The Normalization of Time |
The phenomena of frequency, oscillatory and circular motion has discrepancies that stem from the inherent ambiguity in the dimensionality of radians. An overlap in time defining the Tangent (tan (Ɵ)) and the timed derivative of the angular displacement (dƟ/dt) exists - illustrated by the fact that Ɵ = arctan y/x and is only valid between -π/2 < Ɵ <π/2.
These discrepancies account for the slight misalignment in frequency and field concepts pertaining to time, present in the current angular-velocity and frequency congruences in circular motion. The discrepancies are resolved by using harmonics to separate the concept of frequency from angular-velocity.
Measuring frequency in hertz (1/s), naturally implies another dimension of change. However, the connotation of frequency infers the sameness in change implying a lack of change, when it actually defines the change involved in sameness denoting an additional repetition in change. Therefore, harmony is used to accurately relate sphered change forming the field as depth relates to volume.
Frequency, when contrasted with angular-velocity, can be surmised as follows: A frequency is a definition of repetition; an angular-velocity is a definition of uniform circular motion defining rotation and implying repetition. However, repetition does not have to be uniform to occur – thus, the term "harmonics" is used to capture the true essence of repetition visible in a Fourier Synthesis.
Definable repetitive motion is assumed to involve harmonic frequencies defined as a derivative of the acceleration separating them from angular velocity and validated by Fourier Analysis. “Harmonics” is assumed to add depth to change differentiating music from noise or matter from light via resonant frequency patterns.
Any uniform rotating body has components of a tangential velocity (m/s), a tangential acceleration (m/s2), and a frequency (1/s). Of course the field is defined as (1/s3) - which can provably be considered a component of an existing harmony of (m/s3) in uniform rotation. To be perfectly clear: the terms for acceleration and frequency will change as time is normalized.
The radians, being an arc length divided by a radius, are a dimensionless quantity. However, in geographical notations the radians are congruent to arc-seconds (a subdivision of degrees). This interesting fact highlights a key to normalizing time – All change can be depicted as an angle- removing time from motion.
Though radians are dimensionless, they are really an abstraction of seconds or a measure of change itself, with the minimal angle of change representing the smallest detectable time change.
The angle Theta (Ɵ) is a measurement of change, similar to degrees, minutes, and “seconds” in geographical notations. "Harmonics" utilizes angles and arc lengths to relate space and time. Classically, radians are a dimensionless quantity, but hypothesized to relate an arc-length or wavelength to time in seconds (s).
The non-normalized hypothesis of angular-frequency is postulated to have the units of m/s3 providing the angular acceleration as an integral and accounting for density simulation in light's gamma-ray range.
However, by normalization, the angular-velocity, angular acceleration and angular-frequency correspond to wave-length, wave-width, and wave-depth multiplying time to form density.
Using spherical change to characterize frequency also allows cubic equations to define a frequency – deriving an angular-velocity as a second integral. Thus, the harmonic frequency is logically considered to be: how fast the phenomena will repeat at the specific instant under consideration when time is held constant – implying that frequency, though repetitive need not be constant and has partial derivatives.
However, it must be noted that: there are only 3 dimensions of time. Even a variable frequency can be defined in the 3rd dimension of time (repetition).
Time can be normalized by expressing circular motion as a parametric equation. Instead of expressing angular velocity in terms dƟ/dt it is expressed in terms of dx/dƟ, dy/dƟ and dz/dƟ - with x, y and z being related by their dependence on the angle Ɵ as well as the cartesian geometry of the spherical spiral.
Circular motion under a constant angular velocity is provably congruent to simple harmonic motion. Supposing that dampened harmonic oscillation is congruent to Circular motion under a constant angular acceleration, completes the relation between rotational kinematics and harmonic motion.
The angle (Ɵ) not only becomes the the common factor in the parametric equations, but by being dimensionless it normalizes time! With time being normalized, the angular-velocity is not expressed by a change in the angle Ɵ with respect to the change in time but as a change in the positions of x and y with respect to the change in the angle Ɵ.
By inspecting the above parametric concept, it is obvious that the angular acceleration as well as the frequency can be expressed parametrically and independently of time. In fact, the physics to normalize time already exist. By allowing (Ɵ) to equal time (t), velocity = ((dr/d(Ɵ))2 + r2)1/2 and acceleration = (d2r/d(Ɵ)2-r) x ur + 2 dr/d(Ɵ) x uƟ where ur and uƟ are unit vectors.
Accordingly, if a constant angular acceleration (aƟ) is assumed in rotational kinematics, the linear acceleration is equivalent to the square root of the addition of the squared tangential and radial accelerations: (at2 + ar2)1/2 with at = raƟ and ar= rvƟ2.
This makes the linear acceleration equal to: r(aƟ2 + vƟ4)1/2, which by differentiation makes the change in acceleration (jerk) equal to the cubical equation: a = 2r x (aƟ + 2vƟ3) x (aƟ2 + vƟ4)-3/2 - where angular velocity is represented by vƟ.
However, in rotational kinematics the motion of each point along the radial path varies making (m/s) by itself a meaningless term. To reconcile this discrepancy, the parametric equations must depict the radial distance (r) in the numerator, and the angle in radians in the denominator.
By assuming the radial distance to be 1 unit as in trigonometry, the angular velocity becomes a meaningful term congruent to harmonic motion. With any other radial distance, the appropriate distance must be used to accurately depict nature and convert from angular to linear quantities.
In The Shape of Time and following sections torioidal and spiral functions are defined to enhance the unity of the circle. This only relates the various aspects of time in circular, linear, and harmonic motions and does not normalize time. The angular velocity is traditionally measured in terms of radians/seconds coming to 1/s in SI units – which is synonymous with frequency.
With a radian representing an abstraction of change equivalent to a second, the UFT places it in the denominator and by normalizing, the angular velocity, angular acceleration and frequency all have units of meters/radian (m/s) - conceptualizing that the radians2 represents seconds as well.
This concept must be understood because the current concept of acceleration is serialized and uses s2. However, physics never explained the (s2) as an abstraction of change. Ask any physicist what does s2 mean? The meters squared (m2) is the area of space - why is the s2 an acceleration and not an area of time.
When viewed in this manner, the "s2" does not represent an acceleration, but an activity used to define the Velocity Distribution in the Molecular Theory of Gases or any other time distributed phenomena (the Area of Time). Time normalization means that all forms of motion can be expressed parametrically as meters/second (m/s).
The direction of space (length, width, or depth) must be stipulated to designate what extension is being referenced. However, regardless of the extension the unit is meters (m). Likewise the direction of time (duration, variance, repetition) must be stipulated to designate what extension is being referenced. And regardless of the extenseion the unit is seconds (s) completing the concept of 3-d time.
To graphically display this concept, picture a position-time graph (with the x-axis being position and the y-axis being time). A constant acceleration is a parabolic equation of x squared and a changing acceleration becomes a cubical, curved equation of some complexity. The acceleration is the Area under the squared or cubical curves.
To normalize time, picture a velocity-acceleration time graph (with the x-axis being velocity and the y-axis being acceleration - both having units of seconds). Conceptually, a constant acceleration becomes a horizontal line and a changing acceleration is a non-verticle line - though not necessarily a parabolic curve.
This representation of time in 2 dimensions seems enormously complex until the mathematical essence is portrayed. Knowing that the velocity is related to the acceleration by the differential, no verticle or horizontal lines can exist.
In fact this phenomena can only be characterized as the differential equation: y = dy/dx (actually the Integral Equation: y = ∫ x dx) but the solutions are the same: y = kex (having far reaching reprocussions).
This means that all motion can be shown to be a function of the current motion. Newton's Laws of Motion are only approximations already proven to be invalid at relativistic speeds.
It will be proven that the first law of motion: an object that is in motion will not change its velocity (accelerate) until a net force acts upon it; is incorrect. The relativistic mass is: m = 1/c2 x (E2 - pc2)1/2 where p = momentum = (mass x acceleration).
However, the relativistic momentum: p = (1-v2/c2)-1/2m0v is based on the invariant mass m0. This means that as an object is accelerated, not only does its velocity increase, but its relativistic mass also increases.
However, F = ma (a = F/m and m = F/a) stipulating that even when an object is not acted upon by an external unbalanced force, its acceleration decreases based on the increase in mass. A constant velocity in the absence of all friction, will require external application of force because the initial impulse causing the motion must increase the mass resulting in a deceleration.
Though retarding frictional and gravitational forces always exist in motion, if an object were thrown in a vacuum, absent of all external forces, it will eventually come to rest because the initial increase in velocity generates a negative acceleration.
This dampening or retarding acceleration must be related to the current velocity, which relates all motion to kex. Physically, an oscillation between mass and velocity occurs eventually reaching an equilibrium at some non-zero velocity. This is defined as matter and is reserved for latter discussion.
Polar coordinates must be used to base time on the circle or spiral (r = f(Ɵ)) and the Logarithmic spiral (r=keaƟ) when combined with the Archimedean spiral (r = aƟ) should depict all motion. The frequency becomes the z-axis of this 3-d time coordinate system based on the logarithmic (equiangular), Archimedean and spherical spiral - which has the unique property of winding around each pole an infinite number of times but reaching the pole in a finite distance.
A general equation combining the Archimedean with the Logarithmic spiral will have the form of: r = aƟ + kebƟ. When k=0, the Archimedean spiral is represented; when a=0 the logarithmic spiral is represent; and when a and b = 0 a circle is represented.
With the concept of time being solidified, The Shape of Time should clarify any misconceptions about its nature.
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© 2001 By Daryl E. Waite (dewaite1@hotmail.com) (All Rights Reserved worldwide)