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Accident reconstruction research Research / Linear Momentum Exploring the Relativistic Energy-Momentum Relationship 1.
Complementary Time Dependent Coordinate Transformations It is interesting
to examine in more detail the expression of the quantity II.1. Complementary Time Dependent Coordinate Transformations We distinguish
between ordinary time dependent coordinate transformations (OTs)
and complementary time dependent coordinate transformations (CTs).
The OTs are simply obtained by changing angles and lengths in time
independent coordinate transformations into time dependent quantities.
They are represented by spatial rotations and translations. CTs
are related to the tracing of radii vectors by physical signals
traveling through space with constant velocity u . This tracing
is required by our need of knowing the length and the direction
of any radius vector before drawing and projecting it onto the coordinate
axes. There are in nature stationary" subspaces in uniform translatory
motion and the space at absolute rest (see also Sect.I.1.1). The
radii vectors of the geometrical points are defined with respect
to coordinate systems in space (K) and such subspaces (k). CTs are
established for space points coinciding with points of a subspace
at an instant of time. Unlike OTs which can be written whenever
after the radii vectors of a geometrical point were traced by a
pencil, the CT can be written only after the radii vectors we trace
by a pencil have previously been traced by physical signals of identical
nature. Depending on the nature of the physical signals tracing
radii vectors, we have a CT or another. For light signals we have
LT as a particular CT in the three-dimensional space. The preference
for LT is related to the large value of c in comparison with the
speeds of all the known physical signals, to the propagation of
the electromagnetic and gravitational fields at speed c, and especially
to the fact, pointed out in Sect.III.2.1, that c is also a subquantum
velocity. The equations of any CT are those of LT with c changed
to the speed u of the used physical signal. Specific to all CTs
is their time equation obtained in their preliminary form as the
time equivalent of their spacial equation written along the direction
of motion of k relative to K. The manner in which we use the physical
signals to establish a CT is just that used to obtain LT in Sect.I.4.
Like LT, any CT reduces to GT in the "low-velocity" approximation.
This only means that in such a situation OO’ becomes negligible
in comparison with OP1 (OP’) and O’P1 (O’P’)
in the diagram in Fig.5 (10), ct* reduces to ct and, implicitly,
t* reduces to the time t on the time axis. As concerns the homogeneity
of the CTs, it originates in the initial superposition of the coordinate
systems k and K required to obtain the geometry in Figs.5 and 10.
The most simple CT is that given by Eqs.(I-38). It follows from
the first of Eqs.(I-5) and (I-21) related to the upper diagrams
in Figs.1 and 2. The raising of Eq.(I-21) was largely discussed
in Sect.I.4.1. Like LT, Eqs.(I-38) form a group. For v=c, Eqs.(I-38)
reduce to x’=x-ct, t’=t-x/c.
(II-6)
Fig.14 Eqs.(II-6) are related to the diagram in Fig14. Since k is carried by the tip of a light signal, only geometrical points P(x’,x)Î (O’,O), where O’ and O are, respectively, the origins of k and K, can be joined by light signals. Naturally, Eqs.(II-6) do not form a group; this because, carried by light signals leaving simultaneously O, the coordinate systems kA and kB are always superposed to each other. Moreover, the time component of Eqs.(II-6) should not be identified with the time relation t’=t - r/c which, connecting two synchronous clocks, does not belong to a coordinate transformation (for consequences of CT see Sect.III.7 below). Final remark: Discovering the class of complementary time dependent coordinate transformations and showing that the Lorentz transformation belongs to this class, we proved that the non-co-linear LTs form group without requiring rotations of inertial coordinate systems in this aim. The full correctness of the LT is that enabling further to develop Einstein's theory of relativity into a physical theory. II.2. Transformation Laws for Energy and Linear Momentum Assume for the beginning that we do not know that the energy and the linear momentum form a four-vector. Also assume that we do not know the transformation laws satisfied by the covariant and contravariant components of a four-vector. So that we propose to establish the transformation laws of the two from the invariance of the action E’t’- p’x’ = Et - px (II-7) under Eqs.(II-4) and (II-5), connecting the coordinate system K at absolute rest to the parallel coordinate system k in uniform rectilinear motion along the common x’, x axis of coordinates. Denote by E, p and E’, p’ the energies and linear moments of a free particle in relation to K and k, respectively. Substituting Eqs.(II-4), (II-5) and their inverses in Eq.(II-7), we get, respectively, the equations E = E’+ p’v, p = p’+ E’v/c2, (II-8’) E = b (E’+ p’v), p = b (p’+ E’v/c2) (II-8") E’= E - pv, p’ = p - Ev/c2, (II-9’) E’= b (E - pv), p’= b (p - Ev/c2). (II-9") Eqs.(II-8) and (II-9) constitute the searched laws of transformation of the energy and linear momentum under the CT Eqs.(II-4) and (II-5). Each of these laws is analogous to the inverse of the CTs taken into account as a consequence of the last. II.3. Contravariant and Covariant Four-Vectors It is well-known that the contravariant and covariant components of a four-vector, respectively Am and Am , are mathematically given by the transformation laws18 Am =(¶ xm /¶ x’n )A’n Am =(¶ x’n /¶ xm )A’n ,
(II-10) df=(¶ f/¶
x)dx+(¶ f/¶ xo)dxo=[(¶ f/¶
x’)(¶ x’/¶ x)+(¶ f/¶ x’o)(¶
x’o/¶ x)]dx+ [(¶ f/¶ x’)(¶ x’/¶
xo)+ With the notations ¶ f/¶ x=A, ¶ f/¶ xo=Ao, ¶ f/¶ x’=A’, ¶ f/¶ x’o=A’o, we regain the first of Eqs.(II-10). This result is worthwhile because it infers that the components of any four-vector are always derivatives of a function which must be identified for its physical meaning and consequences to be well-determined. Unfortunately, there is the common tendency of endowing the four-vectors with a mysterious physical existence which, by their transformation law analogous with LT, extends onto the last. II.4. Four-Momentum, Proper Frame The four-momentum was defined by18 pm = mocum, pm = mob dxm /dt = (mob v, mob
c), II.5. The Relativistic Energy-Momentum Relationship Let us write the first of Eqs.(II-8) and (II-9) in relation to the proper frame of a free particle. Assuming p’=0, they are E = E’, (II-11’) E = b E’, (II-11") | p| =Ev/c2. (II-13) Since Eqs.(II-4) and (II-5) [the inverses of Eqs.(II-4) and (II-5)] connect a coordinate system k (K) in uniform rectilinear motion with respect to a coordinate system K (k) at absolute rest, whenever k represents a moving (rest) proper frame, the energy E’ appearing in Eqs.(II-11)[(II-12)] (see also Sect.II.4) is E’ = b moc2 [E’ = moc2]. Thus Eqs.(II-11) and (II-12) become E = b moc2, (II-14’) E= b 2moc2 (II-14") E = b 2moc2,
(II-15’) The quantity (II-1) reduces by Eq.(II-13) to E2/c2 - p2 = b -2E2/c2. Further, by Eqs.(II-14) and (II-15) it takes the forms E2/c2 - p2 = mo2c2 (II-16’) We recognize in Eq.(II-16’) the standard relativistic energy-momentum relationship. We also see that Eq.(II-16"), which is b 2 times Eq.(II-16’) and embodies a change of origin on the energy scale, has previously been missed by assuming that E’=moc2 for particles at rest in their proper frames, irrespective of the state of rest or uniform translatory motion of the last. Therefore, obtained by Eqs.(II-14) and (II-15) as well, Eqs.(II-16) do not depend on the presence of b in the CT taken into account. Implicitly, the dependence on b of Eqs.(II-2) and (II-3) is, in accord with the experiment, not determined by LT. The coincidence of Eq.(II-16’) [(II-16")] with that Eqs.(II-2) and (II-3) raise for a free particle moving relative to a K at absolute rest [in uniform rectilinear motion] assures the invariance of pm pm in relation to LT. II.6. The True Derivation of Standard Energy-Momentum Relationship The true derivation of the standard energy-momentum relationship is related to a particle at absolute rest with respect to a stationary coordinate system K. Its suitable energy is E=moc2. Its linear momentum is p=0. Inserting these values in Eq.(II-8") we obtain p’= -E’v/c2, E=b -1E’. Thus the energy and the linear momentum of this particle relative to a coordinate system k in uniform translatory motion with respect to K, as well as those of a particle moving with the same velocity relative to K, are E’=b moc2, ÷ p’ê =b mov. The relationship (II-16’) is immediate. Observe that, by replacing the stationary coordinate system K by a "stationary" coordinate system K, and denoting the energy and the linear momentum of a free particle respectively by mc2 [with m given by Eq.(II-3)] and p=0, we also deduce by Eq.(II-8") the energy-momentum relationship (II-16"). Unlike their derivation by means of Eqs.(II-11) and (II-12), Eqs.(III-16) have now a precise physical significance.
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