But this is definitely not the case or the time for being corrected.
That doesn't change the fact you haven't got a clue about relativity and differential geometry. You should have put in some effort to learn them before writing books on the subject and trying to con people out of money.
First, you are only partially correct here, and it is this sort of mathematic that you are describing here that keeps physicist away from GR.
I did a degree in a maths department and a PhD in a physics one. I worked along side people researching GR, so I am perfectly aware of the general views of the communities on such things as GR. If you think the level of mathematics I've given is excessive you're just naive. The sort of stuff I've said to you, about tensor ranks and indices, is the stuff 1st year physics students learn about. It is the
basics, required knowledge for everyone doing a physics or maths degree.
The reason that the Super Principia Mathematica was written was to simplify the rigorous mathematics
In order to write a book which simplifies complicated stuff you must first understand the complicated stuff.
You don't.
that people could sink their teeth into and understand
Plenty of people already can sink their teeth into this stuff, plenty of people take physics and maths degrees. Sure, it'd be nice if more of the general public understood more physics and maths but your approach is only going to hinder that because its important to understand things correctly. I'd rather 1 person knows something complicated correctly than 10 people think they know something simpler correctly but it actually be wrong.
Your attitude is "I'm helping people who don't grasp this" but
you don't grasp this.
Then some 100 or more years later Einstein working on his Field Equations wanted to design a mathematical system for an isolated Net Inertial Mass body, applied the concepts of Euler's Moment of Inertia for a spherical body to his model and came up with 9 different terms components, Axes, or "Ranks;" and one for time for a total of 10 components or "Ranks" to his Field Equation.
You don't know how the number of components in the EFE arise.
Rank and components are not the same. A vector in 3 dimensions is rank 1 but has 3 components. More generally a vector in N dimensions is rank 1 but has N components. A general rank M tensor in N dimensions has $$M^{N}$$ components. This is then reduced by symmetries. In the case of a rank 2 tensor in 4 dimensions there are 16 components, ie you have a tensor $$M_{ab}$$ where a,b can range over 1,2,3,4. In relativity its generally written as 0,1,2,3 where 0 is the time index. If the tensor is symmetric then $$M_{ab} = M_{ba}$$. This reduces the number of independent components from $$N^{2}$$ to $$\frac{N(N+1}}{2}$$. If N=4 you get 10 independent components.
That is now 10 components arise in a symmetric 4 dimensional rank 2 tensor, which $$G_{ab}$$, $$g_{ab}$$, $$T_{ab}$$ and $$R_{ab}$$ all are.
If you're going to go down the route of "Lets review some history" at least get it right. You're obviously used to dealing with people who don't know any relativity and whom you can bamboozle with bullshit. That isn't going to work here, I (and others here) have a working understanding of this area of physics and I've got no qualms in demonstrating it.
When Einstein satisfied himself with his 10 terms to his field equation, he was forced to use the 10 terms because he chose to work with "Cartesian Coordinates" (x, y, and z) as his metrics.
No, the (x,y,z) coordinates are
coordinates, not metrics. You should look up the role coordinates play in general relativity. You'll find it in the first chapter of a GR book because they are dealt with in the definition of 'manifold', which you obviously don't know.
Besides, GR is formulated in a coordinate independent way, that's part of its power. Tensors are used because tensor equations are independent of what (valid) coordinate system you use. $$G_{ab} = 8\pi T_{ab}$$ is true in Cartesian coordinates or polar coordinates or Rindler coordinates or Kruskal coordinates or any valid coordinate system you care to make up. The 10 terms are nothing to do with angular momentum axes. Besides, even if they were the angular momentum tensor $$I_{ab}$$ is symmetric too, $$I_{ab} = I_{ba}$$ so there's only 6 independent components in it, you're still 4 short.
Properties of the angular momentum tensor are another thing 1st year physicists have to learn.
However, shortly after Einstein's success, Karl Schwarzschild decided to use Spherical Coordinate and determined that this removed the necessity for having to carry around 10 terms, or "Ranks." And that the angles for latitude and longitude was enough to accomplish what Einstein's 10 terms was trying to accomplish.
No, he didn't. Unlike you I actually know the derivation of the SC metric via Birkhoff's theorem's application to $$R_{ab} = 0$$. The typical form of the metric is written in spherical-like coordinates but that doesn't mean the metric is the result of that choice. People use Kruskal or Eddington-Finklestein coordinates in the SC metric all the time. There are still 16 components in the SC metric, with 10 of them possibly independent. The use of spherical coordinates helps to highlight an underlying symmetry of the metric, that it is spherically symmetric. As such it has fewer independent components than a general metric but it still have more than just latitude and longitude. In fact the whole point of spherical symmetry is that its
independent of latitude and longitude, the system doesn't change. I'd tell you to compute the geodesic arc length joining two points on the sphere and then apply an SO(3) rotation and repeat to get the same result but I know you're incapable of doing such a calculation.
I'll add spherical symmetry to the list of things you don't understand.
and I am being nice here!!
No, you're being ignorant. You don't understand the role of coordinates, the coordinate independent nature of tensors, the notion of tensor rank or the difference between rank and components. I'd expect someone starting their first course in GR to know those things, because they come up in prerequisite courses to study GR. The fact you've written books on this stuff makes your errors inexcusable. Did you even open a book on GR before writing them or did you just 'know' how it all works?
Now I will give you some credit here because Modern Mathematians are using the shorthand of Riemnann where he uses three dimensions of space and one dimension of time and created a four (4) vector or tensor Riemann curvature tensor is $$R^{a}_{bcd}$$. But once again this is using the Einstein concepts of 10 terms in one equation and reducing them down to 4 terms in one equation.
You haven't understood the curvature tensor either. The rank of the curvature tensor (ie the number of indices) is
always 4, whether you're in 1 dimension or 50. If you knew the definition of the tensor you'd know that. This is something anyone studying GR should know, because there's nice little results like the fact the Riemann curvature tensor has only 1 independent component in 1+1 dimensions, which equates to the Ricci curvature scalar.
The curvature tensor can be computed from the metric using
this equation and the definition of
the connection. The way you describe it is as if its used instead of the metric. You continue to confuse rank, components and independent components.
Let's make this even simpler. Let $$M_{ab}$$ be a general rank 2 tensor in 4 dimensions. Then you have
$$M \sim \left( \begin{array}{cccc}
M_{11} & M_{12} & M_{13} & M_{14} \\
M_{21} & M_{22} & M_{23} & M_{24} \\
M_{31} & M_{32} & M_{33} & M_{34} \\
M_{41} & M_{42} & M_{43} & M_{44}
\end{array} \right)
$$
16 different entries or
components. Suppose it is symmetric, so that $$M_{ab} = M_{ba}$$, then you get
$$M \sim \left( \begin{array}{cccc}
M_{11} & M_{12} & M_{13} & M_{14} \\
M_{12} & M_{22} & M_{23} & M_{24} \\
M_{13} & M_{23} & M_{33} & M_{34} \\
M_{14} & M_{24} & M_{34} & M_{44}
\end{array} \right)
$$
Now there's only 10 different components. That's how it works for the metric and the Einstein tensor. What about if its antisymmetric, $$M_{ab} = -M_{ba}$$?
$$M \sim \left( \begin{array}{cccc}
0 & M_{12} & M_{13} & M_{14} \\
-M_{12} & 0 & M_{23} & M_{24} \\
-M_{13} & -M_{23} & 0 & M_{34} \\
M_{14} & -M_{24} & -M_{34} & 0
-\end{array} \right)
$$
Now there's only 6. That's how it works for the Maxwell tensor. Notice how 16 = 10+6? That's because every rank 2 tensor splits into a symmetric part and an antisymmetric part. This is linear algebra 101. You can't even get onto a GR course if you can't do this.
This not wrong on your part per se, just not user friendly.
No, its right, you're the one utterly wrong.
Would you please explain how this term is mechanized into the mathematics of General Relativity; more specifically, what equations is term apart of ?
You thought that the event horizon was a real singularity, where curvature is maximum. I explained its only due to the choice of coordinates, different coordinates remove the problem. In order to see this rigorously you have to compute curvature dependent quantities which are scalars. $$R^{abcd}R_{abcd}$$ is such a thing, so if you're right then the value of that should be maximum at the event horizon. If you crunch through the algebra for the SC metric you get $$R^{abcd}R_{abcd} = \frac{48M^{2}}{r^{6}}$$. Therefore maximum value of this quantity is at r=0, not r=2M, thus your claim is false. If you don't know how to go about working that out for yourself then you illustrate you don't have even a basic grasp of the working of relativity. It's little more than a tedious homework problem.
And if anyone want's a good read, that is user friendly to the Conceptualist or Mathematicians on the subject of General Relativity, I recommend the Super Principia Mathematica, by Robert Louis Kemp.
I'd wager you're him (since I've accused you of that several times and you haven't denied it) and even if you aren't your utter ignorance of the workings of GR illustrates you didn't learn anything useful from those books. Besides, commercial advertising of books isn't allowed, particularly when they are full of ignorant crap. You, Farsight, Anita Meyer, all of you plug books which
failed to meet even basic scientific standards and hence why you're plugging them on websites (or in Farsight's case batshit crazy TV shows on backwater satellite channels).
Short conversations with any of you are enough to expose such profound stupidity and ignorance that its clear that the books aren't as amazing and insightful as you'd like us to believe.
