San José State University |
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The Ambiguity of Relativistic Mass |
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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
m_{1} and m_{2} being

where s is separation distance and G is a constant.

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 l_{0},
a clock that runs at a rate of t_{0} and a mass of
m_{0}. 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

t

m

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.

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

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

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

where v=(dx/dt).

The relativistic kinetic energy is given by

or, equivalently

K = m

Evaluation of (∂K/∂v) gives

The expression m_{0}/(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

This quantity came to be called *longitudial mass* m_{L}.
Thus relativistic momentum p can be computed as

However just because m_{L} is used for mass
in the momentum equation does not mean m_{L}
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

But for computing momentum mass is

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.,

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