Dr. George J. Marklin

DERIVATION OF THE LORENTZ ETHER THEORY

This is a summary of the first talk I gave at the NPA meeting in College
Station, Texas, May 1997, on the derivation of the Lorentz ether theory:

ABSTRACT --- The Lorentz ether theory (LET) reproduces all
the equations of special relativity (SR) and makes the same experimental
predictions.[A. I. Miller, "Albert Einstein's Special Theory of Relativity,"
Addison-Wesley (1981).]  The differences between it and SR are in the
interpretations placed on those equations, and the starting assumptions used
to derive them. This talk will review the derivation of the LET and compare
it with SR. It will be shown that all of the counter-intuitive notions for
which SR is famous can be understood as resulting from the adoption of
mutable standards of measurement. The Lorentz transformation will be shown
to follow from the contraction of the length unit, the dilation of the time
unit, and the adoption of local time zones according to the Einstein
synchronization convention. Relationships between the relative coordinates
of SR and the absolute coordinates of the LET will be derived and the reason
for the importance of the space-time interval will be explained. Expressions
for absolute time, position and velocity will be presented and shown to be
Lorentz invariant.

ASSUMPTIONS

1. There exists ether, a physical substance that fills the entire universe
   and acts as the medium for the transmission of light waves.  It will be
   assumed to be rigid.  [Allowing deformability and differential motion
   in the ether leads to gravitational effects which will not be considered
   here.]  In principle, the ether can be detected and absolute velocities
   determined with respect to it, but the means of accomplishing this has
   not yet been discovered.

2. There exist absolute standards of measurement for distance and direction.
   From this it can be proven that the geometry of space is Euclidean.
   [This proof was given in my geometry talk at a New Mexico Objectivist
   Club meeting in 1997 and will eventually be published somewhere.]

3. There exists an absolute standard of measurement for time.  From this,
   and the fact that motion is relative, it can be proven that the
   kinematics of space-time is Galilean [not Lorentzian].  That is,
   absolute space and time coordinates in different frames of reference
   in relative motion must be related by the Galilean transformation
   equations (GTE's), not the Lorentz transformation equations (LTE's).
   [I will publish this proof somewhere eventually - but for now I will
   just take it to be an assumption.]

By absolute space and time coordinates I mean coordinates measured with
the absolute standards which are assumed to exist in principle but whose
actual construction in terms of concrete materials is not yet specified.
Absolute coordinates will be denoted by capital letters.  Let the absolute
coordinates measured in the frame of the ether be called X and T. Absolute
coordinates measured in a frame moving with constant absolute velocity U
in the X direction will be called X' and T'.  They are related to X and T
by the GTE's:

X' =  X - U T    and    T' =  T                                          (1)

and the relationship between the absolute velocities measured in the two
frames may be obtained by differentiating this, giving:

V' =  V - U                                                              (2)

This also applies to the velocity of light, for which

C' =  C - U                                                              (3)

Ordinary rulers and clocks are not absolute standards of measurement.
They experience length contractions and time dilations as follows:

4. Length Contraction
   The length of a ruler varies as a function of its velocity relative to
   the ether according to the formula L = L0 sqrt(1-V^2/C^2) where L0 is
   the length of the ruler when it is at rest.  This only applies when the
   ruler is aligned with the direction of motion.  If it is perpendicular
   to this direction there is no effect.

5. Time Dilation
   The time interval between ticks of a clock varies according to the
   formula T = T0 / sqrt(1-V^2/C^2) where T0 is the time interval when the
   clock is at rest. [This assumption can be omitted if one is willing to
   accept the notion that all clocks are equivalent to a "light clock"
   which consists of a light pulse perpetually bouncing between two
   mirrors attached to the ends of a ruler.]

Space and time coordinates measured with these mutable standards are called
relative coordinates and will be denoted by lower case letters: x and t in
the ether frame, and x' and t' in the moving frame.  Since there is no
length contraction or time dilation in the ether frame we must have

x = X   and   t = T                                                       (4)

and also for velocities,

v = V,   u = U   and   c = C.                                             (5)

The relative coordinates in the moving frame are related to the relative
coordinates in the ether frame by the LTE's, which we can now derive with
the help of one more assumption:

6. Local Time Synchronization
   Clocks at different locations are to be "synchronized" using the Einstein
   Synchronization Convention (ESC) as follows:  Let one particular master
   clock emit a light pulse when it reads t=0.  All other clocks will be set
   to d/c when they receive the pulse, where d is their distance from the
   master clock measured with relative units (ie. actual rulers).  Notice
   that this procedure will produce absolute synchronization if the master
   clock (and the rulers) is at rest in the ether, but not otherwise.

The meaning of local time is the same as for local time zones on the earth.
Instead of synchronizing clocks to each other, they are synchronized to some
locally observable objective event.  For example, "set your clock to noon
when the sun is directly overhead" roughly gives the ordinary kind of local
time.  Clocks that are 15 degrees of longitude apart will read an hour time
difference.  They are synchronized to the sun, not to each other.  With the
ESC, clocks are "synchronized to light" rather than to each other.  This is
the best that can be done if you are unable to detect the ether.  We will
see later that absolute synchronization is possible in principle, but it
requires knowledge of the absolute velocity of the clocks, which we do not
yet know how to measure.

DERIVATION OF THE LORENTZ TRANSFORMATION EQUATIONS

The LTE's relate x' and t' to x and t.  Since the relative length unit
in the moving frame is contracted by a factor of sqrt(1-U^2/C^2), the
relative coordinate will be expanded by the same factor.  The product of
the coordinate times the unit is equal to the objective length, which
must be independent of the chosen units.  Thus

x' L  =  X' L0                                                           (6)

so we must have

x' =  (L0/L) X' =  G X'                                                  (7)

where G = 1/sqrt(1-U^2/C^2) > 1.  In other words, if the rulers are
shorter, then it takes more of them to span the same objective distance.
Applying (1), (4) and (5) to (7) then gives:

x' =  g (x - u t)                                                        (8)
                 
where g = 1/sqrt(1-u^2/c^2).  This is the first LTE.

In an exactly similar manner we may also find that

t' =  T'/G  =  t/g                                                       (9)

but this is not yet the second LTE.  Since this equation has no dependence
on position, moving clocks at different locations will read the same time.
But this is not right.  By assumption 6, the clocks must be offset to
different local time zones by using the ESC.  We must add an offset:

t' =  t/g  +  o(x',u)                                                   (10)

where o(x',u) depends only on the location of the clock (x') and on the
velocity u, but not on the time (t') so that rate will not be altered.
The ESC is applied as follows:  Let the clock at the origin of the moving
frame be the master clock.  It emits a light pulse at t'=0, which is later
received by another clock at x'.  The absolute time of reception TR' is
the absolute distance X' divided by the absolute velocity of light C', and
using (7) and (5) this becomes

       X'     x'/ g      x'        c + u  
TR' =  --  =  -----  =  --- sqrt(  -----  )                             (11)
       C'     c - u      c         c - u

and TR' = TR = tr.  The ESC says that at this time, the clock at x' must
be set to x'/c, so applying (10) we may write:

tr/g  +  o(x',u)  =  x'/c                                               

and then using (11) we can solve for the offset and find:

o(x',u)  =  - u x'/ c^2

So the second LTE becomes

t' =  t/g  - u x'/ c^2                                                  (12)

In this form [(8) and (12)] the meaning of each term in the LTE's is clear.
The g factor in (8) is from length contraction, the 1/g factor in (12) is
from time dilation, and the -ux'/c^2 term in (12) is the amount by which
each clock must be set back so that it will be "synchronized to light"
rather than to the other clocks.

Equation (12) can be put into a more familiar form by using (8) to replace
the x' term and then simplifying, to get:

t' =  g ( t - u x/ c^2 )                                                (13)

The symmetrical appearance of (8) and (13) is related to an important
property of the LTE's called the group property.  If the LTE's are applied
twice, to go from frame 1 to frame 2 and then to frame 3, the net result
can be expressed as a single LT directly from 1 to 3.  LT's can thus be
"multiplied" as if they were numbers, and they have algebraic properties.
What this means is that even though in the derivation of the LTE's, the
starting frame had to be the ether; in the application of the LTE's the
ether does not have to appear at all.  They can be used to connect any
two frames and can be used even if you can't detect the ether (which we
can't).  However, it needs to be emphasized that while the ether is not
required in order to use the LTE's, it IS required in order to understand
what they mean.

The transformation for velocities can be found by differentiating (8) and
(13) and is given by:

     dx'    g ( dx - u dt )               v - u
v' = --- = ----------------------  =  -------------                     (14)
     dt'    g ( dt - u dx / c^2 )     1 - u v / c^2

whete v = dx/dt.  Note that

          c - u            c - u
c' =  -------------  =  c  -----  =  c                                  (15)
      1 - u c / c^2        c - u

so that the speed of light in relative units is invariant.

AN ALTERNATIVE DERIVATION

Without the offset term the LTE's would not have the group property.  An
alternative derivation (perhaps the one that Lorentz used) would be to
start with equations (8) and (10) and ask what offset would be required
in order to make the transformations form a group.  This can easily be
solved and will be left as an exercise for the reader. [Hint:  Write
the equations in matrix form and set the inverse of L(u) equal to L(-u),
where L(u) is the transformation matrix.]

RELATIONSHIP BETWEEN ABSOLUTE AND RELATIVE COORDINATES

Given an arbitrary frame with relative coordinates x and t, and absolute
coordinates X and T, it is easy to find the formula that relates them if
we assume that the ether can be measured and has some velocity ve or VE.

Use the LTE's to transform the relative coordinates and velocity to the
ether frame:

x0 = ge ( x - ve t )                                                    (16)

t0 = ge ( t - ve x / c^2 )                                              (17)

v0 = ( v - ve ) / ( 1 - ve v / c^2)                                     (18)

where ge = 1/sqrt(1-ve^2/c^2).  The appendix "0" denotes that the
attached quantity belongs to the ether frame.

Use the GTE's to transform the absolute coordinates and velocity to the
ether frame:

X0 = X - VE T                                                           (19)

T0 = T                                                                  (20)

V0 = V - VE                                                             (21)

Then use the fact that in the ether frame, relative and absolute units
are the same, so we can equate x0 = X0, t0 = T0 and v0 = V0 to get the
following three equations:

X - VE T = ge ( x - ve t )                                              (22)

T = ge ( t - ve x / c^2 )                                               (23)

V - VE = ( v - ve ) / ( 1 - ve v / c^2 )                                (24)

Applying (24) to the starting frame itself, which has v = V = 0, we
immediately find that VE = ve.  It does not matter whether the ether
velocity is measured in relative or absolute units, it must have the
same numerical value eigther way.  Using this fact, (22) and (24) can
be simplified to the following:

X = x / ge                                                              (25)

which just indicates that the absolute and relative position coordinates
are related by the length contraction factor as in equation (7), and

          c^2 - ve^2
V  =  v  ------------                                                   (26)
          c^2 - ve v

Applying (26) to the velocity of light we find

C = c + ve                                                              (27)

as one would expect.

INVARIANTS

The quantities in equations (16) through (21) are all invariants.  They
are the same for all observers in different frames of reference, moving
with respect to each other; and they are even the same for observers who
use units of measurement which are relative or absolute.  The formulas
depend on which type of unit is being used, but the numerical value does
not.  Equations (16) through (18) give the expressions for the invariant
position, time and velocity in terms of relative coordinates and they are
all Lorentz invariant.  Equations (19) through (21) give the expressions
in terms of absolute coordinates and they are all Galilean invariant.

Example 1.

Eq. (19) is a Galilean invariant expression for the absolute position.
Proof:  Starting with

X0 = X - VE T

use the GTE's to transform to another frame moving with velocity U:

X' = X - U T

T' = T

VE' = VE - U

Therefore:

X0' =  X' - VE' T'

    =  X - U T - ( VE - U ) T

    =  X - VE T

    =  X0

Q.E.D.

Example 2.

Eq. (17) is a Lorentz invariant expression for the absolute time.
Proof:  Starting with

t0 = ge ( t - ve x / c^2 )

use the LTE's to transform to another frame moving with velocity u:

x' = gu ( x - u t )   [where gu = 1/sqrt(1 - u^2/c^2)]

t' = gu ( t - u x / c^2 )

ve' = ( ve - u ) / ( 1 - u ve / c^2 )

c' = c   [ see eq. (15) ]

ge' = 1 / sqrt( 1 - ve'^2 / c'^2 )

                         1
    = ---------------------------------------
                      ve - u          
      sqrt( 1 - ( -------------- )^2  / c^2 ) 
                  1 - u ve / c^2

                     ( 1 - u ve / c^2 )
    = -------------------------------------------------
      sqrt{ ( 1 - u ve / c^2 )^2 - ( ve - u )^2 / c^2 }

                      ( 1 - u ve / c^2 )
    = ---------------------------------------------------
      sqrt( 1 + u^2 ve^2 / c^4 - ve^2 / c^2 - u^2 / c^2 )

                   ( 1 - u ve / c^2 )
    = --------------------------------------------
      sqrt{ ( 1 - u^2 / c^2 ) ( 1 - ve^2 / c^2 ) }
    
    = gu ge ( 1 - u ve / c^2 )

Therefore:

t0' =  ge' ( t' - ve' x' / c'^2 )

                                          ve - u
    =  gu ge ( 1 - u ve / c^2) ( t' - -------------- x' / c^2 )
                                      1 - u ve / c^2

    =  gu ge { ( 1 - u ve / c^2 ) t' - ( ve - u ) x' / c^2 }

    =  gu^2 ge { (1 - u ve/c^2)(t - u x/c^2) - (ve - u)(x - u t)/c^2 }

    =  gu^2 ge ( t + u^2 ve x / c^4 - ve x / c^2 - u^2 t / c^2 )

    =  gu^2 ge ( 1 - u^2 / c^2 ) ( t - ve x / c^2 )

    =  ge ( t - ve x / c^2 )

    =  t0

Q.E.D.

The four remaining proofs will be left as an exercise for the reader.

A special case of the invariant velocities is the invariant velocity of
light.  Using (18), the Lorentz invariant absolute velocity of light is

c0 = (c - ve)/(1 - ve/c) = c                                            (28)

And using (21), the Galilean invariant absolute velocity of light is

C0 = C - VE                                                             (29)

These must, of course, be equal: c0 = C0.

THE SPACE-TIME INTERVAL

The Lorentz invariant absolute position, time and velocity all depend
explicitly on the ether velocity.  Thus, they cannot be evaluated until
a method of measuring the ether velocity is discovered.  However, any
combination of these invariants will also be an invariant, and we may
look for some special combination in which the ether velocity cancels
out.  This special combination is called the space-time interval and is
given in terms of relative coordinates by:

I = x0^2 - c0^2 t0^2

  = ge^2 ( x - ve t )^2 - c^2 ge^2 ( t - ve x / c^2 )^2

  = ge^2 ( x^2 + ve^2 t^2 - c^2 t^2 - ve^2 x^2 / c^2 )

  = ge^2 ( x^2 - c^2 t^2 ) ( 1 - ve^2 / c^2 )

  = x^2 - c^2 t^2

The invariance of this quadratic form in the relative coordinates is what
makes possible the four-dimensional geometry of Minkowski space-time.  It
allows us to do calculations and solve problems without referring to the
ether.  It even allows the philosophically naive to pretend that there is
no ether.  But it must be remembered that relative coordinates are only a
mathematical device.  They are not real distances and times because they
are not based on immutable standards of measurement.

The interval may also be expressed in terms of absolute coordinates as

I = X0^2 - C0^2 T0^2

  = ( X - VE T )^2 - ( C - VE )^2 T^2

but the dependence on VE does not cancel out in this case, so this
expression isn't particularly useful.

EXTENSIONS TO THE LET

The LET can be extended to include gravitational effects by allowing the
ether to move differentially.  The first such theory was by R. L. Kirkwood
[Phys. Rev., vol. 92, pp. 1557 (1953), and vol. 95, pp. 1051 (1954)].

The second talk I gave at the NPA meeting in May 1997 was on the Kirkwood
ether theory, but I am not going to bother writing up a summary of it
because I don't believe the theory is correct - although it is quite
interesting and does illustrate some of the features that a correct
theory would have to have.  His theory gives the ether a velocity but
no density or pressure and it does not allow the conservation of ether.

A better theory has been found recently by Ilja Schmelzer.  He calls it
the General Ether Theory.  It is mathematically equivalent to general
relativity but uses Euclidean space and absolute time.  He gives the
ether a density, velocity and pressure tensor and satifies all the
appropriate conservation equations, but there are still many unresolved
issues concerning the proper way to interpret what his theory means.  I
will not discuss his theory further here, but those who are interested
can get more information, including copies of his papers, from his web
site:

http://get.ilja-schmelzer.net/

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