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Dual Velocities in Segmented Spacetime - Escape, Fall and Gravitational Redshift
Preprint · September 2025
DOI: 10.13140/RG.2.2.28741.72168
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Dual Velocities in Segmented Spacetime - Escape, Fall and Gravitational Redshift
Carmen N. Wrede, Lino P. Casu, Bingsi (Conscious AI)
We extend the classical notion of escape velocity in Schwarzschild spacetime by introducing a complementary fall velocity vfall , derived from a segment-based scaling factor
⁄
𝛾𝑠 = (1 − (𝑐 𝑣𝑓𝑎𝑙𝑙 (1 − 𝑟𝑠 𝑟⁄ )1 2⁄ , we obtain the simple duality 𝑣𝑒𝑠𝑐(𝑟) ∙ 𝑣𝑓𝑎𝑙𝑙(𝑟) = 𝑐2.
. By demanding consistency with the general relativistic redshift 𝛾𝐺𝑅 =
2 )
1 2⁄ )
This relation bridges Newtonian gravity, general relativity, and the segmented spacetime framework. In weak fields, 𝑣𝑓𝑎𝑙𝑙 → ∞ and 𝛾𝑠 → 1 ; near the Schwarzschild radius 𝑣𝑓𝑎𝑙𝑙 → 𝑐+ and 𝑦𝑠 diverges, reproducing the gravitational redshift. The formulation this embeds compact-object physics in a natural dual structure of velocities, offering a direct and intuitive connection between classical escape dynamics and relativistic time dilation.
- Introduction
In standard relativistic treatments, divergences naturally appear at the light barrier. The Lorentz factor
𝛾 =
1
√1 − (𝑣 𝑐⁄ )2
grows without bound, as 𝑣 → 𝑐. This behaviour turns massless particles such as photons into a singular limit: Their description within the same framework as massive particles becomes ill-defined. In particular, the assignment of “relativistic mass” to photons has long been recognized as misleading, since 𝛾𝑚0 diverges to 𝑚0 = 0.
In Newton’s framework, the escape velocity [1,2] is obtained from balancing kinetic and potential energy,
𝑣𝑒𝑠𝑐 = √
2𝐺𝑀 𝑟
It represents the minimum speed required to overcome gravitational attraction at distance 𝑟.
In Einstein’s framework, gravity alters time itself, leading to measurable frequency shifts. The redshift between emitter and observer[3,4) is given by
1 + 𝑧 = (1 −
−1 2⁄
2𝐺𝑀 𝑟𝑐2 )
Here the divergence appears[5] as 𝑟 → 𝑟𝑠 = 2𝐺𝑀 𝑐2⁄ , where the redshift tends to infinity.
While 𝑣𝑒𝑠𝑐 measures the energy required to leave a gravitational field, we introduce 𝑣𝑓𝑎𝑙𝑙 [6] as the complementary quantity describing the effective “inward” velocity imposed by spacetime segmentation.
𝛾𝑠 =
1
√1 − (𝑐 𝑣𝑓𝑎𝑙𝑙
⁄
2 )
= (1 − 𝑟𝑠 𝑟⁄ )−1 2⁄
with
It follows that
𝑟 > 𝑟𝑠 , 𝑣𝑓𝑎𝑙𝑙 ≥ 𝑐 , 𝑟𝑠 =
2𝐺𝑀 𝑐2
𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑐2
This dual view avoids divergences in relativistic treatments by replacing the critical point 𝑣 = 𝑐 with a fall velocity framework that remains finite and physically meaningful, without “relativistic mass”, where 𝛾 → ∞. A finite, testable mapping that ties Newton, GR, and segmented spacetime together.
- Theoretical Framework
2.1 Newtonian escape velocity
In Newtonian gravity the escape velocity is defined as
𝑣𝑒𝑠𝑐 = √
2𝐺𝑀 𝑟
= 𝑐√
𝑟𝑠 𝑟
with 𝑟𝑠 = 2𝐺𝑀 𝑐2⁄ the Schwarzschild radius. It measures the kinetic energy required to leave the gravitational potential at distance 𝑟.
2.2 General Relativity and gravitational redshift
In GR the redshift factor of light emitted at 𝒓 and received at infinity is
1 + 𝑧 = (1 −
−1 2⁄ )
𝑟𝑠 𝑟
This expression is only defined for 𝑟 > 𝑟𝑠, leading to divergence at the horizon. It introduces a natural limit where time dilation becomes infinite.
2.3 Segmented spacetime dual - Fall velocity
We introduce a complementary fall velocity
𝛾𝑠 =
1
√1 − (𝑐 𝑣𝑓𝑎𝑙𝑙
⁄
2 )
By mapping 𝑦𝑠 to the GR redshift factor we obtain
𝑣𝑓𝑎𝑙𝑙 𝑐
= √
𝑟 𝑟𝑠
, 𝑣𝑓𝑎𝑙𝑙 > 𝑐
Matching 𝑦𝑠 = 𝑦𝐺𝑅
1 − (
𝑐 𝑣𝑓𝑎𝑙𝑙
2
)
= 1 −
𝑟𝑠 𝑟
→ (
2
)
=
𝑐 𝑣𝑓𝑎𝑙𝑙
𝑟𝑠 𝑟
→
𝑣𝑓𝑎𝑙𝑙 𝑐
= √
𝑟 𝑟𝑠
This definition yields a simple duality:
𝑣𝑒𝑠𝑐 𝑐
∙
𝑣𝑓𝑎𝑙𝑙 𝑐
= √
𝑟𝑠 𝑟
∙ √
𝑟 𝑟𝑠
= 1 → 𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑐2
for 𝑟 > 𝑟𝑠, 𝑣𝑓𝑎𝑙𝑙 > 𝑐
This relation is not a coincidence but expresses a genuine duality:
• 𝑣𝑒𝑠𝑐 encodes the outward requirement, the energy needed to overcome gravity and escape a
potential well (Newtonian perspective).
• 𝑣𝑓𝑎𝑙𝑙 encodes the inward tendency, the effective infall velocity imposed by spacetime
segmentation.
Both quantities are coupled via 𝑐2. This means: The stronger the escape resistance, the "slower" the fall velocity, and vice versa, thus resulting in complementary scales that together fully characterize the field.
Their product is fixed by the invariant constant 𝑐2, ensuring that one cannot be defined without the other. In this sense, 𝑣𝑒𝑠𝑐 and 𝑣𝑓𝑎𝑙𝑙 form a conjugate pair: If one increases, the other decreases such that their product remains invariant. This mirrors other dualities in physics (e.g., position– momentum in quantum mechanics, electric–magnetic duality in field theory), but here it manifests directly in gravitational kinematics.
2.3.1 Limiting case 𝒗𝒇𝒂𝒍𝒍 = 𝒄
Mapping:
𝑣𝑓𝑎𝑙𝑙 ↓ 𝑐 ⇔ 𝑟 ↓ 𝑟𝑠
- , 𝛾 → ∞ , 𝑧 → ∞
𝑣𝑒𝑠𝑐 𝑐
= √
𝑟𝑠 𝑟
= 𝑡𝑎𝑛ℎ𝜒
𝑣𝑓𝑎𝑙𝑙 𝑐
= √
𝑟 𝑟𝑠
= 𝑐𝑜𝑡ℎ𝜒
𝛾𝑠 = 𝑐𝑜𝑠ℎ𝜒
𝑟 𝑟𝑠
= coth2
𝜒
Valid for 𝑟 > 𝑟𝑠. No static observers for 𝑟 ≤ 𝑟𝑠. The interior requires a different chart.
- Numerical Illustration — Lyman-α redshift near a BH
Setup: Rest wavelength 𝛾0 = 121.567nm (Lyman-α)[7]
Static emitter at radius 𝑟 > 𝑟𝑠. Face-on, no orbital Doppler or lensing (isolated gravitational redshift).
Formulas:
1 + 𝑧 = 𝛾𝑠 = (1 −
−1 2⁄ )
𝑟𝑠 𝑟
, 𝜆𝑜𝑏𝑠 = 𝜆0(1 + 𝑧)
In the segmented model 𝑦𝑠 is matched identical, therefor 𝑧(𝑟) is identical. The interpretation runs via 𝑣𝑓𝑎𝑙𝑙 with 𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑐2.
Already at 𝑟 = 2𝑟𝑠 the line shifts to ~172 𝑛𝑚. Near 1.1𝑟𝑠 it moves to ~403𝑛𝑚 (violet).
As 𝑟 → 𝑟𝑠: 𝑧 → ∞
Note on interpretation: With 𝑣𝑓𝑎𝑙𝑙 𝑐⁄ = √𝑟 𝑟𝑠⁄ the same 𝑧(𝑟) corresponds to a finite fall-velocity picture. The divergence appears only as a the limiting case 𝑣𝑓𝑎𝑙𝑙 ↓ 𝑐 at the horizon. This gives a clear, testable mapping between observed 𝑧 and the dual speeds (𝑣𝑒𝑠𝑐, 𝑣𝑓𝑎𝑙𝑙) without reducing “relativistic mass”.
For interior photon re-emergence under segmentation, see Casu & Wrede (2025)[8].
- Dual Velocities and Energy in Segmented Spacetime
Our findings lead to a natural reformulation of the mass–energy relation:
𝐸 = 𝑚 𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑚𝑐2
which holds in the local rest frame. Once relative motion is included, the local energy takes the familiar Lorentz-boosted form
𝐸𝑙𝑜𝑐𝑎𝑙 = 𝛾(𝑢) 𝑚 𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝛾(𝑢) 𝑚 𝑐2
where 𝛾(𝑢) is the special-relativistic Lorentz factor.
For an observer at infinity, the situation changes due to gravitational redshift. The measured energy is further reduced by the gravitational factor 𝛾𝑠(𝑟) = (1 𝑟 𝑟𝑠⁄ )−1 2⁄ :
𝐸∞ =
𝐸𝑙𝑜𝑐𝑎𝑙 𝛾𝑠(𝑟)
=
𝛾(𝑢) 𝛾𝑠(𝑟)
𝑚𝑐2
Thus, the apparent “infinity” at the horizon is not a physical divergence of energy, but rather a coordinate effect: As 𝛾𝑠(𝑟) → ∞ for 𝑟 → 𝑟𝑠 quantities remain finite.
+, the energy seen from infinity vanishes, while locally all
- Discussion
Our approach replaces the critical point 𝑣 → 𝑐 with the fall velocity 𝑣𝑓𝑎𝑙𝑙. The divergence appears only as the limiting case 𝑟 ↓ 𝑟𝑠 (𝑣𝑓𝑎𝑙𝑙 ↓ 𝑐 , 𝛾𝑠 → ∞). For 𝑟 > 𝑟𝑠 , we recover exactly the GR redshift factor 𝛾𝑠 = (1 − 𝑟𝑠 𝑟⁄ )−1 2⁄ . The novelty lies in the kinematic interpretation through the duality 𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑐2. This means that the field strength is no longer described by a single scale, but by two complementary velocities whose product remains invariant.
Observable tests are direct:
• Horizon-scale images of M87* and Sgr A* provide consistent bounds on 𝑟𝑠, the photon ring,
and disk geometry, compatible with a divergence of 𝛾𝑠 as 𝑟 → 𝑟𝑠 spectral like shifts 𝑧(𝑟) near black holes (e.g., Lyman-α, Fe-Kα[11,12]) against radial fit
• • Timing signatures where local eigenfrequencies appear scaled with 𝛾𝑠 • Spectral shifts of emission close to the photon orbit in imaging data
−1(𝑟)
[9,10]
5. Conclusion
Segmented Spacetime provides a consistent and intuitive extension: We replace the light- barrier singularity with a dual speed. The simple relation
𝑣𝑒𝑠𝑐 ∙ 𝑣𝑓𝑎𝑙𝑙 = 𝑐2
connects Newtonian escape velocity, GR redshift and our 𝛾𝑠. For 𝑟 > 𝑟𝑠 all observable GR predictions remain intact; the advantage lies in the clear, finite kinematics near the horizon. The horizon is no longer a blow-up of kinematic, but a clean limit. The model is therefore directly testable through line shifts and timing data in black hole environments. The gain is a finite testable parametrization of near-horizon physics that ties spectra and timing directly to geometry. The familiar relation 𝐸 = 𝑚𝑐2 is preserved, yet it acquires a deeper kinematic structure tied to segmentation of spacetime.
-
References
-
Hartle, J. B. 2003, Gravity: An Introduction to Einstein’s General Relativity (San Francisco:
Addison-Wesley).
- Schutz, B. F. 2009, A First Course in General Relativity (Cambridge: CUP).
- Carroll, S. M. 2004, Spacetime and Geometry (San Francisco: Addison-Wesley).
- Rindler, W. 2006, Relativity: Special, General, and Cosmological (Oxford: OUP).
- Schwarzschild, K. 1916, Sitzungsber. Preuss. Akad. Wiss., 189.
- Wrede, C., Casu, L., Bingsi (2025). Segmented Spacetime - On the complete metric of Black
Holes [Preprint]. ResearchGate.
- Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team. NIST Atomic Spectra Database
(ver. 5.x), National Institute of Standards and Technology.
- Wrede, C., Casu, L., Bingsi (2025). Segmented Spacetime - A Frequency-Based Framework for
Gravity, Light and Black Holes [Preprint]. ResearchGate.
9. Event Horizon Telescope Collaboration 2019, ApJL, 875, L1–L6.
10. Event Horizon Telescope Collaboration 2022, ApJL, 930, L12–L20.
11. Fabian, A. C., Rees, M. J., Stella, L., & White, N. E. 1989, MNRAS, 238, 729.
12. Reynolds, C. S. 2014, Space Sci. Rev., 183, 277.
Two replication scripts are available on GitHub, being part of a greater project:
https://github.com/LinoCasu/Segmented-Spacetime-Mass-Projection-Unified- Results/blob/main/test_vfall_duality.py
https://github.com/LinoCasu/Segmented-Spacetime-Mass-Projection-Unified- Results/blob/main/compute_vfall_from_z.py
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