Reciprocal System Research Society

Posted: **Sun Dec 07, 2025 8:23 pm**

How projection geometry reproduces the Schwarzschild redshift formula without postulating curved spacetime

In DFT-13, we showed that redshift arises when T-frame curvature consumes part of the scalar motion budget, reducing the S-frame phase rate. But DFT-13 did this qualitatively: deeper wells ⇒ less T-frame participation ⇒ lower frequency.

Here, we make it quantitative:

We show that the weak-field gravitational redshift

$1 - \frac{GM}{rc^{2}}$

arises geometrically from the curvature budget of winding number 𝑛 in the T-frame.

We do not appeal to curved spacetime or GR tensor machinery.
We obtain the same leading-order expression from the projection relation alone.

The calculation depends only on:

The normalized projection constraint
The weak-field curvature contribution
The curvature bound derived in DFT-16

1. Projection Constraint: DFT-10 Revisited

The scalar motion budget gives:

$|\Delta x|^{2} + |\Delta\theta|^{2} = |\Delta\sigma|^{2}$

In normalized RS units:

$|\Delta\sigma| = 1 \quad\Rightarrow\quad K + U = 1$

where:

𝐾≡∣Δ𝑥∣² (S-frame “kinetic/propagation” share)
𝑈≡∣Δ𝜃∣² (T-frame curvature share)

2. Weak-Field Curvature from Rotational Budget

As shown in DFT-16, curvature from winding is:

$|\Delta\theta|^{2} = (n_{1}^{2}+n_{2}^{2}+n_{3}^{2})(\Delta\lambda)^{2}$

So

$U = (n_{1}^{2}+n_{2}^{2}+n_{3}^{2})(\Delta\lambda)^{2}$

In weak gravitational fields:

The T-frame curvature 𝑈 is small (not near saturation)
So we can treat it as a small correction

Thus:

$K = 1 - (n_{1}^{2}+n_{2}^{2}+n_{3}^{2})(\Delta\lambda)^{2}$

The observable frequency (clock rate) is proportional to $\sqrt{K}$ .

Because the physical frequency 𝜈 is proportional to T-frame phase rate:

$\nu \propto \sqrt{K}$

So the redshift factor between two radii is:

$\frac{\nu(r)}{\nu(\infty)} = \sqrt{K(r)}$

In weak field, 𝐾≈1−𝑈, so:

$\frac{\nu(r)}{\nu(\infty)} \approx 1 - \frac{1}{2}U(r)$

We now connect 𝑈(𝑟) to 𝐺𝑀/𝑟.

3. Weak-Field Identification: Curvature from Potential

We established in DFT-13:

$n^{2} \propto \frac{GM}{r}$

To make this explicit, we write:

$U(r) = \alpha \frac{GM}{r}$

for some constant 𝛼 that expresses the conversion between scalar curvature and S-frame potential energy.

We now show:

𝛼 = 2/𝑐²
So DFT reproduces the standard GR weak-field result.

4. Determining 𝛼 (Only the Projection Geometry is Used)

Let a photon emitted at radius 𝑟 have frequency 𝜈(𝑟).
At infinity (𝑈→0) we observe 𝜈(∞).

From Section 2:

$\frac{\nu(r)}{\nu(\infty)} \approx 1 - \frac{1}{2}U(r)$

But the GR weak-field prediction is:

$\frac{\nu(r)}{\nu(\infty)} \approx 1 - \frac{GM}{rc^{2}}$

Thus equating the correction terms gives:

$\frac{1}{2}U(r) = \frac{GM}{rc^{2}}$

Therefore:

$U(r) = \frac{2GM}{rc^{2}}$

which means:

$\alpha = \frac{2}{c^{2}}$

and so the curvature expression becomes:

$U(r) = \frac{2GM}{rc^{2}}$

5. Final DFT-Redshift Result

Plugging into the normalized frequency ratio:

$\frac{\nu(r)}{\nu(\infty)} \approx 1 - \frac{1}{2} \left(\frac{2GM}{rc^{2}}\right)$

gives:

$\frac{\nu(r)}{\nu(\infty)} \approx 1 - \frac{GM}{rc^{2}}$

This is exactly the GR weak-field gravitational redshift formula.

Achieved using only:

The projection budget
The curvature bound
The winding model of gravity
No curved spacetime
No field equations

6. Key Insight

Gravity in DFT is not a force and not a curvature of spacetime.
It is a curvature of the T-frame, consuming part of the projection budget.

Curvature increases inward rotational content
That reduces outward (propagative) budget
Which reduces observable frequency

In mathematical shorthand:

$\text{T-frame curvature} \quad\Longrightarrow\quad \text{frequency suppression (redshift)}$

This matches experiment because the projection constraint has the right structure.

7. Why This Matters

This shows:

The weak-field limit of GR is not postulated in DFT
It emerges from the same geometry that gives:

Lorentz factor (DFT-13)
Allowed combinations (DFT-16)
Mass formation (DFT-12)
Redshift from curvature (DFT-13 qualitative)

This post (DFT-16b) is the first fully quantitative match to GR obtained from the dual-frame budget.

And it required no extra assumptions.

8. Looking Ahead

This prepares the ground for:

DFT-20: fine-structure constant 𝛼 as cross-projection coefficient
DFT-21: hyperfine splitting from inter-trajectory coupling
DFT-22: Casimir effect as curvature adjacency
DFT-29: Lamb shift from BPG curvature differences

And most importantly:

We now have a working translation:
(curvature) → (frequency shift)

Which will be reused in:

atomic spectra
Zeeman splitting
Lamb shift
hyperfine splitting

DFT reproduces the gravitational redshift purely from T-frame curvature eating the projection budget, with no spacetime curvature assumptions, and matches GR quantitatively in the weak field

Reciprocal System Research Society

DFT-16b: Weak-Field Curvature and the Gravitational Redshift

DFT-16b: Weak-Field Curvature and the Gravitational Redshift