**2. Application of Lorentz transformation for a group of mass.**

To make it clear what rest mass is, it is possible to suppose that a lot of masses are moving in a coordinates A as shown in Fig.2-1. As a first step, the coordinates A is supposed to be t-x dimension.

Among a group, mi is a mass and vi is the velocity of mass in the preceding coordinates A. Individual mass and momentum are expressed by following expression in coordinates A.

A group of mass is observed from a different coordinates B. Coordinates B is moving with velocity V in coordinates A.

There is always a special coordinates B where a group of mass is considered to be stationary. According to the formula of velocity composition, vi is converted to vi’ in coordinates B. Velocity vi’ is expressed by equation (2-3).

Using equation (2-3), mi’ and pi’ are expressed by equation (2-4), (2-5) in coordinates B.

There is always a special coordinates B where a group of mass is considered to be stationary. According to the formula of velocity composition, vi is converted to vi’ in coordinates B. Velocity vi’ is expressed by equation (2-3).

Using equation (2-3), mi’ and pi’ are expressed by equation (2-4), (2-5) in coordinates B.

Equation (2-6) is always true because it is a mathematically identical equation.

We could not know simple relation between (mi, pi) and (mi', pi') if interesting equation (2-6) was not found.

We could not know simple relation between (mi, pi) and (mi', pi') if interesting equation (2-6) was not found.

By substituting equation (2-1), (2-2), (2-6) into equation (2-4), (2-5), equation (2-7), (2-8) are obtained.

These equations mean that (mi,pi) is converted to (mi', pi') according to Lorentz transformation. The sum of mi and pi are also expressed by following equations. In order that the sum of pi' equals zero, we can choose a special coordinates B whose velocity is V (=sum of pi/sum of mi).

We are approaching the concept of rest mass.

We are approaching the concept of rest mass.

A group of mass and momentum are defined as following. These equations mean that a group of mass is the summation of relativistic mass (mi or mi'). A group of momentum is the summation of relativistic momentum (pi or pi').

(M, P) is a vector in coordinates A, and (M', P') is the one in coordinates B. P' equals zero when observer chooses a special coordinates B.

(M, P) is a vector in coordinates A, and (M', P') is the one in coordinates B. P' equals zero when observer chooses a special coordinates B.

Equations (2-9), (2-10) are expressed by a matrix equation (2-12).

Equation (2-12) is exactly Lorentz transformation of a group of mass and momentum similar to fundamental equation (1-3).

This equation implies that a group of mass has an invariant as a rest mass as well as a single mass. A group of mass, M'

becomes a rest mass when its momentum P' is zero in a special coordinates B. V is representative velocity of a group of mass. When we choose a coordinates B which makes P' equals zero, P=MV is obtained from the second line of matrix equation (2-12). P=MV seems natural, but we must notice that P equals MV only when P' equals zero. Therefore, V is defined as representative velocity of a group. P is not MV when we choose other velocity V. P will be a complicated equation according to equation (2-7), (2-8) if you do not choose a special coordinates B.

Using the relation of P=MV, the first line of matrix equation (2-12) is following.

This equation implies that a group of mass has an invariant as a rest mass as well as a single mass. A group of mass, M'

becomes a rest mass when its momentum P' is zero in a special coordinates B. V is representative velocity of a group of mass. When we choose a coordinates B which makes P' equals zero, P=MV is obtained from the second line of matrix equation (2-12). P=MV seems natural, but we must notice that P equals MV only when P' equals zero. Therefore, V is defined as representative velocity of a group. P is not MV when we choose other velocity V. P will be a complicated equation according to equation (2-7), (2-8) if you do not choose a special coordinates B.

Using the relation of P=MV, the first line of matrix equation (2-12) is following.

Referring to the relation P=MV, equation (2-14) is obtained.

M’ can be considered as an equivalent rest mass M0. M0 is the summation of relativistic mass mi'. In this case, relativistic

mass is important to define the rest mass of a group of masses. From equation (2-15), equivalent rest mass M’ is larger than the total of rest mass mi0. Despite the fact that mi' is moving, M' is considered to be stationary in a special coordinates B which makes P’ equal zero. This means that M0 (rest mass) contains inner kinetic energy and rest energy. In other words, without increasing of individual rest mass mi0, rest mass (M0) of a group increases when individual kinetic energy increases. The second equation of 2-15 indicates that relativistic mass is a moving mass which forms a group of masses.

mass is important to define the rest mass of a group of masses. From equation (2-15), equivalent rest mass M’ is larger than the total of rest mass mi0. Despite the fact that mi' is moving, M' is considered to be stationary in a special coordinates B which makes P’ equal zero. This means that M0 (rest mass) contains inner kinetic energy and rest energy. In other words, without increasing of individual rest mass mi0, rest mass (M0) of a group increases when individual kinetic energy increases. The second equation of 2-15 indicates that relativistic mass is a moving mass which forms a group of masses.

Equation (2-16) is a famous equation, which indicates a single mass m has an invariant m0. From equation (2-13), (2-14),

equation (2-17) is obtained. Equation (2-17) also indicates a group of mass M has an invariant M0. Only when a group of rest

mass is confined within M0, M0 is an invariant in any reference system. From equation (2-17), M' equals M0 when P' equals zero.

equation (2-17) is obtained. Equation (2-17) also indicates a group of mass M has an invariant M0. Only when a group of rest

mass is confined within M0, M0 is an invariant in any reference system. From equation (2-17), M' equals M0 when P' equals zero.

Fig.2-2 is the result of calculation that indicates M0 is an invariant in any reference system. Three masses are moving in coordinates A. Coordinates B is moving with the velocity of VB(VB=0.4). Va is the velocity of a group of mass in coordinates A. Vb is the velocity of a group of mass in coordinates B. Coordinates C is moving with the same velocity(VC=Va). Therefore, momentum of a group of mass is zero in coordinates C, and a group of mass Mc is equal to equivalent rest mass M0.

Fig.2-3 indicates the relation between a group of mass and coordinates A,B,C. Velocity VB is 0.4 according to an example of calculation table in Fig.2-2.

Simple example of two moving masses can be considered. Fig.2-4 indicates two models of moving mass. Although the speed is the same in coordinates A, The one case is moving mass in opposite direction, the other is parallel motion.

In case of opposite motion, equations (2-18) indicates relativistic mass(M) is equal to rest mass(M0)=2m0/sqrt(1-v^2) in coordinates A. Equations (2-22), (2-23), (2-24) indicate the rest mass(M0) can be logically considered to be 2m0/sqrt(1-v^2) in the coordinates B. In case of parallel motion, relativistic mass and momentum in coordinates B is expressed by equations (2-25), (2-26), and total of rest mass is the same as 2m0 in both coordinates A and B.

In case of opposite motion, equations (2-18) indicates relativistic mass(M) is equal to rest mass(M0)=2m0/sqrt(1-v^2) in coordinates A. Equations (2-22), (2-23), (2-24) indicate the rest mass(M0) can be logically considered to be 2m0/sqrt(1-v^2) in the coordinates B. In case of parallel motion, relativistic mass and momentum in coordinates B is expressed by equations (2-25), (2-26), and total of rest mass is the same as 2m0 in both coordinates A and B.

It seems to be sure that rest mass(M0) is 2m0/sqrt(1-v^2) in the coordinates A. However, it must be confirmed that rest mass(M0) is the same as 2m0/sqrt(1-v^2) even in the coordinates B.

In case of the second model, although the velocity of the mass is different between coordinates A and B, rest mass(M0) is the same as 2m0 in both coordinates A and B.