Physics Modeling

Relativistic Mechanics

When a dynamic situation implies bodies moving at speeds close to the speed of light, the physical modeling that should be used is called relativistic mechanics. If there are large accelerations involved or if there are extremely large masses as found in neutron stars, the way is to work in one of the systems of mechanics related to the general theory of relativity:

✅Special Relativity Theory.
✅Theory of General Relativity.

What may seem complicated is alleviated by requiring that the masses involved be of ordinary values ​​and that any velocities involved, however large, be of constant or uniformly varying values. Such requirements represent the limit of application of the theory of special relativity.

In this present text we will describe about the theory of special or special relativity.

The great agent that boosted the development of relativistic mechanics was the primary result of the Michelson-Morley experiment, whose purpose was to seek information about a possible medium under which electromagnetic waves would propagate.

A basic assumption that followed was that the speed of a packet of light in vacuum is the same for all inertial observers, even when these may be moving relative to each other with an arbitrary constant speed.

On the other hand, it is already quite evident that all observations of natural events are ultimately carried out, in some way, by the mediation of electromagnetic waves. Therefore, with a view to algebraically articulating this problematization, a coordinate transformation different from the Galilean one must be used, that is, the Lorentz transformation.

Based on the scheme in the figure, where two coordinate systems are represented, the corresponding Lorentz equations will then be:

where γ is the Lorentz factor

The ratio of the speed v and the speed c of light is called the speed factor and is indicated by the symbol β:

As the velocity v becomes very small compared to c the equation

reduces to

since, using a polynomial expansion the Lorentz factor can be written as

for example, for a velocity in the range of the Earth’s escape velocity, for example, 12km/s, we have

Still regarding the same configuration of reference frames, the Lorentz transformation or relativistic transformation of corresponding velocities is given by

Obviously, when

then we have

A consequence of the Lorentz transformation is that the difference between the coordinate values ​​of, say, a length measurement along the X axis depends on the relative velocity between that particular axis and the observer measuring that length.

This is seen in the equation

where γ is a common value for any value of x₁, and thus multiplies any length value along the X coordinate axis.

So it is easy to deduce that the transformation law for length measurements is

And, since the value of γ is always greater than unity, then L₁ is always less than L₂, and so we say that this corresponds to a length contraction.

A similar consequence follows from

because chronological intervals between t₁ values ​​are also affected by relative velocity. The resulting transformation law for time intervals is then

and as such we say that this corresponds to a time dilation.

Rewriting the description of the principle of classical relativity, we can do so by declaring that the laws of nature are invariant in form for all inertial frames moving relative to each other with constant and arbitrary velocity.

Through this postulation, the law of conservation of momentum given by the equation

remains valid in the form in which it is presented there, both in classical mechanics and in relativistic mechanics. This fact recommends it as a starting point in a model for Nature. This law, when taken together with the Lorentz transformation, results in a definition of mass that in turn becomes a quantity dependent on relative velocity,

where m₀ is taken as the mass measured by an observer at relative rest. It is the smallest possible value for the mass of any object, a rest mass.

In the case of the quantity force, it is then defined in relativistic mechanics as well as in classical mechanics in the form

equal to Newton’s second law of classical mechanics, except that m is now a function of v according to the expression m=γ·m₀. And this expression can be used to relativistically define the kinetic energy K.

Assuming the force is acting parallel to the velocity direction, we have

result that also extends to a curvilinear movement.

Expanding the integral expression, we have

and

or even doing

we get

when v≪c, ​​then γ→1 and

Also if we do

then we get the expression

which is the definition of total energy, where E₀ is the rest energy.

It is also defined, as a consequence of universalization by relativistic mechanics, the expression of the relativistic moment

or

The principle of conservation of relativistic momentum and the definition of total energy lead to the principle of conservation of mass-energy, which states that, for an isolated system,