# Inertial frame of reference

An inertial frame is a coordinate system defined by the non-accelerated motion of objects with a common direction and speed.

## Introduction

In physics, an object has inertial motion if no external forces are being applied to it, famously stated as Newton's first law of motion. When such an object’s state of motion is extrapolated over a region of space to take in all other possible objects in the region with the same state of motion, and these are used to define a common coordinate system, this system is referred to as a frame

## Use of inertial frames

Inertial frames of reference are relevant to Newtonian relativity and Einstein's special theory of relativity.

• Under Newtonian mechanics, all inertial states of motion are considered to be equivalent: if two inertial observers, A and B have a relative velocity, then the laws of physics should be the same regardless of whether we take A as our “stationary” reference and say that B is moving, or if we take B as our fixed reference and say that A is moving. Included in these rules of physics is the explicit assumption that time progresses at the same rate for all observers, meaning that clocks calibrated in one inertial coordinate system will not become uncalibrated due to one of them being moved into another inertial frame of reference.
• Under special relativity, this equivalence of different inertial states of motion still applies. However, the assumption of constant progression of proper time in all frames of reference is replaced by the assumption that the speed of light is constant, and that this is equally true for every inertial observer. This required the use of a set of protocols created by Einstein (Einstein synchronisation) that allows observers to define apparent distances and times according to the assumption of fixed light speed in their own frame, and then build an extended coordinate system for labeling the times and distances of distant events. Observers using different reference frames will derive different nominal distance and time separations between the same two events. The formulas for converting, or "transforming" values between different frames of reference allow each observer to calculate how the physics taking place appears for another observer. As seen from different points of view the nominal distance and time separation between two events differs, but the combined spacetime interval is unchanged: it is "frame-independent", or "invariant".

## Transformations

The way that nominal distances and times are converted from one coordinate system to another is referred to as a transformation.

In classical mechanics the kinetic energy of a system depends on the inertial frame of reference. It is lowest with respect to the center of mass, i.e., in a frame of reference in which the center of mass is stationary. In another frame of reference the additional kinetic energy is that corresponding to the total mass and the speed of the center of mass.

Einstein argued that if we only assume that light propagates at c in a single preferred frame (i.e., if we assume an absolute fixed aether, classical theory), transformation of space and time coordinates is performed using Galilean transformations, whereas with special relativity we obtain Lorentz transformations, which only coincide with the earlier results for relative velocities that are reasonably small in comparison with the speed of light.

## Einstein’s general theory of relativity

Einstein’s general theory removes the distinction between nominally "inertial" and "noninertial" effects, by replacing SR’s "flat", Euclidean geometry with a curved non-Euclidean metric. Where the SR metric imposes behavior on matter without spacetime being altered in any way, the metric of GR is more dynamic (“space tells mass how to move, mass tells space how to bend”).

In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a given rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play. Consequently, modern SR is now sometimes described as only a “local theory”. (However, this refers to the theory’s application rather than to its derivation.)