Symmetry and homogeneity.
glird
>
>glird wrote: < Consider a spherical object moving along the X axis at v. Suppose it deforms by phi(v)q in X and by phi(v) in Y and Z. >
Tom: <Then in GR you must include the mechanism of deformation in the
energy-momentum tensor. That gives you a completely different problem
to study, and it's not obvious how to compare two such different
situations....>
In the LTE of SR the mechanism was spelled out by Lorentz in his
1904 paper.
glird: <For any value of q, there are only two values of phi(v)
for which the object will deform asymmetrically. >
Tom: <Huh? If q != 1 that deformation is not spherically symmetric,
for any nonzero phi(v). >
RIGHT!!! I should have said, There are only two values
for which the object will deform asymmetrically.
Tom: <There are many different types of symmetry.
You must be more specific. >
So must YOU. To me, in order for a deformation to be symmetrical,
the general shape of the object must remain constant regardless of its
size. for example, if it is an oval that expands for any reason, it
mut remain an oval with the same proportion between its long and short
radii no matter how large or small it ends up.
Tom: <For instance, for any nonzero values of q and phi(v)
that deformation is cylindrically symmetric with axis along X.
For q=1 is it also spherically symmetric. >
For phi(v) = 1, then q = sqrt(1-v^2/c^2); which means that the
deformed object IS cylindrical. BUT!! if v changes, then the length
of the cylinder changes in X but NOT in Y or Z. The deformation is
therefore Asymmetrical.
For q = 1, then phi)v = 1/sqrt(1-v^2/c^2); which means that the
deformed object is cylindrical. BUT if v changes, then the length of
the cylinder changes in Y and Z but NOT in X! The deformation is
asymmetrical again!!
glird wrote: <One of them is phi(v) = 1. THAT is its value in the LTE.
>
Tom: <Huh?? Lorentz transforms do not involve the deformation of any
object. They describe variations in geometrical projections, not
deformations. >
You are old enough to remember that prior to circa 1980, when
Wheeler invented the {uncaused} "rotations" of the X and t axes of
moving systems, the LTE not only involved deformations of moving
objects, they were based on and imposed them.
Tom: <Just THINK about it: how could an observer moving past an object
possibly affect the object itself? >
Think about this, Tom. A few years before Wheeler invented
rotations as the cause of the projected length and rate changes I was
talking to a friend who was a Math Professor at Rutgers -- which is
only a few miles from Princeton where Wheeler lived. When my friend
said something about the mass of an object being a function of its
velocity, I scoffed, "It is impossible for the quantity of matter to
be a function of who is looking at it". The Professor, who smilingly
agreed, probably asked his fellow profs at Rutgers that question;
which may have gone on to Princeton etc etc etc.
Tom: <How could choice of coordinates possibly affect any
physical phenomenon, including shapes of objects? .
Thank you for asking that question in the way that you did, Tom.
It shows me that you are one of the myriad contributors to these
newsgroups who doesn't understand Einstein's 1905 STR paper
nor the LTE he failed to derive.
Tom: <Looking at an object cannot possibly "deform" the object, and
that's what different inertial frames are -- different ways of looking
at THE SAME world. Lorentz transforms merely relate measurements
in one inertial frame to those in another such frame. >
The key to your failure to understand how the LTE work is the word
"merely". Delete it, and we have,
"Lorentz transforms relate
measurements
in one inertial frame to those in another such frame".
The new key word is "measurements". Put that into your head '
and rewrite your statement like this:
Looking at an object cannot possibly "deform" the object, although
that's what different inertial frames do -- they use different ways of
MEASURING the SAME world. The Lorentz transforms relate measurements
by one inertial esynched frame to those a differently moving such
frame would obtain for a given event.
Glird: <Note, then, that "reasons of symmetry" are ruled out 'by phi
(v) = 1!
Similarly, since the space and time of STR and GR deform
asymmetrically, the space-time continuum cannot be homogeneous. >
Tom: <You are confused. In neither SR nor GR do space and time
"deform", symmetrically or asymmetrically. But yes, in GR the
spacetime manifold is not homogeneous (except for unphysical
situations like the Minkowski spacetime of SR)..
I'm glad you understand that the space-time invented by Minkowski is
an "unphysical" mathematical abstraction that doesn't physically exist
in the real world.
Tom: <In {WHEELER'S} SR, observers at rest in different inertial
frames see different PROJECTIONS of the world as their personal views
of space and time. To each observer, space is isotropic and
homogeneous and time is homogeneous. But they see space and time
DIFFERENTLY. That is, for each observer the points that form the locus
of 3-space at a given instant in time are different for the two
observers, but each sees their locus as both isotropic and
homogeneous. Similar remarks apply to time. IOW: These two observers
are foliating {FOLDING?} 4-d spacetime manifold into DIFFERENT
submanifolds of (3-d) space and time. Here's an example:
Consider 3-d Euclidean space, and construct Cartesian coordinates
{x,y,z} on it. The 2-d projection onto x=0 is isotropic and
homogeneous. Now construct Cartesian coordinates {x',y',z'} rotated
relative to the first set. The 2-d projection onto x'=0 is isotropic
and homogeneous. These two projections are QUITE different, yet are
both isotropic and homogeneous. A similar thing happens for those
observers making 3-d projections onto t=0 and t'=0 -- both 3-d
submanifolds are isotropic and homogeneous, yet they are different. >
From steps 1 and 3 we have a Cartesian co-ordinate system (x,y,z)
and a Cartesian cs (x',y',z') whose axes are rotated relative to the
first one.
NOW TRY THIS: Rotate the axes of cs 2 back enough that they are
COINCIDENT with those of cs 1; and THEN let them measure given events.
Since you stipulated that that these are Cartesian systems, the
transformations will be the same Euclidean ones Newton used. If you
want to understand how the non-Euclidean, non-Cartesian co-ordinate
systems of the LTE work, try this:
We have from _modified_ steps 1 and 3, a non-Euclidean co-ordinate
system (x,y,z) and a non-Cartesian cs (x',y',z') whose axes are
rotated relative to the first one, Now rotate the axes of cs 2 back
enough that they are COINCIDENT with those of cs 1; and THEN let them
measure given events.
For ANY value of phi(v) the following transformations will hold
good: x' = phi(v)b(x-vt), y' = phi(v)y; z' = phi(v)z;
and t' = phi(v)b(t-vx/c^2)
in which b = 1/(c^2-v^2)^.5.
There are an infinity of groups that satisfy those equations. The
LTE Group, in which phi(v) = 1 thus is invisible, is only one of
them.
The Galilean group of Cartesian-Euclidean transformations of
classical
physics do not fit, thus are ruled out by the above transformations --
which are
those Einstein DID almost-correctly derive in his 1905 paper.
glird: <Therefore Einstein's assertion that his 1905 paper's
"equations must be linear on account of the properties of homogeneity
which we attribute to space and time" requires that phi(v) = q = 1;
which rules out the LTE!! >
Tom: <I repeat: you are confused, but don't seem to know it. That is
not at all what SR says (GR is irrelevant here, as Lorentz transforms
do not apply in it). The symmetry for motion along X/X' requires that
the
transforms relating Y and Z to Y' and Z' be the same. >
"Reasons of symmetry" presumably require "that the transforms
relating Y and Z to Y' and Z' be the same" AS THOSE RELATING
Y' and Z' to Y and Z. Put simply, they require that the inverse
transformations have the same form as the direct ones.
Einstein, and presumably all of his followers including Tom Roberts,
mistakenly thought that the only value of phi(v) that allowed that to
happen is
"phi(v) = 1".
Tom: <But symmetry puts no constraint on the relationship between that
and the transform relating X to X' -- the motion itself INHERENTLY
breaks spherical symmetry. Einstein's paper shows that the motion need
not break homogeneity (see my Euclidean example above). >
Your Euclidean example, all by itself, PROVES that you don't
understand the LTE nor the STR either. Similarly, your last sentence
'[
PROVES that you don't understand that a "homogeneous" system has
identical units of length and time in all directions, thus is and
remains
"symmetrical" ; while one that "breaks spherical symmetry: is
automatically Asymmetrical.
Fortunately for all of us, other than that it too requires that
successive clocks be esynched via Einstein's postulated method, the
mathematics of General Relativity is entirely different than the
restricted theory's LTE.
glird
xxein
xxein: You have both explained nothing of the physic. All that was,
was different theories of belief emanated (perhaps enamated) by your
heroes. The study of the physic should not be 'copy and paste' from
belief.
If you two cannot understand the physic from scratch, you are just old
farts bragging to each other about your past experiences over the
backyard fence.