Re: (d/dt) ACCELERATION

[John Denker <jsd@MONMOUTH.COM>]

At 06:22 AM 10/12/00 -0400, David Bowman wrote:
> This gravitational radiation formula for the radiant power P is:
>
> P = (1/5)*(G/c^5)* tr(d^3q_i_j/dt^3)                            [1]

1a) That doesn't pass the dimensional-analysis test.
1b) It doesn't express the following "product" operation:

> The trace tr is over the average of the matrix product of the tensor with
> itself.

2) I think you need to take the time derivatives before taking that product.

> Here d^3q_i_j/dt^3 is the third time derivative of the traceless mass
> quadrupole tensor q_i_j.

3) Uhhh, if it's traceless, why are we taking its trace in equation [1]?

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

How about this:

        P = (1/5) (G/c^5) average([ (d/dt)^3 Itick ]^2)         [2]

where Itick would be typeset as an I with an overstrike tick-mark, and
represents the "reduced quadrupole moment":

        Itick_j_k := I_j_k - (1/3) delta_j_k trace(I)

and where I is just the ordinary second moment of the mass
distribution.  Reference: Misner, Thorne, Wheeler equations 36.1 and 36.3.

> The constants G and c are Newton's gravitational constant and the speed
> limit of causation respectively.

Speed limit of causation.  I like that.  I wonder if it will catch on.

> ...  Similarly, a uniformly accelerated massive
> particle in an otherwise empty spacetime doesn't emit gravitational
> radiation in GR since its motion can be transformed away to a comoving
> frame for which the external gravitational field vanishes.

4) That's hauntingly close to the right explanation, but not quite there.

4a) A uniformly-accelerated particle has a position that goes like t^2,
hence a quadrupole moment that goes like t^4.  This is not entirely
annihilated by the jerk operator (d/dt)^3.  So equation [2] says that a
uniformly-accelerated particle can radiate ... assuming equation [2]
applies;  see item (4c) below.

> In this frame there is no gravitational radiation, so there is not any
> such radiation in any other frame either.  This is why the jerk rate is so
> important in inducing gravitational radiation for massive particles.

A better argument is to say that the jerk operator is necessary in equation
[2] to ensure that two masses moving past each other in uniform
_un_accelerated motion do not radiate.  Such motion has a position that
goes like t and a quadrupole moment that goes like t^2, which is entirely
annihilated by the jerk operator and not by any lower-order derivative.

Note that the analogy to electromagnetic radiation is misleading in some
ways:
 -- The EM source term is the dipole moment, which is linear in the
coordinates, but the gravitational source term is the quadrupole moment,
which is nonlinear in the coordinates.
 -- In EM, we usually assume overall charge neutrality.  This muddies the
equivalence principles, since if we are calculating the radiation from a
negatively-charged object, we get to ask if it is moving (or accelerating)
relative the the corresponding positive charges.

[again]
> ... a uniformly accelerated massive
> particle in an otherwise empty spacetime doesn't emit gravitational
> radiation in GR since its motion can be transformed away to a comoving
> frame for which the external gravitational field vanishes.

4b) "Comoving" isn't the right word.  One would need a "co-accelerating"
frame to make that argument work.  Consider the contrast:
 -- Galileo's principle of equivalence says we are always free to
transform to a comoving frame of our choice, and the physics is unchanged.
 -- We are not quite so free to transform to a co-accelerating frame of
our choice.  The transformation produces a gravitational field in the new
frame.

It's not clear that equation [2] is valid in an accelerated frame (or the
equivalent gravitating frame).  The corresponding argument about
co-accelerated frames applied to the EM case would predict zero Bremsstrahlung.

4c) It is 100% clear that equation [2] is _not_ valid for a uniformly
accelerated particle.  The equation was derived under the assumption of a
closed system with "internal" motions.  To see that it is not valid, let
the particle have a velocity as well as an acceleration: r = vt + .5 a t^2
and show that the amount of radiation depends on velocity -- clearly a no-no.

I haven't worked out the details, but my intuition is that there _can_ be
such a thing as gravitational Bremsstrahlung, i.e. an accelerated particle
_can_ radiate... and somebody in a co-accelerated reference frame should
come to the same conclusion.

This is what I was referring to in my earlier post when I opined that the
laws of gravitational radiation limit our ability to apply a large
acceleration to a particle.

OTOH I said the laws don't clearly forbid such accelerations.  I was
thinking it may be possible to arrange that Itick is zero (perhaps by
accelerating something else in the opposite direction to compensate).  But
this is not the general case.

> Thus it seems that whatever reasoning that would forbid a charged
> particle from undergoing an infinite acceleration because of its emitted
> EM radiation would also seem to apply to any localized distribution of
> mass trying to undergo an infinite jerk among its parts with the only
> difference being that the mass emits gravitational radiation and the
> charged particle emits EM radiation instead.

I don't buy that.  I think it doesn't account for the fact that Itick is
nonlinear in the coordinates, as mentioned above.