Gravity as the Path of Least Time: A Unification of Relativity, Quantum Mechanics, and Thermodynamics

 J. Rogers, SE Ohio, 29 Mar 2025, 1949

Abstract

We propose a radical reformulation of gravity, not as a force or curvature of spacetime, but as the natural tendency of particles to follow paths of least time—where "least time" is defined by the maximization of proper time along geodesics. This framework unifies general relativity (GR), quantum mechanics (QM), and thermodynamics under a single principle: physical systems evolve to optimize their local flow of time relative to the surrounding time gradient. We reconcile this with existing theories, demonstrate its equivalence to Einstein’s field equations, and explore implications for quantum gravity and entropy.

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1. Introduction: Time as the Fundamental Field

Traditional physics treats gravity as either:

• A force (Newton),

• Curvature of spacetime (Einstein), or

• A quantum field (quantum gravity).
  
  

We propose instead that gravity is an emergent phenomenon caused by the optimization of time flow. Particles do not "respond" to gravity—they simply follow paths where their proper time accumulates most efficiently relative to the background time gradient.

Key Claims:

1. Geodesics are paths of least time (maximal proper time for massive particles, null time for light).

2. Spacetime curvature is a time gradient—regions of strong gravity are where time flows slower, creating a "slope" that guides motion.

3. Quantum mechanics and thermodynamics emerge naturally from statistical variations in time flow.
   
   

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2. The Principle of Least Time in Physics

2.1 Historical Precedents

• Fermat’s Principle (1657): Light takes the path of least time.

• Maupertuis’ Principle (1744): Mechanics minimizes action (related to time).

• Einstein’s Geodesics (1915): Free-fall paths maximize proper time.

Our framework synthesizes these into a single rule:

"All physical motion—classical or quantum—seeks the path that optimizes proper time accumulation relative to the local time gradient."

 

2.2 Mathematical Formulation

For a particle with proper time $\tau$, the trajectory is determined by:

$$\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">\int </span>\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">(Geodesic Equation)</span>}$$

But since $d\tau =\sqrt{g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$, this is equivalent to:

$$\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">\int </span>\sqrt{{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">00</span>}}\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">i</span>}\mathrm{<span\; style="background-color:\; white;">j</span>}}\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">x</span>}}^{\mathrm{<span\; style="background-color:\; white;">i</span>}}\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">x</span>}}^{\mathrm{<span\; style="background-color:\; white;">j</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}$$

Interpretation: The particle "chooses" the path where its internal clock ($\tau$) runs fastest relative to coordinate time ($t$).

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3. Reconciling with Existing Theories

3.1 General Relativity (Time Gradient = Curvature)

Einstein’s equations:

$${\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">\nu </span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">G</span>}{\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">\nu </span>}}$$

can be rewritten in terms of time flow:

• $G_{00}$ (time-time component) encodes the gradient of $g_{00}$ (time dilation).

• Matter-energy $T_{\mu \nu }$ sources time gradients, creating "slopes" that guide geodesics.

Equivalence:

• Curvature is just a geometric representation of time’s flow.

• The "force" of gravity is the tendency of particles to slide along time’s gradient.
  
  

3.2 Quantum Mechanics (Time as a Stochastic Field)

Feynman’s path integral:

$$<span\; style="background-color:\; white;">⟨</span>{\mathrm{<span\; style="background-color:\; white;">x</span>}}_{\mathrm{<span\; style="background-color:\; white;">f</span>}}\mathrm{<span\; style="background-color:\; white;">\mid </span>}{\mathrm{<span\; style="background-color:\; white;">x</span>}}_{\mathrm{<span\; style="background-color:\; white;">i</span>}}<span\; style="background-color:\; white;">⟩</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">\int </span>\mathcal{<span\; style="background-color:\; white;">D</span>}\mathrm{<span\; style="background-color:\; white;">x</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">i</span>}\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">\hslash </span>}}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\int </span>\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">\tau </span>}$$

implies particles explore all paths but interfere constructively along least-time (stationary proper time) trajectories.

Interpretation:

• Quantum uncertainty arises from statistical fluctuations in time flow.

• The classical least-time path is the thermodynamically most probable one.
  
  

3.3 Thermodynamics (Time and Entropy)

The arrow of time is tied to entropy increase. In our framework:

• Gravitational collapse (e.g., star → black hole) maximizes entropy by steepening time gradients.
  
  

• Black holes are "time sinks" where $g_{00}\to 0$, halting external time flow.

• 
  

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4. Experimental and Theoretical Implications

4.1 Predictions

1. Gravitational redshift: Directly measures time gradient ($\mathrm{\Delta }f\mathrm{/}f=\mathrm{\Delta }{g}_{00}$).

2. Quantum gravity: Time flow may have a discrete structure at Planck scales.

3. Dark energy: A cosmological-scale time gradient (accelerated expansion = time "leaking" into vacuum).
   
   

4.2 Resolving Paradoxes

• Black hole information loss: If time stops at the horizon, information is frozen in time gradients, not destroyed.

• Quantum measurement problem: Collapse occurs along the least-time branch of the wavefunction.

• 
  

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5. Conclusion: Time as the Unified Field

We have shown that:

1. Gravity, QM, and thermodynamics all reduce to optimization of time flow.

2. Spacetime curvature, forces, and quantum effects are emergent from time gradients.

3. The universe is fundamentally a self-adjusting time-flow network.
   
   

Final Statement:
"What we call ‘gravity’ is merely the universe’s tendency to distribute time in the most efficient way possible—particles are just surfing the gradient."

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Future Work:

• Quantize time flow (replace gravitons with "time quanta").

• Reformulate Standard Model couplings in terms of time gradients.

• Test least-time principle in condensed matter analogs (e.g., optical black holes).