Clear writing about relativity
- From: Uncle Ben <ben@xxxxxxxxxxx>
- Date: Fri, 26 Jun 2009 09:41:14 -0700 (PDT)
Through an unsatisfactory exchange with colleagues
here recently I was reminded of the casualness of
expression of most of us who are not
mathematicians in using mathematical variables in
algebraic discussions of physical things.
Example: When I say, "Let x = vt, then ---", I
will be understood to mean one thing, until I go
on to say, "where x represents the mass of an
object of density v and volume t." Most physicists
in this newsgroup would be astonished and
displeased. A mathematician or logician wouldn't
care, but a physicist might insist on "M = dV" or
"M=\rho\ v.
Our conventions let us abbreviate our discouse and
remember our definitions. They cause no trouble in
simple cases.
But when we start talking about
several frames of reference and need symbols for
the coordinates in each, we have to improvise
symbols that fit our habits and yet distinguish
different versions of similar things. Nowdays we
use primed and double-primed variables, whereas in
earlier times when classical learning was assumed
among the intelligensia, we would use greek
letters or even hebrew or arabic letters.
Einstein's 1905 paper on relativity was translated
into english more than once with more than one
degree of accuracy. Some translations even
improved on the orginal by correcting small errors
or oversights. The paper is not difficult to read,
although what is said is quite unconventional to
the ordinary mind.
If we focus just on length contraction in Section
4, we find the derivation quite unfamilar to
students using modern textbooks. But if we edit
Einstein's words, using memorable terms and modern
rigor to resolve normally insignificant ambiguities --
in the minds of naive readers -- we may help these
readers comprehend the astonishing simplicity of
Einstein's demonstration.
For an example of better choice of terms, let us
describe a sphere moving with respect to a
laboratory. To describe the sphere we need
coordinates in which the sphere is at rest: the
"sphere coordinates" of S, of which there are four if
we include sphere time for greater generality.
We say that an event in S occurs as
described by an ordered set of four real numbers:
S( a, b, c, d), where the slots have physical
reference to time, east position, north, and up).
Thus in given units and an origin for S and a
compass, the expression S(1, 2, 3, 4) refers
unambiguously to a possible event.
The reader will appreciate that one might use
letters t, x, y, and z instead of a, b, c, and d,
but that is a human consideration, not a logical
one.
The sphere of radius R at rest in S can be
described as follows:
"For all real numbers t,x,y,z, and condition xx +
yy + zz = RR, if and only if the four variables
meet the condition, then the event they describe
in S occurs on the surface of the sphere."
Note that any t will do.
The same event can be described in the lab frame
L(,,,). In doing so, there is no reason to require
different variable letters. The variable letters
in the paragraph above are "bound" by the phrase
"For all t,x,y,z". The letters can be re-used in a
different context.
It is convenient to require that the coordinate
axes of S and L coincide at defined time zero in
both frames, and that the sphere is moving due
east, without loss of generality.
When we describe the relation between the
coordinates of an event in S and those of the same
event in L, we need eight different variables and
a real parameter v, the speed of the sphere in the
lab. (We use units in whch c=1.).
The relation between the coordinates of S and L is
the Lorentz Transformation, in which -1 <= v <= +1
is a parameter. Given the four coordinates of an
event in S, the LT will give us the coordinates of
the same event in L. I assume that the reader knows
this relation, derived by Einstein in Section 3.
With this preamble as an interpretive guide, we
can finish this example tersely.
Using the LT with parameter v, we have that the
object that is a sphere in S becomes the following
geometrical object in L, where g = 1/sqrt(1-vv).
The object, as a locus of events, is described in
L at lab time t as follows:
"For all real numbers t,x,y,z and condition
[g(x-vt)]^2 + yy +zz = RR,
if and only if the numbers satisfy the condition,
then the event L(t,x,y,z) occurs on the surface of
the object."
The object is moving, so to see the shape of the
object, we need a snapshot to freeze its position.
Without loss of generality, we choose t=0. And
since the only change in the condition from S to L
is in the second, or east, slot of L, let us set y
and z also to zero.
Thus in L,
x = R/g,
whereas in S under the same
conditions t=y=z=0,
x = R.
This is the Lorentz contraction.
What we have done is to remove the useless
distraction of different symbols for coordinates
in different frames, and have reduced the demand
on memory to say which frame is which.
I hope that at least one reader is helped by this
lengthy analysis of a simple problem. Since
absolute rigor is unattainable. I am sure some
professional logician can improve upon it.
Uncle Ben
.
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