Extending complex number to spaces with 3, 4 or any number of dimensions

Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.

In about 500 years after the birth of complex number, there were several attempts to extend complex number to more than 2 dimensions, for example we have theories such as quaternions, tessarines, coquaternions, biquaternions, and octonions. But none has reached the success of the classical complex number in 2 dimensions. Among these theories the most famous is quaternion which has found use in computational geometry. But quaternion is a 4 dimensional complex number but is used in 3 dimensional vector space, which is somewhat awkward.

In this article we will show that multidimensional complex number with 3, 4 or more dimensions exist and will explain how to construct them. Like classical complex number system, a multidimensional complex number system possesses algebraic operations in its complex space that have geometrical meaning in the corresponding vector space.

In the following exposition, spaces with 3, 4 or n dimensions will be referred to as 3D, 4D and nD spaces and the corresponding complex numbers as 3D, 4D and nD complex numbers. Since a complex number corresponds to a vector, a complex number will be referred to as a vector when convenient. We will begin with constructing 3D complex number system. Then we will generalize to spaces with 4 and more dimensions.

The 3D complex number system is constructed from a 2D complex number system which is the classical complex number system. So, let us see how classical complex number works.

Classical complex number

Classical complex space is a plane with two orthogonal axes, see Figure 1:
The axis of real numbers which is labeled as h.
The axis of imaginary numbers which is labeled as i.

This plane is labeled as (h, i). On this plane a complex number is both a point and a vector, for example the vector u in Figure 1. u makes the angle  with the axis h and and its length is |u|. As complex number, u’s argument is  and its modulus is |u|. In polar coordinate system the complex number u is expressed in equation (1), where i is the imaginary unit, see (3). Equation (1) is referred to as the trigonometric form of u.

We develop (1) into (2) in which we introduce (4) and obtain (5) where the numbers ‘a’ and b are the Cartesian coordinates of u. So, equation (5) is referred to as the Cartesian form of u.

Equation (6) is the Euler’s formula for  and i, and is introduced into (1) which becomes (7). Equation (7) expresses u in the form of an exponential function and is referred to as the exponential form of u. So, a classical complex number can be expressed in Cartesian, trigonometric or exponential form and has a geometrical meaning which is the vector u in Figure 1.

3D complex number

3D space and vector
A 3D complex number is also a vector, which we will construct from the 2D plane (h, i). For doing so, we add the axis j perpendicularly to the plane (h, i) and obtain the 3D space whose axes are labeled as h, i and j, see Figure 2. We refer to this space as (h, i, j).

We attach the unit vectors eh, ei and ej to the axes h, i and j respectively. The 3D space based on these vectors is referred to as (eh, ei, ej). We have then two 3D spaces: the complex space (h, i, j) and the vector space (eh, ei, ej). We will create a vector labeled as v in (eh, ei, ej) which corresponds to a 3D complex number in (h, i, j) labeled also as v.

With the help of Figure 2 we create the vector v in the desired form by starting with a vector u which is expressed in (8) with |u| being its modulus and  the angle it makes with the axis h. So, u is in the horizontal plane (eh, ei). Dividing u by |u| gives the unit vector eu, see (9). The unit vectors eu and ej are the basis vectors of the vertical plane (eu, ej), see Figure 2. The vector v is created by rotating the vector u in this plane toward the axis j. The angle of rotation is , so v is expressed with the angle  on the basis vectors eu and ej in (10).

As the length of u stays the same during the rotation, the modulus of u and v are equal, see (11). Introducing the expression of eu (9) into (10) gives (12) which is developed into (13) using (11). The vector v is expressed with its modulus |v| and the angles  and  on the basis vectors eh, ei and ej, see (13).

Notice that the angle  is between the vector v and the horizontal plane (eh, ei), see Figure 2, which is different from the usual spherical coordinate system where the angle  is between the vector v and the axis j. So, when u is horizontal the angle  equals zero rather then /2.

See the article with figures and equations here

https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions   https://pengkuanonmaths.blogspot.com/2022/02/extending-complex-number-to-spaces-with.html

Step by step rotation in normal and high dimensional space and meaning of quaternion

The orientation of body in space is defined 3 by angles. The step by step rotation process and chain of three-dots multiplication give an easy way to compute pile of rotations in 3D and high dimensional space and give a general orientation system. A visualization of quaternion is proposed.

The orientation of a rigid body in space is defined by 3 angles, for example the angles of pitch, roll and yaw of an airplane. The most commonly used orientation systems are Euler angles and Tait–Bryan angles. Quaternion is often used for computing rotation of body.

In this article we will present a new system for computing rotation in space that consists of rotating the proper frame of reference of the rigid body with respect to that of the space. This system is more general than Euler angles and Tait–Bryan angles and easier to use than rotation matrix and quaternion.

Below we will refer 2-dimensional space as 2D and 3-dimensional space as 3D. First, we will explain our approach using 2D body, then apply it to 3D body and body of even higher dimensions. We will use our system to give a visualization of the geometrical meaning of quaternion in 3D space.

Let us begin with defining the frame of reference of a rotated body and that of the space. A frame is defined by its basis that is a set of orthogonal unit vectors. The frame of the space is fixed. The frame of the rotated body is solidary to the body and rotates with it. The orientation in space of the body is defined by the angles its frame makes with that of the space.

We refer the basis of the frame of the space as space basis and that of the body the proper basis. The set of unit vectors of the space basis is (ex, ey) for 2D space and (ex, ey, ez) for 3D space. The set of unit vectors of the proper basis is (exi, eyi) and (exi, eyi, ezi) when it has gone i rotations. i=0 at the start.

Figures and equations are in the article below:

«Step by step rotation in normal and high dimensional space and meaning of quaternion»



Relativistic dynamics: force, mass, kinetic energy, gravitation and dark matter

Special relativity does not deal with acceleration, general relativity does not deal with non gravitational acceleration, which leave the theory of relativity imperfect. We will demonstrate some relativistic dynamical laws that specify relativistic acceleration, force and kinetic energy. Also, based on equivalence principle does gravitational mass vary with inertial mass?

Newtonian kinematics defines motions of objects with velocity and acceleration, Newtonian dynamics defines force with acceleration and mass, which makes Newtonian mechanics the most complete theory in physics. Special and general relativity are extremely successful, but they lack the capability of dealing with acceleration and force. For relativity mass increases to infinity when u=c, which makes energy and momentum to become incorrectly infinity. Also, general relativity is based on equivalence principle according to which inertial mass is equivalent to gravitational mass. Then does gravitational mass increases when inertial mass increases? So, relativity needs new laws to deal with acceleration and force.

In previous studies of relativity [1][2][3][4][5], we have already correctly treated acceleration, inertial mass, kinetic energy. Below we will demonstrate the laws that describe them. For setting the demonstrations on a strong base, we begin with rigorously proving the equality of differential momentum in 2 relatively moving frames of reference.

In relativity, a change of velocity has different value in relatively moving frames. However, differential momentum has the same value in such frames, which we call equality of differential momentum and have explained in « Velocity, mass, momentum and energy of an accelerated object in relativity » [2].

For rigorously proving this equality, let us take 2 identical objects labeled a and b. The object b moves at the velocity u with respect to a, the frame of reference of the object b is labeled frm. b, see Figure 1. If the object b gets an infinitesimal impulse,it gets a differential momentum and a differential change of velocity labeled dub.

Notice that in the frame frm. b the velocity of b is constantly zero. Then, how can its change of velocity dub be nonzero? In fact, dub is with respect to an inertial frame, not to frm. b. For defining dub we have to create a new type of inertial frame that coincides with b. We name this type of frame “Proper inertial frame”.

For example, the object b moves at the instant velocity ut at a given time t. At this time we create the proper inertial frame of b labeled Ref. b which moves at constant velocity that equals ut. The trajectory of Ref. b is a straight line while that of b is a curve, see Figure 2. After the infinitesimal impulse b moves at the instant velocity u’t and the change of velocity equals dub = u’tut with ut being the velocity of the inertial frame Ref. b. In the same way the proper inertial frame of the object a is labeled Ref. a.

Figures and equations are in the article below:

«Relativistic dynamics: force, mass, kinetic energy, gravitation and dark matter»



How galaxies make their rotation curves flat and what about dark matter?

The rotation curves of disc galaxies are flat and dark matter is speculated as explanation. Alternatively, the gravity of material disk could explain the flat curves. Using the gravitational force that a disk exerts on a body in the disk, we have computed the the rotation curves of disc galaxies and the curve of their mass densities. The numerical result fits the flat curves and the observed mass densities of galaxies. This theory gives a new way to measure the masses of galaxies using their rotation velocities and shape.

  1. Rotation curve

In a disc galaxy stars orbit the center of the galaxy at velocities that depend on the radial distance from the centre. The measured orbital velocities of typical galaxies are plotted versus radial distance in Figure 1, which are the rotation curves of these disk galaxies. Figure 1 shows that the observed rotation curves are flat for large radial distance, which means that these velocities are roughly constant with respect to radial distance [1].

The flat aspect of the rotation curves is puzzling because the Newtonian theory of gravity predicts that, like the velocities of the planets in the solar system, the velocity of an orbiting object decreases as the distance from the attracting body increases. Since the centers of galaxies are thought to contain most of their masses, the orbital velocities at large radial distance should be smaller than those near the center. But the observed the rotation curves clearly show the contrary and the observed orbital velocities are bigger than expected.

The masses of galaxies estimated using the luminosity of visible stars are too low to maintain the stars to move at such high speed. So, large amount of matter is needed to explain the observed velocity, but we do not see this matter. The missing matter should act gravitational force because it should hold the flying stars, but should not radiate light because invisible. So, it is dubbed as dark matter. However, dark matter has not been observed directly despite the numerous actively undertaken experiments to detect it. Several alternatives to dark matter exist to explain the rotation curve.

We propose to model galaxy as regular matter disk. Because of its shape, the resultant gravitational force of material disk within the plane of a galaxy is different from that of the galaxy taken as concentrated masses. But, is this resultant force bigger?

  • Gravity of material disk

Galaxies are made of stars that move in circular orbits. Let us take out one circular orbit with all its stars and put them in free space without an attracting body at the center. These stars form a circle and attract each other gravitationally, which would make them to fall toward the center if they were stationary, see Figure 2.

Figure 2 shows a simplified image of stars in a circle with F_A being the combined gravitational force that the other stars act on the star A. For the star A not to fall out of the circle, it must rotate at a nonzero velocity which is labeled as v_A. By symmetry, all the stars feel the same gravitational force and must rotate at the same velocity to stay in the circle. So, this circle of stars should rotate although no attracting body is at the center. It is the proper mass of all the stars of the circle that maintains the stars rotating.

In the explanation, we will often use the case of a single object in a circular orbit around an attracting body. For referring to this case, we give it the following name.

Definition: The Newtonian orbital acceleration and velocity of a single object in a circular orbit around an attracting body are named Single-Orbit acceleration and velocity.

If a star orbits in circle around a central mass M, the gravitational force on it is from M only and is labeled as F_M. The star would move at the Single-Orbit velocity which is v_s, see equation (1).

Now, let us add the mass M at the center of the circle of stars, see Figure 3. The gravitational force acted on the star A by the other stars of the circle is still F_A. But in addition it feels the force F_M from the central mass M. So, the total force on the star A is F_A+F_M which is bigger than F_M. In consequence, to stay in the circle the star A should orbit at a velocity bigger than v_s, say at v_s+∆v_A with ∆v_A being the contribution of the force F_A. So, if a circle of stars orbit an attracting body at the center, their orbital velocity should be bigger than the Single-Orbit velocity of one star around the same body, which is kind of like the case of the bigger than expected orbital velocity in galaxies.

Now, let us smear the stars of the circle into a disk of dust around the central mass M which is not held by cohesion but by gravitational force, see Figure 4. Like the stars of the circle, the gravitational force on a dust is from the mass M and the proper mass of the disk. So, the orbital velocity of the dust, v_d in Figure 4, will be bigger than the Single-Orbit velocity around the mass M like the stars in the disk of a galaxy.

This is the working principle of our model that explains the faster than expected orbital velocity in galaxies. Now, let us see if the gravity of material disk could make the orbital velocity constant for large radial distance.

  • Force in a disk

For computing the gravitational force that the entire disk exerts on a chunk of material of mass m_1, we use the Newtonian law of gravitation which expresses the gravitational force that an elementary mass dm exerts on the chunk of material, see equation (2) where dF_d is the gravitational force, G the universal gravitational constant, R_3 the distance between m_1 and dm, e_3 the unit vector pointing from m_1 to dm, see Figure 5.

Figures and equations are in the article below:

How galaxies make their rotation curves flat and what about dark matter?



Gravitational time dilation and black hole

In this article, we derive the gravitational time dilation factor in a new manner, which allows us to identify the mathematical cause of Schwarzschild radius, to give a theoretical way to avoid it and to compute properties of black hole. General relativity effects are computed as simple as in special relativity. Observability of black hole is discussed.

The wonderful work of Karl Schwarzschild describes the spacetime curved by a spherical gravitational field, which agree precisely with observation as shown by the orbital precession of the planet Mercury. In addition, Schwarzschild metric possesses a completely new feature: a critical radius named Schwarzschild radius that marks the lower limit of describable space. This feature is weird because it splits space into 2 regions, the normal space is outside this radius and the region inside it cannot be mathematically defined. This region is called a black hole because no information, not even light, could escape from it.

The funding principle of physics is that the entire universe is governed by physical laws whatsoever. Such weird region out of the reach of physics hints that unknown physical laws are there. So, let us try to understand deeper Schwarzschild radius and find out where it comes from.

Comparison with the time dilation in special relativity
What makes Schwarzschild radius?
Using relativistic transformation of acceleration
Gravitational relativistic dynamics
Below Schwarzschild radius
About black hole, Observational evidences

Figures and equations are in the article below:
Gravitational time dilation and black hole

General equation for Space-Time geodesics and orbit equation in relativistic gravity

  • Orbit equation and orbital precession

General Relativity explains gravity as Space-Time curvature and orbits of planets as geodesics of curved Space-Time. However, this concept is extremely hard to understand and geodesics hard to compute. If we can find an analytical orbit equation for planets like Newtonian orbit equation, relativistic gravity will become intuitive and straightforward so that most people can understand.

From gravitational force and acceleration, I have derived the analytical orbit equation for relativistic gravity which is equation (1). Below I will explain the derivation of this equation. Albert Einstein had correctly predicted the orbital precession of planet Mercury which had definitively validated General Relativity. Equation (2) is the angle of orbital precession that this orbit equation gives, which is identical to the one Albert Einstein had given [1][2].

If this orbit equation gave the same result than Space-Time geodesics, then everyone can compute the orbit of any object in gravitational field which obeys General Relativity using personal computer rather than big or super computer. Also, everyone can see how gravity leads to Space-Time curvature without the need of knowing Einstein tensor.

The derivation of the orbit equation is rather tedious and lengthy. So, for clarity of the reasoning and explanation, I have collected all the mathematical equations in the last section “Derivation of equations”, in which full details are provided to help readers for checking the validity of my mathematics.

  • Relativistic dynamics
    a) Velocity in local frame

Take an attracting body of mass M around which orbits a small body of mass m, see Figure 1. We work with a polar coordinate system of which the body M sits at the origin. The position of the body m with respect to M is specified by the radial position vector r, of which the magnitude is r and the polarangle is q.

Let the frame of reference “frame_m” be an inertial fame that instantaneously moves with m. Frame_m is the proper frame of m where the velocity of m is 0. So, Newton’s laws apply in this frame. Let am be the acceleration vector of m in frame_m and the inertial force of m is m·am, see equation (3). The gravitational force on m is given by equation (4). Equating (4) with (3), we get equation (5), the proper acceleration of m caused by gravitational force in frame_m.

Let “frame_l” be the local frame of reference in which M is stationary. In frame_l m is under the effect of gravity of M, the velocityvector of m is vl and the acceleration of m is a l. As frame_m moves with m, it moves at the velocity vl in frame_l.

The acceleration of m in frame_m and frame_l are respectivelyam and al. To transform al into am we use the transformation of acceleration between relatively moving frames which is the equation (18) in «Relativistic kinematics and gravity»[3][4], in which we replace a1 with al, a2 with am and u with vl. Then the transformation between am and a l is equation (6).

Equating (5) with (6) we get equation (7), both sides of which are then dotted by the vector vl·dtl, see equation (8). On the left hand side of (8), we find dr the variation of the radial distance r, see (9). On the right hand side, we find the variation vector of velocity dvl, see (10), which, dotted by the velocity vector vl, gives dvl2/2 in (11).

Plugging (9) and (11) into (8), we get (12), both sides of which are differential expressions, see (13) and (15). Then, plugging (13) and (15) into (12) gives (16) which is a differential equation. (16) is integrated to give (17), with K being the integration constant. Then, we rearrange (17) to express vl2/c2 in (18), which relates the local orbital velocity vl to the gravitational field of M.

The value of eK is determined at a known point 0 at which the velocity is v0 and the radial distance is r0, see (19).

General equation for Space-Time geodesics and orbit equation in relativistic gravity https://www.academia.edu/44540764/Analytical_orbit_equation_for_relativistic_gravity_without_using_Space_Time_geodesics


Explaining Oumuamua and Pioneer anomaly using Time relativity

I find that in theory the weird Speed Boost of the interstellar object ‘Oumuamua should be
0.217 mm/s above the prediction and that ‘Oumuamua should slow down less than prediction, in proportion of which the difference is 4.28×10-8 near the Sun. For Pioneer anomaly I have computed the gap between real and predicted acceleration and found the value 8.70×10-10 which is very close to the observation (8.74±1.33)×10−10 m/s2.

The mysterious interstellar object ‘Oumuamua confuses scientists because of its Speed Boost, which is an excess of velocity with respect to the expected one. In the past, the manmade Pioneer spacecrafts were also found to deviate from expected Newtonian trajectory.

One thinks the velocity of ‘Oumuamua is too high because it is faster than the expected velocity that the mass of the Sun allows. But if we have used a mass for the Sun slightly different from the real one, then the expected velocity would be not correct. So, let us see how the mass of the Sun is determined

Read the article below.

Relativistic kinematics and gravitation

Like in Newtonian kinematics, the relativistic change of reference frame must be a vector system of transformation laws for position, velocity and acceleration.

In special relativity, when changing the reference frame the coordinates of a moving point is transformed using Lorentz transformation. But the velocity-addition formula that transforms velocity is in a too different mathematical form than the Lorentz transformation. And for acceleration, there is not a transformation at all. The theory of Time relativity that I develop provides a coherent vector system of transformation laws for position, velocity and acceleration
Read the article below.

‘Oumuamua, Pioneer anomaly and solar mass with Time Relativity

The theory of Time relativity explains well the weird behavior of the interstellar object ‘Oumuamua. I find that the real solar mass is slightly higher than today’s value, which caused the mysterious Speed Boost of which the value should be 0.217 mm/s above the prediction at perihelion. Time relativity confirms that ‘Oumuamua should slow down less than prediction, in proportion of which the difference is 4.28 ×10-8 near the Sun. For Pioneer anomaly I have computed the gap between real and predicted acceleration and found the value 8.70 ×10-10 which is very close to the observation (8.74±1.33)×10−10 m/s2.

The mass of the Sun is not measured by weighting, but derived from the parameters of Earth’s orbit which is nearly circular. Let rM be the radius of the Earth’s orbit and uE its orbital velocity. By equating the orbital acceleration of the Earth (see equation (36)) with its Newtonian gravitational acceleration (see equation (34)), we obtain equation (48) which gives the today’s used value of the mass of the Sun, M0, in equation (49).

Read the article below.

Velocity, mass, momentum and energy of an accelerated object in relativity

Analytical derivation of relativistic velocity, mass, momentum and kinetic energy of an accelerated object. For Special relativity the momentum of an object of rest mass m0 and velocity u is expressed by equation (1) which is infinite when u equals c. Is it physically meaningful that the momentum of an object becomes infinite while its velocity stays finite? On the other hand, the principle of mass–energy equivalence proposed by Albert Einstein in his article “Does the Inertia of an object Depend Upon Its Energy Content?” has not been rigorously demonstrated, hence it is called a principle not a law. In the contrary, in the theory of Time relativity which is been developed here, momentum and kinetic energy are derived by direct integration and stays limited when u=c.…

The expression of velocity (equation (13)) is directly integrated and thus is mathematically exact. In the contrary, the velocity-addition formula in Special relativity cannot be analytically integrated and one had to make an approximation to compute the velocity of an object which is thus not exact (see section 5.3 of « Introduction to Special Relativity » by James H. Smith).

In Special relativity the expression of relativistic mass is derived with the help of a shock between 2 objects (see section 9.4 of « Introduction to Special Relativity » by James H. Smith). For Time relativity relativistic mass is the derivative of momentum with respect to velocity, which is exactly the definition of mass.

In Special relativity the expression of momentum was derived with the help of a shock between 2 objects (see section 9 of « Introduction to Special Relativity » by James H. Smith) and is infinite when the velocity equals c (see equation (1)). For Time relativity momentum is the integral of infinitesimal change of momentum (see equation (25)). When the velocity of the object equals c its momentum equals the constant π/2 m_0 c, which gives a negative answer to the question of the beginning: “Is it physically meaningful that the momentum of an object becomes infinite while its velocity stays finite? ”

For Time relativity the total kinetic energy of an object is the integral of the work done on it and thus, its expression is mathematically exact. Moreover, when the velocity of the object equals c, its expression equals m0c2 (see equation (45)), which is a proof for the the principle of mass–energy equivalence, while in Special relativity mass–energy equivalence does not has mathematical proof.

At the end, we have derived the momentum-kinetic energy relation for Time relativity, which reduces to the expression of kinetic energy for classical mechanics for small velocity, while the momentum- energy relation in Special relativity does not. In the contrary, this relation is infinite when the velocity equals c.

Read the article below.