Looking at a copy of "Problem book in relativity and gravitation",
1975, Princeton U.P., "...almost 500 problems and solutions in the
fields of special relativity, general relativity, gravitation,
relativistic astrophysics, and cosmology".
"The collection is motivated by a simple premise: that the
most important content of this field does not lie in its
rigorous axiomatic development, nor, necessarily, in its
intrinsic aesthetic beauty, but rather does lie in computable
results, predictions, and models for phenomena in the
real universe."
The authors "also try to show the reader ''good'' ways
to compute things, methods and tricks which can vastly
reduce the labor of a plug-in and grind-away approach,
but we also try to avoid the opposite pitfall of introducing
too much confusing but powerful formalism for an easy problem."
"... unless noted otherwise throughout this book, we use units
in which c, the speed of light, is unity."
Problem 1.6: "Tachyons are hypothetical particles whose velocity is
faster than light. Suppose that a tachyon transmitter emits particles
of a constant velocity u > c in its rest frame. [...] Show that ... the
reply can be received before the signal is sent!"
[That appears to depend on the solution to 1.2
"Show that the boosts, in x then y, performed in
the reverse order would give a different transformation."]
Problem 1.11 refers to Sagnac.
1.23: "There is a special class of Lorentz transformations -
called the ''little group of p'', which leave the components of
p (p_0 = p^x = E, p_y = p^z = 0) unchanged, e.g. a pure rotation
through an angle alpha in the y-z plane [example] is such a
transformation. Find a sequence of pure boosts and pure
rotations whose product is _not_ a pure rotation in the y-z plane,
but _is_ in the little group of p."
1.28: "What is the least number of pure boosts which generate
an arbitrary Lorentz transformation?"
Answer 1.23 says "make an arbitrary boost", then "a pure rotation
lines up the coordinate frame again", but that "p^x doesn't
have its original magnitude", then "boost along p^x again".
"You can easily convince yourself that the product of these
transformations is not a pure rotation, there is in general
a net boost left over."
But, in 1.2 there isn't "a pure rotation
lines up the coordinate frame again".
About 1.28, matrix multiplication, and gimbal lock,
https://en.wikipedia.org/wiki/Gimbal_lock , about
the irreversibility of the application of boost, and some
pathological Lorentz transformation, is the "chasing its tail".
(I.e. stop hitting yourself.)
Einstein tensor
Riemann tensor
Weyl tensor
Ricci tensor
10.8: A metric is ''stationary'' if and only if it has a
Killing vector field E which is timelike at infinity
(the ''time'' direction is d/dt). There are [at least]
two ways to define a ''static'' metric:
(i) stationary and invariant under time reversal, d/dt = -d/dt, or
(ii) stationary and d/dt is hypersurface orthogonal (see Problem 7.23)
Show that the two definitions are equivalent.
https://en.wikipedia.org/wiki/Wave_vector
The wiki article describes wave vectors about
SR and crystallography, where, these days,
Christoffel is about working up what happens
when space relaxes as about some infinitely-fine
lattice gauge theory.
"The direction in which the wave vector points
must be distinguished form the ''direction of
wave propagation''. The ''direction of wave
propagation'' is the direction of a wave's energy
flow, and the direction that a small wave packet
will move, i.e. the direction of the group velocity.
In other words, the wave vector points in the normal
direction to the surfaces of constant phase,
also called wavefronts."
Looking at a copy of Louck's "Augmented Plane
Wave Method", (Frontiers in Physics, a Lecture
and Preprint Series, 1967), from the beginning
of the foreword, "The problem of communicating
in a coherent fashion the recent developments in
the most exciting and active fields in physics
seems particularly pressing today. The enormous
growth in the number of physicists has tended to
make the familiar channels of communication
considerably less effective. It has become increasingly
difficult for experts in a given field to keep up with
the current literature; the novice can only be confused.
What is needed is both a consistent account of a field
and the presentation of a definite ''point of view''
concerning it."
"The [Augmented Plane Wave] method was originated
in 1937 by J.C. Slater. Of course, at that time there were
no high speed computers available."
"Descriptions of the other methods commonly used in
energy band calculations [...] include the tight-binding
method, the cellular method, the orthogonalized plane
wave (OPW) method, the pseudopotential method
and the Green's function method."
https://en.wikipedia.org/wiki/Pseudopotential
Pseudopotential: not quite all the potential.