signals and are not addressed in standard physics[8]. In a Euclidean space, the separation between two points is measured by the distance between the two points. A distance is purely spatial, and is always positive. In space-time, the separation between two events is measured by the interval between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation. The interval between two events is defined as

∆s^2=∆X^2-c^2 ∆t^2   (68)


Where ∆X^2=∆x^2+∆y^2+∆z^2, and Δt and ΔX denote differences of the time and space coordinates, respectively, between the events. This equation is invariant under the Lorentz transformation, based on the constant of speed of light by Einstein’s postulate. In the other hand, for a case that frame S' moves with constant velocity u relative to S, we can write

∆s^2=∆X^2-c^2 ∆t^2=∆〖X'〗^2-c^2 ∆〖t^'〗^2     (69)


But we know that speed of light isn’t constant relative to S and S’ frames under ‘’’Imaginary Relativity’’’, and therefore we must replace follow equations instead equation (69)

∆X^2-c^2 ∆t^2=∆〖X'〗^2-C^2 ∆〖t^'〗^2          (70)


Or

∆〖X_C〗^2-cC^* ∆t^2=∆X_C'^2-cC∆〖t^'〗^2     (71)


Where C^* is the complex conjugate of C and ∆〖X_C〗^2defined as

∆〖X_C〗^2=c/C ∆x^2+∆y^2+∆z^2      (72)


Equation (71) is invariant under Lorentz imaginary transformation (9), waive of C transform to C^*, and so in this space distance ∆s^2 is invariant and given as

∆〖s_C〗^2=∆〖X_C〗^2-cC∆t^2      (73)


Since C is complex, above equation presented at least a complex six-dimensional Minkowski space with a purely geometrical model formulated in terms of space and time coordinates. This complex metrical space includes the three real dimensions of space and the usual dimension of time; it also includes one imaginary dimensions of space and one imaginary dimension of time. In the six spaces, the real components comprise the elements of the space defined by Einstein and Minkowski. The standard Minkowski metrical space is constructed so that all spatial components are real. But, the square of the temporal component differs by a -c^2, which is formulated from ict_Re, yielding a component -c^2 t_Re^2. In Minkowski complex space each spatial component has an ix_Im component, yielding the square component -x_Im^2. The corresponding temporal component is +c^2 t_Im^2. This is the basis upon which the eight spaces allows apparent zero spatial and temporal separation. Rauscher and Newman[9] expressed the complex eight-space metric using along lines of the detailed formalism of Hansen and Newman[10] expressed in general relativistic terms and extended usual four-dimensional Minkowski space into a four complex dimensional space-time. This new manifold (or space-time structure) is analytically expressed in the complexified eight spaces. They wrote in general for real and imaginary space and time components in the special relativistic formalism

∆s^2=∆X_Re^2+∆X_Im^2-c^2 (∆t_Re^2+∆t_Im^2 )          (74)


We can rewrite equation (73) as the format of (74)

∆〖s_C〗^2=c/C (∆X_Re^2+∆X_Im^2 )-cC(∆t_Re^2+∆t_Im^2 )    (75)


The light cone metric this space may imply superluminal signal propagation between subject and event in the real four spaces, but the event-receiver connection will not appear superluminal in some eight-space representations. We can consider that our ordinary four-dimensional Minkowski space is derived as a four-dimensional cut through the complex eight spaces[11]. Recall that the normalized value of equation (108) leads to the Minkowski space equation (101).Then

|∆〖s_C〗^2 |=∆s^2       (76)


## Conclusion

To get a better understanding of the concept of imaginary relativity, the equations of imaginary relativity and special relativity have been compared in Table 2.

## References

1. Imaginary Relativity, Journal WSEAS TRANSACTIONS on COMMUNICATIONS, Volume 9 Issue 2, February 2010
2. Wang, Kuzmich & Dogariu, Detailed statement on faster-than-c light pulse propagation, NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA
3. Feinberg, G., Possibility of faster-than-light particles, Physical Review, 159, 1089. 1967
4. Resnick, Robert, Introduction to Special Relativity, Wiley Fastern Private Limited, 1972
5. Krane, Kenneth S., Modern Physics, John Wiley & Sons, 1996
6. Enge ,Harald A.& Wehr , M. Russell and Richards , James A., Introduction to Atomic physics, Addison Wesley, 1972
7. Rolhf, James William, Modern Physics, John Wiley & Sons, 1994
8. Rauscher A. & Targ R., The Speed of Thought: Investigation of a complex Space-Time Metric of describe Psychic Phenomena, journal of scientific exploration, Vol. 15, No.3, 331-354, 2001
9. Rauscher A. & Targ R., The Speed of Thought: Investigation of a complex Space-Time Metric of describe Psychic Phenomena, journal of scientific exploration, Vol. 15, No.3, 331-354, 2001
10. Hansen, R. O. & Newman, E. T., A complex Minkowski space approach to twisters, General Relativity and Gravity, 6, 361–385, 1975
11. Newman, E. T., Hansen, R. O., Penrose, R., & Ton, K. P., The metric and curvature properties of H-space, Proceedings of the Royal Society of London, A363, 445–468, 1978
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