Mordred

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)

Im going to a phenomena in SR in terms of symmetry that will likely blow your mind and quite frankly that of many other forum members. If one takes a manifold and In general, manifold is any set that can be continuously parameterised. The number of independent parameters required to specify any point in the set uniquely is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. so lets use a simple manifold a sheet of graph paper. Now on that graph paper place two points label one point P and the other Q. The actual label doesn't matter. Now measure the distance between P and Q. Then roll up the paper into a cylinder. Now this is the tricky part. Following the surface of the paper the distance does not change between P and Q. The geometry is still Euclidean but now in cylindrical coordinates. So now we introduce two terms " intrinsic geometry and extrinsic geometry. The above case the ds^2 (seperation distance ) is unchanged so it is invariant. The geometry intrinsically is identical to Euclidean flat. Aka the laws of physics is the same regardless of inertial reference frame. There is no intrinsic curvature in this case. The curvature itself is the extrinsic geometry ( the cylinder viewed from the outside) in the first case think of an ant embedded on its surface. The above is essential to understand symmetry relations in SR and GR unfortunately when you combine time dilation using the Interval (ct) and apply the Lorentz transformations the example above becomes more complex as the above is a 3 dimensional manifold while spacetime is described by 4 dimensional manifolds. However there is no limit to the number of ubique parameters that can be used as unique coordinates on a manifold. Aka higher dimensions. ( a parameter can be any set) ie set representing time or charge or temperature etc etc. The above is something you need the math skills to properly understand and the above is also needed to understand SR ( Minkowskii metric) GR via the field equations including its tensors and gauge gauge groups. A common term for the above is local vs global geometry. For others the above is an example of coordinate basis. However the parameters used and subsequent coordinates can be under others "basis". The above should also give a very strong hint of why covariant and contravariant vectors become useful on manifolds 4d and up. (Kronecker delta ) first case 4d needing ( Levi-Cevita ) the above is also useful with regards to Hilbert spaces aka QM. The above is obviously a 2d manifold mathematical the extrinsic dimensions however requires the z axis to (curl) the 2d plane. ( curl equates to rotational symmetry)

Lets try a simple example with regards to symmetry relations involved for spacetime. The Minkowskii metric for example has a specific mathematical statement defining orthogonality which must also be symmetric. \[\mu \cdot \nu= \nu \cdot \mu\] This directly applies to vectors more specifically covariant and contravariant vectors. You wont find any image that will teach the above key relationship. In your first link where is the length contraction as applied under the Lorentz transformations? To give another example

I wouldn't rely on images to understand gravity. They can often be more misleading. For example describe how either image shows the equivalence principle between inertia mass and gravitational mass or show how either image describes time dilation when neither image contains a spacetime diagram Lets put it this way and only you can honestly answer the following. In the time frame since your last post in the your other thread has your understanding of gravity significantly improved ? In that same time frame would your understanding of gravity improved significantly more if you had instead studied an introductory GR textbook such as Lewis Ryder's General relativity even if you only spend 2 to 3 hours on it a week ?

In order to understand gravity especially under GR you need to have a good grasp of kinematics. GR uses the 4 momentum and its symmetry relations are freefall states with no force acting upon the object or particle ( which directly applies to the conservation of momentum). Newton treats gravity as a force acting upon the falling object instead of freefall. GR uses spacetime curvature instead of treating gravity as a force. Curvature is easily understood if you take 2 or more freefall paths. For example take 2 laser beams in parallel. If the two beams remain parallel spacetime is flat. If the two beams converge it is positively curved. If the beams diverge ( move apart) = negative curvature. To better understand freefall study the Principle of equivalence. https://webs.um.es/bussons/EP_lecture.pdf You can see under the section " Local inertial frames" the freefall paths are approaching one another as the elevator is freefalling toward a center of mass ( positive curvature). Indeed the equation of GR employ geodesics to describe these paths for photons they are null geodesics and how parallel null geodesics remain parallel converge of diverge are used to describe the curvature terms. At a more advanced level this is the basis of the Raychaudhuri equations. Which is a good formalism to understand how spacetime geometry affects multiparticle paths with regards to curvature terms. As mentioned Newton described gravity as a force so the falling objects have the gravitional force acting upon them. Under GR they are in freefall but the spacetime paths become curved. Hence gravity is treated as the result of spacetime curvature . In terms of geometry the Newton case the geometry is Euclidean and unchanging. This isn't the case under GR. In GR the geometry itself changes resulting in what we describe as gravity.