# Dictionary Definition

inertial adj : of or relating to inertia

# User Contributed Dictionary

## English

### Pronunciation

inertial
1. of, relating to, or depending on inertia

# Extensive Definition

\mathbf^ = \mathbf - \mathbf_ - \mathbf t
t^ = t - t_
where \mathbf_ and t_ represent shifts in the origin of space and time, and \mathbf is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time between two events (t_ - t_) is the same for all inertial reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, \left| \mathbf_ - \mathbf_ \right|) is also the same.

## Einstein's theory of special relativity

Einstein's theory of special relativity, like Newtonian mechanics, assumes the equivalence of all inertial reference frames, but makes an additional assumption, foreign to Newtonian mechanics, namely, that in free space light always is propagated with the speed of light c0, a defined value independent of its direction of propagation and its frequency, and also independent of the state of motion of the emitting body. This second assumption has been verified experimentally and leads to counter-intuitive deductions including:
These deductions are logical consequences of the stated assumptions, and are general properties of space-time, not properties pertaining to the structure of individual objects like atoms or stars, nor to the mechanisms of clocks.
These effects are expressed mathematically by the Lorentz transformation
x^ = \gamma \left(x - v t \right)
y^ = y
z^ = z
t^ = \gamma \left(t - \frac\right)
where shifts in origin have been ignored, the relative velocity is assumed to be in the x-direction and the Lorentz factor γ is defined by:
\gamma \ \stackrel\ \frac \ \ge 1.
The Lorentz transformation is equivalent to the Galilean transformation in the limit c0 → ∞ (a hypothetical case) or v → 0 (low speeds).
Under Lorentz transformations, the time and distance between events may differ among inertial reference frames; however, the Lorentz scalar distance s2 between two events is the same in all inertial reference frames
s^ = \left( x_ - x_ \right)^ + \left( y_ - y_ \right)^ + \left( z_ - z_ \right)^ - c_0^ \left(t_ - t_\right)^
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.)

## References

inertial in Arabic: إطار مرجعي عطالي
inertial in Bengali: জড় প্রসঙ্গ কাঠামো
inertial in Belarusian: Інерцыяльная сістэма адліку
inertial in Bosnian: Inercijski referentni okvir
inertial in Catalan: Sistema inercial
inertial in Czech: Inerciální vztažná soustava
inertial in Danish: Inertialsystem
inertial in German: Inertialsystem
inertial in Modern Greek (1453-): Αδρανειακό σύστημα αναφοράς
inertial in Spanish: Sistema de referencia inercial
inertial in Basque: Erreferentzia-sistema inertzial
inertial in Persian: دستگاه مرجع لخت
inertial in French: Référentiel galiléen
inertial in Galician: Sistema inercial
inertial in Korean: 관성 좌표계
inertial in Croatian: Inercijski referentni okvir
inertial in Indonesian: Kerangka acuan inersial
inertial in Italian: Sistema di riferimento inerziale
inertial in Mongolian: Инерциал тооллын систем
inertial in Dutch: Inertiaalstelsel
inertial in Japanese: 慣性系
inertial in Norwegian: Treghetssystem