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.

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

In so far as a particle origin for DM it would be easier to apply the virial theorem to DM with regards to galaxy rotation curves the weakly interactive characteristics for example well suits neutrinos. Or any other particle that is weakly interactive. When you study the NFW profile which utilizes Virial theorem for galaxy rotation curves you find that its power law relations show that in order to obtain the measured galaxy rotation curves instead of the Keplar decline you need a greater amount of mass in fairly uniform distribution surrounding the galaxy. The DM halo distribution while being weakly interactive DM is still subject to gravity. One of the primary reasons its expected to drop out of thermal equilibrium early is that its also considered responsible for initializing mass density anistropy for early large scale structure formation. Obviously we cannot measure directly DM but we can certainly infer its existence in some form or another as a pressureless ( matter) component via the equations of state in cosmology. Some of these articles describe secondary effects that can be measured involving DM. Example below. The double beta decay in the above articles should also allow for some interactions that we can hopefully measure https://arxiv.org/pdf/1402.4119

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.

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.

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

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.

I must admit this is the first time Ive heard of this particular possibility. Thank you for bringing it up. Lol knowing me I will dig considerably deeper into related articles to get a better feel for the status of Strangeness as DM +1 They would certainly drop out of thermal equilibrium early enough to form DM seeding for large scale structure formation

Lately I have been seeing numerous articles on right hand neutrinos contributing to dark matter. There are several different proposals. Those proposals involve whether or not neutrinos follow the terms of Dirac mass or Majorana mass https://arxiv.org/abs/2008.02110 here is a breakdown into singlets and doublets SU(2) \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\] \(\psi_L\) doublet \[D_\mu\psi_L=[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^+\tau^-W_\mu^-)-i\frac{g}{2}\tau^3W^3_\mu+i\acute{g}YB_\mu]\psi_L=\]\[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^-)+ieQA_\mu-i\frac{g}{cos\theta_W}(\frac{t_3}{2}-Qsin^2\theta_W)Z_\mu]\psi_L\] \(\psi_R\) singlet \[D_\mu\psi_R=[\partial\mu+i\acute{g}YB_\mu]\psi_R=\partial_\mu+ieQA_\mu+i\frac{g}{cos\theta_W}Qsin^2\theta_WZ_\mu]\psi_W\] with \[\tau\pm=i\frac{\tau_1\pm\tau_2}{2}\] and charge operator defined as \[Q=\begin{pmatrix}\frac{1}{2}+Y&0\\0&-\frac{1}{2}+Y\end{pmatrix}\] \[e=g.sin\theta_W=g.cos\theta_W\] \[W_\mu\pm=\frac{W^1_\mu\pm iW_\mu^2}{\sqrt{2}}\] \[V_{ckm}=V^\dagger_{\mu L} V_{dL}\] The gauge group of electroweak interactions is \[SU(2)_L\otimes U(1)_Y\] where left handed quarks are in doublets of \[ SU(2)_L\] while right handed quarks are in singlets the electroweak interaction is given by the Langrangian \[\mathcal{L}=-\frac{1}{4}W^a_{\mu\nu}W^{\mu\nu}_a-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\overline{\Psi}i\gamma_\mu D^\mu \Psi\] where \[W^{1,2,3},B_\mu\] are the four spin 1 boson fields associated to the generators of the gauge transformation \[\Psi\] The 3 generators of the \[SU(2)_L\] transformation are the three isospin operator components \[t^a=\frac{1}{2} \tau^a \] with \[\tau^a \] being the Pauli matrix and the generator of \[U(1)_\gamma\] being the weak hypercharge operator. The weak isospin "I" and hyper charge \[\gamma\] are related to the electric charge Q and given as \[Q+I^3+\frac{\gamma}{2}\] with quarks and lepton fields organized in left-handed doublets and right-handed singlets: For neutrinos involving Majorana mass an overview of the related mathematics is below including links to relevant papers \[m\overline{\Psi}\Psi=(m\overline{\Psi_l}\Psi_r+\overline{\Psi_r}\Psi)\] \[\mathcal{L}=(D_\mu\Phi^\dagger)(D_\mu\Phi)-V(\Phi^\dagger\Phi)\] 4 effective degrees of freedom doublet complex scalar field. with \[D_\mu\Phi=(\partial_\mu+igW_\mu-\frac{i}{2}\acute{g}B_\mu)\Phi\]\ \[V(\Phi^\dagger\Phi)=-\mu^2\Phi^\dagger\Phi+\frac{1}{2}\lambda(\Phi^\dagger\Phi)^2,\mu^2>0\] in Unitary gauge \[\mathcal{L}=\frac{\lambda}{4}v^4\] \[+\frac{1}{2}\partial_\mu H \partial^\mu H-\lambda v^2H^2+\frac{\lambda}{\sqrt{2}}vH^3+\frac{\lambda}{8}H^4\] \[+\frac{1}{4}(v+(\frac{1}{2}H)^2(W_mu^1W_\mu^2W_\mu^3B_\mu)\begin{pmatrix}g^2&0&0&0\\0&g^2&0&0\\0&0&g^2&g\acute{g}\\0&0&\acute{g}g&\acute{g}^2 \end{pmatrix}\begin{pmatrix}W^{1\mu}\\W^{2\mu}\\W^{3\mu}\\B^\mu\end{pmatrix}\] Right hand neutrino singlet needs charge conjugate for Majorana mass term (singlet requirement) \[\Psi^c=C\overline{\Psi}^T\] charge conjugate spinor \[C=i\gamma^2\gamma^0\] Chirality \[P_L\Psi_R^C=\Psi_R\] mass term requires \[\overline\Psi^C\Psi\] grants gauge invariance for singlets only. \[\mathcal{L}_{v.mass}=hv_{ij}\overline{I}_{Li}V_{Rj}\Phi+\frac{1}{2}M_{ij}\overline{V_{ri}}V_{rj}+h.c\] Higgs expectation value turns the Higgs coupling matrix into the Dirac mass matrix. Majorana mass matrix eugenvalues can be much higher than the Dirac mass. diagonal of \[\Psi^L,\Psi_R\] leads to three light modes v_i with mass matrix \[m_v=-MD^{-1}M_D^T\] MajorN mass in typical GUT \[M\propto10^{15},,GeV\] further details on Majorana mass matrix https://arxiv.org/pdf/1307.0988.pdf https://arxiv.org/pdf/hep-ph/9702253.pdf Now in order to account for the mass terms of DM the mass terms must be in or above the Kev range. Below are some related articles involving DESI. The Kev range would readily fall under the mentioned warm dm models. However there is also papers that place right hand neutrinos being in the GeV range through double beta decay. DESI constraints https://www.osti.gov/servlets/purl/3011043 Has a particular section to follow up on massive neutrinos behaving as dark matter described in above link. https://arxiv.org/abs/2507.01380 double beta decay primer https://arxiv.org/abs/2108.09364 In a nutshell the possibility is there so I started this thread to explore various examinations and starting a discussion on the the pros and cons of such a proposal. Naturally I would be interested in any related papers including counter arguments. This is not my own model proposal but a discussion on models presented by others. It doesn't suit a mainstream forum not yet anyways lol. As for myself I see the potential but I question whether or not the mass terms will meet the required DM mass distribution. There was a fairly recent study that placed constraints on any simple Dirac mass term for right hand neutrinos in that examinations of the energy sector did not have any relevant findings. Still digging up that study hopefully I can find it however if I recall it constrained 5 KeV or less if memory serves. other related papers https://arxiv.org/pdf/1911.05092.pdf https://arxiv.org/pdf/1901.00151.pdf https://arxiv.org/pdf/2109.00767v2.pdf https://arxiv.org/abs/1402.2301 https://arxiv.org/pdf/0708.1033 Located the light neutrino constraint paper via MicroBoone https://arxiv.org/abs/2512.07159

Well having gone through this thread as well as the OPs main paper. I dont see any practicality behind this. No vector fields, no spinor relations, Redefining standard physics terminology to suit the OP ( energy as primary example) Use of geometry relations without defining any geometry. Trying to replace GR without actually understanding GR.... Trying to apply energy to geometry when spacetime by itself has no energy. ( it's simply a mathematical object) a mathematical construct. A field is also a mathematical construct. It is the SM particles that reside in spacetime and how they interact with one another that tells spacetime how to curve. So having an energy equivalence to the invariant mass only fills the \{T^{00}\} component of the stress energy momentum tensor. Leaving all other components of that stress energy momentum tensor unfulfilled. As the OP doesn't understand GR its useless pointing that out. How the OP plans on dealing with stress and shear components of a multiparticle field without use of any geometry is something I find utterly impractical. I may have missed this but I also didn't see any treatments of how angular momentum factors in let alone linear and angular force... something which GR fully describes. After all physics includes the study of forces. And force is a vector quantity right along with acceleration which is both change in speed and direction Ola another vector

High school classrooms I attended had windows on both left and right hand side in different rooms of the same school

Here's an ontology question for you why does GR use calculus and not algebra could it have something to do with rate of change ?

My response has nothing to do with memorization from a textbook. The definition for energy has been the sane for well over 5 centuries. Perhaps you should study a classical physics textbook and see how energy ties into the work equations then learn how it ties into the kinematic equations under GR and SR Those definitions are used in all physics regardless of any ontology. Unlike yourself I do not rely on AI Do you even consider anywhere in your article the inner product of vectors or the outer product or the cross product of vectors which is incorporated into the tensors your trying to replace? Nor have I seen anything regarding bilinear forms needed for curvature I certainly haven't seen anything related to parametric equations which GR incorporates Looking through your article you completely ignored all symmetry relations with regards to first order, second order and higher relations. Specifically the symmetry relations with regards to freefall velocity (first order terms) used with conservation of energy momentum. I didn't see much in regards to acceleration (second order terms) Nor does your normalization of energy to invariant mass have much practicality when it comes to distinguishing potential energy and kinetic energy when applying the four momentum. good luck with your article. As I read it and can honestly say it will never get far as it is written. Is the circle the only curvature form you have examined ? Ie just positive curvature? How do you plan to deal with energy measured being relative to the observer when you normalized energy to invariant mass ? Your article deals primarily with first order scalar quantities not very practical when you require vector fields including higher order time differentials perhaps that's something you look into

Why dont we start with some very basic classical physics definitions which apparently you never learned. Space is volume, spacetime is a geometry that uses the Interval (ct) to give Time dimensionality of length. Energy is the ability of a system/ state etc to perform work. Spacetime does not equal energy by any mainstream physics application. The above definitions apply to all main stream physics theories if your not following the above definitions then this thread definitely belongs in Speculations. Particularly how those tensor entries apply to the Kronecker delta and Levi-Civitta connections.

Lol ain't that the truth Good point, many models and theorems are continously evolving as new data becomes available.

Useful +1 was wondering a few times how to do chem latex

I concur +1

Lol there's some debate on whether math is a science or not.

It could also be argued there is no hard and fast truth in science. There is truth to the best of current understanding. Good example that everyone is familiar with in physics is Newtons laws of inertia. Everyone firmly believed the equations applied regardless of the measured objects inertia. Later findings showed its only valid for non relativistic inertia hence GR. I also wonder why this thread is in politics.

There's another key detail when it comes to the intrinsic curvature it is independent of any higher dimensional embedding. Very useful for invariant functions. Particularly when it comes to applying the tangent vector to the line element ds^s. Of key note is the basis vectors. Taking the infinisimal distance between P and Q (local) this can be shown independent on coordinate transformations. So the basis vectors are independent. Subsequently this equates to the covariant and contravarient vectors. As well as the Christoffel connections.

Its the set that can be continuously parameterized where each parameter is a coordinate. Line segment is one example. The association of points/coordinates with their measured values can be thought as the mappings of the manifold. However you may not be able to parameretize the entire manifold with the same parameters. Some manifolds are degenerate. Simple case a finite set of R^n in Euclidean space is non degenerate. However in Cartesian coordinates involving angle the origin or center is degenerate as at zero the angle is indeterminate. This is where the use of coordinate patches get involved. A manifold can have different coordinate systems as per above on the same manifold. With no preference to any coordinate system. The set of coordinate patches that covers the entire manifold is called an atlas. The saddle shape for negative curvature would be a good example. Edit scratch that last example it can be continously parameterised under the same coordinate set. The Cartesian coordinate requires 2 sets.For reasons provided above. Hyperbolic paraboloid \[z=x^2-y^2\] can be parameterized by one coordinate set. Though multiple sets can optionally be used it isn't required.

In the first example when you set the lines on a graph paper prior to bending this is intrinsically flat ( it is independant ) Once you curl the paper your curve is extrinsic as you need an extra dimension in order to curl the plane. Im not sure you missed anything tbh. Cylinder can simply be described as Eucludean flat is the internal geometry with extrinsic curvature. A sphere for example however has an intrinsic positive gaussian curvature ie circumference of the sphere. Intrinsic curvature K=1/r^2. With extrinsic curvature you need a higher dimension embedding. the 2 principle curvatures being \(k_1=K_2=1/R\) with mean curvature being \(H=1/2 (k_1+k_2)=1/R\). with \(K_{a,b}\) being the second fundamental form \[K_{\theta\theta}=R\] \[K_{\phi\phi}=Rsin^2\Theta\] \[k_{\theta\phi}=0\] under GR the extrinsic curvature tensor is the projection of the gradient of the hypersurface. \[K_{a,b}=-\nabla_\mu^\nu\] \[K_{\theta\theta}=\frac{r}{\sqrt{g(r)}}\] \[K_{\phi\phi}=\frac{r\sin^2\theta}{\sqrt{g(r)}}\] mean curvature bieng \[k=h^{a,b}k_{a,b}=\frac{2}{r\sqrt{g(r)}}\] K being a surface of a hypersphere where all affine normals intersect at the center above ties into n sphere aka hypersphere https://en.wikipedia.org/wiki/3-sphere edit: I was at work earlier decided when I got home to go into greater detail further detail in same format as above https://en.wikipedia.org/wiki/Gaussian_curvature https://faculty.sites.iastate.edu/jia/files/inline-files/gaussian-curvature.pdf https://arxiv.org/pdf/1209.3845

Nice explanation @KJW +1

Nice thread I may look into including Fock and Hilbert spaces into this thread might be handy to have specific spaces inclusive.

Good point ( pun intended)