Mordred

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)