d4 natural or Planck units Name Quantity Planck

d=4: “natural” or Planck units Name Quantity Planck length Length (L) Planck mass Mass (M) Planck time Time (T) Planck charge Electric charge (Q) Planck temperature Temperature (Θ) Reduced Planck: 8 p G = 1 G = h = c = k. B = 1 Expressions

Parameterized post-Newtonian (PPN) formulation (classical gravity: metric theory that obey EP) γ How much space curvature gij is produced by unit rest mass ? β How much nonlinearity is there in the superposition law for gravity g 00 ? β 1 How much gravity is produced by unit kinetic energy ? β 2 How much gravity is produced by unit gravitational potential energy ρ0 / U ? β 3 How much gravity is produced by unit internal energy ρ0Π ? β 4 How much gravity is produced by unit pressure p ? ζ Difference between radial and transverse kinetic energy on gravity η Difference between radial and transverse stress on gravity Δ 1 How much dragging of inertial frames g 0 j is produced by unit momentum ρ0 v ? Δ 2 Difference between radial and transverse momentum on dragging of inertial frames

Parameter Bound Experiment γ− 1 2. 3 x 10 - 5 Cassini tracking β− 1 2. 3 x 10 − 4 Nordtvedt effect ξ 0. 001 Gravimeter data α 1 10 − 4 Lunar laser ranging α 2 4 x 10 - 7 Sun axis' alignment with ecliptic α 3 4 x 10 − 20 Pulsar spin-down statistics ζ 0. 02 Combined PPN bounds ζ 2 4 x 10 - 5 PSR 1913+16 ζ 3 10 - 8 Lunar acceleration ζ 4 0. 006† Kreuzer experiment α 1 = 7Δ 1 + Δ 2 − 4γ − 4 ζ 2 = 2β + 2β 2 − 3γ − 1 ζ 3 = β 3 − 1 α 2 = Δ 2 + ζ − 1 ζ 4 = β 4 − γ α 3 = 4β 1 − 2γ − 2 − ζ