Inertial frame of reference

An inertial reference frame is a coordinate system in which Newton's first and second laws of motion are valid. In particular, a body viewed from an inertial frame accelerates only when a physical force is applied; in the absence of a net force, a body at rest remains at rest and a body in motion continues to move uniformly, i.e., in a straight line at constant speed (Newton's first law of motion).

Equivalence of inertial reference frames

A fundamental principle of all physics is the equivalence of inertial reference frames. In practical terms, this equivalence means that scientists living inside an enclosed box moving uniformly cannot detect their motion by any experiment done exclusively inside the box.

By contrast, bodies are subject to so-called fictitious forces in non-inertial reference frames; that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames. Therefore, scientists living inside a box that is being rotated or otherwise accelerated can measure their acceleration by observing the fictitious forces on bodies inside the box.

Inertial frames in classical mechanics

Classical mechanics assumes the equivalence of all inertial reference frames, and makes one additional assumption, namely, that time flows at the same rate in all reference frames. This corresponds to Newton's concepts of absolute space and absolute time, which in turn were founded on Newton's idea that space and time were a kind of sense organ for God. Given these two assumptions, the coordinates of the same event (a point in space and time) described in two inertial reference frames are related by a Galilean transformation

\mathbf{r}^{\prime} = \mathbf{r} - \mathbf{r}_{0} - \mathbf{v} t

t^{\prime} = t - t_{0}

where $\mathbf{r}_{0}$ and $t_{0}$ represent shifts in the origin of space and time, and $\mathbf{v}$ is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time between two events ( $t_{2} - t_{1}$ ) is the same for all inertial reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, $\left| \mathbf{r}_{2} - \mathbf{r}_{1} \right|$ ) is also the same.

Einstein's special theory of relativity

Einstein's theory of special relativity likewise assumes the equivalence of all inertial reference frames, but makes a different additional assumption, namely, that the speed of light is the same when measured in all inertial reference frames. This second assumption leads to counter-intuitive effects that have been verified experimentally, including:

time dilation (moving clocks tick more slowly)
length contraction (moving objects are shortened in the direction of motion)
simultaneity breakdown (simultaneous events in one reference frame are not simultaneous in any frame moving relative to the first).

These effects are expressed mathematically by the Lorentz transformation

x^{\prime} = \gamma \left(x - v t \right)

y^{\prime} = y

z^{\prime} = z

t^{\prime} = \gamma \left(t - \frac{v x}{c^{2}}\right)

where shifts in origin have been ignored, the relative velocity is assumed to be in the $x$ -direction and the factor $\gamma$ is defined

\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{c}{\sqrt{c^2 - v^2}} \ge 1

The Lorentz transformation is equivalent to the Galilean transformation in the limit $c \rightarrow \infty$ or, equivalently, $v \rightarrow 0$ (low speeds).

Under Lorentz transformations, the time and distance between events may differ among inertial reference frames; however, the Lorentz scalar distance $s^{2}$ between two events is the same in all inertial reference frames

s^{2} = \left( x_{2} - x_{1} \right)^{2} + \left( y_{2} - y_{1} \right)^{2} + \left( z_{2} - z_{1} \right)^{2} - c^{2} \left(t_{2} - t_{1}\right)^{2}

where $c$ is the speed of light. From this perspective, the speed of light is only accidentally a property of light, and is rather a property of spacetime, a conversion factor between conventional time units (such as seconds) and length units (such as meters).

Einstein’s general theory of relativity

Einstein’s general theory modifies the distinction between nominally "inertial" and "noninertial" effects, by replacing special relativity's "flat" Euclidean geometry with a curved non-Euclidean metric. 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 particular 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 special relativity is now sometimes described as only a “local theory”. (However, this refers to the theory’s application rather than to its derivation.)

External links

Stanford Encyclopedia of Philosophy entry

References

Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics 2nd ed. (Freeman, NY, 1992)
Albert Einstein, Relativity, the special and the general theories, 15th ed. (1954)
Poincaré, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaises, V, 253-78.