Frame of reference
Two observers may choose to use different frames of reference to investigate a common system. The measurements that an observer makes about a system generally depend on the observer's frame of reference (see examples below). The concept of "axes" can be generalized to deal with curvilinear coordinate systems; for example in cylindrical coordinates, there is a z-axis, and the radial coordinate is measured from it, but the angular coordinate is defined by circular measure from an arbitrarily chosen plane passing through the z-axis. The usual reference frames for Newtonian physics are rigid, but in Newtonian fluid mechanics, one can use coordinates carried with the fluid motion, defining a "Lagrangian reference frame," as in . Returning to the special case of a frame with rectangular coordinates, one can define translations, rotations, and velocity transformations (those that carry one to a moving frame) as transformations of the reference system to another. The time is not transformed, except sometimes by a constant offset. Translations and velocity transformations (i.e. to moving frames) commute.
These definitions apply to "classical" physics, i.e. before the special theory of relativity and general theory of relativity. In special relativity, time becomes a coordinate on a nearly equal footing with the space coordinates. The primary rigid reference frames are the inertial reference frames, which can be mapped to each other via the Lorentz transformations. The Lorentz transformations again include displacement and velocity, but rotation is not as cleanly separated any more. If one applies an "x-boost" (transformation changing the velocities in the x direction, and next a y-boost, one finds that the final coordinate axes are not parallel to the original ones. This non-commutivity leads to the Thomas precession. The principle of relativity states that, even though a set of measurements may depend on an observer's particular frame of reference, the observed physical events still must follow the same physical laws in all inertial frames of reference.
For example, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).
Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving - for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.
Now assume Candace is driving her car in the opposite direction. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same - in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a' = a - A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a' = a + A in the negative y-direction - a larger value than Alfred's measurement.
Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another reference frame. The speed of light is considered to be the only true constant between moving frames of reference.
Nomenclature and notation
When working a problem involving one or more frames of reference it is common to designate an inertial frame of reference.
An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x' , y' , a' .
The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r'. From the geometry of the situation, we get
Taking the first and second derivatives of this, we obtain
where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.
These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as
When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in tangential direction, the coriolis effect). In actual fact the force exerted on the object that keeps the object's motion in sync with the rotating frame elicits manifestation of inertia. If there is insufficient force to keep the object's motion in sync with the rotating frame, then seen from the perspective of the rotating frame there is an apparent acceleration. Whenever manifestation of inertia appears to act as a force it is labeled as a fictitious force. Inertia is very much real, of course, but unlike force it never accelerates an object. In General Relativity, fictitious forces due to acceleration are indistinguishble from gravity in the small (local region); even in the large, the two kinds of force can be distinguished only in special cases, such as static reference frames or reference frames asymptotic (at large distances) to Minkowskian, or at least static ones.
A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation
Multiplying through by the mass m gives
Particular frames of reference in common use
- Mach's principle
- Principle of relativity
- Theory of relativity:
Distortions can vary from place to place, with gravity appearing to be the common cause. (See, for example #fn 1). Research demonstrates experimentally that the rotation of the Earth pulls inertial reference frames near it in a circular motion whose rotational speed must fall off at large distances. Simply put, a set of locally inertial reference frames at varying distances from the Earth's axis are twisted like molasses stirred by a central rotator.)