N-DIMENSIONAL TIME AND 1-TIME DIMENSIONED HISTORIES - A Structure for Time

Our world is not a movie. This is the basic assertion of this study. Another
assertion is that Time is more complicated than what is supposed by the
usual thinking. Some regions of spacetime are sufficiently handled by the
current spacetime theories, with their unique { 1-Time Variable } for each
[ R 3+1 ]. However, not all regions of space and time are addressed. Therefore,
we obligingly explore those regions here. The author’s viewpoint is that the
currently assumed time dimensionality, i.e. one, is insufficient for explaining a
large class of phenomena. This conviction was affirmed by proving that a set
of time variables is not always a set of “reducible” (i.e. mutually dependent)
time variables. In general, since local space is measured by a composition of
a set of rods having lengths, local time is defined by a composition of a set
of generalized clocks having 1-time variables.
The same understanding of the distinction between space variables and time
variables is in need of some improvement. It was long ago observed that
the standard rods and clocks approach relies on a set of artificial devices
rather detached from the usual world. Here, it is taken to attention the more
primitive and natural approach via a set of “static elements”(as perceived by
observers) versus a set of “mobile elements”. In this study, it is proven that
their interactions yield time variables of many kinds. In short, this study is
on the nature of time.
Given that the argument of this study is the time structure of the observer’s
world, we may suppose one primitive “Observed Observer”≡ {OΨ ` } possess-
ing intelligence, memories, etc., but wanting of the concepts of space and
time insofar as such concepts are sophisticated sensations. Therefore,
n a { set
o
of diligent observers possessing the concepts of space and time } ≡ { O }
is supposed to study {OΨ ` }. A necessary preamble is the investigation of
{OΨ ` }’s perceptive system. In this study, this investigation is no more than
an ordered list of definitions, being outside its argument. {OΨ ` }’s field of
observation is a [ World Part ] ≡ [ W ]. Here, its description is schematized
and simplified. In this mimicking of the animal sensory processes, information
1112
deemed unnecessary is dropped.
The structure of this study is as follows.
Chapter 1 introduces the set terminology of perceptive systems, including
the direct and indirect senses, as well as the mapping between an observer
and an observed object (i.e. the observation map).
Chapter 2 introduces the terminology of generic observed objects, the
subset of “important objects (for the observer)”, the set of signals with its
classes: particle signals and wave signals.
h
i
In Chapter 3, the “Set of Generalized Clocks” ≡ [ GC ] is introduced
as a subset of “World Parts” ≡
h
i
[ W ] . A [ GC ] is defined as a [ W ]
with some noticeable features generating an [ N-Time Variable ], N ∈ [ Z + ]
(note that the cases wherein N ∈
/ [ Z + ] are left to the reader). By its name,
a [ GC ] is not necessarily a sophisticated device. Also, rarely a [ GC ]
is a practical useful device; Generalized Clocks are introduced here only
for defining and clarifying Time. The principal components of a [ GC ] are
succinctly presented in advance. The convenience of a separation by an animal
observer of the sensed [ Generalized Spacetime ] ≡ [ GS ] into Real Space
and Time is elucidated. Accordingly, a theoretical distinction between Real
Space variables and Time variables is introduced ( more extensively discussed
in chapter 4 ). It becomes necessary, by a “Partition Factor”, to make the
distinction between Space variables ( let us say that here Space variables are
assumed to be Real following Euclid ) and Time variables ( here assumed
to be Imaginary ).
In Chapter 4, the set of generalized clocks, of which common clocks are a
subset, is examined more diffusely. By definition, a [ GC ] mingles with the
ordinary physical systems and is apt to yield a [ N-Time Variable ]. Some of
the arguments are as follows: [ GC ]’s subclocks; { Set of Levels } i.e. any class
or succession of generalized clocks having some common components; main
components and eventual components of a [ GC ]; elementary clocks; simple
clocks; main generalized clock. Again and by other words, a formulation
of time: a time variable being expressed as the intersection of pointers
indexes signals, dial, and correlator. Some types of intersections are: ( History
defining Intersection ) and ( Value defining Intersection ). To specify a ( Time
Value of a chosen [ GC ] ): a { value defining intersecant } acting within a
( Value defining Intersection ). [ Function Time ], the time variable being
function of a chosen Real space variable, and its classes: [ Position Time ],
[ Absolute Travel Time ], [ Counter Time ], [ Envelope Time ]. It is said
that not always and not necessarily a { 1-Time Variable } is produced by an
integration. Subspaces of a [W] ( hence, also of a [GC] ) are listed: { History },13
{ Hypothesis }, ( State ), and ( Pointlike Event ). In order to simplify
this study, an assumption: linear time for { History } and { Hypothesis }.
[ Quantized Time Variable ]. Inversion of a { Time Interval } of a { 1-Time
Variable }. Inversion of a { Time Interval } of a [ N-Time Variable ]; N > 1.
In Chapter 5, two of the many possible time concepts are analyzed and
compared, namely [ Widespread Time ]=[ PT ], which is of current use in
the author’s spacetime interval, and [ Natural Time ]=[ NT ], which is more
primitive, i.e. more spontaneous to an unlearned observer.
Chapter 6 analyzes map correlation, i.e. the correlation of a set of variables,
in particular time variables, by a set of maps that the observer knows. In
short: map-correlated, feebly map-correlated, and map-uncorrelated time
variables are encountered in a [ World Part ], by a specified observer. An
example of a map-uncorrelated time variable is annexed. In an analysis not
restricted to a history { World Part } but addressed to a sample space [ World
Part ], an assumed unique { 1-Time Variable } is generally insufficient for
the time-labeling of said [ World Part ], since not all other time variables
are dependent of this chosen independent time variable. On the contrary, a
history { World Part } is temporally linear by our definition, but rarely a
[ World Part ] is reducible to a known history { World Part }.
Chapter 7 describes how a [ Reference Time Variable ] may be chosen
for a [ World Part ] and map-correlated to each element of the set of
dependent time variables. This must be done cautiously; not in all cases
this choice is satisfactory. Ideally, a one-to-one correspondence between any
[ Time Variable ] of the observed [ World Part ] and the [ Reference Time
Variable ] should be necessary. It is unnecessary to remember that in general
the assumption of a suitable [ Reference Time Variable ], for example a
{ Reference 1-Time Variable }, is a practical and useful method.
Chapter 8 introduces ( Value Defining Intersection ) and { value defining
intersecant }, which are concerned with the separation of a ( Time Value )
from a [ Time ].
Chapter 9 describes how linear velocity, angular velocity, acceleration, . . .
may be considered time variables of generalized clocks.
Chapter 10 details how in a set of levels, a change of level yields change
of time variable. Restrictions of a [ GC ], i.e. the assumption of another
(similar) [ GC ] of a lower level, may be achieved by many modalities; the
case examined here is by restriction of the [ Dial ].
Chapter 11 contains examples of generalized clocks and comparisons
between them, revealing possible time stops, repetitions of time values, time
discontinuities, etc.
Chapter 12 describes how by a ( Value Defining Intersection ), the observer14
creates a correspondence between the state ( GC ) of [ GC ] at his or her
disposal and the state ( W ) of [ W ] under observation. Necessarily, this
state ( W ) depends on the type of [ GC ]. The process of specifying ( W )
by [ GC ] is explained, as well as the “quality” of the separation of ( W )
from [ W ], i.e., time-slicing and time-defining sharpness. As an example, it
is shown that a [ GC ] with an [ N-Time Variable ], large N, is convenient
for a thin Time slicing.
In Chapter 13, since a ( World Part ) state is relative to a specified chosen
[ GC ], it is considered the largest [ GC ] at the observer’s disposal, namely
his or her [ Main Generalized Clock ]. By this temporally powerful [ Cm G ],
in a ( Value Defining Intersection ) the ( Observer ) state may extract a
very thinly sliced state ( W ) of [ W ]. Obviously, this is a primitively and
locally defined state, and therefore it lacks conceptual interpretations. The
last section of this chapter discusses two alternate specifications in Real
space and time of a ( World Part ) state.
In Chapter 14, a ( Pointlike Event ) being a ( State ) of a Real spatial
point, is introduced as relative to a given set of generalized clocks ( that may
be a one-element set ) and labeled by a ( Time
Value ) corresponding to them
all. Pointlike Event in regards to [ C A ] does not necessarily correspond
to
n


Pointlike Event in regards to [ C B ] ; instead, it may correspond to an
o
Interval in regards to [ C B ] . It is a comparison between [ C A ] and [ C B ]
that proves the relativity of the concept of ( Pointlike Event ).
In Chapter 15, some specific cases of generalized clocks are introduced to
discuss the argument of time dimension.
In Chapter 16, the [ Generalized Spacetime ] introduced here acknowl-
edges not only the existence of map-correlated time variables but also the
presence of map-uncorrelated and feebly map-correlated time variables. The
latter variables are difficult to deal with, lacking or with scarce associative
bounds with other time variables; nevertheless, they cannot be theoretically
ignored. It is proposed that a [ World Part ] = [ W ], suitably referred to a
[ C ` ; i T ` ] where [ i T ` ] is an [ N-Time Variable ] with N > 1, be mapped
onto [ Generalized Spacetime ] ≡ [ GS ], with [ R 3+1 ] being a subcase of
[ GS ]. The introduction here of [ GS ] has implications. For example, by the
definition of History used in this study, the observer may sense a History
{ W; i t ` } with a { 1-Time Variable } and necessarily with its boundary
states. However, its development outside { W ; i t ` } toward the past and
future must not be regarded as a line in any mathematical representation.
Another implication here is the approximate symmetry in the observer’s15
knowledge of past ∆ P [W ] and future ∆ F [ W ] , the first joined to an end,
the second to the other end of said history { W ; i t ` }.
Chapter 17 shows the merging of our proposals with [ R 3+1 ] theory.
Though some arguments here are well-known, they are included to ensure
the continuity of exposition. This chapter’s contributions are listed below:
The parametric { 1-Time Variable } of [ R 3+1 ] masks the presence of other
1-time variables map-connected with it. The { 1-Time Variable } of any
Elementary [ GC ] is assumed to be associated to either a wave signal or
a particle signal. The natural time system [ NT ] is applied to [ R 3+1 ].
{ Main Frame } is defined. The ( Metric Equation ) ≡ ( ME ) is generalized,
that is applied to non-electromagnetic signals. Our kind of complexification
technique. A suitable tetrad. Correlation in the [ R 3+1 ] subcase. Partition
Factor in the [ R 3+1 ] subcase. For a learned observer, an “Extended Correla-
tion” that is dubbed Representation. ( ME ) is shown to be in accordance
with both widespread time system [ PT ] and natural time system [ NT ].
( ME )-derived Equations: Real space equation ( SE ) , time equation ( TE ).
Both have uniformity of sign; hence, no perplexities about signature. The
classes of a ( Sq. Spacetime Interval ) studied in [ NT ] System. In ( TE ), a
symmetry exists between its–dutifully defined–(sq. wave time component)
and (sq. particle time component).