Calculations and Interpretations of The Fundamental Constants

Calculations and Interpretations of The Fundamental Constants "The only true wisdom is in knowing you know nothing." ― Socrates By Manjunath.R #16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore560010, Karnataka, India *Corresponding Author Email: manjunath5496@gmail.com *Website: http://www.myw3schools.com/ Throughout all of the formulations of the basic equations of gravitation, quantum mechanics, electromagnetism, the nuclear physics and their application to the real world, there appear again and again certain fundamental invariant quantities called the fundamental physical constants – which are generally believed to be both universal in nature and have constant value in time. This book discusses the calculations and Interpretations of the Fundamental Constants which consistently appear in the basic equations of theoretical physics upon which the entire scientific study rests, nor are they properties of the fundamental particles of physics of which all matter is constituted. The speed of light signifies a maximum speed for any object while the fine-structure constant characterizes the strength of the electromagnetic interaction. An accurate knowledge of fundamental constants is therefore essential if we hope to achieve an accurate quantitative description of our physical universe. The careful study of the numerical values of the fundamental constants − as determined from various experiments − can in turn determine the overall consistency and correctness of the basic theories of physics themselves. I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Isaac Newton A set of fundamental invariant quantities that describes the strengths of all the interactions and the physical properties of all the particles observed in nature and appearing in the basic theoretical equations of physics "It Takes Fundamental Constants To Give Us Our Universe, But They Still Don't Give Everything." The speed of light (c) (ultimate speed limit) The conversion factor between the time dimension and the three space dimensions in our 4 dimensional space-time If particles with intrinsic mass exceed the speed of light, then c loses its special 186,000 miles per second status, giving rise to a host of other problems elsewhere in the world of physics, where c has been used in calculations, such as the equation in Albert Einstein's theory of special relativity that expresses the equivalence of mass and energy: E=mc2 Planck's constant (h) One of the smallest constants used in quantum mechanics that sets the {tells about the behavior of the particles and scale for quantum phenomena the waves on the atomic scale} (6.626 070 15 ×10−34 J Hz−1) Planck's constant defines the amount of energy that a electromagnetic radiation photon can carry − according to the frequency of the electromagnetic wave in which it travels Newtonian gravitational constant (G) One of the earliest fundamental constants that defines the strength of gravitational force The basis of our understanding of nonrelativistic gravity The constant relating the force of gravitational attraction between two objects to the product of their masses and the inverse square of the distance between them in Sir The Boltzmann constant (kB) relates temperature to energy. It is a fundamental Isaac Newton's universal law of gravitation: constant of physics occurring in nearly every F= statistical formulation of both classical and quantum physics. It is named after Austrian physicist and philosopher Ludwig Boltzmann, 6.673 × 10–11 N m2 kg–2 one of the pioneers of statistical mechanics. 1 𝐆m1 m2 r2 The idea of Quantum foam was PLANCK FORCE: devised by John Wheeler in 1955 The amount of force required to accelerate one Planck mass by one Planck acceleration: Planck force = Planck mass × Planck acceleration c4 G = √ ℏc G ×√ c7 ℏG FPlanck = 1.2103 × 1044 N The maximum force value that can be observed in nature appears in the Albert Einsteinian field equations describing the properties of a gravitational field surrounding any given mass: Einstein tensor = 8π × energy–momentum tensor 𝐏𝐥𝐚𝐧𝐜𝐤 𝐟𝐨𝐫𝐜𝐞 The Planck force describes how much or how easily space-time is curved by a given amount of mass-energy. The amount of energy possessed by a Schwarzschild Black Hole is equal to its mass multiplied by the square of the speed of light: E =Mc2 , where: c is not just the constant namely the maximum distance a light can travel in one second in vacuum but rather a fundamental feature of the way space and time are unified to form space-time. E= FPlanck 2 This means: Half of the Planck force is responsible for confining × rS the energy E =Mc2 of the Black Hole to a distance rS= The value of h is about 0.6 trillionths of a trillionth of a billionth of 1 joule-second. 2 2GM . c2 Any object with a physical radius < △p △x ≥ 2GM ℏ c2 will be a Black Hole. △E △t ≥ 2 Planck momentum × Planck length = ℏ ℏ 2 Planck energy × Planck time = ℏ The Planck mass is so large because the The Planck time is the time it takes for light to traverse a Planck length. gravitational force in this universe is very weak The Planck mass is approximately the mass of a black hole where quantum and gravitational effects are at the same scale: where its reduced Compton wavelength and half of its Schwarzschild radius are approximately the same. If √ ℏc5 G is confined to the volume of a cube of size √ ℏG c3 it will form a black hole. In fact, this is thought to be the smallest possible mass limit for a black hole and at Distance = √ Time = √ ℏG ℏG c3 it is thought that quantum gravitational effects will be very significant. c5 Energy = √ ℏc5  G Space-time would become chaotic quantum foam. Matter and antimatter would be constantly created and destroyed.  at Space-time would become quantized (which would cause violations of Lorentz invariance). 3 The attempt to understand the Hawking radiation has a profound impact upon the understanding of the Black Hole thermodynamics, leading to the description of what the black hole entropic energy is: Black Hole Entropic Energy = Black Hole Temperature × Black Hole Entropy ES = TBH × SBH = Mc2 2 This means that the entropic energy makes up half of the mass energy of the Black Hole. For a Black Hole of one solar mass (M☺ = 2 × 1030 kg), we get an entropic energy of 9 × 10 46 joules – much higher than the thermal entropic energy of the sun. Given that power emitted in Hawking radiation is the rate of energy loss of the black hole: P = − dMc2 dt = 2 ×− dES dt . The more power a black hole radiates per second, the more entropic energy being lost in Hawking radiation. However, the entropic energy of the black hole of one solar mass is about 9 × 10 46 joules of which only 4.502 × 10 –29 joules per second is lost in Hawking radiation. ES = This means: 1 4 FPlanck 4 × rS th of the Planck force is responsible for confining the entropic energy E S = (TBH × SBH) of the Black Hole to a distance rS= 2GM c2 . A photon sphere or photon ring is an area or region of space where gravity is so strong that photons are forced to travel in orbits. The radius of the photon sphere for a Schwarzschild Black Hole: r = 3GM . This equation entails that c2 photon spheres can only exist in the space surrounding an extremely compact object (a Black Hole or possibly an "ultracompact" neutron star). E = hυ The first "quantum" expression in history − stated by Max Planck in 1900 This means: E= FPlanck 3 4 ×r 1 rd of the Planck force 3 times the radius of the photon sphere equals the amount of energy possessed by a Schwarzschild Black Hole. "Nature shows us only the tail of the lion. But there is no doubt in my mind that the lion belongs Radiation Constants: with it even if he cannot reveal himself to the eye all at once because of his huge dimension. We see him only the way a louse sitting upon him would." — Albert Einstein Fundamental physical constants characterizing black body radiation. The first radiation constant is c1 = 2πhc2 = 3.7417749 × 10−16 Wm2, the second is c2 = hc kB = 1.438769 × 10–2 mK, where: h is the Planck constant c is the speed of light in vacuum and k B the Boltzmann constant. Radiation Heat flows through space by means of electromagnetic waves (elementary charge )2 (Planck charge)2 Fine structure constant: Sommerfeld's constant α= 1 4 α= e2 4πε0 ℏc = 1 4 Z0G0 𝐞𝐥𝐞𝐦𝐞𝐧𝐭𝐚𝐫𝐲 𝐜𝐡𝐚𝐫𝐠𝐞 = = 1 4 Z0G0 Planck charge 2 √Z0 G0 × impedance of free space × conductance quantum = 0.0072973525693 expresses the strength of the electromagnetic interaction between elementary charged particles. (elementary charge ) (Planck charge)2 2 When I die my first question to the Devil will be: What is the meaning of the fine structure constant? — Wolfgang Pauli 5 If e2 4πε0 ℏc were greater than 0.1, stellar fusion would be impossible and no place in the cosmos would be warm enough for carbon-based life as we know it. The ultra-high-energy cosmic ray observed in 1991 had a Planck units ℏc5 measured energy of about 2.5×10−8 √ G Planck mass Planck length Planck time Planck temperature Planck charge ℏc ℏG LPlanck = √ 3 c ℏG tPlanck = √ 5 c mPlanck = √ mPlanckc = mPlanckc2 = ℏ tPlanck mPlanck c2 Planck force LPlanck  interactions at this time interval. At this temperature, the wavelength of emitted ℏG ℏc5 G = 2.6121 × 10−70 m2 c3 c9 LPlanck =√ = ℏ thermal radiation reaches the Planck length. = 4.2217 × 10−105 m3 ℏc3 =√ G = 6.5249 kg-m/s = 1.9561 × 109 J ℏ LPlanck tPlanck = c4 G At which quantum effects of gravity become strong. = 1.2103 × 1044 N It is the gravitational attractive force of two bodies of one Planck mass each that are held one Planck length apart Planck scale are  undetectable with Quantum effects of gravity dominate physical = 1.875546 × 10−18 C ≈ 11.7e ℏ3 G3 L3Planck = √ Planck energy unified. = 1.416784 × 1032 K L2Planck = Planck momentum At which all the fundamental forces are = 5.391247 × 10−44 s qPlanck = √4πε0 ℏc Planck volume = 2.176434 × 10−8 kg = 1.616255 × 10−35 m ℏc5 TPlanck = √ 2 GkB Planck area Events happening at the G It is the electrostatic attractive or repulsive force of two Planck units of charges that are held one Planck length apart. current scientific technology 6 mPlanck c2 Planck power mPlanck Planck density L3Planck Planck acceleration Planck frequency = tPlanck c5 ℏG2 = c5 G = 3.628 × 1052 W The density at which the Universe can no longer be described without quantum gravity c7 =√ = 5.5608 × 1051 m/s2 tPlanck ℏG c tPlanck qPlanck tPlanck Planck voltage t2Planck = = 5.1550 × 1096 kg/m3 1 Planck current ℏ =√ ℏG = 1.8549 × 1043 s−1 4πε0 c6 =√ = 3.479 × 1025 A G mPlanck c2 qPlanck ℏc5 For energies approaching or exceeding√ G c5 =√ c4 4πε0 G = 1.43 × 1027 V = 1.22 × 1019 GeV , gravity is problematic and cannot be integrated with quantum mechanics. A new theory of quantum gravity is necessary. Approaches to this problem include:  String theory (point-like particles are replaced by one-dimensional infinitesimal vibrating strings − smaller than atoms, electrons or quarks)  M-theory (The Mother of all theories or Mystery − an 11 dimensional theory in which the weak and strong forces and gravity are unified and to which all the string theories belong) A theory that extends general theory of relativity by quantizing spacetime—predicts that black holes evolve into white holes  Loop quantum gravity (a theory of quantum gravity which aims to merge quantum mechanics and general theory of relativity)  Non-commutative geometry (a branch of mathematics concerned with a geometric approach to noncommutative algebra)  Causal set theory (an approach to quantum gravity that tries to replace the continuum spacetime structure of general relativity with the spacetime that has the property of discreteness and causality)  The study of how things influence one other  The study of how causes lead to effects 7 Martin Bojowald The idea of quantum foam arises out of Albert Einstein's idea that gravity is caused by the warping and curving of spacetime A German physicist who developed the application of loop quantum gravity to cosmology The incorporation of a standard model into the framework of the quantum gravity Loop Quantum Gravity String Theory (quantized space and time) Does not attempt to unify fundamental Attempts to unify all four fundamental interactions interactions Approaches the quantum gravity assuming Approaches the quantum gravity assuming the aspects of general relativity the aspects of quantum theory Does not require a super-symmetry Expanding matter Grand unification theory Fundamental symmetries existed at the The Coulomb constant White hole 1 4πε0 " is a proportionality constant in electrostatics equations. It was beginning of the universe and then broke Quantum transition as the temperature dropped − just as H2O which looks the same in every " Black hole named after the French physicist Charles-Augustin de Coulomb who introduced direction, freezes into ice, which has Coulomb's law. distinct directions. Contracting matter 8 Newton's law of gravitation: FG = Gm1 m2 FG ∝ r2 m1 m2 G → Proportionality constant r2 m1 = m2 = 1kg The universal gravitational constant is numerically r = 1m FG = G equal to the Force of attraction between two unit masses placed at a unit distance apart. Because E=mc2: FG = FG ∝ GE1 E2 c4 r2 E1 E2 r2 E1 = E2 = 1J r = 1m FG = 1 FPlanck 1 FPlanck → Proportionality constant The reciprocal of Planck force is numerically equal to the Force of attraction between two unit energies placed at a unit distance apart. (Stoney mass)2 Fine structure constant: α = Stoney mass = e2 Z0 2RK 4π × ℏc × gravitoelectric gravitational constant = impedance of free space 2 × von Klitzing constant (Stoney mass)2 √4πε (Planck mass) 0G 9 2= e2 q2Planck elementary charge Stoney mass = Planck mass × ℏ △x △p ≥ 2 ℏ △E △t ≥ △x △p ≥ Planck charge 2 Planck length × Planck momentum △E △t ≥ 2 Planck energy × Planck time 2 △p Planck momentum △E Planck energy Gravitoelectric gravitational constant: εg = ≥ Planck length △𝑥 Planck time △t 1 4πG Gravitomagnetic gravitational constant: μg = ≥ 4πG c2g 10 The speed of gravitation: cg = 1 √εg μg The Schwarzschild radius of the Stoney mass: rS = Planck force = 2GmS c2 2G = c2 e2 √4πε 0G 4ℏc × fine structure constant r2S Optical Telescope A telescope that is designed to collect visible light If we take the mass of electron as m, when it is moving with velocity v, then m= me 2 √1−v2 c where me is the rest mass of the electron and m is the relativistic mass. m2 = e2 m2e v2 1− 2 c : If m = Stoney mass =√ 4πε0 G v = c √1 − Hypernova an exploding star that produces even more energy and light than a supernova Schwarzschild radius of electron 2 × Classical electron radius Velocity a electron must travel so that its relativistic mass to be equal to Stoney mass 11 The Compton wavelength of the Stoney mass: λC = λC = h mS c 4πε0 G h = ×√ c e2 2π×Planck length √Fine structure constant The time it takes for a planet to complete one spin around its axis is called its rotation period. The Hawking radiation temperature is: TBH = e2 If M = Stoney mass =√ : 4πε0 G TBH = TBH = ℏc3 8πGMkB ℏc3 8πGkB 4πε0 G √ e2 Planck energy 8πkB √Fine structure constant TBH = Planck temperature 8π√Fine structure constant Observatory: A place where telescopes and other astronomical instruments are housed and used. 12 ℏc5 If a hot body were to reach the temperature of√ 2 , the radiation it would emit would have a GkB ℏG wavelength of√ 3 , at which quantum gravitational effects become relevant. c Planck temperature which equals about 100 million million million million million degrees, ℏc5 denoted by TPlanck =√ 2 , is the unit of temperature in the system of natural units known as GkB Planck units. The Planck temperature is thought to be the upper limit of temperature that we know of according to the standard model of particle physics − which governs our universe. In physics the Stoney units form a system A fundamental limit of quantum theory in of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881 where: LPlanck = ℏG √ c3 ℏc5 combination with gravitation − first c2 TPlanck =√ 2 = GkB 2πLPlanck introduced in 1899 by German physicist Max Planck together with his introduction of what today is known as the Planck length, the Planck mass and Planck time. is the Planck length and c2 is the second radiation constant. This means: TPlanck × LPlanck can never be less than or greater than c2 c2 but = . 2π 2π When the gold particles were smashed together, for a split second, the temperature reached 7.2 trillion degrees Fahrenheit. That was hotter than a supernova explosion. That was the hottest temperature that we have ever actually encountered in the Large Hadron Collider (the world's largest and most powerful particle accelerator). The universe was Ge2 Stoney length = √ = √α × Planck length 4πε0 c4 TPlanck × Stoney length = 13 at TPlanck about 10−43 seconds after the big bang explosion. √α c2 2π At this time, the entire universe was roughly one-billionth of the diameter of a proton. Planck density c5 ℏG2 Hagedorn temperature is very large − {1.7×1012 K} about equivalent to 10 23 solar masses No temperature → No heat exchange. squeezed into the space of a single atomic nucleus. At Planck time after mass density was thought to have been m= approximately 5.1550 × 10 kg/m .  is no longer stable and must either "evaporate" or convert into quark matter − the Big Bang explosion, the cosmic 96 The temperature at which hadronic matter 3 m0 as such − it can be thought of as the 2 √1−v2 c "boiling point" of hadronic matter . When the velocity of the particle v is very small compared to velocity of light c, then negligible compared to one. Therefore, v2 is c2 m = m0  If the velocity of the particle v is comparable to the velocity of light c, then √1 − one, then v2 c2 is less than m> m0  If the velocity of a particle v is equal to velocity of light c, then it possesses infinite mass. FG = FG = Gm1 m2 m1 m2 m2Planck 2 m1m2 = mPlanck FG = ℏc r2 r2 qPlanck = × FE = ℏc r2 FE =  m1= m2 = mPlanck  m1 > mPlanck and m2 < mPlanck elementary charge √fine structure constant 14 2 q1q2 = q Planck q1 q2 4πε0 r2 q1 q2 2 qPlanck  q1= q2 = qPlanck  q1 > qPlanck and q2 < qPlanck × ℏc r2 FE = ℏc r2 The rest mass energy of any particle is defined by the Albert Einstein's mass energy equivalence relation: Erest = m0c2 = kBTthreshold, where: m0 is the mass of a stationary particle, also known as the invariant mass or the rest mass of the particle and Tthreshold implies the threshold temperature below which that particle is effectively removed from the universe. All particles have an intrinsic real internal vibration in their rest frame: ʋC = m0 c2 h = c λC , where: υC and λC denote the quantum mechanical properties of a particle (i.e., the Compton frequency and Compton wavelength of the particle). hʋC = hc = kBTthreshold λC λC × Tthreshold = c2 where: c2 is the second radiation constant and is related to the Stefan–Boltzmann constant (also known as Stefan's constant) by: σ = π4 c1 15c42 . This means: λC ∝ 1 Tthreshold The Compton wavelength of the particle is inversely proportional to the threshold temperature below which that particle is effectively removed from the universe. TPlanck × LPlanck = TPlanck × LPlanck = c2 2π λC × Tthreshold 2π (λC × Tthreshold) > (TPlanck × LPlanck) rS × λC = 2 × L2Planck = 2 × Planck area, where: λC = ℏ m0 c is the reduced Compton wavelength of the particle. This means: The Schwarzschild radius of the particle times the reduced Compton wavelength of the particle is never smaller than a certain quantity, which is known as Planck area. 15 If the reduced Compton wavelength of the particle = Stoney length: Ge2 =√ m0 c 4πε0 c4 ℏ m0 = mPlanck √Fine structure constant Mass a particle must possess so that its reduced Compton wavelength to be equal to Stoney length 2 Erest = m0c2 × hυC Erest = Planck energy √ Erest = Stoney mass × c2 √ π × Schwarzschild radius of the particle Compton wavelength of the particle π × Schwarzschild radius of the particle Fine structure constant × Compton wavelength of the particle Sunspot If the Schwarzschild radius of the particle = Stoney length: A cooler region of the Solar surface − which looks dark in comparison to the 2Gm0 c2 m0 = Ge2 =√ 4πε0 c4 hotter material around it. √Fine structure constant ×Planck mass 2 Mass a particle must possess so that its Schwarzschild radius to be equal to Stoney length Planetary Nebula A shell of gas ejected by a relatively low-mass star that is in the process of dying and becoming a white dwarf 16 Planck temperature = mPlanck c2 kB Planck area = L2Planck Planck area = Planck volume = Planck energy = Planck energy = Planck force = ℏ tPlanck mPlanck c2 ℏ Planck force = LPlanck tPlanck Planck momentum = mPlanck c Planck power =  mS = Stoney mass  LS = Stoney length  tS = Stoney time  α = Fine structure constant Planck density = L3Planck Planck frequency = c 3 √α ℏ √α tS αℏ LS tS c LPlanck Planck power = tPlanck  mS = elementary charge  LS = elementary charge  tS = celestial body passes directly between a larger body and the observer. 17 Planck charge Planck charge elementary charge Planck charge √α L3S Planck frequency = mPlanck c2 mS c α mS Planck acceleration = tPlanck Astronomical transit is a phenomenon when a L3S Planck momentum = mPlanck Planck acceleration = α m c2 Planck force = LS S LPlanck Planck density = √α kB L2S Planck volume = L3Planck Planck force = mS c2 Planck temperature = c √α tS c √α LS mS c2 tS × Planck mass × Planck length × Planck time ℏc PLANCK MASS: mPlanck = √ = 2.17647 × 10−8 kg, where: c is the speed of light in a vacuum, G G is the gravitational constant, and ħ is the reduced Planck constant. mPlanck =n m0 mPlanck c2 m0 c2 = Number of particle masses that make up one Planck mass. kB TPlanck kB Tthreshold =n TPlanck = n × Tthreshold λC × Tthreshold = c2 λC = c2 TPlanck ×n λC × TPlanck n = c2 λC ∝ n This means: The Compton wavelength of the particle is directly proportional to the number of particle masses that make up one Planck mass. Planck charge Planck mass ε = √4πε0 G = √ 0 = √ Gravitoelectric gravitational constant εg electron charge electron mass Vacuum permittivity = − 1.75882001076×1011 C/ kg proton charge proton mass 18 = + 9.58 × 107 C/ kg When negatively charged electrons move in electric and magnetic fields the following two laws apply:     F = e (E + v × B) → Lorentz force law F = mea = me dv dt me → Newton's second law of motion Felectric = eE Fmagnetic = eBv When equal: v = = e electron mass electron charge = (E + v × B) a The Planck length ≈ 1.616255 ×10−35 m E B is the scale at which classical ideas about gravity and space-time cease to be valid and quantum effects dominate. Fine structure constant = μ0 c 2RK = Vacuum permeability × Planck speed 2 ×von Klitzing constant Stoney mass = √ Vacuum permeability × Planck speed 2 × von Klitzing constant × Planck mass The gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. αG is typically defined in terms of the gravitational attraction between two electrons. More precisely, αG = Gm2e ℏc = m2e m2Planck where: me is the invariant mass of an electron αG = α × m2e m2S Number of electrons that make up one Planck mass = n= 1 √αG 19  mS = Stoney mass  α = Fine structure constant mPlanck me = 1 √αG The Compton wavelength of electron: c2 λC,e = n × λC,e ∝ 1 λC,e = TPlanck 1 √αG × c2 TPlanck The Compton wavelength of the electron is inversely proportional to the √αG square root of gravitational coupling constant. Gravitational characteristic impedance of free space = 4πG cg Speed of gravitation Quantum of circulation: Half the ratio of the Planck constant to the mass of the electron. Q0 = h 2me = 3.636 947 5516 × 10−4 m2 s−1 Erest = mec2 = Erest ∝ Q0 = λC,e = h 2me = h 2√αG mPlanck =√ c1 = first radiation constant 4πQ0 The intrinsic energy of the electron is 1 Q0 α αG h c1 × inversely proportional to the Quantum of circulation h 2mS  mS = Stoney mass  α = Fine structure constant is the cutoff below which quantum field theory (which can describe particle creation and me c annihilation) becomes important. 2Q0 2h λC,e = = = 2Q0 √ε0 μ0 2me c c 20 The classical electron radius is sometimes known as the Compton radius or the Lorentz radius or the Thomson scattering length is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. The classical electron radius is defined by equating the electrostatic potential energy of a sphere of charge e and radius re with the intrinsic energy of the electron: re = e2 4πε0 re e2 4πε0 me c2 = Fine structure constant × reduced Compton wavelength of the electron re = e2 4πε0 √αG mPlanck c2 re = √ α αG △x △p ≥ △x △p ≥ △p m0 c = mec2 √αG For an electron, the Thomson cross-section is numerically given by: 2 σT = λC × m0 c 2 ≥ α ×LPlanck × Stoney length = 2.8179 × 10−15m ℏ λC = σT = 8π 3 × α αG 8πr2e 3 × (Stoney length) 2 Classical electron radius = Bohr radius × (Fine structure constant) 2 △x Fine structure constant = √ 21 Classical electron radius Bohr radius Bohr radius: a0 = 4πε0 ℏ2 me e2 = ℏ = me c α a0 = ℏ √αG mPlanck c α The mean radius of the orbit of an electron LPlanck around the nucleus of a hydrogen atom at its √αG × α ground state (lowest-energy level) 5.29177210903×10−11 m a0 = a0 = ε0 h2 π me e2 = ε0 π ×2( h 2me )× h e2 2ε0 × Quantum of circulation × von Klitzing constant π a0 = 2 × Quantum of circulation × von Klitzing constant πμ0 c2 Wien's Displacement Law The product of the peak wavelength and the temperature at λPeak × T = b which a blackbody radiates is constant − which means the peak of the radiation shifts to shorter wavelengths as the temperature increases. Wien's constant: b = hc 4.9651kB = c2 4.9651 c2 = 4.9651 b The second radiation constant is 4.9651 times the Wien's constant 22 Radiation density constant: a= 4σ c = 8π5 k4B 15c3 h3 = a= 8π5 kB 15c32 4σ c = = 7.5657 × 10−16 J m−3 K− 4 4π4 c1 √μ0 ε0 15c42 where: μ0 is the absolute permeability of free space and ε0 is the absolute permittivity of free space. 8π5 kB 15c32 kB = c1 2πc2 = 4π4 c1 15c42 √μ0 ε0 √μ0 ε0 = 1.3807 × 10−23 J/K kB = c1 31.180 b √μ0 ε0 Magnetic flux quantum: Φ0 = h 2e Conductance quantum: G0 = 2e2 h Φ0 × G0 = e 23 where: e is the elementary charge. Φ0 × G0 = √Fine structure constant × qPlanck Planck charge = Magnetic flux quantum × Conductance quantum √Fine structure constant von Klitzing constant: RK = h e2 = h RK = e 2 = h ϕ20 G20 h α × q2Planck Conductance quantum: G0 = 2e2 h = 2α × q2Planck h = 2 RK The magnetic coupling constant: β= β= Bohr radius is about 19,000 times bigger than the classical electron radius ε0 hc 2e2 = 1 4α = ε0 hc 2e2 = m2S 4m2Planck A fundamental physical constant πℏ cμ0 e2 = characterizing the strength of the magnetic force interaction L2S 4L2Planck = t2S 4t2Planck Bohr radius β=√ 16 ×classical electron radius 24 c= Time is relative 1 cg = √μ0 ε0 1 √μg εg μ0 ε0 = μg εg μ0 It changes with speed and in the presence μg of gravity = If Gravity travel at the Speed of Light εg ε0 The Planck charge √4πε0 ℏc is approximately 11.706 times greater than electron charge. Φ0 × G0 × RK = h e Magnetic flux quantum × Conductance quantum × von Klitzing constant = Quantum / Charge Ratio Φ0 × G0 × RK = h √Fine structure constant ×qPlanck ϕ0 × G0 × RK 2Q0 = Electron mass-to-charge ratio Planck charge: qPlanck = √4πε0 ℏc q2Planck = 4πε0 ℏc = 2h √ qPlanck tPlanck ε0 × qPlanck = 4π√ μ0 ε0 μ0 × ℏ tPlanck ε0 Planck conductance = 4π√ μ0 ε0 Planck current × qPlanck = 4π√ × Planck energy μ0 ε0 Planck current × qPlanck = 4π√ × (qPlanck × Planck voltage) μ0 25 1 Planck resistance = 4π√ ε0 μ0 Admittance of free space: Impedance of free space: Y0 = μ0 Z0 = μ0c = √ ε0 q2Planck = 2h Z0 q2Planck = qPlanck = e √ 1 Z0 = 2h × Y0 2RK e2 Z0 2R 2RK = ϕ0 G0√ K Z0 Z0 Stefan–Boltzmann law: The radiative power of a black body is proportional to the surface area and to the fourth power of the black body's temperature P = εσT4A Emissivity Stellar Planck constant: hS = 2 × M × R × CS  M : mass of the neutron star  R: radius of the neutron star  CS: the characteristic speed of the particles in the neutron star  For all substances: ε < 1  For a perfect black body: ε = 1 Stellar Stefan–Boltzmann constant: ΣS = Luminosity of the galaxy Area of the galaxy × (Effective kinetic temperature of the stellar gas of the galaxy)4 26 Rydberg constant: R∞ = R∞ = 1 me e4 8ε20 ch3 = Fine structure constant 4π ×Bohr radius = 10 973 731.6 m−1 Fine structure constant μ 0 × √ von Klitzing constant 4 ε 0 × Compton wavelength of the electron Rydberg energy: hc R∞ = me c2 4 μ 0 √ε × 0 Fine structure constant von Klitzing constant Rydberg frequency: c R∞ = Compton frequency of the electron 4 μ 0 √ε × 0 Fine structure constant von Klitzing constant Rydberg wavelength: 1 R∞ =4√ ε0 μ0 × von Klitzing constant × Compton wavelength of the electron Fine structure constant Hartree energy: Eh = 2R∞ hc = me c2 2 μ 0 √ε × 0 Fine structure constant von Klitzing constant 27 = 4.3597447222071 × 10−18 J R∞ = Fine structure constant 4π ×Bohr radius Fine structure constant = qPlanck = R∞ = e2 q2Planck = 4π × Bohr radius × R∞ e √4π × Bohr radius × R∞ Fine structure constant 4π ×Bohr radius Classical electron radius Fine structure constant = √ = 4π × Bohr radius × R∞ Bohr radius R∞ = 1 R∞ = Fine structure constant = Classical electron radius √ 4π (Bohr radius )3 Fine structure constant 4π ×Bohr radius Conductance quantum × impedance of free space R∞ = 4 = 4π × Bohr radius × R∞ Conductance quantum × impedance of free space 16π × Bohr radius △S0 + SBH ≥ 0 The sum of the entropy outside the black hole and the total black hole entropy never decreases and typically increases as a consequence of generic transformations of the black hole. 28 Nernst-Simon statement The entropy of a system at absolute zero temperature either vanishes or becomes independent of the intensive thermodynamic parameters The Bohr magneton is defined in SI units by: μB = eℏ 2me = Faraday constant ×Planck angular momentum 2 × molar electron mass μB = 9.27400968 ×10 −24JT−1 = √α ×qPlanck × Q0 2π ϕ0 ×G0 × Q0 2π Conductance quantum = eℏ 2mp = ϕ0 ×G0 μN The Nuclear magneton is defined in SI units by: μN = 2π μB μB Faraday constant ×Planck angular momentum 2 × molar proton mass = 5.050783699 ×10−27JT−1 Fine structure constant μN = √ × Planck charge × reduced Compton wavelength of proton 4μ0 ε0 Planck angular momentum = mPlanck × c × LPlanck = ℏ Planck angular momentum = 29 mS × c × LS α = me mp Black Hole: A great amount of matter packed into a very small area where gravity is intense enough to prevent the escape of even the fastest moving particles. Not even light can break free. Temperature → TBH = ℏc3 8πGMkB Evaporation time of a black hole: TBH TPlanck Density → ρBH = where: ρPlanck = M 4πr3 s 3 c5 = = tev = 8πM tev tPlanck 3c6 32πG3 M2 ρBH ρPlanck ℏG2 mPlanck = 480c2 V ℏG = 480 × V L3Planck m2Planck 32πM2 is the Planck density. If the star core's mass is more than about three times the mass of the Sun, the force of gravity The rate of evaporation energy loss of the black hole: P=− dMc2 dt = P where: PPlanck = c5 G Entropy → SBH = PPlanck ℏc6 overwhelms all other forces and produces a black hole. 15360πG2 M2 = m2Planck 15360πM2 is the Planck power. 4πkB M2 m2Planck SBH SPlanck = 4πM2 m2Planck 30 where: SPlanck = kB is the Planck entropy. ℏc If M = √ → Planck mass: G   TBH = ρBH = TPlanck 8π If V = ρPlanck 32π L3Planck → Planck volume: tev = 480 × tPlanck = PPlanck  P=  SBH = 4π × SPlanck 15360π 480tS √α Compton shift: △λ = If △λ = Stoney length: h me c √α × LPlanck = (1−cosθ) h me c (1−cosθ) √α α θ = cos−1 (1 − 2πG ) √α α The wavelength shift of the scattered photon in an angle of θ = cos−1 (1 − 2πG ) is equal to the Stoney length. Second radiation constant: c2 = 2π ℏc kB = 2π × Planck angular momentum × Planck speed Planck entropy 31 If △λ = classical electron radius: e2 4πε0 me c2 = h me c θ = cos−1 (1− (1−cosθ) α 2π ) The wavelength shift of the scattered photon in an angle of θ = cos−1 (1− If △λ = Bohr radius: 4πε0 ℏ2 me e2 = h me c θ = cos−1 (1− α 2π ) is equal to the Classical electron radius . (1−cosθ) 1 2πα ) The wavelength shift of the scattered photon in an angle of θ = cos−1 (1− 1 2πα First radiation constant: c1 = 4π2ℏc2 ) is equal to the Bohr radius . c1 = 4π2 × Planck angular momentum × (Planck speed) 2 32 Spin-statistics connection theorem:  Fermions (such as electrons and protons) having a half integer spin must be described by Fermi-Dirac statistics  Bosons (such as photons and helium-4 atoms) having an integer spin must be described by Bose-Einstein statistics. 2GM : c2 The time it takes for light to travel a distance equal to 2GM τ1 = E= c2 PPlanck 2 where: E is the energy of the black hole and P Planck = 1 × c × τ1 c5 G is the Planck power. The time it takes for light to travel a distance equal to Stoney length: τ2 = LS c = √α × LPlanck c τ2 =√α × t Planck The time it takes for light to travel a distance equal to τ3 = h me c τ3 = × 1 c h me c2 33 = h : me c h me c2 = 1 υC = 2Q0 × μ0 × ε0 c1 = 2πhc2 c2 = c1 c2 c1 c2 hc kB = 2πckB = 2π × Planck speed × Planck entropy Unruh temperature: TU = ℏa 2πkB c where: ħ is the reduced Planck constant, a is the local acceleration, c is the speed of light and kB is the Boltzmann constant.  a of 2.47 × 1020 m/s2 corresponds approximately to a TU of 1 K.  a of 1 m/s2 corresponds approximately to a TU of 4.06 ×10−21 K. TU = ℏa c2 c1 = Planck angular momentum × a × c2 c1 Hawking–Unruh temperature: TH = where: g is the surface gravity of a black hole. TH = ℏg 2πkB c c2 g √μ0ε0 4π2 34 PCT theorem All interactions are invariant The vacuum energy density or dark energy density is defined as: under the Charge, parity and time reversal symmetry εΛ = c4 8πG Λ = cosmological constant ×Λ The mass density corresponding to the vacuum energy density is expressed as: ρΛ = εΛ c2 The act of tearing space apart resulting in a sort of "reverse singularity" − where space and time If dark energy gets stronger and stronger over time, it can either be reborn or can disappear into will eventually overcome gravitational force of attraction and then everything is torn apart. nothingness. Big Rip The ultimate fate of the universe − in which the matter of the universe and even the fabric of spacetime itself − is progressively torn apart by the expansion of the universe at a certain time in the future − until distances between single atoms will become infinite. Dark energy  the cosmological constant from General theory of Relativity  the zero-point energy inherent to space from quantum field theory maintains a constant energy density and would cause all galaxies to recede from each other at speeds proportional to their distance of separation. h, c Quantum Field Theory and the standard model of particle physics G, c General Theory of Relativity (geometric theory of gravitation) and the standard model of cosmology h, kB Quantum Statistics and Modern quantum physics 35 Second radiation constant: c2 =  hc = kB NA h NA kB ×c NA = Avogadro number (the number of particles that are contained in one mole of a substance) 6.02214076 × 1023 c2 = Molar Planck constant Ideal gas constant F R = NA e = NA kB F R 1 √μ0 ε0 Molar electron charge Ideal gas constant F = = KJ R × 2 e hc × c2 × c2 √μ0 ε0 where: F is the Faraday constant and KJ is the Josephson constant. Quantum of circulation = h 2me = Molar Planck constant 2 ×Molar electron mass The Avogadro number is named after the Italian scientist Amedeo Avogadro – who − in 1811 − first proposed that the equal volumes of gases under the same conditions of temperature and pressure will contain equal numbers of molecules. 36 Q0 × rS = Q0 × rS = 2π Gℏ c2 h 2me = 2π × 2Gme c2 Planck volume Planck time Planck volumetric flow rate = Energy mass Planck Energy Planck mass Q0 × r S 2π = Specific energy = Planck Specific energy = c2 Planck specific energy = (Planck speed) 2 Hawking radiation temperature: TBH = Unruh temperature: ℏc3 8πGMkB = 6 × 10−8 K c4 4GM Mass of the black hole TU = Black hole's gravitational acceleration If a = Solar mass : ℏa 2πkB c TU = TBH The temperature of the vacuum − observed by an isolated observer accelerating at the Black hole's gravitational acceleration of g = c4 m/s2 is Hawking radiation temperature. 4GM 37 rS × TBH = 2GM c2 × rS × TBH = ℏc3 8πGMkB c2 8π2 This means: rS × TBH can never be less than or greater than Unruh temperature = If Unruh temperature = Planck temperature: TPlanck = ℏa 2πkB c c2 8π2 but = c2 8π2 ℏa 2πkBc → a = 2π × aPlanck If a = Planck acceleration: TU = 2e  Josephson constant: KJ =  Magnetic flux quantum: ϕ0 = KJ × ϕ0 = 1 h h 2e ℏaPlanck 2πkB c → TU = TPlanck 2π  Conductance quantum: G0 =  Resistance quantum: R0 = 2e2 G0 × R0 = 1 38 h 2e2 h . Modified Newtonian dynamics Hypothesis proposing a modification of Newton's Schwarzschild radius of electron: rS = law of universal gravitation to account for observed 2Gme properties of galaxies c2 The threshold temperature below which the electron is effectively removed from the universe: Tthreshold = me c2 kB rS × Tthreshold = rS × Tthreshold = 2Gm2e kB αG × c2 π Irradiation KE = e × V KE = √α × qPlanck × V KE EPlanck If V = Planck voltage: = √α × The process by which an object is exposed to radiation V VPlanck KE = √α × EPlanck Planck voltage: VPlanck = Planck energy Planck charge =√ c4 4πε0G = √Planck force × Coulomb constant Planck current: IPlanck = Planck charge Planck time = 4πε0 c6 √ G Planck force × Planck specific energy =√ 39 Coulomb constant Planck pressure: ΠPlanck = Planck force Planck area = c7 ℏG2 = ℏ L3Planck tPlanck = α2 ℏ L3S tS = 4.633 × 10113 Pa Most of the matter in the Universe is dark Dark Matter → nonluminous and it looks like a matter Why does it gravitate as ordinary matter does, and thus slows the expansion of the universe? ΠPlanck = ΠPlanck = h me c L3S tS classical electron radius Bohr radius Planck acceleration = λC,e = α2 ℏ × ℏ L3S tS Planck frequency √ε0 μ0 = 𝟐 × 𝐌𝐚𝐠𝐧𝐞𝐭𝐢𝐜 𝐟𝐥𝐮𝐱 𝐪𝐮𝐚𝐧𝐭𝐮𝐦× Electron Charge to mass ratio × √μ0ε0 40 Both Albert Einstein's and Sir Isaac Newton's theories of gravitation have a problem when they encounter quantum mechanics and that problem involves the very nature of space and time. ℏG √ c3 = LPlanck → a fundamental limit to space ℏG √ c5 = tPlanck → a fundamental limit to time S = kB lnW This equation takes pride of place on the Ludwig Eduard Boltzmann's grave in the A measure of statistical disorder of a system Zentralfriedhof, Vienna. If S = Planck entropy = kB: W = e = 2.718281828459045 Planck capacitance = Planck charge Planck voltage 4πε0 G = √4πℏcε0 × √ c4 Planck capacitance = Coulomb constant × Planck length 1 4πε0 = μ0 c2 4π = μ0 4π × Planck specific energy= 41 μ0 4π × (Planck speed) 2 Q = ne × e dQ dt = I= dne dt dne dt If I = Planck current = Planck charge : Planck time qPlanck tPlanck ×e dne ×e dt = = dne dt ×e 1 √α tPlanck Rate of flow of electrons = 1 Stoney time "The infinite is nowhere to be found in reality, no matter what experiences, observations, and Standard gravitational parameter: knowledge are appealed to" − David Hilbert μ = GM For Planck mass: μ = GmPlanck = √G × ℏ × c GMsun For Stoney mass: μ = GmS = √α GmPlanck = √α × G × ℏ × c Heliocentric gravitational constant Classical electron radius: re = 1 4πε0 me c2 × e2 The threshold temperature below which the electron is effectively removed from the universe: Tthreshold = me c2 kB re × Tthreshold = 42 α ×c2 2π Bohr radius: ℏ a0 = me cα a0 × Tthreshold = c2 2πα Black Hole Density: ρBH = ρPlanck × e2 : If M = mS = √ 4πε0 G m2Planck 32πM2 ρBH = = ρPlanck × m2S 32π α M2 ρPlanck 32π α The rate of evaporation energy loss of the black hole: P = PPlanck × e2 If M = mS = √ : 4πε0 G Black hole Entropy: SBH = SPlanck × m2Planck 15360πM2 P= 4πM2 = PPlanck × m2S 15360 π α M2 PPlanck 15360 π α m2Planck = 43 4π α M2 m2S e2 If M = mS = √ : 4πε0 G SBH = SPlanck × 4π α P × SBH = PPlanck × P × SBH = m2Planck 15360πM 4πM2 m2Planck Planck power × Planck entropy 3840 ρBH × SBH = ρPlanck × ρBH × SBH = × SPlanck × 2 m2Planck 32πM2 × SPlanck × 8 Rydberg constant = 2 3 λR∞ × Tthreshold = 8εm0he4c × e λR∞ × Tthreshold = 44 2c2 α2 8ε20 h3 c me e4 me c2 kB 3840G m2Planck Planck density × Planck entropy 1 kB c5 4πM2 Rydberg wavelength: λ R∞ = = = kB c5 8ℏG2 rS × re = 2Gme c2 1 × 4πε0 me c2 rS × re = 2αL2Planck rS × re = 2 × Fine structure constant × Planck area rS × a0 = 2Gme c2 × 2 ℏ me cα 2L rS × a0 = Planck α rS × a0 = 2 × Planck area Fine structure constant rS × re = 2L2S rS × a0 = 2L2S α2 Mach's Principle The inertia of the mass is caused by all other masses in the entire universe 45 λR∞ × rS = 2 3 8ε0h c me e4 λR∞ × rS = × 2Gme c2 8πL2Planck α2 8π ×Planck area λR∞ × rS = α2 λR∞ × rS = 8πL2S α3 Science aims at constructing a world which shall be symbolic of the world of commonplace experience. − Arthur Eddington   Stoney length = LS = √ Ge2 4πε0 Stoney time = TS = √ Planck force μ 0 =e√ 6 4π × Planck force 4πε0 c c= Refractive index: n = Ge2 Coulomb constant =e√ c4 c v = LPlanck tPlanck 1 v√ε0μ0 = = LS tS LS = √Planck specific energy v × tS = 46 Planck speed v First radiation constant: (Stoney length)2 c1 = 4π × Planck angular momentum × 2 (Stoney time)2 Bohr's Quantization Rule: L = nℏ n= For n = 1: electron angular momentum Planck angular momentum Electron angular momentum = Planck angular momentum Second radiation constant: c2 = hc kB = molar Planck constant Ideal gas constant × Stoney length Stoney time Radiation Constant = 4 × Stefan-Boltzmann constant × Black hole temperature: TBH = Stoney time Stoney length ℏc3 8πGkB m0 The threshold temperature below which the particle of mass m0 is effectively removed from the universe: Tthreshold = m0 c2 kB TBH × Tthreshold = 47 T2Planck 8π Kardashev scale Classification of alien civilization based on how much energy an extraterrestrial civilization uses  Type I civilization (planetary civilization): A civilization capable of using and storing all of the energy resources available on its planet.  Type II civilization (stellar civilization): A civilization capable of using and controlling all of the energy resources available in its planetary system or all of the energy that its star emits.  Type III civilization (galactic civilization): A civilization capable of accessing and controlling all of the energy resources available in its galaxy. Q0 × T threshold = Q0 × T threshold = h 2me c2 × √4μ0ε0 me c2 kB = c2 LS 2tS White's Energy Formula: C=E×T Culture evolves as the amount  E is a measure of energy consumed per capita per year  T is the measure of efficiency of technical factors utilizing the energy  C represents the degree of cultural development 48 of energy harnessed per capita per year is increased  a∝t ρ∝ 2 3(1+w) Radiation dominated universe (w = 1 ): 3 a∝t  a−3(1+w) ρ∝ Planetary engineering 1 2 a−4 The development and application of technology for the purpose of influencing the environment of a planet Non-relativistic matter dominated universe (w = 0): 2 a ∝ t3 ρ∝  a−3 Dark energy dominated universe (w = −1): a∝ Λ 𝑒 Ht with H = √ 3 Terraforming The hypothetical process of deliberately modifying the Planet's atmosphere, temperature, surface topography or ecology to be similar to those of Earth in order to make it suitable for human life 49 Geoengineering The gravitational force between 2 electrons is: FG = Gm2e r2 The electrical force between 2 electrons is: FE = FG FE = Planetary engineering applied to Earth e2 4πε0r2 αG β → magnetic coupling constant = 4β × αG α The electric field E is related to the electric force F that acts on an electron charge e by: F E= e F = √α qPlanck E  Habitable Planet: A Planet with an environment hospitable to life.  Biocompatible Planet: A Planet possessing the necessary physical parameters for life to flourish on its surface. μB × Tthreshold = eℏ 2me × μB × Tthreshold = me c2 e kB 4π e 4π × c2 × √ μB × Tthreshold = c2 √ 50 = c1 × c2 × c c1 2πh 32π3RK TBH = TPlanck × If M = me: mPlanck 8πM m2Planck P = PPlanck × 8π√αG m2Planck ρBH = ρPlanck × 32πM2 ρBH = m2Planck P= 4πM2 32πm2e ρPlanck 32παG P = PPlanck × 15360πM2 SBH = SPlanck × 8πme TPlanck TBH = ρBH = ρPlanck × mPlanck TBH = TPlanck × m2Planck 15360πm2e PPlanck 15360παG SBH = SPlanck × m2Planck αG → Gravitational coupling constant 4πm2e m2Planck SBH = 4π SPlanck αG v × vPhase = c2 = (Planck speed) 2 = Planck specific energy Since the particle speed v < c for any particle that has mass − according to Albert Einsteinian special theory of relativity, the phase velocity of matter waves always exceeds c, i.e. vPhase > Planck speed vPhase > Planck length Planck time 51   Stoney length vPhase > Stoney time v< Stoney length Stoney time The strong coupling constant One of the fundamental parameters of the Standard Theory of particle physics that defines the strength of the force that holds protons and neutrons together The electrostatic repulsion between 2 electrons is described in quantum electrodynamics as the result of an exchange of a virtual photon between the 2 electrons. A particle with a mass m, when at motion, has an energy of E = √p2 c 2 E = √p2 c 2 + m20 c 4 . But for photons + 0 = pc since they are never at rest; they always move at the speed of light. Energy ∝ momentum 2 mec = e2 4πε0 r mec2 = Gm2e r Requirement for masslessness r = Classical electron radius = r= e2 4πε0 me c2 𝐒𝐜𝐡𝐰𝐚𝐫𝐳𝐬𝐜𝐡𝐢𝐥𝐝 𝐫𝐚𝐝𝐢𝐮𝐬 𝐨𝐟 𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐧 2 = Gme c2 1 eV is the energy that an electron acquires when it is accelerated through a voltage of one volt. me × Hartree velocity × Bohr radius 1 keV = 1000 eV 1 MeV = 1000 keV ℏ 1 GeV = 1000 MeV Rydberg constant sets the magnitude of the various allowed electron energy 1 TeV = 1000 GeV levels in atoms such as {Planck angular momentum} hydrogen. 52 At energy of 14,000 GeV (i.e., 15,000 times the mass of a proton in units of energy): The velocity of the proton is 0.999999998c (so almost equal to c). mpc2 = r= μN μB Gm2e × M r rS 2 Distance between 2 electrons at which mPlanckc2 = Gm2e r r = √electron gravitational coupling constant × rS 2 Distance between 2 electrons at which 𝐜𝟐 Sun 1.99 × 1030 kg 2.95 × 103 m Jupiter 1.90 × 1027 kg 2.82 m Earth 5.97 × 1024 kg 8.87 × 10−3 m Moon 7.35 × 1022 kg 1.09 × 10−4 m Saturn 5.683 × 1026 kg 8.42 × 10−1 m Uranus 8.681 × 1025 kg 1.29 × 10−1 m Neptune 1.024 × 1026 kg 1.52 × 10−1 m Mercury 3.285 × 1023 kg 4.87 × 10−4 m Venus 4.867 × 1024 kg 7.21 × 10−3 m Mars 6.39 × 1023 kg 9.47 × 10−4 m Human 70 kg 1.04 × 10−25 m Planck mass 2.18 × 10−8 kg 3.23 × 10−35 m gravitational potential energy between them is equal to intrinsic energy of proton 𝟐𝐆𝐌 gravitational potential energy between them is equal to Planck energy My studies of the natural sciences have particularly involved that part of physics which looks at the atomic world. (Twice the Planck length) Amedeo Avogadro Stellar gas constant = Avogadro constant × Stellar Boltzmann constant 53 The relativistic energy of an electron can be expressed in terms of its momentum in the expression: E = √p2 c 2 ℏc3 If p = Planck momentum = √ : G E = √p2 c 2 + m2e c 4 2 + αG EPlanck E = EPlanck √1 + αG Relativistic energy a electron must possess so that its momentum to be equal to Planck momentum Stoney energy: ES = mSc = √α 2 mPlanck L2Planck t2Planck ms L2S t 2S Stoney temperature: TS = ES kB = √α × EPlanck TBH = kB TS = √α × TPlanck TPlanck × mPlanck TBH = 54 8πM TS × mS 8π × α × M Today's universe in Planck and Stoney units 13.8 × 109 years Age 8.08 × 1060 tPlanck 8.7 × 1026 m Diameter 5.4 × 1061 LPlanck 3 × 1052 kg Mass Density 9.9 × 10−27 kg⋅m−3 Temperature 2.725 K 8.08 × 1060 5.4 × 1061 approx. 1060 mPlanck 1.8 × 10−123 √α LS √α approx. 1060 mPlanck L3Planck 1.9 × 10−32 TPlanck (Temperature of the cosmic microwave background 1.8 × 10−123 1.1 × 10−52 m−2 2.9 × 10−122 constant Hubble constant Planck charge density = 2.2 × 10−18 s−1 Planck charge Planck volume 1.18 × 10−61 =√ c10 4πε0 ℏ2 G3 Planck charge density = α t2S 55 × = 1 L2Planck 1 tPlanck 1 t2Planck × mS √α α mS L3S T 1.9 × 10−32 S √α radiation) Cosmological tS 2.9 × 10−122 1.18 × 10−61 1 α L2S √α tS √G × Coulomb constant 1 √G × Coulomb constant Planck energy density = Planck energy density = Planck force density = Planck energy = mPlanck = Planck volume LPlanck × t2Planck Planck force Planck volume = c7 G2 ℏ α × mS LS × t2S ℏ L4Planck tPlanck Planck force density = 5 α2 ℏ L4S tS Hartree Energy: Eh = 2 R∞ hc = αhc a0 = Bohr radius 2πa0 Hartree Force: Fh = Hartree Energy Bohr radius Fh = 2πa20 Fh = 56 αhc α5 hc 2πr2e re = classical electron radius Fh = αhc 2πa20 Fh = = αℏc a20 = e2 4πε0 a20  Z0 = impedance of free space  G0 = conductance quantum Z0 G0 hc 8πa20 Hartree Momentum: ph = ℏ a0 = α2 ℏ re RK = von Klitzing constant Z20 ℏ ph = 4R2K re Hartree Time: th = th = ℏ Eh = tPlanck 2R∞ hc 4πR∞ LPlanck 57 ℏ = tS 4πR∞ LS   Eh × t h = ℏ ph × a0 = ℏ Eh × th = ph × a0 Eh = ph × a0 th Hartree velocity: Planck speed = vh = vh = a0 th =α×c α LPlanck tPlanck 2 Erest = mec = = Hartree velocity Fine structure constant α LS tS mev2h α2 The threshold temperature below which the electron is effectively removed from the universe: Tthreshold = me c2 kB Tthreshold = v2h = × 2 α ideal gas constant molar electron mass molar electron mass ideal gas constant 58 × v2h a0 re c1 = 4π2 × Planck angular momentum × (Planck speed) 2 c1 = 4π × Planck angular momentum × 2 c1 = c2 = hc kB = (Hartree velocity)2 (Fine structure constant)2 2πhv2h a0 re molar Planck constant ideal gas constant × Hartree velocity Fine structure constant The Compton wavelength of the electron h me c = 2 × Quantum of circulation × c= 1 √μ0 ε0 vh = α √μ0 ε0 59 Fine structure constant Hartree velocity △x △p ≥ △E △t ≥ △x △p ≥ Sir Isaac Newton's famous Law of ℏ 2 ℏ 2 Bohr radius × Hartree momentum △E △t ≥ 2 Hartree energy × Hartree time 2 Universal Gravitation states that the force of gravitation is proportional to 1 (radius of the planet)2 − which implies that if a radius of the planet shrinks by a factor of 2, then the force of gravitation at its surface must rise by a factor of 4. △p Hartree momentum △E Hartree energy Planck force = Planck power = c5 G 60 ≥ ≥ △𝑥 Hartree time △t a20 v2h c4 = = v5h G Bohr radius Gr2e 5 = Gα a20 v2h Gr2e × Ls ts Black hole surface gravity is given by: gBH = Planck force 4 c4 4GM = Black hole mass × Black hole surface gravity gBH aPlanck = mPlanck 4M ℏc If M = mPlanck = √ : G gBH = aPlanck e2 : If M = mS = √ 4πε0 G gBH = aPlanck 4 4√ α Lorentz factor: γ= 1 2 √1−v2 c A term by which relativistic mass, time and length changes for an object in motion The Lorentz factor is always greater than 1 but it grows towards infinity as the object's velocity approaches the speed of light. 61 If v = Hartree velocity: γ= 1 √1−α2 m0  m=  L = L0 √1 − α2 √1−α2 Fine structure constant = △t0  △t =  KE = m0c2 ( √1−α2 1 √1−α2 1 4 × magnetic coupling constant − 1) The wavelength of a relativistic particle is given by: c2 λ = λC √ 2 v If v = −1 c √2 λ = λC If v = vh: 1 λ = λC √ 2 α a − 1 = λC √ 0 − 1 re λ = λC √16β2 − 1 β → magnetic coupling constant 62 : Hartree Temperature: Th = Eh kB = hc kB Th = × α 2πa0 c2 α 2πa0 Hartree electric potential: Vh = Vh = Eh e = hc e × α 2πa0 = Eh = 4R∞ × c × Φ0 = e 4R∞ Φ0 √μ0 ε0 Vh = Φ0 vh πa0 4R∞ vh Φ0 α Hartree pressure: Ph = Eh a30 = αhc 2πa0 Ph = vh × × 1 a30 ℏ a40 Hartree current: Ih = e ℏ × Eh = 63 e ℏ × αℏc a0 Ih = e × vh a0 Hartree charge density: e a30 = 13 α 2 qPlanck r3e Hartree electric dipole moment: e × a0 = e ×re α2 Hartree electric dipole moment = α − 3 2 × qPlanck × re The gravitational force between 2 electrons: FG = If FG = Hartree Force = αℏc a20 Gm2e r2 : αℏc a20 = Gm2e r2 αG r=√ × a0 = √4 × β × αG × a0 α Distance between 2 electrons at which gravitational force between them is equal to Hartree force 64 The electrical force between 2 electrons: FE = If FG = Hartree Force = αℏc a20 : αℏc a20 e2 4πε0 r2 = e2 4πε0 r2 r = a0 = re α2 Distance between 2 electrons at which electrical force between them is equal to Hartree force Quantum Chromodynamics Units: QCD Length: LQCD = QCD Time: tQCD = ℏ mp c ℏ = reduced Compton wavelength of the proton 2 = mp c 1 Compton angular frequency of the proton QCD mass: mQCD = mp = 1.673 × 10−27 kg QCD energy: EQCD = mpc2 QCD Temperature: TQCD = EQCD kB = the threshold temperature below which the proton is effectively removed from the universe 65 △E △t ≥ △E △t ≥ 2 EQCD × tQCD △E 2 EQCD ≥ LQCD × tQCD × mQCD = LQCD × tQCD × mQCD = ℏ tQCD △t ℏ mp c × ℏ mp c2 × mp mPlanck × LPlanck × tPlanck √Proton gravitational coupling constant Astronomical range Typical units Distances to satellites kilometers Distances to near-Earth objects lunar distance Planetary distances astronomical units, gigameters Distances to nearby stars parsecs, light-years Distances at the galactic scale kiloparsecs Distances to nearby galaxies megaparsecs Solar mass Solar mass 1 Jupiter masses 1048 Earth masses 332950 66 F = eE If F = Hartree force = αℏc a20 : E= LQCD × TQCD = Φ0 vh πa20 ℏ mp c LQCD × TQCD = LQCD tQCD LPlanck tPlanck × mp c2 kB c2 2π =c LS =c tS LQCD tQCD = LPlanck LS = tPlanck tS 67 =c c1 = 2πh × LQCD tQCD × LPlanck tPlanck EQCD = mQCD × c2 EQCD = mQCD × LQCD tQCD × LS tS The electrical force between 2 protons is given by: FE = e2 4πε0 r2 If r = LQCD: FE = e2 4πε0 L2QCD FE = Fine structure constant × The gravitational force between 2 protons is given by: 68 EQCD LQCD FG = Gm2p r2 If r = LQCD: FG = Gm2p L2QCD FG = Proton gravitational coupling constant × EQCD LQCD The critical density of the universe: ρcritical = If ρcritical = Planck density = c5 ℏG2 3H2 8πG : 8π H=√ 3 × tPlanck If the galaxy is taken to be spherical and the mass within the radius R is M, the circular rotational GM velocity at distance R is given by: vrot = √ . Thus, if vrot is constant, it follows that M ∝ R, so that r the total mass within radius R increases linearly with the distance from the centre. 69 me v2 2 v2 = 3 × kB me ×T=3× v2 = = 3kBT 2 ideal gas constant molar electron mass 3v2h α2 √ αG × ×T T TPlanck  vh = Hartree velocity and αG = Electron gravitational coupling constant  α = Fine structure constant and T Planck = Planck temperature me v2 v2 = 2 × e me 2 × V = 2 × electron charge to mass ratio × V v2 = 2 × εg = Gravitoelectric gravitational constant = eV Faraday constant molar electron mass v2 = 2V √ 70 α × ε0 αG × εg ×V Radiation density constant: a= 4σ c = 4σ × tQCD LQCD If I = Hartree current: e ℏ × Eh = dne dt dne = dt ×e Eh ℏ Rate of flow of electrons = 1 Hartree time Space debris Extremophiles Artificial objects in space that are orbiting Earth but no longer Organisms capable of living in extreme environments serve a useful function Precisely because Mars is an environment of great potential biological interest, it is possible that on Mars there are pathogens, organisms which, if transported to the terrestrial environment, might do enormous biological damage. − Carl Sagan 71 The volume of the black hole: VBH = VBH VPlanck = 4πR3S 3 32π 3 × M3 m3Planck If M = mPlanck: VBH = If M = mS = √α × mPlanck: VBH = 32π VPlanck 3 3 32π × α2 × VPlanck 3 The surface area of the black hole: ABH = 4πR2S If M = mPlanck: ABH APlanck = 16π × M2 m2Planck ABH = 16π × APlanck If M = mS: ABH = 16π × α × APlanck 72 c= 1 √ε0 μ0 LQCD = tQCD √ε0 μ0 The Compton wavelength of the electron: λC,e = 2π × α × a0 Eh × λC,e = αℏc a0 × (2π × α × a0) Eh × λC,e = α2hc Eh × LQCD = αℏc a0 Eh × LQCD = α2ℏc × Eh × LQCD = 73 × ℏ mp c electron mass proton mass α2 ℏc 1836.15267343 Hartree energy: Eh = αℏc a0 = αℏc × me cα ℏ Eh = α2mec2 Eh e2 α=√ = = me c2 q2Planck 𝑍0 𝐺0 4 =√ Th Tthreshold ℏG Eh × LPlanck = α2mec2 × √ 3 c αG = electron gravitational coupling constant Eh × LPlanck = α2 √αG ℏc Eh × LS = α2mec2 × ( √α LPlanck ) 5 2 Eh × LS = α √αG ℏc Hartree energy × Rydberg wavelength = 2hc 74 Eh × rS = α2mec2 × 2Gme c2 Eh × rS = 2α2 × αG × ℏc FG = FG = Gm2Planck r2 Gmp me √Proton gravitational coupling constant ℏc FG = 2 √Proton gravitational coupling constant r    Eh × tQCD = r2 α2 ℏ × √Electron gravitational coupling constant × √Electron gravitational coupling constant "there are no arbitrary constants ... nature is 1836.15267343 so constituted that it is possible logically to lay down such strongly determined laws that Eh × tPlanck = α2 √αG ℏ within these laws only rationally determined constants occur (not constants, therefore, 5 2 Eh × tS = = α √αG ℏ whose numerical value could be changed without destroying the theory)." ― Albert Einstein 75 If △λ = LQCD: △λ = h me c LQCD = (1 − cosθ) h me c (1 − cosθ) θ = cos−1 (1− me 2πmp ) The wavelength shift of the scattered photon in an angle of θ = cos−1 (1− QCD length. Ih × tQCD = e ℏ Ih × tQCD = e × α × 2 × Eh me mp = × me 2πmp ) is equal to the ℏ mp c2 e×α2 1836.15267343 electron gravitational coupling constant Ih × tQCD = e × α2 × √ proton gravitational coupling constant 76 e Ih × tPlanck = ℏ × Eh × ℏ mPlanck c2 Ih × tPlanck = e × α2 × √electron gravitational coupling constant Ih × t S = Ih × t S = e × α 5 2× e ℏ × Eh × (√α × tPlanck) √electron gravitational coupling constant Φ0 × Ih = h 2e × e ℏ × Eh Φ0 × Ih = π Eh Hartree energy Planck energy = (Fine structure constant) 2 × √electron gravitational coupling constant 77 Hartree force Planck force = α2 × √electron gravitational coupling constant × Planck length Bohr radius Hartree force: Fh = Rest mass energy of electron = 1 α2 me c2 a0 2πα3 Fh = = 2πα3 = 2πα3 me c2 λC,e × Hartree force × Compton wavelength of electron m2e c3 h = 2πα3 EQCD × tPlanck = mpc2 × kB T2threshold c2 ℏ mPlanck c2 EQCD × tPlanck = √proton gravitational coupling constant × ℏ EQCD × tS = mpc2 × (√α × ℏ mPlanck c2 ) EQCD × tS = √Fine structure constant × proton gravitational coupling constant × ℏ 78 EQCD × th = mpc2 × EQCD × th = ℏ Eh 1836.15267343 × ℏ α2 The energy required to eject the Einstein's Photoelectric Equation: electron from the metal surface E = W0 + KEelectron me v2 2 = h (υ – υ0) v = 2 √Q 0 (υ − υ0 ) eVS = h (υ – υ0) VS = 2 Φ0 (υ – υ0) aPlanck × th = aPlanck × th = c tPlanck × c ℏ α2 me c2 α2 √electron gravitational coupling constant 79 aPlanck × tQCD = aPlanck × tQCD = c × tPlanck mp c2 c √proton gravitational coupling constant Eh × μB = α2mec2 × E h × μB = α2 × e × c1 8π2 eℏ 2me α2 × e × c1 8π2 Eh × μN = α2mec2 × Eh × μN = ℏ × 80 me mp = eℏ 2mp α2 × e × c1 14689.2213874 π2 EPlanck × μB = mPlanckc2 × EPlanck × μB = eℏ 2me e × c1 √electron gravitational coupling constant × 8π2 ES × μB = √α mPlanckc2 × E S × μB = eℏ 2me √Fine structure constant × e × c1 √electron gravitational coupling constant × 8π2 Planck intensity = Planck power Planck intensity = (Planck force)2 Planck power = 1 2π Planck area ℏ = c8 ℏG = 2 = mS c2 tS × α L2S = αmS t3S 4π2 × (Planck power)2 First radiation constant √First radiation constant × Planck intensity 81 Planck Intensity = m2e c4 ℏ  ωC = Compton angular frequency of the electron  rS = Schwarzschild radius of the electron × c4 G2 m2e EQCD × μB = mpc2 × EQCD × μB = = 4ℏω2C rS eℏ 2me √proton gravitational coupling constant × e × c1 √electron gravitational coupling constant × 8π2 EQCD × μN = mpc2 × EQCD × μN = eℏ 2mp e × c1 8π2 EPlanck × μN = mPlanckc2 × EPlanck × μN = eℏ 2mp e × c1 √proton gravitational coupling constant × 8π2 82 ES × μN = √α mPlanckc2 × E S × μN = eℏ 2mp √Fine structure constant × e × c1 √proton gravitational coupling constant × 8π2 Planck Temperature: ℏc5 TPlanck = √ 2 GkB TPlanck = c2 √Gm × e TPlanck = hc 2πkB me c2 ×√ kB c2 × Tthreshold √ πrS "Fine Structure Constant: Fundamental numerical constant of atomic physics and quantum electrodynamics, defined as the square of the charge of the electron divided by the product of Planck's constant and the speed of light." ― Steven Weinberg 83 Eh Fh × th = a0 × ℏ Eh Fh × th = α mec Fh × th = Fine structure constant × √electron gravitational coupling constant × Planck momentum Fh × tPlanck = Eh a0 × ℏ EPlanck Fh × tPlanck = α3 × electron gravitational coupling constant × Planck momentum Fh × tS = Eh a0 × 7 2 √α × ℏ EPlanck Fh × tS = α × electron gravitational coupling constant × Planck momentum 84 Reduced mass of hydrogen atom: μ= μ= me mp (me +mp )  μ ≤ me  μ ≤ mp √electron gravitational coupling constant × √proton gravitational coupling constant × Planck mass (√electron gravitational coupling constant + √proton gravitational coupling constant) Hartree Power: Ph = Fh × vh = Ph = α2 me c2 a0 × αc α4 m2ec4 ℏ Ph = α4 × Electron gravitational coupling constant × Planck power Ph × tPlanck = α4 × Electron gravitational coupling constant × Planck energy 9 2 Ph × tS = α × Electron gravitational coupling constant × Planck energy "The fine-structure constant derives its name from its origin. It first appeared in Sommerfeld's work to explain the fine details of the hydrogen spectrum. ... Since Sommerfeld expressed the energy states of the hydrogen atom in terms of the constant [alpha], it came to be called the fine-structure constant." ― John S. Rigden 85 aPlanck × Q0 = aPlanck × Q0 = c tPlanck × 2me π × c3 √electron gravitational coupling constant c aPlanck × Φ0 = aPlanck × Φ0 = tPlanck h 2e √Fine structure constant E h × Q0 = h 2me α2 c1 4π2 EQCD × Q0 = mpc2 × √proton × πc ×Planck voltage Eh × Q0 = α2mec2 × EQCD × Q0 = h h 2me gravitational coupling constant × c1 4π2√electron gravitational coupling constant 86 EPlanck × Q0 = mPlanckc2 × EPlanck × Q0 = 4π 2me c1 2 4π √electron gravitational coupling constant ES × Q0 = √α mPlanckc2 × ES × Q0 = h 2 √Fine √electron h 2me structure constant × c1 gravitational coupling constant A quantum fluctuation can create an proton antiproton pair with energy △E ≥ 2mpc2 provided the fluctuation lives less than the time △t ≤ ℏ . In that time, the proton and antiproton can separate by a distance of order △E △x = c ×△t. As they separate they gain energy eE ×△x, in the electric field with strength E. If they gain sufficient energy to compensate for their rest mass, they no longer have to annihilate: they can become real particles. The condition for real proton− antiproton pair creation is therefore that the electric field be greater than a critical value, Ecritical given by: e Ecritical × (c × ℏ 2mp c2 Ecritical = ) = 2mpc2 4m2p c3 ℏe A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. — Hermann Weyl 87 Eh Fh ×tQCD = Fh × tQCD = a0 × ℏ mp c2 α3 × electron gravitational coupling constant ×Planck momentum √proton gravitational coupling constant Ph × tQCD = Ph × tQCD = α4 m2e c4 ℏ × ℏ mp c2 α4 × Electron gravitational coupling constant × Planck energy √Proton gravitational coupling constant Number of electron charges that make up one Planck charge: n= Planck charge Electron charge = 1 √α = 2 √ impedance of free space × conductance quantum Any photon orbiting below this distance The radius of photon orbit: r= If M = mPlanck = √ ℏc G : 3GM c2 r = 3 × Planck length 88 will plunge into the black hole, while photon that remains further away will spiral out towards infinity. The electric potential energy between 2 electrons: Ep = If Ep = Hartree energy: e2 4πε0 r α2mec2 = r= e2 4πε0 r re Distance between 2 electrons at which the electric potential energy between α2 them is equal to Hartree energy The gravitational potential energy between 2 electrons: Ep = Gm2e If Ep = Hartree energy: α2mec2 = r= rS r 2 Gme r Distance between 2 electrons at which the gravitational potential energy between them is 2 × α2 equal to Hartree energy "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. " − Richard P. Feynman 89 If e2 4πε0 r = Planck energy = mPlanck c 2 : Distance between 2 electrons at which the electric potential energy between r = √electron gravitational coupling constant × re If e2 4πε0 r them is equal to Planck energy = Stoney energy = √α mPlanck c 2 : Distance between 2 electrons at which the electric potential energy electron gravitational coupling constant r=√ × re Fine structure constant If Gm2e r Gm2e r energy = Planck energy = mPlanck c 2 : r = √electron gravitational coupling constant × If between them is equal to Stoney Distance between 2 electrons at which the rS gravitational potential energy between 2 them is equal to Planck energy = Stoney energy = √α mPlanck c 2 : "Primitive life is very common electron gravitational coupling constant r=√ Fine structure constant × rS 2 and intelligent life is fairly rare. Some would say it has yet to occur on Earth." − Stephen Hawking Distance between 2 electrons at which the gravitational potential energy between them is equal to Stoney energy 90 FG = Because rS = Gm1 m2 r2 2Gm c2 : FG = FPlanck 4 FG ∝ × rS1 × rS2 r2 rS1 × rS2 r2 FPlanck 4 → Proportionality constant Niels Bohr was a Danish physicist who is generally regarded as one of the foremost physicists of the 20th century. He was the first to apply the quantum concept, which restricts the energy of a system to certain discrete values, to the problem of atomic and molecular structure. For that work he received the Nobel Prize for Physics in 1922. His manifold roles in the origins and development of quantum physics may be his most-important contribution, but through his long career his involvements were substantially broader, both inside and outside the world of physics. In 1911, fresh from completion of his PhD, the young Danish physicist Niels Bohr left Denmark on a foreign scholarship headed for the Cavendish Laboratory in Cambridge to work under J. J. Thomson on the structure of atomic systems. At the time, Bohr began to put forth the idea that since light could no long be treated as continuously propagating waves, but instead as discrete energy packets (as articulated by Max Planck and Albert Einstein), why should the classical Newtonian mechanics on which Thomson's model was based hold true? It seemed to Bohr that the atomic model should be modified in a similar way. If electromagnetic energy is quantized, i.e. restricted to take on only integer values of hυ, where υ is the frequency of light, then it 91 seemed reasonable that the mechanical energy associated with the energy of atomic electrons is also quantized. However, Bohr's still somewhat vague ideas were not well received by Thomson, and Bohr decided to move from Cambridge after his first year to a place where his concepts about quantization of electronic motion in atoms would meet less opposition. He chose the University of Manchester, where the chair of physics was held by Ernest Rutherford. While in Manchester, Bohr learned about the nuclear model of the atom proposed by Rutherford. To overcome the difficulty associated with the classical collapse of the electron into the nucleus, Bohr proposed that the orbiting electron could only exist in certain special states of motion called stationary states, in which no electromagnetic radiation was emitted. In these states, the angular momentum of the electron L takes on integer values of Planck's constant divided by 2π, denoted by ℏ = h 2π (pronounced h-bar). In these stationary states, the electron angular momentum can take on values ℏ, 2ℏ, 3ℏ... but never non-integer values. This is known as quantization of angular momentum, and was one of Bohr's key hypotheses. He imagined the atom as consisting of electron waves of wavelength λ = h me v = h p endlessly circling atomic nuclei. In his picture, only orbits with circumferences corresponding to an integral multiple of electron wavelengths could survive without destructive interference (i.e., r = 𝐧ℏ 𝐦𝐞 𝐯 could survive without destructive interference). For circular orbits, the position vector of the electron r is always perpendicular to its linear momentum p. The angular momentum L has magnitude mevr in this case. Thus Bohr's postulate of quantized angular momentum is equivalent to mevr = nℏ where n is a positive integer called principal quantum number. It tells us what energy level the electron occupies. Since λ = h me v = h p For an electron moving in a circular orbit of radius r: ω= (de Broglie relation), pvp = hvp λ = hυ = ℏω 92 v r where ℏ = h 2π phase velocity. is the reduced Planck constant, ω = 2πυ is the angular frequency and vp is the pvp = ℏv r Since nℏ = pr (quantization of angular momentum), v = n × vp The velocity of the electron or the group velocity of the corresponding matter wave associated with the electron is the integral multiple of the phase velocity of the corresponding matter wave associated with the electron. Quantum of circulation: Q0 = By the de Broglie hypothesis, we see that: pvp = hυ pv = hυ λ nλ mevr = nℏ λ v= λ Substituting nλ = 2πr, ω= me v2 r = 2π hυ λ v r = nQ0 nQ0 πr2 πr = → v= 2me 2Q0 λ nQ0 Area of circular orbit The classical description of the nuclear atom is based upon the Coulomb attraction between the positively charged nucleus and the negative electrons orbiting the nucleus. Furthermore, we consider only circular orbits. The electron, with mass me and charge e− moves in a circular orbit of radius r with constant velocity v. The attractive Coulomb force provides the necessary acceleration to maintain orbital motion. (Note we neglect the motion of the nucleus since its mass is much greater than the electron). The total force on the electron is thus 93 h F= where ε0 = 8.854 ×10−12 𝐹 𝑚 Ze2 me v2 = 4πε0 r2 r F = 2π is the permittivity of free space. − Substituting 2πr = nλ, − Ze2 = − 2πr Ze2 = U = − nhυ 4πε0 r 4πε0 r hυ λ hυ λ The potential energy of the electron The negative sign indicates that it requires energy to pull the orbiting electron away from the nucleus. From the equation: KE = me v2 2 pv = 2 we can determine the kinetic energy of the electron (neglecting relativistic effects) Substituting p = nℏ , r KE = nℏv 2r = nℏω 2 = The kinetic energy of the electron hυ is the integral multiple of 2 nhυ 2 The total energy of the electron E = KE + U is thus: E = KE + U = nhυ 94 2 + (− nhυ) E=− nhυ 2 The frequency of photon absorbed or emitted when transition occurs between two stationary states that differ in energy by ΔE, is given by: υphoton = △E h = E2 −E1 h where E1 and E2 denote the energies of the lower and higher allowed energy states respectively. This expression is commonly known as Bohr's frequency rule. υphoton = n hυ n hυ (− 2 2 ) − (− 1 1 ) 2 h 2 n1υ1 – n2υ2 = 2υphoton In physics (specifically, celestial mechanics), escape velocity is the minimum speed needed for an electron to escape from the electrostatic influence of a nucleus. If the kinetic energy me v2 of 2 Ze2 the electron is equal in magnitude to the potential energy 4πε r , then electron could escape 0 from the electrostatic field of a nucleus. Ze2 4πε0 r2 = me v2 r Ze2 vorbital = √ 4πε0 rme vorbital = c√ = me v2 = nhυ 2 Orbital velocity: nhυ =√ Ze2 me v2 2 me 4πε0 r 2nhυ v = vescape = √ me Z ×Bohr radius Z ×classical electron radius = c × Fine structure constant √ r r 95 = √4nQ 0 υ Z ×Bohr radius vorbital = Hartree velocity √ r Total energy of the electron: E=− E υC = me c2 h me c2 nhυ 2 =− Ze2 = meω2r 4πε0 r2 nυ 2υC 4πε0 r2 is the Compton frequency of the electron. = Ze2 4πε0 r2 4π2 T2 me T2 ∝ (standing-wave condition) r + λ= h p (de Broglie relation) = me × = re c2 × 2πr = nλ v2 Ze2 4π2 Z T2 r3 ×r L = nℏ (Bohr’s postulate) where re denote the Classical electron radius r3 Z "The very nature of the quantum theory ... forces us to regard the space-time coordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description, symbolizing the idealization of observation and description, respectively." ― Niels Bohr 96 The moment of inertia of an electron in nth orbit is: λ= particle I = n × mer2 Planetary Model failed to mer = explain stability of atoms in accordance with classical laws h me v wave nℏ v of physics 2 ℏr n ℏ I=n × = v ω 2 The acceleration of the electron: a= a= 2π T v2 r Iω n=√ ℏ =ω×v nhυ √m = e 2π √2nQ0υ Plum Pudding Model failed to T explain large-angle deflections of scattered alpha particles F=Z× e2 4πε0 F = Z × Fine structure constant × Ze2 4πε0 r2 = me r × n2 ℏ2 m2e r2 r= n2 Z × 1 r2 hc 2 × Area of the circular orbit × Bohr radius = 97 n2 re Zα2 Rydberg formula: υphoton = Rydberg frequency × Z 2 For hydrogen atom: Z = 1 υphoton = Rydberg frequency × n1 υ1 − n2υ2 2   n21 n22 n22 −n21 n21 n22 = Rydberg frequency × Rydberg frequency = n22 −n21 n22 −n21 n21 n22 n21 n22 (n1υ1 − n2 υ2 ) 2(n22 −n21 ) n1 n2 Series Name 1 2–∞ Lyman 2 3–∞ Balmer 3 4–∞ Paschen 4 5–∞ Brackett 5 6–∞ Pfund 6 7–∞ Humphreys Area of ellipse (integral form): ∮ Ldφ Area of ellipse (geometrical form): 2πnℏ Bohr-Sommerfeld quantization rule for angular momentum: ∮ Ldφ = 2πnℏ In the case of circular orbits: L is constant and 2π Bohr quantization rule ∮ Ldφ = L ∫ dφ = 2πnℏ → L = nℏ 0 98 Hartree electric potential Total energy of the electron: E = KE + U = − R∞ = hc R∞ n2 n3 υ =− nhυ Planck voltage 2 2c Ionization energy e × qPlanck EPlanck 3 me v2 r = Ze2 4πε0 r2 v= The minimum energy required to binding of nucleus. Eh α2 × √electron gravitational coupling constant Electron charge × Ionization potential liberate the electron from the = Ze2 4πε0 nℏ = Zαc n α= nhυ =√ me 1 Z n3 υ √υ C Separation energy The energy needed to remove a proton or a neutron from an Ground state → Excited state atomic nucleus. First excitation potential = E2 − E1 − hc R∞ n22 + hc R∞ n21 Rydberg formula for the spectrum of the hydrogen atom: = −3.4 + 13 .6 = 10.2 eV λmax = Second excitation potential = E3 − E1 − hc R∞ n23 + hc R∞ n21 = −1.5 + 13 .6 = 12.1 eV n21 n22 λmax λmin E= hυ λmin = (n22 −n21 )R∞ = n21 R∞ n22 (n22 −n21 ) Because E = mc2: Bohr's model does not work for systems with more than one electron. The Planck constant relates mass to frequency. 99 e= Fine structure constant: α= KJ = e2 2ε0 ch = h K2J 8ε0 c 1 = h K2J 8 Magnetic flux quantum μ0 2 KJ RK √ε 0 h= 4 K2J RK RK = von Klitzing constant = Josephson constant Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but a great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet. − Max Planck △α = αprevious − α now If the fine-structure constant really is a constant, then any experiment should show that △α = 0 Any value far away from zero would indicate that α does change over time. So far, most experimental data is consistent with α being constant. Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? ― Stephen Hawking 100 The wavelength associated with an electron is related to the momentum of the electron by the de Broglie relation: λ = p= h λ → dp dt = p2 p dλ ×− h h dt Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis" in 1686. His second law defines a force exerted on the electron to be equal to the rate of change in momentum of the electron: F = F= mrelativistic = p2 h ×− me dt dλ dt 2 The mass of the electron is not constant; it varies with changes in its velocity. √1−v2 c dp m2relativistic c2 – m2relativistic v2 = me2c2 mrelativisticv dv + v2dmrelativistic = c2dmrelativistic On differentiation dmrelativistic (c2 – v2) = mrelativisticv dv dmrelativistic dt F= = mrelativistic va mrelativistic × a v2 1− 2 c mrelativisticc2 = mec2 + KE dmrelativistic c2 (c2 −v2) = dt m3relativistic a = dKE dt = Fv For non-relativistic case (v << c): m2e F= mea 101 Albert Einstein was a German- In no experiment, F= matter exists both as a m3relativistic a m2e particle and as a wave = p2 h ×− dλ dt simultaneously. It is born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity, but he either the one or the also made important contributions other aspect. a= to the development of the theory m2e v2 hmrelativistic ×− dλ of quantum mechanics. dt For nonrelativistic case (v << c): a= me v2 h ×− dλ dt "It was an act of desperation. For six years I had struggled with the blackbody theory. I knew the problem was fundamental and I knew the answer. I had to find a theoretical explanation at any cost, except for the inviolability of the two laws of thermodynamics." − Max Planck Irradiance is power per unit area. Just like Energy, TOTAL MOMENTUM IS ALWAYS CONSERVED Classical Picture Quantum Picture Energy of EM wave ~ (Amplitude) 2 Energy of photon = 102 hc λ The time will come when diligent research over long periods will bring to light things which now lie hidden. A single An 'up' quark has a charge of + 2 e 3 lifetime, even though entirely devoted to the sky, would not be enough for the investigation of so vast a subject... And so this knowledge will be unfolded only through long successive qup = + 1 and a 'down' quark has a charge of − e 3 2 e 3 qdown = − FE = q2up 4πε0 r = 2 ages. There will come a time when our descendants will be amazed that we did not know things that are so plain to them... Many discoveries are reserved for ages still to come, when memory of us will have been effaced. 1 e 3 ― Seneca 4αℏc FE = 9r2 FE = qup × qdown 4πε0 r2 q2down 4πε0 r =− 2αℏc 1 α2 Hartree wave number = a0 = 2 αℏc 9r2 9r2 = re Hartree energy = ℏω0 = 2ℏcR∞ = α2mec2 ℏω0 = 2ℏcR∞ Hartree frequency = 2 × Rydberg frequency ℏω0 = α2mec2 Hartree frequency = α2 × Compton angular frequency of electron 103   Energy density of electric field = ε0 E2 Energy density of magnetic field = Energy density of EM wave: Electromagnetic wave consists of an oscillating electric field with a 2 B2 perpendicular oscillating magnetic field. 2μ0 "What is known of [photons] comes from observing the results of their being created uwave = c= E B = 1 √ε0 μ0 uparticle = number density of photons × hυ ε0 E2 2 + B2 or annihilated." 2μ0 uwave = ε0 E 2 − Eugene Hecht uwave does not depend on the frequency of the wave depend on the frequency of the wave uwave = uparticle number density of photons ∝ E2 "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of Radiation pressure = their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the 4σT4 3c A very small increase in temperature will result in a very large increase in the radiation pressure Hydrostatic Equilibrium: gas and radiation pressure balance the gravity sixth place of decimals." − Albert A. Michelson, 1894 Thermal Equilibrium: Energy generated = Energy radiated 104 Hartree pressure Planck pressure = Eh a30 × L3Planck EPlanck Hartree pressure = α5 × (electron gravitational coupling constant) 2 × Planck pressure 2 2 Erelativistic = p2c2 + Erest 2 Erelativistic − E2rest = p2c2 (Erelativistic − Erest) (Erelativistic + Erest) = p2c2 KE = F= p2 h ×− dλ dt → F= KE = For non-relativistic case: p2 (mrelativistic + mrest ) KE(mrelativistic + mrest ) h ×− hF ha 3kB T 2 = ha dλ 2 × − dt a= 3kB T h dt ×− For non-relativistic case: F = mresta dλ where KJ is the Josephson constant ha 2×− dλ dt a = KJV dλ dt 105 2 mrelativistic = mrest 2 × − dt KE = eV = mrest v2 dλ dλ (mrelativistic + mrest ) × − dt KE = KE = KE = ×− dλ dt Cherenkov radiation is the electromagnetic radiation emitted when a charged particle (such as an electron) travels in a medium with speed v such that: c n <v<c where c is speed of light in vacuum, and n is the refractive index of the medium. We define the ratio between the speed of the particle and the speed of light as: The heavier the charged particle, the higher kinetic energy it must possess to be able to emit Cherenkov radiation. v c = 1 The emission of Cherenkov n × cosθ cosθ = radiation depends on the refractive index n of the medium and the c velocity v of the charged particle in n×v that medium Since the charged particle is relativistic, we can use the relation: c2 λ = λC √ 2 v −1 λ = λC √n2 cos 2 θ − 1 If λ = λC: The wavelength of the charged particle is √2 θ = cos−1 ( n ) equal to its Compton wavelength when Cherenkov angle equals cos−1 ( √2 n ) Pavel Alekseyevich Cherenkov was a Soviet The Cherenkov Effect is used as a tool in: physicist who shared the Nobel Prize in  nuclear physics to detect solar neutrinos  high energy experiments to identify the nature of particles  astrophysical experiments to study the cosmic showers physics in 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934. 106 "The element carbon can be found in more kinds of molecules than the sum of all other kinds of molecules combined. Given the abundance of carbon in the cosmos — forged in the cores of stars, churned up to their surfaces, and released copiously into the galaxy — a better element does not exist on which to base the chemistry and diversity of life. Just edging out carbon in abundance rank, oxygen is common, too, forged and released in the remains of exploded stars. Both oxygen and carbon are major ingredients of life as we know it." ― Neil deGrasse Tyson For a spherical star of uniform density, the The core pressure of a star of mass M and radius R is gravitational binding energy EB is given by given by: the equation: EB = − Pcore = 3GM2 5R 5GM2 4πR4 where G is the gravitational constant, M is the mass of the star and R is its radius. where rS = 2GM c2 − EB 0.3Mc2 = rS R Pcore = − 25EB 9V =− 25 9 × ρB where ρB is the gravitational binding energy density of the star. is the Schwarzschild radius of the star. Any star with Radius smaller than its Schwarzschild radius will form a black hole. Subrahmanyan Chandrasekhar was an Indian-American astrophysicist who spent his professional life in the United If R < rS: |EB| > 0.3Mc 2 States. He was awarded the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and The star will form a black hole evolution of the stars" 107 10EB − 9PcoreV 25 = EB: 2 =− 3Mc Pcore = 0.833ρE where ρE = Mc2 V If R < rS: rS The core density of the star is given by: R ρcore = 3M πR3 The core temperature of the star is given by: rS R Tcore = 5μmH GM 3kB R is the mass energy density of the star. where kB is the Boltzmann constant, μ denotes mean molecular weight of the matter inside the star and mH is The star will form Pcore > 0.833ρE the mass of hydrogen nucleus a black hole. William Alfred Fowler was an American nuclear physicist, later astrophysicist, who, ρcore × Tcore = 4μmH Pcore kB Pcore = ρB = − 9Pcore 25 =− ρcore × Tcore × kB with Subrahmanyan Chandrasekhar won the 4μmH 1983 Nobel Prize in Physics. He is known for his theoretical and experimental research into nuclear reactions within stars and the 9 × ρcore× Tcore × kB energy elements produced in the process. 100μmH The ideal gas equation PV = NkBT does not hold good for the matter present inside a star. Because, most stars are made up of more than one kind of particle and the gas inside the star is ionized. There is no indication of these facts in the above equation. We need to change the ideal gas equation, so that it holds good for the material present inside the star. It can be shown that the required equation can be written as PV = M μmH kBT where μ denotes mean molecular weight of the matter inside the star, M is the mass of the star and mH is the mass of hydrogen nucleus. PV MT = kB μmH = P 4Pcore Pcore ρcore Tcore 108 =4× ρ ρcore × T Tcore Planck force density = Planck force Planck volume = Planck pressure mec2 = r= Planck length = √α Planck pressure Stoney length Gmp me r Distance between proton and Schwarzschild radius of proton electron at which the gravitational 2 potential energy between them is equal to intrinsic energy of electron The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom. mpc2 = ― Isaac Asimov r= Gmp me Schwarzschild radius of electron 2 r Distance between proton and electron at which the gravitational potential energy between them is equal to intrinsic energy of proton Black hole type Description Constraints Schwarzschild has no angular momentum angular momentum = 0 electric charge = 0 and no electric charge Kerr does have angular momentum electric charge = 0 but no electric charge Reissner–Nordström has no angular momentum but does have an electric charge Kerr–Newman has both angular momentum and an electric charge 109 angular momentum = 0 Heat Capacity: C = Mc2 dQ dT Substituting dQ = dMc2 and T = = TBH × SBH Specific heat capacity of a black hole = − 3 ℏc : 8πkB GM Heat capacity of a black hole = − 2SBH M 8πkB GM2 ℏc Specific heat capacity of a black hole = − SBH = 2 8πkB GM ℏc =− c2 Black hole temperature 4πkB GM2 "For the past forty-five years, Stephen and hundreds ℏc On differentiation of other physicists have struggled to understand the precise nature of a black hole's randomness. It is a question that keeps on generating new insights about dSBH = 8πkB GM ℏc3 the marriage of quantum theory with general relativity—that is, about the ill-understood laws of × dMc2 quantum gravity." TBH × dSBH = dMc2 ― Stephen Hawking Black holes are the harmonic oscillator of Mc2 2 quantum gravity. = TBH × SBH On differentiation (A. Strominger) dMc2 = 2 (TBH × dSBH) + 2 (dTBH × SBH) = 2 dMc2 + 2 (dTBH × SBH) − dMc2 dTBH = 2SBH  Neutron Star has a hard surface; the curvature is large - but finite.  Black Hole: No Surface − curvature is infinite at the centre. 110 A photon of higher frequency causes the ejected photoelectron to propagate faster. The energy of photon − converted into the kinetic It is impossible, using the current energy of the electron − is proportional to its frequency. laws of quantum mechanics and the known behavior of gravity, to determine a position to a ℏ, c, G, e, ε0, me, mp ….. ℏG precision smaller than√ 3 c Other constants λC = re = Fundamental dimensionless constants h mp c mp e2 4πε0 me c2 α= me e2 4πε0 ℏc αG = ℏ, c, G, ε0 Gm2e ℏc Magnetic coupling constant = 1 4α Planck units mp ℏc mPlanck = √ G ℏG LPlanck = √ c3 ℏG tPlanck = √ 5 c qPlanck = √4πε0 ℏc me = μB μN proton gravitational coupling constant = √electron gravitational coupling constant The Planck units simplify the expression of physics Theories of proton decay predict laws and are the universal that the proton has a half life on limits beyond which all the the order of at least 1032 years. Till known laws of physics date, there is no experimental break down. In order to comprehend anything beyond it − we need new unbreakable laws of theoretical physics. 111 evidence of proton decay. If you wish to make eℏ μB × r S = an apple pie from 2me scratch, you must 2Gme × c2 first invent the universe. μB × r S = ― Carl Sagan μN × r S = eℏ 2mp A thinker sees his own actions as × √α × c1 × qPlanck 4π2 × FPlanck 2Gmp μB × a0 = experiments and questions--as attempts to find out something. Success and failure are for him eℏ 2me μB × a0 = ― Friedrich Nietzsche FG = FG = FG = 1 4π2 × Planck force Gm2Planck r2 × r2 = GE2rest c4 r2 √α × c1 × qPlanck 4π2 × FPlanck × ℏ me cα eQ20 2π2 cα answers above all. Gm2e = c2 = 2π2 c Planck force × r2 (Second radiation constant)2 × μ0 ε0 T2threshold r2 (First radiation constant)2 = 4π2× Planck force × (Second radiation constant)2 × 112 αeQ20 k2B T2threshold (First radiation constant)2 1 μB × r e = μ0 ε0 T2Planck r2 GE2rest h2 υ2C FG = Gm2e FG = c1 × Planck angular momentum r2 = c4 r2 Planck force × r2 Planck force × λC × r2 FG = FG = = Gm2e r2 = L2Planck e2 c1 16π2 μ2B r2 √ε0 μ0 = G r2 × e2 ℏ2 4μ2B Planck area × e2 × c1 16π2 × μ2B × r2 × √ε0 μ0 Gravitational redshift The change in the wavelength of electromagnetic radiation photon in a gravitational field predicted by general theory of relativity. A heuristic Newtonian derivation gives z= I do not feel obliged to believe that △E E = − GM rc2 the same God who has endowed us of mistakes, but they are with sense, reason, and intellect has intended us to forgo their use. Science, my lad, is made up mistakes which it is useful to Mc = − z × Planck force × r 2 ― Galileo Galilei make, because they lead little by little to the truth. [Letter to the Grand Duchess Christina] ― Jules Verne 113 Gravitational waves are 'ripples' in space-time, generated by accelerated masses that propagate as waves outward from their source at the speed of light. They were proposed by Henri Poincaré (French mathematician, theoretical physicist, engineer and philosopher of science) in 1905 and subsequently predicted in 1916 by Albert Einstein on the basis of his general theory of relativity. Gravitational waves were first directly detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in 2015. Gravitational wave is to gravity what light is to electromagnetism. It is the transmission of variations in the gravitational field as waves. Predicted by Einstein's theory of general relativity, the waves transport energy known as gravitational radiation. Two objects orbiting each other in highly elliptical orbit or circular orbit about their center of mass comprises binary system. This system loses mass by emitting gravitational wave (ripple in the geometry of space and time) whose frequency υ = E h << frequency of electromagnetic radiation and this is associated with an in-spiral or decrease in orbit. Suppose that the two masses are m1 and m2, and they are separated by a distance "r" orbiting each other in highly circular orbit about their center of mass. The rate of loss of energy from the binary system through gravitational radiation is given by: P=− dE dt = 32G4 m21 m22 (m1 +m2 ) 5c5 r5 where G = 6.674 × 10 −11 m3 kg−1 s −2 is the Newtonian gravitational constant and c = 3 × 108 ms−1 is the speed of light in vacuum. Gravitational radiation robs the energy of orbiting masses. As the energy of the orbiting masses reduces, the distance between the masses P=v× Gm1 m2 FG = 2r2 2P v decreases, and they orbit more rapidly. More generally, the rate of decrease of distance between the masses with time is given by: v=− dr dt = 64G3 m1 m2 (m1 +m2 ) 5c5 r3 where FG is the force of gravitation between the two masses orbiting each other in highly circular orbit about their center of mass. The loss of energy through gravitational radiation could eventually drop the mass m1 into the mass m2. The lifetime of distance "r" between the masses orbiting each other in highly circular orbit about their center of mass is given by: 5c5 r4 tlife = r 2P 8P × tlife 256G3m1 m2 (m1 +m2 ) tlife = FG = = 4×v v r 114 The gravitational wave signal was observed by LIGO detectors in Hanford and in Livingston on 14 September 2015. An exact analysis of the gravitational wave signal based on the Albert Einsteinian theory of general relativity showed that it came from two merging stellar black holes with 29 and 36 solar masses, which merged 1.3 billion light years from Earth. Before the merger, the total mass of both black holes was 36 + 29 solar masses = 65 solar masses. After the merger, the mass of resultant black hole was 62 solar masses. What happened to three solar masses? It was turned into the energy transported by the emitted gravitational waves. Using Albert Einstein's equation E = mc2, where E is the energy transported by the emitted gravitational waves, m is the missing mass (3 solar masses) and c is the speed of light, we can estimate the The amplitude of gravitational energy released as gravitational waves: E = (3 × 2 × 1030 kg) × (3 × 108 m/s) 2 waves gets smaller with the distance to the source. E = 5.4 × 1047 J This is roughly 1021 more energy than the complete electromagnetic radiation emitted by our sun. υ= E h = 5.4 × 1047 6.626 ×10−34 = 8.14 × 1080s−1  Gravity → Curvature of 4-dimensional (3 space + 1 time) space-time fabric produced by matter.  Gravitational-waves → Ripples on 4-dimensional space-time produced by accelerated matter. "Newton's law of gravitation. That's all you need (with a spot of calculus to crunch the numbers) to work out how the Earth will orbit the Sun or how an apple will fall if you let it go at a certain height. The only trouble is that Newton had no idea how this gravity thing worked. His model was simply: There is an attraction between bits of stuff, and let's not bother about why." Albert Einstein theorized that smaller masses travel toward larger masses, not because they are "attracted" by a mysterious force called gravity, but because the smaller objects travel through space that is warped by the larger object. 115 ― Brian Clegg References:  http://www.ebyte.it/library/educards/constants/ConstantsOfPhysicsAndMath.html.  The Fundamental Constants: A Mystery of Physics by Harald Fritzsch.  The Constants of Nature By John D. Barrow.  https://en.wikipedia.org/wiki/List_of_physical_constants.  Hawking on the Big Bang and Black Holes by Stephen W. Hawking.  The Universe in a Nutshell by Stephen W. Hawking. Natural science, does not simply describe and explain nature; it is part of the interplay between nature and ourselves. Werner Heisenberg 116