«Abstract The value of a geometric model is to provide an accurate representation of mathematical relationships and help visualize the nature of ...»
The space-time-motion diagram: a relational model
Theodore J. St. John
The value of a geometric model is to provide an accurate representation of
mathematical relationships and help visualize the nature of complex functions. Each
component represents a concept that can be quantified so the quantities are mapped as
lengths, areas, angles, etc. so that their relationships can be compared and interpreted. The
Minkowski diagram of spacetime attempted to represent the relationship between space and time, but it was based on the 4D equation that unfolded one side of the equation (to fit the visual concept of space) without doing the same with the other. Therefore it represents a distorted map of a map, so the advantage of the geometric model is depleted if not destroyed.
The purpose of this paper is to present a perspective on space, time and motion that is not biased by the presumption that time is already known to be 1D and to propose a space-timemotion (STM) model based on that perspective.
Introduction Physicists will tell you that spacetime is a continuum, but if you ask them what that means, you get an answer that describes a mixture rather than a continuum: “Space really is 3 dimensional and time really is 1D. This is not an arbitrary division. Spacetime is unified in that different states of motion cause time and space to "mix", i.e. time moves at different rates to different observers. But a piece of paper is 2D because it takes two numbers to say where a point is. The room is thus 3D (3 numbers to describe position) and time is 1D because it takes only one number (the time) to say where you are in it.” On the other hand, they also admit that they don’t really understand what time actually is. In the January 2013 edition of Foundations of Physics, University of Pennsylvania physics professor Vijay Balasubramanian emphasized that “time remains the least understood concept in physical theory. While we have made significant progress in understanding space, our understanding of time has not progressed much beyond the level of a century ago when Einstein introduced the idea of space-time as a combined entity. (Balasubramanian, 2013)”. He provides extensive references and a brief synopsis of the various perspectives on why there is an arrow of time, including geometric considerations (Minkowski vs. Euclidean), maximally supersymmetric four dimensional Yang-Mills theory, multi-dimensional string theory, and discusses numerous questions to illustrate his conclusion that “We have more questions about time than answers.” One that serves to introduce this paper is the question: “Why is there only one time?” The maieutic answer is: Can we be certain that there is only one time if we don’t even know what it is? True, it only takes one number to describe time, but not because it is a onedimensional entity — it’s because everyone agreed upon a single time standard. Nothing prevents us from using a different clock for each direction of motion, giving time the same 3D character the spatial dimensions. Using the same standard clock has nothing to do with the nature of time; it only synchronizes it allowing a single symbol to represent it in every equation.
Newton’s predecessor, Isaac Barrow stated the assumption about time in his 1735 “Geometrical
“Time is commonly regarded as a measure of motion, and… consequently differences of motion (swifter, slower, accelerated, retarded) are defined by assuming time is known [emphasis added]; and therefore the quantity of time is not determined by motion but the quantity of motion by time: for nothing prevents time and motion from rendering each other mutual aid in this respect.” (Burtt, 2003 p. 158) Einstein emphasized, in his paper “On the Electrodynamics of Moving Bodies (Einstein, 1905), that time is a value used to describe motion and that events are what we judge:
“If we wish to describe the motion of a material point, we give the values of its coordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by ‘time’. We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events.” The purpose of this paper is to consider, as was done by Dan Shanahan (Shanahan), “how Lorentz might have proceeded if informed by later insights as to the underlying wave nature of matter” without the highly specialized language that has separated many physicists and isolated them to their own sub-fields (Mehta, 2008). The approach will be to present a perspective on space, time and motion that is not biased by the presumption that time is already known, as was the Minkowski space-time (ST) diagram, and to propose a space-timemotion (STM) model based on that perspective. The primary differences between the STM model and the ST model are
1) STM is based on the concept of motion so the origin of the coordinate system refers to the at-rest state.
2) The S and T axes of STM represent positive scalar values ( and ) so there is no representation for zero space or zero time.
3) The axes on the STM model are scaled for both the inside and the outside of spherical wavefront. These scales are different as will be explained below.
A particle/wave as a simple vector The key point that was not yet known around the turn of the twentieth century was that a particle may be considered a particle if it is at rest, but as a wave1, the very essence of a particle itself is motion (Morrison, 1990 p. 183). This particle/wave duality means that a complete model must include both the wave aspect and the particle aspect. The Schrödinger wave equation provided that model in 1926. (Anderson, 1982) (Morrison, 1990) (Goswami,
1992) Before quantum physics, the particle proper was modeled geometrically as a point in space, with zero dimensions, to represent its position in space and thus its point-wise particulate nature. The concept of Hilbert space was just introduced in the late 19th century, so state vectors had not been used and although vectors were used to describe particle motion, it was only in the sense that a particle experiences motion in 3D space with respect to inertial reference frames – not that it is motion itself2.
Motion must be represented as having both magnitude and direction. Therefore, as a unit of motion, a particle will be represented in the STM model as an arrow (in standard vector format). However, as an isolated particle at rest (no reference point outside of itself) the word “motion” refers to an internal property. Direction in 3D space is irrelevant and meaningless for an isolated particle so the direction part of the vector will be called “outward” (similar to up spin where “inward” would refer to down spin). Instead of assuming a number of dimensions i.e. unfolding a space axis into 3 dimensions and leaving time as one, spacetime will be illustrated as a “space-time-motion” (STM) model. Similar to the ST model, the independence of space and time will be represented by perpendicular lines but their relation to motion will also be represented – as a third line perpendicular to the space-time “plane” – to accurately represent the definition of velocity as the derivative of space with respect to time. As an isolated particle, there is no reference frame to which motion can be referred, so space and time are only concepts (non-entities) that can be used later when relative motion is considered.
Also, the words speed and velocity are used interchangeable to indicate scaled motion.
Graphically, the particle is represented as vector in the motion dimension with magnitude, see Figure 1.
Louis de Broglie introduced the theory of electron waves in 1924.
Vectors themselves had only recently been discovered. In 1837, William Rowan Hamilton (1805-1865) showed that complex numbers could be considered abstractly as ordered pairs of real numbers. It wasn’t until 1843 when Hamilton realized that a pure number (scalar term) could be added to a set of directed line segments (three rectangular components, or projections on three rectangular axes) that he called a VECTOR to represent a quaternion. The first book on modern vector analysis in English, “Vector Analysis” was written in 1901 and the idea of multidimensional Hilbert space was introduced shortly thereafter.
http://www.math.mcgill.ca/~labute/courses/133f03/VectorHistory.html and http://en.wikipedia.org/wiki/Hilbert_space#History Figure 1 The space-time-motion (STM) model. An isolated particle is represented as a vector, C and its relation to space and time only exists in-potentia as the slope of an imaginary projection of C onto the ST plane, until a separate reference frame is selected from which its relative motion can be measured.
There is no such thing as a truly isolated particle and there exists an infinite number of inertial reference frames in the universe from which the particle could be referenced3.
Therefore, the concepts of space and time exist in potentia and motion can be represented in terms of and as the slope of the diagonal line on the plane. This line is conceptually projected as a potential onto the plane with the same magnitude as, i.e.. It is represented by the diagonal (natural units4) because there is no preferred scale for either space or time.
Actual measurements will require scales (“unnatural” gauges that assign scalar values to and ) to be defined, but they are arbitrary and introduce an artificial skew to the geometric representation. In natural units (Jaffe, 2007), the value of is 1 and.
Any one of the infinite moving reference frames, whose velocity is represented by the vector with slope and magnitude in Figure 1, would be located outside of the particle’s surface, at the tip of. The smallest possible distance in space, from the particle’s center represents the surface of the particle at, which of course is the particle’s at-rest reference There is no need for an absolute reference frame or an ether.
see http://stuff.mit.edu/afs/athena/course/8/8.06/spring08/handouts/units.pdf and http://superstringtheory.com/unitsa.html for an explanation of natural units.
frame5, i.e.. All possible moving reference frames6 (relative to the particle) are represented by and the maximum value of, which would be oriented along the diagonal, is.
This model represents measurable quantities; therefore numerical values (measurement scales represented by the dummy variables and ) are inherently dependent upon the natural internal motion of the particle. Because the slope of the diagonal represents the change in space (i.e. displacement) per unit time, it is numerically equal to the magnitude of one unit of space, as shown in Figure 2a. Therefore, because relative velocity is referenced to the same gauge (unit time), the scale for external motion is numerically assigned to the same change in space (per unit time), which is c.
The Lorentz factor therefore translates relative velocity, to the natural scale defined by, via the angle shown in Figure 2a and b, as follows:
Figure 2 (a) The speed of light (internal motion of a particle) sets the scale for space/time in terms of length per unit time. In natural units, light travels one unit of space, c per unit of time so the vector C is constant in magnitude and direction (out of the page) while can vary from 0 to c. (b) The Lorentz factor translates a measure of relative velocity to a fractional value of the maximum possible motion in one unit of time.
The fact that the at-rest reference frame is one (out of an infinite number) that seems to be shared by every visible object gives rise to the notion that there should exist some preferred ether. However, each object, no matter how small or large, has its own unique at-rest frame. Superimposing frames is a perspective that allows one to see the unity or continuity in discreteness.
Even though Figure 1 represents an isolated particle at rest, the point which represents the surface of the particle, could be interpreted as a separate point particle that moves with time but remains at a constant distance from the origin. If it is measured, it appears to be a point particle, an electron, orbiting (i.e. motion at a constant distance) its own center.
As a separate system, in Figure 2a, the magnitude of relative velocity is labeled as and the displacement in one unit of time is. If expressed as a fraction of its motion ( ), the numeric value of displacement is the same at slow speed as it is at the maximum speed,.
Therefore, the same fraction, written in terms of and, (i.e. ) provides the inherent gauge
to which relative motion can be referenced:
The two terms on the right can also be squared and written as () () Because both time scales are synchronized to a common time scale, one unit of time is the same for both vectors. Thus the sides of the small triangle (relative velocity) are related by or Substituting for Thus and