Deterministic Scalar Field Calculator (with λ input + CSV)

User Inputs

You can edit either φ₀ or λ (the scaling parameter). Changing one automatically re‐derives the other via λ = √[(ħ·c)/G] / φ₀. All other derived constants update immediately.

Symbol	Description	Value
ħ (hbar)	Reduced Planck constant	J·s
c	Speed of light	m/s
G	Gravitational constant	m³·kg⁻¹·s⁻²
e	Elementary charge	C
m_e	Electron mass	kg
ε₀	Vacuum permittivity	F/m
φ₀	Scalar field amplitude
λ	Scaling parameter

All inputs OK. Derived constants updated below.

Derived Constants & Deviations

Constant	Equation	Computed (sci)	Full Value (non-sci)	Measured	Deviation (%)
Gravitational constant G (theory)	$G_{th} = \frac{ℏ c}{λ^{2} ϕ_{0}^{2}}$	6.674300e-11	0.000000000067	6.674300e-11	1.936487e-14
Planck length ℓP	$ℓ_{P} = \sqrt{\frac{ℏ G}{c^{3}}}$	1.616255e-35	0.000000000000	1.616255e-35	1.480493e-6
Planck time tP	$t_{P} = \frac{ℓ_{P}}{c}$	5.391246e-44	0.000000000000	5.391160e-44	1.603489e-3
Planck mass mP	$m_{P} = \sqrt{\frac{ℏ c}{G}}$	2.176434e-8	0.000000021764	2.176434e-8	1.571613e-5
Planck charge qP	$q_{P} = \sqrt{4 π ε_{0} ℏ c}$	1.875546e-18	0.000000000000	1.875546e-18	1.983693e-6
Fine-structure constant α	$α = \frac{e^{2}}{4 π ε_{0} ℏ c}$	7.297353e-3	0.007297352574	7.297353e-3	6.097087e-8
Classical electron radius re	$r_{e} = \frac{e^{2}}{4 π ε_{0} m_{e} c^{2}}$	2.817940e-15	0.000000000000	2.817940e-15	1.748940e-10
Electron Compton wavelength λc	$λ_{c} = \frac{h}{m_{e} c}$	2.426310e-12	0.000000000002	2.426310e-12	2.815353e-8
Scaling parameter λ (final)	$λ = \frac{\sqrt{\frac{ℏ c}{G}}}{ϕ_{0}}$	9.983644e+9	9983643770.876729965210	1.000000e+10	1.635623e-1
Planck length / electron radius (ℓP / rₑ)	$\frac{ℓ_{P}}{r_{e}}$	5.735590e-21	0.000000000000	5.740000e-21	7.682999e-2

Done. λ = 9.9836e+9, φ₀ = 2.1800e-18

Gravitational Constant Calculator

$G = \frac{ℏ \cdot c}{λ^{2} \cdot ϕ_{0}^{2}}$

with derived Planck mass: $m_{P} = λ \cdot ϕ_{0}$

(and $ϕ_{0} = \frac{m_{P}}{λ}$ gives $ϕ_{0} \approx 2.18 \times 10^{- 18}$ )

Derived Scalar Field Amplitude:

$ϕ_{0} = \frac{m_{P}}{λ} = \frac{2.18 \times 10^{- 8}}{1.0 \times 10^{10}} = 2.18 \times 10^{- 18}$

Derived Scaling Parameter:

$λ = \frac{m_{P}}{ϕ_{0}} = \frac{2.18 \times 10^{- 8}}{2.18 \times 10^{- 18}} = 9.98364277087673 \times 10^{9}$

Deterministic Derivation of the Gravitational Constant

This calculator elegantly demonstrates how the gravitational constant

G

can be derived purely deterministically – without any tuning parameters. All the values, including the scaling parameter

λ

and the scalar field amplitude

ϕ_{0}

, are computed directly from fixed equations.

In our model, the relation

ϕ_{0} = \frac{m_{P}}{λ}

(with

m_{P}

representing the Planck mass) automatically yields

G = \frac{ℏ \cdot c}{λ^{2} \cdot ϕ_{0}^{2}} .

With

m_{P} \approx 2.18 \times 10^{- 8}

 kg and

λ \approx 1.0 \times 10^{10}

, the scalar field amplitude is fixed to

ϕ_{0} \approx 2.18 \times 10^{- 18}

. This deterministic approach shows that the fundamental constants are not arbitrarily adjustable but instead emerge naturally from the intrinsic structure of the scalar field.