No thank you I don't visit forums to deal with attitude I read your paper that was enough for me. Take my opinion or not couldn't care one way or another

Just reading through this thread its obvious your lacking in areas that others have pointed out as well.

Of course you could have instead shown where your applying the vectors etc but you chose attitude instead of showing where my statement is in error.

( hint tangent vectors for slope curve fitting) commonly used for SR and GR... how is your methodology replacing them and giving the same detail ya know basic calculus curve fitting....

After all not all spacetimes are Maximally symmetric like Euclidean or Cartesian.

Edited Wednesday at 10:47 AM5 days by Mordred

18 minutes ago, Mordred said:

Of course you could have instead shown where your applying the vectors etc but you chose attitude instead of showing where my statement is in error.

This is getting ridicules! All my results are publicly open but instead of reading you accuse me in not showing. We have nothing to discuss because you never engaged with the arguments provided.

What's ridiculous is whenever I mention something in textbooks Im met with scorn.

There is good reasons the stuff I mentioned exist in textbooks. Its a known methodology proven to work...

For example you could take a constant accelerating twin and plot the curve after following the rextbook methodology and fully describe the curve by \{\frac{g^4€{c^2}\} which will return the hyperbolic geometry produced via a spacetime graph of the travelling twins worldline...

I won't waste my time showing how that equation is the resultant see Lewis Ryders General relativity textbook

Edited Wednesday at 11:21 AM5 days by Mordred

1 hour ago, Anton Rize said:

So for nearly 3 months this post was in Relativity section without a word about moving it and then after your public humiliation you suddenly remembered the the rules. I see...

Not at all. You didn’t confirm that this wasn’t just GR with the equations presented differently until the middle of November. We had some discussion of the details. The thread was moved shortly after; it’s been in speculations since before Dec 1st.

Another hypothesis bites the dust.

2 hours ago, Anton Rize said:

This is getting ridicules! All my results are publicly open but instead of reading you accuse me in not showing. We have nothing to discuss because you never engaged with the arguments provided.

Read the forum rules material needs to presented here Im not about to go through a bunch of different links. This requirement has already been mentioned. I did the one exception by reading your main article. I will stick to that.

From that main article I do not believe you can give the proper seperation distance between two inertial reference frames ds^2 without being able to curve trace the worldline between the two events.

Particularly when the Lorentz transformations include not just time dilation but also length contractions.

Try this without considering geometry try more than two events say 3 different reference frames and what each observer sees relative to each observer at 3 different coordinate locations.

Then try it in a non Maximally symmetric spacetime such as one in rotation...ie Sagnac effect.

Edited Wednesday at 05:14 PM4 days by Mordred

@Mordred , we achieved incredible results in physics so far but there's fundamental challenges still remain. By blindly repeating textbooks we cant achieve any progress. That's why challenging anthropocentric beliefs is the main engine in scientific progress. Unfortunately you still have no clue what are you talking about. Spacetime interval is a mathematical tool we invented, Its just one of many possible interpretations. Same as fundamentality of mass and G. But look, when we ignore interpretations and dealing with directly measured phenomena we can see that neither mass or G needed to predict Mercury's presession. Here's Desmos project explicitly showing this https://www.desmos.com/calculator/wimnrykbvy How would you explain this?

If your going to attempt to do that then perhaps some consistency might be in order. Take for example your statement there is no momentum later on in your article yet your B_y which you describe as the inverse of the gamma factor includes velocity terms.

Mass for example is resistance to inertia change. It is a kinematic property. You also claim no need for any geometry yet you discuss two manifolds S_1 and S_2. Which are geometric objects. Your use of a circle and sphere are both geometric objects.

Now consider this any invariant quantity in physics does not rely on a metric one can arbitrarily choose any metric without changing the value of that invariant quantity. Ive read your paper several times over and I see no clear purpose or ontology view in its written format. However that's just my opinion. The inherent problem of ontology views trying to dictate how physics is done is that they often forget one of our primary jobs is to interpret datasets and graphs produced by experimental apparatus. You need geometry to accomplish that. Yes physics uses mathematics it is a fundamental tool for describing what we measure. Do not be fooled into thinking I believe in any fundamental realism I am well aware that terms such as mass, energy, fields, time etc Are abstract. Ive read countless ontology papers over the past 40 years. I am well aware of the difference between mathematical objects or descriptives vs fundamental realism.

None of that changes the job of a physicist which requires those mathematics you dont feel important in order for that physicist to secure jobs etc.

As far as that last link how did you program it without applying some form of geometry.. can you honestly state no geometry was used. I can readily accomplish the same using the standard methods with all those geometric relations.

Do you honestly believe that simply because you can plot a 2d orbital that this encompasses all possible observer from other angles ?

That was why I mentioned those little challenges. Lets see how well your mathematics work when you have multiple reference frames at any random 3d coordinate. Not just simply a simple case of a 2d plane.

By the way you and I are both aware that link describes a Maximally symmetric spacetime. Lets see how well your calculations work without having a Maximally symmetric spacetime. ( Marcus Hanke already mentioned the relevant killing vectors).

Im choosing to ignore the scalar quantity E as being in any regards a suitable replacement for spacetime geometry.

Edited Thursday at 02:56 AM4 days by Mordred

@Mordred , buddy, hold on. You still having major problems in understanding the most basic things. If you did read my paper as you claim than your reading skills rising concerns:
1. Spacetime is about meters and seconds. Relations are unitless. Its NOT "Maximally symmetric spacetime". Its a relational carrier of conservation.
2. The relational geometry of Sun-Mercury system will not change regardless of your point of view.
3. Lying is bad. You haven't read my paper and by continue lying about it you only make it worse.
4. Im not going to engage in philosophy with you because you have no clue what you talking about.
5. I gave you VERY SIMPLE solution for Mercury's precession with no mass no G and asked you "How would you explain this?" And the unswear you gave me is: "Do you honestly believe that simply because you can plot a 2d orbital that this encompasses all possible observer from other angles ?" - Your answer sounds like you saying that precession will change when viewed from different frame. Im just hoping that this some kind of misunderstanding.

And in general try to remember that before you want to critic anything make sure that you at least can repeat your opponents point of view without twisting it. Without it everything we say is meaningless.

Great so where is the difference between using a unitary basis under GR. Normalization I fully recognize and relate to same goes for dimensionless values.

Still doesn't address where dimensionless values isn't appropriate for specific relations.

Oh Ive read your article its mannerism of writing is rather scattered but that's another issue so dont call me a liar on that.

One of the reasons I had to reread it was I initially thought you declared the Gamna factor being the inverse of the beta function which would be incorrect but your B_y isn't identical to the normal beta function relation. Not that I saw you employing Gamma factor so it was irrelevant to mention.

What Im suppose to be convinced by your graphic ? Simply because you employed dimensionless replacements or using normalized units ? Its fairly rudementary to normalize or make some relation dimensionless. Ive come across numerous articles that make \{8 \pi G \} normalized to one good example is the critical density formula nothing new or exciting about that.

Do you not want to expand on your article for example The Kerr metric isn't a static solution. Perfect arena for testing your method on a rotating frame. You didn't really go into alot of detail in that section of your article.

If you feel your article is a done deal then it amounts to just advertising in which case I lose all interest.

Correct me if Im wrong but you assigned E_0 as invariant energy with M_0 being the invariant mass with E being total energy so explain why you have \{E=m_0\} and not \{E_0=m_0\} ?

Correct me if Im wrong but your thread title does state testing. So add tests you haven't already done.

Solving twin paradox with your methodology might prove a useful challenge as another example. However then you will have to deviate from the symmetry relations of constant velocity to include the rotations involved for acceleration

Edited Thursday at 04:26 AM4 days by Mordred

There is a handy simplification related to the non linearity of the relativistic addition of velocities where the Lorentz transformations matrix comes in handy.

Rapidity using rapidity velocity is replaced by rapidity and becomes linearly additive. The method applies the hyperbolic spacetime diagram of the Minkowskii metric its also useful for a constant accelerating object.

Not sure if that would interest you or not but its a useful simplification on calculations with regards to Lorentz transformations.

14 hours ago, Mordred said:

What Im suppose to be convinced by your graphic ? Simply because you employed dimensionless replacements or using normalized units ? Its fairly rudementary to normalize or make some relation dimensionless.

Did you even open my Desmos project? Out of 3 inputs precession derived:
3 IINPUTS NO MASS NO G NO c PRECESSION DERIVATION
1. Mercury's kinetic projection beta_p at perihelium unitless ( transverse Doppler shift, obtained via spectroscopy or radio signal.) \beta_{p}=0.000196736103348
2. Gravitational Redshift at the surface of the sun (R_sun) (obtain via spectroscopy) z_{sun}=2.1224\cdot10^{-6}
3. Ratio between R_sun and r_p (Mercury radius at perihelium) R_ratio=R_sun/r_p from astrometric data. No absolute scales needed. R_{ratio}=0.0151235185169
THATS ALL OUR INPUTS
if gravitation is the curvature of 4D spacetime induce by mass, then mass and G has to be primary parameters. But in my calculation they are unphysical unmeasurable redundant values:

\kappa_{p}=\sqrt{\left(1-\left(1+z_{sun}\right)^{-2}\right)\cdot R_{ratio}}=0.000253369506895
\Delta_{precession}=\frac{3\cdot\pi}{2}\cdot\frac{\kappa_{p}^{4}}{\beta_{p}^{2}}=5.0175347157\times10^{-7}
after units conversion we getting 42.9710621566 arcsec/100years

DATA SOURCES: \cite{NASA_Eclipse_Mercury}, \cite{UniverseToday_Mercury}, \cite{IAU_2015_ResB3}.

Dare to interpret this result?


P. S. I don't know what article you reading probably not mine. im talking about WILL_GR_I.pdf on my website.

10 hours ago, Mordred said:

Not sure if that would interest you or not but its a useful simplification on calculations with regards to Lorentz transformations.

You still dont get it. When you think about physical process you assuming that there's a flexible "box" (4D manifold) where mass/energy/fields are interacting.
Im NOT MAKING THIS ASSUMPTION. I argue that the hole concept of the "box" thinking is an anthropocentric speculation inbuild silently in foundations of modern physics. If you would red anything that I wrote this would be the first thing you would understand. Its a very old debate going back to Newton vs Leibniz.

I don't get it... Why are you engaging in criticism without a clue about the subject of your critic? Whats the point? To show your arrogance? Thats not something you should be proud about.

Whatever as I mentioned before the rules state that there is no requirement to visit other sites or links and all pertinent information should be here.

Im sorry you do not get that policy but its your full pdf on your opening page

I have absolutely zero interest in opening up any other of your website links.

So good luck with your work .

Im done I have better things to do

Edited Thursday at 07:08 PM3 days by Mordred

13 hours ago, Mordred said:

There is a handy simplification related to the non linearity of the relativistic addition of velocities where the Lorentz transformations matrix comes in handy.

Rapidity using rapidity velocity is replaced by rapidity and becomes linearly additive. The method applies the hyperbolic spacetime diagram of the Minkowskii metric its also useful for a constant accelerating object.

Not sure if that would interest you or not but its a useful simplification on calculations with regards to Lorentz transformations.

This post had nothing to do with your article. I simply thought it was an idea you could make use of.

Good luck.

You want thinking outside the box its simple any mathematical methodology that can accurately describe a system or state has validity.

You dont need tensors to do GR its simply another handy mathematical tool.

You dont need to use 4 parameters to describe spacetime you can use parametric equations to reduce them.

Thats my view point

Edited Thursday at 09:28 PM3 days by Mordred

18 hours ago, Mordred said:

its your full pdf on your opening page

Ahhhh... now it makes more sense... Sorry I forgot that I put a permalink on to opening page. Its a heavily shortened and significantly outdated. The core idea is the same and math too. Its just there's no dynamic systems explicitly described in this version. So now I understand why you were saying:

On 2/12/2026 at 11:29 AM, Mordred said:

Do you honestly believe that simply because you can plot a 2d orbital that this encompasses all possible observer from other angles ?

So you thought Im postulating circular maximally symmetrical spacetime geometry? No Im not.

Its just so upsetting that Im putting so much effort in to making the website easy to read and methodically transparent and you wasting your time reading outdated document. But its my fault I shouldn't put a permalink there in a first place.

On 2/12/2026 at 11:29 AM, Mordred said:

I can readily accomplish the same using the standard methods with all those geometric relations.

Can you elaborate on this please? I genuinely want to understand if its a well known method (as I understood from this comment) why I couldn't find it anywhere? And if we can predict observations that in GR are caused by mass and G without involving mass and G at all what does it tells us about the nature of gravity and spacetime geometry?

1 hour ago, Anton Rize said:

And if we can predict observations that in GR are caused by mass and G

There are some important subtleties here to be aware of. These predictions you are referring to arise from a particular metric, the Schwarzschild metric, which is a solution to the Einstein vacuum equation

\[R_❴\mu \nu❵=0\]

Notice how neither G nor m appear anywhere in this equation - it is simply a geometric statement, a constraint on what form any possible metric can take in vacuum. So those physical quantities aren't part of the theory at all at this stage. It is only when you begin solving these differential equations that there naturally appear integration constants in the process - and to find the physical meaning of those constants, we look at a boundary condition which we impose, namely that in the weak field limit, GR should reduce to Newtonian gravity. It is only through this particular boundary condition that G and m arise - they are thus the result of boundary conditions, not GR itself.

2 hours ago, Markus Hanke said:

These predictions you are referring to arise from a particular metric, the Schwarzschild metric, which is a solution to the Einstein vacuum equation

I appreciate your clarification regarding the vacuum equations. But in this case you are mistaking.

My derivation is not a reverse-engineering of GR. It is a direct result of applying a specific set of methodological principles that strictly forbid the use of a spacetime manifold:

1. Epistemic hygiene: We rely only on directly measurable signal relations (spectroscopy and astrometry), discarding unobservable theoretical entities (like mass).

2. Ontological minimalism: We do not multiply entities beyond necessity. If [math]G[/math], [math]M[/math], and the metric tensor are not required for the calculation, they are excluded from the ontology.

3. Relational origin: Everything originate from direct relations between potentials and kinetic states, not objects residing in a container (the unspoken assumption in modern physics).

4. Mathematical transparency: The connection between observables must be algebraic and direct. Each mathematical object must correspond to explicitly identifiable relation between observers with transparent ontological origin.

5. Simplicity: Everything must be expressed in the simplest form possible. Any unjustified complexity risks reintroducing metaphysical artefacts and contradicts the foundational insight of Epistemic Hygiene.

Here is the proof that RG is a superset of the Schwarzschild solution, not a copy of it.

1. The General Definition of the Horizon (The Limit of Causality)

In standard GR (Schwarzschild), the horizon is defined strictly spatially via mass ([math]r_s = 2GM/c^2[/math]). This describes a static boundary where escape velocity equals [math]c[/math].

In Relational Geometry, the horizon is defined by the saturation of the Total Relational Shift [math]Q[/math]. An observer defines the state of any external system via two orthogonal projections:

1. Gravitational Potential ([math]\kappa[/math])

2. Kinematic State ([math]\beta[/math])

The total state difference is the norm of these projections:

[math]Q^2 = \beta^2 + \kappa^2[/math] ( it is NOT Pythagorizes identity)

The natural causal horizon occurs when this relational difference saturates to unity ([math]Q=1[/math]). This leads to a distinction:

* WILL RG Horizon: [math]\beta^2 + \kappa^2 = 1[/math] (The limit depends on both motion and potential).

* Schwarzschild Horizon: This is merely the static slice of the generalized horizon where [math]\beta \to 0[/math]. In this degenerate case,
[math]\kappa^2 = 1[/math], which we label [math]R_s[/math].

Thus, the Schwarzschild radius is just the "potential-axis intercept" of the true relational horizon.

2. Deriving Precession from the General State [math]Q[/math]

The precession is intrinsic to the accumulation of this state difference [math]Q[/math]. The system accumulates a state mismatch over every closed cycle. The total angular shift is the full phase ([math]2\pi[/math]) scaled by the intensity of the shift ([math]Q^2[/math]), normalized by geometry:

[math]\Delta\varphi = \underbrace{2\pi}_{\text{Cycle}} \cdot \underbrace{Q^2}_{\text{Intensity}} \cdot \underbrace{\frac{1}{1-e^2}}_{\text{Shape Factor}}[/math]

To solve this for a stable orbit, we apply the Closure Condition (proved in the full paper). At the reference scale [math]a[/math], the relation between kinetic and potential states stabilizes as:

[math]Q^2(a) = \frac{3}{2}\kappa^2(a)[/math]

Recognizing that in the static limit [math]\kappa^2[/math] corresponds to [math]R_s/r[/math], we can substitute:

[math]Q^2 = \frac{3R_s}{2a}[/math]

Mapping this to the periapsis ([math]p[/math]) to use direct observables (Doppler [math]\beta_p[/math] and Redshift [math]z_{sun}[/math]), we arrive at the operational equation:

[math]\Delta\varphi = \frac{3}{2}\pi \frac{\kappa_p^4}{\beta_p^2}[/math]

Conclusion

1. Genealogy: The formula is derived from the accumulation of [math]Q[/math], a quantity that generalizes the horizon concept beyond the static Schwarzschild definition.

2. Independence: We define [math]R_s[/math] not as a mass parameter, but as the geometric saturation point ([math]\kappa=1[/math]) of the potential axis.

3. Epistemology: The fact that a model defining the horizon as [math]\beta^2+\kappa^2=1[/math] perfectly predicts a phenomenon traditionally explained by curvature tensors suggests that the tensor formalism is an emergent, albeit limited, map of a deeper relational terrain.

I am not hiding the metric. I am showing that the metric is a static approximation of a broader kinematic-potential relation obtain only from observables:

Inputs:

1. Kinetic projection (Doppler): [math]\beta_p[/math]

2. Gravitational potential (Redshift): [math]z_{sun}[/math]

3. Geometric ratio (Astrometry): [math]R_{ratio}[/math]

The Crucial Question:

If philosophy that categorically denies the existence of a metric tensor naturally yields the exact predictions attributed to the metric tensor, does this not imply that the metric is an epistemological artifact (a map) rather than a fundamental entity (the terrain)? What do you think?

P. S. We already touched this topic with you earlier in this post. It went badly. Maybe this time we will be able to reach some consensus?

P.P.S. I forgot to say thank you to you. Your first question in this post lead me in to developing a full Relational Orbital Mechanics: https://willrg.com/documents/WILL_RG_I.pdf#eq:rom
Thank you for the question.

Edited yesterday at 10:37 AM1 day by Anton Rize

Thank you for the above, its a tremendous help in understanding the purpose of your article. Sorry I was being a bit of a stickler on material needed being presented here. I do have good reasons for that, lol lets just say I've come across one poster in the past that although his ideas were sound. He had dozens of different papers and articles he kept referring to and you literally had to go through them to get any sense of what he was doing in the first place....

That's not the reason of course but its a good extreme example.

In the above you have a statement of avoiding any unnecessary complexity. Obviously scalar relations does indeed simplify the mathematics I would argue that requiring "direction of kinematic relations is a necessary complexity". Which direction an interaction (whatever that kinematic interaction represents) is just as important as the scalar relations.

Obviously we all know any " Field treatment requires geometry" particularly for any mappings of particle or measured quantity distributions".

Depending on what your after those mappings will also give a necessary complexity.

Those are two aspects I would consider as being necessary ( for what I do in physics absolutely necessary)

So the question of what is "necessary complexity" is something I think should be looked into in greater detail.

Side note I will often post added Mainstream relations relating to a thread. I've found in the past this habit is an aid to other readers not involved in the conversation better understand what is being discussed as well as useful for comparisons between methodologies etc

Forgot to add I don't see anything particularly wrong in your treatment above at the moment. In so far as the math relations involved.

I would be curious though if you agree that direction would be an inherent degree of freedom of any underlying state/system being described. Where one state resides in relation to another obviously is related.

Edited 20 hours ago20 hr by Mordred

On 2/15/2026 at 11:33 AM, Anton Rize said:

But in this case you are mistaking.

How am I mistaken in that G and m don’t appear in the vacuum equations?

On 2/15/2026 at 11:33 AM, Anton Rize said:

to Rs/r, we can substitute:

Did you not just say that your solution depends on neither G nor M, yet the above substitution explicitly introduces both of those quantities?

On 2/15/2026 at 11:33 AM, Anton Rize said:

At the reference scale a

How do you find this scale quantity a?

On 2/15/2026 at 11:33 AM, Anton Rize said:

Gravitational potential (Redshift): zsun

This, again, explicitly depends on both M and G. And it implicitly assumes you are in a spacetime that has a time-like Killing vector, and is asymptotically flat, or else no concept of gravitational potential exists.

On 2/15/2026 at 11:33 AM, Anton Rize said:

Kinetic projection (Doppler): βp

How do you find this quantity?

On 2/15/2026 at 11:33 AM, Anton Rize said:

Geometric ratio (Astrometry): Rratio

And where does this come from?