A clock consists of two frames of references. This is seen in an ordinary analogue clock, which is composed of two parts:
Circular Space Frame
The space frame is at rest relative to the observer. The time frame is in uniform angular motion relative to the observer. Measurement of space and time requires both frames. The units marked on the space frame have dual significance: (a) as amounts of space or angles in space, and (b) as amounts of time.
V. I. Arnold’s Mathematical Methods of Classical Mechanics (Springer, 1989) provides a contemporary approach to classical mechanics. We follow the presentation here but modify it to six dimensions of space-time.
1 The principles of relativity and determinancy
A series of experimental facts is at the basis of classical mechanics. We list some of them.
A Geometry and order
Motion is three-dimensional and Euclidean.
B Galileo’s principle of relativity
There exist uniform rectilinear motion (URM) coordinate systems possessing the following two properties:
In other words, if a coordinate system attached to the earth is URM, then an experimenter on a train which is moving uniformly in a straight line with respect to the earth cannot detect the motion of the train by experiments conducted entirely inside their car.
In reality, the coordinate system associated with the earth is only approximately URM. Coordinate systems associated with the sun, the stars, etc. are more nearly URM.
C Newton’s principle of determinancy
The initial state of a mechanical system (the totality of positions and motions of its events) uniquely determines all of its motion.
Four-Dimensional Formulations of Newtonian Mechanics
First we reproduce section 2 from Michael Friedman’s “Simultaneity in Newtonian Mechanics and Special Relativity” in Foundations of Space-Time Theories (ed. Earman et al., UMinn, 1977), p.405-407. Then we provide the dual.
According to the space-time point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R4, the set of quadruples of real numbers, to represent the space-time manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (a0, a1, a2, a3), (b0, b1, b2, b3) are two fixed points in R4, then a straight line is a subset of R4 consisting of elements (x0, x1, x2, x3) of the form
(1) x0 = a0r + b0
x1 = a1r + b1
x2 = a2r + b2
x3 = a3r + b3
where r ranges through the real numbers. A curve on R4 is a (suitably continuous and differentiable) map σ: R → R4. Such a curve σ(u) is a geodesic if and only if it satisfies
(2) x0 = a0u + b0
x1 = a1u + b1
x2 = a2u + b2
x3 = a3u + b3
where (x0, x1, x2, x3) = σ(u) and the ai and bi are constants. So if a curve is a geodesic its range is a straight line. Note that the geodesies are just the curves that satisfy
(3) d2xi/du2 = 0 i = 0, 1, 2, 3.
The importance of straight lines and geodesies is due to the fact that both theories agree that the trajectories of free particles are straight lines in space-time. So we can represent such trajectories as geodesies in R4.
We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.
A transicle is defined as a moving object of insignificant size. The motion of a transicle of vass n at the chronation t is governed by Newton’s Second Law for time-space, R = nb or, more precisely,
R(t; t′) = h′ (1.1)
where R is the release which, in general, can depend on both the chronation t as well as the lenticity t′, and h = nt′ is the fulmentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to stance x. Equation (1.1) reduces to R = nb if n′ = 0. But if n = n(x), then the form with h′ is correct.
General theorems governing differential equations guarantee that if we are given t and t′ at an initial stance x = x0, we can integrate equation (1.1) to determine t(x) for all x (as long as R remains finite). This is the goal of classical dynamics.
Written compositions organized by temporal order are narratives. Items such as descriptions of people, places, or objects are organized as they occur to the narrator, for example, as the narrator takes apart an object or walks through a building or meets various people. This is a common method of composition but there are others.
Spatial order is another method of composition. Items such as descriptions of people, places, or objects are organized by their physical or spatial positions or relationships, for example, starting at the top and proceeding downward. Explanations of a geopolitical matter might proceed in geographic order.
Travel can be described temporally or spatially. An itinerary is usually arranged temporally but telling about it afterwards might be more interesting if arranged spatially. There are other principles of organization such as climactic order (order of importance) and topical order.
In science the independent variable determines the type of organization. If the independent variable is time, the organization is temporal. If the independent variable is space or distance, the organization is spatial. The stance in spatial organization corresponds to the time in temporal organization.
The values of the independent variable are the index to the order of the composition. If the independent variable is time, then the times indicate the steps in the order. If the independent variable is space or distance, then the stances indicate the steps in the order. Once the step is indicated, the composition may be the same: whether it’s Tuesday, so the tour is in Paris or it’s Paris, so the tour is on Tuesday makes no difference.
This is an update and expansion of the post here.
Here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time = t, location = x, initial location = x(t0) = x0, velocity = v, initial velocity = v(t0) = v0, speed = v = |v|, and acceleration = a.
First equation of motion
v = ∫ a dt = v0 + at
Second equation of motion
x = ∫ (v0 + at) dt = x0 + v0t + ½at²
Third equation of motion
From v² = v ∙ v = (v0 + at) ∙ (v0 + at) = v0² + 2t(a ∙ v0) + a²t², and
(2a) ∙ (x ‒ x0) = (2a) ∙ (v0t + ½at²) = 2t(a ∙ v0) + a²t² = v² ‒ v0², it follows that
v² = v0² + 2(a ∙ (x ‒ x0)), or
v² − v0² = 2a ∙ x, with x0 = 0.
Here is a derivation of the time-space equations of motion, in which retardation is constant. Let stance = x, time (chronation) = t, initial time = t(x0) = t0, lenticity = w, initial lenticity = w(x0) = w0, pace w = |w|, and retardation = b.
First equation of motion
w = ∫ b dx = w0 + bx
Second equation of motion
t = ∫ (w0 + bx) dx = t0 + w0x + ½bx²
Third equation of motion
From w² = w ∙ w = (w0 + bx) ∙ (w0 + bx) = w0² + 2x(b ∙ w0) + b²x², and
(2b) ∙ (t ‒ t0) = (2b) ∙ (w0x + ½bx²) = 2x(b ∙ w0) + b²x² = w² ‒ w0², it follows that
w² = w0² + 2(b ∙ (t ‒ t0)), or
w² − w0² = 2b ∙ t, with t0 = 0.
Speed is defined as “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (Dictionary of Physics, 3rd edition, McGraw-Hill, 2002.
This is ambiguous, however. Consider a light beam reflected off a surface:
(1) Since the light returns to its starting point, the total travel distance is zero, so the overall velocity is zero and the speed is zero.
(2) However, the interest is in each leg of the journey. In that case, in the first leg light travels +L in time t, and in the second leg light travels –L in time t. The mean velocity in the first leg is v1 = +L/t, and the mean velocity in the second leg is v2 = –L/t. The mean velocity for both legs is the harmonic mean of these two velocities because what is fixed and independent is the length, not the duration.
1/((1/v1) + (1/v2)) = 1/((1/L) – (1/L)) = 1/0 = ∞.
Thus the mean velocity is infinite, and the mean speed of light is infinite.
(3) Another approach looks at length of each leg apart from direction. In that case, in the first leg light travels L in time t, and in the second leg light travels L in time t. The speed in each leg is L/t, so the mean speed of light is L/t. This is the best known approach to the speed of light.
It’s interesting that (2) leads to the Galilean transformation, and (3) leads to the Lorentz transformation.
Note: as the research develops this post will be updated.
Here is a formulation of Newtonian physics in six dimensions (3+3), three dimensions of space and three dimensions of time, that is effectively either 3+1 or 1+3 dimensions of space and time.
A frame of reference (“frame”) is a six-dimensional physical system relative to which the location of physical bodies can be determined. Frames are composed of two idealized constructions. These frames do not come with clocks.
Start with two idealized physical structures, one static, the other kinetic, with the kinetic structure in constant motion relative to the static structure. Each body or observer has one of each structure associated with it. The structures are dual to one another: (a) a static structure, which is at rest relative to its associated body or observer; and (b) a kinetic structure, which is in uniform motion relative to its associated body or observer at a fixed rate and direction, which are established by convention. The position of a body on a structure is determined by contiguity with the structure and is known universally, without signals, from the universal extent of each structure.
The position of a particle relative to each structure is compared, either the kinetic to the static or the static to the kinetic. Length is a result of comparing a point on the kinetic structure to two points on the static structure:
As I’ve noted before (here etc.) history and science have different aims and methods. Mixing them just confuses both of them. There is no genuine “historical science” or “scientific history”. History narrates particulars among unique events. Science theorizes universals among repeatable events. In physics time is homogeneous: an experiment is the same whether conducted today or 100 years in the past or future. That is not true in history. Time is not homogeneous there.
History and science can and should balance one another. The more science expands its universals, the more history can point out particulars that are overlooked or are important in a particular context. The more history focuses on unique particulars, the more science can point out the significance of universals.
The homogeneous and inhomogeneous aspects of time can both be known only by balancing history and science. One could say something similar about all universals and particulars. The universal and particular aspects of reality can both be known only by balancing history and science.
