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Problem in the relativistic energy and momentum conservation

Shalender Singh* and Vishnu Priya Singh Parmar

Priza Technologies Inc. R&D, 1525 McCarthy Blvd, Ste 1111,

Milpitas, California, USA 95035

Email: shalender@prizatech.com

Date: 6th Sept 2014

Abstract

Lorentz transformations and special theory of relativity have existed for more than a century and

mathematics related to them has been used and applied for innumerous times. Relativistic energy and

relativistic momentum equations have been derived and proven to be conserved if energy/momentum

transaction is seen from different frames of reference. The set of permissible inertial reference frame

velocities from where the energy and momentum of a closed system of particles may be observed to be

conserved forms a ball in the velocity vector space. In this paper we use the existing equations of special

theory of relativity and Lorentz transformations and the mathematical structure of the observation velocity

space to prove that the conservation of kinetic energy implies the conservation of momentum. We also

prove that the conservation of momentum implies the conservation of kinetic energy. The derivation of the

conservation of kinetic energy from the conservation of momentum implies that either potential energy has

a momentum thus made of inertial particles or there cannot be a net conversion of potential energy to kinetic

energy. This is an important result for classical mechanics because it implies that all fields, which lead to

particle interaction must carry momentum in them even in the case of classical mechanics. We further

discuss the case of attractive field leading to increase in momentum. In this paper we also show that a

similar derivation for conservation of Galilean/Newtonian energy leads to conservation of momentum but

applying the derivation on conservation of momentum just leads to conservation of mass.

Keywords: Conservation of energy, Conservation of Momentum, Lorentz Transformation, Special Theory

Of Relativity, Principle of Inertia, Infinite Conservation Equations.

Classification: PACS 03.30.+p

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I. INTRODUCTION

The conservation of energy and the conservation of momentum are considered two different principles,

each of them exists independently of the other [1] [2] [3] [4] [5]. In most of the existing models of particle

interactions and collisions, the conservation of momentum being a part of “principle of inertia” is

considered a more sacrosanct principle than the conservation of mechanical energy as mechanical energy

may be converted to some other forms of energy [1] [2] [3] [4] [6].

The first authoritative change to the long lasting formulae for kinetic energy was given by Einstein who

showed equivalence of mass and energy. He and his other contemporary physicist and mathematician came

up with space-time, energy and momentum transformation as per the new findings about the speed of light

[7] [8] [9] [10] [11] [12] [13].

In this article we prove that the conservation of momentum can be derived from the conservation of kinetic

energy. We also prove that the conservation of momentum implies conservation of kinetic energy. The

derivation is based on looking at conservation of energy of particles from a multiple (continuum of) inertial

reference frames. As per the special theory relativity the conservation of relativistic energy and momentum

remains valid if seen from any inertial frame of reference moving at a speed less that light [14]. The set of

observation inertial frames of reference form a continuum of velocities. The topology of the continuum is

defined only by the velocities so the position of reference frames in the space is irrelevant. A pair of inertial

reference frames with very close velocities but with a separation of very large distance are very close

elements in this continuum. As per the special relativity, the energy and momentum conservation is

consistent across all the inertial reference frames in the continuum. Figure 1 shows the 3D continuum of

velocities.

Figure 1: Continuum of inertial frames of references of observers as defined by velocities

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Assume that an observer in an inertial reference frame A considers himself to be stationary and is able to

get information from observers in various other inertial reference frames moving a speed w.r.t. him. It is

designated as the primary observer who gets data from the other secondary observers in other frames of

reference and then does a data analysis on it. They all are observe a closed system of particles S in universe.

Figure 2 is an illustration of it:

Figure 2: Observers in many inertial reference frames sending observations to the primary observer

The secondary observers measure total energy and momentum of the closed S and send the data to the

primary observer. Special theory of relativity allows any and all inertial frames of reference with velocity

less than light relative to A. So any inertial frame of reference with velocity 3( , , )x y zV V V and

( , , )x y zV V V c is allowed to send total energy and momentum of the closed system of particles S and it

is also supposed to observe them as conserved.

Figure 3: A conserved field in state 𝛼 and 𝛽 where Vx and Vy are velocity components of the continuum of allowed

inertial reference frames

For the ease of illustration assume 2D velocity space with A as origin (0,0) . With A as origin in the velocity

space there exists many allowed inertial frame of references with velocity 2( ),x yV V relative to A and

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( ),x yV V c . The energy and momentum of a closed system of particles is conserved for the observers in

all of them. The primary observer plots conserved quantities energy and momentum w.r.t. to the velocity

of the reference frame. Figure 3 shows a 2D surface plot of a conserved field F (energy or momentum) as

observed by observers in different inertial frames of reference. As we can see in the figure 3, if the field is

conserved with the change of state from α to β, the topology of the plot done by the primary observer

remains exactly the same. As the topology of the system remains exactly the same, various topological

constructs like derivatives along the topology also remain the same, which means conserved. This means

that / , /x yF V F V remains same for the states 𝛼 and 𝛽. Same is true for higher order derivatives. It is

important to note that these derivatives are taken along the topology of the value of the conserved function

as observed by secondary observers in the allowed inertial reference frames and does not in any way mean

the acceleration of the system or the reference frames.

Using the above formulation we given a detailed mathematical derivation of our results in the section II. In

the subsection B of section II we prove conservation of momentum from conservation of kinetic energy for

an arbitrary dimensional space. In the section C we prove the conservation of kinetic energy from the

conservation of momentum. In the section D we further prove that in the case of Galilean/Newtonian

mechanics conservation of kinetic energy does lead to conservation of momentum but conservation of

momentum lead to just conservation of mass.

II. DERIVATION OF CONSERVATION OF MOMENTUM FROM CONSERVATION OF ENERGY

Let’s assume there are n particles in a closed system and there exists a scalar conservation function F, which

solely depends on the magnitude of the velocity of the particles (we look at states of the system where there

is no inter particle potential energy).

Let the velocities of the particles be 1 2{ , ,..., }a a anV V V in a reference frame A. Let us take a continuum of

inertial reference frames as a set BC with velocities h w.r.t. to A where h c . Let h B BC be a

frame of reference in the set of inertial reference frames with velocity h w.r.t. A. Let the resultant velocity

of the particles in frame of reference hB be 1 2{ , ,..., }b b bnV V V . Then as per the Lorentz velocity addition

rule [7] [8] [16]:

22

( ) ( )

2

( 1 / )1

(1 / )

ak parallel ak perpendicular

bk

ak

V h h c VV k n

V h c

… (1)

A. Calculation of energy and momentum of the system for any reference frame along X axis

If the velocity of the particle k in the reference frame A is cos sinak ak k ak kV V i V j and we consider

only secondary reference frames along X axis then for an observer with velocity h hi as in figure (4)

the velocity of the kth particle is:

2 2

2

( cos( ) ) sin( ) 1 /, ,

(1 cos( ) / )

ak ak ak kbk k ak

ak ak

V h i V h c jV h V

V h c

… (2)

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Figure 4: The continuum of observers along the X axis

If , ,bk k akF V h V is the conserved function then per the conservation in the frame of reference B

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( ) , , . Cn

bk k ak b

k

Total B F V h V const

Let and be two states of the system in which the function is conserved then:

1 1

, , , ,n n

bk k ak bk k ak

k k

F V h V F V h V

… (3)

As equation (4) is valid for any h < c and h is a real number thus a derivative with respect to h should also

satisfy the equality both in state and

1 1

, ,, ,n nbk k akbk k ak

k k

F V h VF V h V

h h

… (4)

Equation (4) implies

1

, ,nbk k ak

k

F V h V

h

is also conserved as it remains constant with any arbitrary

change of state to when the F is conserved. … (5)

For calculation of energy and momentum of particle k we first find bk

2 2 2 2 2

2

( cos( ) ) sin ( ) 1 /

(1 cos( ) / )

ak ak ak ak

bk

ak ak

V h V h cV

V h c

2 2 2 2 2 2 2 2 2

2

cos ( ) 2 cos( ) sin ( ) / sin ( )

(1 cos( ) / )

ak ak ak ak ak ak ak

bk

ak ak

V h hV V h c VV

V h c

2 2 2 2 2 2

2

2 cos( ) / sin ( )

(1 cos( ) / )

ak ak ak ak ak

bk

ak ak

V h hV h c VV

V h c

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2 2 2 2 2 2 2 2 22

2

2 2

(1 cos( ) / ) 2 cos( ) / sin ( )

(1 cos( ) / )

ak ak ak ak ak ak akbk

ak ak

c V h c V h hV h c Vc V

V h c

2 2 2 2 2 2 2 2 2 2 22

2

2 2

2 cos( ) cos ( ) / 2 cos( ) / sin ( )

(1 cos( ) / )

ak ak ak ak ak ak ak ak akbk

ak ak

c V h V h c V h hV h c Vc V

V h c

2 2 2 2 2 22

2

2 2

/

(1 cos( ) / )

ak akbk

ak ak

c V h c V hc V

V h c

2 2 2 22

2 2

2 2

1 / 1 /

(1 cos( ) / )

ak

bk

ak ak

h c V cc V c

V h c

2 2 2 22

2

2 2

1 / 1 /1 /

(1 cos( ) / )

ak

bk

ak ak

h c V cV c

V h c

2 2

2 2 2 2 22

1/ cos( )1

1 / 1 /1 /

ak ak

bk

akbk

c c V h

h c V cV c

… (6)

Then energy of the particle k as observed by an observer in frame B is:

2

2 2 2 2

cos( )

1 / 1 /

k ak ak

bk

ak

m c V hE

h c V c

… (7)

And the momentum is:

2 2 2 2

22 2 2 2

1/ cos( ) ( cos( ) ) sin( ) 1 /

(1 cos( ) / )1 / 1 /

ak ak ak ak ak akbk bk bk k

ak akak

c c V h V h i V h c jP m V m

V h ch c V c

2 2

2 2 2 2

( cos( ) ) sin( ) 1 /

1 / 1 /

ak ak ak akbk k

ak

V h i V h c jP m

h c V c

2 2

2 2 2 2

( cos( ) ) sin( ) 1 /

1 / 1 /

ak ak ak akbk k

ak

V h i V h c jP m

h c V c

… (8)

Components of momentum in X and Y directions:

2 2 2 2

( cos( ) )

1 / 1 /

ak akbkx k

ak

V hP m

h c V c

… (9)

2 2

sin( )

1 /

ak akbky k

ak

VP m

V c

… (10)

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B. Conservation of mass energy implies conservation of momentum

If F in equation (5) is the energy function then:

1

nbk

k

E

h

is conserved. This implies

2

2 2 2 21

cos( )

1 / 1 /

nk ak ak

kak

m c V h

h h c V c

is conserved.

Simplifying the above expression:

2

2 2 2 21

cos( )

1 / 1 /

nk ak ak

kak

m c V h

h h c V c

2 2

3/22 2 2 2 2 2 2 21

cos( ) 2 /cos( ) 1

21 / 1 / 1 / 1 /

nk ak akk ak ak

kak ak

m c V h h cm V

h c V c h c V c

For 0h

2 2

1

cos( )

1 /

nak ak

kak

m V

V c

The above equation is the equation of conservation of momentum in X direction.

Similarly if we take observers along the Y axis with h hj then we get the conservation of momentum

along the Y axis.

Hence conservation of mass energy implies conservation of momentum.

C. Conservation of momentum implies conservation of mass energy

If F in equation (5) is the momentum in X direction function then:

1

nbkx

k

P

h

is conserved. This implies

2 2 2 21

( cos( ) )

1 / 1 /

nak ak

k

kak

V hm

h h c V c

is conserved.

Simplifying the above expression:

2 2 2 21

( cos( ) )

1 / 1 /

nak ak

k

kak

V hm

h h c V c

2

3/22 2 2 2 2 2 2 21

( cos( ) ) 2 /1

21 / 1 / 1 / 1 /

nk ak ak

k

kak ak

m V h h cm

h c V c h c V c

For 0h

8

2 21 1 /

nk

kak

m

V c

2

2 2 21

1

1 /

nk

kak

m c

c V c

As above expression is the conservation of energy.

D. Newtonian/Galilean case

Galilean velocity addition: If a particle is going with velocity aV w.r.t. a inertial reference frame A then

it’s velocity as seen from a reference frame B which is moving with velocity h is [7] [8] [16]:

b aV V h

For an observer with speed h hi (velocity (-h, 0)) as per Galilean velocity addition is:

( cos( ) ) sin( )bk ak ak ak akV V h i V j

2 2 2( cos( ) ) sin ( )bk ak ak ak akV V h V

2 2 2 cos( )bk ak ak akV V h hV

So the energy of the particle is:

2 212 cos( )

2bk k ak ak akE m V h hV

And momentum of the particle is:

( cos( ) ) sin( )bk bk k ak ak ak akP mV m V h i V j

Momentum in X and Y directions:

( cos( ) )bkx k ak akP m V h

sin( )bky k ak akP m V

If F in equation (5) is the Newtonian energy function then:

1

nbk

k

E

h

is conserved. This implies 2 2

1

12 cos( )

2

n

k ak ak ak

k

m V h hVh

is conserved.

1

cos( )n

k ak ak

k

m h V

is conserved.

For 0h

9

1

cos( )n

k ak ak

k

m V

is conserved.

The above equation is the equation of conservation of momentum in X direction.

Similarly if we take observers along the Y axis with h hj then we get the conservation of momentum

along the Y axis.

Hence conservation of mass energy implies conservation of momentum.

Now we take If F in equation (5) is the Newtonian momentum in X direction then:

1

nbkx

k

P

h

is conserved. This implies

1

( cos( ) )n

k ak ak

k

m V hh

is conserved.

1

n

k

k

m

is conserved, which is the conservation of mass.

III. ANYALYSIS OF THE RESULT

The result of derivation of conservation of relativistic mass energy from the conservation of momentum

implies that either the potential energy has a momentum thus composed of inertial particles or there

cannot be any net conversion of potential energy to the kinetic energy in a closed system. This implies

that all interaction fields are composed of momentum carrying particles even in the classical mechanics.

This concept is becomes especially interesting when the field imparting energy to the particles is an

attractive field, which means these field carrying particles of interaction lead to decrease of overall field

energy and the particles under attraction gain kinetic energy. In the figure (5) Particle 1 and Particle 2

have a mutual attraction. If the momentum transfer due to mutual attraction is being done by field

particles we have to assume that these particles are originating from a point in-between of Particle 1 and

Particle 2 and these particles approach Particle 1 and Particle 2 in the opposite direction of the motion. As

these lead to transfer of momentum of attraction then we can conclude that the mutual field generated by

this attraction is made up of particles with momentum in opposite direction of the motion.

Figure 5: Mutual attaraction of particles made from special field particles

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