Complex Numbers for Relativistic Operations

: When presenting special relativity, it is customary to single out the so-called paradoxes. 10 One of these paradoxes is the formal occurrence of speeds exceeding the speed of light. An 11 essential part of such paradoxes arises from the incompleteness of the relativistic calculus of 12 velocities. In special relativity, the additive group is used for velocities. However, the use of 13 only group operations imposes artiﬁcial restrictions on possible computations. Naive expansion 14 to vector space is usually done by using non-relativistic operations. We propose to consider 15 arithmetic operations in the special theory of relativity in the framework of the Cayley–Klein 16 model for projective spaces. We show that such paradoxes do not arise in the framework of the 17 proposed relativistic extension of algebraic operations.


Introduction
Thus, a situation arise when relativistic operations are used to add the velocities, 26 and in other cases, in our opinion, non-relativistic operations are used. This leads to the 27 emergence of the so-called paradoxes. 28 The authors suggest that it is necessary to extend the relativistic operations with 29 velocities to algebra. It is assumed that this extension will remove some of the so-called 30 paradoxes of the special theory of relativity. 31 Several attempts have been made to describe relativistic operations consistently. 32 Most of these systems are quite complex [1][2][3]. The authors propose not to create a 33 new algebra, but to use in this role the systems of complex numbers based on the reader's attention that this implementation describes only collinear movements in one 48 spatial dimension. This is done on purpose to simplify the perception of the proposed 49 formalism. In section 8, the basic relativistic operations are written out explicitly.       The simplest model of superluminal motion can be an oblique incidence of plane waves on a certain flat interface between the media (also called a screen) [14-17]. Let ϕ be the angle of incidence of the wave on the screen, that is, the angle between the wave vector and the normal to the screen. Then the light spot on the screen moves across this screen with the speed: Here n is the refractive index of the medium in which the light pulse propagates (the medium above the screen). Since sin ϕ 1, then the speed of the light spot with a decrease in the angle of incidence ϕ can be made greater than the speed of light c. When considering the case of wave propagation in a vacuum, this becomes most obvious: The velocity v can be arbitrarily large since as the light pulse tends to normal incidence 118 (ϕ → 0), then the velocity tends to infinity (v → ∞). The phase velocity of a wave is the speed of propagation of the surface of a constant phase along a given direction. The phase velocity in the direction of the wave vector is set as follows: where ω is the circular frequency, k is the wavenumber. In a vacuum for an electromag-121 netic wave, the phase velocity along the vector is equal to the speed of light c.

122
When deviating from the wave vector by an angle ϕ, the phase velocity will be equal to: From the equation (2), it is seen that the phase speed can be greater than the speed of 123 light [18].  The elliptic measure of lengths is defined on the line o as follows (see Fig. 2). We define outside o some point Q and assume that the distance between the points A and B of the straight line o is equal to the angle ∠AQB: For consecutive points A, B, C (see Fig. 2) we have the following equality: The elliptic geometry may serve as an example of an elliptic measure of lengths. For consecutive points A, B, C (see Fig. 3) the following equality is true:  The hyperbolic measure is defined on the line o as follows. Between two points I and J of the line o, two points A and B of this line are specified (see Fig. 4). Then the hyperbolic distance will have the form: The κ coefficient specifies the distance units (and the base of the logarithm). The angles 142 are considered to be directed. The entire hyperbolic line is represented by the segment 143 I J.

144
For consecutive points A, B, C (see Fig. 4) the following equality is true: An example of a hyperbolic measure of lengths is the geometry of Lobachevsky.
An example of an elliptic measure of angles is the geometry of Euclid, which is 151 familiar to us.  Obviously, from (3) and (5) we may write: For the straight lines of the bunch a, b, c (see Fig. 6) we have the following equality: An example of a parabolic measure of angles is Galileo's geometry. 5  The hyperbolic measure of angles in a bundle with center O is defined as follows (see Fig. 7). We fix two straight lines i and j of the bundle and for any two other straight lines a and b we set the angular distance between them: δ + ab = κ ln sin ∠ai sin ∠aj sin ∠bi sin ∠bj .
The κ coefficient specifies the distance units (and the base of the logarithm). The angles 159 are considered to be directed.

160
Since the double ratio from the equation (7)   For the straight lines of the bunch a, b, c (see Fig. 7) the following equality is true: The geometry of Minkowski may serve as an example of a hyperbolic measure of 164 angles.   170 Points in the Euclidean plane can be identified with complex numbers by associating a point with Cartesian rectangular coordinates (x, y) or with polar coordinates (r, ϕ) to a complex number:

Cayley-Klein model and complex numbers
where i 2 = −1. The quantities x and y are called, respectively, the real and imaginary parts of the number z: x = Re{z}, y = Im{z}.
However, you can give a general definition of complex numbers. Let's set a quadratic equation in the form: The determinant of the quadratic equation (8) will have the form: Depending on the sign of the determinant, we can obtain the following systems of

176
Only ordinary complex numbers have a field structure. Dual and double complex 177 numbers have a ring structure because they contain nontrivial zero divisors.

178
Here are the basic representations and equations for complex numbers. 7 We will 179 denote elliptic complex numbers by the symbol "−", parabolic complex numbers by the 180 symbol "0", and hyperbolic complex numbers by the symbol "+". Algebraic representation is the most common representation for complex numbers. This representation is most convenient for recording additive operations (addition, subtraction). z = a + ib; z = a + εb; z = a + eb.

Trigonometric representation 184
In trigonometric representation, the real and imaginary parts of the complex number z are expressed in terms of the modulus r = |z| and the argument ϕ = Arg z of the 6 Note that in the literature there is also a reverse name for complex numbers: • ∆ < 0: hyperbolic complex numbers; • ∆ = 0: parabolic complex numbers (dual numbers); • ∆ > 0: elliptic complex numbers.
given complex number. This representation is most convenient for writing multiplicative operations (multiplication, division, exponentiation, root extraction). z = r(cos ϕ + i sin ϕ); We also need to define the modulus and argument of a complex number.

185
Complex number module 186 The value r = |z| is called the modulus of the complex number z: Complex number argument

187
The value ϕ = Arg z is called the argument of the complex number z: , |a| < |b|.

Exponential representation 188
The exponential representation is related to the trigonometric representation by Euler's formula. When writing arithmetic operations, these representations are interchangeable. z = r exp(iϕ);  For additive operations, we will use the algebraic representation of complex numbers. Let's set two complex numbers z 1 and z 2 , then we get: z 1 + z 2 = (a + eb) + (c + ed) = (a + c) + e(b + d).

193
Similarly, we write down for the subtraction of complex numbers z 1 and z 2 :

195
In the exponential representation, multiplication is reduced to the product of moduli and the sum of the arguments of complex factors.
The last expression looks somewhat cumbersome due to the form of Euler's formula 197 for hyperbolic complex numbers.

206
The hyperbolic complex number z = r exp{eϕ} corresponds to a point in the Minkowski space. The argument ϕ of the complex number z is the angle between the tangent to the particle's world line and the time axis in the base frame. The argument is related to speed by the following: relationship: It also has the name rapidity [29,30].

207
The sequence of actions is as follows (we will consider only time-like intervals):

208
• As part of the operational part of preparing the system, the usual values are converted into the form of hyperbolic complex numbers C + . Since the speeds are converted to rotations in the time plane (boosts), the following operation must be performed: v c → tanh ϕ, that is, we get the corresponding complex number: Here ϕ is the argument of the corresponding complex number, and we neglect the 209 module of the complex number.

210
• In the theoretical part, we perform calculations on the resulting complex numbers.

211
• Within the framework of the measuring operational part, we convert expressions in hyperbolic complex numbers C + into expressions in real numbers describing relativistic relations Λ: The corresponding relativistic velocity will be: For convenience, within the framework of one-dimensional motions, we can introduce the symbol of Einstein operations E, which directly transforms the operation Op Gal in the Galilean space into the operation Op Λ in the Lorentz space: This operation masks the full cycle of transition from non-relativistic expressions Gal to relativistic Λ by using hyperbolic complex numbers. We may write this transition in the form of a commutative diagram, which will have the following form:

Basic algebraic operations 212
Let's write down the main operations. transformation E. For other operations, for brevity, we will only use the E operation.

217
The Lorentz transformation is determined by multiplying by a hyperbolic complex number with unit module exp{eψ}, as a result, the Minkowski plane is rotated by the angle ψ: Replacing rapidity with speed, from (44) we get: Following the expression (7), let's pass on to real relativistic velocities: Let's demonstrate the same with the E operation. Based on the diagram (43), we may write the addition operation: We may write this operation for an arbitrary number of operands. For example, for three operands, we will get the following expression: It can be seen from the above relations that the addition operation in the proposed 218 formalism coincides with the generally accepted one, but at the same time, it is easier to 219 use. The operation of multiplication of velocities is usually not used in relativistic calculations.

Multiplication by a number 222
Consider the multiplication of the velocity vector by the number k ∈ R in the proposed representation: Obviously, the expression (47) will never exceed the speed of light c: lim k→∞ E(kv) = c.
As an example, we may write the expression for E(2v) using (45) and (47). From (45) we get: From (47) we obtain: We may see that the results of (48) and (49) are the same. Thus, the proposed representa-223 tion does not contradict the speed addition procedure. The procedure for multiplying 224 the speed by a number is consistent with the procedure for adding speeds.

225
Similarly, we can write the expression for E(3v) using (46) and (47). From (46) we obtain: From (47) we get: Obviously, the results of (50) and (51) are the same. Based on the generally accepted definition of multiplication by a number, from the equation (2) we concluded that the phase velocity might exceed the speed of light. However, if we replace the usual multiplication with the relativistic one according to the formula (47), we obtain: Let's write the limit of the equation (52): The phase speed is not greater than the speed of light.