Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-L] phase velocity, one kind of charge, and other intangibles



On 05/09/2017 08:08 PM, Stefan Jeglinski wrote:

Two spaceships travel toward one another at speeds of 0.7c and 0.8c,
each wrt to an inertial Earth observer – at what speed does each
spaceship observer see the other spaceship approach? Answer is 0.96c,
maintaining the necessary result that “no object can travel at the
speed of light.”

OK.

However, wrt Earth observers, the closing speed of
the two spaceships is in fact 1.5c.

I would play those cards differently.

1) I suggest adding the /rapidities/, as follows:

rapidity velocity
θ tanh(θ)

0.87 ← 0.70
1.10 ← 0.80
-----
1.97 → 0.96

(velocities in units where c=1)
(rapidities in radians)

The physics point is, the rapidities just add linearly. There's
no muss, no fuss, and no questions about the interpretation.

The pedagogical point is, I like to start out by emphasizing
what works. Later we can discuss various things that don't work.

2) We can back that up with the following parable. It involves
wedges that have an /angle/ as well as a slope.

2a) Suppose we have a very keen wedge with a slope of 1-in-100
aka 0.01, and another wedge with a slope of 2:100 aka 0.02. If
we stack one on top of the other, the result has a slope of 3:100
aka 0.03, very nearly. So you might imagine that slopes are
additive. HOWEVER.....

2b) Suppose we have a wedge with a slope of 1-in-4 aka 0.25, and
another wedge with a slope of 1:2 aka 0.5. If we stack one on
top of the other, the resulting slope is not 0.75 but rather
0.86. There is some nonlinearity.

angle slope
θ tan(θ)

0.245 ← 0.25
0.464 ← 0.50
-----
0.71 → 0.86

(angles in radians)

2c) Now suppose we have two wedges with unit slope (45°) and stack
them atop one another. The slope is not 2:1. It's rather bigger
than that. It's infinite.

angle slope
θ tan(θ)

0.785 ← 1.00
0.785 ← 1.00
-----
1.57 → ∞

You can demonstrate this in class, hands-on, using wooden wedges.

The point here is simple: The idea that slopes are additive is a
misconception. You might get away with it if the slopes are very
small, but not otherwise. If you're not careful, you could be
wrong by a factor of infinity. The idea of "slope plus slope"
is just a bad idea.

In contrast, the idea of "angle plus angle" works just fine. Given
some slopes, convert to angles and add the angles. If necessary,
convert back to slope at the end.


By the same token, the idea that velocities should be additive
is a misconception. You might get away with it if the velocities
are very small, but not otherwise. There is no such thing as
"closing velocity" for the same reason there is no such thing
as "slope plus slope".

In contrast, the idea of "rapidity plus rapidity" works just fine.
Given some velocities, convert to rapidities and add the rapidities.
If necessary, convert back to velocity at the end.

Let's be clear:
rapidity is to velocity
as
angle is to slope.

There is nothing accidental about this analogy. A boost in the
x-direction (i.e. a change in x-velocity) is just a rotation in
the xt plane. You have to use hyperbolic trig functions instead
of the more familiar circular trig functions, but that's all.

You can approximate tan(θ) by θ for small angles, but not otherwise.
Rotations (including boosts) are linear when the angle is small,
but not otherwise.

3) As a separate matter, if you have some skill with vectors,
nine times out of ten you can structure the calculation so that
you never have to calculate a slope or an angle. You can just
work with the vectors. The orientation of the vectors can be
expressed in terms of dot products, wedge products, et cetera.

To say the same thing another way: Trig functions show up when
you use polar coordinates. You can (mostly) avoid them by using
Cartesian coordinates, or by using no coordinates at all, just
abstract vectors.

I was taught
"The goal is not to learn how to do Lorentz transformations;
the goal is to learn how to avoid doing Lorentz transformations."

This is the modern (post-1908) way of doing relativity.

==========

Bottom line: The idea of "closing velocity" is tempting, but it's
based on a misconception. It's the sort of temptation that ought
to be resisted. If you want to add something, add the rapidities
instead (or add the spacetime vectors).

For more on all of this, see
https://www.av8n.com/physics/spacetime-welcome.htm
especially
https://www.av8n.com/physics/spacetime-welcome.htm#sec-steady-accel