Reformulation, in temporal physics (again)

Reformulating Physics: The Temporal Index Approach

Introduction

Physics, as traditionally understood, treats space and time as distinct but interwoven concepts. However, what if space is not fundamental at all? In this model, space is just an emergent sequence of time values, and all physical equations must be rewritten in terms of this fundamental temporal index.

By reformulating relativity, quantum mechanics, classical mechanics, electromagnetism, and thermodynamics, we uncover a deeper coherence in how reality operates. The speed of light is no longer a velocity limit but a maximum rate at which index values can change. Mass is measured in seconds, momentum is a rate of index change, and energy is a function of sequencing constraints.

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Reformulating Relativity in Temporal Index Notation

1. Lorentz Transformation

$$t'_{\text{index}} = \gamma \left( t_{\text{index}} - \frac{(t_{\text{index value change}})}{c} \right)$$

2. Time Dilation

$$\Delta t_{\text{index}}' = \gamma \Delta t_{\text{index}}$$

3. Length Contraction

$$L_{\text{index}}' = \frac{L_{\text{index}}}{\gamma}$$

4. Mass-Energy Equivalence

$$E = (t_{\text{index mass}})(c^2)$$

5. Relativistic Energy-Momentum Relation

Traditional Form:

$$E^2 = p^2c^2 + m^2c^4$$

Reformulated:

$$(t_{\text{index energy}})^2 = (t_{\text{index value change}})^2 c^2 + (t_{\text{index mass}})^2 c^4$$

Invariant interval and Lorentz transformations:

• Invariant Interval:

  $$s_{\text{index}}^2 = c^2\,t_{\text{index}}^2 - \Bigl(t_{\text{index value change}}\Bigr)^2$$
  

• Lorentz Transformation:

  $$t'_{\text{index}} = \gamma \left( t_{\text{index}} - \frac{t_{\text{index value change}}}{c} \right), \quad t'_{\text{index value change}} = \gamma \left( t_{\text{index value change}} - c\,t_{\text{index}} \right)$$

Temporal Index (Discrete Time Progression):
$$t_{\text{index}}=n{t}_P$$

where $n$ is the index (an integer) and $t_P$ is the Planck time.

Rate of Change (Spatial Emergence):
The rate at which the temporal index evolves over a given index range is what we’ll use to describe spatial displacement. This rate can be thought of as the change in the index values relative to the time progression, expressed as:

$$\text{Rate of Change}=\frac{\mathrm{\Delta }{t}_{\text{index}}}{\mathrm{\Delta }n}\mathrm{.}$$

Here, $\mathrm{\Delta }{t}_{\text{index}}$ represents the difference between temporal index values (which, when measured in Planck units, reflects time) and $\mathrm{\Delta }n$ is the index range. This rate captures how quickly the index evolves relative to the range of $n$.

This formulation ensures that the maximum allowed rate of change (analogous to $c$) is invariant—thus preserving Lorentz invariance in the temporal index model.

let's express the Lorentz transformation using the rate of change:

1. Temporal Index Transformation:

   $$t'_{\text{index}} = \gamma \left( t_{\text{index}} - \frac{ \text{Rate of Change}}{c} \right)$$
   

2. Rate of Change Transformation (Spatial Emergence):

   $$\text{Rate of Change}' = \gamma \left( \text{Rate of Change} - \frac{c\, t_{\text{index}}}{1} \right)$$

This way, the "Rate of Change" reflects how quickly the temporal index evolves and correlates directly with space via the rate of temporal evolution. This preserves the symmetry in Lorentz transformations—both time and space are connected via the "rate of change" over the discrete index range, making the model fit well with Lorentz invariance.

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Reformulating Quantum Mechanics in Temporal Index Notation

1. Schrödinger Equation

$$i\hbar \frac{\partial \Psi}{\partial t_{\text{index}}} = -\frac{\hbar^2}{2(t_{\text{index mass}})} \nabla^2_{t_{\text{index}}} \Psi + V(t_{\text{index}}) \Psi$$

2. Uncertainty Principle

$$\Delta t_{\text{index}} \Delta (t_{\text{index value change}}) \geq \frac{\hbar}{2}$$

3. Momentum Operator

$$\hat{p} = -i\hbar \nabla (t_{\text{index}})$$

4. Kinetic Energy

$$T = \frac{(\nabla (t_{\text{index}}))^2}{2(t_{\text{index mass}})}$$

5. de Broglie Wavelength

$$\lambda = \frac{h}{t_{\text{index value change}}}$$

6. Quantum Tunneling Probability

$$T \approx e^{-2 \int_{\text{barrier}} \sqrt{2(t_{\text{index mass}})(V(t_{\text{index}})-E)} \, d(t_{\text{index}}) / \hbar}$$

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Reformulating Classical Mechanics in Temporal Index Notation

1. Newton’s Second Law

Traditional Form:

$$F = ma$$

Reformulated:

$$F_{\text{index}} = (t_{\text{index mass}}) \frac{d^2 (t_{\text{index}})}{dt_{\text{index}}^2}$$

2. Work-Energy Theorem

Traditional Form:

$$W = \int F dx$$

Reformulated:

$$W_{\text{index}} = \int F_{\text{index}} d(t_{\text{index}})$$

3. Lagrangian Mechanics

Traditional Form:

$$L = T - V$$

Reformulated:

$$L_{\text{index}} = \frac{(\nabla (t_{\text{index}}))^2}{2(t_{\text{index mass}})} - V(t_{\text{index}})$$

∇(tindex​): The discrete change in the temporal flow value between two adjacent time indices (this represents how the flow changes, analogous to velocity).

$t_{\text{index mass}}$: The mass associated with the system at the given index (could be a discrete "particle mass" at each time step).

$V(t_{\text{index}})$: The potential energy arising from the interactions at each time index (could depend on the spatial distance that emerges between flows).

($T=\frac{1}{2}m{v}^2$) but now express this in terms of the discrete change in flow.

Lagrangian function as:

$$L_n = T_n - V_n = \frac{1}{2} m \left( \frac{\Delta F_n}{\Delta t} \right)^2 + \frac{G m_1 m_2}{r_n}$$

This Lagrangian reflects the discrete nature of time evolution, where the rate of change of temporal flows determines the kinetic energy, and the interaction between flows generates a potential energy.

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Reformulating Electromagnetism in Temporal Index Notation

Since electric and magnetic fields are traditionally defined in space, we need to rewrite them in terms of temporal flows.

1. Gauss’s Law for Electric Fields

$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$

Reformulated:

$$\nabla (t_{\text{index}}) \cdot E_{\text{index}} = \frac{\rho_{\text{index}}}{\epsilon_0}$$

2. Faraday’s Law of Induction

$$\nabla \times E = - \frac{\partial B}{\partial t}$$

Reformulated:

$$\nabla (t_{\text{index}}) \times E_{\text{index}} = - \frac{\partial B_{\text{index}}}{\partial t_{\text{index}}}$$

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Reformulating Thermodynamics in Temporal Index Notation

1. First Law of Thermodynamics

$$dU = dQ - dW$$

Reformulated:

$$d(t_{\text{index energy}}) = d(t_{\text{index heat}}) - d(t_{\text{index work}})$$

2. Entropy and the Second Law

$$S = k_B \ln \Omega$$

Reformulated:

$$S_{\text{index}} = k_B \ln (\Omega_{\text{index}})$$

3. Heat Capacity

$$C = \frac{dQ}{dT}$$

Reformulated:

$$C_{\text{index}} = \frac{d(t_{\text{index heat}})}{d(t_{\text{index temperature}})}$$

The causal structure of spacetime can be written as a causal set of events, where the relationships between events (points in the index) determine the structure of spacetime. The causal ordering is governed by the relationship of flows across indices. In this, the events are indexed by time steps, and the interactions are defined by the rates of change between these indices.$$\text{Causal Set:}{E}_n\to E_{n+1}\text{if}{F}_n\text{ interacts with }{F}_{n+\mathrm{1}}$$

Where:

• $E_n$ and $E_{n+1}$ are two events at time indices $n$ and $n+1$, respectively.
• The interaction between the events is governed by their temporal flow magnitudes and the distance between them (in terms of time).

Consdiering Smooth trajectories, field and waves in discrete steps for the discrete derivative approximation:

$$\frac{dF}{dt} \approx \frac{F(n+1) - F(n)}{t_P}$$

This captures the idea that changes in the field $F$ at discrete time steps can be approximated by the difference between adjacent index values ( $F(n+1)$ and $F(n)$ ), divided by the temporal scale $t_P$, which represents the Planck time. Ensuring that the Planck time 

$t_P$ is frame-invariant. If $t_P$ is not invariant across reference frames, the model would lose the consistency required by Lorentz invariance. Given that $t_P$ represents a fundamental unit of temporal change, this invariance would be a key assumption.*A point of irreducability.

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Conclusion: A Unified View of Physics Through Temporal Indexing

By converting all fundamental equations into a single evolving temporal index, we eliminate artificial separations between space, time, mass, and energy. Instead of treating position and momentum as distinct, they are now seen as aspects of indexed sequencing flow.

This reformulation provides a more fundamental perspective on:

• The speed of light as a sequencing constraint
• Quantum mechanics and relativity as natural indexing effects
• Space as an emergent perception of temporal flows

This temporal index framework offers a deeper foundation for understanding physics—not as a collection of separate equations, but as a coherent, evolving system of sequencing constraints.