Mordred

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.

I agree pipelines to both coasts only makes economic sense provided the relevant safeties vs leakage and detection are properly installed. Lol though in our times detection of theft would also be required Nice descriptive lol

OMG now he's threatening a 100 % tariff on Canada if we make a trade agreement with China. All he's doing is motivating my Countries resources and trade with other countries. Lol after all I already canceled my plans to goto Florida this Summer and my wife and I chose to visit friends in Britain instead. If I were to send a message to Trump it would be somewhere along the lines of " that's cute, go play with the other children. The adults are talking"

I believe you may be referring to the antisymmetric affine connections and anti symmetric stress energy tensor vanishing in the Newton limit spacetimes or in spacetimes with zero torsion ( some literature state in the vacuum) but near or in BH regions is non vanishing if I recall this was a means to solve the singularity problem. However it's been an incredibly long time since I looked at Godel, Gravitoelectromagnetism or Einstein Cartan. All 3 the above involve intrinsic spin couplings in some fashion or other

there have been studies into intrinsic spin couplings to gravity example below https://arxiv.org/pdf/2502.07604 to date as far as I know there are no measured couplings and the article mentions the key violations that would result including those pertaining to freefall differences. There are numerous papers pertaining to this in the Godel Universe for spin gravity couplings which more often that not employ Einstein Cartan spacetimes. One of the factors against a rotating universe tight bounds is the lack of any measured spin gravity couplings which experimental data places a very tight bound against any universe rotation. Mashoom has a paper on it pertaining to Godel universe https://arxiv.org/pdf/2304.08835 the papers typically employ the magnetic moment of intrinsic spin with their couplings

Angular diameter distance integral \[d_A(z)=\frac{c}{\sqrt{|\Omega_{k,o}|H_o(1+z)}} \cdot S_k[H_o\sqrt{|\Omega_{k,o}|} \int^z_0 \frac{dz}{H(z)}\] \[S_k(x)=\begin{cases}sin(x)&k>0\\x&k=0\\sinh&k<0\end{cases}\] commoving distance \[D_c =\frac{c}{H_0} \int^z_0 \frac{d\acute{z}} {E(\acute{z}) }\]

Luminosity distance cosmographic approach. https://arxiv.org/abs/2307.08285 Equation 61 and 62 below https://arxiv.org/pdf/hep-ph/0004188v1

second order Luminosity distance full integral \[D_L(z)=(1+z)\cdot D_M(z)\] where \(D_M(z)\) is the transverse commoving distance Universe with arbitrary curvature \[d_L(z)=\frac{c}{H_0}\frac{(1+z)}{\sqrt{|\Omega_k|}}[sinn \sqrt{|\Omega_k|}]\int^z_0\frac{\acute{z}}{E(\acute{z})}\] sinn(x) defined as sin(x) when \(\Omega_k<0\), sinh(x) when \(\Omega_k>0\), x when \(\Omega_k=0\) Expansion function (dimensionless Hubble parameter) \[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z^2)+\Omega_\Lambda}\] modern times radiation is negligible, and for k=0 simplifies to \[D_L(z)=\frac{c(1+z)}{H_0}\int^z_0 \frac{d\acute{z}}{\sqrt{\Omega_M(1+\acute{z})^3+\Omega_\Lambda}}\] angular diameter distance reprocity relation \[D_A(z)=\frac{d_L(z)}{(1+z)^2}\]

https://arxiv.org/pdf/2411.11328 Article contains proposed neutrino mass mixing matrix.

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

Interesting study, hadn't seen that one before and your judgement on the paper is accurate. Another factor to consider is the particle spin itself. One known problem with the particle view is say the electron in the particle view it's angular momentum exceeds c. However in the QFT field excitation view this is resolved. Im still digging into how the Pilot wave theory deals with that (assuming solutions have been presented) more for my own curiosity lol. If I do find some decent literature in that regard I will share here

On the early time measurements ie measurements using the CMB in regards to improved calibration. The process involves calibrating the Baryon acoustic oscillations (BAO) https://arxiv.org/pdf/2405.20306 This paper includes references to the Hubble tension. For reference here are the 2024 DESI constraints. https://www.osti.gov/servlets/purl/3011043 Many of the regular forums members have often seen me refer to the equations of state and how matter, radiation evolve over time and seen me post a key formula to determine Hubble rate in the past compared to Hubble rate today as a function of redshift. https://www.osti.gov/servlets/purl/3011043 See equation 2.2 of article. this post I show how equation 2.2 comes about in the key equations and includes the look back time adjustment based on the same procedure.

Ive been off and on watching for papers researching the hubble contention between early and late time measurements. Figured I would share this interesting article here. https://arxiv.org/abs/2408.06153 One of the factors involved being Leavitts law https://www.astro.gsu.edu/lab/website/4LeavittLab-1.pdf The main point however is the paper expresses no new physics is required and supports the need for tighter constraints on luminosity to distance relations including tighter calibrations. Last link above is just a quick overview of the law. Below is a more recent calibration papers. https://arxiv.org/abs/2509.16331 A simplistic way of describing the law is The longer the period of the star, the higher its absolute luminosity. Local cephieds however are used to calibrate this law however as distance increases other factors may become involved hence the last link is further studies increasing calibration constraints

I recall my high school physics, it basically covered rudimentary SR. They certainly didn't cover gravity waves let alone GR. Lol several of us had better knowledge of GR than the instructors.

This video is not too bad. I usually dont subscribe to videos other than lectures but this one isn't too bad. David Bohm’s Pilot Wave Interpretation of Quantum MechanicsScience News, Physics, Science, Philosophy, Philosophy of Science She touches on the key points between the Copenhagen interpretation ( non locality in particle physics terms specifically aa applied to interactions). The issue this causes with Lorentz invariance. The 3 principle equations of the the theory. Some details to add is that the Hamilton-Jacobi usage is nonlocal and non linear as opposed to the linearity of the Schrodinger equation. The other key point is there is no testable means of showing its more or less accurate than the Copenhagen interpretation aka standard QM. It makes no predictions that will differ from those of QM. The other issue being the non locality when it comes to QFT. There are papers available of course presenting possible solutions to this problem but it's still in the works so to speak. This should help answer sone of your questions on our cross post lol. I would like you to consider the following. A wavefunction under QM and QFT as you are aware is a probability graph. All functions are graphs but not all graphs are functions. Those wavefunctions do have relevant constraints. For example causality is a constraint with regards to time dependent wavefunctions. Example the Dirac equations or Klein Gordon, Schrodinger etc Other constraints applied include conservation laws for probability wave functions applicable to closed groups. From those constraints anything not allowed is not included in the probability wavefunction. This is a very technical article describing boundary conditions as applicable to quantum mechanics included in the article is the Borne approximation or Borne condition. "Quantum boundary conditions" https://dottorato.fisica.uniba.it/wp-content/uploads/2018/03/tesiPhD-Garnero-compressed.pdf all boundary conditions is a form of constraint. All finite groups are also constrained. What many laymen or those not mathematically versed in physics often do not realize is every statement under physics is mathematically defined or described. This includes the axioms of a physics theory, group etc. Simple example symmetry ie laws of physics must be the same regardless of reference frame in the Minkowskii group is mathematically defined via \[\mu\cdot\nu=\nu\cdot\mu\] Constraints and boundary conditions are also mathematically defined.

I saw that on FB as well lmao. Lot of hoaxes floating around lately. Naturally I had to comment its falsehood on the FB post I came across