Re: [Phys-L] Question on special relativity

[John Denker <jsd@av8n.com>]

On 12/20/2015 11:11 AM, Ken Caviness wrote:

> The easiest way to solve the problem is to do it in the mirror's rest
> frame, and then find what the angles in the car's rest frame will be.
> We can be assured that the beam will go through both pipes, according
> to all observers.

Yes, that is certainly the "easiest" way to figure out what's
going on.

However ... if I understand the point of the original question,
the goal was to solve the problem in the frame of the car,
without switching (even temporarily) to another frame.  An
expert should be able to work in more than one frame, to see 
things from more than one point of view, and to reconcile the
various viewpoints.

This means we need to derive a formula for what happens when
something reflects off a moving mirror.  There is precedent
for this.  Similar equations in physics are known to be useful.
As a non-relativistic example, there is a formula for what it
looks like (in the lab frame) when a ball is hit by a bat that
is moving (relative to the lab frame).  Also all of the Doppler
formulas and aberration formulas tell us what things look like
in the lab frame, even when they would be simpler in some other
frame.

It turns out that the formula for reflection is astonishingly
simple when expressed in terms of geometric algebra.  The
reflection of any vector v is

        w = -a v a                 [1]

where a is a unit vector normal to the mirror.  The reason
for using geometric algebra is that you can play around with
equation [1] in three dimensions and build up confidence in
it ... and then extend it into four-dimensional spacetime and
mirabile_dictu, it just works.  As I have said before:  special 
relativity is the geometry and trigonometry of spacetime;  
nothing more and nothing less.

Just now I worked out about 80% of the details at:
  https://www.av8n.com/physics/rotations.htm#sec-reflection

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

You may say this is bad pedagogy, insofar as it violates the
principle that learning proceeds from the known to the unknown.
You can't explain relativity in terms of geometric algebra to
kids who don't know any geometric algebra.

Well, that may be a valid objection in the short run.  However:

 a) This is how I think of it, and it would it would take a 
  ton of work to solve the problem using less-powerful tools,
  which is more work than I feel like doing right now.

 b) Furthermore, geometric algebra really isn't that tricky.
  Just convince yourself that it's OK to multiply vectors,
  and start turning the crank.
     https://www.av8n.com/physics/clifford-intro.htm
  Normalization:
     x^2 = y^2 = z^2 = 1    but   t^2 = -1 in spacetime
  Mixed products are antisymmetric:
     xy = -yx  xz = -zx  tx = -xt  et cetera.

  That funny normalization in the timelike direction is 
  essentially the *only* thing that makes spacetime different
  from ordinary Euclidean space.
    https://www.av8n.com/physics/rotations.htm
    https://www.av8n.com/physics/spacetime-welcome.htm

 c) You get to use geometric algebra for lots of things,
  not just relativity.  For example, I can teach somebody 
  about bivectors *and* explain gyroscopic precession in 
  terms of addition of bivectors far more easily than I
  I could explain just gyroscopic precession, even assuming
  the students already had a good understanding of cross
  products and the right-hand rule, which they probably don't.
    https://www.av8n.com/physics/clifford-intro.htm#fig-add-bivectors
  If I never see another cross product, that would be just
  fine with me.

  The idea of having a /unified/ view of rotations, in any
  dimensionality from D=2 on up, is just too sweet to pass
  up.  And then there is the /unified/ view replacing both
  electric and magnetic fields by /one/ electromagnetic
  field bivector......
    https://www.av8n.com/physics/maxwell-ga.htm

  And the fundamental symmetry issues:
    https://www.av8n.com/physics/pierre-puzzle.htm