Inertial frame of reference

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Figure 1: Two frames of reference moving with relative velocity . Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.
Figure 1: Two frames of reference moving with relative velocity \stackrel{\vec v}{}. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

An inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid.[1]

Newton viewed the first law as valid in any reference frame moving with uniform velocity relative to the fixed stars;[2] that is, neither rotating nor accelerating relative to the stars.[3] Today the notion of "absolute space" is abandoned, and an inertial frame is defined as:[4][5]

An inertial frame of reference is one in which the motion of a particle not subject to forces is a straight line.

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—that is, 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 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. With this in mind, we can define inertial frames collectively as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

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[edit] Equivalence of inertial reference frames

The basic principle of relativity states: "Only relative motion is observable; there is no absolute standard of rest". According to this principle, the Laws of Physics are the same in all inertial frames (otherwise the differences would set up an absolute standard). In practical terms, this equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment.

By contrast, bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); 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.

[edit] Inertial frames in Newtonian mechanics

Classical mechanics, which includes relativity, assumes the equivalence of all inertial reference frames. Newtonian mechanics makes the additional assumptions 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

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

where \mathbf{r}_{0} and t0 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 (t2 − t1) 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.

[edit] 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^{\prime} = \gamma \left(x - v t \right)
y^{\prime} = y
z^{\prime} = z
t^{\prime} = \gamma \left(t - \frac{v x}{c_0^{2}}\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{\mathrm{def}}{=}\ \frac{1}{\sqrt{1 - (v/c_0)^2}} \ \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^{2} = \left( x_{2} - x_{1} \right)^{2} + \left( y_{2} - y_{1} \right)^{2} + \left( z_{2} - z_{1} \right)^{2} - c_0^{2} \left(t_{2} - t_{1}\right)^{2}

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).

[edit] 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.)

[edit] See also

[edit] External links

[edit] References

  1. ^ C Møller (1976). The Theory of Relativity, Second Edition, Oxford UK: Oxford University Press, p. 1. ISBN 019560539X. 
  2. ^ The question of "moving uniformly relative to what?" was answered by Newton as "relative to absolute space". As a practical matter, "absolute space" was considered to be the fixed stars.
  3. ^ Resnick, Robert and Halliday, David. PHYSICS. (Section 1-3 Reference Frames) John Wiley & Sons, Inc. New York (1960). Library of Congress Catalog Card Number: 66-11527
  4. ^ RG Takwale (1980). Introduction to classical mechanics. New Delhi: Tata McGraw-Hill, p. 70. ISBN 0070966176. 
  5. ^ NMJ Woodhouse (2003). Special relativity. London/Berlin: Springer, p. 6. ISBN 1-85233-426-6. 

[edit] Further Reading

  • 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