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Tag Archives: Physics

Clocks and frames

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.

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Mathematical methods of classical mechanics

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.

Geometry and order

Motion is three-dimensional and Euclidean.

Galileo’s principle of relativity

There exist uniform rectilinear motion (URM) coordinate systems possessing the following two properties:

  1. All the laws of motion are in all cases the same in all URM coordinate systems.
  2. All coordinate systems in uniform rectilinear motion with respect to a URM one are themselves URM coordinate systems.

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.

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.

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4D Formulations of Newtonian Mechanics

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.

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Newtonian mechanics in time-space

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.

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Equations of Motion Generalized

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 = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (xx0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (xx0)), or

v² − v0² = 2ax, 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 = ww = (w0 + bx) ∙ (w0 + bx) = w0² + 2x(bw0) + b²x², and

(2b) ∙ (tt0) = (2b) ∙ (w0x + ½bx²) = 2x(bw0) + b²x² = w² ‒ w0², it follows that

w² = w0² + 2(b ∙ (tt0)), or

w² − w0² = 2bt, with t0 = 0.

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Speed of light

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:

light clock at rest(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.

A theory of 6D space-time

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:

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What is a clock?

What is a clock? it is a device that measures time, but what are the essentials of a clock? I submit these are the essentials of a clock:

(1) A clock requires a uniform motion. Because only the kinematics (not the dynamics) are significant, a uniform rotation is acceptable. But because the result will be represented as a line – a timeline or time axis – a linear uniform motion has a more direct connection with what is measured, so let us take the first essential as a uniform linear motion.

(2) In order for clocks to be measuring alike, it is necessary that there be a standard rate for all clocks. In addition, clocks should have a standardized beginning point, so that clocks are interchangeable.

(3) A clock requires a pointer which indicates the present time on a time scale as it moves at the standard uniform rate. This would be the hands and dial on a common analogue clock. On a linear clock it is a part whose position in motion is interpreted as the present value of time. The pointer and scale are essentials of a clock.

Furthermore, a clock must be interpreted as showing the present time of the observer’s rest frame.

Figure 1

All the essentials of a clock can be represented by a frame in standard uniform motion relative to the observer’s rest frame. In that case, a clock should be definable in terms of frames of reference: one rest frame and one frame in uniform motion relative to the observer’s rest frame, as in the following.

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Space, time, and dimension

The post continues the ones here, here, and here.

There are three dimensions of motion. The extent of motion in each dimension may be measured by either length or time (duration). There are three dimensions of length and three dimensions of time (duration) for a total of six dimensions.

But there is no six-dimensional metric. Why? Because a metric requires all dimensions to have the same units, which requires a ratio to convert one unit into the other unit. The denominator on a ratio is a one-dimensional quantity, which means either the length or time dimensions need to be reduced by two dimensions.

This ratio is a conversion factor that is either a speed, which multiplied by a time equals a length, or a pace, which multiplied by a length equals a time. In general a speed is the ratio Δdr²/|Δdt²| = Δdr²/(Δt1² + Δt2² + Δt3²)1/2, and a pace is the ratio Δdt²/|Δdr²| = Δdt²/(Δx² + Δy² + Δz²)1/2. The denominator is a distance or distime, which is a linear measure of length or time (duration).

The conversion factors required are the speed of light in a vacuum, c, or its inverse, the pace of light in a vacuum, k. The resulting four-dimensional metric is either c²dt² − dx² − dy² − dz² (with time reduced to one dimension) or dt1² − dt2² − dt3² − k²dr² (with space reduced to one dimension).

These metrics are often simplified by taking c = 1 and k = 1 so that symbolically they are the same. Their units are not the same, however.

Each metric may be further reduced by separating space and time, as in classical physics. Then the space metric is |Δdr²| = (Δx² + Δy² + Δz²)1/2 and the time metric is |Δdt²| = (Δt1² + Δt2² + Δt3²)1/2. In the classical (3+1) of three space dimensions and one time dimension, time is replaced by its metric, and in the classical (1+3) of one space dimension and three time dimensions, space is replaced by its metric.

Space and time as frames

An observer is a body capable of use as a measurement apparatus. An inertial observer is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer.

An observer makes measurements relative to a frame of reference. A frame of reference is a physical system relative to which motion and rest may be measured. An inertial frame is a frame in which Newton’s first law holds (a body either remains at rest or moves in uniform motion, unless acted upon by a force). A frame of reference here shall mean an inertial frame.

A rest frame of observer P is a frame at rest relative to P. A motion frame of observer P is a frame in uniform motion relative to P. Each observer has at least one rest frame and at least one motion frame associated with it. An observer’s rest frame is three-dimensional, but their motion frame is effectively one-dimensional, that is, only one dimension is needed.

Space is the geometry of places and lengths in R3. A place point (or placepoint) is a point in space. The space origin is a reference place point in space. The location of a place point is the space vector to it from the space origin. Trime (3D time) is the geometry of times and durations in R3. A time point (or timepoint) is a point in trime. The time origin is a reference time point in trime. The chronation of a time point is the trime vector to it from the time origin.

A frame of reference is unmarked if there are no units specified for its coordinates. A frame of reference is marked by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A space frame of observer P is a rest frame of P that is marked with units of length. A time frame of observer P is a motion frame of P that is marked with units of duration.

Speed, velocity, and acceleration require an independent motion frame. Pace, lenticity, and retardation require an independent rest frame. These independent frames are standardized as clocks or odologes so they are the same for all observers.

Let there be a frame K1 with axes a1, a2, and a3, that is a rest frame of observer P1, and let there be a motion frame K2 with axes 1, 2, and 3, that is a motion frame of P1 along the coincident a1-a´1 axis. See Figure 1.

Two frames

Figure 1

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