The Pythagorean Curvature Correction Theorem - a modified Pythagorean Thereom

[mikelewis]

4. You're welcome... here's a tougher one..

(Estimating the correct terms for flat space is harder than it looks. This equation expresses both the hyperbolic and spherical aspects of a geodesic so figuring out the correct spacing is tough. This example as just an example of the math not the true distances. So keep in mind, this is a conceptual illustration; in reality, interplanetary distances involve many nuances, but our example gives a flavor of how curvature can tweak distance measurements.)

So let's walk through another example that estimates the geodesic distance from Earth to Saturn using our Pythagorean Curvature Correction Theorem.


The Scenario

Imagine we model the Earth-to-Saturn distance as the hypotenuse c of a right triangle, where the two legs a and b are orthogonal components of the displacement between Earth and Saturn. For our example, we’ll assume:

- a = 8.0 × 10⁸ km
- b = 6.0 × 10⁸ km

These values are chosen to roughly represent cosmic distances on the order of a billion kilometers.

Next, we need an effective curvature radius R for the region of space we’re considering. Although cosmic space is nearly flat, let’s assume (for demonstration purposes) that we have a curvature scale of:

- R = 2.0 × 10⁹ km

Finally, we choose the chirality parameter h. For a spherical (positively curved) geometry, we set h = -1.

Our Formula

a² + b² + h (a²b² / R²) = c²

Step-by-Step Calculation

1. Compute the Classical (Euclidean) Sum

First, calculate a² + b²:

a² = (8.0 × 10⁸ km )² = 6.4 × 10¹⁷ km²,
b² = (6.0 × 10⁸ km )² = 3.6 × 10¹⁷ km²,
a² + b² = 6.4 × 10¹⁷ + 3.6 × 10¹⁷ = 1.0 × 10¹⁸ km².

2. Compute the Correction Term

Next, calculate the correction term h (a²b² / R²):

- Calculate a²b²:
a²b² = (6.4 × 10¹⁷ ) × ( 3.6 × 10¹⁷ ) = 2.304 × 10³⁵ km⁴.

- Calculate R²:
R² = (2.0 × 10⁹ km )² = 4.0 × 10¹⁸ km².

- Form the ratio:
(a²b² / R² ) = (2.304 × 10³⁵ ) / (4.0 × 10¹⁸ ) = 5.76 × 10¹⁶ km².

- Apply the chirality h:
Since h = -1 (for spherical curvature), the correction term becomes:
-1 × 5.76 × 10¹⁶ km² = -5.76 × 10¹⁶ km².

3. Combine to Find c²

Now, add the Euclidean term and the correction term:

c² = 1.0 × 10¹⁸ km² - 5.76 × 10¹⁶ km² = 9.424 × 10¹⁷ km².

4. Solve for c

Take the square root:

c = √ ( 9.424 × 10¹⁷ ) = 9.71 × 10⁸ km.

Interpretation

- Classical (Flat) Calculation:
Without any curvature correction, the Euclidean distance would be:
c = √ ( 1.0 × 10¹⁸ ) = 1.0 × 10⁹ km.

- Curvature-Corrected Distance:
Our tool gives a slightly shorter distance, c = 9.71 × 10⁸ km, which is about 971 million km.

The correction reduces the distance by roughly 2.9% ( about 29 million km in this example). In the vast scales of the cosmos, even such “small” adjustments can have significant implications for understanding light propagation, gravitational lensing, and cosmic structures.

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Wrapping Up

This mocked up Earth-to-Saturn example illustrates how our Pythagorean Curvature Correction Theorem accounts for curvature by introducing a correction term. Even if the numbers we used are somewhat idealized, the process remains the same:

1. Measure your sides a and b.
2. Determine the effective curvature radius R of the region.
3. Choose the appropriate h based on the type of curvature.
4. Plug these values into
a² + b² + h ( a²b² / R² ) = c²,
and solve for c.

This tool not only refines static distance measurements but also plays a vital role in understanding how waves (like light or radio signals) traverse a curved space. And again... this was just me playing with numbers, there was no real science here, just math.