﻿ Ambiguity of Relativistic Mass
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The Ambiguity of
Relativistic Mass

In physics the notion of mass arises in three different contexts: 1. In the relation between force and acceleration, 2. In the relation between momentum and velocity, 3. In the relation between kinetic energy and velocity squared.

In Newtonian mechanics these relations are embodied in the following formulas F is force, a is acceleration, p is momentum, v is velocity, and K is kinetic energy:

• In the relation between force and acceleration

#### m = F/a

• In the relation between momentum and velocity,

#### m = p/v

• In the relation between kinetic energy and velocity squared

#### m = 2K/v²

.

Because the same value of m satisfies all three relationships it is taken to be an intrinsic property.

There is also the force between two bodies of mass m1 and m2 being

#### F = Gm1m2/s²

where s is separation distance and G is a constant.

## Intrinsic Properties and Contextual Characteristics

Under Special Relativity a situation involves an observer at rest and a vehicle traveling at a velocity v with respect to the observer. In the vehicle there are; three objects a stick of length l0, a clock that runs at a rate of t0 and a mass of m0. These values were established at a time when the vehicle was motionless with respect to the observer. When the vehicle is moving at a velocity of v with respect to the observer the characteristics of the three objects appear to be

#### l0(1−β²)½ t0/(1−β²)½ m0/(1−β²)½

where β is velocity with respect to the speed of light v/c.

These quantities are not intrinsic properties of the object; they are characteristics of the situation in which they are observed.

## Relativistic Mechanics

In Relativistic Mechanics it is said that the mass of a moving body is ⁪

#### m = m0/(1−β²)½

as though this an intrinsic property of the object. This is only a contextual characteristic with respect an observer.

It is then said that that the relativistic momentum is

#### p* = mv = m0v/(1−β²)½

But Lagranian analysis says the momentum with respect to a location variable x is

#### p = (∂K/∂v)

where v=(dx/dt).

The relativistic kinetic energy is given by

#### K= mc² − m0c² or, equivalently K = m0c²[1/(1−β²)½ −1]

Evaluation of (∂K/∂v) gives

#### p = mv/(1−β²) = m0v/(1−β²)3/2

The expression m0/(1−β²)3/2 has a historical precedence. In cerca 1881 J.J. Thomson noticed a resistance of charged bodies to acceleration. This resistance to acceleration he found was as though its mass was

#### m0v/(1−β²)3/2

This quantity came to be called longitudial mass mL. Thus relativistic momentum p can be computed as

#### p = mLv

However just because mL is used for mass in the momentum equation does not mean mL should be used for mass in the kinetic energy equation or other quantities. Mass and other variables under relativity are not intrinsic quantities. They depend upon the context of the situation.

Thus for computing relativistic kinetic energy mass is equal to

#### m0/(1−β²)½

But for computing momentum mass is

#### m0/(1−β²)3/2

This maintains the formula of momentum being equal to mass times velocity. One can equally maintain the mass concept and vary the formula; i.e.,