2.0  Newton's Laws of Motion - Relativistic Domain Theory Applicability.

2.1  Within the Relativistic Domain D0, (Pseudo-Euclidean Space-Time).

It is important to note that in the discussion of these laws, all applied forces referred to are defined as purely spatial. The corresponding temporal forces generated as a result of the rotation of the Existence Velocity vector in the X0 - R plane, cancel each other as is demonstrated in [6]. However, it is still necessary to perform all analytical derivations here within the full spatial-temporal dimensions in order to include the correct mass rate effects.

2.1.1  Newton's First Law.

In [1] Newton's first law is stated as follows:-

Ëvery body preserves in its state of rest or of uniform motion in a straight line except insofar as it is compelled to change that state by impressed forces."

In [4] and [5] this has been given mathematical form in spatial Cartesian co-ordinates thus,


d 2x
dt 2
    ;   d 2y
dt 2
   ;   d 2z
dt 2
 = 0
(2.1)

However, by way of introducing the mass, it is considered that the following is a more suitable expression of this law,


m0 dv
dt
 = 0
(2.2)

Where,

v    = The velocity vector.

m0 = The rest mass

References [4] and [5] have discussed both four dimensional and relativistic modifications to (2.1), but the important concept of Existence Momentum is absent from these discussions. Examination of Existence Momentum is considered necessary in the relativistic evaluation of these laws because it involves the mass of the body and in relativistic motion the mass is a variable.

The spatial-temporal Existence Momentum of a mass in D0 is, in [6], Eq.(3.1) given by,


M = m V= m ì
í
î 		v + jc æ
è 		1- v2
c2
 ö
ø 		1/2

 
 ü
ý
þ
(2.3)

Where

M = The spatial-temporal Existence Momentum vector.

V  = The spatial-temporal Existence Velocity vector.

m  = The energy mass.

j    = The temporal unit vector.

c   = The spatial terminal velocity in D0 , ( » the velocity of light).

Accordingly, in studying the Relativistic Domain Theory applicability of this law, the relationship to be analysed is,


dM
dt
 = 0
(2.4)

The criteria satisfying (2.4) are determined via the following simple process. The energy mass m, of a body in motion, is given by [6], Eq.(3.6), viz.


m =

m0
æ
è 		1- v2
c2
 ö
ø 		1/2

 
(2.5)

Where

v = The magnitude of v.

Inserting (2.5) into (2.3) gives,


M = m0 ì
í
î 		v
æ
è 		1- v2
c2
 ö
ø 		1/2

 
 + jc ü
ý
þ
(2.6)

so that


dM
dt
 =
m0
×
v
 
æ
è 		1- v2
c2
 ö
ø 		1/2

 
 +
m0 v
×
v
 
c2 æ
è 		1- v2
c2
 ö
ø 		3/2

 
(2.7)

and is all spatial.

Eq.(2.7) shows that the only requirement for (2.4) to be met is that the spatial velocity be constant in both magnitude and direction. This is identical to the condition of (2.2) and therefore Newton's first law, as stated above, is without change, fully applicable to relativistic motion within D0. Consequently, its mathematical expression, (2.2), is also fully applicable. However, it will become apparent from later analysis in relation to the third law, that (2.4) may be a more appropriate expression. It is also of interest to note that, from (2.6), temporal momentum is always constant irrespective of the state of spatial motion.

2.1.2  Newton's Second Law.

In [1] Newton's second law of motion is stated as follows,

"Change of momentum is proportional to the impressed force and takes place along the line of action of that force".

Mathematically it is stated in [1] to be represented by the equation,


d
dt
 ( mv ) µ F
(2.8)

which assumes that the mass is a variable. The more generally accepted non-relativistic relationship, also quoted in [1], is


m dv
dt
 µ F
(2.9)

in which the mass is implicitly defined as a constant, (of course this relationship is well known to be an equality rather than a proportionality).

In [4] and [5] et al the relativistic modification of this law is discussed. However, once again, despite (2.8) above, none of these discussions deal with Existence Momentum or, more importantly, in relativistic curvi-linear motion, that the applied force vector, the spatial acceleration vector and the spatial velocity vector, all lie in different directions, (see [6]). The relativistic mathematical expression to be analysed here, is given simply by,


d M
dt
 = F
(2.10)

Where

F = An applied spatial force vector.

Eq.(2.10) was analysed in some depth in [6] where the statement above concerning the non-alignment of the force, acceleration and velocity vectors was demonstrated as, viz. [6], Eq.(3.24),


tany = dvy/dt
dvxdt
 = sinx - v2/c2sinh cos( x - h )
cosx - v2/c2cosh cos( x - h )
(2.11)

Where

y = The angle between the acceleration vector and the X axis.

x = The angle between the applied force vector and the X axis.

h = The angle between the velocity vector and the X axis.

The geometry is depicted in [6], Fig.4.1 and clearly, from (2.11) these three vectors are only co-incident when,


x = h
(2.12)

i.e. for pure rectilinear motion.

As a result of the above discussion it is clear that the definition of Newton's second law is not adequate to rigorously cover full relativistic curvi-linear motion. To do so requires a re-statement thus,

"Change of momentum is proportional to the impressed force and takes place such that the direction of the acceleration vector is proportional to the directional vectors of the impressed force and the velocity".

which clearly covers both rectilinear and curvi-linear motion. The mathematical relationship applicable to this revised statement is that of (2.10) as augmented by (2.11).

2.1.3  Newton's Third Law.

In [1] Newton's third law is stated as follows,

Äction and reaction are always equal and opposite; that is to say, the actions of two bodies upon one another, are equal and directly opposite"

Mathematically, this law is also represented by (2.9) because the term on the LHS of this relationship represents the reaction to the applied force.

In relativistic motion it can be stated without further review that this law, as defined above, does not apply. This is because it takes no account of the spatial mass rate reaction terms. Spatial mass rate reactions, for artificially accelerated motion is discussed in [6], Section 4. It is here that the non-applicability of this law is most apparent. As an example, [6], Eq.(4.31) and [6], Eq.(4.32) show that a small reaction force is generated at right angles to the applied force due to the combination of the mass rate effect and the velocity. This effect is demonstrated in [6] by the relationships,


dvx
dt
 = - Fy v2
m0 c2
  æ
 è 		1 -  v2
c2
 ö
ø 		1/2

 
 sinhcosh

max = 0
(2.13)

Where

dvx /dt  = Acceleration in the X direction.

   Fy      = Applied force in the Y direction.

   v        = Magnitude of the total velocity vector.

   c        = Terminal velocity in D0, ( » the velocity of light).

   h       = The angle between v and the X axis.

   max    = Inertial mass in the X direction.

It is precisely this effect that is the cause of the non-alignment of the force, acceleration and velocity vectors discussed in the relation to the second law above. For applicability to motion in a relativistic domain, this law has to be completely restated as follows,

"To every action causing acceleration of a mass there are always two reactions. The mass reaction will always act in opposition along the acceleration vector, while the mass rate reaction will always act in opposition along the velocity vector.

Note that this automatically covers rectilinear motion where all force and motion vectors are co-incident. A formula specifically describing this law in relativistic terms is then,


m dv
dt
 + v dm
dt
 = F
(2.14)

This is evident from [6], Eqs.(3.10), (3.11), (3.12) and (3.13) and the associated discussion.

2.2  Consolidation.

The original three laws as expressed by (2.2) and (2.9), can be consolidated into one, that of (2.9) and the first law obtained from (2.9) by putting F to zero. In the relativistic case, this consolidation can only strictly be maintained for the first and third laws, i.e. (2.10), wherein the first relativistic law is obtained by putting F to zero, {note that for the third law, in (2.14) the LHS is simply the time differential of (2.10)}. In addition to (2.10), the second relativistic law also requires the more detailed but explicit relationship of (2.11).

2.3  The Incorporation of Gravitation - (The Gravitational Space-Time D1).

All of the subject analysis in the above Section has been carried out in the relativistic domain D0, (Pseudo-Euclidean Space-Time). The three laws of Newton in their original non-relativistic form, and in their modified relativistic form, have all been expressed within the characteristics of that domain. However, all motion of whatever kind takes place within a gravitational field, the domain D1, and the final and proper form of the relativistic laws should be expressed within such a domain.

Gravitational motion, both free and as augmented by artificially applied forces, were analysed in detail in [2] and [7] respectively. Accordingly, from these it is a simple matter to combine [2], Eq.(3.17) and (2.10) to give,


d M
dt
 = F - mc2u du
ds
(2.15)

where co-ordinates of space and time, s and t, are those of D1, Gravitational Space-Time, and u is the temporal rate of D1 with respect to D0. The relationship of (2.15) is in fact quoted in [7], {[7]. Eq.(3.16)} and analysed in depth there.

Eq.(2.15) is the most general relationship which, when appropriately expanded, expresses Newton's third law of relativistic motion within either the Gravitational Space-Time D1, or the Pseudo-Euclidean Space-Time D0, in the latter by putting u = 1, so that du/ds = 0 and t = t.

In the former domain however, the equivalent first law has to be restated because in putting F to zero in (2.15) leaves,


dM
dt
 = - mc2u du
ds
(2.16)

i.e. free gravitational motion as derived in [2], Eq.(3.17). The difference between (2.16) and (2.4) is very clear and expresses the fact that the natural state of existence in D0 is different from that in D1. In D0 the natural state of existence is to be spatially at rest, while in D1 it is to spatially accelerate towards the gravitational source. This was fully demonstrated and explained in [2], [6], and [7]. Within D1 therefore, an appropriate first law of (gravitational) motion would be stated as follows,

"Within a gravitational field, every body preserves in its state of acceleration towards the gravitational source, except insofar as it is compelled to change that state by artificially impressed forces."

Newton's second law of motion within D1 remains in its relativistic form as stated above, because the addition of the gravitational effect merely augments the artificial force term already present. Mathematical representation is by (2.15) and (2.11), the latter as modified with the change to gravitational spatial and temporal parameters.



R6 Version 1.0.0
Ó P.G.Bass, November 2008

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