Physics:Mass-Entropy Equivalence in the Theory of Entropicity(ToE)

Abstract

This article derives the analogue of Einstein’s [math]\displaystyle{ E=mc^{2} }[/math] within the Theory of Entropicity (ToE). In ToE, entropy is elevated to a local, dynamical field whose constrained content gives rise to mass and energy. We show that the energy of a system can be expressed as [math]\displaystyle{ E=\Theta_{*}S }[/math] with [math]\displaystyle{ \Theta_{*} }[/math] an entropic potential (not the ambient temperature), leading to a mass–entropy relation [math]\displaystyle{ M=\Theta_{*}S/c^{2} }[/math]. Two worked examples are presented: (i) horizon calibration with a Schwarzschild black hole, and (ii) weak-field calibration using solar light bending.

1. Entropy Variables and Units

We introduce a dimensionless entropy field:

[math]\displaystyle{ \sigma(x,t)=\frac{S(x,t)}{k_{B}} }[/math]

where [math]\displaystyle{ S(x,t) }[/math] is the physical entropy content and [math]\displaystyle{ k_{B} }[/math] is Boltzmann’s constant. The physical entropy density and total content are:

[math]\displaystyle{ s(x,t)=\frac{\partial S}{\partial V}\quad \text{[J\,K}^{-1}\text{m}^{-3}\text{]}} }[/math]

[math]\displaystyle{ S_{\Omega}=\int_{\Omega}s(x,t)\,\mathrm{d}^{3}x\quad \text{[J\,K}^{-1}\text{]}} }[/math]

Energy and mass densities:

[math]\displaystyle{ \varepsilon(x,t)\,[\text{J\,m}^{-3}],\qquad \rho_{m}(x,t)=\frac{\varepsilon(x,t)}{c^{2}}\,[\text{kg\,m}^{-3}] }[/math]

We define an entropic potential [math]\displaystyle{ \Theta_{*}(x,t) }[/math] with units of kelvin (K). This is not the ambient temperature but the conjugate field to constrained entropy.

2. First-Law-Type Statement in ToE

For quasi-static changes at fixed constraints:

[math]\displaystyle{ \boxed{\mathrm{d}E=\Theta_{*}\,\mathrm{d}S}} }[/math] (1)

This defines [math]\displaystyle{ \Theta_{*} }[/math] operationally as the energy cost of increasing the constrained entropy by one unit.

3. Local and Global Energy Relations

If [math]\displaystyle{ \Theta_{*} }[/math] varies slowly across [math]\displaystyle{ \Omega }[/math], the local and global energy densities are:

[math]\displaystyle{ \varepsilon(x)=\Theta_{*}(x)\,s(x)} }[/math] (2)

[math]\displaystyle{ E_{\Omega}=\Theta_{*}\,S_{\Omega}} }[/math] (3)

4. Mass–Entropy Equivalence (ToE Analogue of [math]\displaystyle{ E=mc^{2} }[/math])

Dividing Eqs. (2)–(3) by [math]\displaystyle{ c^{2} }[/math] gives:

[math]\displaystyle{ \rho_{m}(x)=\frac{\Theta_{*}(x)}{c^{2}}\,s(x)} }[/math] (4)

[math]\displaystyle{ M_{\Omega}=\frac{\Theta_{*}}{c^{2}}\,S_{\Omega}} }[/math] (5)

Tagline equations:

[math]\displaystyle{ \boxed{E=\Theta_{*}\,S,\qquad M=\frac{\Theta_{*}}{c^{2}}\,S} }[/math] (6)

These express that mass is constrained entropy scaled by [math]\displaystyle{ \Theta_{*}/c^{2} }[/math].

5. Calibration of [math]\displaystyle{ \Theta_{*} }[/math]

[math]\displaystyle{ \Theta_{*} }[/math] emerges from the equations of motion of [math]\displaystyle{ \sigma }[/math] in the ToE Lagrangian. In limiting cases:

5.1 Horizon Calibration (Black Holes)

For a Schwarzschild black hole:

[math]\displaystyle{ T_{\mathrm{H}}=\frac{\hbar c^{3}}{8\pi GMk_{B}},\qquad S_{\mathrm{BH}}=\frac{k_{B}c^{3}A}{4G\hbar},\quad A=4\pi\left(\frac{2GM}{c^{2}}\right)^{2}} }[/math] (7)

Smarr relation:

[math]\displaystyle{ Mc^{2}=2\,T_{\mathrm{H}}\,S_{\mathrm{BH}}} }[/math] (8)

Comparing (6) with (8):

[math]\displaystyle{ \boxed{\Theta_{*}=2\,T_{\mathrm{H}}}\quad \text{(horizon limit)}} }[/math] (9)

Numerically for [math]\displaystyle{ M=M_{\odot} }[/math]:

[math]\displaystyle{ T_{\mathrm{H}}=6.17\times10^{-8}\,\text{K},\quad \Theta_{*}=1.23\times10^{-7}\,\text{K}} }[/math]

and [math]\displaystyle{ (2T_{\mathrm{H}})S_{\mathrm{BH}}=Mc^{2} }[/math] to numerical precision.

5.2 Weak-Field Calibration (Solar Light Bending)

Entropic mass density:

[math]\displaystyle{ \rho_{m}(x)=\frac{\Theta_{*}(x)}{c^{2}}\,s(x)} }[/math] (10)

Potential sourcing:

[math]\displaystyle{ \nabla^{2}\Phi_{e}(x)=4\pi G\,\rho_{m}(x)=4\pi G\,\frac{\Theta_{*}(x)}{c^{2}}\,s(x)} }[/math] (11)

For a nearly uniform [math]\displaystyle{ \Theta_{*} }[/math] inside a spherical star:

[math]\displaystyle{ \Phi_{e}(r)=-\frac{GM_{\odot}}{r},\qquad M_{\odot}=\frac{\Theta_{*}}{c^{2}}\int_{\text{Sun}} s(x)\,d^{3}x=\frac{\Theta_{*}}{c^{2}}\,S_{\odot}} }[/math] (12)

Light deflection for a null ray with impact parameter [math]\displaystyle{ b }[/math]:

[math]\displaystyle{ \alpha_{\text{ToE}}=\frac{2}{c^{2}}\int_{-\infty}^{+\infty}\nabla_{\!\perp}(\Phi_{e}+\Psi_{e})\,dl =\frac{4}{c^{2}}\int_{-\infty}^{+\infty}\nabla_{\!\perp}\Phi_{e}\,dl =\frac{4GM_{\odot}}{c^{2}b}} }[/math] (13)

With [math]\displaystyle{ b\simeq R_{\odot} }[/math]:

[math]\displaystyle{ \alpha_{\text{ToE}}=\frac{4GM_{\odot}}{c^{2}R_{\odot}} =8.48\times10^{-6}\,\text{rad}=1.75''} }[/math] (14)

matching the observed 1.75 for grazing rays. Hence Eqs. (10)–(14) reproduce the standard weak-field test.

Appendix A: From the Line Integral to [math]\displaystyle{ \alpha=4GM/(c^{2}b) }[/math]

Let the unperturbed light path be the [math]\displaystyle{ z }[/math]-axis, with closest approach [math]\displaystyle{ b }[/math] along [math]\displaystyle{ x }[/math]. At [math]\displaystyle{ (x=b,z) }[/math]:

[math]\displaystyle{ r=\sqrt{b^{2}+z^{2}},\quad \Phi_{e}(r)=-\frac{GM}{\sqrt{b^{2}+z^{2}}}} }[/math] (A1)

Transverse gradient:

[math]\displaystyle{ \nabla_{\!\perp}\Phi_{e}=\frac{\partial \Phi_{e}}{\partial x}\Big|_{x=b} =-GM\,\frac{\partial}{\partial b}\left(\frac{1}{\sqrt{b^{2}+z^{2}}}\right) =\frac{GM\,b}{(b^{2}+z^{2})^{3/2}}} }[/math] (A2)

Insert into the weak-deflection integral:

[math]\displaystyle{ \alpha=\frac{4}{c^{2}}\int_{-\infty}^{+\infty}\nabla_{\!\perp}\Phi_{e}\,dz =\frac{4GM}{c^{2}}\int_{-\infty}^{+\infty}\frac{b\,dz}{(b^{2}+z^{2})^{3/2}}} }[/math] (A3)

Using:

[math]\displaystyle{ \int_{-\infty}^{+\infty}\frac{dz}{(b^{2}+z^{2})^{3/2}}=\frac{2}{b^{2}}} }[/math] (A4)

gives:

[math]\displaystyle{ \alpha=\frac{4GM}{c^{2}}\left(b\cdot \frac{2}{b^{2}}\right)=\frac{4GM}{c^{2}b}} }[/math] (A5)

Setting [math]\displaystyle{ b=R_{\odot} }[/math] yields [math]\displaystyle{ \alpha=1.75'' }[/math].

Conclusion

By elevating entropy to a local, dynamical field and introducing an entropic potential [math]\displaystyle{ \Theta_{*} }[/math], the Theory of Entropicity leads naturally to a mass–entropy equivalence relation:

[math]\displaystyle{ \boxed{E=\Theta_{*}S,\quad M=\Theta_{*}S/c^{2}} }[/math]

The horizon and weak-field examples show how [math]\displaystyle{ \Theta_{*} }[/math] can be calibrated from well-tested phenomena, making the ToE formulation predictive rather than heuristic.