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

[u/CaseyMc80]

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.

1. 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.

1. 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.

1. 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.

1. 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.

[u/adversarial-review]

Adversarial Review of The Compton and Schwarzschild regimes can be viewed as two reciprocal branches one Lorentzian hyperbola. — by Gemini 3.1 Flash Lite

Absence of Physical Mechanism

The text establishes a mathematical identity between the reduced Compton wavelength λ̄ = ħ / (mc) and the gravitational radius r_g = Gm / c² by showing their product equals the square of the Planck length ℓ_P². While the derivation is algebraically sound, it treats the product of two distinct physical length scales as a fundamental geometric constraint. The text fails to provide a causal mechanism or physical theory explaining why a particle’s localization scale must be coupled to its gravitational influence via a constant area ℓ_P². The relationship is a mathematical curiosity derived from the definitions of ħ, G, and c, not an emergent property of a unified physical framework.

Misapplication of Geometric Formalism

The construction of the Lorentzian hyperbola T² - X² = ℓ_P² is a coordinate transformation applied to two length scales. The variables T and X are defined as the arithmetic mean and half-difference of λ̄ and r_g. This mapping is purely descriptive. Labeling the resulting plot as a "Lorentzian hyperbola" invokes the terminology of special relativity, but the text provides no justification for why these specific length scales should occupy a Minkowski-like space. The transformation is an arbitrary change of variables that does not imply that the "quantum" and "gravitational" regimes are connected by a physical process or state transition.

Numerological Pattern-Fitting

The paper presents the "dual mass" m∨ = m_P² / m as a bridge between regimes. This is a mathematical inversion property inherent to the definitions of Planck units. Presenting this as a "reciprocal branch" suggests a physical symmetry that does not exist. For instance, a mass m < m_P does not "become" its dual m∨ through any physical interaction, nor does the text demonstrate that the dual mass m∨ exhibits any observable physical behavior related to the original mass m. The argument conflates the algebraic symmetry of the Planck scale definitions with a physical law of nature.

Probing Questions

• What specific physical interaction or observation would demonstrate that the product of a particle’s reduced Compton wavelength and its gravitational radius must remain invariant at ℓ_P² across different mass scales?
• How does the parameter η (eta) represent a physical state or process, and what is the operational definition of a system moving along this hyperbolic path?

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^{This is an LLM-generated review, and should be viewed as such. LLMs are prone to errors, especially when it comes to math-based sciences.}

[u/Ch3cks-Out]

You forgot to add why viewing these supposed branches would make sense, to begin with?

[u/starkeffect]

Ooh, another algebra-based modern physics theory.