Forces and Mass in Temporal Physics

Forces and Mass in Temporal Physics

In this model, the fundamental building blocks of reality are "temporal elements" Ei that exist at each instance of time t. These are discrete entities that make up the fabric of time itself. Each temporal element Ei carries a "temporal flow" property vi(t) that can take values of either +1 or -1 at any given time t.

A value of vi(t) = +1 represents a positive temporal flow, indicating an increase or strengthening of the inherent "temporal attributes" like density, flux, energy and information content associated with that element at time t. Conversely, vi(t) = -1 corresponds to a negative temporal flow, signifying a decrease or weakening of these attributes.

The temporal attributes themselves - density D(t), flux F(t), energy E(t) and information I(t) - are proposed as fundamental quantities that characterize the state of the temporal fabric at each instance. For example, density D(t) represents the concentration of temporal elements, while flux F(t) quantifies the rate of change or dynamics within the temporal domain.

Interactions and Forces

These temporal elements interact with each other based on their individual flow values vi(t). The strength of the interaction between any two elements Ei and Ej is encoded in a "temporal dynamics matrix" Tij(t). The force Fij(t) arising from the interaction between elements i and j depends on the difference between their flows:

Fij(t) = α * (vi(t) - vj(t))

Where α is a proportionality constant. This temporally-induced force acts to influence the motion and evolution of elements within the fabric.

Mass, Density and Inertia

Within this framework, the mass m of an object is identified as being directly proportional to its temporal density D(t) - the concentration of temporal elements composing it. Additionally, an object's inertia or resistance to changes in its temporal flow scales with its mass or temporal density.

Dynamical Evolution

The temporal flows vi(t) themselves evolve dynamically according to an equation coupling them to the temporal attributes and inter-element interactions:

dvi(t)/dt = f(vi(t), D(t), F(t), E(t), I(t), Tij(t))

This means the rate of change of an element's flowvi(t) depends on its current flow value, the local temporal density, flux, energy, information content and its interaction strengths Tij(t) with other elements.

Similarly, the temporal attributes like density D(t) and flux F(t) also evolve based on the flows vi(t) and interactions between elements via equations of the form:

dD(t)/dt = g(vi(t), Tij(t))

dF(t)/dt = h(vi(t), Tij(t))

This symbolic coupled system of differential equations governs the intricate co-evolution of discrete temporal flows vi(t) and continuous temporal attributes like D(t), F(t), E(t), I(t) within the fabric.

Symmetry Breaking and Fluctuations

Deviations from symmetric or equilibrium temporal evolution are captured by "temporal fluctuations" δO(t), which oscillate around the mean value <O>. These fluctuations play a crucial role in spontaneously breaking temporal symmetries and driving emergent phenomena. The fluctuations evolve as:

dδO(t)/dt = -γδO(t) + η(t)

The -γδO(t) term makes the fluctuations decay over time in the absence of a driving force η(t). However, the presence of non-zero driving η(t) can sustain ongoing symmetry-breaking temporal fluctuations δO(t). The driving term η(t) represents influences from the underlying dynamics of temporal elements, their interactions, or external factors.

Spatial Dimensionality

Remarkably, in this model the very notion of spatial dimensions ΔS(t) is proposed to emerge from the dynamical interplay between temporal fluctuations δO(t) and the differences in positions of elements over time:

ΔS(t) = Σ((ri+1 - ri) · (1 + δO(t)))

Here ri represents the spatial position vector of an element at time i. This equation proposes that spatial extents arise as a consequence of temporal symmetry breaking encoded in the fluctuations δO(t).

Non-Linear Dynamics and Interactions

The specific functions f, g, h, ... governing the evolution of flows vi(t) and attributes like D(t), F(t) can incorporate non-linear scaling with respect to time t:

vi(t) ~ t^bi

Tij(t) ~ t^aij

This allows for non-linearly scaled interaction terms of the form:

vi(t)*Tij(t) ~ t^(bi + aij)

When summed over all elements, such terms can naturally give rise to overall non-linear dynamical evolution within the temporal fabric, potentially explaining emergent non-linear phenomena.

Limits and Connections

For mathematical consistency, the model can be derived from an overarching action principle formulation. Investigating the model's symmetries would reveal associated conservation laws. Importantly, in certain limits or regimes, the temporal physics framework is designed to reduce to the predictions of well-established theories like relativity and quantum mechanics.

Maximum Speed Limit

Finally, the model also incorporates a fundamental speed limit or maximum bandwidth constraint, akin to the speed of light in traditional physics. This arises from the intrinsic relationships governing information propagation and interactions within the temporal fabric dynamics. This speed limit regulates the rates at which temporal fluctuations and influences can propagate through the fabric.