A Physicist's Derivation of Special Relativity
Tom 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 article will, 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
Lucent Technologies / Bell Laboratories
tjro...@lucent.com
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 (raised to the 2nd power == 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 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.
Identifying 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
Lucent Technologies / Bell Laboratories
tjro...@lucent.com
Dan Evens
Tom Roberts wrote:
> Choosing q<0 yields the Lorentz transformation.
I have never seen special relativity derived in just this way.
I like it a lot.
Is there any useful or interesting result from choosing q to be
other than real? It seems obvious that you won't get the geometry
we find ourselves in, but will you get anything even vaguely
sensible as a geometry?
--
Standard disclaimers apply.
My usual and customary fee for bouncing unwanted junk e-mail
advertising is $500 U.S. per message. Sending me such e-mail
is a contract which acknowledges and accepts my fee schedule.
Dan Evens
Philip Gibbs
Tom Roberts wrote:
>
> The Mapping Postulate and the Homogeneity postulate imply that the
> transformation equations are linear, with coefficients independent
> of position. That is
>
> x' = A(u)*x + D(u)*t + E(u) 1
> t' = B(u)*x + C(u)*t + F(u) 2
>
Tom, you gave a nice derivation but there is a flaw.
The above step cannot be justified. I believe that
there are actually spaces which satisfy your
postulates which are not flat. Examples are the
de Sitter space-times. The transformations
would have to be non-linear in those cases.
I agree that once you have selected linear
transformations you will get Newtonian or
Minkowski space-time as the only physically
reasonable solutions but it would be better
to have a complete list of possible solutions
for more general transformations.
--
====================================================
Phil Gibbs p...@pobox.com http://pobox.com/~pg
glird
In article <54jfst$g...@ssbunews.ih.lucent.com> Tom Roberts wrote:
>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
Not necessarily. A word to the wise being sufficient, let the
middle system be moving to the right at v and let it plot two other
systems as moving at -u and u relative to it. Then calculate what
happens when the clocks of S' are affected exclusively by their
absolute velocities in Einstein's "empty space" rather than merely
their relative velocities as plotted by the moving system in the
middle.
>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).
For the record: The "full Poincare group" does NOT include
Minkowski "rotations" nor "time reversal (t' = -t)". It merely
included symmetrical classical rotations of the spatial axes of a
given coordinate system that thereupon placed the direction of
relative v upon the now-coinciding X axes, AFTER which the lorentz
transformations then can be used.
>Tom Roberts
>Lucent Technologies / Bell Laboratories
A VERY good company. Especially its subsidiary. :-)
glird
Christopher R Volpe
> >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
> middle system be moving to the right at v and let it plot two other
> systems as moving at -u and u relative to it. Then calculate what
> happens when the clocks of S' are affected exclusively by their
> absolute velocities in Einstein's "empty space" rather than merely
> their relative velocities as plotted by the moving system in the
> middle.
Einstein does not use the term "empty space" to denote an absolute rest
frame.
--
Chris Volpe Phone: (518) 387-7766
GE Corporate R&D Fax: (518) 387-6560
PO Box 8 Email: vol...@crd.ge.com
Schenectady, NY 12301 Web: http://www.crd.ge.com/~volpecr
.
Tom Roberts
In article <54qfqn$5...@news-e2d.gnn.com>, glird <gl...@gnn.com> wrote:
>
>In article <54jfst$g...@ssbunews.ih.lucent.com> Tom Roberts wrote:
>>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
YES! CORRECT! You exposed a hidden assumption -- that time and space
are orthogonal. Thanks!
Fortunately, it is easy to correct:
From eqns 17 and 19:
B(-u)C(u) = -C(-u)B(u) 19a
As this is true for all u, it can only occur if EITHER
B(u) = 0 19b1
OR
C(u) = 0 19b2
OR
B(u) = B(-u) AND C(u) = -C(-u) 19b3
OR
B(u) = -B(-u) AND C(u) = C(-u) 19b4
19b2 and 19b3 are inconsistent with the values given for u=0.
If we assume 19b1, then eqn 18 gives C(-u) = 1/C(u), which
can only be consistent with the isotropy postulate if C(u)=1.
This results in the same transformation equations as selecting
q=0 in the topological choice below. The only nontrivial choice
is 19b4. [there is probably a better way to dismiss 19b1, but
it will take some more thought...]
I omitted your bogus objection because you invoked "absolute velocities"
in "empty space", concepts which do not apply to the mathmatical
environment in which this derivation takes place. They are inconsistent
with the four postulates. Yes, I realize that the contents of your
bogus objection is actually the essence of your objection (and I suspect
you did not realize the kernel of truth in it reflected above).
You clearly have a different worldview that that of this derivation;
and clearly do not accept the relativity postulate. That's OK -- reality
will be the ultimate arbiter.
Note, however, that the relativity postulate as stated in my original
derivation is MUCH stronger than what is actually used in the derivation.
Only the second sentence of my wording is actually used: "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'." -- I suspect that
it is very difficult to build a consistent worldview which violates that.
I recognize that the essence of your objection does violate it.
Tom Roberts tjro...@lucent.com
Wayne Throop
: gl...@gnn.com (glird)
: A word to the wise being sufficient, let the middle system be moving
: to the right at v and let it plot two other systems as moving at -u
: and u relative to it. Then calculate what happens when the clocks of
: S' are affected exclusively by their absolute velocities in Einstein's
: "empty space" rather than merely their relative velocities as plotted
: by the moving system in the middle.
OK. Choose a frame, any frame. To make glird happy, we'll choose a
frame, and call it "empty space", and it'll be really and for true
the case that lightspeed is anisotropic with value c in the coordinate
system we call "empty space". We'll have another system we'll call
"middle", and it'll move "to the right" WRT "empty space". Finally,
we'll introduce two more systems, which in the "middle" system have
velocities -u and +u; we'll call these the "-u" and "+u" systems.
Now, glird tells us to "calculate what happens when the clocks of S'
[I'll go glird one better; "the clocks of *all* the systems] are
affected exclusively by their absolute velocities in Einstein's "empty
space" rather than merely their relative velocities as plotted by the
moving system in the middle".
And what happens is exactly, precisely, and identically,
the same thing that happens if clocks are "affected exclusively"
by their relative velocities as measured WRT any arbitrary standard.
That is, no matter what v, the relationship of clocks in terms of
the -u, middle, and +u coordinates is exactly and precisely the same.
In other words, both "v" and the "empty space" frame are superfluous.
The properties of interest of the "empty space" frame are identically
the properties of the other frames in this case; specifically here, the
"middle" frame. As Einstein said, "[...] the view to be developed here
will not require an absolutely stationary space [...]".
That is, after all, the reason most physicists decided to stop worrying
about finding the "absolute frame", and decided to get on with the
business of doing physics instead of outcome-less mental mas.. uh,
consequence-free speculation.
--
Wayne Throop thr...@sheol.org http://sheol.org/throopw
thr...@cisco.com
gl...@gnn.com
In article <8465...@sheol.org> Wayne Throop wrote:
>Now, glird tells us to "calculate what happens when the clocks of
>S'[I'll go glird one better; "the clocks of *all* the systems] are
>affected exclusively by their absolute velocities in Einstein's
>"empty space" rather than merely their relative velocities as
>plotted by the moving system in the middle".
>
>And what happens is exactly, precisely, and identically,
>the same thing that happens if clocks are "affected exclusively"
>by their relative velocities as measured WRT any arbitrary
>standard. That is,
From there on Throop merely states his opinions rather
than trying to actually ""calculate what happens", as requested.
How come, Wayne? Don't you know how to DO this?
glird
Wayne Throop
::: gl...@gnn.com ()
::: let the middle system be moving to the right at v and let it plot
::: two other systems as moving at -u and u relative to it. Then
::: exclusively by their absolute velocities in Einstein's "empty space"
::: rather than merely their relative velocities as plotted by the
::: moving system in the middle.
:: And what happens is exactly, precisely, and identically, the same
:: thing that happens if clocks are "affected exclusively" by their
:: relative velocities as measured WRT any arbitrary standard.
: gl...@gnn.com ()
: Throop merely states his opinions rather than trying to actually
: "calculate what happens", as requested.
: How come, Wayne? Don't you know how to DO this?
Well, if glird will state something specific to calculate, I'll
calculate something specific. But since glird only said "calculate
what happens" in general, I can only state generally what such
calculations will show.
Does glird want to specifiy something more specific to calculate?
In the meantime, we can show that, say, the number of clock ticks
between two specific events is the same when we offset from "middle" to
"absolute". The square of some interval in the middle system
s(m)^2 = t(m)^2-x(m)^2. In the absolute system, that's
s(a)^2 = t(a)^2 - x(a)^2
[lorentz xform] = ((t(m)+vx(m))^2 - (x(m)+vt(m))^2) / (1-v^2)
[distribute] = ( t(m)^2 + 2t(m)vx(m) + v^2x(m)^2
- (x(m)^2 + 2t(m)vx(m) + v^2t(m)^2) ) / (1-v^2)
[cancel] = ( t(m)^2 + v^2x(m)^2 - x(m)^2 - v^2t(m)^2 ) / (1-v^2)
[factor] = (1-v^2)(t(m)^2-x(m)^2)/(1-v^2)
= t(m)^2-x(m)^2
And since elapsed times of clocks in SR are intervals[1], we see that
the behaviors of clocks as derived WRT the velocity in the absolute
frame is exactly the same behavior that was derived WRT the velocity in
the middle frame. [2]
Would glird also like to challenge me to show that you get the
same distance between two points after you rotate the coordinate
system by some angle? I think I can remember enough high school
algebra to do that. Though I might have to count on my fingers
and toes and chant "soh, cah, toa" a lot.
I'll admit my math is rusty, but come ON folks, the assertions I'm
making about SR are high-school geometry and algebra stuff.
You are no more likely to find a case where SR-predicted behaviors
are different when refered to the middle frame instead of the absolute
frame than you are likely to find a way to square the circle.
--
Wayne Throop thr...@sheol.org http://sheol.org/throopw
thr...@cisco.com
--
[1] If you doubt that elapsed clock times are proper intervals,
I suppose that derivation can be shown also:
show from t' = (t-xv)/(1-v^2)^(1/2) that (t')^2 = (t^2-x^2)
t' = (t-vx)/(1-v^2)^(1/2)
(t-x^2/t)/(1-(x/t)^2)^(1/2) [ subst v=x/t ]
(t^2-x^2)/(t*(1-(x/t)^2)^(1/2)) [ subst (x/t)=v & t' ]
(t')^2 = (t^2-x^2)
[2] Well, glird might have meant that the velocity v was not
colinear with +u and -u; he might have meant to introduce
non-colinear lorentz boosts, and is asking for a proof that things
still behave the same. That demonstration is tedious enough to be a
peril for somebody as rusty as me; it's sort of analogous to the
above, but there's lots more substitution and simplification to go
through... as I say, "tedious", and more room for trivial errors to
creep in. Especially when it's obvious by inspection of a 3D
spacetime chart (2d space + time). But if anybody else wants to run
through the addition of spatial rotations to lorentz boosts, and
show everything is still consistent, and/or choose a nice notation
for this, fire away. I'd be interested to see it.