The Spacetime Response Constant kSEG

This note introduces the constant

k_SEG = 4 π G / c³ = 4 π ℓ_P² / ħ,

and shows that it provides a unified constants-explicit normalization across gravitational physics. Expressed in terms of k_SEG, the Einstein coupling (8 π G / c⁴), the black-hole first law, the critical density of flat FRW cosmology, the standard Planck units, and the Einstein–Hilbert action all take compact and transparent forms.

The analysis is purely algebraic: all results follow from exact substitutions within standard general relativity and semiclassical black hole thermodynamics. No new field equations or assumptions are introduced.

The constant k_SEG has physical dimension time per mass (s kg⁻¹). In Einstein’s equations, 8 π G / c⁴ can be written as (2/c) k_SEG, so k_SEG acts as a flux–response coefficient that converts stress-energy (energy density) into spacetime curvature with the correct units. Rewriting the black hole first law shows that the energy–area term can be written as (κ / 2c)(1 / k_SEG), where κ is the surface gravity. In cosmology, the critical mass density can be written as ρ_crit = 3 H² / (2 k_SEG c³), with H the Hubble parameter.

The Planck hierarchy can also be expressed using k_SEG alone: the Planck length and time are

ℓ_P = sqrt[ (ħ k_SEG) / (4 π) ],

t_P = sqrt[ (ħ k_SEG) / (4 π c²) ],

while the Planck mass and Planck temperature are

M_P = sqrt[ (4 π ħ) / (k_SEG c²) ],

T_P = sqrt[ (4 π ħ c²) / (k_SEG k_B²) ].

Thus ℓ_P and t_P scale as sqrt(k_SEG), whereas M_P and T_P scale as 1 / sqrt(k_SEG). This dual scaling shows that k_SEG simultaneously sets (i) a minimal causal grain for spacetime structure and (ii) the mass/temperature threshold at which gravitational and quantum effects become comparable.

Finally, the Einstein–Hilbert Lagrangian density can be written in the form

L_EH = (1 / 4 k_SEG) R sqrt(-g),

so that the overall prefactor becomes 1 / (4 k_SEG), which has units kg s⁻¹. This reveals the inverse of k_SEG as an effective stiffness of spacetime curvature, in the same sense that material response coefficients relate stress to strain.

In summary, k_SEG is a universal response constant of spacetime: it controls the normalization of the Einstein equations, black hole mechanics, cosmological expansion, the Planck scale, and the Einstein–Hilbert action, all through exact constants-explicit identities.