The only metric that is spherically symmetric and solves the Einstein field equations is the Schwarzschild metric, or so sayith Jørg Tofte Jebsen aka Birkhoff's theorem. [Correction: The theorem applies for a spherically symmetric, non-rotating, and uncharged source for the vacuum field equations.]

Study an alternative, that will have both the right answer infinitely far away (the flat Minkowski metric), could pass weak field tests like the precession of the perihelion of Mercury, but is not a solution to the Einstein [vacuum] field equations.  I like to call this the exponential metric, but it is known as the Rosen metric in the literature:
$d \tau^2 = e^{-2 \frac{G M}{c^2 R}} d t^2- e^{2 \frac{G M}{c^2 R}}\frac{dR^2}{c^2} - \frac{R^2}{c^2}(d \theta^2 + \sin^2 \theta d \phi^2)$
Get to step 4 of the above derivation.

Background
Misner of Misner, Thorne, and Wheeler has written a paper or two using this for number crunching simulations because it makes the calculations go quicker.  Exponential functions are computation friendly.

Multiply through by an exponential, one with a minus two:
$e^{-2 \frac{G M}{c^2 R}} d \tau^2 = \left(e^{-2 \frac{G M}{c^2 R}}\right)^2 d t^2- \frac{dR^2}{c^2} - e^{-2 \frac{G M}{c^2 R}} \frac{R^2}{c^2}(d \theta^2 + \sin^2 \theta d \phi^2)$
Use the first term of the Taylor series expansion for the three exponential:
$\inline \left(1-2 \frac{G M}{c^2 R}\right) d \tau^2 = \left(1-2 \frac{G M}{c^2 R} \right)^2 d t^2- \frac{dR^2}{c^2} - \left(1-2 \frac{G M}{c^2 R}\right) \frac{R^2}{c^2}(d \theta^2 + \sin^2 \theta d \phi^2)$
Divide by the factor of tau:
$\left(1-2 \frac{G M}{c^2 R}\right) = \left(1-2 \frac{G M}{c^2 R} \right)^2 \left(\frac{d t}{d \tau} \right)^2- \frac{1}{c^2}\frac{dR^2}{d \tau^2} - \left(1-2 \frac{G M}{c^2 R}\right) \frac{R^2}{c^2}\left(\left(\frac{d \theta}{d \tau} \right)^2 + \sin^2 \theta \left(\frac{d \phi}{d \tau}\right)^2\right)$
Proceed as before.

Discussion
I constructed this link to the standard derivation of the precession of the perihelion of Mercury pretty quickly.  It was the other 20 steps I kept struggling with on the back burner for several years.

May 14, 4-Parameter Analytic Animations, Solid Man

Snarky problem
Imagine what a 10 parameter quaternion expression would look like, then push the knob to 11.  Does it make any difference after 4?

Background
Folks who work with strings have so much more freedom.  What do they do up there?

One can use an arbitrarily high number of parameters.  What is interesting is the least number of parameters needed.  It is like golf, not bowling.  I have created situations that had four parameters, but they did not look like a dynamic solid, only a dynamic membrane, because two of the parameters were effectively "doing the same thing".

Discussion
Dynamic solids is as complicated as quaternion animations can get.  That requires a minimum of four parameters, but there is not an upper bound.

June 4, Scalars, Vectors, and Quaternion [Scalars and Quaternion Vectors]: Definitions (1 of 2)

Snarky Puzzle
I learned if your cross product is zero, then you have nothing but dot product, and visa versa.  Is that always true?  Worry about the 2 parameter situation...

<start ???>
I am too new to thinking about any and all quaternion expressions being able to be viewed as dynamic dots, strings, membranes and solids based on the parameter space used to evaluate the quaternion expression to answer this one with much confidence.  I can picture two strings lining up perfectly, which would have no chance for a cross product term.  Two membranes would seem to require more than one angle to describe the relationships between all the locations in the two membranes.  Once there is a requirement for more than one angle, the angles cannot all be zero or 90 degrees.  But perhaps I am looking at that wrong because a membrane is a bunch of lines bundled together, and if each of those had no cross product, then the sum of them wouldn't either.  So I will keep worrying about this one.
<end ???>

June 11, Quaternion Scalars, Quaternion Vectors: Life Without An Orthonormal Basis (2/2)

Snarky puzzle

Take one basis, boost it along three axes with the same velocity.  Form the product with an unboosted basis.  Report back.
Background

I almost decided to punt this question.  The pair of blogs had not gone well.  I always think of quaternions the same darn way: a 4x4 matrix over the real numbers.  I was unclear how to map that view with all the other ways one can look at 4-tuples.  In my little Boston suburb, there is another person who has dabbled with quaternions, so we got together for breakfast on Saturday.  He followed up with a link to a chapter from a book by John D. Norton, "Geometries in Collision: Einstein, Klein and Riemann."

The Geometrically Real in Klein's Erlangen Program and Riemann's Inaugural Address

The difference between Klein's and Riemann's approaches is usually understood in terms of the different types of geometries they best addressed.  Klein's approach flourished with geometries of uniform spaces, that is (in more modern language), spaces with non-trivial symmetries.  These included projective geometries and the geometries of metrical spaces of zero or constant curvature.  Riemann's approach extended the methods Gauss developed to deal with surfaces of variable curvature.  Such spaces in general admit no non-trivial symmetries [Blogger's note: that was a double negative, so Riemannian geometry only has trivial symmetries like the identity.]  Beneath this prominent difference lies another.  The two approaches employed opposing strategies to determine the geometrically real.  Klein employed a subtractive strategy: we would over-describe the space and then direct which parts of the over-description should be accepted as geometrically real.  Riemann employed an additive strategy: he would begin with an impoverished description and then only carefully add in further structure in an effort to ensure that all his structures had geometric significance.

This paper is worth the time to read.  I plan on rereading it a few times since it touches on a number of issues I am struggling with.  It is a minor game to try an slot various commenter into the Klein versus Riemann camps.

\begin{align*} q_b = (&-t + 2 t \cosh(2 \alpha) - (x + y + z) \sinh(2 \alpha), \\& - t \sinh(2 \alpha) +x - y - z + (y + z) \cosh(2 \alpha),\\&- t \sinh(2 \alpha) -x + y - z + (x + z) \cosh(2 \alpha), \\&- t \sinh(2 \alpha)-x - y + z + (x + y) \cosh(2 \alpha) ) \end{align*}
\begin{align*} q \times q_b = (&t^2-x^2+-y^2 -z^2 +2(xy + xz + yz)(1 - \cosh(\alpha)),\\&2 (t x \cosh^2(\alpha)-(t^2+t (y-z)+x (x+y+z)) \cosh(\alpha) \sinh(\alpha)+(t+y-z) (x+y+z) \sinh^2(\alpha)),\\&2 (t y \cosh^2(\alpha)-(t^2+t (z-x)+y (x+y+z)) \cosh(\alpha) \sinh(\alpha)+(t-x+z) (x+y+z) \sinh^2(\alpha),\\&2 (t z \cosh^2(\alpha)-(t^2+t (x-y)+z (x+y+z)) \cosh(\alpha) \sinh(\alpha)+(t+x-y) (x+y+z) \sinh^2(\alpha)) \\ \\ =&(t^2-x^2-y^2-z^2,2 t x,2 t y,2 t z) \quad if \; \alpha = 0 \end{align*}