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Re: [Phys-L] gravitational waves



The link to the "Gravitational Waves" article:

http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/GravitationalWaves.pdf

was shared with us today (see below). Let me confess:

1) My eyes scanned it.

2) My brain was complaining that nothing is clear.

Am I the only one to in this way?

Ludwik

================================================


On Apr 12, 2016, at 11:57 AM, John Denker wrote:

On 04/11/2016 02:15 PM, Ludwik Kowalski wrote:

Why is calling g(x,t) "wave intensity" is not appropriate? Because
the quantity g(x,t) is gravitational acceleration, whose unit is
(m/s^2). But the unit of "wave intensity" (radio, light, sound,
etc.) is usually (W/m^2).

We agree that g(x,t) is not the intensity. We agree that
it doesn't even have the right dimensions for intensity.
Note that for present purposes it is better to think in
terms of dimensions rather than units. Units are important,
but it's a lot easier to change units than change dimensions.

One can make even stronger statements: Even if the vector
field g(x,t) did have the right dimensions, it still wouldn't
be the correct wavefunction, because it cannot possibly be
a solution to the gravitational equations of motion.

By way of analogy, consider an electromagnetic plane wave.
Focus attention on the E component. If you rotate your
point of view 180 degrees around the direction of propagation,
then E turns into -E. In other words, E is a vector.

To say the same thing the other way, if you imagined the
ordinate of the wavefunction to be a scalar, any attempt
to understand such a rotation would get the wrong answer.
It would be an impossible polarization.

Turning now to gravitational waves, the wavefunction can
be written h(x,t), where h is a second-rank tensor. If
you rotate your point of view 90 degrees (not 180), then
h turns into -h.

To say the same thing the other way, if you imagined
the ordinate of the wavefunction to be a scalar *or*
a vector, any attempt to understand such a rotation
would get the wrong answer.

Over the years (before and after GR came along) there have
been attempts to construct scalar and/or vector theories
of gravity, but nobody has managed to come up with one
that is consistent with observations.
https://en.wikipedia.org/wiki/Alternatives_to_general_relativity#PPN_parameters_for_a_range_of_theories

Again I say: Just because you can construct a model does
not mean it is faithful to reality. You can construct a
model where the earth is flat and Helios carries the sun
from west to east in a golden cup each night, but that
does not mean the model is consistent with observations.

Every minute spent studying the flat-earth model is at
least two minutes wasted, because it will have to be
unlearned.

Possibly useful lecture notes:
http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/GravitationalWaves.pdf

Those notes are not particularly pedagogical at the
introductory level. They skip more than 90% of the steps
in the derivations. If you want to see the derivations,
diagrams, and explanations, grab a copy of the Misner/
Thorne/Wheeler _Gravitation_ book. OTOH those notes might
be useful as a map for tourists, identifying the main
tourist attractions. For example, they tell you that
the wavefunction for a gravitational plane wave is a
tensor, and has transverse polarization.

Every minute studying the g(x,t) wavefunction is at least
two minutes wasted, because it will have to be unlearned.

==========

A subset of the steps in a reasonable pedagogical sequence
might include:

*) Basic rule: Show the work. Check the work. Show the checks.

*) Scaling laws. Application to strength of materials and
100 other things.

*) Conservation in general. Application to conservation of
charge, energy, momentum, angular momentum, et cetera.
Approximate conservation of chemical species.

*) Basic vectors. Applications to velocity vector, momentum
vector, acceleration vector, et cetera.

*) Longitudinal sound waves e.g. on a slinky, with a scalar
wavefunction, with application to acoustics.

*) Transverse sound waves e.g. on a slinky or Shive machine,
with a vector wavefunction, with application to seismology.

*) Basic bivectors and tensors, e.g. tensor of inertia,
angular momentum bivector, electromagnetic field bivector,
imaginary numbers as bivectors, quaternions, et cetera.
Non-commutative multiplication. Non-commutative rotations.
Application to gyroscopic precession.

*) Special relativity i.e. the geometry and trigonometry of
spacetime, including the idea or rotations in the xt plane.
Four-vectors. Spacetime diagrams. Review then extend basic
geometry, basic trig, and basic vectors.

*) Basic electromagnetism.

*) Electromagnetic plane waves in otherwise-empty space.
Bivector wavefunction, transverse polarization.

*) Geodesics in general. Connection between straight path
and shortest path. Principle of least action. Application
to navigation, classical physics, physical optics, quantum
mechanics, et cetera.

*) Gravitation in general. Emphasis on frame-independent
tidal stress as distinct from frame-dependent acceleration.
Principle of equivalence.

*) Connection between curved space and geodesic deviation.

*) Gravitational plane waves in otherwise-empty space.
Tensor wavefunction. Transverse polarization. The ordinate
of the wavefunction looks like tidal stress.

*) Wave equations with dispersion. Application to waveguides.

*) Spherical coordinates in general.

*) Outgoing sound waves from a point source. Scalar wavefunction.
Dispersion, even in media where plane waves would be non-dispersive.

*) Outgoing electromagnetic waves from a localized more-or-less
pointlike source. Liénard-Wiechert potentials. Larmor formula.

*) Outgoing gravitational waves from a localized more-or-less
pointlike source.

=============

That skips a lot of steps, and one could quibble about the details,
but the point remains that it would be madness to talk about a
point source of gravitational waves (disk-like or otherwise)
without sufficient foundation. A *lot* of foundation is required.

In physics it is common to have well-posed simple questions
that do not have simple answers. Asking about "gravitational
waves" is in this category. It may sound simple, but it's not.
It took Einstein several decades to figure this stuff out, even
with the help of the rest of the scientific community. Unless
you're waaaay smarter than everybody else, you're not going to
figure it out by guessing.

Also keep in mind that gravitational waves are quite low on the
list of priorities, because they are so very weak. Insofar as
they are a topic of current interest, they can be used as a hook
to motivate discussion ... but even so, the discussion should
emphasize concepts and principles that are useful in general,
rather than narrowly focusing on gravitational waves.
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