LORENTZ TRANSFORMATION FROM INFORMATION
Derivation of Lorentz transformation from principles of statistical
information theory
Revised 2/28/2016
Thomas E. Butler
1 Woodruff Way, Columbia, New Jersey 07832
e-mail address: astraclara@embarqmail.com
The Lorentz transformation is derived from invariance of an information quantity related to statistical
hypothesis testing on single particle system identification parameters. Invariance results from recognition
of an equivalent observer as one who reaches the same conclusions as another when the same statistical
methods are used. System identity is maintained by parameter values which minimize discrimination
information, given by a Kullback-Liebler divergence, under a constraint of known shift in observation time.
Deviation of discrimination information from the minimum value gives the difference in information
between an observed system under a constraint shift and the expected system that maintains identity under
the same constraint. System observation states are represented by parametric probability distributions of
particle system measurement values.
Keywords: Lorentz transformation, special relativity, information, Kullback-Liebler discrimination information
PACS: 03.30.+p;89.70.Cf;01.55.+b
I. INTRODUCTION
The observation state of a free single particle system at a
constant velocity is represented by a probability distribution
of observation measurement values.
Change in the
observation time results in a change of the observed system
state to a state that retains the system identity of the initial
system. The system identity [1] [2] [3] is preserved by a
final observation state that minimizes the number of bits of
discrimination information given by the Kullback-Liebler
discrimination information [4] under the constraint of an
observation time shift. Excess discrimination information
above the minimization value shows how close an observed
system state is to a state which preserves system identity,
and is an input to identity hypothesis testing methods.
Different observers are expected to obtain the same system
identity hypothesis conclusions when using the same
statistical methods. Then the state associated with a
different observer must keep the excess discrimination
invariant.
Invariance of the excess discrimination
information for a free single particle system gives the
Lorentz transformation, not of random variables which do
not transform, but of the observed system state as
represented by parameters of the probability distribution.
Derivations of the Lorentz transformation from within
other areas of physics, for example quantum information
and communication theory [5], make the transformation
dependent on theories within those branches of physics.
The derivation based on a discrimination invariant shares a
similar dependence, but primarily on concepts of statistical
information theory, independent of other branches of
physics beyond representation of simple Galilean motion.
Although the statistical discrimination derivation depends
on specific probability models, the resultant transformation
is the same for all similar probability models for small
shifts in space and time parameters. The information based
derivation also supports generalization of the Lorentz
transformation to a concept of equivalence transformations
of distributions of arbitrary quantities.
Another characteristic of the statistical information
derivation of the Lorentz transformation is that there is no
assumption of an invariant speed [6] [7]. Domain and
range set properties of equivalence transformations imply
existence of an upper bound on the magnitude of the
velocity.
Then transformations that preserve excess
discrimination invariance require all boundary velocities to
map to the boundary.
II. PARTICLE OBSERVATION STATE
Observations of the motion of a particle in an experiment
produce a set of times of observation and associated
particle positions.
In a repeatable experiment, the
information contained by an ensemble of sets of positiontime value pairs can be concisely represented by a
probability distribution which normalizes over both
observation time and position random variables, as shown
in the random sample of Figure 1. Each experiment is
independent, so the probability distribution represents the
expected distribution of a single experiment. No special
LORENTZ TRANSFORMATION FROM INFORMATION
100 Point Normal f(x,t) Sample: Velocity=8.325 std deviation(x,t)=(16.667,2)
4
2
O b s e rv a t io n T im e t
interpretations are required beyond the probability
distribution as a simple representation of measurement data,
given that a measurement exists. The probability
distribution is the observation state of the particle system.
An observation time and position density that represents
measurement data is applicable to many variants of a
position versus time experiment. A single experiment can
consist of dynamic observations in which the particle time
and position are recorded as the particle moves. An
alternative experiment might consist of a single timeposition measurement, with apparent motion from an
ensemble of repeated experiments with different
observation times. The same density represents
measurements for both interpretations.
-2
A. Example probability density
The observed particle state is a probability density in
position and time random variables which provides a
realistic summary of observation data from a controlled
experiment. This is made evident by a plot of data
generated by a simulated random sample from a normal
time and space probability density, shown in Figure 1.
The probability density of Figure 1, which is normalized
over both observation time and position random variables,
generates a random sample that appears to be an ordinary
collection of observation data from a repeated constant
velocity experiment of finite duration that might have come
from a physics class lab. The same result could also be
produced by a single experiment with a large number of
observations. In either case the measurement position
standard deviation of 16.667 is not an error, but primarily
the width of the range of the main concentration of position
observations. Position error is given by the conditional
position given time standard deviation of 0.745. Similarly,
the measurement time standard deviation of 2 is not an
error but primarily the extent of the range of observed
times. Peak frequency of observations increases without
limit for a dynamic interpretation of the model, as the
number of observations rises, without any effect on the
probability density. In this respect the normal density
exhibits a classical physics behavior of independence from
arbitrarily small time intervals.
0
-4
-40
-20
0
20
40
x Position Observation
Figure 1
B. A model of uniform motion
The example probability density of Figure 1 is a two
dimensional normal probability density in observation
position x and time t random variables as a model of
experimental data from non-quantum mechanical uniform
particle motion. Extension to a normal density in four
dimensions models a three dimensional velocity. The four
dimensional normal density f is
f (R : R, Σ) =
with
random
parameters
1
det (2πΣ)
variable
mean
V = σt
(Σxt
Σyt
vector
Σzt
(
R= x
vector
covariance matrix Σ .
−2
T
1
− (R −R ) Σ−1 (R −R )
2
e
(
R= x
y
z
,
y z t
t
)
T
,
Define the vector V
)
T
(1)
)
T
,
and
by
. Then the position-time block
matrix form of Σ is
C + σ 2VV T

t
Σ = 
 σ 2V T

t
σt 2V 
 ,
σt 2 
(2)
with C the 3x3 position given time co-variance matrix and
σt 2 is the observation time variance. Define the vector of
random variable observation position components by
LORENTZ TRANSFORMATION FROM INFORMATION
(
X = x
y z
)
T
with
mean
(
X = x
y
z
)
T
The
density f is the product of the marginal probability density
g in the observation time random variable and the
conditional probability density h in observed position
given observed time, which for (1) are defined by
1
h (ε : C) =
det (2πC )
g (δt : σt ) =
1
−
e
e
1
− εTC −1ε
2
,
and
A. Parameter classification
δt 2
2σt 2
equation of state an isentropic gas produces a different
relation between pressure and density than that of an
isothermal gas. Parameters can be classified as either
constraints, controlled by an experiment, or as
unconstrained, left to be determined as parameters of a
distribution of measurements after an experiment is
performed.
.
(3)
2πσt
with ε = δX −V δt , δX = X − X , δt = t − t and where
f (R : R, Σ) = m (δt : σt ) h (ε : C)
(4)
. Although the position standard deviation of (4) is the
width of the range plus position error, and is not a position
error, the diagonal components the position covariance
matrix C are an indication of position error. The four
dimensional probability density with parameters
X , t , V,C and σt is selected as a model of single free
particle motion.
III. PARAMETERS
Parameters of the probability distribution are a
representation of the particle observation state. A theory
provides the general form of the distribution as a collection
of possible allowed distributions, with parameter values to
select a particular distribution. All inputs allowed by the
applicable theory which select a particular f (R : R, Σ)
from the set of all possible densities supported by the
theory are parameters. Parameters are constructed such that
there is a one to one relationship between a parameter value
and a probability distribution. Initial, boundary and
environmental conditions of an experiment are parameters,
as are numerical parameters. Statistical methods are used
to estimate parameter values from the output data of an
experiment.
Parameters of probability distributions are a necessary
and common component of physics. Pressure, temperature,
entropy and density are thermodynamic parameters
associated with probability distributions of the underlying
statistical mechanics of very large degree of freedom
composite systems. Some parameters are designed to be
controlled by an experiment, and others set free to vary in
response to the controlled parameters. Thermodynamics
provides the similar example that for the same ideal gas
All parameter values are extracted from experimental
data. Prior to execution of an experiment the design of an
experiment can select planned values of some parameters,
based on prior theoretical considerations independent of the
outputs of the experiment. Mean and extent of the range of
the observation time are two parameters which might be set
by the design of an experiment, and confirmed by
experimental data, since they can be regarded as under the
control of the observer and independent of the object
particle of the experiment. Parameters are classified as
constraint or responsive in a given experiment:
1. Constraint Parameters
Constraint parameters are parameters with planned
values input to an experiment which exhibit high levels
of repeatability and are independent of the subject of
an experiment. The value of a constraint associated
with a constraint parameter value is given by a subset
of the set of allowed parameters to which the
constraint confines distribution selection.
2. Responsive
Parameters
Responsive parameters have values which are not
planned input but are outputs from the execution of an
experiment and may depend on the subject of the
experiment.
(
Let P = X V
t
σt
C
)
represent the parameters
of the probability density so that (4) is condensed to
f (R : P ) = m (δt : σt ) h (ε : C)
(5)
Some parameters or transformations of parameters might be
selected to be either constraints or responsive, for different
experiment designs. Other parameters can have intrinsic
properties that effectively classify the parameters as
constraint or responsive parameters. Time parameters have
such intrinsic properties.
LORENTZ TRANSFORMATION FROM INFORMATION
{
Let Q = {∀P } be the set of all probability distribution
parameters allowed by a theory. An experiment design
defines constraints which restrict parameter values to a
subset C ⊂ Q . The value of a constraint is given by subset
C . Denote the set of all valid constraint sets as Λ (Q ) , so
C ∈ Λ (Q ) .
Define the set of constraint sets ϒ (C ) relative to a set
C to be the set of all subsets of C which are also valid
constraint sets, so that ϒ (C ) = Λ (Q ) ∩ Ρ (C ) where Ρ (C )
is the set of all subsets of C . All constraints associated
with a particular type of parameter constraint are contained
in an adjustable constraint relative to constraintD which is
defined to be a set Α (D ),
Α (D ) = {C 1, C 2 , C 3 , …} ,
(6)
with elements C i ⊂ D that satisfy
elements satisfies
C1 ∪ C 2 ∪ C 3 ∪ … = D
(8)
Properties (7) and (8) of an adjustable constraint imply that
every element of D is contained within one and only one
element of Α (D ) .
Then there exists a function
)
for each
P ∈ D which returns the
constraint set element of Α (D ) that contains P .
For
example, if one of the co-ordinates of P is the mean
observation time t parameter constraint, then index i = t
and M P : Α (D ) = C t . If a type of constraint is defined
(
)
by known values of parameter components denoted by ζ
(
)
then ζ can be used as the index and C ζ : Α (D ) is the
corresponding constraint set element of the adjustable
constraint set.
Let C i ∈ Αc (D ) be an element of an adjustable constraint
and Bi ∈ Αb (D ) be an element of a different adjustable
constraint.
Then
the
B. Confirmation Sets
Verification of a theory is rarely accomplished by a
single type of experiment represented by the constraint
choices in a one adjustable constraint set. More typical is a
variety of types of experiments to provide a stronger
verification of a theory.
Each type of experiment
corresponds to a different adjustable constraint set. Define a
confirmation set of adjustable constraints to be the set of all
adjustable constraints used in experiments to verify a
theory.
ɶ given by
Designate A
{
}
ɶ = Α (D ), …, Α (D ), …
A
1
n
(9)
as the confirmation set for the single particle theory.
(7)
for distinct elements C i ≠ C j , and where the union of all
(
Α(b,c ) (D ) is an adjustable constraint relative to D .
C. The Time Postulate
Ci ∩ C j = ∅
M P : Α (D )
}
Α(b,c ) (D ) = Bi ∩ C j ∀Bi ≠ C j satisfy (7) and (8) so that
4. Constraint Sets
elements
of
set
In an experiment set up to measure the particle position
about a specified time the mean observed time and the
extent of the range of observed times give all parameters –
mean and standard deviation - of the normal marginal time
density. Since all of the time measurements are under
control of the observer, all of the marginal observation time
density parameters are constraint parameters.
Set up a different experiment in which a clock is triggered
to measure the time the particle passes a detector at a
known position. This experiment can be used to measure
mean velocity, and provides an example where the mean
position is a constraint parameter and the mean time is not a
constraint.
A verifiable theory with a time and spatial components
includes adjustable constraints corresponding to both types
of experiments, which are elements of the confirmation set
of the theory. A prominent characteristic of observation
time distributions is the tendency whenever possible to
regard the time distribution parameters as experimental
constraints. This tendency is prevalent because time values
are generally considered to be under the control of the
observer to the maximum extent possible.
These
consideration suggest the time postulate, which is
1.
Time Postulate
Every parameter that selects a marginal distribution of
an observation time independent of any other random
LORENTZ TRANSFORMATION FROM INFORMATION
observation variables is a control parameter, and thus
an experimental constraint, in at least one Adjustable
Constraint set element of a Confirmation Set of a
theory with time dependence, and it is this quality of
the necessity of control that distinguishes time from all
other quantities.
To the extent that every mechanism is repeatable and
controlled, all mechanisms are clocks. Under the time
postulate, parameters which describe a clock’s data must all
be under the control of an observer in at least one
verification experiment, and are therefore constraint
parameters for that experiment.
Each confirmation set of a time dependent theory must
contain at least one adjustable constraint in which all
parameters of the marginal observation time density are
constraint parameters. The design of the experiment must
select one of these time parameter adjustable constraints as
a primary constraint, to serve as the focus of statistical
decisions. Only the primary constraint is used to determine
the excess discrimination invariant. This is because only a
constrained time parameter marginal density provides
potential operator selection of all possible time
measurement scenarios that are under control of the
observer with certainty. Parameters in other adjustable
constraints in a confirmation Set are incomplete in that they
do not provide certainty of access to all possible
measurement times, for if they did the time postulate
requires that they also be time, potentially as a parameter of
a clock mechanism.
B. Discrimination Information
A sequence of experiments which measure position and
observation time of a single particle produces a sequence of
observation data and a corresponding sequence of
observation data probability density parameters. Parameter
values which are far from values expected for the uniform
motion of a free particle are not representative of the
uniform motion of the originally observed free particle.
When trajectory information is the only available particle
identity information then only parameters along an initial
trajectory maintain the identity of the particle system.
Let P0 be the initial particle system observation state
parameter, and P1 the observation state of a subsequent
experiment with new constraint parameter values. The
discrimination information available in the density in favor
that analysis of data selects P1 over P0 is
f (R : P1 )
∫ dxdydzdt f (R : P1 ) ln f (R : P )
0
∀R
∞
m (δt1 : σt 1 )
= ∫ dt m (δt1 : σt 1 ) ln
+
δ
σ
m
t
:
(
)
t
0
0
−∞
h (ε1 : C1 )
δ
σ
ε
dxdydzdt
m
t
:
h
:
C
ln
(
)
(
)
∫
t1
1
1
1
h (ε0 : C0 )
∀R
I (P1, P0 ) =
(10)
which evaluates to
σ
1 (∆t )
1
=
− ln t 1 − σt 12 ∆σt −2
2
σt 0 2
2 σ
2
I 10
t0
IV. CHANGE OF OBSERVATION STATE
A. Parameter Shifts
A single particle probability density in observation time
and position random variables models data from an
experiment of finite duration. Motion can continue outside
the range of data from an experiment, which presents the
opportunity for observation data collected by additional
experiments.
The density for each experiment is
represented by different parameter values.
Particle observation state is given by the probability
density which represents data from an experiment.
Different densities represent different experiments, and
different particle observation states. Motion of a particle
through different experiments occurs with a change of
particle observation states. Transitions between observation
states of the particle are represented as changes in
parameters of the probability density.
where
detC 1
1
1
− ln
− C 1∆C −1
(11)
2 detC 0
2
1
1
+ σt 12 ∆V TC 0−1∆V + δΠTC 0−1δΠ
2
2
∆ indicates a component of P1 − P0 and
δΠ = ∆X −V0∆t .
C. Preservation of identity
An initial observation state parameter P0 shifts to a
parameter P1 for a single particle system when controlled
constraint parameters, such as the mean observation time
component of P1 , shift to planned values. Adjustable
constraint sets provide structures to constraints which can
be used by all observers. A particular adjustable constraint
set defines possible constraint values for a particular type of
constraint associated with an experiment.
LORENTZ TRANSFORMATION FROM INFORMATION
Parameters R1 which preserve identity of a system with
observation state parameter P0 under a shift in constraint
value must minimize discrimination information and so
must satisfy
I (R1, P0 ) =
(
min
(
)
p∈C ζ1 :Αζ (Q )
I (p, P0 )
(12)
) = M (R1 : Αζ (Q )) .
where C ζ1 : Αζ (Q )
The time postulate requires t1 and σt 1 be constraint
parameters.
Position error in pre-quantum theory C 1 is
under the control of the observer, so C 1 is also a constraint
)
as constraint
parameter, only parameters X1 and V1
remain to be
parameter.
With
(
ζ1 = t1
σt 1 C 1
adjusted in (11) to satisfy (12). The identity preservation
parameters which solve (12) are
X1 = X 0 +V0 ∆t
V1 = V0
(13)
with t1 , σt 1 and C1 as constraint parameters with known
values. Thus the parameters that preserve identity continue
motion along a trajectory with the same initial velocity on
the same line.
V. EQUIVALENT OBSERVERS
A. Experiment Verification
Acceptance of the outcome of an experiment demands an
independent method of verification. The demand can be
met in traditional physics where there are implicit
assumptions of independent, identifiable, characteristics of
the physical system outside the scope of a theory. One
example is the Kepler-Newton theory of orbital motion of
the planet Mars about the sun, where deviations from the
orbit can be observed since the planet is identifiable by
characteristics, such as surface features and diameter,
outside the scope of the orbital theory. When the theory
contains all of the identity information of the physical
system, verification cannot be based on system
characteristics independent of the theory. An example of
such a theory is the bosonic theory of photons, which are
indisitnguishable particles fully identified by symmetric
wave functions, where no single photon experiment can be
repeated using exactly the same photon [8].
Exclusion of verification based on independent system
characteristics elevates the significance of verification
restricted to statistical analysis of repeatability which
underlies the probability densities of the theory.
Repeatability can be verified by observers outside the scope
of the theory. Thus by keeping the theory incomplete in the
sense that it does not bring all observers within the scope of
the theories physical descriptions, the requirement of
verification with separate system identification moves to a
requirement of independent observer verification,
especially when the entire physical system description is
contained within the theory. Verification of a bosonic
photon experimental result never involves preparation of
the same photon by a different observer, but instead
repetition of the same experimental conditions by an
independent observer. Any theory that fully contains the
identity of a system cannot support validation based on
independent characteristics of the system.
In the theory of special relativity independent
observers are represented by Lorentz transformations of
space time co-ordinates. Lorentz transformations in one
approach arise from quantum consideration [5].
In
quantum theory the wave function becomes the object of
the transformation through invariant wave equations,
associated with corresponding transformations of
observables. An information theoretic approach to define
the reference frame of equivalent observers has the
opportunity to generalize the transformation to another
observer to be a transformation of the probability
distribution for any type of random variable and theory,
without limitation to only space and time quantities. Define
an equivalence transformation to be a generalized
transformation of the distribution, and therefore of the
distribution parameters, to the frame of an equivalent
observer. An identity transformation of the probability
distribution of a theory defines the local observer and is
also an equivalence transformation.
A verification
capability requires that at least one other observer exist that
is not the local observer. A complete and independently
verifiable scientific theory has the property of:
1.
Independent Observer Scrutiny
Every independently verifiable theory must support at least
one independent observer other than the local observer,
and must necessarily have at least one equivalence
transformation that is not the identity transformation.
B. Equivalence Transformations
An equivalent observer must reach the same conclusions
from data available to the observer as the local observer of
an experiment reaches. Each equivalent observer then has
the same information to accept or reject the conclusion of
identity preservation under a shift in constraints. The
design of the experiment makes the transformation of all
initial parameters available to all observers, and execution
LORENTZ TRANSFORMATION FROM INFORMATION
of the experiment results in estimates of final parameter
values for all observers. An equivalence transformation E
transforms parameters P in use by an observer into
parameters P ′ = E P used by an equivalent observer. If
E is the equivalent observer transformation, the final
discrimination information determined by the equivalent
observer is I (E P1, E P0 ) , but the same observer would
determine that discrimination as I (E R1, E P0 ) if identity
preservation parameters were observed as expected by the
theory, where R1 is given by (13). The larger the
magnitude of the difference between the two discrimination
values the more likely is a rejection of the conclusion that
identity is preserved by the tested theory, while a very
small magnitude difference supports the conclusion. The
difference is proportional to the number of bits input into
statistical decision methods and for the local observer is
equal to
(
)
where
(
ζ1 = t1
the
R1
min
(
)
p ∈M P1 :Αζ (Q )
constraint
C. Equivalent Observer Collections
as a set
Define an equivalent observer collection C
containing a set E = {E 1, …, E n , …,} of equivalence
transformations using primary adjustable constraint Αζ
which is also a member of C , a parameter domain D ⊆ Q
and range R ⊆ Q common to all transformations withinE ,
{
I (p, P0 )
parameter
(
)
required
of
an
are:
Containment Structure
Every equivalence transformation E is an element
of a set E contained in some equivalent observer
collectionC , E ∈ E ∈C . A theory may produce
more than one equivalent observer collection.
2.
choose constraints from an adjustable constraint Αζ
(15)
which must be the primary constraint of a
confirmation set of adjustable constraints. The time
postulate implies that all parameters of the model
marginal probability distribution of observation time
within Αζ are constraint parameters.
In the invariant equation (15) the same adjustable constraint
Αζ is used by all equivalent observers of the experiment.
Equivalence transformations are dependent on the primary
adjustable constraint.
An equivalent observer transformation operates on all
in a subset P ⊂ Q of the set of
parameters P ∈ D
parameters Q . As every equivalent observer is also an
equivalent observer to any other equivalent observer, the
inverse E −1P exists for every P in the range of E and is
an equivalence transformation.
An equivalence transformation defines observers in the
context of a theory. Equivalence transformations depend
on properties of the Primary Constraint of a Confirmation
Set. In the Primary Constraint all parameters of the
marginal time distribution are constraint parameters; and
the equivalence transformations for that adjustable
constraint define the transformations which must be
associated with all remaining adjustable constraints in the
Confirmation Set.
Equivalence transformations are defined by properties,
such as invariance, of each transformation. Properties of
Time Postulate Primary Constraint
All equivalent observers of an experiment on a
probability model of a physical system, with each
observer represented by an element E ∈ E ∈C
K is input to statistical methods which determine rejection
or acceptance of identity preservation, K is invariant:
)
properties
values
)
(
The
equivalent observer collectionC
(14)
σt 1 C 1 are also component values in P1 . Since
K P1, P0 : Αζ = K E P1, E P0 : Αζ
}
C = E ,D , R , Αζ .
1.
K P1, P0 : Αζ = I (P1, P0 ) − I (R1, P0 )
= I (P1, P0 ) −
collections of transformations, such as the verifiable theory
property of independent observer scrutiny, also contribute
to the definition of equivalence transformations.
3.
Invariant Excess Identity Discrimination
Every E ∈ E preserves
(
)
K P1, P0 : Αζ .
identity
discrepancy
Let S = {B1, …, Bn , …,} be the
set of all transformations which preserve K . Then
∀Pi ∈ D , K P1, P0 : Αζ = K B i P1, B i P0 : Αζ ,
(
)
(
)
and E ⊆ S .
4.
Independent Observer Scrutiny
Every E contains at least one E not the identity.
5.
Transitivity
LORENTZ TRANSFORMATION FROM INFORMATION
{
If C ′ = E ′,D , R , Αζ
}
{
and C = E , R , H , Αζ
}
are
equivalent
operator
collections,
then
C ′′ = E ′′,D , H , Αζ must also be an equivalent
{
}
operator collection with every equivalence
transformation E ′′ ∈E
E ′′ equal to the composition
E and E ′ ∈E ′ , where
E ′′ = E ′E for some E ∈E
H can be any domain or range in an equivalent
operator collection. This property implies there must
always exist a collection for which the
transformation domain equals the range.
6.
makes (16) a third order invariant in parameter shifts.
The Equivalent observer collection property of Invariant
Excess Identity Discrimination requires invariance of K
K =0
then
and
implies
that
if
so
that
identity
K ′ = K E P1, E P0 : Αζ = 0 ,
(
E
−1
exists
for
E ∈ E ∈C = {E ,D , R , Αζ } , and each E
all
−1
is a
valid equivalence transformation and an element of
an
equivalent
observer
collection
C
−1
E
−1
{
= {E
= E
−1
}
, R ,D , Αζ ,
Identity preservation equation (13) transforms to
X1′ = X 0′ +V0′∆t ′
V1′ = V0′
}
. Every E is a one to
one map.
µ = X −Vt
η = σt
Maximal Transformation Set
{
Every E ∈C = E ,D , R , Αζ
} contains all
possible E that have properties 1 through 6
and produce only valid probability models.
b. Maximal Parameter Set
{
}
Every D ∈C = E ,D , R , Αζ and G ∈C
contain all possible parameter values P
that give valid probability models for all
transformations which satisfy the Maximal
Transformation Set property.
These properties define an equivalent observer collection,
and the equivalent transformations contained within the
collection. There can be more than one observer collection
for each probability model.
D. K Invariance
Invariant K is defined by (14), and for the norm density
space and time observation model is equal to
K=
T
1
∆X −V0∆t ) C 0−1 (∆X −V0∆t )
(
2
1
+ σt 12 ∆V TC 0−1∆V
2
(16)
τ =t
ε =C
υ =V
(18)
−1
so that the identity preservation equations (13) and (17)
become
Maximally Inclusive
a.
(17)
Each component of P ′ potentially depends on components
Represent components of parameter P by the
of P .
invertible transform
where
−1
, …, E −1n , …
1
)
preservation
is
invariant.
With
parameter
′
′
′
P = (X ,V , t , σt ,C ) set E P = P = (X ,V , t ′, σt′,C ′)
Inverse Existence
Unique
7.
The presence of σt 12 = σt 02 + ∆σt 2 in the second term
µ*1 = µ0
µ1′* = µ0′
υ*1 = υ0
υ1′* = υ0′
with τ1, η1, ε1 as constraint parameters and
(19)
*
to indicate
identity preservation parameters. In the following analysis
assume the transformations are continuous functions of
parameters and continuous for all first through third
derivatives with respect to components of P0 . Differentiate
µ1′* and υ1′* in (17) with respect to τ1, η1 and ε1 , and use
(19) to get
∂
∂
υ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
υ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂τ1
∂τ1
∂
∂
υ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
υ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂η1
∂η1
∂
∂
υ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
υ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂ε1
∂ε1
∂
∂
µ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
µ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂τ1
∂τ1
∂
∂
µ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
µ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂η1
∂η1
∂
∂
µ ′ (µ0 , υ0 , τ1, η1, ε1 ) =
µ ′ (µ0 , υ0 , τ 0 , η0 , ε0 ) = 0
∂ε1
∂ε1
which imply that µ ′ and υ ′ can be written as
LORENTZ TRANSFORMATION FROM INFORMATION
υ ′ = υ ′ (µ, υ)
µ ′ = µ ′ (µ, υ)
(20)
Next differentiate (21) with respect to τ0 :
In the P = (µ, υ, τ, η, ε) representation of parameters the
invariance equation K = K ′ becomes
η1
T
1
∆υT ε0∆υ + (∆µ + τ1∆υ) ε0 (∆µ + τ1∆υ) =
2
2
(21)
η1′
T
1
T
′
′
′
′
′
′
′
′
′
′
∆υ ε0∆υ + (∆µ + τ1∆υ ) ε0 (∆µ + τ1∆υ )
2
2
Differentiate (21) with respect to η0 to get
η1′
∆υ ′T
∂ε0′
∆υ ′
2
∂η 0
T ∂ε0′
1
+ (∆µ ′ + τ1′∆υ ′)
(∆µ ′ + τ1′∆υ ′)
2
∂η 0
0=
η1′
∂ε0′
∆υ ′
2
∂τ 0
T ∂ε0′
1
+ (∆µ ′ + τ1′∆υ ′)
(∆µ ′ + τ1′∆υ ′)
2
∂τ 0
0=
∆υ ′T
∂ε ′
= 0.
∂τ
Take the second derivative of (21) with respect to η1 :
Analysis similar to that for (22) results in
0=
(22)
1 ∂2
η ′ + τ1′2 ∆υ ′T ε0′ ∆υ ′
2 ∂η 2 1
(
)
1
Extraction of valid equivalence transformations E begins
with determination of transformations B which keep K
invariant, with initial assumption that the domain D and
range R are the set of all parameters. Then D and R are
adjusted to satisfy the properties of equivalent observer
collections.
Components of P1′ appear in (22) but not components of
(24)
+
∂2 τ1′
∂η12
(25)
∆υ ′T ε0′ ∆µ ′
. Components of P0′ appear in (25) but not components of
P0 , so P0′ can be any valid parameter. Select µ0′ = µ1′ to
Since ε0′ is an inverse co-
remove the second term.
P1 , so the initial assumption that range and domain equal
variance matrix, ∆υ ′T ε0′ ∆υ ′ is almost positive definite,
D implies P1′ can be any valid parameter in the D . Then
and positive for some ∆υ ′ so that
∆µ ′ , ∆υ ′ and τ1′ can take on an infinity of values.
To establish a procedure that reveals the impact of the
infinity of P1′ values that satisfy (22), define unit vectors
(
)
jˆ1T = 1 0 0
,
(
)
(
jˆ2T = 0 1 0 ,
(
)
1 ˆ ˆ
1
kˆij =
( ji + jk ) and lˆij = jˆi − jˆj .
2
2
3 x 3 symmetric matrix
(
)
jˆ3T = 0 0 1 ,
If S is any
)
equations to replace a ∆υ ′T S ∆υ ′ term with a 2Sik .
Select µ1 such that ∆µ ′ = 0 in (22), and select any small
scalar a . Adjust υ1′ to apply the procedure to use (23) for
∂ε0′
∂ε ′
a
= 0.
, so that
η1′ + τ1′2
∂η
∂η0
2
(
)
with the result
2
∂2 η ′
∂η 2
=
∂2 τ ′
∂η 2
= 0.
The third derivative of (21) with respect to τ1 is:
(23)
to akˆij for some scalar a and then to alˆij and subtract the
0=
(η ′ + τ ′ ) = 0 .
1 ∂3
η ′ + τ1′2 ∆υ ′T ε0′ ∆υ ′
2 ∂τ 3 1
(
)
1
If i = k , (23) becomes jˆiT Sjˆi = Sii . In (22), set ∆υ ′ first
2
∂η 2
Values of ∆µ ′ exist for which ∆υ ′T ε0′ ∆µ ′ is not zero,
0=
1 ˆT ˆ
k Skij − lˆijT Slˆij = Sik
2 ij
each index pair i, j with indices 1… 3 .
∂2
The result is
+
∂ 3 τ1′
∆υ ′T ε0′ ∆µ ′
∂τ13
so that by following a similar analysis to that used for
η,
derivatives
with
respect
to
∂3
3
(η ′ + τ ′ ) = ∂∂ττ
2
′
=
∂3η′
= 0 . Presence of a non3
∂τ 3
∂τ 3
zero quadratic terms in the expression integrals violates the
∂2 τ ′ ∂2 η ′
=
= 0.
unique Inverse Existence property, so
∂τ 2
∂τ 2
The second derivative of (21) with respect to τ1 is then:
LORENTZ TRANSFORMATION FROM INFORMATION
 ∂τ ′ 2
∆υT ε0 ∆υ =  1  ∆υ ′T ε0′ ∆υ ′
 τ1 
∂
(26)
that follows (23) to get
Integrate the η ′ equations to get η ′ = a τ + g η + bτη + h
where a, b, g, h do not depend on τ or η . Since τ is a
mean time with potential unbounded values that can be
negative or positive, non-zero a and b allow a negative
variance η ′ and invalid probability model.
Thus
a = b = 0 and η ′ = g η + h . Integrate the τ ′ derivatives
to obtain τ ′ = q τ + pτη + s η + r where q, p, s, r do not
depend on τ or η . Substitute this expression for τ ′ into
(21) and isolate the η12 and η12 τ12 terms which appear
only on the right transformed side of the equation. The
resultant equation requires that p = s = 0 and
τ ′ = q τ + r , with q, r independent of τ or η . A first
derivative
of
(21)
∆υT ε0 (∆µ + τ1∆υ) =
implies
∂τ1′
∂τ1
τ1 results
by
∆υ ′T ε0′ (∆µ ′ + τ1′∆υ ′)
in
and
let P1 approach P0 , P1 → P0 = P
get
∂η ′
= 0. Then select ∆µ ′ = 0 to
∂εij
∂τ ′
= 0 . Next differentiate (21) by η1 to get
∂εij
∆υT ε0 ∆υ =
∂η1′
∂η1
∆υ ′T ε0′ ∆υ ′
The derivative of (29) by µ0 is 0 =
and
implies
0=
∂2 η1′
∂η1∂µ1
∂ε ′
= 0.
∂µ
∂η1′
∂η1
∆υ ′T
Differentiation
∆υ ′T ε0′ ∆υ ′ .
∂2 η ′
∂η 2
by
∂ε0′
∂µ0
µ1
∂2 η ′
=0
∂η∂µ
The
combined with the prior
(29)
∆υ ′
is
result,
= 0 result integrates to
η ′ = g (υ ) η + h (υ ) for some positive functions g, h .
Differentiate (21) first by ε0ab and then by ε0kl to get
∂τ ′
=q ≠ 0.
∂τ
Using results obtained so far, apply
Adjust µ0′ so that ∆µ ′ = −τ1′∆υ ′ and use the procedure
∂2
∂µ12
0 = η1′∆υ ′T
to (21) and then
∂ε0′
∂ε0ab ∂ε0kl
(∆µ ′ + τ1′∆υ ′)
T
to get
∆υ ′ +
∂ε0′
∂ε0ab ∂ε0kl
(∆µ ′ + τ1′∆υ ′)
(30)
T
ε = η′
∂υ ′
∂υ ′
ε′
∂µ
∂µ
T
 ∂µ ′
 ∂µ ′
∂υ ′ 
∂υ ′ 
′
 ε ′ 

+ 
+ τ′
+
τ
 ∂µ
 ∂µ
∂µ 
∂µ 
Differentiate by τ twice to obtain
(27)
∂ε0ab ∂ε0kl
, differentiation of (30) by η1′ as an
τ1′ = 0 so that ∆µ ′T Q0 ∆µ ′ = 0 .
Then (30) becomes
∆µ ′T Q0 ∆υ ′ = 0 . Since µ1′ and υ1′ , and thus ∆µ ′ and
∂υ ′
= 0 and, with (20) that υ ′ is a
∂µ
function of only υ , υ ′ = υ ′ (v ) .
∆υ ′ can be arbitrarily and independently selected, only
Q = 0 can satisfy the equation for all possible P ′
0
1
component values. Thus
∂εij′
∂εab ∂εkl
= 0 , each εij′ depends
linearly on the εab components, and the inverse co-variance
The derivative of (21) with respect to εij 1 is
1 ∂η1′
∆υ ′T ε0′ ∆υ ′
0=
2 ∂εij 1
∂τ1′
+
∆υ ′T ε0′ (∆µ ′ + τ1′∆υ ′)
∂εij 1
∂ε0′
independent variable yields ∆υ ′T Q0 ∆υ ′ = 0 . Next select
 ∂τ ′ 2 ∂υ ′T ∂υ ′

0 = 
ε′
 ∂τ  ∂µ
∂µ
which shows that
With Q0 =
matrix has the form
(28)
εij′ = Tijkl (υ ) εkl + Dij (υ )
(31)
LORENTZ TRANSFORMATION FROM INFORMATION
for some T , D . Substitute (31) into (21) and let
ε = φ (υ )
2
components of matrix ε0 approach zero, ε0 → 0 , or
equivalently isolate terms with no εkl factors, to get
η1′
∆υ ′T D (υ0 ) ∆υ ′
2
(32)
T
1
+ (∆µ ′ + τ1′∆υ ′) D (υ0 )(∆µ ′ + τ1′∆υ ′)
2
0=
Variation of η1′ gives ∆υ ′T D (υ0 ) ∆υ ′ = 0 , selection of
τ1′ = 0 gives ∆µ ′T D (υ0 ) ∆µ ′ = 0 .
arbitrarily
and
εij′ = Tijkl (υ ) εkl .
∆µ ′ and
independently,
Any
linear
P1
Let
∆υ ′ can be selected
so
D (υ0 ) = 0 and
transformation
ε ′ = U (υ ) εU (υ )
T
of
(33)
into (21) , remove the η1 factor terms, select µ0′ so that
∆µ = −τ1∆υ to get
∆υ ′T ε0′ ∆υ ′
T
1
+ (∆µ ′ + τ1′∆υ ′) ε0′ (∆µ ′ + τ1′∆υ ′) = 0
2
(34)
T
Negative
h (υ) allows
improper probability model so that
negative η ′ in an
h (υ ) = 0
and
η ′ = g (υ ) η = φ (υ ) η .
2
Multiply (29) by η1 to get
1
1 ∂η1′
η1∆υT ε0 ∆υ =
∆υ ′T ε0′ ∆υ ′
2
2 ∂η1
2
1
(35)
= η1φ (υ ) ∆υ ′T ε0′ ∆υ ′
2
1
= η1′ ∆υ ′T ε0′ ∆υ ′
2
Take the second derivatives of (35) by components of υ1
and let P1 → P0 = P to get
ε=
T
T
∂µ ′
∂µ ′ ∂µ ′
∂µ ′
=
ε′
U (υ) εU (υ )
∂µ
∂µ
∂µ
∂µ
(38)
Then
T
T
∂
∂2 µ ′
∂µ ′
ε=2
U (υ) εU (υ)
=0
2
∂µ
∂µ
∂µ
so
∂2 µ ′
that
∂µ2
=0,
and
µ′
is
linear
in
µ,
µ ′ (µ, υ ) = S (υ ) µ + l (υ ) for some matrix S (υ ) and vector
l (υ ) .
Apply
∂2
∂µ12
ε0 =
to (37) to get
∂2 τ1′
∂µ12
∆υ ′T ε0′ (∆µ ′ + τ1′∆υ ′)
 ∂µ ′ ∂τ ′
T  ∂µ ′ ∂τ ′


1
1
+ 
+
∆υ ′ ε0′  1 + 1 ∆υ ′


 ∂µ1 ∂µ1

 ∂µ1 ∂µ1

The almost positive definite property of ε0′ in (34) implies
h (υ ) ≤ 0 .
P1 → P0 = P in the second
approach P0 ,
derivative of (37) with respect to components of µ to get
a
Substitute the derived form for η ′ , η ′ = g (υ ) η + h (υ ) ,
2
(37)
As µ1′ does not
covariance matrix can take the form U T εU for some
orthogonal matrix, so there exists matrix U (υ) such that
h (υ )
(36)
Subtract (35) from (21) to get
T
1
∆µ + τ1∆υ) ε0 (∆µ + τ1∆υ) =
(
2
T
1
∆µ ′ + τ1′∆υ ′) ε0′ (∆µ ′ + τ1′∆υ ′)
(
2
depend on τ1′ , then the only term that remains is
∆µ ′T D (υ0 ) ∆υ ′ = 0 .
T
∂υ ′
∂υ ′
ε′
∂υ
∂υ
Variation δµ1′ gives
∂2 τ ′
∂µ 2
(39)
= 0 so that τ ′ depends linearly
on µ .
Since every co-variance inverse co-variance matrix is
almost positive definite and symmetric, there exists matrix
Ζ such that ε = ΖT Ζ , and (38) can be written
T

∂µ ′ −1 
1 = ΖU (υ)
Ζ 
∂µ


Then Ζ (ε)U (υ)

∂µ ′ −1 
Ζ 
ΖU (υ)
∂µ


(40)
−1
∂µ ′
Ζ (ε) = O for some orthogonal
∂µ
matrix O .
Analysis performed to this point implies the equivalence
transformation
components
show
the
following
dependencies:
LORENTZ TRANSFORMATION FROM INFORMATION
S (υ ) = L − υ ′lT
η ′ = η ′ (η, υ) = φ (υ) η
µ ′ = µ ′ (µ, υ) = S (υ) µ + d (υ)
2
(
ε ′ = ε ′ (ε, υ) = U (υ) εU (υ)
τ ′ = τ ′ (τ, υ, µ) = φ (υ) τ + l (υ) µ + b (υ)
T
Differentiate (37) by µ1 , then by τ1 and finally by υ1 and
with the sign of φ chosen to make +φ be the τ coefficient of τ ′ .
Using (41), equations (38) , (36)
orthogonal matrices Ra , Rb .
become, for some
−1
Ζ (ε)U (υ) S (υ ) Ζ (ε) = Ra
RaT Ra = 1
(42)
−1
∂υ ′
φ (υ) Ζ (ε)U (υ)
Ζ (ε) = Rb RbT Rb = 1
∂υ
Using index notation with double index summation
convention, the first of these equations can be written as
Ζil (ε)U lm (υ ) Smj (υ ) = Rail Ζlj (ε)
(43)
Any non-singular matrix Ζ results in a non-singular almost
positive definite matrix ε . As there are no limitations on
possible values of ε in the model beyond those required of
an inverse covariance matrix, Ζ can be any non-singular
3x3 matrix. Variation of Ζij in (43) shows that
U (υ ) S (υ ) = Ra = r 1
equation of (42) shows that
φ (υ)U (υ)
∂υ ′
= ±1
∂υ
∂υ ′
= ±S (υ)
∂υ
Differentiate (37) by µ1 and then by µ0 to get
φ (υ )
(44).
+
∂µ1
(
)
vector κ and constant matrix Q such that
∂φ
∂υ ′
= −w κ φ
− w υ ′κT = Q
∂υ
∂υ
with solutions
uυ + j
υ −s
φ = w 1 − κT v
υ′ =
=Q
w 1 − κT v
w 1 − κT v
(
)
(
)
(
)
with constant scalar w and constant vector j = −Qs .
And since
∂υ ′
φ (υ )
= Q + w υ ′κT = ±S (υ) = ± L − υ ′lT
∂υ
Q = ±L, l = ∓wκ .
(
)
0=
T
1
∆d + b1∆υ ′) ε0′ (∆d + b1∆υ ′)
(
2
Since the right side cannot depend on P0 , b (υ ) = r for
constant vector u .
With these results the τ1 term of (37) is
(45)
τ1∆υT ε0 ∆µ =
( (
) (
) )
φ (υ1 ) τ1∆υ ′T ε0′ ∆ S (υ ) µ + lT µ1 ∆υ ′
∆υ ′ = S (υ1 ) + υ1′l (υ1 ) − υ0′l (υ1 )
T
becomes
some constant scalar r and d (υ ) = u − r υ ′ (υ ) for some
 ∂µ ′ ∂τ ′
T
∂µ0′
ε0 =  1 + 1 ∆υ ′ ε0′
 µ1 ∂µ1

∂
∂µ0
∂µ1
terms
to
get
T
T

−1
 ∂υ ′
∂φ
∂φ 
ε0′−1ε0 LT − lv0′T
= φ1 1 + υ1′ 1 − υ0′ 1 
 ∂υ1
∂υ1
∂υ1 


The right side of this equation must not depend on
components of P1 , which requires that there exist constant
which requires
∆d + b1∆υ ′ = 0
d (υ1 ) + b (υ1 ) υ ′ (υ1 ) = d (υ0 ) + b (υ1 ) υ ′ (υ0 )
and thus
T
re-arrange
In (37) set µ1 = µ0 = 0 and τ1 = 0 so that the equation
is the product of scalar and the identity matrix. Since
Similar analysis of the second
RaT Ra = 1 , ra = ±1 .
∂τ1′
τ ′ = φ (υ ) τ + l T µ + b (υ )
(41)
υ ′ = υ ′ (υ )
so that
∂µ1′
)
µ ′ = L − υ ′lT µ + d (υ )
T
must not depend on any component of P1 . This requires
that l be constant and that for some constant matrix L ,
The µ1 derivative of the τ1 factors is
(
∆υT ε0 = φ (υ1 ) ∆υ ′T ε0′ S (υ1 ) + ∆υ ′lT
= φ (υ1 ) ∆υ ′T ε0′S (υ0 )
and a subsequent υ1 derivative is
)
LORENTZ TRANSFORMATION FROM INFORMATION
T
 ∂φ (υ )T
∂υ1′

1 
ε ′S (υ0 )

 + φ (υ1 )
∂υ1 0
 ∂υ1 
 ∂φ (υ )T
T

1 

 + ±S (υ1 ) ε0′S (υ0 )
 ∂υ1 
space and time parameter transformations, but also
transformations of velocity, time variance and position
given time co-variance from an invariant that is
conceptually simpler than the Minkowski space-time
separation.
To see this consider that for the case
2
C = σc 0 1, the invariant is
Let P1 approach P0 to get ε = ±S (υ ) ε ′S (υ ) . Since ε
2
2
1 −2
1
σc 0 (∆X −V0∆t ) + σt 12 σc 0−2 (∆V ) .
2
2
Analysis leading up to transformation (47) shows that K is
the sum of two invariant terms, namely
2
1
K X = σc 0−2 (∆X −V0 ∆t )
2
2
1 2 −2
KV = σt 1 σc 0 (∆V )
2
(
)
ε0 = ∆υ ′ ε0′S (υ0 )
T
(
)
= ∆υ ′T ε0′S (υ0 )
T
and ε ′ are positive definite, only the positive sign is valid
and ε = S (υ ) ε ′S (υ ) .
T
These results allow (41) to be written as
(
η ′ = w 2 1 − κT v
υ′ = Q
(υ − s )
(
)
2
η
(
(
)
ε ′ = Q + w υ ′κT
(
)
A Lorentz transformation extended to include
transformation of σt , σc , and V keeps these terms
)
w 1 − κT v



(υ − s ) T 
µ ′ = Q 1 +
κ  µ + u − rυ′


1 − κT v


T −1
)
(
ε Q + w υ ′κT
K=
(46)
invariant, and so maintains an elliptic, rather than a
hyperbolic invariant, although in a parameter space of up to
21 dimensions.
−1
)
E. Transformation Selectors
τ ′ = w 1 − κ v τ − wκ µ + r
T
T
a. Transitivity, Inverse and Composition
Restoration of the original parameter representation makes
the solution of the invariance equation be
σt′2= w 2θ (V ) σt 2
2
V′ = P
V −s
θ (V )
X ′ = w P (X − st ) + u
(
) (
C ′ = w 2 P +V ′κT C P +V ′κT
(
)
(47)
)
T
t′ = w t −κ X + r
T
Q
and θ (V ) = 1 − κTV . The
w
transformations given by selection of values of w, k, s, P, r
and u in (47) are transformations that keep K invariant
and satisfy the Invariant Excess Identity Discrimination
property of equivalence transformations under stated
assumptions of continuity of derivatives. If w = γ ,
with P defined by P =
S = (w, κ, P, s, u, r ) is a transformation selector, which
picks a single transformation from the set of all
transformations defined by (47), and acts as an index of
transformations that preserve identity discrepancy.
Remaining equivalence transformation properties determine
the allowed selectors contained within equivalent observer
collections, along with allowed domain and range
parameter sets. There is a one to one correspondence
between selectors and invariant preserving transformations,
so an equivalent observer collection set of transformations
E can be represented by a set of selectors S . This also
implies that C = E ,D , R , Αζ can be represented by
C
{
= {S ,D , R , Αζ } .
Exclusion of invalid probability models is implicitly
required by the Maximally Inclusive properties. In (47),
valid probability models require non-zero time variance and
non-zero covariance matrix determinant, so that
k = −s γc−2 , P = 1 , r = 0 and u = 0 are selected, with
γ=
1
−
2 −2 2
1−s c
(
)
w≠0
θ (V ) = 1 − κTV ≠ 0
, then the X ′ and t ′ transformations of
(47) correspond to a simple instance of the Lorentz
transformation with moving frame velocity s . The set of
transformations defined by (47) contains but is larger than
the Poincare group. Invariance of K produces not only
}
T
κ V ≠1
and
(48)
LORENTZ TRANSFORMATION FROM INFORMATION


 V − s T 
det P +V ′κT = det (P ) det 1 +
κ  ≠ 0


θ (V )


det (P ) ≠ 0
(49)


T
 V − s T 
V
s
s
κ
−
−
1
κ  = 1 + κT
det 1 +
=
≠0


V
θ
1 − κTV
1 − κTV
(
)


κT s ≠ 1
(
)
In the selector representation, the Transitivity property
requires
that
if
and
C = S ,D , R , Αζ
{
{
C ′ = S ′, R , H , Αζ
}
}
(
εɶ = Pɶ 1 − sɶκɶT
λɶ = 1 − κɶ sɶ
w ′ = λɶ−1wɶ −1
κ ′ = −PɶT −1κɶ
{
}
collection with selectors formed from all compositions of
This implies that given selector
S ′ and S .
with associated invariant
S ′ = (w ′, κ ′, P ′, s ′, u ′, r ′)
transformation E ′ , and selector S = (w, κ, P, s, u, r ) with
E , the transformation E ′E P = E ′′P for P ∈ D must be
of
the
form
of
S ′′ = (w ′′, κ ′′, P ′′, s ′′, u ′′, r ′′) .
(47)
with
selector
ε = P + s ′κT and
Define
λ = 1 + κ ′ Ps . Work through the algebra to get
T
w ′′ = λw ′w
(
κ ′′ = λ−1 κ + PT κ ′
u ′ = −wɶ −1εɶ−1 (uɶ + rɶPɶsɶ)
r ′ = −λɶ−1wɶ −1 rɶ + κɶT Pɶ−1uɶ
(
(
Next form the composition of a transformation E from
D → R , represented by S = (w, κ, P, s, u, r ) , with the
inverse of any transformation Eɶ from D → G , Eɶ −1 from
R → D . The composition E ′′ = Eɶ −1E is an equivalence
transformation from D → D . With Sɶ = (wɶ, κɶ, Pɶ, sɶ, uɶ, rɶ) to
represent Eɶ and S ′′ = (w ′′, κ ′′, P ′′, s ′′, u ′′, r ′′) to represent
(
−1
r ′′ = r ′ + w ′ r − κ ′T u
as the inverse transformation selector. The existence of an
inverse (51) requires existence of Pɶ−1 so det (Pɶ) ≠ 0 .
εɶ = Pɶ 1 − sɶκɶT
)
(
εˆ = P 1 − sɶκT
)
λˆ = 1 − κɶ s
T
T
λɶ = 1 − κɶ sɶ
s ′′ = ε−1 (s ′ + Ps )
u ′′ = w ′P ′ (u − rs ′) + u ′
)
E ′′ , use (51) to form the inverse selector and (50) to form
the composition selector with result
)
P ′′ = λ P ′ε
(51)
ɶɶ−1
P ′ = λε
s ′ = −Pɶsɶ
are equivalent observer collections,
then C ′′ = S ′′,D , H , Αζ is an equivalent observer
)
T
(50)
)
as the selector formed by the composition of equivalence
transformations.
The Inverse Existence Property of equivalent observer
collections requires the inverse of an equivalence
transformation to be an equivalence transformation within
an associated inverse collection. Then there must be a
transformation selector for the inverse transformation. Let
represent
an
equivalence
Sɶ = (wɶ, κɶ, Pɶ, sɶ, uɶ, rɶ)
transformation E from domain D to range R , and let
S ′ = (w ′, κ ′, P ′, s ′, u ′, r ′) represent E −1 . Using the form
(47) for both transformations, and E −1E = 1 , work
through the algebra to get
ˆ ɶ−1wɶ −1w
w ′′ = λλ
κ ′′ = λˆ−1 (κ − κɶ)
ɶɶ−1εˆ
P ′′ = λˆ−1λε
s ′′ = εˆ−1P (s − sɶ)
(
(52)
)
u ′′ = wɶ −1εɶ−1 u − uɶ + (r − rɶ) Pɶsɶ
(
)
r ′′ = λɶ−1wɶ −1 r − rɶ + κɶT Pɶ−1 (u − uɶ)
b. Range equals domain case
Consider the case R = D . Then, given the same
adjustable constraint so that all inputs into formation of the
observer collection are identical, the Maximal
Transformation Set property implies Sɶ and S represent
transformations in the same set. Since Sɶ = S is then
possible, making (52) the identity transformation1 , the
identity transformation is always an element of every
equivalent observer collection for which range and domain
are the same.
In (52) valid probability models require
LORENTZ TRANSFORMATION FROM INFORMATION
(
)
ɶɶ−1εˆ =
det (P ′′) = det λˆ−1λε
λɶ3 det (εˆ)
λˆ3 det (εɶ)
(
)) λɶ (1 − κ sɶ) det (P ) (53)
(
=
=
λˆ det (Pɶ (1 − sɶκɶ )) λˆ (1 − κɶ sɶ) det (Pɶ)
λɶ3 det P 1 − sɶκT
3
=
T
λɶ2 det (P )
3
T
3
T
Reasoning similar to that used in (54) through (57) can be
≠ 0, ±∞
λˆ2 det (Pɶ)
applied to (48), κTV ≠ 1 .
Since the inverse of (52) must exist, (53) can be neither
zero nor infinite. Thus λɶ ≠ 0 and λˆ ≠ 0 , or
κɶT sɶ ≠ 1
κT sɶ ≠ 1
(54)
So if D = R then κ and sɶ can be independently chosen
from the same set of transformations. This implies that the
second inequality in (54) is true for any κ and any s in
the transformation set, even if from different
transformations and an interdependence exists.
The Maximal Transformation Set property implies that
all possible κ and s be included in the selectors for the
observer collection transformation set. Since there is no
imposition of a favored direction, there exist κ and s
vectors in all spatial directions in the set. Assume that the
magnitude κ is unbounded in the set. Given any κ with
magnitude at least s
−1
, since s can be in any direction, s
can always be chosen close enough to perpendicular to κ
that
 −1

κT s = κ s cos θ =  s + δ κ  s cos θ


π


= 1 + s δ κ cos  + δθ  = 1

 2
(
)
with upper bound of ac = 1 . κT s can take any values
with magnitude less than one, including values very near
one, and the largest value ac must therefore equal 1:
1
a=
(57)
c
κT s = βκT βs
(55)
If the magnitude of V is
1
, the angle
V > c , and κ of any magnitude κ <
c
π
formed by κ and V can be chosen close enough to
2
radians to make the product equal 1. Thus the magnitude of
V is bounded by c :
(58)
V <c
Different values of c result in different parameter domain
sets and therefore different equivalent observer collections.
Transformations to a range not the same as the domain are
to observers which use different values of c . The value of
c acts as an index selector for equivalent observer
collections, and is a meta selector, providing data about
collections of transformations that primarily results from
equivalence transformation properties other than
invariance.
c. Transformation of velocity bound
Assume R = D , so both V and V ′ are bounded by c .
V
V′
and β ′ =
so β < 1 and β ′ < 1 . The
c
c
transformation of velocity in (47) is, in ς , β, βr and c
Define β =
terms,
β′ = P
β − βs
(59)
1 − βκT β
which violates (54) . Thus the magnitude of κ must be
bounded in the transformation set. Switch κ and s in the
argument to show that s must also be bounded in a
transformation set. Lack of any favored direction implies
that the boundaries are spheres. Designate c as the upper
bound of s and a as the upper bound of κ in a
s
κ
and βκ = , with
c
a
magnitudes less than 1. Then (54) is written
transformation set.
Define βs =
κT s = ac βκT βs ≠ 1
(56)
Let r be the magnitude of β − βs , and q̂ be the unit
vector in the direction of β − βs , so
β − βs = rqˆ with
r ≥ 0 . Define the square magnitude of the transformed
normalized velocity β ′ as a function Π (β ) = β ′T β ′ so
that
Π ( β ) = r 2 θ −2 R
with
θ = 1 − βκT β = 1 − βκT βs − r βκT qˆ .
R = qˆT PT Pqˆ
and
The minimum
value of Π is Π = 0 , which occurs at β = βs , a
stationary point of Π . Fix q̂ so that the derivative
LORENTZ TRANSFORMATION FROM INFORMATION
(
)
(
d
d 2 −2
Π (β ) =
r θ R = 2 r θ −1
dr
dr
)
with A = S Τ S , and S = ζ −1P . Since (61) restricts only
the velocity magnitude and not direction, equation (62)
must be true for all directions of unit vector ϒ . Reverse the
direction
of
the
unit
vector
to
obtain
 −1

θ − r θ−2 d θ  R


dr 
( )(
)
= 2 (r θ )(θ + r β qˆ) R = 2 (r θ )(1 − β
= 2 r θ−1 θ−1 + r θ−2 βκT qˆ R
−3
Since R > 0 ,
−3
T
κ
r > 0,
βκT βs < 1 ,
and
T
βs
κ
βκT β ,
)R
the
d
Π (β ) > 0 for all possible directions q̂ . Thus
derivative
dr
there can be no other stationary points besides β = βs .
Then the maximum value of Π , which must equal 1 if
R = D , must occur for β with velocity at the boundary of
T
D , where also β β = 1 is a maximum. Therefore the
boundary of D maps into the boundary of R , which by
assumption R = D , is also the boundary of D , and (59)
must map all magnitude one β to magnitude one β ′ .
Every range R can also be a domain, so if R ≠ D , R
and D are characterized by different values of the velocity
bound (58). Designate the bound for D by c and for R
c′
by c ′ . Define ζ = . Then (59) becomes
c
β − βs
V′ V′
β′
=
=
=P
(60)
c′
ζc
ζ
1− β Tβ
κ
The maximum value of the magnitude right side of (60) is
the maximum value of Π , which equals 1, so the
maximum value of the magnitude of β ′ is ζ . Since the
maximum magnitude of Π occurs at the boundary of D ,
the boundary of D maps into the boundary of R .
(1 + β ϒ) = (ϒ + β )
2
Τ
V′ = P
Τ
1−κ V
=P
Τ
1−κ V
= ζc ϒ ′
Τ
(c ϒ − s )
P P
(1 − β ϒ)
Τ
2
(β
Since these equations must be true for any unit vector ϒ ,
(
)
(
A = βκ βκ Τ + 1 1 − βs Τ Aβs
(
(
Τ
)
βκ = Aβs = βκ βκ + 1 1 − βs Τ Aβs
Re-arrange the second equation to get
(
)
(
Aβ ) β
βκ 1 − βκ Τ βs = βκ 1 − βs Τ Aβs
(
= 1 − βs
Τ
s
)) β
(63)
s
)
(64)
s
If the expressions in parenthesis in (63) were to vanish,
then A = βκ βκ Τ for which det (A ) = 0 and therefore,
det (P ) = 0 , in violation of the Inverse Existence
Property. Thus the terms in parenthesis do not vanish.
s
Consequently (64) requires βκ = βs , or κ =
, and βs
c2
is an eigenvector of A . The definition of A makes A a
positive definite matrix, so it is possible to write
A= MΤ DM for some diagonal matrix D and some
orthogonal matrix M , MΤ M = MMΤ = 1 . With A
expressed in this manner the first line of (63) becomes
(
Define γ = 1 − βs Τ βs
(
))
1
−
2
(
)
.
Then
)
the
(65)
component
equations of (65) are
Dii δij = (Mβs ) (Mβs ) + γ −2δij
j
which imply that only one of the three (Mβs ) can be noni
zero. Select the index of the non-zero component of Mβs
as i = 1 . In the following use of either of the other two
axis for the non-zero component yields the same A . Since
the matrix M is orthogonal, Mβs = βs = βκ ,
(Mβs )1
2
Τ
− βs Τ A ϒ = 0
i
(61)
(c ϒ − s ) = ζ 2c2
= (ϒ − βs ) A (ϒ − βs )
)
two
)) ϒ = 0
= (Mβs )(Mβs ) + 1 1 − βs Τ βs
which can be written as
2
Τ
κ
Τ
κ
1 − βκ Τ ϒ
(
(
for some unit vector ϒ ′ . From this equation form the
squared magnitude of the transformed velocity as
Τ
(
ϒ Τ A − βκ βκ Τ − 1 1 − βs Τ Aβs
These
D = M βs βs Τ + 1 1 − βs Τ Aβs M Τ
Since the analysis of the domain structure shows that an
equivalence transformation maps any velocity on the
boundary of D ,V = c ϒ , with unit vector ϒ , to a velocity
also magnitude c ′ = ζc on the boundary of R .
cϒ − s
A (ϒ + βs ) .
equations imply that
F. Lorentz Transformation
V −s
Τ
s
κ
(62)
= βs Τ βs and
D11 = βs Τ βs + γ −2 = 1
D22 = D33 = γ −2
LORENTZ TRANSFORMATION FROM INFORMATION
Selection of any other index as the non-zero component
results in a corresponding permutation of the D indices.
Matrix M rotates βs to align with a coordinate axis. The
definition of A implies
Τ
A = S Τ S = M Τ DM = (ΛM)
with
( ΛM )
1
2
ɶΛM for some
Λij = Dij . Consequently, S = O
ɶ ,O
ɶ ΤO
ɶ= 1 .
orthogonal matrix O
Define a position unit scale factor d by reference to an
axis perpendicular to βs as
(
ɶ Τ SMΤ
d = wζ O
)
33
= w ζΛ33 = w ζγ −1
Selector w is then given by
d
w =  γ
ζ 
With d as position scale factor and ζ as velocity scale
factor, the equivalence transformation (47) becomes
X ′ = d γ S (X − st ) + u
d 
sT X 
t ′ = γ t −
 + r
ζ 
c 2 
V −s
V ′ = ζS
(66)

sTV 

1 − 2 

c 
2
d 2 2 
sT  2
2

σt′ =   γ 1 − V  σt

 ζ 
c 2 
T 


V − s ) sT  
s (V − s )  T

(
2 2 
C ′ = d γ S 1 +
C 1 + 2
 S

c 2 − sTV  
c − sTV 



1 0

ɶ 0 γ −1
with S = O

0 0

1
0 
−
Τ  2


s s 
0  M , and γ = 1 −
 , for


c 2 

−1 
γ 
ɶ and M , such that matrix M
some orthogonal matrices O
rotates s
into a vector which lies on the x axis
ɶ = OMT .
ɶ
so that O
Ms = + s xˆ . Define O = OM

1 0

Then S= OMT 0 γ −1

0 0
0 

0  M = OW

γ −1 
with W

0 
1 0


0  M . W first rotates
defined by W = MT 0 γ −1


−1 

0 0 γ 
s to lie along the x axis, next multiplies the components
of V perpendicular to s by γ −1 and then returns s to the
original orientation. The matrix M which rotates V to
the x axis is
s ⊥ = sy 2 + s z 2
Mυ = s xˆ s = sx 2 + sy 2 + sz 2
s
sy
sz 
 x


s
s 
 s
1 0
0 




−
s
s
s


−sx sz 
 ⊥

x y
 Γ−1 = 0 γ −1
0 
M = 

 s
s s⊥
s s⊥ 

−1 



0 0 γ 
−sy 

sz

 0
s s⊥
s s⊥ 

ssT
W = MT Γ−1M = γ −1 1 + (1 − γ −1 )
sT s
Application of W to the velocity transformation term
results in the traditional relativistic velocity transformation.
W is the same no matter which axis the frame velocity is
first rotated into, and the matrix can be expressed as
W = γ −11 + (1 − γ −1 )
ssT
sT s
With these results the equivalence transformation (66) is

ssT 

X ′ = dO 1 + (γ − 1)
 (X − st ) + q

sT s 
d  
sT X 
t ′ =   γ t −
 + r
 ζ  
c 2 

ssT  V − s

V ′ = ζγ −1O 1 + (γ − 1)


sT s  
sTV 
1 −


c 2 
2
d 2 2 
sTV  2
2


′

(67)
σt =   γ 1 −
 σt
 ζ 

c 2 

ssT 

C ′ = d 2O 1 + (γ − 1)


sT s 
Τ



(V − s )s Τ   s (V − s ) 
1 +
C 1 + 2


c 2 − sTV  
c − sTV 


T 

ss

 Τ
1 + (γ − 1) T  O

s s 
LORENTZ TRANSFORMATION FROM INFORMATION
The first three equations of (67) are a combination
of time and velocity units scaling transformations and
Lorentz transformations in time and three spatial
dimensions [9]. These are relativistic transformations which
account for the possibility of different measurement scales.
In addition to conventional relativistic Lorentz
transformations, the equivalence transformation defines
precisely the transformation of the observation time
variance and the spatial covariance given time.
(
)
1
∆W T FRW FRR−1 FRW ∆W
2
−∆W T FRW FRR−1 FRW ∆W
1
+ ∆W T FWW ∆W
2
Now the excess discrimination invariant is
I P1* , P0 ≈
(
K (P1, P0 ) = I (P1, P0 ) − I P1* , P0
=
G. Small Parameter Shift
Consider a general continuous probability model with
continuous derivatives, vector R of real scalar responsive
parameters, and vector W of real scalar constraint
parameters. Designate P as the vector of parameters with
components from X ,V and W . Under sufficiently small
∆Pi = P1i − P0i the discrimination information [10] is, to
second order in ∆P ,
1
∆PT F∆P
2
where F is the Fisher information matrix defined by
I (P1, P0 ) ≈
Fij =
∫ dvR f (R : P0 )
(68)
∂ ln f (R : P0 ) ∂ ln f (R : P0 )
∀R
F in block matrix form is
F

F =  RR
FWR
∂P0i
∂P0 j
which makes (68) become
1
∆RT FRR ∆R + ∆RT FRW ∆W
2
1
+ ∆W T FWW ∆W
2
The variational equation
∂
I (P1, P0 ) = 0
∂R1
(69)
defines the stationary point, and minimum, with solution
R1* :
∆R * = R1* − R0 = −FRR−1 FRW ∆W
Substitution into (69) with result
)
)
T
1
∆R + FRR−1 FRW ∆W FRR
2
∆R + FRR−1 FRW ∆W
(
(72)
)
In the simple constant velocity probability model of (10)
the response, or un-constrained component vector R is
composed of components of X and V , while W is
composed of components of t , σt ,C . The second order
excess discrimination (72) for the constant velocity model
is
T
1
K (P1, P0 ) = (∆X −V0 ∆t ) C 0−1 (∆X −V0 ∆t )
2
(73)
1 2
T
−1
+ σt 0 ∆V C 0 ∆V
2
The correct value of K given by (16) differs from (73) by
replacement of σt 0 with σt 1 . Since σt 1 / σt 0 is not
invariant under the equivalence transformation, the
transformation does not exactly keep (73) invariant, but
only to a second order approximation. The discrepancy
also implies that the transformation that preserves
invariance of (73) is only an approximation to the correct
transformation. Extension of (68) to third order terms
recovers (16). Despite initial appearances (16) is actually a
third order invariant since σt 12 = σt 02 + ∆σt 2 .
FRW 

FWW 
I (P1, P0 ) ≈
(
(71)
(70)
H. Equivalence Transformations Recap
Equivalence transformations are defined on
parameters of a probability model by properties which
include invariance of information in a deviation from
parameter values expected after a controlled parameter
component shift. The invariant information can be
used to determine if parameter estimates confirm to
expected
system
behavior,
so
equivalence
transformations keep invariant related statistical
decision processes.
Remaining properties of equivalence transformations
define relations between the transformations and
structures of transformation domain and range needed
to generate valid probability models for all elements of
the domain.
Some of the significant concepts,
properties and mathematical tools developed to support
them are
LORENTZ TRANSFORMATION FROM INFORMATION
1.
The property of independent observer
scrutiny requires that collections of
equivalence transformations contain
transformations other than identity.
Independent observers are needed to
confirm a theory that does not support
identification of the observed system
independent of the applied model.
2.
Adjustable parameter constraints allow
parameter shifts to be defined in an
indexed set theory context. A parameter
value acts as index to an adjustable
constraint set element, which is a
constraint set and the value of a
constraint.
3.
The time postulate requires that
observation
time
parameters
be
constraints in an equivalent observer
collection. Under the postulate time as
control is the essential characteristic
which distinguishes time from other
quantities.
Statistical samples on a high correlation coefficient
probability model show that a multi-variate normal
density in observed position and time random variables
is a reasonable representation of uniform motion.
Application of the properties of equivalence
transformations to this probability model results in
equivalence transformations which are
Lorentz
transformations of mean observed position and time,
Lorentz transformation of velocity, and specific
transformations of the co-variance matrix and standard
deviation of observation times. The elliptic information
invariant is conceptually simpler than the Minkowski
space-time interval, though in higher dimension than
four since there are more parameter components.
Structural properties on parameter domain and range
require that velocity magnitude be bounded, without
introduction of any external light speed concept.
LORENTZ TRANSFORMATION FROM INFORMATION
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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