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Intro To Gravitational Field Theory (Warp Mechanics 101)

  1. Gravity ≡ a field (not a force) which causes distortions (i.e., curvatures and dilation) in spacetime.
    • Corollary 1: Gravity is a wave with λ = v/f. Specifically, it is a low frequency (103Hz), long wavelength (λ ≥ 105m) field that is observed at macroscale levels. (f ≈ 1KHz, λ ≈ 100-1000 km)
    • Corollary 2: Gravitational waves travel at the speed of light. Therefore λgravity = c/f
    • Corollary 3: Gravitational waves behave similar to electromagnetic waves.
    • Corollary 4: The closer one moves to the point-source of the gravity field, the “strength” of the field (amplitude of the gravity wave) increases.
    • Corollary 4b: Distortions (curves and expansion-dilation) in space and time increase (become more pronounced) as one moves closer to the point source of the gravitational wave
  2. Gμν=(8πG/c4)×Tμν
  3. Location of an arbitrary point called s in 2-d (x, y), 3-d (x, y, z), and 4-d (x, y, z, t):
    • 2-d: s2 = x2 + y2
    • 3-d: s2 = x2 + y2 + z2
    • 4-d: s2 = x2 + y2 + z2 – c2t2
      • Past influences the present, and present influence the future (Minkowski lightcone)
      • General Relativity tells us we live in a spacetime geometry. It applies to accelerating frames of reference.
      • Special relativity applies to inertial, constant velocity, non-accelerating frames of reference
  4. Strong Nuclear Force: Within the nucleus of atoms (i.e., at subatomic, nanoscale levels) the force field that binds protons together within a nucleus (via neutrons) is multiple orders of magnitude “stronger” than gravitational fields.
    • Corollary 1: spacetime distortions (and constraints) don’t exist at the subatomic level. At the subatomic level all effects of gravity are canceled out. Thus particle theory (i.e., quantum mechanics) takes over.

Where:

  • ≡ defined as
  • ≈ equivalent to
  • f: frequency in Hertz
  • λ: wavelength
  • v: velocity
  • c: velocity of light = 2.998×108m/s
  • x, y, z: arbitrary coordinates in 3-dimensional space
  • t: time
  • Fμν: electromagnetic field tensor, all the information about the electric and magnetic field
  • Tμν: energy-stress tensor, describes the mass-energy, momentum, and stress density
  • Gμν: Einstein tensor, curvature of space-time
  • (8πG/c4)Tμν: coupling of energy, momentum, stress and strain from matter to curvature
  • 8πG/c4: matter-gravity coupling (coupling of energy-momentum) to curvature = 2.077×10-43 1/Newton
  • G: Newton’s Gravitational Constant = 6.674×104 Newton⋅meter2/kilogram2