The Compton and Schwarzschild regimes can be viewed as two reciprocal branches one Lorentzian hyperbola.

The complete relationship is:

λ̄(m) r_g(m) = ℓ_P²

together with:

m∨ = m_P² / m

and:

λ̄(m) = r_g(m∨)

r_g(m) = λ̄(m∨)

In logarithmic scale coordinates, the transformation is simply a reflection about the Planck point.

The Compton and gravitational regimes can therefore be viewed as two reciprocal branches of one Planck-centred Lorentzian hyperbola.

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Full derivation

1. The Two Characteristic Scales

For a given invariant mass (m), define two characteristic length scales.

Reduced Compton radius

This represents the quantum localization scale:

λ̄ = ħ / (mc)

Gravitational radius

This represents the gravitational or horizon scale:

r_g = Gm / c²

The Schwarzschild radius is usually defined as:

r_s = 2Gm / c² = 2r_g

Here, (r_g = Gm/c²) is used because it gives an exact Planck-scale inversion relation.

2. Planck Inversion and the Geometric Mean

The Planck length and Planck mass are defined by:

ℓ_P = √(ħG / c³)

m_P = √(ħc / G)

Multiplying the reduced Compton radius by the gravitational radius eliminates the mass:

λ̄ r_g
= [ħ / (mc)] [Gm / c²]
= ħG / c³
= ℓ_P²

Therefore:

λ̄ r_g = ℓ_P²

Taking the geometric mean gives:

√(λ̄ r_g) = ℓ_P

The Planck length is therefore the exact geometric mean of the quantum and gravitational scales.

Because their product is fixed, the two radii are reciprocal under inversion through the Planck scale:

r_g = ℓ_P² / λ̄

λ̄ = ℓ_P² / r_g

As one scale increases, the other decreases by exactly the inverse amount.

3. The Lorentz Hyperbola

Define new coordinates using the sum and difference of the two radii:

T = ½(λ̄ + r_g)

X = ½(λ̄ − r_g)

Now calculate the Lorentzian interval:

T² − X²
= [½(λ̄ + r_g)]² − [½(λ̄ − r_g)]²
= λ̄ r_g
= ℓ_P²

Therefore, the two scales lie on the hyperbola:

T² − X² = ℓ_P²

This gives a single continuous geometric track connecting the quantum and gravitational regimes.

A convenient parameterization is:

T = ℓ_P cosh(η)

X = ℓ_P sinh(η)

which implies:

λ̄ = T + X = ℓ_P e^η

r_g = T − X = ℓ_P e^(−η)

Their product remains fixed:

λ̄ r_g = ℓ_P²

The parameter (\eta) therefore moves scale between the quantum and gravitational sides without changing the underlying Planck-area product.

4. The Self-Dual Point

At the Planck mass:

m = m_P

the two radii become identical:

λ̄ = ħ / (m_P c) = ℓ_P

r_g = Gm_P / c² = ℓ_P

Therefore:

λ̄ = r_g = ℓ_P

This is the unique self-dual point of the inversion.

For masses below the Planck mass:

m < m_P → λ̄ > ℓ_P > r_g

The quantum scale dominates.

For masses above the Planck mass:

m > m_P → r_g > ℓ_P > λ̄

The gravitational scale dominates.

5. Reciprocal Mass Inversion

Define the dual mass:

m∨ = m_P² / m

Because:

m_P² = ħc / G

the gravitational radius of the dual mass is:

r_g(m∨)
= Gm∨ / c²
= Gm_P² / (mc²)
= ħ / (mc)
= λ̄(m)

Similarly:

λ̄(m∨)
= ħ / (m∨c)
= Gm / c²
= r_g(m)

Therefore:

λ̄(m) = r_g(m∨)

r_g(m) = λ̄(m∨)

The inversion:

m → m_P² / m

exactly exchanges the quantum and gravitational length scales.

A sub-Planckian particle is therefore paired mathematically with a super-Planckian mass whose gravitational radius equals the particle’s reduced Compton radius.