Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
(26-41)
Relativistic Rule of Multiplication of Velocities
Consistent with Lorentz – Einstein Law of
Addition and Derivation of the Missing
Equations of Special Relativity
M.O.G. Talukder1, Mushfiq Ahmad2
1
Varendra University, Rajshahi-6204, BANGLADESH
Department of Physics, Rajshahi University, Rajshahi-6205, BANGLADESH
2
Abstract
In this paper, we present the rule for relativistic multiplication of a velocity by a number. We have
reasoned on the basis of a thought experiment and we have taken into consideration the L-E law of
addition of velocities. The formalism gives the result of repeated L-E addition just as ordinary
multiplication gives the effect of repeated Galilean addition. In the classical limit, it complies with
the Galilean law of multiplication. The formalism presented here can extend the horizon of
relativity. The thought experiment also reveals the values of both the relative length and time in the
longitudinal direction. Further, it has been demonstrated, as implications, that each relative
quantity has two values - one in the longitudinal and the other in the transverse directions. As a
consequence, we have found out the missing equations which are necessary to make Einstein’s
theory of special relativity self-consistent and complete. Moreover, we use the geometric mean to
get the mean value of the relative quantities. The justification of doing so is also demonstrated in
this paper. Finally, in the appendix, we present the relativistic multiplication rules for the relative
quantities like velocity, mass, time and length by a number. We also present the general rules for
the product of two relative quantities of the same entity.
Keywords: Relativistic addition and multiplication, velocity, mass, time, length,
special relativity, missing equations, geometric mean.
INTRODUCTION
The relativistic addition of velocities using Lorentz-Einstein transformation1,2 is given
by the well known formula
Vr  u  v 
uv
uv
1 2
c
(1)
where, Vr is the relative velocity, observed from an inertial reference frame S, of a
body moving with a speed u in another frame S'; when S' moves uniformly with
speed v relative to S and c is the speed of light. We have introduced the symbol  to
mean L-E addition. This means that velocity v should be added to the velocity u by LPage 26
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
E addition
()
ISSN 2305-915X(p); 2307-9584(e)
and not by Galilean addition (+). Now, if there is N number of equal
velocity u to be added relativistically, then
Vr  u  u  u .......... ...  u  N  u
(2)
where, for the L-E sum of N terms we have used the expression N
u .
In other
words, the symbol  has been used for L-E multiplication.
The relativistic addition of velocities have also been derived without using the L-E
transformations but using thought experiments and the formulas that account for time
dilation and length contraction3,4, from the invariance5 of c and using the time dilation
formula6. We have used the formalism, presented in this paper, to find the relativistic
expression of momentum conservation law7. Further, in a recent paper8, we have shown
the wave representation of particle kinematics and the equivalence between continuous
and discrete time using the same. On the other hand, Mr. Ahmad9,10 has studied the
discrete and continuous representations of the same motion employing this formalism.
The main objectives of this work are (a) to widen the scope of special relativity (to
accommodate quantum mechanics) by including some missing equations into Einstein’s
theory of SR and (b) to understand the role of the speed of light.
DERIVATION OF THE FORMALISM
Let us consider that the frame of reference S' moves along the X-axis with a uniform
velocity v relative to a stationary frame of reference S as shown in Fig. 1. There is a light
beam clock, with its two parallel mirrors placed horizontally along X-axis, in the frame of
reference S'. The clock traps a light pulse between two parallel mirrors that bounces off
the mirrors at perfectly regular intervals of time. The light pulse takes time t0/2 to travel
from one mirror to another. Suppose, initially the origins of S and S' are coincident.
Y
Y΄
l0
vt0/2
vt0/2
vt0/2
ct0/2 – vt0/2
ct0/2 + vt0/2
S
S΄
M1′
M1
M2′
M2
X, X΄
Z
Z΄
Fig. 1: S (X, Y, Z) is a Stationary and S' (X', Y', Z') is moving frame of reference. M1', M2'
are the horizontally placed mirror positions, of a light beam clock, after time t0/2, with
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
(26-41)
M1 and M2 being their positions after time t0. The separation between the mirrors is l0.
The solid line with arrows represents the path of light beam as seen from S.
Then an observer in S will see that in time t0, the frame S' will move a distance (d)
given in terms of the path of light by
 ct vt   ct vt 
d   0  0    0  0   vt0
2   2
2 
 2
(3)
During the same time interval, the light pulse traverses a path (d') given by
 ct vt   ct vt 
d 0  0   0  0 
2   2
2 
 2
d c  v   c  v  v



d  c  v   c  v  c
(4)
(5)
Multiplying both sides by c and rearranging the terms,
 1 v

1 v
v
c vc 
c
 1 v

 1 v
c
 1
c 
c
 1
c 
(6)
Hence, we can conclude that a velocity can be represented as a fraction of c by the
above relation. Thus, the L – E law (Eq. 1) can be written as
u cv c
u v
Vr  c    c
 c c  1  u c v c 
(7)
Hence, following Eq. (6), we can write
1u

1u
Vr  c 
1u

 1 u
c  1  v c 

 1
c  1 - v c 
c  1  v c 

 1
c  1  v c 
(8)
Therefore, for v = c
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
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2
 1 u / c 

 1
1  u / c 

Vr  u  u  2  u  c
2
 1 u / c 

  1
 1 u / c 
(9)
where, the symbol  indicates L-E multiplication. Similarly, we can show that
1 u

1 u
N u c 
1 u

1 u
N
c
 1
c 
N
c
 1
c 
(10)
Where, N is a number. The above equation represents the L-E multiplication of the
velocity u by the number N. It is equivalent to the L-E sum of N number of equal
velocity u. It has some advantages over the conventional form as follows. Suppose
there is N number of equal velocities to be added relativistically. If the conventional
L-E law is used for this purpose, N – 1 number of steps is needed to get the final
result. But the operation becomes cumbersome after 3 or 4 steps. Whereas, using the
present form, the final result can be obtained in just a single step even for large values
of N. The conventional form can be found suitable for certain cases, whereas, the
present form can be used to uncover different aspects of physical phenomena. As a
result, the domain of relativity is expected to be expanded beyond its horizon.
The above equation can also be written as
N u  c
1 u / c N  1 u / c N
1 u / c N  1 u / c N
(11)
Hence, in the classical limit (u << c)
N u  c
1  Nu / c   1  Nu / c   Nu
1  Nu / c   1  Nu / c 
(12)
which is the Galilean multiplication of the velocity u by the number N.
PROPERTIES OF
N u
We would like to verify if the expression for N  u , as given by Eq. (10), correctly
represents the L-E sum given by Eq. (2). The correct representation has to have the
following properties.
0 u  0
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(13)
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
1 u  u
(m  u)  (n  u)  (m  n)  u
(m u)  (m  v)  m  (u  v)
(26-41)
(14)
(15)
(16)
m  n  u   mn  u
(17)
Where, m and n are any numbers and u and v are velocities. For example, Eq. (16) can
be written, following Eq. (10), as
m
m
 1 u / c 
 1 v / c 

  1

 1
1 u / c 
1  v / c 


( m  u )  ( m  v)  c
c
m
m
 1 u / c 
 1 v / c 

  1

  1
 1 u / c 
 1 v / c 
m
(18)
m
 1 u / c   1 v / c 

 
 1
1  u / c   1  v / c 

c
 m  (u  v)
m
m
 1 u / c   1 v / c 

 
  1
 1 u / c   1 v / c 
(19)
Thus, it can be shown that all conditions of Eqs. (13) – (17) are fulfilled. Hence, we can
conclude that the formalism given in Eq. (10) correctly represents N
u .
RELATIVE LENGTH
As shown in Fig. 1, the length between the two mirrors is l0. Now in time t0, the light
travels twice between the mirrors,

ct0  2l0
(20)
2l0
c
(21)
or
t0 
As observed by an observer in S, the distance (d+) traveled by light beam from mirror M1
to M2 in the forward direction is
d 
Page 30
ct0
1  v / c 
2
(22)
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Again the distance (d-) traveled by the same beam from M2 back to M1 (i.e. in the
backward direction) is
d 
ct0
1  v / c 
2
(23)
Hence, the relative length (l) between the mirrors will be equal to the geometric mean
(justification is given in Section 6) of d+ and d- expressed as follows:
ct0
v2
v2
l d d 
1 2  l0 1 2
2
c
c


Thus, the length is contracted in the longitudinal direction by the factor
(24)
1
v2
.
c2
RELATIVE TIME
Further, as observed by an observer in S, the time taken by the light beam (t+) to travel
from M1 to M2 (i.e. in the forward direction) is from Eq. (22),
d  t0
t 
 1  v / c 
c
2

(25)
Again, the time taken by the same light beam (t-) in traveling from M2 back to M1 (i.e. in
the backward direction) is from Eq. (23),
d  t0
t 
 1  v / c 
c 2

(26)
Hence, the relative time (t/2) taken by the light beam to travel from one mirror to other is
the geometric mean of t+ and t-.

t
t
v2
 t t   0 1 2
2
2
c
 t  t0 1 
v2
c2
(27)
(28)
where, t is the total relative time for the light beam to travel from M1 to M2 and back to
M1. The same relation can also be obtained from Eqs. (21) and (24) as
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
t
2l
v2
 t0 1  2
c
c
(26-41)
(29)
Thus, time is contracted in the longitudinal direction.
JUSTIFICATION OF USING GEOMETRIC MEAN FOR RELATIVE QUANTITIES
Let us consider a different velocity u instead of v in Eq. (23), that is
d 
ct0
1 u c 
2
(30)
Then, from Eqs. (22) and (30), the arithmetic average of d + and d- is
d av 
Where,
1 ct0
2  v c  u c 
2 2
(31)

ct0  v  u  c 
1 

2 
2

(32)

ct0  1 wG 
1 

2  2 c 
(33)
wG  v  u 
(34)
is the Galilean addition of velocities. However, we need relativistic addition which
can be achieved as follows. Let us take the geometric mean of d + and d-:
d gm  d  d  
1
ct0
1 v c 1 u c 2
2
(35)
1
ct  uv v u  2
 0 1  2   
2  c
c c

Page 32

ct0
1  uv c
2
(36)
 1  1c  1vuvuc
1
2 2


2

 

1
2
(37)
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
ISSN 2305-915X(p); 2307-9584(e)
1

1
ct
 w 2
 0 1  uv c 2 2 1  r 
2
c 

where,
wr  v   u  
v u
1  uv c 2
(38)
(39)
is the relativistic addition of v and –u according to the Lorentz – Einstein law. When v
= u, wG = wr = 0 and
d av 
ct0
2
d gm 
(40)

ct0
1 v2 c 2
2

1
2
(41)
Hence, we can conclude that geometric mean is relativistic and arithmetic average is
Galilean in nature.
IMPLICATIONS OF THE RESULTS
(I) RELATIVE TIME
However, according to Einstein’s theory of relativity2, time is dilated in the transverse
direction which can be derived by considering the light beam clock placed perpendicular
to the direction of motion. Let us denote it by tET (in the suffix, E stands for Einstinian
relativity and T for transverse), then it can be expressed as follows:
t ET 
t0
v2
1 2
c
(42)
where, t0 is the time interval during a complete round trip of a pulse in a stationary light
beam clock (proper time). Now, since Eq. (28) represents contraction of time in the
longitudinal direction, we will denote the relative time by tEL (L stands for longitudinal).
Henceforth, all relative quantities will be denoted with suffixes EL for longitudinal and
ET for transverse values. Then, Eq. (28) can be written as
v2
t EL  t0 1 2
c
(43)
From the above two equations, we get
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
t ELt ET  t 0
2
(26-41)
(44)
That is the product of the longitudinal and transverse relative times is equal to the square
of the proper time. The above equation also indicates that the proper time is an invariant
quantity and is equal to the geometric mean of the relative times.
(II) RELATIVE LENGTH
Using Eq. (21) in the above equation, we can write
4l02
t ELt ET  2
c
(45)
Hence, using Eqs. (42) and (43), we obtain
t
0


 4l 2
t0
 0
1 v2 c 2 
 1  v2 c2  c2


(46)
or
 ct0
 ct0 2
1  v 2 c 2 

 2
 1  v 2 c 2
or
l
0


 l2
 0


 2
l0
 l
1  v2 c 2 
 1  v2 c 2  0


(47)
(48)
or
l ELlET  l02
(49)
lEL  l0 1  v 2 c 2
(50)
where,
and
lET 
l0
1  v2 c 2
(51)
where, lEL and lET are longitudinal and transverse lengths respectively. Equation (50)
indicates that the length is contracted in the longitudinal direction as in Eq. (24). On the
contrary, Eq. (51) indicates that the length is dilated in the transverse direction. However,
Eq. (49) shows that the product of the longitudinal and transverse lengths is equal to the
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
ISSN 2305-915X(p); 2307-9584(e)
square of the length at rest. It means that the length at rest is an invariant quantity and is
equal to the geometric mean of the relative lengths.
(III) RELATIVE VELOCITY
Rearranging the terms, Eq. (46) can also be written as

  2l
2l0
  0 1  v 2 c 2 
c2  

2
2
 t 1  v c   t0

 0



(52)
Hence, using Eq. (20), we get




c

 c 1  v2 c2  c2
 1  v2 c2 


(53)
 v ET v EL  c 2
(54)
where,
v ET 
c
(55)
1  v2 c2
and
v EL  c 1  v 2 c 2
(56)
vET and vEL are the transverse and longitudinal velocities, respectively. It is clear from the
above equations, both of them become equal to the speed of light when v→0. However,
for 0 < v < c, c > vEL > 0 and c < vET < ∞. That is, for increasing v, vEL decreases but vET
increases from c. That is vEL and vET change in opposite directions with increasing
velocity. So that for v → c, vEL→ 0 and vET→ ∞. Moreover, the product of these two
velocities is equal to the square of the speed of light. That is the speed of light is an
invariant quantity, in conformity with the postulate of special relativity, and is equal to
the geometric mean of the relative velocities.
(IV) RELATIVE MOMENTUM AND MASS
Multiplying both sides of Eq. (53) by m02, we get




m0c

 m c 1  v 2 c 2  m c 2
0
 1  v2 c2  0


(57)
or
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
(26-41)
pET pEL  p02
(58)
p0  m0 c
(59)
where,
pET 
and
m0c
(60)
1  v2 c2

pEL  m0c 1  v 2 c 2

(61)
In the above equations, p0 is the momentum when v→0. pEL and pET are the momentums
in the longitudinal and transverse directions, respectively. It is clear from Eq. (58) that the
product of the transverse and longitudinal momentums is equal to the square of the
momentum when v→0. It means the momentum p0 is an invariant quantity and is equal
to the geometric mean of the relative momentums.
Moreover, from Eq. (57), we can write




m0

 m 1  v2 c 2  m2
0
 1  v2 c2  0


(62)
mET mEL  m02
(63)
or
where,
mET 
m0
1  v2 c 2
(64)
and
mEL  m0 1  v2 c2
(65)
In the above equations, m0 is the rest mass; mET and mEL are relative masses in the
transverse and longitudinal directions, respectively. It should be pointed out here that
mET is the relative mass presented by Einstein in his theory of special relativity.
Moreover, Eq. (63) shows that the product of the relative masses is equal to the square of
the rest mass. It means the rest mass is an invariant quantity and is equal to the
geometric mean of the relative masses.
The expressions for longitudinal time given by Eq. (43), transverse length given by Eq.
(51), Transverse velocity given by Eq. (55), longitudinal velocity given by Eq. (56) and
longitudinal mass given by Eq. (65), respectively, are the missing equations in Einstein’s
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
ISSN 2305-915X(p); 2307-9584(e)
theory of special relativity. Further, the expressions for the invariance of time given by
Eq. (44), the invariance of length given by Eq. (49), the invariance of the speed of light
given by Eq. (54) and the invariance of mass given by Eq. (63) are also necessary to make
the total set of equations self-consistent. Thus, we conclude that these equations along
with the existing ones make Einstein’s theory of special relativity self consistent and
complete.
The complete set of self consistent equations for the relative mass, time, length and
velocity in the case of Einstein’s theory of Special Relativity (SR) has been given in the
following Table 1. The table includes both the existing and missing equations of SR.
For each of the relative quantities, both the longitudinal and transverse values and the
value of their product are given. The relative values in the transverse and longitudinal
directions are denoted by the suffixes ET and EL, respectively.
Table 1: Equations of relative quantities in Einstein’s theory of special relativity
Relative
Quantity
Transverse
Mass
mET 
Time
t ET 
Length
l ET 
Velocity
v ET 
m0
1 v 2 c 2
t0
1 v 2 c 2
l0
1  v2 c2
c
1  v2 c2
Longitudinal
Product
mEL  m0 1  v 2 c 2
mEL mET  m02
t EL  t 0 1  v 2 c 2
t ELt ET  t02
l EL  l0 1  v 2 c 2
lELlET  l02
v EL  c 1  v 2 c 2
vEl vET  c 2
Where, the symbols have their usual meanings. The equations in yellow color are the
missing equations.
CONCLUSIONS
Through a thought experiment based on L-E law for the addition of velocities, we
have found:
(a) A relativistic rule for multiplication of a velocity by a number.
(b) That both the length and time contract in the longitudinal direction.
(c) Geometric mean is relativistic and arithmetic average is Galilean in nature.
Further, as implications of the results obtained, we have found:
(a) The relative time contracts in the longitudinal direction but dilates in the
transverse direction. Their product is equal to the square of the proper time.
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
(26-41)
(b) The relative length contracts in the longitudinal direction but dilates in the
transverse direction. Their product is equal to the square of the proper length.
(d) The relative mass increases in the transverse direction but decreases in the
longitudinal direction. Their product is equal to the square of the rest mass.
(e) The relative velocity decreases in the longitudinal direction but increases in
the transverse direction. Their product is equal to the square of the speed of
light c.
REFERENCES
A. Einstein, Annalen der Physik, 17, 891 (1905).
H.A Lorentz, KNAW, Proceedings, Amsterdam, 6, 809 (1904).
L. Sartori, Am. J. Phys., 63, 81 (1995).
M. Ahmad and M.O.G. Talukder, Phys. Essays, 24, 593 (2011).
M. Ahmad and M.O.G. Talukder, Sent for publication in Phys. Essays (2011).
M. Ahmad, J. of Sc. Research 1, 270 (2009). DOI: 10.3329/jsr.v1i2.1875
M. Ahmad, Phys. Essays. 22, 44 (2009)
M. S. Greenwood, Am. J. Phys., 50, 1156 (1982).
N.D. Mermin, Am. J. Phys., 52, 1119 (1984).
W. N. Mathews Jr., Am. J. Phys., 73, 45 (2005).
APPENDIX
MULTIPLICATION RULES FOR THE RELATIVE QUANTITIES:
A. Product Rule
Equations (44), (49), (54) and (63) can be expressed as the following general rule:
X EL X ET  X 02
(A1)
 X EL  X ET


 X 0  X 0
(A2)
or

 1

where, X is any relative quantity with X0 being its value at rest; XEL and XET are its
values in the longitudinal and transverse directions, respectively.
B. Multiplication by a Number
(i) Relative Velocity
Putting u = vEL in Eq. (10), we get
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
 1 v EL

 v EL   1 v EL
N 

 c   1 v EL

 1 v EL
ISSN 2305-915X(p); 2307-9584(e)
N
c
 1
c 
N
c
 1
c 
(B1)
Now, from Eq. (54), we can write
v EL c

c v ET
(B2)
Using the above value of vEL/c in Eq. (B1), we get
 1 c

 c   1 c
N  
 
 v ET   1 c

 1 c
N
v ET 
 1
v ET 
N
v ET 
 1
v ET 
(B3)
(ii) Relative length
From Eq. (20), we can write
c
2l0
t0
(B4)
Putting this value of c in Eq. (59), we get
v EL 

2l0
2l
1  v 2 c 2  EL
t0
t0
v EL lEL

c
l0
 ct0  l0 
(B5)
(B6)
Hence, from Eqs. (B1) and (B6), we can write
 1 lEL

 lEL   1 lEL
N    
 l0   1 lEL

 1 lEL
N
l0 
 1
l0 
N
l0 
 1
l0 
(B7)
Similarly, it can be shown that c/vET = l0/lET. Putting this value in Eq. (B3), we can write
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Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz – Einstein Law of Addition and Derivation …
 1 l0

 l0   1 l0
N    
 lET   1 l0

 1 l0
lET
lET
lET
lET
(26-41)
N

 1

N

 1

(B8)
C. Relative time
Equation (B5) can also be expressed as follows:
v EL 

2l0
2l
1  v2 c2  0
t0
t ET
(B9)
v EL t0

c t ET
(B10)
Hence, from Eqs. (B1) and (B10), we can write
 1  t0

 t0   1 t0
N  
 
 t ET   1 t0

 1  t0
t ET
t ET
t ET
t ET
N

 1

N

 1

(B11)
Similarly, it can be shown that c/vET = tEL/t0 and hence Eq. (B3) can be written as
 1 tEL

 t   1 tEL
N   EL   
 t0   1 tEL

 1 tEL
N
t0 
 1
t0 
(B12)
N
t0 
 1
t0 
D. Relative mass
Now, multiplying both sides of Eq. (56) by m0, we get
m0 v EL  m0c 1 v 2 c 2  mELc

v EL mEL

c
m0
(B13)
(B14)
Hence, using Eq. (B14) in Eq. (B1), we can write
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Asian Journal of Applied Science and Engineering, Volume 2, No 1/2013
 1 mEL

 mEL   1 mEL
N  
 
 m0   1 mEL

 1 mEL
ISSN 2305-915X(p); 2307-9584(e)
N
m0 
 1
m0 
(B15)
N
m0 
 1
m0 
Similarly, it can be shown that c/vET = m0/mET and hence Eq. (B3) can be written as
 1 m0

 m0   1 m0
N  
 
 mET   1 m0

 1 m0
N

 1

N
mEt 
 1
mEt 
mEt
mEt
(B16)
Hence, the general expressions for the multiplications of relative quantities by any
number N can be written as:

N  

 1  X EL

X EL   1  X EL
 
X 0   1  X EL

 1 X EL
N
X0 
 1
X 0 
(B17)
N
X0 
 1
X 0 
and
1 X0

 X  1 X0
N   0   
 X ET   1  X 0

 1 X 0
X ET
X ET
X ET
X ET
N

 1

N

 1

(B18)
where, X is any relative quantity with X0 being its value at rest; XEL, XET are its relative
values in the longitudinal and transverse directions, respectively.
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