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Re: [Phys-L] Question about Gravitons



Let's work up to it, by a sequence of analogies:

0) Suppose we have a spherical balloon in the middle of a large
room. Now suppose the balloon suddenly gets bigger, perhaps
because of a chemical reaction occurring inside. This will
launch a spherically-symmetrical sound wave into the room.

If we rotate the balloon by some angle φ, the result will be
the same in every respect. Therefore if we describe the
wavefunction as being proportional to exp(i m φ), the value
m=0 is appropriate.

0.5) Suppose we have a fermion such as an electron. If we
rotate it by 2π radians, the wavefunction changes sign. We
would need to rotate it by a total of 4π radians to get the
wavefunction back to where it started. More generally, we
can write a wavefunction that is proportional to exp(i m φ)
where the value m=0.5 is appropriate.

1a) If we return to scenario (0) and put some static electric
charge on the balloon, expanding the balloon will /not/ launch
an electromagnetic wave. Absolutely not. It's impossible by
symmetry.

1b) As a partly-related but partly-different argument leading
to the same conclusion, you cannot launch an electromagnetic
wave by changing only the monopole moment of the charge
distribution, because there is no way to change only the
monopole moment without violating conservation of charge.

1c) In the multipole expansion, the lowest-order term that
can possibly radiate is the /dipole/ moment. Even that is
subject to restrictions, because two equal-and-opposite
charges moving past each other (each moving in a straight
line through spacetime) would have a nonzero rate-of-change
of the dipole moment, but cannot possibly radiate. This
should be obvious by application of Galileo's principle
of relativity (plus superposition) if nothing else. So
now the lowest-order thing that could possibly radiate is
the /second/ time derivative of the dipole moment.

If you rotate the dipole by 2π radians, you get back the same
dipole, and also the same radiation field. We can write a
wavefunction proportional to exp(i m φ), where the value m=1
is appropriate.

2a) You can't have a mass distribution in the form of an
oscillating dipole. That would correspond to an oscillating
center of mass. That would violate conservation of momentum.

2b) A mass moving in a straight line through spacetime cannot
radiate gravitational waves. This should be obvious as a corollary
of Galileo's principle of relativity if nothing else. Similarly,
two masses moving past each other (each moving in a straight line)
cannot radiate. Note that in such a system, the quadrupole moment
of the mass distribution is changing. It is a quadratic function
of time.

2c) So ... the lowest-order thing that could possibly radiate is
the /third/ time derivative of the reduced /quadrupole/ moment of
the mass distribution. This radiates just fine. It radiates
gravitational waves.

If you rotate a quadrupole π/2 radians, it changes sign. If you
rotate it π radians, you get back what you started with. We can
write exp(i m φ), with m=2.

2d) As another line of argument leading to the same conclusion,
let's expand the gravitational potential in a Taylor series
(instead of in a multipole expansion). The term with vector
symmetry (i.e. the garden-variety gravitational field) can be
made to go away by exercise of Einstein's equivalence principle.
The lowest-order thing that could possibly have any frame-independent
meaning is the next term in the Taylor expansion, i.e. the tidal
stress term. It has quadrupole symmetry. The equations for this
are spelled out at
http://www.av8n.com/physics/tides.htm#eq-Taylor-pot

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

That's basically it. You could make it more complicated than that,
but there's no real need to.

The astute nitpicker will have noticed that I have glossed over the
distinction between the eigenvalues of S^2 (conventionally called "the"
spin) and the eigenvalues of Sz (also conventionally called "the" spin).
This is fixable.