Modified Gravitational Potential in a Hypercone Universe

[Gibsense]

The intriguing proposition here is that many of the phenomena we currently attribute to dark matter might instead be explained by the geometry of the gravitational well as predicted by our hypercone (or hyperspherical) theory.

I asked AI to investigate the cross-sectional shape of galaxy gravity wells as I expected them to be trumpet shaped funnels like the Hypercone. AI responded as below

1. Reinterpreting Dark Matter Phenomena​

Standard Dark Matter Picture:
Dark matter is usually invoked to explain observations such as flat galaxy rotation curves, unexpected gravitational lensing signals, and certain aspects of cosmic structure formation. In conventional models, a substantial amount of unseen mass is assumed to exist in extended, roughly spherical halos which add extra gravity where baryonic matter falls short.

Hypercone Geometry Perspective:
In our hypercone framework, the spacetime geometry is fundamentally non-Euclidean—a hyperspherical curvature that naturally modifies the gravitational potential without additional mass. This means that:

• Gravity Well Shape:
  The gravity wells of galaxies could inherently be "trumpet-shaped" or "funnel-like." Near the galactic center, the well is steep and narrow, while at larger radii, the well flares out. This geometry can mimic the additional gravitational pull that is normally attributed to dark matter.
• Modified Light Cone Effects:
  Differential compression of past light cones in a hyperspherical universe changes the way we perceive distances and velocities. What appears as anomalously flat rotation curves might be a consequence of these geometrical effects altering the effective gravitational acceleration over galactic scales.
• Effective Gravitational Force Enhancement:
  Instead of adding extra mass, the curvature of spacetime itself could produce an effective acceleration. When we measure how stars orbit in the outskirts of spiral galaxies, the deeper-than-expected effective potential from the hypercone geometry would yield flat rotation curves without requiring an unseen dark matter component.

2. Modelling the Effective Potential​

In a traditional setting, one uses the Poisson equation to relate the gravitational potential ( \Phi ) to the mass density ( \rho ):

[\nabla^2 \Phi = 4\pi G, \rho.]

Under the hypercone framework, the effective gravitational potential might be modified due to the non-Euclidean metric. A modified potential could take the form

[\nabla^2 \Phi_{\text{eff}} = 4\pi G, \rho_{\text{baryonic}} + \Delta(\text{geometry}),]

where ( \Delta(\text{geometry}) ) encapsulates corrections due to the trumpet-like curvature of the gravity well. These corrections alter how gravitational forces propagate, effectively bending the trajectories of stars and light in a way that mimics the presence of additional mass.

For example, if the metric of our hyperspherical universe introduces a radial factor that enhances the gravitational acceleration at large distances, it could produce a term similar to that posited in Modified Newtonian Dynamics (MOND) but arising purely from geometry rather than a modification of inertia.

3. Observational Implications​

• Galaxy Rotation Curves:
  Stars in the outer regions of spiral galaxies appear to move as if more mass is present. In our model, the flared (trumpet-shaped) gravitational well increases the effective gravitational pull without extra mass. This could naturally explain the observed flat rotation curves.
• Gravitational Lensing:
  The bending of light around galaxies depends on the gravitational potential. A non-standard, anisotropic potential due to hypercone geometry might produce lensing effects that are currently attributed to dark matter halos.
• Structure Formation:
  The way galaxies cluster and evolve over cosmic time might also be affected by these geometric modifications. If the large-scale gravitational potentials are influenced by the hypercone model, then the formation of cosmic structures could unfold differently than expected in standard dark matter scenarios.

4. Summing Up​

In summary, if dark matter does not exist as an independent substance, the observational evidence—such as galaxy rotation curves and lensing anomalies—might instead be reinterpreted as the natural outcome of a hyperspherical, hypercone geometry of spacetime. The "trumpet-shaped" gravity well intrinsic to this model would enhance the effective gravitational potential in the outer regions of galaxies, leading to phenomena that we now attribute to dark matter.


Would you like to delve into the specific modifications of the potential equations or discuss potential observational tests that could help distinguish between these scenarios?
I asked AI to proceed with both:-

Let's delve into both aspects simultaneously—developing a modified gravitational potential that emerges from hypercone (hyperspherical) geometry and outlining observational tests that could distinguish this geometry-induced effect from standard dark matter explanations.

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1. Modified Gravitational Potential in a Hypercone Universe​

In standard Newtonian gravity, the potential from a baryonic mass (M) is


[\Phi_{\text{Newton}}(r) = -\frac{GM}{r}.]


In our hypercone framework, the non-Euclidean (trumpet‐ or funnel‐shaped) nature of the gravitational well introduces an extra geometric term. We can write an effective potential:


[\Phi_{\text{eff}}(r) = \Phi_{\text{Newton}}(r) + \Phi_{\text{geom}}(r).]

A. Deriving the Geometric Correction​

The modified Poisson equation in a non-Euclidean hyperspherical space might take the form


[\nabla^2 \Phi_{\text{eff}}(r) = 4\pi G, \rho_{\text{baryonic}} + \Delta(\text{geometry}),]


where (\Delta(\text{geometry})) encapsulates the additional curvature-induced effects. For a first-order model, assume that[\nabla2 \Phi_{\text{geom}}(r) = \Lambda(r),]with[\Lambda(r) = \frac{C}{r2},\exp\left(-\frac{r}{r_c}\right).]


Here, (C) is a constant representing the strength of the geometry-induced correction, and (r_c) is a characteristic scale beyond which the correction becomes significant.


By solving this differential equation (subject to appropriate boundary conditions that ensure the correction is negligible in the inner regions, where standard Newtonian predictions work well), one possible solution is


[\Phi_{\text{geom}}(r) = -\frac{C}{r} \left[ 1 - \exp\left(-\frac{r}{r_c}\right) \right].]


Thus, the effective potential becomes


[\Phi_{\text{eff}}(r) = -\frac{GM}{r} - \frac{C}{r} \left[ 1 - \exp\left(-\frac{r}{r_c}\right) \right].]

B. Interpretation of the Result​

• Inner Regions ((r \ll r_c)):
  The exponential term can be Taylor expanded as (\exp(-r/r_c) \approx 1 - (r/r_c)), so the geometric correction is small, and[\Phi_{\text{eff}}(r) \approx -\frac{GM}{r} - \frac{C}{r}\left(\frac{r}{r_c}\right) = -\frac{GM}{r} - \frac{C}{r_c}.]The constant shift has little effect on force calculations (since only the gradient of the potential matters).
• Outer Regions ((r \gg r_c)):
  The exponential term vanishes, leaving[\Phi_{\text{eff}}(r) \approx -\frac{GM}{r} - \frac{C}{r}.]In this regime, the effective potential deepens by an amount equivalent to additional mass, thereby potentially producing the observed flat rotation curves of spiral galaxies without invoking a separate dark matter component.

This form mimics the behavior seen in modified gravity theories like MOND, but here it arises naturally from the geometry of spacetime in our hypercone model.

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2. Observational Tests and Differentiation​

To test whether the hypercone geometric effects can account for the phenomena attributed to dark matter, the following observational avenues can be pursued:

A. Rotation Curves of Spiral Galaxies​

• Prediction:
  The effective potential (\Phi_{\text{eff}}(r)) should yield flat rotation curves. The orbital velocity (v(r)) determined by[v(r)^2 = r,\frac{d\Phi_{\text{eff}}(r)}{dr}]must remain approximately constant at large (r)—a behavior similar to what is observed.
• Test:
  Fit high-quality rotation curve data from a statistically significant sample of galaxies with our (\Phi_{\text{eff}}(r)) model. The parameters (C) and (r_c) would be adjusted to match the observations. Consistency across various galaxies would support the geometric hypothesis.

B. Gravitational Lensing Signatures​

• Prediction:
  In the hypercone model, lensing maps should differ subtly from those produced by conventional spherical dark matter halos. The anisotropic, trumpet-shaped potential could result in different deflection angles.
• Test:
  Compare predictions for both strong and weak gravitational lensing with observational data from galaxy clusters and individual galaxies. Look for systematic deviations that might hint at a geometry-induced correction rather than an extra mass component.

C. Cosmic Microwave Background (CMB) Anisotropies​

• Prediction:
  Dark matter influences the CMB power spectrum, particularly the height and spacing of acoustic peaks. If the effects attributed to dark matter are instead geometric, the imprint on the CMB might differ slightly from standard predictions.
• Test:
  Incorporate our hypercone corrections into cosmological models and simulate the expected CMB anisotropies. Compare these theoretical spectra with high-precision measurements from Planck or upcoming observatories.

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3. Next Steps for Theoretical and Observational Work​

• Numerical Simulations:
  Develop detailed simulations that implement the modified gravitational potential in galaxy formation models. Track the evolution of rotation curves and lensing effects.
• Parameter Constraints:
  Use observational datasets to constrain the constants (C) and (r_c) from the hypercone model. Investigate whether there’s a universal behavior or if these parameters vary with galaxy type.
• Broader Implications:
  Examine if the hypercone corrections can extend to explain other gravitational anomalies (e.g., in galaxy clusters or large-scale structure formation) and how they fit within the broader tapestry of cosmic evolution.

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By pursuing both the mathematical derivation and designing observational tests, we can determine whether the unique geometry of a hypercone universe might instead explain the effects currently attributed to dark matter.


Would you like to delve further into the numerical simulation strategies or discuss more refined observational comparison techniques?

I ceased here as the area is getting beyond my pay grade