Re: (d/dt) ACCELERATION

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

I'm sorry to resurrect a thread that seems to have been buried for a few
weeks.  However, when I was looking back through some old posts I came
across an old post of John Denker's from 4 weeks ago.  It then stimulated
me to make the following comment below concerning the question of the
possibility of a discontinuity in an object's acceleration which would
imply in an infinite jerk rate for the object.

On Sept. 16, 2000 John Denker quoted JLU:
> At 10:14 AM 9/14/00 -0500, Jack Uretsky wrote:
> > Sudden change of velocity implies infinite acceleration, which is infinitely
> > destructive and therefore unphysical.  Sudden change of acceleration
> > does not violate any physical principle, although it may be difficult
> > to achieve in particular cirmumstances.  So the displacement-time graph
> > need be only twice differentiable.
>
> I'm skeptical of this argument, for dimensional reasons among others.

He then gave his reasons which I skip here.  John continued:
> ...
> I know of no physical law that allows infinite jerk but forbids infinite
> acceleration.  The laws of electromagnetism essentially prohibit infinite
> acceleration of charged particles, but have nothing to say about neutral
> particles.  As far as I can tell, the laws of gravitational radiation do
> not forbid infinite acceleration, although they place severe restrictions
> on it.
>
> If somebody knows of a law that clearly forbids infinite acceleration (in
> the limit of small particles), please explain.

As I understand John's quotation here he seems to argue that a charged
particle cannot undergo an infinite acceleration because that would
result in an infinite EM radiation power emitted from the particle--power
which is simply not available for any physical process because (to
leading order in a localized charge distribution's multipole expansion)
the emitted radiant power in EM waves tends to be proportional to the
square of the second time derivative of that distribution's dipole
moment.  In the case of a single charged particle this means this 2nd
derivative of the dipole moment is just the particle's charge times its
acceleration, thus giving the radiant power a dependence on the square of
the particle's acceleration and on the square of its charge (i.e. the
Larmor formula for a nonrelativistic accelerated charge).  After
claiming that neutral particles would be exempt from this radiation-
induced restriction he expressed his doubts that gravitational radiation
would forbid an infinite acceleration.

Now, finally, to my comment.  We can use general relativity to calculate
the gravitational radiation from a time dependent localized distribution
of stress/energy/momentum.  As long as that localized distribution is
internally moving at a speed slow compared to c (and the relevant
gravitational potential is small compared to c^2) we can derive an
expression for the gravitational radiation from it which is exact to
leading order in multipoles and in powers of 1/c (actually this leading
order is 1/c^5 and the lowest order multipole contributing is the mass
quadrupole moment).  This formula is the gravitational analog of the
dipole formula for the EM radiation from a fluctuating electric dipole
whose parts are moving at nonrelativistic speeds (which occurs at order
1/c^3 for electromagnetic radiation).  This formula gives the power of
the emitted gravitational radiation from the particle as proportional to
the square of the *third* time derivative of the traceless mass
quadrupole moment tensor.  This gravitational radiation formula for the
radiant power P is:

P = (1/5)*(G/c^5)* tr(d^3q_i_j/dt^3)

Here d^3q_i_j/dt^3 is the third time derivative of the traceless mass
quadrupole tensor q_i_j. The trace tr is over the average of the
matrix product of the tensor with itself.  The constants G and c are
Newton's gravitational constant and the speed limit of causation
respectively.

The contribution to the 3rd time derivative of the mass quadrupole moment
by the motion of a given particle will have a part that is proportional
to the particle's jerk rate (in addition to other terms that go like
its acceleration, velocity, etc.).  This is analogous to how a charged
particle's acceleration contributes to the 2nd time derivative of its
electric dipole moment.  The result is that a particle undergoing an
infinite jerk rate will cause an infinite contribution to the radiated
power.  BTW, a uniformly moving charged particle does not radiate EM
radiation because it has no acceleration.  Uniform motion can be
transformed away by transforming to a comoving inertial frame in SR, but
radiation is present/absent in all inertial frames if it is present/
absent in any one of them.  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.  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.

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.

Of course this whole line of reasoning ignores any possible relevant
quantum effects that might serve to limit the jerk and/or the
acceleration in addition to the classical radiation effects discussed
here.  The whole picture presented here is one of classical continuous
matter & classical continuous fields interacting with it.

David Bowman
David_Bowman@georgetowncollege.edu