The Inertial Reference Frame

When doing physics, it’s always possible to choose the reference frame from which events will be measured, and these frames can differ both in orientation and relative motion. Among all possible reference frames, there is a special class that allows us to do physics as we know it: the inertial reference frames. A reference frame is said to be inertial when it satisfies Newton’s first law, which states that in the absence of external agents, particles maintain their state of motion and therefore:

\displaystyle \frac{dx^2}{dt^2} = \frac{dy^2}{dt^2} = \frac{dz^2}{dt^2} = 0.

From this, it follows that, in the absence of gravity, if two frames S and S^\prime are inertial, then S^\prime can only differ from S in:

• A translation,
• A rotation,
• Relative motion between both frames at a constant speed.

The concept of inertial frame is fundamental to the principle of relativity, which states that the laws of physics have the same form in all inertial frames. This principle applies equally in both Newtonian physics and special relativity.

The Principle of Relativity in Newtonian Physics and Special Relativity

Newtonian and special relativistic descriptions differ in how the coordinates of an event, relative to an inertial system, relate to those of another inertial system.

Consider two Cartesian inertial frames S and S^\prime in “standard configuration,” that is, where S^\prime moves along the \hat{x} axis of S at a constant speed \vec{v}_{ss^\prime} = v_{ss^\prime_x} \hat{x}, and the respective axes of S and S^\prime are aligned and coincide at t=t^\prime = 0.

Then it is found that, if there is a linear transformation relating the coordinates of an event seen from S and S^\prime, then they are related through the following system of linear equations

\begin{array}{rl} t^\prime &= At + Bx,\\ x^\prime &= Dt + Ex,\\ y^\prime &= y, \\ z^\prime &= z, \end{array}\;\;\;[\triangle]

where A, B, D, and E are constants to be determined.

Simplifying Transformations Between Inertial Reference Frames

These transformations can be simplified if we make the following observations:

• Since the transformations must hold for any x^\prime, it follows that if we place the event at the origin of S^\prime, we will have x^\prime =0. This implies that the event moves along with S^\prime and its position relative to S will be x=v_{{ss^\prime}_x}t.

  Substituting x=v_{{ss^\prime}_x}t. into the second equation of [\triangle] results in D=-Ev_{{ss^\prime}_x}.

  Similarly, the transformations also hold for any x, so if we place the event at the origin of S, we will find that, seen from S^\prime, its position is x^\prime = -v_{{ss^\prime}_x}t^\prime.

  Substituting this into the first and second equations of [\triangle] leads to t^\prime=At and -v_{{ss^\prime}_x}t^\prime =Dt. Dividing these two equations concludes that D=-v_{{ss^\prime}_x}A.

• Therefore, the only way to reconcile the previous points is to impose that A=E,, and with this, the transformations are reduced to:

  \begin{array}{rl} t^\prime &= At + Bx,\\ x^\prime &= A(x - v_{ss^\prime_x} t), \\ y^\prime &= y, \\ z^\prime &= z. \end{array}\;\;\;[2]

Lorentz and Galileo Transformations

In the case of Newtonian physics, we have Galilean relativity, where time elapses in the same way for all inertial frames of reference, and therefore t=t^\prime. As a consequence, A=1 and B=0. This leads to the well-known Galilean transformations which allow transforming observations between two inertial frames.

\begin{array}{rl} t^\prime &= t\\ x^\prime &= x - v_{ss^\prime_x} t, \\ y^\prime &= y, \\ z^\prime &= z. \end{array}\;\;\;[3]

On the other hand, in the case of relativistic physics, we have Einstein’s Principle of Relativity, which instead considers the speed of light in vacuum to be the same in all inertial frames. This leads to the well-known Lorentz Transformations of special relativity, which, as we will see in later entries, take the following form:

\begin{array}{rl} ct^\prime &=\gamma_x \left( ct - \beta_x x \right) \\ x^\prime &= \gamma_x(x - \beta_x ct) \\ y^\prime &= y \\ z^\prime &= z \end{array}

where \beta_x=v_{ss^\prime_x}/c and \gamma= 1/\sqrt{1-\beta_x^2}.

Conclusions

The Principle of Relativity not only revolutionizes our understanding of the universe but also challenges our most fundamental perceptions of time and space. Through the analysis of inertial frames of reference, we have seen how the laws of physics maintain their constant form, regardless of the observer, in both Newtonian physics and special relativity. The Lorentz and Galileo transformations uniquely illustrate the subtle and profound differences between these two approaches. This principle, which lies at the heart of modern physics, is essential not only for the theoretical understanding of physical phenomena but also for practical applications ranging from GPS technology to space exploration. By unraveling the complexities of the Principle of Relativity, we take one step closer to understanding the intricate fabric of the cosmos and our place within it.