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Lorentz transformation derivation

The following derivations are similar to here.

Lorentz transformations for space with time

Let unprimed x and t be from inertial frame K and primed x′ and t′ be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, C, and D:

x′ = Ax + Bt
t′ = Ct + Dx

A body at rest in the K′ frame at position x′ = 0 moves with constant velocity v in the K frame. Hence the transformation must yield x′ = 0 if x = vt. Therefore, B = −Av and the first equation becomes

x′ = A (x – vt).

Using the principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing v with −v:

x = A (x′ − (−vt′)) = A (x′ + vt′).

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Equivalence principle

Inertia is the property of a body that resists changes in its motion. Inertial mass of a body is the ratio of the applied force divided by the body’s acceleration. Gravitational mass is the mass of a body as measured by its gravitational attraction to other bodies.

The Equivalence Principle takes several forms. The Newtonian version of the equivalence principle:

The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing. Then, gravitational force is proportional to inertial mass, and the proportionality is independent of the kind of matter. Ref.

The Einsteinian version of the Equivalence Principle:

All objects fall the same way under the influence of gravity; therefore, locally, one cannot tell the difference between an accelerated frame and an unaccelerated frame. Ref.

These can be expressed in terms of facial vass and levitational vass as follows:

Facilia is the property of a body that does not resist changes in its motion. Facilial vass of a body is the ratio of the applied release divided by the body’s retardation. Levitational vass is the vass of a body as measured by its levitational attraction to other bodies.

The Equivalence Principle then states:

Facilial vass and levitational vass are the same thing. Levitational release is proportional to facilial vass, and the proportionality is independent of the kind of matter.

All bodies rise the same way under the influence of levity; therefore, locally, one cannot tell the difference between a retardated frame and an unretardated frame.

Galilean transformations derived

This derivation of the Galilean transformations is similar to that of the Lorentz transformations here.

Since space and time are assumed to be homogeneous, the transformations must be linear. The most general linear relationship is obtained with four constant coefficients: A, B, C, and D:

x′ = Ax − Bt
t′ = Ct − Dx

Without loss of generality, if t = 0, let A = 1, and if x = 0, let C = 1. Then

x′ = x − Bt
t′ = t − Dx

The inverse transformation for the position from frame R′ to frame R should have the same form as the original but with its motion in the opposite direction, as is confirmed by algebra:

x = x′ + Bt′
t = x′ + Dt′

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Independent and dependent motion variables

Independent variables are measured first, independent of other variables. They may be either set to a fixed value or allowed to change at a fixed rate. An example of the former is a race in which the distance is the independent variable set for the race, and of the latter is a time variable, which increases with each tick of a clock.

Dependent variables are functionally dependent on an independent variable. A dependent variable may or may not be causally dependent on the independent variable. Dependent variables are measured relative to the independent variable. For example, given a time of four minutes (the independent variable), how far can someone run (the dependent variable)? Dependent variables are known by many names, including target variable.

The extent of motion

The extent of a motion is measured by its time intervals (“times”) and its space intervals (“spaces”). Let T represent an interval of time, and let S represent an interval of space. Uniform motion may be stated as a proportion in either of two ways:

(1) Given the ratio of two elapsed times, the corresponding ratio of two traversed spaces are in the proportion:

S1 : S2 :: T1 : T2

(2) Given the ratio of two elapsed spaces, the corresponding ratio of two traversed times are in the proportion:

T1 : T2 :: S1 : S2

In case (1) time is the independent variable and space is the dependent variable (the time speed). In case (2) space is the independent variable and time is the dependent variable (the space speed). Let us adopt the convention that the variable on the right side of the proportion is independent, and the variable on the left side is the dependent variable.

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Interchangeability of space and time

The extent of a motion is measured in two ways: by its time (duration) and by its space (length). The relation between these two measures is the subject here.

Although a definition of uniform motion was given by Archimedes, Galileo was the first to give a complete definition:

Equal or uniform motion I understand to be that of which the parts run through by the moveable in any equal times whatever are equal to one another. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 148)

Archimedes first stated a proportion for uniform motion:

If a point move at a uniform rate along any line, and two lengths be taken on it, they will be proportional to the times of describing them. (Archimedes, The Works of Archimedes, ed. by T. L. Heath, Dover, p.155.)

In other words, given a uniform motion and on it any two lengths, L and M, then the lengths and the corresponding times of motion, T and U, will satisfy the proportion L : M :: T : U.

Galileo gave the converse proportion:

If a moveable equably carried [latum] with the same speed passes through two spaces, the times of motion will be to one another as the spaces passed through. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 149)

In other words, given a uniform motion and on it any two spaces, S1 and S2, then the times of motion, T1 and T2, for the corresponding lengths will satisfy the proportion T1 : T2 :: S1 : S2.

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Classical dynamics

The following presents the space-time and time-space versions of Newton’s laws based on the book Classical Dynamics of Particles and Systems by Thornton and Marion (Fifth Edition, 2008).

Start with page 49, section 2.2:

2.2 Newton’s Laws [for space-time]

We begin by simply stating in conventional form Newton’s laws of mechanics:

I. A body remains at rest [in space] or in uniform motion unless acted upon by a force.

II. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

III. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

To demonstrate the significance of Newton’s Third Law, let us paraphrase it in the following way, which incorporates the appropriate definition of mass:

III′. If two bodies constitute an ideal, isolated system [in space], then the accelerations of these bodies are always in opposite directions, and the ratio of the magnitudes of the accelerations is constant. This constant ratio is the inverse ratio of the masses of the bodies.

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Ballistics

Ballistic table based on launching from a height and angle with coasting ascent and descent (no drag, no thrust). Note the handy trigonometry identity for range: 2 sin θ cos θ = sin 2θ. This table is in pdf form here.

Space-time

Time-space

Initial space angle = θ Initial time angle = φ
Initial height distance = y0 Initial height distime = b0
Elapsed time interval = t Elapsed stance interval = s
Distance downrange or horizontal location = x Distime downrange or horizontal chronation = a
Altitude distance or vertical location = y Altitude distime or vertical chronation = b
Gravitational acceleration = g Levitational retardation = h
Initial velocity = v₀ Initial lenticity = w₀
Initial horizontal velocity = v0x = v0 cos θ Initial horizontal lenticity = w0a = w0 cos φ
Initial vertical velocity = v0y = v0 sin θ Initial vertical lenticity = w0b = w0 sin φ
Horizontal velocity = vx = v0x Horizontal lenticity = wa = w0a
Vertical velocity = vy = v0y – gt Vertical lenticity = wb = w0b – hs
Velocity at apex point: vy = 0 Lenticity at apex instant: wb = 0
Horizontal location x = v0x t Horizontal chronation a = w0a s
Vertical location y = v0yt – ½ gt2 Vertical chronation b = w0bs – ½ hs2
Vertical location at impact point: y = 0 Vertical chronation at impact instant: b = 0
Time of flight to apex tapex = v0y/g Stance of flight to apex sapex = w0b/h
Total time of flight ttotal = 2tapex = 2v0y/g Total stance of flight stotal = 2sapex = 2w0b/h
Distance range to apex xapex = vox voy/g Distime range to apex aapex = woa wob/h
Total distance range xtotal = 2vox voy/g Total distime range atotal = 2woa wob/h
Max altitude distance yapex = ½ v0y2/g Max altitude distime bapex = ½ w0b2/h
Trajectory formula: y = y0 + x tan θ − ½ gx²/v0x2 Trajectory formula: b = b0 + a tan φ − ½ ha²/w0a2

Abstract classical mechanics

The following builds on the book Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006).

Basic Principles of Classical Mechanics (cf. Chapter 1)

Space and Time

The space where the motion takes place is three-dimensional and Euclidean with a fixed orientation. We shall denote it by E3. We fix some point o ∈ E3 called the “origin of reference”. Then the position of every point s in E3 is uniquely determined by its position vector os = r (whose initial point is o and end point is s). The set of all position vectors forms the three-dimensional vector space ℝ3, which is equipped with the scalar product 〈 , 〉.

The time in which motion takes place has the same structure as the abstract space above. The combined vector space is ℝ3 × ℝ3. The abstractions for space and time are unconnected unless there is defined a fixed relationship between them. Examples of such a fixed relationship include a default rate of motion or a maximum rate of motion. Let us begin without such a relationship.

Position in space is called location and in time is called chronation. The Euclidean metric for space is called length and for time is called duration (or time).

A frame of reference (“frame”) is a method to assign every particle a unique position in a coordinate system of points in ℝ3. Such assignment is known continually and universally, without signals, from the universal extent of the frame. The coordinate system is commonly Cartesian.

A system of reference (“reference system”) is a method to assign every event a unique position in a coordinate system of points in ℝ3 × ℝ3. A reference system is composed of dual frames of reference, one called the space frame and the other called the time frame, such that the time frame is in standard uniform motion relative to the space frame. This requires that given the magnitudes s1 and s2 of any two intervals of the curve of motion in the space frame, then the corresponding intervals of the time frame, t1 and t2, relative to the space frame satisfy the proportion: s1:s2 :: t1:t2.

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Rest in time-space

Speed is the travel distance per unit of duration (or time interval). Rest in space is a speed of zero. That is, there is no change in location per unit of time. A body does not change location (relative to an inertial observer) while time continues.

But rest in time seems different. It cannot be zero pace because that would mean it takes no time to go a positive distance, right? No, that is not what zero pace means.

Pace is the travel time per unit of distance (or stance interval). Time is the dependent variable and distance is the independent variable.

Consider a race that is about to begin. The runners are in place waiting for the signal to start. The official timer is set to begin. In terms of motion, the runners are at rest with speed of zero. They are not making any distance, but time continues as usual.

What is the pace of the runners in that case? There is no change on the official timer. But the stance continues as usual. For example, if stance is related to the distance from the Sun of a Voyager spacecraft (see here), it continues to increase as usual.

A map with a time scale shows a point for a pace of zero. Despite the distance made by an odologe, a body with a pace of zero remains in the same place in time. It is at rest in time.

Runner about to start

What about an infinite value for pace in time? The Galilean transformation implicitly has an infinite speed of information in space, which makes information spatially ubiquitous since it travels an infinite distance in a finite time. The symmetric Galilean transformation implicitly has an infinite pace of information in time, which makes information temporally ubiquitous since it takes an infinite time to travel a finite distance.

The Galilean group in time-space

The following is based on A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres (Cambridge UP, 2004) starting with Example 2.29 on page 54 and modifying it for time-space.

The Galilean group. To find the set of transformations of space and time that preserve the laws of Newtonian mechanics we follow the lead of special relativity and define an event to be a point of R4 characterized by four coordinates (t1, t2, t3, s). Define Galilean time G4 to be the time of events with a structure consisting of three elements:

  1. Distance intervals Δs = s2s1.
  2. The duration intervals Δt = |q2q1| between any pair of simulstanceous events (events having the same stance coordinate, s1 = s2).
  3. Motions of facilial (free) particles, otherwise known as rectilinear motions,
    q(s) = ws + q0,                 (2.19)
    where w and q0 are arbitrary constant vectors.

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