Re: Light Postulate Invalidated
From: Oriel36 (geraldkelleher_at_hotmail.com)
Date: 09/10/04
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Date: 10 Sep 2004 03:38:54 -0700
Tom Roberts <tjroberts@lucent.com> wrote in message news:<chl00j$as0@netnews.proxy.lucent.com>...
> Cadwgan Gedrych wrote:
> > Surprisingly, simple observation without measurements can
> > disprove the basis of SR (aka Einstein's light postulate).
>
> No. You goofed.
>
> [I ignore the obvious impossibility of "disproving"
> a physical theory via a gedanken -- in truth that requires
> a real experiment.]
>
>
> > Consider any two entities which are on a collision course,
> > as shown below:
> > E(a)-------------><--------------------E(b)
> > The location of their meeting point depends only upon their
> > relative motion,
>
> Not true. Your "analysis" is flawed. See below.
>
> > and this depends on their individual speeds,
>
> That is true, as long as "meeting point" and "individual speeds" are
> measured wrt a given inertial frame (e.g. your "frame A" below).
>
>
> > Suppose the two entities E(a) and E(b) always leave frame A
> > points (0,0,0) and (10,0,0) respectively at the same time (per
> > frame A's clocks at the given points). E(a) and E(b) will of
> > course always move toward each other so that they eventually
> > collide. Now suppose that after one such collision point is
> > marked on frame A, E(a) and E(b) return to their starting
> > points, and again leave on a collision course, except this
> > time E(a) moves at a different speed. This change in entity
> > E(a)'s speed will change the location (wrt frame A) of the
> > entities' collision point. Indeed, a change in either or both
> > of the entities' speeds will make their collision point shift.
>
> Yes.
>
> [Let me consider only inertial motions of E(a)and E(b),
> and discuss the situation between their starting and
> their collision.]
>
>
> > Clearly, any shift in the location of the collision point
> > indicates a change in the entities' relative motion.
>
> No. Such a shift indicates a change in the entities' MOTION RELATIVE TO
> FRAME A. You have shown nothing at all about the motion of E(b) relative
> to E(a) -- your entire discussion is in frame A, not in the frame of E(a).
>
> [Your notation sucks, as the distinction between "a" and "A"
> is crucial here.]
>
>
> > For example, the conclusion holds in the case where entity
> > E(a) is an inertial observer, and entity E(b) is a light ray.
>
> Not true. When E(b) is a light ray, then it moves at c wrt frame A, and
> also moves at c wrt E(a) [#] -- changes in the speed of E(a) wrt frame A
> will indeed change the location of the collision without any change in
> the relative velocity of E(b) wrt E(a).
>
> [#] you are attempting to "disprove" SR via a "contradiction",
> so the theoretical context must be that of SR. Remember that
> E(a) and E(b) are moving inertially (see above).
>
>
> > In other words, the shifting meeting point proves that light's
> > one-way speed varies with observer speed,
>
> No. You only showed that varying E(a)'s speed RELATIVE TO FRAME A
> changes the collision point RELATIVE TO FRAME A. You attempt to make an
> unwarranted conclusion here, about speed relative to a frame you never
> discussed -- your argument is flawed. And wrong.
>
>
> Tom Roberts tjroberts@lucent.com
Do you know who wrote this in 1991 ? -
[It may be a surprise to some readers that no postulate includes
a statement that light has the same speed in all frames; such
a statement is not required. This is just one example of the
power of group theory.]
I can't imagine anyone who has talked so much about so little and for so long.
A Physicist's Derivation of Special Relativity
From: att!ihlpl!tjrob (Thomas J. Roberts)
Many criticisms of Special Relativity center on the "assumption" that
the speed of light is constant in all reference frames. The derivation
given here does not make that assumption; the existence of a universal
speed (c) is a natural consequence of the Postulates forming the basis
of the derivation. General symmetry properties of space-time are
sufficient to determine the equations of the Lorentz Transformation
[to within a topological choice - see below]. The bottom line is that
it is IMPOSSIBLE to formulate an alternative to Special Relativity,
while obeying the observed symmetries of space-time and agreeing with
the experimental evidence [see below about the limitations of the
symmetry postulates used in this derivation].
This message was originally inspired by the recent posting of a
note by Dr. S. R. Kaip, purporting "Paradoxes of Special Relativity
(all) resolved". As several fundamental mistakes were made in that
article, I resolved to provide a definitive derivation of the equations
of Special Relativity, based upon unimpeachable First Principles.
This article will also, I hope, show why physicists believe in
Special Relativity (within its applicable domain), and are extremely
sceptical of "alternative descriptions". Historically, it took a long
time for physicists to accept Special Relativity. Even today, the
compelling derivation given here is usually not presented in textbooks;
I don't know why.
I claim no originality for this derivation; I do not know who originally
discovered it; I have re-created it based upon dimly-remembered ideas
from graduate school.
Written by:
Tom Roberts
AT&T Bell Laboratories
att!ihlpl!tjrob
original date: sometime 1989-1990
Colloquially, a Lorentz Transformation is called a "boost".
This derivation will be heavy going, in algebra; I hope it will be
understandable to most people with a good understanding of elementary
algebra, and a smattering of common sense. This is NOT a rigorous
mathematical derivation, but one at the level of rigor common to physicists.
NOTATION:
F(x) F is a function of x
a*b a multiplied by b
A**2 A squared (== A*A)
== "is identical to", or "is the same as"
= mathematical equality (NOT the FORTRAN meaning)
First, four Postulates will be given, with a brief discussion.
Then, the general form of the transformation equations will be derived,
followed by a brief discussion of their implications.
THE MAPPING POSTULATE
When two observers observe the same physical space-time, they assign
individual coordinate systems to THE SAME points of space-time. There is
a relation between the assignments they (separately) make, which is
called a coordinate transformation, usually expressed as a consistent
set of mathematical formulas relating the coordinates of one observer
to the coordinates of the other. The coordinate transformation from one
system to the other MUST be one-to-one and onto the other, BECAUSE THEY
ARE DESCRIBING THE SAME PHYSICAL SPACE-TIME; the transformation must
be invertible (see Relativity Postulate, below).
[Mathematicians worry about a lot of conditions for this, and
for the other Postulates; this is a Physicist's derivation, and
will assume that physical systems satisfy the mathematical
conditions necessary (continuity, etc.).]
THE ISOTROPY/HOMOGENEITY POSTULATE
Space is isotropic, in that there is no "preferred direction" in
space. The transformation must have the same mathematical form
for a boost in any (spatial) direction. Space is also homogeneous,
in that there is no "preferred position" in space. The
transformation must have the same mathematical form for any origin
of coordinates; this applies to time, as well.
THE RELATIVITY POSTULATE
There is no "preferred velocity", or "Preferred coordinate system" -
only relative velocities between coordinate systems are observable.
If coordinate system S' is moving with velocity v, as observed in
coordinate system S, then S is moving with velocity -v, as observed
in S'.
[This is Einstein's fundamental departure from classical
physics. Today, it seems natural.]
THE GROUP POSTULATE
The collection of all possible Transformations must form a
group under composition by successive application of transformations.
This is the key postulate, and the one that makes a general derivation
of the transformation equations possible; it imposes severe constraints
on the form of the equations. It has four important implications:
1. An identity transformation exists, which maps a coordinate system
to itself.
2. Any transformation has an inverse, which is also a transformation.
3. The result of applying two transformations in succession is itself
a transformation.
4. The application of three transformations in succession follows the
law of associativity [ABC = (AB)C = A(BC)].
[This is a more modern approach to the subject than was common
in Einstein's day; Einstein was instrumental in pointing out
how important symmetries are in physics, which leads naturally
to group theory.]
Those are the Postulates; make sure you understand and believe in them
now, because they are sufficient to derive the general form of the
transformation equations.
[It may be a surprise to some readers that no postulate includes
a statement that light has the same speed in all frames; such
a statement is not required. This is just one example of the
power of group theory.]
Now for the math...
This derivation will be done in "1+1" dimensions, that is, for one space
coordinate and one time coordinate. The derivation would follow similar
lines in "3+1" dimensions, but the extra complexity would exhibit no
additional features.
There are three frames of reference (or coordinate systems) of interest;
they will be called S, S', and S"; their coordinates will be called
x and t, x' and t', and x" and t", respectively. They are constructed
so that their x, x', and x" axes are all collinear, with the origins
of coordinates coincident (i.e. in the exact same place in the real
(i.e. physical) space-time); that is, the coordinates x=0,t=0 and
x'=0,t'=0 and x"=0,t"=0 all refer to the same point (event) in the
real space-time. The Homogeneity Postulate guarantees that no special
significance arises from their coincident origins of coordinates.
All three frames will use the same scales for length and time (these
simplifications are not necessary, but relaxing them would add
unenlightening complexity).
The difference between the three frames is their relative velocities.
We will call the velocity of S' as measured in S, u; S" as measured
in S' is v; S" as measured in S is w. The physical situation ensures
that these assignments can all be made. Implicit is the assumption
that the relative velocities are constant (but arbitrary).
The Mapping Postulate and the Homogeneity postulate imply that the
transformation equations are linear, with coefficients independent
of position. That is
x' = A(u)*x + D(u)*t + E(u) 1
t' = B(u)*x + C(u)*t + F(u) 2
The coefficients (A,B,C,D,E,F) can depend upon the relative velocity
between S' and S (i.e. upon u), but there is no other quantity that
can have physical relevance. Thus, eqn 1 & 2 are the most general
possible transformation equations satisfying the postulates.
This is important; if there were other powers of x or t on
the right-hand side, the transformation would not be one-to-one
everywhere. If the coefficients depended upon x or t (as they
do in General Relativity - see below), then space-time would
not be homogeneous and isotropic.
[Some other derivations use a postulate that straight lines
are transformed into straight lines to deduce the linearity
of the transformation equations.]
The translation terms (E(u) and F(u)) can easily be calculated,
based upon the construction of the systems S and S'; they are both 0.
They are not functions of u, because we arranged for coincident coordinate
origins independently of u (i.e. for each value of u, the origins were
individually arranged to be coincident). This is true also for the
other transformations (S' to S", and S to S"). The Homogeneity Postulate
guarantees that this choice has no physical significance.
Since S' is moving with velocity u relative to S, the point x'=0
is moving with velocity u (with respect to S); this allows us to
solve for D(u), with no loss in generality:
x' = A(u) * (x - u*t) 3
t' = B(u)*x + C(u)*t 4
Note: u=0 is certainly possible, in which case S'==S, so A(0)=1,
B(0)=0, C(0)=1 (i.e. x'=x and t'=t). In the following, u and v will
be assumed to be non-zero, but w will have no such restriction.
The transformations S' to S", and S to S" follow similarly:
x" = A(v) * (x' - v*t') 5
t" = B(v)*x' + C(v)*t' 6
x" = A(w) * (x - w*t) 7
t" = B(w)*x + C(w)*t 8
We will now use the Group Postulate to compose Eqns 3 and 4 with
Eqns 5 and 6, to get 7 and 8 (i.e. u and v are arbitrarily fixed,
and w will be determined from them).
Substituting 3 and 4 into 5 and 6:
x" = [A(v)*A(u) - A(v)*v*B(u)]*x -
[A(v)*v*C(u) + A(v)*A(u)*u]*t 9
t" = [B(v)*A(u) + C(v)*B(u)]*x +
[(-u)*B(v)*A(u) + C(v)*C(u)]*t 10
Comparing 9 and 10 with 7 and 8, and equating coefficients of
x and t (Eqns 7-10 are each valid for ALL x and ALL t), we conclude:
A(w) = A(v)*A(u) - A(v)*v*B(u) 11
w*A(w) = A(v)*v*C(u) + A(v)*A(u)*u 12
B(w) = B(v)*A(u) + C(v)*B(u) 13
C(w) = C(u)*C(v) - u*B(v)*A(u) 14
Now, let's consider the special case v=-u. Then 5&6 will be the
inverse of 3&4 (Relativity Postulate), so w=0. 11-14 become:
1 = A(-u)*A(u) + u*A(-u)*B(u) 15
0 = A(-u)*(-u)*C(u) + A(-u)*A(u)*u 16
0 = B(-u)*A(u) + C(-u)*B(u) 17
1 = C(u)*C(-u) - u*B(-u)*A(u) 18
Assuming u*A(-u) is non-zero (see below), eqn 16 says:
C(u) = A(u) 19
[Note this is true in general (not just for v=-u); it is
a mathematical statement about the two functions, valid
for all u.]
The Isotropy postulate requires that C(-u) = C(u) [if I boost S'
in a different direction (i.e. backwards), the clocks of S' must
be affected exactly the same as before]. This plus eqn 17 gives:
B(-u) = -B(u) 20
[Note that the stated symmetries of A(u), B(u), and C(u) are
all consistent with their values at u=0 given above.]
Eqns 15-18 reduce to:
1 = A(u)**2 + u*A(u)*B(u) 21
Returning to the general case (arbitrary v), Eqns 11-14 become:
A(w) = A(v)*A(u) - v*A(v)*B(u) 22
w*A(w) = v*A(u)*A(v) + u*A(u)*A(v) 23
B(w) = B(v)*A(u) + A(v)*B(u) 24
A(w) = A(u)*A(v) - u*B(v)*A(u) 25
[Note the symmetry of Eqns 22-25 under interchange of u <-> v
(interchange 22 and 25); this is expected, as adding collinear
velocities should not depend upon their order.]
Eqns 22 and 25 yield:
v*A(v)*B(u) = u*B(v)*A(u) 26
or (assuming u*A(u) and v*A(v) are both non-zero):
B(u)/(u*A(u)) = B(v)/(v*A(v)) 27
Since Eqn 27 must hold for all u and for all v, eqn 27 must
be a universal constant; call it q:
q == B(u)/(u*A(u)) = B(v)/(v*A(v)) 28
or
B(u) = q*u*A(u) 29
Substituting 29 into Eqn 21 gives:
1 = A(u)**2 + q*u**2*A(u)**2 30
Solving for A(u) gives:
A(u) = 1/sqrt(1+q*u**2) 31
Combining Eqns 3, 4, 19, 29, and 31, we have the general form of
the transformation equations:
A(u) = 1/sqrt(1+q*u**2) 31
x' = A(u) * (x - u*t) 32
t' = q*u*A(u)*x + A(u)*t 33
By solving Eqn 22 for w, we get the rule for composition of velocities:
w = (u + v) / (1 - q*u*v) 34
The choice of q is arbitrary. There are three basic choices that
have significantly different behavior: zero, negative, and positive.
This is the topological choice mentioned above.
Choosing q=0 yields the Galilean transformation:
x' = x - u*t 35
t' = t 36
w = u + v 37
Note the universal time; velocities simply add.
These are the "familiar" transformation equations that are
approximately true (to very high accuracy) in our ordinary
lives where velocities are small.
Choosing q<0 yields the Lorentz transformation. By convention,
define a constant, c, by q==-1/c**2 [manifestly negative], and let
G(u/c)==A(u), we have:
x' = G(u/c) * (x - (u/c)*ct) 38
ct' = -(u/c)*G(u/c)*x + G(u/c)*ct 39
G(u/c) = 1/sqrt(1-(u/c)**2) 40
w/c = (u/c + v/c) / (1 + (u/c)*(v/c)) 41
Here, ct and ct' are the time coordinates multiplied by c
(which gives them the same units as x and x': length).
Normally, G(u/c) is called gamma, and u/c is called beta.
In the limit u/c -> 0, 38-41 reduce to 35-37, the Galilean
transformation. Here, velocities do not simply add, but have a
more complicated composition rule; an object moving with
velocity c in one frame moves with velocity c in all frames.
Note, however, that the transformation equations are not
well-behaved when transforming to a frame moving with
velocity c; the velocity c serves as a limiting velocity,
because G(u/c) goes to infinity as u/c goes to 1. Eqn 41
guarantees that the composition of two velocities will be
less than c, as long as the individual velocities are each
less than c. If u/c > 1, imaginary numbers appear, leading
most physicists to be sceptical of the physical applicability
of such velocities.
Choosing q>0 yields a third transformation;
here q==+1/c**2 [manifestly positive], and H(u/c)==A(u):
x' = H(u/c) * (x - (u/c)*ct) 42
ct' = (u/c)*H(u/c)*x + H(u/c)*ct 43
H(u/c) = 1/sqrt(1+(u/c)**2) 44
w/c = (u/c + v/c) / (1 - (u/c)*(v/c)) 45
This transformation can be cast into more familiar form by substituting:
u = c * tan(k) (where k,l,m are in the range -PI/2 < k,l,m < +PI/2 )
v = c * tan(l)
w = c * tan(m)
Then the transformation becomes (with some analytic continuation and
trigonometric identities):
x' = x * cos(k) - ct * sin(k) 46
ct' = x * sin(k) + ct * cos(k) 47
H(u/c) = 1/sqrt(1+tan(k)**2) = cos(k) 48
m = k + l 49
This transformation is clearly a simple Euclidean transformation,
in which the time coordinate behaves just like the spatial
coordinate, and boosts are simple rotations. The limit
(u/c) -> 0 still yields the Galilean transformation.
The velocity c serves as a velocity "scale", but nothing
dramatic happens to the transformation when (u/c) = 1
(i.e. k = PI/4), or when (u/c) > 1. The singularity
of Eq. 45 disappears when "velocities" are viewed as "angles"
in Eq. 49. Note that two positive velocities greater than c
are composed into a NEGATIVE velocity (Eq. 45), which is explained
by Eq. 46-49 as simply going more than halfway around a circle.
Note that there is no velocity that is the same in all frames,
and that causality is not necessarily preserved by a coordinate
transformation (ct' can run BACKWARDS with respect to ct).
It seems very difficult to build a world view based upon
Eqs 42-45 (or 46-49).
Before discussing the implications of these transformation equations,
let me suggest the following exercises:
Exercise for the reader: At several places in the derivation,
the velocities u and v were assumed to be non-zero, as well
as some other functions of u or v were assumed to be non-zero.
Verify that all such assumptions are valid.
Exercise for the reader: Re-do the derivation while retaining
the translation terms of Eqns 1 and 2; show that their presence
doe not change the conclusions.
Exercise for experts: As you know, the full Poincare group
includes not only the boosts derived here, but also spatial
rotations and two point transformations: parity inversion
(x' = -x) and time reversal (t' = -t). Don't bother deriving
the equations for the full Poincare group (adding rotations
is trivial, but tedious). Instead, note that Eqns 31-33 were
derived from very general considerations, BUT DO NOT INCLUDE
THE POINT TRANSFORMATIONS. Point out exactly where they were
left out, and modify the derivation to retain them.
Choosing the actual topology of space-time can only be done by resorting
to physical observations of phenomena in the real world (i.e. by doing
an experiment). There is a tremendous body of experimental evidence that
shows that the speed of light is independent of the velocities of either
the source or observer (there are also many other, equivalent observations).
This compels us to choose the Lorentz Transformation (Eqns 38-41), and
to identify the arbitrary constant "c" with the speed of light. No other
choice is possible, while satisfying the four Postulates and the
experimental evidence.
This is why most (if not all) physicists today believe in Special
Relativity - it is IMPOSSIBLE to construct an alternative description
without violating one of the postulates or disregarding a very large
body of experimental evidence. If you truly believe that Special
Relativity simply must be false (for whatever reason), go back and
review the four Postulates, and find a hole in them.
Einstein DID find a hole in the four Postulates, and brought us
General Relativity. He was, in a very real sense, the first fish to
see the water, and to describe it.
Einstein's departure was in the Isotropy/Homogeneity Postulate -
he proposed that space-time is isotropic and homogeneous only within
an infinitesimal region of any given point in space-time; that is,
in the presence of matter, space-time itself is NOT homogeneous,
but its geometry is affected by the presence of matter.
[Before General Relativity, Cartesian coordinates were used as
a matter of course, and their applicability to the real world
was never challenged (Lagrangian mechanics is very different
from this). Physical theory had two basic parts, in which the
Laws of Physics possess many symmetries (such as isotropy and
homogeneity of space-time), while the initial conditions rarely
possess the same symmetries (this remains true today, but the
lesson has been learned to be careful). Implicitly, these
symmetries were applied GLOBALLY, to the entire space-time
(e.g. as in the statements of the Postulates above). After
General Relativity, Cartesian coordinates have been replaced
by general curvilinear coordinates, and the symmetries are LOCAL
in nature (i.e. apply only within each infinitesimal region of
space-time). Unfortunately, this generality causes enormous
complexity in the mathematics; curvilinear coordinates and
general coordinate transformations have not yet been successfully
applied to the other great advancement in physics of the
Twentieth Century - Quantum Mechanics.]
Tom Roberts
AT&T Bell Laboratories
att!ihlpl!tjrob TJROB@IHLPL.ATT.COM
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