🚀 go-pugleaf

On Jun 9, 4:28 am, "Bill Hobba" <rubb...@junk.com> wrote in
sci.physics.relativity:
> Hi All
>
> On a whim I purchased the following book (basically to see what they do in
> an introductory honours mechanics course at Harvard) http://www.amazon.com/Introduction-Classical-Mechanics-Problems-Solutions/dp/0521876222/
>
> What a fantastic book.  I have long been a huge fan of Landau - Mechanics
> liking it even over Goldstein which I never really saw the need to actually
> buy.  Morins book is in many ways better.  It includes most of the material
> Landau does and relativity all done at a level that an advanced high school
> student could understand (it explains gauss theorem etc - all you really
> need is introductory calculus).  Even includes the statement and proof of
> Noethers theorem that Landau omits.  The treatment of relativity is just as
> good - it includes a treatment omitting the second postulate which is my
> favourite method.  And for what you get the price is great.  Professors
> could do a lot worse than use this book in any introductory mechanics
> course - even at HS level.
>
> I love it.
>
> Thanks
> Bill

You should have thought before falling in love. Consider this:

http://www.amazon.com/Introduction-Classical-Mechanics-Problems-Solutions/dp/0521876222/
Chapter 14: "The equivalence principle has a striking consequence
concerning the behavior of
clocks in a gravitational field. It implies that higher clocks run
faster than lower
clocks. If you put a watch on top of a tower, and then stand on the
ground, you will
see the watch on the tower tick faster than an identical watch on your
wrist. When
you take the watch down and compare it to the one on your wrist, it
will show more
time elapsed."

Now remember:

http://aps-ca.lbl.gov/abstracts-2008.html
"The prediction of a new relativistic gravitational effect that can be
measured by the NASA Lunar Reconnaissance Orbiter (LRO) Spacecraft.
Alexander Mayer Pritzker Fellowship
In Volume II of the Feynman Lectures on Physics (Chapter 42-6) Feynman
presents a gedanken experiment in which clocks in the nose and the
tail of an accelerating rocket demonstrate the phenomenon of
gravitational time dilation. The argument rests on only two
principles: THE CONSTANT VELOCITY OF THE SPEED OF LIGHT and the
Equivalence Principle."

So you can "demonstrate the phenomenon of gravitational time dilation"
if the speed of light in a gravitational field is CONSTANT. Is it? See
this:

http://www.physlink.com/Education/AskExperts/ae13.cfm
"So, it is absolutely true that the speed of light is not constant in
a gravitational field [which, by the equivalence principle, applies as
well to accelerating (non-inertial) frames of reference]. If this were
not so, there would be no bending of light by the gravitational field
of stars....Indeed, this is exactly how Einstein did the calculation
in: 'On the Influence of Gravitation on the Propagation of Light,'
Annalen der Physik, 35, 1911. which predated the full formal
development of general relativity by about four years. This paper is
widely available in English. You can find a copy beginning on page 99
of the Dover book 'The Principle of Relativity.' You will find in
section 3 of that paper, Einstein's derivation of the (variable) speed
of light in a gravitational potential, eqn (3). The result is,
c' = c0 ( 1 + V / c^2 )
where V is the gravitational potential relative to the point where the
speed of light c0 is measured."

http://www.speed-light.info/speed_of_light_variable.htm
"Einstein wrote this paper in 1911 in German (download from:
http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1911_35_898-908.pdf).
It predated the full formal development of general relativity by about
four years. You can find an English translation of this paper in the
Dover book 'The Principle of Relativity' beginning on page 99; you
will find in section 3 of that paper Einstein's derivation of the
variable speed of light in a gravitational potential, eqn (3). The
result is: c'=c0(1+phi/c^2) where phi is the gravitational potential
relative to the point where the speed of light co is measured......You
can find a more sophisticated derivation later by Einstein (1955) from
the full theory of general relativity in the weak field
approximation....For the 1955 results but not in coordinates see page
93, eqn (6.28): c(r)=[1+2phi(r)/c^2]c. Namely the 1955 approximation
shows a variation in km/sec twice as much as first predicted in
1911."

http://www.mathpages.com/rr/s6-01/6-01.htm
"In geometrical units we define c_0 = 1, so Einstein's 1911 formula
can be written simply as c=1+phi. However, this formula for the speed
of light (not to mention this whole approach to gravity) turned out to
be incorrect, as Einstein realized during the years leading up to 1915
and the completion of the general theory. In fact, the general theory
of relativity doesn't give any equation for the speed of light at a
particular location, because the effect of gravity cannot be
represented by a simple scalar field of c values. Instead, the "speed
of light" at a each point depends on the direction of the light ray
through that point, as well as on the choice of coordinate systems, so
we can't generally talk about the value of c at a given point in a non-
vanishing gravitational field. However, if we consider just radial
light rays near a spherically symmetrical (and non- rotating) mass,
and if we agree to use a specific set of coordinates, namely those in
which the metric coefficients are independent of t, then we can read a
formula analogous to Einstein's 1911 formula directly from the
Schwarzschild metric. (...) In the Newtonian limit the classical
gravitational potential at a distance r from mass m is phi=-m/r, so if
we let c_r = dr/dt denote the radial speed of light in Schwarzschild
coordinates, we have c_r =1+2phi, which corresponds to Einstein's 1911
equation, except that we have a factor of 2 instead of 1 on the
potential term."

Pentcho Valev
pvalev@yahoo.com