Mass-Energy Relation
 Mass

Accordingly with Newton's second law of motion, if a net external force F acts over an object of mass m, this object accelerates with the acceleration a obtained from the formula where the arrows over the quantities means that the force and acceleration are in the same direction. Understanding that the directions are the same, the acceleration can be obtained from . Notice that in this classical expression when the force is constant, the acceleration is also constant.

For an object accelerating with constant acceleration, the velocity at any instant of time can be calculated from which in terms of the force applied to the object becomes . Therefore, if the object accelerates for a sufficient long time, the velocity of the object can increase above the speed of light. However, if at the same time that the object increases its velocity, it gets harder to increase the object velocity; then, even when a force is still acting on the the object, the object can not exceed the speed of light. This result is achieved by redefining the mass (notice that the force depends on an external agent and it should not be redefined). The redefinition of the mass when moving at a velocity v with respect to an observer is:

The value of m0 is the measurement of the mass of the object with respect to an observer at rest with respect to the mass (measured in a traditional scale). The mass m represents the value measured for the mass of the object by an observer for whom the mass is moving at the velocity v.

 Understanding the Relativistic Factor or Gamma (g) Factor.

Most of the relativistic quantities are related to their classical counter parts through the relativistic factor g,

As it can be observed on the graph below, this factor is relevant for speeds greater than about 75% of the speed of light.

 The graph above shows the behavior of the relativistic multiplication factor gamma as a function of the square of the ratio between the speed of the object and the speed of light. Since the factor depends of the square of the velocity, the factor gamma is independent of the direction of the velocity. Therefore, an extreme value is obtained when the velocity is minimal in magnitude, velocity zero. When the velocity increases to values approaching the speed of light, this ratio becomes closer and closer to one which makes the g factor greater and greater reaching infinity, ∞. For velocities greater than the speed of light, the  ratio between the velocities will be greater than 1. The radical to be calculated will be the radical of a negative number resulting in a complex quantity result. As a consequence of these mathematical results, the speed of any mobile can not be greater or equal to the speed of light. The graph above represents a comparison between the exact factor g  and approximation of the factor for low velocities. Notice that for velocities up to about 25% of the speed of light, v = 7.53 × 107 m/s, the two expressions result in about the same value for the factor As a way of comparison, the speed of sound is only 3 × 102 m/s. The previous approximation shows the first relativistic correction to classical values for the different physical quantities.

 Energy

Derived from electromagnetism, an electromagnetic wave (light) carrying an amount of energy E has a linear momentum of .

 The diagram on the left represents two instance for a system made out of a spaceship and a pulse of light. The top portion of the diagram shows the instant when the pulse of light is emitted at the back of the spaceship. The coordinate system selected moves with the center of mass of the system. After the pulse of light reaches the front of the spaceship the following relations are valid: Distance travel by the pulse ; and Distance traveled by the spaceship center of mass . Since there is not external force acting on the system, the center of mass of the system must be unchanged. In this discussion, the pulse of light is assigned a contribution to the change in the center of mass of the system in the amount of  . This amount much the change in the position of the center of mass of the system coming from the change in the center of mass of the spaceship given by . For the position of the system center of mass to be unchanged due to this emission, the two changes must be equal to each other, . Thus,                    . On the other side, the linear momentum of the system must be conserved in any inertial frame of reference, including the frame of reference attached to the system center of mass. In this case, the linear momentum of the spaceship is just , to the left of the diagram while the linear momentum of the pulse of light is , to the right of the diagram.  Equating these two momentums, the following relation is established    with   or

The previous discussion is not a derivation of the formula that relates the energy with the mass of the object and the square of the speed of light.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com