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Re: [Phys-L] neutrons

On 2/9/21 10:14 AM, David Ward wrote:

Here's the student question: If a neutron were made to move, would
its electromagnetic nature result in EM wave propagation?

In addition to my previous answer, here is how I actually think
about it. Depending on the student, this may or may not be how
I explain it. So let me explain it to *you*, and delegate to you
the decision of what to say to the students.

Short answer: The question almost answers itself. The key concept
is *electromagnetic* field.

That's key because there is no such thing as space or time; there is
only spacetime. By the same token, there is no such thing as an
electric or magnetic field; there is only the electromagnetic field.

To set up an analogy, let's review some basic geometry: Start out
with a long thin needle, represented by a vector aligned with the
X direction.

0) To zeroth order, it just sits there. It has some extent in the X
direction, but no extent in the Y direction.

1) Now if you rotate it a tiny bit in the XY plane, to first order it
picks up some extent in the Y direction. To first order the extent
in the X direction is unchanged.

2) To second order, the extent in the X direction changes also.

The three numbered items are the first three terms in the Taylor
series for the rotation matrix.

Tangential remark: I could write a couple of pages about all
the fun you can have with rotation matrices, the Lie derivatives
thereof, the generator of rotations, and exponentials and power
series thereof. It's ridiculously simple if you do it right. It's
a package of several ideas that hang together nicely.


Now apply this to the spacetime momentum vector of a particle.
This is just the mass times the spacetime velocity:
p = m u = m d(position)/dτ

where τ (tau) is the proper time. This could not possibly be any simpler.

0) To zeroth order, in our lab frame, the particle just sits at rest.
It has some energy proportional to mass, and that's it. The components
in our [t, x, y, z]@lab directions are:

p = m [1, 0, 0, 0]@lab

1) If you give it some velocity, to first order it picks up some
momentum in the spacelike direction. To first order, the energy
is unchanged.

Seriously: The old-fashioned spacelike momentum is just mass, rotated
a little bit in the xt plane.

2) To second order, there is a change in energy. We call this the
/kinetic/ energy.

These are the first three terms in a Taylor series......


The same goes for electromagnetism:
— The thing we formerly called a electric field in the x direction
is really a bivector in the xt plane.
— The thing we formerly called a magnetic field in the x direction
is really a bivector in the yz plane.
— These are the *same thing* except for direction.

There is only one electromagnetic field, F. It exists as a physical
thing unto itself. It *might* have components such as F_xt and/or
F_yz and/or others, but the components exist only relative to some
chosen frame. F itself exists independently of whatever frame
(if any) we have chosen.

If you have something that has a F_xt component (which we formerly
called Ex) and you give it some velocity in the y direction, that
velocity is just a rotation in the ty plane. The field *will* pick
up a component in the xy direction (which we formerly called Bz).
In symbols xt ⋅ ty = xy.

And now (!) to answer the question that was actually asked: If you
start out with something that has a F_yz component (which we formerly
called magnetism) and give it some changing velocity in the y direction,
it *will* pick up some extent in the tz direction (which we formerly
called an electric field). yz ⋅ tz = - yz. So there will be a changing
electromagnetic field. It will radiate.

The math here is related to gyroscopic precession, i.e. changes to
the angular momentum bivector. A useful diagram is here:

I can visualize these things in my head, but when explaining them
to others it reeeeally pays to draw the diagram ... or even to bring
a hands-on demo, i.e. a couple pieces of cardboard so you can add the
bivectors edge-to-edge (just as you add vectors tip-to-tail).


Note that the scalar part of (F F~) is a relativistic invariant,
which we formerly called E² - B². For a static field, it will tell
you whether the field is "mostly timelike" or "mostly spacelike"
i.e. mostly electric or mostly magnetic.

However ... for the radiation field, it's zero. The radiation is
exactly 50/50 electric/magnetic.

This means that for launching EM radiation, a changing electric
field is exactly as good as a changing magnetic field. Provided
you measure things in the appropriate units, the equations do not
play favorites.


Special relativity is the geometry and trigonometry of spacetime,
nothing more and nothing less.

Von Stund’ an sollen Raum für sich und Zeit für sich
völlig zu Schatten herabsinken
und nur noch eine Art Union der beiden
soll Selbständigkeit bewahren.

Henceforth, space of itself and time of itself
shall sink into mere shadows
and only a kind of union of the two
shall maintain its independence.

— Hermann Minkowski (1908)