Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets

An analysis of Cantor’s theory of uncountable sets: The logic of his proofs has some weaknesses. Cantor assumes for both his proofs that all real numbers (set R) are in a list (list L). Considering L as a set this assumption assumes R belongs to L. This makes the claim “a real number is constructed but is not in the list L” questionable. We propose a solution to this problem, an axiom for counting infinite sets and a solution to continuum hypothesis.

«Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets»

https://www.academia.edu/86410224/Examination_of_Cantors_proofs_for_uncountability_and_axiom_for_counting_infinite_sets

Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets

I do a detailed analysis of Cantor’s theory of uncountable sets. The logic of his proofs has some weaknesses. I propose an axiom and a solution to continuum hypothesis.

The main idea is:
Assumption of Cantor’s proofs: All real numbers (set R) are in a list (list L).

This assumption means R=L, considering L as a set. This makes the claim “a real number is created but is not in the list L” wrong. Indeed, if a number is outside L, it is outside R too. So, the statement “the created real number is not in the list L” means it is not in R and is not a real number, which is equivalent to claim that a real number is not real number. This is absurd but Cantor’s diagonal argument and nested intervals proof both claim that a real number is not in the list L and thus, is not a real number, which make them wrong.

On the other hand, Cantor’s both proofs search for contradiction. Can “this real number is not a real number” be the contradiction? No. The contradiction of the proofs is in the third step: sout “is not in the list”. By failing to create sout the second step collapses before the third step declares the contradiction.

See the paper
https://www.academia.edu/86410224/Examination_of_Cantors_proofs_for_uncountability_and_axiom_for_counting_infinite_sets

Radius of a black hole for relativity and Newtonian mechanics

For Newtonian mechanics

The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics, which is explained below. Consider a big mass M which creates the gravitational acceleration a for a small mass m at the distance r from M, see figure 1 and equation (1) for gravitational acceleration a=GM/r2. For computing v the radial velocity of m in the gravitational field of M we integrate equation (1).

We compute for the case where m freefalls from infinitely far starting with zero velocity, see (7). With these conditions the radial velocity of m at the distance r2 from M is computed in (8), v2 = 2GM/r2. Reversing (8), r2 = 2GM/v2. The Schwarzschild radius of the event horizon of M is rs such that the Schwarzschild factor equals infinity, see (10).

When v2 equals the speed of light c, we apply v2 = c into (9) and we obtain (11) where r2 = rs. So, r2 equals rs and the Schwarzschild radius is computed with Newtonian mechanics.

For relativity

Although the Schwarzschild radius rs is a relativistic quantity, in the above it is derived completely with Newtonian mechanics, which is somewhat weird. What will be its value if we apply relativistic principle?

In the following derivation we will use the formula for relativistic transformation of acceleration which is derived in the paper « Relativistic kinematics » linked here: https://www.academia.edu/44582027/Relativistic_kinematics

The formula is the equation (18) of the paper.

Here, this formula is given by (12) in which the gravitational acceleration of m is a and the acceleration in space is ar and the radial velocity is computed by integration. With the same conditions as (7), the constant of integration k equals 0, see (16). Then, using k = 0 in (15), v is expressed with r in (17).

In the case where the small mass m approaches M, the distance r approaches 0, the radial velocity of m approaches the speed of light c, see (18). So, v the radial velocity of m does not become bigger than the speed of light c for r > 0. The Schwarzschild radius rs is the radius such that v = c. So, rs = r = 0, see (19).

Conclusion

When the relativistic principles are applied correctly, the Schwarzschild radius rs equals zero. The gravitational force on m approaches infinity near M. But the speed of m never reaches the speed of light c, which is true for any force however strong it is and for time of acceleration however long it is. The Schwarzschild radius rs must obey relativistic principle and is shown to be zero.

That the Schwarzschild radius rs equals zero means that the geometrical size of a black hole should be zero and thus, a black hole should not have an interior. I have reached this conclusion and explained it in the paper « Gravitational time dilation and black hole » in which I have also shown that point masses could not coalesce to form a black hole whose volume is zero and that observations support this conclusion. This paper is linked here:

https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

The equations and figure are in the paper below

https://www.academia.edu/84798805/Radius_of_a_black_hole_for_relativity_and_Newtonian_mechanics

For more detailed information, I invite you to read the two cited papers:

« Gravitational time dilation and black hole » https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

« Relativistic kinematics » https://www.academia.edu/44582027/Relativistic_kinematics

Computing orientation with complex multiplication but without trigonometric function

Today’s methods for computing orientation are quaternion and rotation matrix. However, their efficiencies are tarnished by the complexity of the rotation matrix and the counterintuitivity of quaternion. A better method is presented here. It uses complex multiplication for rotating vectors in 3D space and can compute orientation without angle and trigonometric functions, which is simple, intuitive and fast.

  1. Basic orientation 1
  2. Rotation using complex multiplication 1
    a. Complex multiplication 1
    b. Mixed multiplication 2
  3. Reference frames 2
    a. Ground frame and proper frame 2
    b. Direction frame 3
  4. Base vectors of the direction frame 3
    a. Rotation around the z axis 3
    b. Rotation around the y axis 3
    c. The 3 axes of the direction frame 4
    d. Direction frame and 3d complex number 4
  5. Roll, Pitch and Yaw 4
    a. Roll 4
    b. Pitch and Yaw 4
  6. Determination of the angles 5
    a. Direction angles 5
    b. Roll angle 5
    c. Direction frame angles without trigonometric functions 5
    d. Roll angle without trigonometric functions 6
  7. Computation for oriented points 6
    a. Position of one point 6
    b. Computation without trigonometric functions 6
    c. Procedure of computation without trigonometric function 7
  8. Discussion 7
    See the article with figures and equations here
    https://www.academia.edu/80277267/Computing_orientation_with_complex_multiplication_but_without_trigonometric_function
    For downloading the paper
    https://pengkuanonmaths.blogspot.com/2022/05/computing-orientation-with-complex.html
  9. Basic orientation
    A rigid body can rotate around 3 orthogonal axes in space, see Figure 1. The state of these 3 rotations is the orientation of this body. The commonly used orientation systems are Euler angles and Tait–Bryan angles[1]. Two methods are usually used to compute orientation: quaternion [2] and rotation matrix [3]. But they have their drawbacks. For quaternion, the computation for rotating a vector p needs to multiply p on the left by the rotation vector q and on the right by its conjugate q-1, which implies two 4D multiplications with p in between, see equation (1). The weird thing is that the rotation vector q is not computed with the rotation angle  but its half /2, see (2), [4]. The half angle and the “sandwich multiplication” make the quaternion method counterintuitive.

For rotation matrix, the 3 angles of orientation are mingled in the 9 elements of the matrix where one gets easily lost. For example, in «Step by step rotation in normal and high dimensional space and meaning of quaternion»[5], the transformation matrix is equation (3) which is confusing with the messed trigonometric functions.

Can we find a better method? In fact, I have constructed a 3D complex number system in «Extending complex number to spaces with 3, 4 or any number of dimensions» [6] which computes easily the rotation of a vector in 3D space. The combination of this system with the step by step rotation described in «Step by step rotation in normal and high dimensional space and meaning of quaternion»[5] gives birth to a new method. This method can use directly the coordinates of points to rotate an object without using trigonometric function. We will explain first this method that uses complex multiplication and trigonometric function.

  1. Rotation using complex multiplication
    a. Mixed multiplication
    We notice that when a 3D vector rotates around an axis that is perpendicular to it, the vector rotates in a plane and we call it the rotation plane. For using 2D complex multiplication in 3D space, we consider the rotation plane as a complex plane and use the 2D complex multiplication to rotate a vector in this plane.

Let (e1, e2) be a plane in 3D space, e1 and e2 the base vectors of the plane, so they 3D vectors. We make this plane equivalent to the complex plane using equation (7), that is, e1 corresponds to the real axis and e2 to the imaginary axis. u is a vector in the plane and is expressed in (8), with a and b being its components. Because the plane (e1, e2) is equivalent to the complex plane, u has an equivalent complex number u, which is expressed in (9). Let v be an other complex number which is expressed in (10).

We multiply u with v and the complex product uv is given in (11). The real and imaginary parts of uv are written in (12). Because the complex plane is equivalent to the plane (e1, e2), we replace the 1 and i that are in (11) with e1 and e2 and obtain uv in (13) which is a vector in the plane (e1, e2).

So, we have created a new type of multiplication: the vector u multiplied by the complex number v. The vector u is a 3D vector because e1 and e2 are 3D vectors. The product of this multiplication is uv which is a 3D vector too. We call this multiplication “mixed multiplication” and the product “mixed product”. We state the Definition 1.

Definition 1: Mixed multiplication and mixed product
The plane (e1, e2) is equivalent to the complex plane. u is a vector in this plane and equals ae1+be2. u is a complex number and equals a+bi. v is an other complex number. u is multiplied by v and the result of this multiplication is denoted as uv and equals real(uv)e1+ imag (uv)e2, with real(uv) and imag (uv) being the real and imaginary parts of the complex product uv. uv is a vector in the plane (e1, e2). This multiplication is called the mixed multiplication and uv the mixed product.

Let us use the newly defined mixed multiplication to rotate the vector u given in (14). The complex number u given in (4) is the complex equivalent of u and the complex number v is the ei given in (5). The complex product uv is computed in (6). Then the mixed product uv is given in (15) whose components equal the real and imaginary parts in (6). So, uv is well the vector u rotated by the angle . We see that mixed multiplication is a very easy way to compute the rotation of a 3D vector in the rotation plane.

See the article with figures and equations here
https://www.academia.edu/80277267/Computing_orientation_with_complex_multiplication_but_without_trigonometric_function
For downloading the paper
https://pengkuanonmaths.blogspot.com/2022/05/computing-orientation-with-complex.html

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»

https://www.academia.edu/52628458/Step_by_step_rotation_in_normal_and_high_dimensional_space_and_meaning_of_quaternion

https://pengkuanonmaths.blogspot.com/2021/09/step-by-step-rotation-in-normal-and.html

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»

https://www.academia.edu/49921891/Relativistic_dynamics_force_mass_kinetic_energy_gravitation_and_dark_matter

https://pengkuanonphysics.blogspot.com/2021/07/relativistic-dynamics-force-mass.html

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?

https://www.academia.edu/46903516/How_galaxies_make_their_rotation_curves_flat_and_what_about_dark_matter

https://pengkuanonphysics.blogspot.com/2021/04/how-galaxies-make-their-rotation-curves.html

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
https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole
https://pengkuanonphysics.blogspot.com/2021/03/gravitational-time-dilation-and-black.html

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

https://pengkuanonphysics.blogspot.com/2020/11/analytical-orbit-equation-for.html