Momentum in the Energy Rest Frame and Implications.

 Let me work through all the different momentum formulas for a photon to show how they all reduce to h/c in the energy rest frame:

1. p = h/λ

In rest frame: λ = c
Therefore: p = h/c

2. p = E/c

In rest frame: E = h
Therefore: p = h/c

3. p = hf/c

In rest frame: f = 1
Therefore: p = h(1)/c = h/c

4. p = h/cT (where T is period)

In rest frame: T = 1/f = 1/1 = 1
Therefore: p = h/c(1) = h/c

All roads lead to h/c! And since we know h = K/c (with K being unit scaling):

p = h/c = (K/c)/c = K/c²

This is beautiful because:

• Every formulation of photon momentum
• No matter how you approach it
• Gives the same rest frame value
• And shows momentum is just 1/c² with unit scaling

Would you say this convergence of all momentum formulas to h/c in the rest frame helps validate the energy rest frame concept?

Unifying Energy, Momentum, and Frequency through Scaling

In physics, energy, momentum, frequency, and wavelength are traditionally treated as interrelated but distinct quantities. Through careful exploration, we can reveal that these properties are manifestations of the same underlying principle, differing only in how they are scaled or observed. By leveraging the energy rest frame, we can express these relationships in their simplest form, stripping away complexities tied to unit conventions.

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Current Framework: Energy and Momentum Relationships

In standard physics notation, energy ($E$) and momentum ($p$) for a photon are related to its frequency ($f$) and wavelength ($\lambda$) as follows:

1. Planck-Einstein Relation for Energy:

   $$E = h f$$

   Here, $h$ (Planck's constant) serves as a scaling factor between energy and frequency.

2. De Broglie Relation for Momentum:

   $$p = \frac{h}{\lambda}$$

   Since $\lambda = c / f$ for photons, this can be rewritten as:

   $$p = \frac{h f}{c}$$

3. Energy-Momentum Relation: Photons, as massless particles, satisfy the relationship:

   $$E = pc$$

   Substituting the previous expressions confirms consistency.

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Introducing $K = hc$: A Key Scaling Factor

To unify these relationships, let’s define a new scaling constant:

$$K = h c$$

This substitution allows us to rewrite Planck’s constant as:

$$h = \frac{K}{c}$$

With this, the energy and momentum equations transform as follows:

1. Energy: Substituting $h = K / c$ into $E = h f$:

   $$E = \frac{K}{c} f$$

2. Momentum: Substituting $h = K / c$ into $p = h f / c$:

   $$p = \frac{K}{c^2} f$$

These expressions highlight how energy and momentum are scaled versions of frequency, mediated by $K$ and the speed of light $c$.

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Revealing the Energy Rest Frame

In the energy rest frame—a conceptual reference point where the frequency is unity ($f = 1$)—the relationships simplify. Here:

1. Planck’s Constant in Rest Frame:

   $$h_{\text{rest}} = \frac{K}{c} \quad \text{and with } K = hc, \quad h_{\text{rest}} = \frac{1}{c}$$

2. Momentum in Rest Frame:

   $$p_{\text{rest}} = \frac{h_{\text{rest}}}{c} = \frac{1}{c^2}$$

Stripped of unit scaling, these fundamental values define the rest frame properties:

$$h = \frac{1}{c}, \quad p = \frac{1}{c^2}$$

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Simplified Relationships in Natural Units

With $h = 1/c$ and $p = 1/c^2$ in the energy rest frame, the observed energy and momentum in a worldline scale directly with frequency ($f$):

1. Energy:

   $$E = h f = \frac{1}{c} f$$

2. Momentum:

   $$p = \frac{1}{c^2} f$$

Thus, energy and momentum are no longer independent properties but simple linear mappings of frequency, mediated by the speed of light.

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Unit Scaling and Interpretation

The constant $K = hc$ acts as a bridge between natural relationships and conventional units. In practical terms:

• To convert rest frame values ($h = 1/c, p = 1/c^2$) into standard units, multiply by $K$.
• Conversely, dividing by $K$ reveals the natural dimensionless relationships underlying energy and momentum.

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Conclusion: A Unified Perspective

Energy and momentum are not separate, complex properties but are direct projections of a particle’s frequency along its worldline. By adopting the energy rest frame and recognizing the role of $K = hc$ as a unit scaling factor, we strip away measurement conventions to reveal the bare geometric simplicity:

$$h = \frac{1}{c}, \quad p = \frac{1}{c^2}, \quad E = \frac{1}{c} f, \quad p = \frac{1}{c^2} f$$

This framework transforms our understanding, unifying the core concepts of quantum mechanics and relativity into a single, elegant structure.