Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.
Inertial frames and rotation In an inertial frame,
Newton's first law, the
law of inertia, is satisfied: Any free motion has a constant magnitude and direction.
Newton's second law for a
particle takes the form: :\mathbf{F} = m \mathbf{a} \ , with
F the net force (a
vector),
m the mass of a particle and
a the
acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force
F is the
vector sum of all "real" forces on the particle, such as
contact forces, electromagnetic, gravitational, and nuclear forces. In contrast, Newton's second law in a
rotating frame of reference (a kind of
non-inertial frame of reference), rotating at angular rate
Ω about an axis, takes the form: :\mathbf{F}' = m \mathbf{a} \ , which looks the same as in an inertial frame, but now the force
F′ is the resultant of not only
F, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics): :\mathbf{F}' = \mathbf{F} - 2m \mathbf{\Omega} \times \mathbf{v}_{B} - m \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{x}_B ) - m \frac{d \mathbf{\Omega}}{dt} \times \mathbf{x}_B \ , where the angular rotation of the frame is expressed by the vector
Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation
Ω, symbol × denotes the
vector cross product, vector
xB locates the body and vector
vB is the
velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer). The extra terms in the force
F′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the
Coriolis force, the second the
centrifugal force, and the third the
Euler force. These terms all have these properties: they vanish when
Ω = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of
Ω; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular,
matter. Also, fictitious forces do not drop off with distance (unlike, for example,
nuclear forces or
electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the real forces,
F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present. In Newton's time the
fixed stars were invoked as a reference frame, supposedly at rest relative to
absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars,
Newton's laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of
fictitious forces, for example, the
Coriolis force and the
centrifugal force. Two experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking
two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a
rotating bucket. In both cases, application of
Newton's second law would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket). As now known, the fixed stars are not fixed. Those that reside in the
Milky Way turn with the galaxy, exhibiting
proper motions. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to
expansion of the universe, and partly due to
peculiar velocities. For instance, the
Andromeda Galaxy is on
collision course with the Milky Way at a speed of 117 km/s. The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based on the simplicity of the laws of physics in the frame. The laws of nature take a simpler form in inertial frames of reference because in these frames one did not have to introduce inertial forces when writing down Newton's law of motion. In practice, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center. To illustrate further, consider the question: "Does the Universe rotate?" An answer might explain the shape of the
Milky Way galaxy using the laws of physics, although other observations might be more definitive; that is, provide larger
discrepancies or less
measurement uncertainty, like the anisotropy of the
microwave background radiation or
Big Bang nucleosynthesis. The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If its apparent rate of rotation is attributed entirely to rotation in an inertial frame, a different "flatness" is predicted than if it is supposed that part of this rotation is actually due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example,
dark matter is invoked to explain the
galactic rotation curve. So far, observations show any rotation of the universe is very slow, no faster than once every years (10−13 rad/yr), and debate persists over whether there is
any rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity. When
quantum effects are important, there are additional conceptual complications that arise in
quantum reference frames.
Primed frames 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 : \mathbf r = \mathbf R + \mathbf r'. Taking the first and second derivatives of this with respect to time : \mathbf v = \mathbf V + \mathbf v', : \mathbf a = \mathbf A + \mathbf a'. 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,
Newton's second law can be written as : \mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'. 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 a direction orthogonal to an object's motion, the
Coriolis effect). 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 (see
Fictitious force for a derivation): : \mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0, or, to solve for the acceleration in the accelerated frame, : \mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0. Multiplying through by the mass
m gives : \mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{Euler} + \mathbf F'_\mathrm{Coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0, where : \mathbf F'_\mathrm{Euler} = -m\dot{\boldsymbol\omega} \times \mathbf r' (
Euler force), : \mathbf F'_\mathrm{Coriolis} = -2m\boldsymbol\omega \times \mathbf v' (
Coriolis force), : \mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') = m(\omega^2 \mathbf r' - (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega) (
centrifugal force). ==Separating non-inertial from inertial reference frames==