Re: [Phys-L] GR waves

[John Denker <jsd@av8n.com>]

On 04/23/2016 06:48 AM, Ludwik Kowalski wrote:

> Thanks for the link, Alex. and for the introductory comment. 

> Yes, the book is not written for those who want to learn G.R.

Still, though, there are some nice tidbits hiding in there.  For one
thing, it calls attention to the supermirrors.  All to often this
point is not explained;  indeed it is all too easy to find descriptions
of the LIGO machine that are so simplified that they don't mention the
etalon at all;  they just imagine a plain-vanilla Michelson interferometer,
for example:
  https://www.learner.org/courses/physics/unit/text.html?unit=3&secNum=7

The mirrors are a very big deal.  One might imagine that increasing the
reflectivity from 99% to 99.9% would be less than a 1% improvement, but
really it is a lot more than that;  it is a tenfold improvement.

The finesse of the LIGO resonator is something like 400, which is just
insanely hard to do.

========

OTOH there are dozens of details that those articles get wrong.  It just
cracks me up every time I read something like this:

> > The light waves are carefully misaligned so that their troughs 
> > and crests interfere to create dark fringes, ensuring that no light 
> > hits LIGO’s photon counter.

That appears to be part of the "standard" description of the machine.

OTOH, nobody I know is dumb enough to operate an interferometer that
way.  Think about it:  If you're sitting on a dark fringe and the 
mirror moves in the x direction, you get a signal.  But you get the
same signal for +x and -x.  So it's an /even/ function of x.  Now if
x is small, the signal is small squared, which is insane.  Also the
signal is at twice the frequency of the physics, which is also insane.

Everybody operates off resonance, so there is some signal at the
resting x value.  A change in the -x direction makes the signal
smaller, and a change in the +x direction makes the signal bigger.
 
> 1) The caption on page 3, below the left-hand illustration, refers to
> the "Earth’s gravity warped the space-time, presumably 4-dimentional
> [?]"

If you cannot make sense of that diagram, good for you;  it shows you
are paying attention.

There are several problems:
  a) The figure is trying to represent the static gravitational field
   (not waves).  It cannot easily be generalized to represent waves,
   for the same reason that the Coulomb law cannot be extrapolated
   to explain electromagnetic waves.

  b) The figure doesn't even do a very good job of representing the
   static field.

> 2) I can imagine three mutually perpendicular space axes x, y and x
> axes, (on that illustration) but not the time axis t.

Figuring out what's going on in the timelike direction is absolutely
essential.  It's the key.  For the static gravitational field, e.g.
for particles in orbit around the earth, the only curvature that
matters is the curvature in the timelike direction.

This is diagrammed at
  https://www.av8n.com/physics/geodesics.htm#fig-darts
and almost equivalently
  https://www.av8n.com/physics/geodesics.htm#fig-computer-darts-xt-16

I added a long discussion of why "pit" diagrams and "trampoline" models
produce misleading ideas about orbits.  The pit diagram by itself is
not necessarily wrong -- certainly a Flamm paraboloid is not wrong --
but it's mostly irrelevant and highly misleading.  For one thing, the
spacelike orbital circles that you would like to be geodesics are not
geodesics.

At a more fundamental level, the whole idea of things moving in the xy
plane makes no sense in relativity.  For particles in terrestrial orbit,
the t-component of the 4-velocity is enormous compared to the spacelike
components.  Again:  For particles in orbit, the only curvature that
matters is the curvature in the t-direction ... which the "pit" diagram
does not even attempt to show.

An orbit in spacetime is a helix.  If you project that onto the xy 
plane, you get a circle ... but the circle is not a geodesic.  In
general, the projection of a geodesic is not itself a geodesic.

This is reason #374 why students need to get up to speed on modern
(post-1908) concepts of special relativity before they go anywhere
near general relativity.  Archaic (pre-1908) notions of clocks that
can't be trusted, rulers that can't be trusted, velocity-dependent
mass, etc. are not gonna get it done.  Just because Einstein did
things that way in 1905 doesn't make it OK.

> 3) Should we think that the shape of the space-time surface (the
> shape of the grid) is not static, that it changes, as a function of
> time?

For a static gravitational field, e.g. for particles in an ordinary
orbit, the field is not changing in time.  That's what we mean by
static.

OTOH gravitational waves, by definition, are time dependent.  They
are very much more complicated than the static gravitational field
... and cannot be figured out by extrapolating from the static field.

So the pit diagram is at least two jumps removed from telling you
anything about the propagation of gravitational waves.

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

One more bit of advice:  Don't even try to visualize four-dimensional
spacetime, at least not at first.  Start with two dimensions (x and t).

Most people have a hard time visualizing spatial relationships in
three spacelike dimensions.  Going directly to four dimensional
spacetime would be terrible pedagogy.

Also don't try to visualize the xt plane in your head. Draw lots of
spacetime diagrams.

Gravitational waves are inherently four dimensional (time, direction
of propagation, and transverse polarization).  However, to a first
approximation you can separate the propagation from the polarization,
so it's not as bad as it could be.

The waves run along 45 degree diagonals in spacetime diagrams.