Most are familiar with Newton's second law, , in which the mass of an object relates the force and the acceleration vectors. In Newton's theory, the acceleration is always parallel to the force, since is a scalar (just a number). In relativity, however, our concept of velocity is radically different, and the corresponding equation takes a significantly different form. As I will demonstrate, acceleration is not always parallel to the force. But, before doing this, let's look at the relativistic version of in one dimension.
In one dimension...
To begin, we will use Newton's second law in it's original form, where ordinary force on an object is defined as the rate of change of momentum.
The relativistic version of momentum is
where is the speed of a particle with non-zero rest mass and is the Lorentz factor given by
Note that for simplicity, I will be letting the speed of light for the remainder of this post. Since the the Lorentz factor is changing with time, we need to find its derivative.
where is the acceleration. Now we can calculate the force.
In two dimensions...
In two dimensions, the Lorentz factor becomes
Taking the derivative, we find that
In two dimensions, force is a vector given by
For a particle traveling in the direction, is initially zero. So the Lorentz factor derivative becomes
The force vector is then given by
It should be clear that we could do the same thing in three dimensions and find that
Clearly, force and acceleration need not be parallel in relativity.
Transverse and Longitudinal Mass
In the early days, around 1900, Hendrik Lorentz defined mass as the ratio of force to acceleration rather than the ratio of momentum to velocity. Using such a definition, we see from the results above that mass would directional! That is, an object would have more mass in the direction of travel (longitudinal) than it would perpendicular to travel (transverse).
where m_0 is the rest mass. Today, these concepts are no longer used. Instead, mass without a subscript is always assumed to be the rest mass unless otherwise stated. For more on the subject of the concept of mass in relativity, see Relativistic mass [Fact or Fiction].