The nature of Space-Time
About a century ago Poincaré arrived at the conclusion that, “No matter what observational facts are found, the physicist is free to ascribe to physical space any one of the mathematically possible geometrical structures….”[3, p. v]. In this paper, I shall take the liberty afforded by Poincaré and propose a geometrical structure for space-time that is not consistent with present theory.
In present relativity theory, space and time are combined
into what is called a 4 vector [1]. A
presumed and/or agreed upon direction for 
, 
and 
is essential to
coordinate this mathematical representation to any real physical
phenomenon. The distinction of an
opposite sign in a metric or the multiplication by the square root of negative
one, are taken as adequate reasons for considering time to be non-directional
[3].
In this paper, time will be treated as a vector. The direction of the time vector relative to any spatial axis will be related to the velocity of the object or system under consideration. Only in the case of stationary objects will the time vector be perpendicular to all spatial axes.
I will use the new term ‘space/time’ to represent space and time treated in the manner described. A vector space representation of space/time will be developed in this paper. The older term ‘space-time’ will refer to the conventional four-dimensional or four-vector representation.
Time and the Geometric Representation of Motion
The physical definition of time varies substantially from reference source to reference source, and the particular definition in a scientific context is often expressed in terms of relativity theory. When referring to time in this paper, its definition will not be limited to or associated with the common physical definition. Prior to making epistemological assumptions about the nature of Time a brief discussion is in order.
For every change in a measurable property of an observable
there is an associated change in time.
For example, take the two statements: I am in
In his landmark book, Reichenbach states: “Every lapse of
time is connected with some process, for otherwise it could not be perceived at
all.” [3, p. 116]. A process by its very
definition implies a change. Therefore
we can conclude: Any change in an
observable is associated with a time change and any time change is associated
with a change in some observable.
If time stopped, all clocks would certainly stop. If time were to speed up or slow down, one would expect all clocks to speed up or slow down so as to track the rate of time. It is reasonable then to call the observed change on a clock’s display the effect of time. Time is the cause and the observed change on the display of the clock is the effect.
Assume that a clock is put into linear motion. It does not seem as reasonable to say that time is the cause of the linear motion. However, moving the clock linearly involves some sort of process not unlike the process that moves the hands or increments the display. It is the designer of the clock who decided just how to move the hands. The same designer could use a very similar process to propel the clock in linear motion. One could infer that the linear motion is just as reasonably the effect of time. Time is the cause and linear motion is the effect.
If time is the cause of both the change in the display of the clock and the clock’s linear motion, then time might best be treated as having components. This is the first motivation for the vector representation of time introduced here. We also believe from experiment that the rate at which the clock increments time is not independent of the linear motion. As the clock moves faster, its hands rotate slower. This effect is not in the design of clocks, but fundamental to the nature of time. There is a maximum time rate that the clock can achieve and this occurs when the clock has zero linear velocity. There is also a maximum speed (the speed of light) that the clock can achieve, and as this speed is approached, the time rate is reduced to zero. Proper choice of units/conversion factors can make these maximum values of clock rate and light speed numerically identical. It would appear then that time also should be treated as having a magnitude. This is the second motivation for the vector representation of time introduced here. In mathematics, something having both a magnitude and components is called a vector; so in this paper, time will be treated as a vector.
I now make the following epistemological assumptions regarding the nature of time:
• There exists a cause and effect relationship between time and change.
• Time is the cause, and change is the effect.
• Any and all observable change is the effect of time.
• The quantity of time available to affect any and all changes in any and all properties of an observable is fixed.
Motion is a perceivable change and therefore is the effect of time.
Linear space/time motion:
To translate the above assumptions into a mathematical representation of linear space/time motion, I make the following assumptions:
1) Space and time
are related as proposed, by axiom, by N.
Vivian Pope and Anthony Osborne, specifically “(A2) Observational
distance and time have a constant ratio of units, 
….” [2, p193]. The value c is simply a conversion factor.
2) Space/time is a vector space with each
dimension having units of distance.
3) The path that an object experiencing linear motion follows in space/time is a line in space/time. The line shall be referred to as “the axis of motion”.
4) Any perceived
change in time by an observer, multiplied by the speed of light, is a vector in
space/time and is designated by
.
5) The vector representing the objects
motion in space/time is the vector projection of 
onto the axis of motion.
6) The vector projection of an observer’s vector onto a line
perpendicular to the axis of motion, herein referred to as “the orthogonal component”,
is the observed progression of time for the observable multiplied by.
7) All objects moving at the same space/time velocity will move along the same axis of motion.
The above
assumptions do not fixin any particular direction. However from 5, it is clear that
as the space/time velocity approached zero the angle between and the axis of motion
approaches 90 degrees. When the
space/time velocity is zero (i.e. a stationary object) we will call the axis of
motion x and consider it to have the
properties of all space. The axis on which the 
vector lies for zero space/time velocity shall be called the 
axis. From 7, it is
clear that these axes are the same for all observables and hence can be used as
coordinate axes. Any vector in space/time can be described in terms of its x and components.
For now, I shall
limit the discussion to two dimensions using the plane of x and .
The following definitions and conventions will be used in the discussion of linear space/time motion:
• The symbol
designating a vector in space/time will be in bold type (i.e. 
) and its magnitude in italic (i.e. 
).
• The vector
representing the component of projected onto the
axis of motion shall be designated 
.
• The axis of motion shall be called the p axis.
• The vector
representing the space/time velocity shall be designated
, and it is defined by the formula
.
• The orthogonal
component shall be designated
.
Figure 1 is a geometric representation of linear motion in
the vector space. There the vector of observer A
has been projected onto the axis of motion to represent the motion of the observable
B. The orientation with respect to x
andis not important at this point in the discussion.
From Figure 1 and simple geometry it is clear that 
. Multiplying each
term by 
and simplifying yields:
1.1
It is also apparent that:
1.2
To understand the significance of 
, we will evaluate an object at rest and then in motion. We wish to have a device that has moving
parts but does not move on the p
axis. We will use a clock. When the clock is stationary, lies on the 
axis and is
perpendicular to the x axis. The clock, however, is changing and hence
requires a component of time. This is
represented in Fig. 2.
The time that increments the clocks display is represented as
a small component of time perpendicular to the plane of 
. If it weren’t
perpendicular the clock would have translational motion. The component of time the clock requires is a
function of its internal construction and is proportional to 
.
Next the clock is put into motion. This is illustrated in Fig. 3. There, the clock is moving relative to
A. The axis is in general, no longer the
x axis but the axis of motion p.
The amount of time available to run the clock from the point of view of
A is now . The stationary
observer A would then see the clock run slower.
Its time would be dilated. The
time on the clock relative to the stationary observer’s time would be:
1.3
Next, I shall develop space/time transform equations using
two separate assumptions. First I will
assume that the rest observer’s time direction is fixed on the axis. I will then assume that the rest observer’s
direction of motion is fixed on the x
axis.
Fixed Time Direction Space/Time Transforms
In the following discussion, assume that the direction of the
observer’s vector is fixed for
all possible values of motion and lies on the axis. The linear motion of an observable will be
represented by rotating the axis of motion toward the fixed axis.
It is common practice to use a primed coordinate system to
refer to the moving object or system and an unprimed coordinate system to refer
to the rest object or system. In Fig. 4,
the unprimed system is represented by the solid lines and the primed by the
dashed lines. The center of the unprimed
system is labeled A. The point 
represents the moving
object or the center of the moving system.
For an observer in the moving system, the point is stationary and
hence the p axis is also the p’ axis.
The x’ axis represents the
motion of A relative to . The intersection of
the x’ axis with the axis yields another
time 
and it is clear that 
.
It is important to note that in this representation, the two coordinate systems are symmetrical. If one were to flip the primed coordinate system around, its motion and time axis could be aligned to lie directly on the motion and time axis of the unprimed system. The only difference would be the direction of positive displacement on the respective motion axes.
We now develop the space/time transform for the positions on
the motion lines. Refer to Fig. 5. As discussed above, the moving object’s zero
motion axis is just the p axis and
hence the transformation equation for any point 
moving with is simply:
1.4
Eq. 1.4 is a Galilean transformation
Now, if the observer at rest (A), believes that his/her
spatial direction (zero motion line) must be the same as an observer traveling
with 
, he/she would use the x’ axis as the spatial axis for .
If asked to develop a transformation equation, he/she would
take a random point P on the p axis
and draw a line parallel to ’s time line and find the intersection with the 
axis. This would be the point the stationary
observer would assign (as the spatial point) to P based on the observers zero
motion space line. It is clear from Fig.
5 and Eq. (0.1) that:
1.5
The above equation can be written in the more recognizable form:
The above transform is identical in form to the Lorentz
transform for spatial position with 
substituted for and 
substituted for 
.
To get the complete form of the space/time time transformation,
refer to Fig. 6. is now the time associated
with a random point P on the p
axis. It is clear that 
. But is just 
. Using Eq. 1.1 we
conclude that:
Now 
, and it is obvious that 
. Substituting yields:
or 
Substituting and rearranging yields the space/time time transform
1.6
The above transform is identical to the Lorentz time transform
with replacing and replacing .
In the above discussion it was assumed that the observer’s
time was fixed in direction. The axis of
motion was rotated towards time. This
representation shall be referred to as 
space/time representation.
Fixed spatial direction Space/time transforms
In this Section the p
axis shall be fixed and assumed to lie along the x axis. Motion shall be
represented by rotating toward it. The representation shall be referred to as 
space/time
representation. Figure 7 represents two
observers labeled A and B. We let A be
at rest and B is moving away from A. 
represents a random
point on the x axis. 
is the time that B
passes or will pass the point . Since the motion is
along a spatial x axis, it will be represented
by the conventional symbol . B’s motion must be
represented by projecting a component of , 
on the x axis.
The position of B is then given in general by 
. From our previous definition of velocity, B’s
position can also be written as 
.
The time available to run B’s clock is 
. As before, the
rest observer A will have a different space/time representation (different time
directions) for objects moving at different velocities.
To get the transformation equations for a specific point,
refer again to Fig. 7. Assume that both
clocks read zero at the starting point and call this the 
point. We are interested in finding the time and position
transformation equations for the point . Clearly:
1.7
Multiplying the numerator and denominator by 
yields:
Substituting 
for the second in the above, and
eliminating the subscript, yields the popular form of the Lorentz transform:
1.8
We see that 1.7 is identical to 1.8.
To get the position transform we again refer to Fig. 7. From A’s point of view the distance between B
and , at any time , is simply 
. It is not so obvious
what the distance 
is for B from A’s
point of view. What is clear is that B
must see the point moving toward it at speed . In order to
represent this (again from A’s perspective), B’s time, must be projected onto
the x axis at the same angle 
and its speed is 
, which is identical to 
. This speed is obviously
dilated as viewed form A, although B would measure it as . A would measure it
as 
, and the following speed ratio is apparent:
1.9
Let 
be the time on A’s
clock that it takes to reach B. The time on B’s clock is then 
. B’s distance from from A’s perspective
is 
.
The distance that B would report is simply B’s perceived
speed times the lapsed time
on B’s clock . B’s distance then is
. But is just the distance
from A’s perspective, 
. Hence we conclude
that:
1.10
As before, we arrive at a Galilean transformation for spatial distance.
In the 
representation we
found that a Lorentz-like transform resulted from the rest observer assuming
that his/her spatial direction was fixed for all moving objects.
In the above scenario, we would then expect to get a
Lorentz-like transformation if the rest observer uses his/her time in both
calculations. If we fix for both
measurements, then A’s assumed distance is given by the same formula 
. B’s distance would
now be given by 
. From this we get 
. Using 1.9 we get
1.11
1.11 is exactly the Lorentz transform for spatial position.
In the 
representation 
cannot equal 
It should be clear that, in the representation, no one
single spatial dimension (, or ) can equal 
. Assume that a
straight ruler is placed in a direction and an object is sent moving down the
ruler at constant velocity. See Fig.
8. The ruler is not moving and lies
exactly perpendicular to the observer’s time line. To find the position one would measure on the
ruler, a line perpendicular to the ruler intersecting the point P is drawn (The
point of intersection with the ruler is labeled 
). From the previous
discussion, it is clear that this yields the Fitzgerald length contraction formula.
In Fig. 8, a line labeled ST
connects a point 
on the x axis with a point on the p axis.
The line is labeled ST because
it is in general a combination of space and time. In the case above, where ST
is perpendicular to the x axis we can write
Energy and Momentum
Both representations of motion in space/time result in the
same space/time triangle. To develop an
energy/momentum representation, we redraw the triangle using vector notation as
shown below in Fig. 11. 
represents a unit vector in the 
direction, 
a unit vector in the 
direction and 
a unit vector in the 
direction.
Next we multiply each side by 
where 
is the rest mass of
the moving object. This yields the diagram
in Fig. 12. Clearly 
. Letting 
and substituting
yields Fig. 13. All three sides have
been transformed into momentum.
Next we define the following variables:
, 
, 
, 
, 
In terms of these variables, the momentum and energy triangles are depicted below in Fig. 14. It is obvious from Fig. 14, that:
2.1
We have developed a vector representation of energy. To understand exactly what this means we must first look at exactly what making a measurement entails. As we discussed previously, any spatial measuring device (ruler for example) will lie on a line perpendicular to the observer’s time line.
Any device for measuring energy will be stationary relative
to the observer and must lie on a line perpendicular to the observer’s time
line as well. It is the measuring
device, means or method that communicates the measurement to the observer. The situation is illustrated in Fig. 15. There the moving object encounters the measuring
device on what is called the actual line of measurement. The time of the encounter with the measuring
device differs from the observed time.
The energy the observer would associate with this measurement is 
. The significance of 
will become clear
shortly.
If the observer’s time direction is fixed, energy has both a scalar and a vector representation. The scalar representation is just the vector projections onto the observer’s time line. The vector magnitude is given by:
2.2
Eq. 2.2 can be written in the more familiar form:
2.3
The scalar equation can be written as:
2.4
and 
. But and hence:
2.5
In terms of the measurable energy we can write:
2.6
In Equation 2.6, the second term is the amount of energy that
cannot be directly measured. It ranges
from minus infinity to 0. It has the
exact same form as the potential energy of an object in a circular orbit (
) with a 
potential well.
We can write (1.4) in terms of 
by noting that 
and 
to yield:
2.7
For small angles we can approximate 2.7. Using the first order expansion terms 
, 
, and 
. Substituting for the second term
and noting that 
yields:
2.8
Equation 2.8 is the classic Newtonian energy.
In present theory, the vector components and scalar components are often used interchangeably leading to much confusion. It has led some to speculate that the energy formulas and/or the theory cannot be correct [5].
Which representation to use?
We have developed two different representations of motion in
space/time, which we call and . In the representation the
direction of time is the same for all objects.
If energy is to be treated as a scalar, the representation must be
used. We shall therefore use it in the
following discussion to explain the Dirac matrices.
The Dirac Matrices
Before showing how the geometry developed previously can
explain the Dirac matrices, we have to determine just how to add the and dimensions.
It is commonly accepted that we live in, or at least
perceive, three spatial dimensions.
Mathematically these three dimensions are represented as orthogonal or
perpendicular. They are considered
independent. That is to say, one can
often calculate an event along one of the three without considering the
others. Motion or velocity is one
property that is considered independent.
, and (or some
transformation of them) are treated as independent variables. It can be shown that the three dimensions
cannot be truly independent.
In present theory the Lorentz transforms are assumed to be
independently applicable to 
, 
and 
. Each individual
component can be transformed to any velocity less than . The vector sum of
these transforms 
could exceed .
In classical relativity, this problem is generally handled by
choosing a coordinate system such that one axis, say , lies exactly along the direction of motion.
A quantum object does not move in just one direction and
therefore using the concept of independence of spatial dimensions is not
compatible with QM. One would calculate
states that have velocity components in each direction that are near the speed
of light and hence the total velocity could exceed .
We wish to find a three-component representation that meets the following two criteria.
A) 
for all possible
values of , and .
B) The representation is compatible with
standard mathematical transformations, specifically 
.
Criterion A is obvious if one hopes to achieve any sensible results. Criterion B simply makes the task
easier. First we must note that there is
not a one to one correspondence between the perpendicularity of , and and the common
transformation equation .
Clearly if , and are real and mutually
perpendicular, then . But given that ,
one cannot assume that the vectors are mutually perpendicular. Assume equals 
where 
is a real vector and 
is the square root of
negative one. Now 
, and cannot be perpendicular
to both 
and 
.
Criterion A can be satisfied if the three spatial components
add as vectors to 
, and represents the
space-time velocity in the representation. 
and as approaches , approaches zero, and
hence 
.
Consider the diagram in Fig. 11 with each vector given a direction
as shown in Fig. 12. 
now represents the
amount of momentum that must be added to 
to equal 
.
.
The following vector equations are apparent from the diagram in Fig. 12.
3.1
3.2
Next a two vector representation of is created. This is depicted in Fig. 13. Two vectors labeled 
and 
are created. The two meet at a point above the 
plane as illustrated
below in Fig. 14. comes directly out of
the plane and is perpendicular to . on the other hand is
not perpendicular to . Obviously 
and:
3.3
The choice of the two vectors may seem arbitrary. What is important is that they sum to and that the magnitude
of is equal to the square
root of the sum of the squares of the magnitudes of three components.
The magnitude of is clearly 
and hence can be considered to have components 
. The subscript 
is used to indicate
that these ‘vector components’ are valid in so far as they yield the proper
magnitude of their vector sum (using the square root of the sum of the
squares).
From Fig. 15, one can see that and are in fact orthogonal,
and can thus be treated as a conventional complex number.
If is combined with then the magnitude of the resultant is
determined simply by the Pythagorean Theorem. On the other hand, if is combined with the magnitude of the
resultant is determined by multiplying 
by its complex conjugate. Not surprisingly, and do always appear together as a single variable
in Dirac’s equations, and in solving the equations, conventional complex
methods are employed. Also note that will always have the
opposite sign of which again is
consistent with the Dirac formulas.
Combining Eqs. 3.1, 3.2, and 3.3 yields the following vector equation:
3.4
It is clear that Equation 3.4 is valid irregardless of the
plane that 
and 
lie in. is orthogonal to and hence it can be
considered orthogonal to all other components by choice of dimensions.
If is chosen to be
perpendicular to, all components can be treated as orthogonal (for the
purpose of determining the total magnitude).
In this representation 
is always in the same
direction and hence we can write:
3.5
is taken to be a
scalar and all the components on the right are independent vector components.
From the above discussion there are two obvious facts, alluded to earlier.
1) 
and 
should always appear
as a single complex variable. Treating
them as such will maintain their orthogonality as well as their compatibility
with quantum mechanics.
2) 
will always have the
opposite sign of .
All four of Dirac’s equations meet the requirements of 1 and 2 above as shown below.
In terms of Dirac matrices, 
, where the Dirac Matrices [7] are listed below.
, 
, 
So far it is not clear as to why there are four coupled equations
in Diracs set. One explanation is the
result of the direction of . This is illustrated
in Fig. 16. It is clear that just
negating does not give the
correct vector equation. It is necessary
to define an up direction 
and a down direction 
. The net energy and
total momentum are unaffected by weather one uses the up or down direction, but
the momentum and hence angular momentum in the case of orbiting electrons, for
example, should be distinguishable. At
any rate, a total quantum mechanical solution must include both distinguishable
states and we can now write two equations:
, 
To get the remaining two equations refer to Fig. 17. It is apparent that negating results in two distinguishable
states for which are labeled 
and 
, and hence, the two equations must be further split each into
two distinguishable states. The final
four equations are:
, 
, 
Quantum Mechanics in the 
space-time representation
In quantum mechanics, one replaces the components of momentum
with differential operators [4]. What
about and? One might attempt to
find a separate differential operator but using symmetry considerations, it
makes more sense to replace them with the same operator and define separate
wave functions. The same argument
applies to 
and 
.
It is apparent that four separate wave equations will be required. The following eigenvector equations can be
written. The ’s now refer to operators and 
is treated as a single
complex variable.
3.6
3.7
3.8
3.9
Clearly for the same states of energy and momentum there will be relationships between the wave functions.
The geometric relationship between the spatial components of
momentum is not affected by 
(
). The equivalent
states for the positive energy and negative energy solutions should have the
same magnitude of and the same
sign. As was clear from the previous
discussion, it is not possible to simply negate . (you can’t replace a
with a 
)
The 
terms in Eqs. (2.6)
and (2.8) should be identical for identical states. Similarly, the 
terms in Eqs 3.7 and 3.9
should be identical for identical states.
Substituting yields:
3.10
3.11
3.12
2.13
There is insufficient coupling in the above equations to achieve a unique solution for each wave function for any given state. Additional substitutions are required.
A similar substitution based on 
would result in no
additional coupling. Referring to Fig.
15, it is clear that reversing the sign of 
, which results in reversing the sign of 
creates an exact
mirror image using the plane of 
and 
as the mirror. The solutions will have identical values of
energy and momentum.
Limiting the solutions to those that are symmetrical with regards
to a reversal of we can conclude that
we can exchange wave functions terms in any two equations, provided that we
replace one operated on by with one operated on
by 
, if has the opposite sign
in the two equations.
Considering the above restriction, a substitution with
regards to the wave function operated on by in Eqs. 3.13 and 3.10
is valid as well as a substitution of the same terms in Eqs. 3.11 and 3.12. Making the substitutions yields:
The above equations are exactly the four coupled differential equations that result from the Dirac matrices formulization of relativistic quantum mechanics.
Some consequences of this theory
1) The spatial
dimensions in this representation are not homogeneous. The variables do not all range from minus infinity
to plus infinity. It is not difficult to
show that in the representation, 
when 
is perpendicular to 
. Also, 
. This may explain the
cut off points [7] that are commonly required in present quantum field theory
to achieve only valid results.
2) The direction of for anti matter is not
the opposite direction as it is for matter.
One obvious consequence of this is the fact that mater and anti matter
do not have perfect symmetry with regards to energy. One would expect that using devices and
material made purely of matter, and used to measure anti matter, would not give
results that are a ‘mirror image’ of the results for matter. Experiments seem to bear this out [6].
Measuring anti matter with devices and material made entirely of anti matter should give the exact ‘mirror image’ results.
3) The (square root of
negative one) is required only for the purpose of making the vector magnitude
equation work. It should be possible to
formulate this space-time geometry with all real vectors. The author is working on formulating this
representation in a real homogeneous space/time.
4) If momentum in space/time is conserved, certain phenomenon can be explained. For example: beta decay. A neutron decaying into a proton/ electron pair would not conserve momentum in space/time. The proton’s motion line would be at a much smaller angle to a fixed spatial axis than the electrons. A neutrino may be a requirement of conservation of momentum in beta decay.
Summary
When Dirac developed his matrices he did so for mathematical reasons. There was no tangible physical rationality behind the development. A similar statement could be made in regard to Lorentz. Although the Lorentz transforms were given a degree of physical meaning by Einstein, the Diracs matrices remain a mystery.
The space/time geometry that results by treating time as a vector and motion as a projection of the time vector, results in a simple geometric explanation of both the Lorentz transforms and the Dirac matrices. This theory unifies the two concepts.
There is obviously much yet to explore with this theory. Questions and comments are welcome.
References
[ 1 ] W. Benenson,
J. Harris, H. Stocker, H. Lutz (Eds), “Gravitation and the Theory of
Relativity”, in Handbook of Physics
(Springer Publishing,
[ 2 ] N. Vivian Pope, Anthony D. Osborne, “A new approach to Special Relativity”, International Journal of Mathematical Education in Science and Technology, 18, no. 2 (p193, 1987).
[ 3 ] H. Reichenbach, The Phylosophy of Space and Time (Dover Publications, New York, 1957).
[ 4 ] L Schiff, Quantum Mechanics-3rd Edition (McGraw-Hill Book Company, New York, 1968).
[ 5 ] V.N. Strel’Isov, ‘Incorrectness of the Formula 
’, Galilean Electrodynamics 12, Special Issues 1, GED-East,
20 (Spring 2003).
[ 6 ] P. Weiss, ‘Decays may reflect matter-antimatter rift’, Science News, 155, no. 8, p. 118 (Feb. 20, 1999).
[ 7 ] A. Zee, Quantum Field Theory in a Nutshell (