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2.6.3 Modern Physics 1

2.6.3.1 Overview

As mentioned before, Classical Physics ranged over Newtonian mechanics (related to space, time, mass, and energy), Newton's Law of Gravitation, to Maxwell's Equations (related to electricity and electromagnetic wave including light).
In contrast to that, Modern Physics would range from Planck Constant (involving light and Quantum), Einstein's Theory of Relativity (related to space, time, mass, energy, and gravity), De Broglie's Matter Wave (related to wave), to Schrodinger Equation (related to quantum, electrons, and wave). Thus "Theory of Relativity" and "Quantum Mechanics" were prepared in the era of Modern Physics.
The former part of Modern Physics here (Modern Physics 1) would range to Einstein's Special Theory of Relativity and Mass-Energy Equivalence.

2.6.3.2 Details

2.6.3.2.1 Planck's Law and Quantum Hypothesis in 1900 CE

Background
In the process of German iron industry, iron ores and coals were put into blast furnaces. Workers managed the temperature referring to the color of the inside of furnaces, since the color varies depending on the temperature. However, the reason why the color varies depending on the temperature was unknown. This issue was investigated as Black-Body Radiation in physics.
* "Black-Body Radiation in Wikipedia" http://en.wikipedia.org/wiki/Black-body_radiation
The relation between spectrums of black-body radiation and temperature was researched. Wien and Rayleigh presented approximate solutions. However, the approximations couldn't reach complete explanation.
* "Wien Approximation in Wikipedia" http://en.wikipedia.org/wiki/Wien_approximation
* "Rayleigh Jeans Law in Wikipedia" http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Jeans_law

Planck's Law and Quantum Hypothesis
Planck and his assistant noted that if "-1" is added to Wien's law, it results in complete coordination with the data and complete explanation.
* "Planck Law in Wikipedia" http://en.wikipedia.org/wiki/Planck%27s_law
In addition, Planck noted relation between the energy of light and frequency, Planck Relation, connected by a constant, Planck Constant.
The following is the Planck Relation.
En=n*h*ν (n=1, 2, 3, 4, ... natural number)
En: energy of the light in various cases (from black-bodies)
n: integer (or natural number) varies depending on various cases
*: multiplication sign ("*" here is alternate symbol of "x" tentatively employed here in distinction from "ex" of the 24th letter of alphabet, while multiplication sign is commonly frequently left out in mathematics.)
h: Planck Constant: 6.626*10^(-34) J*s
J: Joule: kg*(m^2)/(s^2); ^: exponentiation sign tentatively employed here
ν (Nu: the 13th letter of the Greek alphabet): frequency of the light
* "Planck Constant in Wikipedia" http://en.wikipedia.org/wiki/Planck_constant
* "Nu in Wikipedia" http://en.wikipedia.org/wiki/Nu_(letter)
It meant that energy of light doesn't vary continuously, but varies discretely. For example, energy of light can be 1hν, 2hν, 3hν, 13hν, 18hν, and so forth. Energy of light can't be like 0.3hν, 1.29hν, 2.56hν, and so forth. In other words, energy of light is integral multiples of "hν" (or 1hν).
Consequently, energy of light consists of minimum quantity element, h*ν.
Planck also introduced the concept of "quantum." Then some physical properties are noted observed or analyzed as integral multiples of a "minimum quantity element" particularly related to microscopic phenomena. Such "minimum quantity element frequently found related to microscopic phenomena" was named "quantum." (Plural of "quantum" is "quanta.") The state composed of integral multiples of the minimum quantity element (quantum) (or one minimum quantity element) is called "quantized."
* "Quantum in Wikipedia" http://en.wikipedia.org/wiki/Quantum

2.6.3.2.2 Rutherford's Discovery of Beta Decay in 1903 CE

Background
Becquerel discovered radioactivity.

Rutherford's Discovery of Beta Decay
Rutherford discovered an incomprehensible phenomenon, Beta Decay.
* "Beta Decay in Wikipedia" http://en.wikipedia.org/wiki/Beta_decay

2.6.3.2.3 Einstein's Light Quantum Hypothesis in 1905 CE (the 1st paper of the Annus Mirabilus Papers)

Background
Maxwell claimed light is a kind of electromagnetic waves.
On the other hand, "photoelectric effect" was reported. When metal such as zinc is exposed to light (such as ultra-violet), the metal emits electrons. In this case, high frequency light exclusively causes electron emission. Low frequency light doesn't cause electron emission. If the frequency of the light increases, momentum of electrons increse, but amount of electrons doesn't change. If the intensity of light increases, amount of electrons increases, but momentum of each electron doesn't change.
The results of photoelectric effect above and light as electromagnetic waves contradicted.

Einstein's Light Quantum Hypothesis
Einstein claimed light consists of a finite number of "energy quanta" (minimum quantity elements like particles associated with energy), succeeded in explaining the photoelectric effect.
The particle of light is later called "photon." The energy of a photon is "E=h*f," where E: energy of a photon, h: Planck Constant, f: frequency of the light (instead of "ν").
*Light in some cases shows particle-like phenomena such as photoelectric effect assiciated with Light Quantum Hypothesis, but light in other cases shows electromagnetic wave-like phenomena. Light has both particle and wave like properties, but details were unclear at that time.

2.6.3.2.4 Einstein's Special Theory of Relativity in 1905 CE (the 3rd paper of the Annus Mirabilus Papers)

2.6.3.2.4.1 Background

According to the Michelson-Morly experiment, the speeds of light on the earth from various directions were the same, though the earth moves in the universe. It conflicts with Galilean transformation based on Newtonian mechanics.

2.6.3.2.4.2 Einstein's Claim on the Special Theory of Relativity

Einstein claimed the Special Theory of Relativity in relation to spaces in translatory movement affected by no gravitation, neither gravity nor acceleration. Then it was a theory representatively involving relation (relativity) between moving spaceships straight at a constant speed respectively equipped with devices to measure the speeds of light at the constant speed respectively. (The original physical term compared to"spaceship" here is "inertial reference frame" in physics.) (Stationary spaceships are included as constant 0-speeded spaceships as well.) (Since the conditions are thus restricted, the theory is called "Special.")
Einstein claimed, for example, "spatial distance between certain 2 objects" and "elapsed time between certain 2 events" for observers in spaceships differ depending on the speeds and directions of the spaceships. Galilei and Newton assumed scales of distance and time are constant and common. However, Einstein denied the constancy of the scales of moving observers (spaceships). The Special Theory of Relativity presents relation (relativity) between scales of translatory moving spaceships (including stationary spaceships) affected by no gravitation (neither gravity nor acceleration).
Specifically, the relationship between the moving spaceships is led by Lorentz Transformation instead of Galilean Transformation.

*The difference between Galilean Transformation and Lorentz Teansformation could be comprehensible illustrated in Minkowski spacetime.
The following 3 figures are simplified Minkowski spacetime involving Galilean Transformation.
In this case, "Ground" represents stationary circumstances. ("Ground" is assumed stationary.)
If a blue streetcar runs to the east at 50m/min, the trace of the blue streetcar could be illustrated as in the following figure, "Ground (G)."
(The starting station located at the 0m point is named "0dam Station" ("da" means "deca" (ten)). The next station located at 180m to the east of the 0m point is named "18dam Station" ("18da" means 180).)

Next, if a purple man on the streetcar walks to the east at 25m/min, the trace of the purple walker from a standpoint of the streetcar could be illustrated as in "Streetcar (G)."

Then the trace of the purple walker on the streetcar from a standpoint of the ground could be illustrated as in figure "Galilean Transformation 1," if Galilean transformation is employed in accordance with ordinary sense. In the Galilean transformation, the vertical axis is inclined and the horizontal axis is horizontally moved.

The following 3 figures are simplified Minkowski spacetime involving Lorentz Transformation.
If a spaceship runs (to the east) at 83,333km/sec (5,000,000km/min = 5Mkm/min, "M" means "mega" (million)), the trace of the spaceship could be illustrated as in the following figure, "Ground (L)." The yellow 45° line shows the trace of a flash of light emitted to the east. It should be noted that the vertical and horizontal scales are arranged so that traces of lights could be expressed by 45° lines. For example, the length of vertical 1min corresponds to the length of horizontal 18,000,000km (1min * 300000km/sec (the speed of light)). (* is multiplication sign) (The scale of time is arranged to correspond to "time * c" (c: the speed of light).)

Next, if a purple bullet is shot to the east from the spaceship, the trace of the purple bullet from a standpoint of the spaceship could be illustrated as in "Spaceship (L)." In addition, the trace of a flash of light emitted to the east from the spaceship is shown as yellow 45° line. Because the speed of light for the spaceship is also 300,000km/sec based on the Michelson-Morley Experiment, the trace of the flash of light is 45° line.

Then the trace of the blue spaceship and the purple bullet could be illustrated as in "Lorentz Transformation 1," if Lorentz transformation is employed. In the Lorentz transformation, both the blue vertical and the blue horizontal axes are symmetrically inclined. In this case, the trace of the flash of the light emitted to the east from the spaceship could be again illustrated as 45° line to express the constant speed of light. Einstein found that the horizontal axis should be inclined to explain the constant speed of light. The Special Theory of Relativity is the theory to explain the constant speed of light.

The following figure shows meanings of the lines, to make sure.

The vertical line (1) shows the passage of time at 0Gm Station (the start point) on the ground (assumed stationary). It could be tentatively called Ground's 0Gm line.
(2) shows the passage of time at 36Gm Station located 36 million km (36 giga m) to the east of the 0Gm Station. It could be tentatively called Ground's 36Gm line.
The horizontal line (3) shows location on the ground to the east from the 0Gm Station at elapsed time of 0.0 min on the ground. It could be called Ground's 0.0min line.
(4) shows location on the ground to the east from the 0Gm Station at elapsed time of 2.0 min on the ground. (Ground's 2.0min line)
Then the intersection point of (2) and (4) is tentatively called here "Ground's point 36Gm-2.0min).
(5) shows the passage of time at the spaceship. (Spaceship's 0Gm line)
(6) shows the passage of time at 36 million km (36 giga m) to the east of the spaceship from a standpoint of the spaceship. (Spaceship's 36Gm line)
(7) shows location to the east from the spaceship from a standpoint of the spaceship at elapsed time of 0.0 min in the spaceship. (Spaceship's 0.0min line)
(8) shows location to the east from the spaceship from a standpoint of the spaceship at elapsed time of 2.0 min in the spaceship. (Spaceship's 2.0min line)
Then the intersection point of (6) and (8) is tentatively called here "Spaceship's point 36Gm-2.0min).
For example, if a flash of light is emitted to the east from the spaceship at elapsed time of 1.0 min for the spaceship, the trace of the flash can be illustrated as the upper yellow line in the following figure "Lorentz Transformation 3."

If speed of a spaceship is faster, it could be illustrated as in the following figure "Lorentz Transformation 4." The blue diamond shape gets sharper. Depending on the change of the diamond shape, the location of the intersection point of "Spaceship's 0.0 min line" and "Spaceship's 18Gm line" (Spaceship's 18Gm-0.0min) moves. The traces of the intersections would be illustrated as the red curves.


*The conclusion is that the Special Theory of Relativity seems correct, since various experiments and observations support it.

Time Dilation
Other than that, Time Dilation in the Special Theory of Relativity could be explained as follows. If a spaceship starts from "Spaceship's point 0Gm-0.0min" and 1.0 min passes in the spaceship, it reaches "Spaceship's point 0Gm-1.0min" (A1). From a standpoint of Ground, "Spaceship's point 0Gm-1.0min" (A1) locates clearly over "Ground's 1.0min line." This is Time Dilation. In this case, it is "Ground's 1.20min." Then if 1.0 min elapses in the spaceship, it corresponds to 1.20 min on the Ground.
In addition in this case, in relation to the Twin Paradox, it should be noted that if 1.0 min elapses on the Ground, it corresponds to 1.20 min in the spaceship.
As mentioned above, "Spaceship's point 0Gm-1.0min" corresponds to "Ground's 1.20min" from a standpoint of the Ground. In this case, "the same time" or "simultaneity" with "Spaceship's point 0Gm-1.0min" is determined based on Ground's coordinate system (horizontal lines). (Ground's coordinate system (brown squares here) could be referred to as Ground's "Reference Frame.")
In contrast, "Ground's point 0Gm-1.0min" corresponds to "Spaceship's 1.20min" from a standpoint of Spaceship, "the same time" or "simultaneity" with "Ground' point 0Gm-1.0min" is determined based on Spaceship's coordinate (lines gently upward to the right). (Spaceship's coordinate system (blue rhombuses here) could be referred to as Spaceship's Reference Frame.)
The time dilation merely comes from "relative deformation of coordinate systems (Reference Frames) including the time axes depending on the relative velocity" and the choice of coordinate systems (Reference Frames) including "simultaneity" (the way to determine the same time) about it.
Aside from that, if the spaceship returns to the earth, the time axis's deformation would be inverted.
* "Twin Paradox in Wikipedia" http://en.wikipedia.org/wiki/Twin_paradox
Consequently, in relation to the Twin Paradox, Ground's time and Spaceship's time accord each other (from a viewpoint of the Special Theory of Relativity).
(Yet, when the Spaceship turns with deceleration and acceleration, Spaceship's time passage is slightly slowed down through the General Theory of Relativity, to be precise.)
On the other hand, it should be noted that Spaceship elapsed 1.0 min is not seen at the Ground elapsed 1.20 min. Because image of the Spaceship is transmitted by light, the light with the image of the Spaceship is transmitted to the Ground (st 0Gm) at A2.

Length Contraction
Length Contraction in the Special Theory of Relativity could be explained as follows. If Spaceship-B starts from Spaceship's point 0Gm-0.0min and Spaceship-C starts from Spaceship's point 18Gm-0.0min (18Gm to the east of Spaceship-B) at the same time at the same speed keeping the interval of 18Gm, they trace on the Spaceship's 0Gm line and on the Spaceship's 18Gm line (keeping the interval of 18Gm). However, from a standpoint of the Ground, for example at 2.0 min on the Ground, Spaceship-B is recognized located at the intersection point of "Spaceship's 0Gm line and Ground's 2.0min line" and Spaceship-C is recognized located at the intersection point of "Spaceship's 18Gm line and Ground's 2.0min line." In this case the distance between Spaceship-B and Spaceship-C is is clearly shorter than 18Gm. This is Length Contraction.
However, it should be noted that Spaceship-B at B1 and Spaceship-C at C1 are not seen at 2.0min on the Ground (at 0Gm) as explained below. The lights (images) of Spaceships emitted from B2 and C2 reach Ground's 0Gm after 3 min as B3 and C3.

Calculation of Lorentz Transformation
Simplification is relevant for specific calculation involving Lorentz Transformation. 18Gm here and 1 min here are converted to "1" unit for the simplification. In addition, the horizontal axis is converted to "x axis" and the vertical axis is converted to "t axis." Then based on the theory of Lorentz transformation, for example, the trace of the intersection starts at "x=1, t=0" is illustrated as "1=x^2-t^2." Similarly, the trace of the intersection starts at "x=0, t=1" is illustrated as "1=t^2-x^2." On the other hand, the inclined blue line shows "x=t*v/c." Then "1=t^2-(t*v/c)^2=t^2*(1-(v/c)^2)." Then "t^2=1/(1-(v/c)^2)."
Then "t of A1" is "1/SQRT(1-(v/c)^2)." In this case, v=10Gm/min=166667km/sec, c=18Gm/min=300000km/sec, v/c=10/18, t=1/SQRT(1-(10/18)^2)=1.20.
"x of A1" is "v/SQRT(1-(v/c)^2)," because the blue line means "x=t*v/c."
1/SQRT(1-(v/c)^2) is defined as "Lorentz Factor," commonly represented by "γ".
* "Lorentz Factor in Wikipedia" http://en.wikipedia.org/wiki/Lorentz_factor
Such mathematics is essential for introduction to further consideration of Mass-Energy Equivalence.

Reference Frames
As mentioned above, a coordinate system would be called "Reference Frame." Specifically, since the Special Theory of Relativity is based on the special conditions on linear motion with constant speed, the reference frames here are Inertial Reference Frames.
* "Frame of Reference in Wikipedia" http://en.wikipedia.org/wiki/Frame_of_reference
* "Inertial Frame of Reference in Wikipedia" http://en.wikipedia.org/wiki/Inertial_frame_of_reference

Insignificance of Absolute Rest Reference Frame
In relation to the above explanation, one might assume an absolute rest (stationary) reference frame in the universe and in contrast a moving reference frame.
However, for example, the earth and the Milky Way Galaxy are moving in the universe, the reference frame based on the earth wouldn't be stationary. The absolute rest (stationary) reference frame in the universe wouldn't be specified. In addition, although the above examples wouldn't be identified as absolute rest, the Special Theory of Relativity well accounted for the constant speed of light. It implies that the absolute rest reference frame at least has no distinctive significance in the universe or possibly the absence of the absolute rest reference frame in the universe.
* "Rest in Wikipedia" http://en.wikipedia.org/wiki/Rest_(physics)

2.6.3.2.5 Einstein's Mass-Energy Equivalence in 1905 CE (the 4th paper of the Annus Mirabilis Papers)

2.6.3.2.5.1 Background

Einstein noted errors of Newtonian mechanics and necessity of Newtonian mechanics' correction. Newtonian mechanics consists of distance, time, velocity, mass, momentum, acceleration, gravity, gravitation, force, and energy. Einstein corrected distance, time, and velocity in the Special Theory of Relativity. Then other elements remained to be corrected. Einstein tried to harmonize Newtonian mechanics's formulas of momentum (p=m*v) and energy (E=1/2*m*v^2) with Lorentz Transformation seeking errors or unnaturalness lurking in Newtonian mechanics.

2.6.3.2.5.2 Einstein's claim of Mass-Energy Equivalence

Einstein claimed the accurate definition of momentum should be " p=γ*m*v " rather than Newtonian mechanics's " p=m*v ." (Lorentz Factor: γ=1/SQRT(1-(v/c)^2) ) (Newtonian mechanics's " p=m*v " can't presume enormous momentum over m*c in extreme cases of particle collisions near the speed of light. In contrast, Einstein's definition " p=γ*m*v " allows enormous momentum in extreme cases of particle collisions near the speed of light.) From this, " E^2=m^2*c^4 + c^2*p^2 " is derived. When a particle is stationary (p=0), " E=m*c^2 " as the reconsideration of the real nature of momentum. Einstein claimed "mass" includes enormous energy represented as E=m*c^2. Then "mass" and "energy" could be interchangeable (to a certain extent).
*Consequently, Mass-Energy Equivalence at least seems roughly correct to a certain extent, as the atomic bombs demonstrated an enormous amount of energy.
*It implies that mass and substances are a form of concentrated energy.
* "Mass-Energy Equivalence in Wikipedia" http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

2.6.3.2.5.3 Derivation

Mass-Energy Equivalence would be derived as follows.

Spacetime consists of 3-dimensional space (distance in 3 directions; x, y, and z) and 1-dimensional time.

Firstly, "w" is defined as " w=c*t ," where "c" is the speed of light, "t" is time. Since the dimension of "w" is "distance," "time" could be dealt with like "distance" through "w".

Secondly, " w^2-x^2-y^2-z^2 " is kept constant (invariant) through various Lorentz Transformations. " w^2-x^2-y^2-z^2 " is defined as "τ^2." "τ" (tau) is called "proper time," while the dimension of τ (proper time) is distance. Since x, y, z are similar, x, y, z could be tentatively represented by "x" for simple explanation. Then the above invariance could be tentatively simplified as " w^2-x^2=τ^2 " and explained as follows.
* "Proper Time in Wikipedia" http://en.wikipedia.org/wiki/Proper_time

In this case, the time axis' 1.0 min is converted to 18Gm, since 1.0 min * speed of light (300,000km/sec) =18,000,000,000 m=18 Gm. ("*" is a multiplication sign.)
L01 locates (0.0*18Gm, 0.0*18Gm). τ^2=(0.0^2-0.0^2)*18Gm^2=0*18Gm^2.
L02 locates (1.87*18Gm, 1.87*18Gm). τ^2=(1.87^2-1.87^2)*18Gm^2=0*18Gm^2.
The 45°line is τ^2=0.0*18Gm^2.
L-11 locates (1.20*18Gm, 0.667*18Gm). τ^2=(0.667^2-1.20^2)*18Gm^2= (0.44-1.44)*18Gm^2=-1*18Gm^2
L11 locates (0.667*18Gm, 1.20*18Gm). τ^2=(0.120*^2-0.667^2)*18Gm^2 =(1.44-0.44)*18Gm^2=1*18Gm^2
L12 locates (1.73*18Gm, 2.00*18Gm). τ^2=(2.0^2-1.73^2)*18Gm^2 =(4-2.99)*18Gm^2=1*18Gm^2
Thus the same τ^2's lie on a curve mentioned before.
The intersections' tracing curves correspond to curves representing invariant τ's.
L31 locates (2.0*18Gm, 3.6*18Gm). τ^2=(3.6*^2-2.0^2)*18Gm^2 =(13.0-4.0)*18Gm^2=9*18Gm^2
τ's would be thus illustrated in Minkowski Spacetime.
If the τ's red curves are regarded as contour lines, it would be like an entrance of a ravine (steep valley) where τ rises depending on the height.

Thirdly, "Four-Velocity" is defined as follows. Since "time" is similar to "distance (or space)" in Minkowski Spacetime, "time" is assumed to be familiarized with "distance" just for mathematical analysis introducing "Four-Velocity" and "Four-Momentum" mentioned later.
"Four-Velocity" in a narrow sense is U⁰, U¹, U², and U³ employing "large U" with numbers in index notation on the right shoulders. In this case, the numbers (0, 1, 2, 3) are not exponentiation or power but mere index notation. (One Index notation on the right shoulder technically signifies "contravariant vector(s)." For example, when the scale of a velocity vector "meter" is transformed into the 1/100 scale "centimeter," 6 meters/second is changed into 600 centimeters/second, then the number 6 is multiplied by 100. In this case, contrary to the scale, which became 1/100, the number contravariantly became 100 times.)
* "Covariance and Contravariance of Vectors in Wikipedia" https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
U⁰=c*dw/dτ
U¹=c*dx/dτ
U²=c*dy/dτ
U³=c*dz/dτ
* "Four-Velocity in Wikipedia" http://en.wikipedia.org/wiki/Four-velocity
dτ^2=dw^2-dx^2-dy^2-dz^2
1=(dw/dτ)^2-(dx/dτ)^2-(dy/dτ)^2-(dz/dτ)^2
c^2=U⁰^2 -U¹^2 -U²^2 -U³^2
On the other hand, (dτ/dw)^2=1-(dx/dw)^2-(dy/dw)^2-(dz/dw)^2
dw=d(c*t)=c*dt
(dτ/dw)^2=1-1/c^2*((dx/dt)^2+(dy/dt)^2+(dz/dt)^2)
(dτ/dw)^2=1-(v/c)^2
dτ/dw=SQRT(1-(v/c)^2)
U⁰=c/SQRT(1-(v/c)^2)=c*γ
U¹=c*dx/dτ=c*dw/dτ*dx/dw=c*γ*dx/dw=γ*dx/dt=γ*v_x
U²=c*γ*dy/dw=γ*dy/dt=γ*v_y
U³=c*γ*dz/dw=γ*dz/dt=γ*v_z

Fourthly, "Four-Momentum" is defined as follows.
P⁰=m*U⁰
P¹=m*U¹
P²=m*U²
P³=m*U³
The numbers on the right shoulders are index notation.
* "Four-Momentum in Wikipedia" http://en.wikipedia.org/wiki/Four-momentum

P¹=m*U¹ =γ*m*v_x
P²=m*U² =γ*m*v_y
P³=m*U³ =γ*m*v_z
P¹, P², and P³ would be summarized as "P=γ*m*v=p." As mentioned before, momentum of Newtonian mechanics (p=m*v) should be corrected to be "γ*m*v" through the Special Theory of Relativity.
These 3 P's proved out meaning momentum.

The question is the meaning of P⁰, while it would be something like momentum heading along time axis.
(m*c)^2=P⁰^2-p^2
P⁰=SQRT((m*c)^2+p^2)=m*c*SQRT(1+(p^2/(m*c)^2))
P⁰ could be approximated through Taylor expansion as follows.
P⁰=m*c*(1+p^2/(2*(m*c)^2))+....) =m*c+p^2/(2*m*c)+...
Since p^2/(2*m)=m*v^2/2 is kinetic energy in Newtonian mechanics, P⁰ seems meaning energy/c="E/c."
Consequently, the presumed derivation is " E^2=m^2*c^4 + c^2*p^2 ."

*As seen in the equation, momentum (p) would be somehow a primary variable for now.
*As explained before, the Special Theory of Relativity consists of space and time. The Special Theory of Relativity is not originally associated with mass. Mass-Energy Equivalence comes from " p=γ*m*v ."

2.6.3.2.6 Sundman's Three-Body Problem and Approximate Calculation in 1909 CE

In planetary orbit calculation of 2 planets, when 2 planets are simply assumed to be heavy 2 points in a plane space (neglecting their radiuses, high mountains, and deep valleys), their trajectories or orbits can be mathematically calculated through simultaneous equations. However, in planetary orbit calculation of 3 planets, when 3 planets are simply assumed to be heavy 3 points in a plane space, their trajectories or orbits would not be generally mathematically calculated through simultaneous equations. This is the Three-Body Problem.
Karl Frithiof Sundman proved that when the number of heavy points are 3, their trajectories or orbits would not be basically mathematically calculated in 1909 CE.
Then it was realized that in many problems of physics, approximate calculation should be reluctantly employed. (Assuming 2 planets to be heavy 2 points is already approximation, though.)
A useful way of approximate calculation would be firstly presuming the situation (allowing minor errors) at slightly elapsed time and then accumulating the presumptions slightly changing elapsed time. For example, firstly based at the elapsed time of t=0 second, the location at the elapsed time of t=0.1 second is presumed integrating inertial motion and gravitational attractive forces from other heavy points. Secondly, based at the elapsed time of t=0.1 second, the location at the elapsed time of t=0.2 second is presumed. Then thirdly, based at the elapsed time of t=0.2 second, the location at the elapsed time of t=0.3 second is presumed.
The approximate calculation is also called perturbation theory. Automated calculation devices would be practical for approximate calculation or perturbation theory.


Motion of Three Bodies
* "Three-Body Problem in Wikipedia" https://en.wikipedia.org/wiki/Three-body_problem
* "N-Body Problem in Wikipedia" https://en.wikipedia.org/wiki/N-body_problem
* "Perturbation Theory in Wikipedia" https://en.wikipedia.org/wiki/Perturbation_theory








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