Inertial frame of reference

In physics, an inertial frame of reference is a frame of reference in which the laws of physics take on their simplest form. In Newtonian mechanics, and in special relativity, an inertial frame of reference is one in uniform translation with respect to the fixed stars, so far as present observations can determine. In general relativity an inertial frame of reference applies only in a limited region of space small enough that the curvature of space due to the energy and mass within it is negligible.

Simplest form of physical law
One could argue that "simplest" is a description that meant the Earth-centered universe at one time, so exactly what is simplest is subject to some evolution in meaning.

Today, the primary simplification of physical laws found in inertial frames is the absence of any need to introduce inertial forces, forces that originate in the acceleration of a noninertial frame. Such inertial forces can be identified by their lack of originating sources like charges or other fundamental particles, and their unusual dependence upon the observer's state of motion.

According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation: Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

The two points emphasized here are (i) simplicity of the laws and (ii) the same form for the laws when transformed between inertial frames.

Galilean frames of reference
In Newtonian physics, the idea of absolute time is introduced, which is the same for all observers. Inertial frames translate relative to one another, but share the same time. For example, they agree upon the simultaneity of two events. The laws of mechanics take the same form in inertial frames, specifically, Newton's laws of motion are the same for all observers. This behavior is referred to as Galilean invariance, and mathematically means that Newton's laws are not changed if the positions of all objects in space, say {rj (t)}, all are replaced by the positions seen in a translating frame moving with steady velocity v, namely {rj (t)−v t}. In particular, the famous second law F=m a is preserved:
 * $$\mathbf{F} = m\frac{d^2}{dt^2}\left(\mathbf{r_j}(t) - \mathbf v t \right) = m\frac{d^2}{dt^2}\mathbf{r_j}(t) \ . $$

In summary, within Newtonian mechanics, inertial frames are those related by Galilean transformations.

Lorentz frames of reference
In special relativity, the idea of absolute time is dropped. If we are to agree that all the laws of physics must apply in all inertial frames, Maxwell's equations of electromagnetism must appear the same in all inertial frames, not just the laws of mechanics. Moreover, the speed of light is postulated to be the same. The changes in coordinates that leave these equations unchanged are not those of Galilean transformations but those of the Lorentz transformation. In particular, the forces between charged particles depend upon their velocities, so Galilean invariance will not work. Both space and time must be transformed. In one dimension, for two inertial frames in uniform translation in (say) the x-direction, with their x-axes lined up (primes distinguish one frame from the other):
 * $$ x' = \frac{x-vt}{\sqrt{1-v^2/c^2}} \, $$
 * $$ t'=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}} \ . $$

The coordinates y and z are unaffected. Laws that do not change under such transformations are called Lorentz invariant. A generalization of the Lorentz transformation that includes an arbitrary displacement of the space and time origins is called a Poincaré transformation.

In summary, within special relativity, inertial frames are those related by Lorentz transformations.