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An inertial frame of reference, or inertial reference frame, is one in which Newton's first and second laws of motion are valid. Newton's laws are valid in any reference frame that is neither rotating nor accelerating relative to the sun and other stars.

Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—i.e. in a straight line and at constant speed. It is useful to picture this situation as if we are situated in a zero-gravity condition, but it is not impossible to establish an inertial frame in which gravity exists. For example, an observer confined in a free-falling lift will assert (if this is what he concerns at that moment) that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. Alternatively, we might consider inertial frames as a set of frames which are stationary or moving at constant velocity with respect to each other.

Contents 1 Equivalence of inertial reference frames 2 Inertial frames in classical mechanics 3 Einstein's theory of special relativity 4 Einstein’s general theory of relativity 5 See also 6 External links 7 References

A fundamental principle of all physics is the equivalence of inertial reference frames. In practical terms, this equivalence means that scientists within an enclosed box moving uniformly cannot determine their velocity 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 within a box that is being rotated or otherwise accelerated (except by gravity) can measure their acceleration and angular velocity by observing the motion of an un-restrained body inside the box.

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. 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

where and t₀ represent shifts in the origin of space and time, and is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time between two events (t₂ − t₁) is the same for all inertial reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, ) is also the same.

Einstein's theory of special relativity likewise assumes the equivalence of all inertial reference frames, but makes a different additional assumption, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. 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)
• relativity of simultaneity (simultaneous events in one reference frame are not simultaneous in almost all frames moving relative to the first).

These effects are expressed mathematically by the Lorentz transformation

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

The Lorentz transformation is equivalent to the Galilean transformation in the limit or, equivalently, (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

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 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.)

• Galilean invariance

• Stanford Encyclopedia of Philosophy entry

• 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)
• Henri Poincaré, (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaises, V, 253–78.
• Albert Einstein, On the Electrodynamics of Moving Bodies, included in The Principle of Relativity, page 38. Dover 1923