Riemannian geometry etc.

On Apr 10, 8:22 am, shuba wrote:

Ah, I see. So Riemann, Gauss, and the thousands of others who have
rigorously developed and studied the field are failures, and an unknown
relativity crank on usenet is the lone genius who has the record
straight. Very plausible.

No, not all of them. Perhaps the following Relativity play will shed
so light on your confusion.

Newton: I came up with the laws of gravity to describe what gravity
does by observing a falling apple under the influence of gravitation.

Einstein: I have no idea of what I am doing, but I can tell you that
I personally have derived the so-called Lorentz transform through two
assumptions which I have proudly speculated. The first speculation is
the principle of relativity, and the second one is the constancy in
the speed of light.

Galileo: Excuse me. I have already discovered the principle of
relativity.

Voigt: Excuse me as well. I have already suggested the necessary
mechanism to explain the null results of the MMX as the constancy in
the observed speed of light regardless if the principle of relativity
holds or not. That is how I derived the Voigt transformation. This
transformation does not satisfy the principle of relativity but
explains the null results of the Michelson-Morley experiment through
the constancy in the speed of light.

Larmor: Well, the Voigt transformation is certainly not the only one
that explains these null results. I have discovered another one by
dividing one side of the Voigt transformation by the square root
quantity. It also does not satisfy the principle of relativity in
general. Just like the Voigt transformation, one of the two observers
must be the absolute frame of reference. All observations must
reference back to this absolute frame of reference.

Galileo: What good are the Voigt and the Lorentz transformations when
neither satisfies the principle of relativity? Mind you that the
Galilean transformation allows the two observers to be anyone. All
observations are relative.

Maxwell: There is no scientific axiom that requires the principle of
relativity to hold. In fact, Maxwell’s equations explain the
propagation of light without the principle of relativity. The
absolute frame of reference must exist to allow for the propagation of
light.

Lorentz: While Mr. Maxwell was explaining it is unnecessary for the
validity of the principle of relativity, I have come across an
infinite such transformations on top of what Mr. Voigt and Mr. Larmor
have discovered. They all satisfy the null results of the Michelson-
Morley experiment but not the principle of relativity.

Maxwell: Good point, Mr. Lorentz, Mr. Larmor, and Mr. Voigt. These
null results actually prove the existence of the Aether.

Michell: Hold it, gentlemen. The ballistic theory of light explains
these null results and satisfies the principle of relativity.

Einstein: Hooray! My speculation which is based on farce can turn
out to be correct after all, and it is all mine.

Galileo: whispering to Newton After stealing two of my
discoveries, this idiot does not realize his second speculation fails
the ballistic theory of light.

Newton: Although I am the founding father describing light as
classical particles, I have to disagree with Mr. Michell. The
ballistic theory of light cannot explain light propagation and
electromagnetic phenomena.

Poinca Then among all these infinite numbers of transformations
that Mr. Lorentz discovered, which one is valid?

Poinca Wait, gentlemen. With the way Mr. Larmor wrote down his
transformation, the absolute frame of reference vanishes for this
special case.

Michelson: Yes, Mr. Poincare. This is a special case where both
observers are moving in parallel relative to the absolute frame of
reference.

Poinca Nevertheless, we can *******ize Mr. Larmor’s transformation
into a new one where the two observers can be anyone just like the
Galilean transformation. In this case, the principle of relativity is
preserved. Let’s now call Mr. Larmor’s transformation the Lorentz
transformation.

Michelson: That is foul. The Lorentz transformation reflects no
experimental bases. It is created in the minds of man. We deal with
physics. We don’t play God, OK?

Einstein: Shut up, Mr. Michelson. You don’t understand relativity.
My groundless speculations have finally paid off. I will now attempt
to fudge the Lorentz transformation into the Maxwell’s equations.

Maxwell: Where did they get this clown from? shaking his head

Einstein: Well then, I have personally discovered spacetime.

Minkowski: Wait! I am the one who wrote down all the equations of
the Lorentz transform into a single, concise one. Spacetime then only
becomes very obvious from then on.

Voigt: The Voigt transformation can be written into a single, concise
equation as well.

Larmor: So is the transformation I have discovered and the infinite
others that Mr. Lorentz have discovered.

H. G. Wells: Well, Mr. Einstein, I don’t know if you have read my
book “The Time Machine”. In it, I have already described time and
space forming a single entity in which a time traveler can travel from
one set of time and space to another.

Einstein: OK, let’s forget about Special Relativity and talk about
General Relativity. I personally discovered the principle of
equivalence by picturing myself as that falling apple trapped in the
gravitational field.

Newton: That is very stupid. Gravity can only be characterized by
observing how an object would behave under the influence of
gravitation not through how you experience it.

Galileo: Not only that, I had already discovered the principle of
equivalence.

Einstein: Well, my discovery of the equivalence principle actually
came after I have finally understood the Newtonian law of gravity.

Newton: whispering to Galileo You are correct. This guy is an
idiot!

Einstein: I am the one who first suggested gravity as a curvature in
spacetime.

Riemann: I have already suggested that gravity is caused by curved
space. But since the mathematical concept of time and space forming
into a single set of coordinate was not yet discovered, I went
nowhere.

Hilbert: That is correct. Space can curve as much as it likes, but
as long as there is no curvature in the temporal dimension or
gravitational time dilation, there is no gravity.

Einstein: Never mind the curvature of spacetime, then. I personally
have derived the field equations and beat Mr. Hilbert to it.

Grossmann: Mr. Einstein, you know nothing about mathematics. There
is no way in hell you can come up with the field equations without a
massive amount of help --- helps like what I gave you without a single
ounce of gratitude while we, more like I alone, were developing the
“entwurf” to explain the laws of gravity through rigorous coordinate
transformations.

Einstein: Well, you failed, Mr. Grossmann. shrug

Christoffel: Before Mr. Einstein claim more credits, I was the one
who came up with the Christoffel symbols.

Ricci: Yes, all thanks to Mr. Christoffel, I was able to invent
something called the covariant derivative. Then, by taking the double
covariant derivative of the spatial distance between two points in
either space or spacetime, I was able to invent a 4-dimensional matrix
now called the Riemann curvature tensor. In space, it is a 3x3x3x3
matrix with 81 elements; in spacetime, it is a 4x4x4x4 matrix with 256
elements.

Riemann: I just want to clarify this. Although I was the first to
mathematically describe what the curvature of space is, the Riemann
tensor and Riemannian geometry have nothing to do with me.

Grossmann: Mr. Christoffel, besides the way you have grouped the
connection coefficients in which now are called the Christoffel
symbols of the second kind, there is another anti-symmetric
arrangement, and this other arrangement of the connection coefficients
would result in a different Riemann curvature tensor through a
different covariant derivative defined by Mr. Ricci.

Ricci: Ooops. There is another possible covariant derivative
different from what I have invented. Hey, nobody is perfect. shrug

Levi-Civita: But who cares? As long as the metric is diagonal, they
are the same. For no apparent reasons, I was able to reduce the 4-
dimensional Riemann tensor into a 2-dimensional matrix which is now
called the Ricci tensor.

Koobee Wublee: You guys are just playing in the sand box. On top of
these two covariant derivatives that yield two different Riemann
tensors, there are four ways to arrange the connection coefficients in
the results to the double operations in covariant derivative. Mr.
Ricci’s Riemann tensor is only one such possibilities even if the
metric is symmetric. shrug

Nordstrom: I agree with Mr. Levi-Civita. As long as we are only
discussing the diagonal metric, all these tensors should be
identical. So, I suggest that the null Ricci tensor would fully
describe gravitation.

Newton: clearing his throat

Hilbert: Well, it certainly would work in vacuum, thus the Laplace
equation. However, it does not explain the more general Poisson
equation. I have a better idea. I will throw in the square root of
the negative of the determinant to the metric into the Ricci scalar.
Demanding the action resulted from this Lagrangian to be stationary,
the result would be the set of field equations.

Einstein: No, the field equations are derived by me only.

Minkowski: Shut up, Albert. Remember that you were my worst
student. Yeah, instead of studying physics, you were thinking with
your dick and chasing after the only skirt in that class.

Weber: Oh, in my class of electromagnetism, Albert was drawing
woman’s blouse in his notes. I am deeply insulted by that.

Einstein: Hey, I had an idea about woman’s blouse. I even had a
patent on one.

Minkowski: I have a question for David. Why is the stationary
condition to this action necessary? How did you pin the significance
of the square root of the determinant to the matric to your
Lagrangian?

Hilbert: I don't know that myself. I fudged it through desperation
after Mr. Einstein told me that he had already derived the perihelion
advance of Mercury.

Einstein: What I had done had nothing to do with the field
equations. I was just using the same method as Mr. Gerber did by
modifying the Newtonian gravitational potential except that I modified
it differently.

Gerber: Yes, I merely modified the Newtonian gravitational potential
with speed dependent terms.

Hilbert: I did not know that. I should have known better. shrug

Newton: I could not believe that was how you clowns did physics in
the early twentieth centuries.

Klein: You have to excuse us, Mr. Newton. We were completely baffled
by the null results of the Michelson-Morley experiment.

Schwarzschild: Since the field equations are derived whether they are
valid or not, having a metric with a determinant of -1 would result in
drastically simplified field equations and thus the Ricci tensor. In
doing so, I have transformed the common spherically symmetric polar
coordinate into one that would yield -1 to the determinant of its
metric. Thus, trivially, I have discovered the very first solution to
the field equations that is static, spherically symmetric, and
asymptotically flat.

Hilbert: That is great, Karl. Here is another such solution which is
now named after you as the Schwarzschild metric. Since there are an
infinite such solutions to the field equations, what I have done must
be total nonsense. I regret getting involved such deeply in this
fiasco.

Einstein: Great! Now, I can claim sole proprietorship to the field
equations.

Self-styled physicists: Wow, folks. Do you see the Schwarzschild
metric manifests black holes? We could get a lot of attention and
grant money to perpetuate our welfare by elaborating on these heavenly
objects created through mathemagical nonsense that we do not even
understand ourselves.

Schwarzschild: But my original metric does not manifest black holes,
and so are other infinite solutions to the field equations.

Self-styled physicists: Who cares about how the math shows. Since
the metric fully describe the geometry regardless any coordinate
system. All solutions to the field equations must be the same.
shrug

Riemann: That is not what I have described of curved space. You guys
are just a whole bunch of clowns. The concept that the metric is the
geometry is fatally flawed. The mathematical description of the thing
called the metric alone cannot possibly be enough to describe the
invariant geometry. You need to specify the choice of coordinate
system as well. Each set of coordinate system would require another
unique metric to describe the same, invariant geometry.

Hilbert: applaud

Grade school children: Even, we can all understand what Mr. Riemann
is talking about.

College dropouts: Come on. The self-styled physicists have PhDs.
Although we cannot think for ourselves, we believe in the self-styled
physicists. Whatever they say must be correct despite we don't know
what they are saying.

Grade school children: But that is all wrong, we can still think for
ourselves without any poison from Einsteinian mysticism.

Einstein: While you guys are arguing about something I have no
understanding of, I have discovered the Cosmological constant. By
adding the Cosmological constant to the field equations, I can halt
the gravitational collapse.

Poisson: Yeah, I thought about the negative mass density in vacuum
too but dismissed it as stupidity of the utmost degree.

Newton: I have to agree with Mr. Poisson here.

Self-styled physicists: But the Cosmological constant deals with
energy not mass. So, you have negative energy, and that is OK we
think. shrug

Einstein: Being no good at all in mathematics, even I see the
stupidity in the Cosmological constant since (E = m c^2). My God, it
was the biggest blunder in my life. I am indeed a nitwit, a
plagiarist, and a liar. shrug

Michelson: Mr. Einstein, the Cosmological constant is the only
blunder in your life. shrug

Self-styled physicists: Even if Mr. Einstein is a nitwit, a
plagiarist, and a liar, we would continue to worship him. We love
these groundless speculations. We are still bedazzled by how he can
start with two equations equating zero with zero and pull out the
Lorentz transformation from these. Einsteinian mysticism must
continue, and Voodoo mathematics rules.

College dropout: Goody! Now we can have empty space that expands
itself, branes, multiverse, wormholes, black holes, budding universes,
etc.

Orwell: I told you so.

** FAITH IS THEORY
** LYING IS TEACHING
** NITWIT IS GENIUS
** OCCULT IS SCIENCE
** PARADOX IS KOSHER
** FUDGING IS DERIVATION
** BULL**** IS TRUTH
** BELIEVING IS LEARNING
** MYSTICISM IS WISDOM
** IGNORANCE IS KNOWLEDGE
** CONJECTURE IS REALITY
** PLAGIARISM IS CREATIVITY
** MATHEMAGICS IS MATHEMATICS

Self-styled physicists: Never mind Mr. Orwell. In the meantime, the
GPS represent the most triumphant prediction of General Relativity.
In order to allow the GPS to function, the satellite time must be in
total synchronization with the ground. That means the clocks on board
the satellite must tick a tiny bit slower to allow the slower ground
clocks to keep up.

Engineers: Gee! This is a myth perpetrated by the self-styled
physicists to promote the nonsense of General Relativity. It is not
the clock that has to be synchronized. The only requirement for the
GPS is to synchronize the chronological time among the satellites even
if the clocks on board the satellites are different, and even if the
ground clock or the ground chronological time is different. However,
it is must easier to synchronize the satellite clocks to achieve
universal synchronization of satellite chronological time.

College dropouts: Although we don’t understand all that, we have to
believe in the arm-chair designers of the GPS, namely the self-styled
physicists.

Engineers: Hey, look, punks. Any GPS receiver is receiving almanac
information from the satellites at a slow pacing rate of 50 bits per
second. The almanac information contains the position and the
chronological time (relative to all the satellites) of one satellite.
It takes several seconds to complete one record of information. With
an acquisition of four satellites, the GPS receiver can build a set of
four equations with the known positions and chronological times of the
satellites and the unknown position and the unknown chronological time
(relative to the satellites) of the receiver itself. With these 4
independent equations, all you have to do is to solve for these 4
unknowns. The chronological time of the ground does not come into
play in determining a person’s position. However, we do provide an
accurate universal time using the good old technique of “at the time
of the beep, the time will be blah blah blah”, and this should not be
extorted to promote the nonsense of General Relativity.

Einstein: Well, I don’t really understand any engineering
applications, but anyhow now I am worshipped as a god which not bad
for being a nitwit, a plagiarist, and a liar. Creativity is to know
how to hide your sources, and it pays off. wink