1 hour ago, Anton Rize said:

For a massive body (partitioned budget):

[math]\Delta\varphi_m = 2 \arcsin\left(\dfrac{\kappa_p^2}{2\beta_p^2 - \kappa_p^2}\right)[/math]

For light (unpartitioned budget, [math]\beta=1[/math]):

[math]\Delta\varphi_\gamma = 2 \arcsin\left(\dfrac{\kappa_p^2}{1 - \kappa_p^2}\right)[/math]

Thank you very much.

I can now reveal where you have made your error. I don't know what the actual formula is for the non-zero mass object, but I can see that the above two formulae do not agree for [math]\beta = 1[/math]. For a non-zero mass object travelling at 0.999999999c, the deflection should be the same as for light. That is, your error was to assume that the factor of 2 arose because light is massless. The trajectory of an object in a gravitational field does not depend on the mass of a test mass (a mass that is sufficiently small as to not affect the surrounding spacetime) but depends on the speed of the object. Thus, the trajectory of a non-zero mass object travelling at 0.999999999c will be negligibly different to the trajectory of light, and therefore have deflections that are the same (negligibly different).

As I see it, the ratio of the actual deflection angle to that predicted by the equivalence principle will itself depend on [math]\beta[/math]. For [math]\beta \approx 0[/math], the spacetime trajectory is mostly governed by time, and therefore the deflection will correspond to that predicted by the equivalence principle. But for [math]\beta \approx 1[/math], the spacetime trajectory is governed more-or-less equally by time and space, and the contribution by space equals the contribution by time, thus doubling the deflection corresponding to that predicted by the equivalence principle.

Edited March 6Mar 6 by KJW

3 hours ago, KJW said:

I don't know what the actual formula is for the non-zero mass object

It is:

\[\varphi =\int_{r_{\min}}^{\infty}\frac{dr}{r^2 \sqrt{\dfrac{E^2}{L^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1}{L^2} + \dfrac{1}{r^2} \right)}}\]

For non-relativistic speeds and weak fields, this reduces to the Newtonian scattering formula. For v=c and massless test particles, you get the Schwarzschild light deflection formula. For strong fields and massive particles, the integral can be evaluated numerically.

Edited March 6Mar 6 by Markus Hanke

1 hour ago, Markus Hanke said:

  5 hours ago, KJW said:

I don't know what the actual formula is for the non-zero mass object

It is:

\[\varphi =\int_{r_{\min}}^{\infty}\frac{dr}{r^2 \sqrt{\dfrac{E^2}{L^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1}{L^2} + \dfrac{1}{r^2} \right)}}\]

For non-relativistic speeds and weak fields, this reduces to the Newtonian scattering formula. For v=c and massless test particles, you get the Schwarzschild light deflection formula. For strong fields and massive particles, the integral can be evaluated numerically.

Thanks. +1

39 minutes ago, KJW said:

Thanks. +1

My pleasure.

As a little exercise, I’ve reworked the integral to something more explicit (I personally never really liked the notation with E and L), and if I’m not mistaken this is what we get:

\[\Delta \varphi = 2 \int_{r_{\min}}^{\infty} \frac{dr}{r^2 \sqrt{\dfrac{1}{b^2 v_\infty^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1 - v_\infty^2/c^2}{b^2 v_\infty^2} + \dfrac{1}{r^2} \right)}} - \pi\]

So the deflection angle depends only on initial speed far away, impact parameter, and mass of the central object - as one would expect.

1 hour ago, KJW said:

That's because I see value in it. Nevertheless, I'm still scrutinising you and your work. And the nature of your work has made it very difficult to scrutinise in the way I feel is necessary. I'm not sure that what value I see is what you intend me to see as I'm not in full agreement with your philosophy, although some of it does align with relativity. What I would like to see is some form of mathematical proof that your theory fully agrees with general relativity. What you have provided so far is not such a proof, and I'm not sure what such a proof would look like.

Thanks. This looks more detailed than what I saw earlier. I want to look through it to see if you have addressed the problem I see in your explanation. Bear in mind that I know why the defection of light under general relativity is twice that of Newtonian theory. It's not straightforward and I believe you have chosen the wrong explanation. I don't wish to reveal what I believe to be your mistake because I think it is important for your theory that it be able to derive things without the guidance that exists when deriving preexisting results.

I feel that what I'm asking for may be too difficult for the piece of information I'm looking for from you. While I asked for a complete formula, in fact I want to know how you handle a specific scenario that I don't wish to reveal, but maybe I should. I apologise for that. Anyway, if you can preempt why I am asking you about non-zero mass, perhaps you can address my concern without deriving the formula.

First, thank you for your honesty. Hearing "I see value in it" from someone who is actively and rigorously scrutinizing the math means a great deal to me. I am not looking for easy agreement; ruthless mathematical scrutiny is exactly what RG needs.

1 hour ago, KJW said:

Thank you very much.

I can now reveal where you have made your error. I don't know what the actual formula is for the non-zero mass object, but I can see that the above two formulae do not agree for β=1. For a non-zero mass object travelling at 0.999999999c, the deflection should be the same as for light. That is, your error was to assume that the factor of 2 arose because light is massless. The trajectory of an object in a gravitational field does not depend on the mass of a test mass (a mass that is sufficiently small as to not affect the surrounding spacetime) but depends on the speed of the object. Thus, the trajectory of a non-zero mass object travelling at 0.999999999c will be negligibly different to the trajectory of light, and therefore have deflections that are the same (negligibly different).

As I see it, the ratio of the actual deflection angle to that predicted by the equivalence principle will itself depend on β. For β≈0, the spacetime trajectory is mostly governed by time, and therefore the deflection will correspond to that predicted by the equivalence principle. But for β≈1, the spacetime trajectory is governed more-or-less equally by time and space, and the contribution by space equals the contribution by time, thus doubling the deflection corresponding to that predicted by the equivalence principle.

This is exactly what I have been looking for, and it is the highest form of respect one researcher can show another. Thank you!

You are 100% correct. My assumption that the factor of 2 arose simply because light is "massless" (and hardcoding the 1/2 partitioning factor for massive bodies) was a critical error. As you rightly pointed out, a massive object traveling at 0.999999*c must asymptotically approach the deflection of light, which my previous discrete geometric separation failed to do. Further more without the graduate rebalancing of the weight of itch axis the hole geometric structure goes haywire and energy comes from nothing go nowhere.

So I spend pretty much all day deriving it. It was way harder than I expected but in the end something incredible happened. I don't know how to prove it but guys I swear I did not try to copy GR results. You will see my ontology and derivation logic has nothing to do with GR. And when I was going to send the results here, I saw your message @Markus Hanke and decided to test my result against this GR equation. Any way you'll see the results below. I don't know how did it happened... Here's the new section on gravitational diflection:

---

The Unified Interaction Gradient and Relativistic Deflection

Energy projections distribution among axis of relational carriers has to obey the conservation law (distribution between axis does not create nor destroy energy). There must exist a strict, algebraically closed gradient connecting all states, governed entirely by the kinematic projection [math]\beta[/math].

Theorem (Unified Interaction Gradient):

The gravitational interaction capacity of any entity is determined by its available phase buffer [math]\beta_Y[/math]. The geometric scaling factor [math]\Gamma[/math] that dictates the distribution of the potential projection [math]\kappa^2[/math] onto the spatial trajectory is strictly defined by the arithmetic mean of the saturated carrier [math]S^1[/math]:

[math]\Gamma(\beta) = \frac{1 + \beta^2}{2} = 1 - \frac{\beta_Y^2}{2}[/math]

Proof:

By the [math]S^1[/math] closure invariant ([math]\beta^2 + \beta_Y^2 = 1[/math]), an object at rest ([math]\beta = 0[/math]) possesses maximum internal phase
([math]\beta_Y = 1[/math]). This phase acts as a geometric buffer, absorbing half of the relational gradient, yielding the classical partitioning [math]\Gamma = 1/2[/math].

As the spatial projection [math]\beta[/math] increases, the internal clock slows, and the phase buffer [math]\beta_Y[/math] depletes. At the topological limit [math]\beta \to 1[/math] (light), the internal phase collapses ([math]\beta_Y \to 0[/math]). The buffer is exhausted, forcing the entity to absorb the full, unpartitioned gravitational gradient, yielding [math]\Gamma = 1[/math].

Using this gradient, we define the Unified Closure Defect for any trajectory—from a slow asteroid to a photon—as:

[math]\delta_{unified} = \frac{\kappa_p^2}{\beta_p^2} \Gamma(\beta_p) = \frac{\kappa_p^2 (1 + \beta_p^2)}{2\beta_p^2}[/math]

The geometric shape parameter (eccentricity) of the trajectory is derived directly from this defect:

[math]e_{unified} = \frac{1}{\delta_{unified}} - 1 = \frac{2\beta_p^2}{\kappa_p^2(1+\beta_p^2)} - 1[/math]

Applying the exact algebraic transit equation for a distant observer ([math]\kappa_o \to 0[/math]), we have [math]\cos(o_\infty) = -1/e_{unified}[/math]. Using the same trigonometric extraction as established for light ([math]\sin(\frac{\Delta\varphi}{2}) = \frac{1}{e_{unified}}[/math]), we arrive at the absolute, unified equation for gravitational deflection:

[math]\Delta\varphi_{unified} = 2 \arcsin(\frac{\kappa_p^2 (1 + \beta_p^2)}{2\beta_p^2 - \kappa_p^2 (1 + \beta_p^2)})[/math]

Verification of Topological Limits:

* Newtonian Limit ([math]\beta_p \ll 1[/math]): The phase buffer is full ([math]\Gamma \to 0.5[/math]). The eccentricity is dominated by [math]2\beta_p^2 / \kappa_p^2[/math]. The deflection reduces to the classical Rutherford/Newton scattering: [math]\Delta\varphi \approx \frac{\kappa_p^2}{\beta_p^2}[/math].

* Relativistic Limit ([math]\beta_p \to 0.99[/math]): The phase buffer is nearly depleted ([math]\Gamma \to 0.99[/math]). The trajectory stiffens, and the deflection angle approaches the photonic maximum, smoothly capturing the post-Newtonian factor without Taylor expansions.

* Photonic Limit ([math]\beta_p = 1[/math]): The phase buffer is completely exhausted ([math]\Gamma = 1[/math]). Substituting [math]\beta_p = 1[/math] yields [math]e = \frac{2}{\kappa_p^2(2)} - 1 = \frac{1}{\kappa_p^2} - 1 = \frac{\kappa_{Xp}^2}{\kappa_p^2}[/math]. The equation resolves perfectly into the exact light deflection identity: [math]\Delta\varphi = 2 \arcsin(\frac{\kappa_p^2}{\kappa_{Xp}^2})[/math].

Elimination of the "Massless" Myth:

This formulation proves that light is not a distinct ontological category governed by separate physical laws. There are no "massless particles" in RG. Light is simply the rightward topological limit of matter, where the internal phase [math]\beta_Y[/math] reaches zero. The historical factor of 2 in gravitational lensing is not an anomaly of curved spacetime, but the inevitable geometric consequence of an exhausted phase buffer.

---

Here's my python script comparing the results of my derived arcsin against the scary equation that @Markus Hanke posted:

import math
from scipy.integrate import quad

z_sun = 2.1224e-6
kappa_p_sq = 1 - 1 / ((1 + z_sun)**2)

def gr_deflection(beta_p, kp2):
    def S(u):
        term1 = (1 - kp2) / (beta_p**2)
        term2 = (1 - kp2 * u) * ((1 - beta_p**2) / (beta_p**2) + u**2)
        return term1 - term2

    def integrand(u):
        s_val = S(u)
        if s_val <= 1e-15: return 0.0
        return 1.0 / math.sqrt(s_val)

    phi, _ = quad(integrand, 0, 1)
    return (2 * phi - math.pi) * (180.0 / math.pi) * 3600.0

def will_deflection(beta_p, kp2):
    b2 = beta_p**2
    num = kp2 * (1 + b2)
    den = 2 * b2 - num
    ratio = num / den
    
    if ratio > 1.0 or ratio < -1.0: return float('nan')
    
    return 2 * math.asin(ratio) * (180.0 / math.pi) * 3600.0

betas = [0.01, 0.5, 0.9, 0.99, 0.9999, 1.0]

print("Sun kappa_p^2: {:.6e}".format(kappa_p_sq))
print("-" * 75)
print("{:<10} | {:<20} | {:<20} | {:<15}".format('beta_p', 'GR Integral (arcsec)', 'WILL RG (arcsec)', 'Diff (%)'))
print("-" * 75)

for b in betas:
    gr_val = gr_deflection(b, kappa_p_sq)
    will_val = will_deflection(b, kappa_p_sq)
    
    if math.isnan(will_val):
        print("{:<10.4f} | {:<20.6f} | {:<20} | {:<15}".format(b, gr_val, 'Captured', '-'))
    else:
        diff = abs(gr_val - will_val) / gr_val * 100 if gr_val != 0 else 0
        print("{:<10.4f} | {:<20.6f} | {:<20.6f} | {:<15.4e}".format(b, gr_val, will_val, diff))
print("-" * 75)

Output:

Sun kappa_p^2: 4.244786e-06
---------------------------------------------------------------------------
beta_p     | GR Integral (arcsec) | WILL RG (arcsec)     | Diff (%)       
---------------------------------------------------------------------------
0.0100     | 8946.966119          | 8946.971578          | 6.1021e-05     
0.5000     | 4.377794             | 4.377797             | 5.9810e-05     
0.9000     | 1.956485             | 1.956485             | 1.6577e-05     
0.9900     | 1.768885             | 1.768885             | 2.0381e-05     
0.9999     | 1.751282             | 1.751283             | 3.3894e-05     
1.0000     | 1.751107             | 1.751108             | 1.3785e-05     

Output with z=

z_sun = 0.0901

Sun kappa_p^2: 1.584744e-01
---------------------------------------------------------------------------
beta_p     | GR Integral (arcsec) | WILL RG (arcsec)     | Diff (%)       
---------------------------------------------------------------------------
0.0100     | -648000.000000       | Captured             | -              
0.5000     | 275119.856251        | 295243.875599        | 7.3146e+00     
0.9000     | 88523.699944         | 89458.026859         | 1.0555e+00     
0.9900     | 78536.915426         | 79109.733546         | 7.2936e-01     
0.9999     | 77621.268245         | 78162.819773         | 6.9768e-01     
1.0000     | 77612.175741         | 78153.418327         | 6.9737e-01     
---------------------------------------------------------------------------

I don't know about you guys but Im shocked.

37 minutes ago, Markus Hanke said:

My pleasure.

As a little exercise, I’ve reworked the integral to something more explicit (I personally never really liked the notation with E and L), and if I’m not mistaken this is what we get:

\[\Delta \varphi = 2 \int_{r_{\min}}^{\infty} \frac{dr}{r^2 \sqrt{\dfrac{1}{b^2 v_\infty^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1 - v_\infty^2/c^2}{b^2 v_\infty^2} + \dfrac{1}{r^2} \right)}} - \pi\]

So the deflection angle depends only on initial speed far away, impact parameter, and mass of the central object - as one would expect.

Thanks again. Yes, I wasn't sure what E and L were, so this reworked formula is much better. I also come across this formula:

[math]\Phi_C = \dfrac{2GM}{v_0^2 b} [1 + \dfrac{v_0^2}{c^2} + \dfrac{3\pi}{2} \dfrac{GM}{c^2 b} + \dfrac{3\pi}{8} \dfrac{GM}{c^2 b} \dfrac{v_0^2}{c^2} + 9(\dfrac{GM}{c^2 b})^2][/math]

from https://ui.adsabs.harvard.edu/abs/2002CQGra..19.5429A/abstract

Finding such formulae for massive particles is difficult because everyone wants to tell you about the formula for light.

1 hour ago, Markus Hanke said:

\[\Delta \varphi = 2 \int_{r_{\min}}^{\infty} \frac{dr}{r^2 \sqrt{\dfrac{1}{b^2 v_\infty^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1 - v_\infty^2/c^2}{b^2 v_\infty^2} + \dfrac{1}{r^2} \right)}} - \pi\]

I think there's an error in this formula. It's not dimensionally correct. Should that [math]\dfrac{1}{r^2}[/math] be [math]\dfrac{1}{r^2 c^2}[/math]?

Actually, on closer look, the two [math]b^2 v_{\infty}^2[/math] should be [math]b^2 v_{\infty}^2/c^2[/math].

Edited March 6Mar 6 by KJW

2 hours ago, Anton Rize said:

There are no "massless particles" in RG.

Once a gain,a few several days ago have been repeatedly been talking of your model requiring " prior knowledge"...mmmm...

The question of why and how,concerning your fundamental tone,has lead to above discussions...leading you to conclude in your methodology there is no massless particle...mmmm....I don't know what to say..sometimes I wonder what predictive power means; is it when the math agree with something or is it talking of something expected at a future date ?...anyway..🤔.

If its true then you conclusion"""There are no "massless particles" in RG.""" Is very significant....mmmm....I don't want to start discussion that will divert this thread.

This thread discussion is enriching to my perspectives...

2 hours ago, KJW said:

I think there's an error in this formula. It's not dimensionally correct. Should that 1r2 be 1r2c2?

Actually, on closer look, the two b2v2∞ should be b2v2∞/c2.

Oh my, you are absolutely correct! My apologies. I took the expressions for E and L from my personal notes, without realising that they were in natural units, whereas the integral was in SI units. Silly amateur mistake on my side. Let’s try again - we have, this time in SI units,

\[E=\gamma c^{2},\ L=\gamma v_{\infty}b\]

with

\[\gamma =\frac{1}{\sqrt{1-\frac{v_{\infty}^{2}}{c^{2}}}}\]

Popping this into the original E-L integral, I get, in slightly different form

\[\varphi =\int_{r_{\min}}^{\infty}\frac{\gamma v_{\infty} b dr}{r^2 \sqrt{\gamma^2 c^4 - \left(1 - \dfrac{2GM}{rc^2}\right) \left( c^2 + \dfrac{\gamma^2 v^{2}_{\infty}b^2}{r^2} \right)}}-\pi\]

The units should be correct now, with the result being in radians - but perhaps it’s wise if you double check, since I’m doing all this pen-on-paper.

Edited March 6Mar 6 by Markus Hanke

3 hours ago, Anton Rize said:

The historical factor of 2 in gravitational lensing is not an anomaly of curved spacetime

The factor of 2 is the result of the curvature of the three-dimensional space of the Schwarzschild spacetime (as a static spacetime). Notice that in my explanation of your error, I shifted from double the Newtonian result to double the equivalence principle result. When Einstein first calculated the deflection of light, he obtained a result that was half the correct result. This is the result one obtains from applying the equivalence principle. But this doesn't mean that the equivalence principle is invalid because the equivalence principle is only valid locally, and it is valid locally. For example, the gravitational deflection of a laser beam across a room does correspond to that expected in an accelerated frame of reference. And this occurs at all locations along the entire trajectory of light. But when one joins together all the many local deflections along the entire trajectory of light, the result is no longer a local deflection corresponding to the equivalence principle. The extra deflection, which would not occur in an ordinary flat space in which Pythagoras theorem is obeyed, is the result of the curvature of the three-dimensional space.

Earlier in this thread, I gave a formula for familiar gravity in terms of time dilation. This formula actually describes an accelerated frame of reference and applies the equivalence principle in describing familiar gravity. Thus, familiar gravity and the equivalence principle is about time dilation and what happens in the time dimension. What happens in the space dimensions do not contribute to familiar gravity. For one thing, space isn't amplified by the speed of light like time is. Thus, we don't notice the curvature of three-dimensional space because it is so small, whereas we notice the effects of curvature in the time dimension because they give rise to impossible to ignore gravity. Basically, what I'm saying is that spatial curvature is an entirely new thing in physics. There is nothing in Newtonian physics that is a manifestation of spatial curvature. Spatial curvature is a purely general relativistic effect without any correspondence in Newtonian physics.

20 minutes ago, Markus Hanke said:

The units should be correct now

No, the [math]c^4[/math] should be [math]c^2[/math]. Then the units will be correct.

27 minutes ago, KJW said:

No, the c4 should be c2. Then the units will be correct.

You’re absolutely right, and it was meant to be that, I once again forgot the conversion. This is what happens when you don’t do this stuff every day. Thanks for picking up on it 👍

3 hours ago, KJW said:

I also come across this formula:

I’m not entirely sure what “to second post-Newtonian order” actually means, but I presume this is an approximation of some kind? The full integral looks elliptic, so there shouldn’t be a closed-analytic form for the exact result.

22 minutes ago, Markus Hanke said:

  48 minutes ago, KJW said:

No, the [math]c^4[/math] should be [math]c^2[/math]. Then the units will be correct.

You’re absolutely right, and it was meant to be that, I once again forgot the conversion. This is what happens when you don’t do this stuff every day. Thanks for picking up on it 👍

Thanks @Markus Hanke. I'll write out the corrected formula:

\[\varphi =\int_{r_{\min}}^{\infty}\frac{\gamma v_{\infty} b dr}{r^2 \sqrt{\gamma^2 c^2 - \left(1 - \dfrac{2GM}{rc^2}\right) \left( c^2 + \dfrac{\gamma^2 v^{2}_{\infty}b^2}{r^2} \right)}}-\pi\]

22 minutes ago, Markus Hanke said:

  3 hours ago, KJW said:

I also come across this formula:

I’m not entirely sure what “to second post-Newtonian order” actually means, but I presume this is an approximation of some kind? The full integral looks elliptic, so there shouldn’t be a closed-analytic form for the exact result.

The formula itself was said to be "classical general relativity". I'm not sure what the authors meant by “to second post-Newtonian order”. My initial thought was that they were applying modified theories to the problem and that "classical general relativity" was just one theory that was applied, since they also mentioned "semiclassical general relativity". But I suspect you may be right about the formula I gave being an approximation of some kind.

1 hour ago, KJW said:

Thus, familiar gravity and the equivalence principle is about time dilation and what happens in the time dimension. What happens in the space dimensions do not contribute to familiar gravity.

Why should they be not familiar? Or in what sense....I think spacetime is a unified thing,what happens in either dimensions should affect the other...If it's not like that then there might be some aspect of either dimension that we don't fully understand...and according to me specifically what time is(nature of time).

2 hours ago, Anton Rize said:

Elimination of the "Massless" Myth:

This formulation proves that light is not a distinct ontological category governed by separate physical laws. There are no "massless particles" in RG. Light is simply the rightward topological limit of matter, where the internal phase βY reaches zero. The historical factor of 2 in gravitational lensing is not an anomaly of curved spacetime, but the inevitable geometric consequence of an exhausted phase buffer.

---

Careful on this statement invariant mass is directly related to how strongly a particle couples to its respective fields it interacts with.

To be honest I wish I had time to go through your math. I simply don't, hence been using generalizations.

That stupid contract where I am working on filtration software pertinent to a change in grating for the spectographic equipment of that telescope I previously mentioned has been a major headache.

Turns out the software employs Horndenski gravity which its been a while since I even studied that variation. Needless to say problematic and head is mush at the end of the world day.

Glad to see KJW spotted the error well done

The OP states that this is about algebra, relational geometry and some other concepts are introduced.

So let us take stock of the consequences this implies.

algebra, geomtery and relations fall within the province of Mathematics.

A relation is defined as a set of members each comprised of two elements, one element from each of two sets.

The two sets may be actual copies of the same set so the relation is then between elements of one set.

This is the case for geometric relations (relational geometry), where the elements are points of a geometric manifold.

the process of creating/defining sets can be carrier further by taking the pairing set and another set to establish further relations.

a metric is such a relation.

But the OP has disbarred the use of such a relation so we are left with simpler geometry to work with.

Less onerous geometric relations are still possible but we must look to other physical justification to provide the necessary rules.

I'm sorry but the expression energy = space just will not cut it.

However I do wonder if some process akin to the derivation of the Madelung Constant might suffice.

Such a relation could yied a non metric, yet numeric (ie algebraic), statement of the relation between any two points of the geometric manifold.

But no meaning can be attached to the statements "The distance between point a and point b is" or "point c is further from point a than point b is", since there is no metric.

42 minutes ago, studiot said:

I'm sorry but the expression energy = space just will not cut it.

Point of correction,it's spacetime being equivalent to Energy.... That equivalency for I, I totally 100% support it,up to to that point.

1 hour ago, Mordred said:

Careful on this statement invariant mass is directly related to how strongly a particle couples to its respective fields it interacts with

On 2/27/2026 at 4:49 AM, MJ kihara said:

According to your formulation how does matter or rather how does things acquire mass?

On 2/27/2026 at 9:34 AM, MJ kihara said:

Again how do things acquire mass?

On 2/27/2026 at 12:33 PM, Anton Rize said:

3. "How do things acquire mass?"

Short answer: I don't know. But I'm working on it

Would help to seperate invariant mass ( coupling strength) (old term rest mass) from variant mass ( relativistic mass) affected by observer

Lol ever wonder where the 19 parameters come from for the standard model. Covering all the invariant mass couplings would definitely be outside the scope of this thread.

1 hour ago, MJ kihara said:

Point of correction,it's spacetime being equivalent to Energy.... That equivalency for I, I totally 100% support it,up to to that point.

Point taken, thanks.

Spacetime it is.

+1

16 minutes ago, MJ kihara said:

  1 hour ago, KJW said:

Thus, familiar gravity and the equivalence principle is about time dilation and what happens in the time dimension. What happens in the space dimensions do not contribute to familiar gravity.

Why should they be not familiar? Or in what sense....I think spacetime is a unified thing,what happens in either dimensions should affect the other...If it's not like that then there might be some aspect of either dimension that we don't fully understand...and according to me specifically what time is(nature of time).

Yes, spacetime is a unified thing. But that's not how we experience reality. Our experience of space and time are very distinct. Firstly, there are three space dimensions and one time dimension. The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature. This creates a notion called the speed of light which acts as an impenetrable barrier between space and time, forcing us to exist as timelike worldlines in spacetime. Thus, we exist in space but experience time.

At human scale, one second in spacetime is very close to three hundred thousand kilometres in the time direction. This disparity in the perceived magnitudes of space and time intervals is connected to why the speed of light seems so fast to us. Actually, why the speed of light is so fast is an interesting question in its own right, but the consequence of this is that the time components of quantities have an exaggerated existence compared to the space components of the same quantities. This can also lead to the notion of magnetic quantities, space components that arise due to the motion of time components. For example, the electric field vector is the time components of the electromagnetic field tensor. But when an electric field vector is put into motion, that motion produces space components of the electromagnetic field tensor that is the magnetic field pseudovector.

With regards to curvature, it is my understanding that when considered in terms of the same units, the space and time components of the curvature of the Schwarzschild spacetime are essentially the same in magnitude. But we do not see the space curvature. Pythagoras theorem is assumed to be true. Yet for time, we see objects fall to the ground, planets orbiting the sun, and moons orbiting their planets. In spacetime, the earth has a helical trajectory around the sun in spacetime. But in spacetime, the trajectory of an object in orbit is a straight line. However, because the time of an orbit of the earth around the sun is one year, the length of the helical trajectory in spacetime is about one light-year, and at this length, no longer seems significantly curved.

Edited March 6Mar 6 by KJW

Sorry, forget about it. OT.

Edited March 6Mar 6 by Genady

@KJW @MJ kihara (and everyone following the mathematical results)

Guys lets put aside our personal preferences and beliefs and engage with the results that this research provides us so far.

The guiding principle of this research is Ontological Minimalism: any physical phenomenon must be interpreted using the absolute minimum number of primitives required for its complete description and prediction.

In standard General Relativity, to explain the the observed phenomena, one must postulate several heavy ontological primitives:

* Mass (as an intrinsic substance).

* A background 4D spacetime manifold.

* The metric tensor.

* Spatial curvature as a distinct geometric entity.

However, the consistent quantitative convergence of this research (including the Python script I just presented) demonstrate a hard, undeniable fact:
these are not necessary primitives. The same empirical predictions are generated purely from the algebraic distribution of relational projections ([math]\kappa[/math] and [math]\beta[/math]) on closed carriers, without ever invoking a spatial metric, a manifold, or spatial curvature.

When a theoretical model relies on constructs beyond the minimal necessary primitives, it inevitably introduces non-physical, artifactual structures. The concept of an "empty 3D space," a "background time axis," or a "curved fabric" are legacy concepts rooted deeply in anthropocentrism.
They are projections of how the human brain biologically parses its environment, not fundamental physical entities.

My methodology acts as a strict epistemic filter against this anthropocentrism.

If WILL Relational Geometry can algebraically generate the same empirical results strictly from the depletion of the internal phase buffer ([math]\beta_Y \to 0[/math] directly measurable transvers Doppler shift [math]z_b=\frac{1}{\beta_Y}-1[/math]) - without using a single metric tensor or spatial curve - then by the fundamental rules of the scientific method (Occam's Razor), spatial curvature is an ontological ghost. It is a mathematical over-parameterization of a purely energetic relation.

We do not need to imagine Earth moving in a "helical trajectory in 4D spacetime." We only need to measure the total relational shift ([math]Q[/math]) between energetic states. The math works. The extra ontology is just cultural baggage.

4 hours ago, KJW said:

Firstly, there are three space dimensions and one time dimension. The spacetime metric itself separates space and time into separate notions by their opposite signs in the signature.

Don't know if am wrong but the opposite signs in spacetime metric... signature -1,+1,+1,+1 or +1,-1,-1,-1 makes sure the spacetime is unified as it's also stated by
□=0. ( To whoever get concerned...Sorry I couldn't not comment in another thread...when experts discuss, sometimes it's reasonable to keep quite).

The three sides of de'Alembert operator magnitude(space axis x,y,z) being equivalent to magnitude of one side of the de'Alembert operator ( square,four sided, x,y,z,t) opposite sign ensure the total magnitude is equal to Zero...each degree of freedom is equivalent...t equivalent to -x,t equivalent to -y,t equivalent to -z and t equivalent to -(x,y,z).

Sorry corrections, each degree of freedom is equivalent...t = x,t = y,t = z and t = (x,y,z).

□=0.

It seems there is no need of correction,it depends on which signature you are using signature -1,+1,+1,+1 or +1,-1,-1,-1.

1 hour ago, Anton Rize said:

@KJW @MJ kihara (and everyone following the mathematical results)

Guys lets put aside our personal preferences and beliefs and engage with the results that this research provides us so far.

The guiding principle of this research is Ontological Minimalism: any physical phenomenon must be interpreted using the absolute minimum number of primitives required for its complete description and prediction.

In standard General Relativity, to explain the the observed phenomena, one must postulate several heavy ontological primitives:

* Mass (as an intrinsic substance).

* A background 4D spacetime manifold.

* The metric tensor.

* Spatial curvature as a distinct geometric entity.

However, the consistent quantitative convergence of this research (including the Python script I just presented) demonstrate a hard, undeniable fact:
these are not necessary primitives. The same empirical predictions are generated purely from the algebraic distribution of relational projections (κ and β) on closed carriers, without ever invoking a spatial metric, a manifold, or spatial curvature.

When a theoretical model relies on constructs beyond the minimal necessary primitives, it inevitably introduces non-physical, artifactual structures. The concept of an "empty 3D space," a "background time axis," or a "curved fabric" are legacy concepts rooted deeply in anthropocentrism.
They are projections of how the human brain biologically parses its environment, not fundamental physical entities.

My methodology acts as a strict epistemic filter against this anthropocentrism.

If WILL Relational Geometry can algebraically generate the same empirical results strictly from the depletion of the internal phase buffer (βY→0 directly measurable transvers Doppler shift zb=1βY−1) - without using a single metric tensor or spatial curve - then by the fundamental rules of the scientific method (Occam's Razor), spatial curvature is an ontological ghost. It is a mathematical over-parameterization of a purely energetic relation.

We do not need to imagine Earth moving in a "helical trajectory in 4D spacetime." We only need to measure the total relational shift (Q) between energetic states. The math works. The extra ontology is just cultural baggage.

Is it onological baggage when you consider the well established detail that no massive object can have a velocity of v=c ?

Yet you conclude there is no massless photons ?

Thats one of the common mistakes we often see here is ppl drawing conclusions based on specific relations whether its their own or not and not looking at other pieces of evidence.

Prime example being the twin paradox. It was never a paradox but was directly related to loss of symmetry between the Lorentz transformation which you are applying albeit through your beta function.

Spacetime curvature your running the same risk. You do get a different result in how you treat time than Newtonian with non variable time.

So ask yourself " why is spacetime curvature needed "

Then look at the axiom " the laws of physics must be the same regardless of reference frame"

This includes Pythagorous theorem which is something you haven't been examining

( it is a geometric relation). See the danger of not taking into consideration other pieces of evidence including those outside your own work ?

Lets try a professional level of examination example.

Encyclopedia Inflationaris

https://arxiv.org/abs/1303.3787.

Every single model in the above paper ( over 70) are still good fits to observational evidence. The order its currently in is best fit onwards.

Yet none of the other fits are invalid. To narrow down the extensive list uses pieces of evidence ( NOT covered within their own models) that leads to invalidation.

I'm positive you do not want others throwing out your hard work because you missed pieces of evidence that may run counter to your conclusions.

Piece of evidence to consider for Hubble constant and expansion,

Did you notice that Hubble constant is decreasing in value while expansion is accelerating ?

1 hour ago, Mordred said:

Is it onological baggage when you consider the well established detail that no massive object can have a velocity of v=c ?

This once again shows your complete lack of engagement with presented materials. For the last 4 pages on this forum I was presenting evidence that mass as a concept is not necessarily primitive. But how would you know if you only glance at my posts and then going back to make the same logically false arguments over and over again never admitting your mistakes and never clearly stating your own falsification conditions. As I told you before:

On 3/5/2026 at 6:41 PM, Anton Rize said:

  On 3/5/2026 at 2:45 PM, Anton Rize said:

Are you saying that you can't see the problem in your logic that I pointed out?

without the clear answer we can't move any further.

Sorry but I will have to ignore you until you will provide a clear unswear.

1 hour ago, Mordred said:

Is it onological baggage when you consider the well established detail that no massive object can have a velocity of v=c ?

This once again shows your complete lack of engagement with presented materials. For the last 4 pages on this forum I was presenting evidence that mass as a concept is not necessarily primitive. But how would you know if you only glance at my posts and then going back to make the same logically false arguments over and over again never admitting your mistakes and never clearly stating your own falsification conditions. As I told you before:

On 3/5/2026 at 6:41 PM, Anton Rize said:

  On 3/5/2026 at 2:45 PM, Anton Rize said:

Are you saying that you can't see the problem in your logic that I pointed out?

without the clear answer we can't move any further.

Sorry but I will have to ignore you until you will provide a clear unswear.

Wow are you ever close minded.

Take a look a look at geometry simply because you replaced geometry with your space equals energy does not mean you didn't have to account for geometry.

Same goes for mass using the energy momentum relation

Go right on ahead keep ignoring me I really couldnt care less. Does not stop other readers from reading my posts.

Those are specific examples.

Edited March 7Mar 7 by Mordred