Gravitation

1. Gravitation

2. Gravity • The natural force that tends to cause physical things to move toward each • other; the force that causes things to fall. – Merriam Webster Dictionary • Gravity is a conservative force field, hence, work performed is independent • of the path taken between initial and final positions of an test mass during • displacement with the central force field of the active mass. • where • DU = Difference in potential energy [Joules] • F(r) = radial force [Newtons] • W = Work (= force x distance) [N·m] • G = Universal Gravitational constant [ 6.67428E-11 N·m2/kg2] • M = central mass [kg] • m = test mass [kg]

3. Gravitational flux

4. Gravitational force Gravitational force Newton’s 1st Law F= GMm/r2·r̂ [Newtons] where G = Gravitational coupling constant relating curvature and energy density gradient [≃ 6.67428E-11 N·m2/kg2] M = active mass [kg] m = passive mass [kg] r = separation distance between masses [m] Newton’s Law of gravitation assumes instantaneous action-at-a-distance independent of time and, hence, is not relativistic. Generalized Newton Eqn: F = GMm/r2·r̂ -GMmv2/2c2r2·r̂ Newtons’s gravitational constant G = c2Sri/mi = Runiv/Muniv [m3/kg·s2]

5. Newtonian gravitational force

6. Gravitational acceleration • Inertial force Newton’s 2nd Law • F = mg [Newtons] • Equating • F = GmM/r2· r̂ = mg [Newtons] • Acceleration of gravity • g = GM/r2 = 2c·Dn [m/s2] • Cogravitation field • K = r x g/cr = (v x g)/c2 [1/s] • F = mg + mi(v x K) [Newtons] • Acceleration is a measure of EM wavefront (geodesic) curvature • which is a function of an energy density gradient. • Inertial mass mi = Gravitational mass mg (Equivalence Principle) • as both arise from the same causal mechanism (i.e., motion • though a region of increased energy density).

7. Gravitational potential energy Gravitational scalar potential (potential energy/unit mass) f = U = -GM/r [J/kg] Gravitational potential is a measure of electromagnetic energy density of mass M of volume V. Near the Earth’s surface, the gravitational potential is linearly approximated as Df = f(R + h) – f(R) = -gh [J/kg] where h = height above Earth’s surface and R = Earth radius Acceleration of gravity g represents the gravitational field strength (= F/m) and is opposite to the scalar gravitational poential f g = -gradf = = -∇f [m/s2] Gravitational force in terms of scalar potential F = -m∇f [Newtons]

8. Normalized gravitational potential Spherical mass M

9. Escape velocity Kinetic energy T = 1/2 mv2 [Joules] Equating gravitational potential energy and kinetic energy -GM/r = 1/2 mv2 yields ve = √(2GM/r) [m/s] where ve = escape velocity of mass m from a central mass M. A black hole is a region of spacetime where gravity is so strong not even light can escape. Gravity is a measure of intrinsic (surface) curvature. The size of a black hole event horizon is described by the Schwarzschild radius rS = 2GM/c2 [m]

10. Schwarzschild radius Mass vs. Schwarzschild radius RS of subatomic and astrophysical objects RS = 2GM/c2

11. Laplacian potential Mass acts as a source of the gravitational field described by Poisson’s equation ∇2f = -4pGr = 4pGM/V [J/kg·m2] where ∇2 = Laplacian operator (divergence of a gradient of a function) [m-2] f = gravitational scalar potential (= -GM/r) [J/kg] M = mass [kg] r = mass density (= mass/volume) [kg/m3] V = mass volume [m3] Gauss’s law of gravity (differential form) ∇·g = -4pGr = -fgr [N/m·kg] where fg = gravitational flux [J/kg] r = mass density [kg·m-1]

12. Polarizable Vacuum In terms of the vacuum refractive index KPV of the polarizable vacuum, the time-indepent form is ∇2f = ∇2c02/(KPV(r,M) = ∇2c02/(1/(1 + 2f/c2) [s-2] A positively curved spacetime corresponds to a converging refractive index (KPV > 1) in which light slows down and material objects contract in size due to increase in EM energy density. For a gravitational potential well, the curvature in tangent space manifold is concave up while the refractive index and frequency hill is concave down. In contrast to GR (with unexplained mechanism for assumed spacetime distortion), gravitational effects in a polarizable vacuum (including length contraction, time dilation, frequency shift, alteration in the speed of light, etc) are EM wave interaction effects due to local variation in the vacuum refractive index KPV(r,w,M)

13. Non-rotating black hole in a polarizable vacuum • Geodesic curvature is produced by gradient in energy density

14. Mass induced EM wavefront curvature • Acceleration is a measure of wavefront curvature induced by an electro- • magnetic spectral energy density gradient in the vicinity of mass

15. Einstein field equation In the Einstein General Theory of Relativity (GR), gravity is represented mathematically as a curvature of spacetime. GR gravitational field equation equates curvature to sources of stress-energy momentum Gmn = Rmn – ½gmnR = -(8pG/c2)Tmn = -kTmn curvature source where: Gmn =Einstein tensor [m-2] Rmn= Ricci curvature symmetrical tensor (contracted from Riemann tensor = Rabcb) [m-2] gmn= Lorentz spacetime metric tensor (= nmn + hmn) [ - ] R = scalar curvature defined as trace of Ricci tensor [m-2] G = Newtonian gravitational constant [≃ 6.67384E-11 nt·m2/kg2] c = velocity of light (= l/f = c0/n = 1/√(e0m0)) [≃ 2.997924E8 m/s] Tmn= stress-energy-momentum tensor [kg/m3] k = Einstein’s constant (= -8pG/c2) [m/kg]

16. Stress-Energy-Momentum tensor In the Einstein General Theory of Relative (GR), no physical mechanism is defined as to how matter is said to ‘bend’ spacetime or how spacetime alters the motion of matter. GR represents a metaphysical mathematical coordinate description of space (relative location of objects) and time (ordering of events) in terms of curvature of geodesics without a quantum mechanical description of the underlying physical vacuum. The Einstein equation is equivalent to a statement that energy density equals pressure, hence, gravitation is related to vacuum energy/pressure. • Tmn = (1/8p)(c4/G)Gmn= k∙FP Gmn • where: • Tmn= stress-energy-momentum tensor [N/m-2] • c = velocity of light (= c0/G = 1/√(e0m0)) [≃ 2.997924E8 m/s] • G = Newton’s Gravitation constant [≃ 6.67428E-11 N·m2/kg2] • Gmn = Einstein tensor [m-2] • k = Einstein constant (= -8pG/c2) [N·m2/kg2} • FP= Planck force (= c4/G = mPlP/tP2) [= 1.210E44 N]

17. Geodesic deviation in a gravity field

18. Vacuum refractive index • In an optical theory of gravity, the vacuum refractive index KPV(r,w,M) is • a measure of the local energy density . The acceleration of gravity g • is a measure of the spectral energy gradient. The Gravitation Constant G • is a constant relating curvature and energy-momentum density. • Gravity represents a frequency arrthymia between mass oscillators as • they attempt to synchronize. The acceleration of gravity g is equivalent to • a frequency shift Dn in a standing wave system restrained from free fall is • given by g = 2cDn. In free fall, the frequency difference is reduced to zero. • Effects of change in gravitational potential on motion of matter in terms • of spacetime curvature may be described equivalently in terms of changes • in frequency and phase of de Broglie matter waves. A moving wave • system undergoes a Lorentz contraction g (= √(1 – v2/c2) and Lorentz- • Doppler shift Dl in the direction of motion. Acceleration is proportional • to the frequency difference Dn while velocity is proportion to the phase • difference Df.

19. Tangent space Gravitational acceleration is equal to the negative of the gravitational potential (g = -∇f) and is proportional to the EM frequency gradient (g = 2cDn·ru)

20. Gravitational potential well • Earth mass ≃ 5.972E24 kg • Earth mean radius ≃ 6,378 km • Acceleration of gravity @ Earth’s surface ≃ 9.8 m/s2 • Escape velocity of Earth ≃ 11.2 km/s

21. Gravitational potential well & Frequency hill

22. Variation in Earth’s gravitational gamma & Vacuum refractive index

23. Acceleration of gravity g and frequency shift Dn vs. distance from Earth

24. Nonuniform gravitational well and frequency hill of a spherical mass

25. Polarized vacuum spectral energy density • Quantum vacuum (QV) spectum (far-field) • Polarized vacuum (PV) spectrum (near-field) • Electro-Gravi-Magnetic (EGM) spectrum (internal)

26. Frequency shift in a gravitational field

27. Gravitational effects on EM fields • Wavelength increase corresponds to apparent time dilation • Frequency increase corrends to apparent space contraction

28. Gravitational field standing wave pattern of a central mass

29. Keplar’s Laws of orbital motion

30. Keplerian & non-Keplarian motion Orbital motion of mass m about a large, central mass M (scalar potential effect) Orbital motion of diffuse, spin density waves (vector potential effect)

31. Lagrange points Lagrange (libration) points are orbital positions where gravitational force equals centrifugal force L1 – L5 = Lagrange points M1 = central mass M2 = orbital mass

32. Gravitational lens

33. Co-gravitation • Mass current generates a gravitomagnetic (co-gravitation) field K (= ∇ x Ag)

34. Motion induced gravitomagnetic field • Mass motion constitutes a mass current with an induced gravitomagnetic • field analogous to an electric current with an associated magnetic field

35. Gravitational field variation @ relativistic velocities

36. Gravitational wave effects • Gravitational waves exhibit quadrapole polarization

37. Graviton interactions

38. Quantum gravity models

39. EM standing wave interference lattice • Interference antinodes act as scattering centers for an incident EM wave

40. Phase conjugate wave reflection • EM Fresnel zones of interacting mass oscillators result in phase • conjugate wave reflection in a polarizable vacuum

41. Standing wave interference • Standing wave interaction of a • pair of oscillators of equal • frequency in an idealized, • nondissipative elastic medium • results in attraction or • repulsion depending on phase • synchronization. • Force imbalance is proportional • to the difference in wave energy • density between oscillators and • inversely to the wave velocity.

42. Quantized wave interference metric • Wavefront interference of mass oscillators result in a quantized field metric

43. Spin 2 Graviton gg* • Graviton formed by coupling of photon and counter-propagating phase conjugate

44. Graviton curvature and torsion • Graviton gg* is of helicoid geometry whereas photon g is a helix

45. Fourier representation of gravitational frequency spectrum

46. Oscillator frequency synchronization of coupled mass pair

47. Vacuum spectral energy density modulation • Mass induces a local recompression of the vacuum spectral energy density. • Acceleration of gravity is a measure of the spectral energy density gradient.

48. Polarized vacuum response due to presence of mass

49. Spectral energy density of Earth’s gravitational field Ref: QE, Storti et al • The spectral energy density rSED(w) represents the energy density per • frequency mode. The number of mode of the PV spectrum increases • with radial distance from a mass object and decreases in energy.

50. Falling mass effects