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Frame of reference

A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer. Or it may refer to both an observational reference frame and an attached coordinate system, as a unit. For example, according to one point of view, the observer, the coordinate system and the observational apparatus all are included in the frame: "Each reference frame must have an observer to record events as well as a coordinate system for the purpose of assigning locations to each event".

Different aspects of "frame of reference"

The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.

In this article the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce the two aspects of a reference frame for the discussion below.

We therefore take the view that observational frames of reference, coordinate systems, and observational equipment are independent concepts:

An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion.

A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations. Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference. This viewpoint can be found elsewhere as well.

Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system. Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.

Here is a quotation applicable to moving observational frames $mathfrak{R}$ and various associated Euclidean three-space coordinate systems [R, R' , etc.]:

{{Quotation|We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted $mathfrak{R}$ , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame $mathfrak{R}$ by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame $mathfrak{R}$ , can be considered to give a physical realization of $mathfrak{R}$ . In a frame $mathfrak{R}$ , coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.| Jean Salençon, Stephen Lyle Handbook of Continuum Mechanics: General Concepts, Thermoelasticity p. 9}} and this on the utility of separating the notions of $mathfrak{R}$ and [R, R' , etc.]: and this, also on the distinction between $mathfrak{R}$ and [R, R' , etc.]:

and from J. D. Norton:

The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani. Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity.

Coordinate systems

Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well.

A coordinate system in mathematics is a facet of geometry or of algebra, in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:

mathbf{r} =[x^1, x^2, dots , x^n] .

In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints. Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

x^j = x^j (x, y, z, dots) ,

j = 1, dots , n

where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:

x^j (x, y, z, dots) = mathrm{constant} ,

j = 1, dots , n .

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e₁, e₂, …, e_n} at that point. That is:

mathbf{e}_i(mathbf{r}) =lim_{epsilon rightarrow 0} frac{mathbf{r}left(x^1, dots, x^i+epsilon, dots , x^n right) - mathbf{r}left(x^1, dots, x^i, dots , x^n right)}{epsilon } ,

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system. If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

An important aspect of a coordinate system is its metric g_ik, which determines the arc length ds in the coordinate system in terms of its coordinates:

(ds)^2 = g_{ik} dx^i dx^k ,

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.

General and specific topics of coordinate systems can be pursued following the See also links below.

Observational frames of reference

An observational frame of reference, often referred to as a physical frame of reference, a frame of reference, or simply a frame, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion. However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and a frame. According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran. This restricted view is not used here, and is not universally adopted even in discussions of relativity. In general relativity the use of general coordinate systems is common (see, for example, the Schwarzchild solution for the gravitational field outside an isolated sphere). There are two types of observational reference frame: inertial and non-inertial. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations. In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest. These frames are related by Galilean transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.

In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the earth's surface. This frame of reference orbits around the center of the earth, which introduces a fictitious force known as the Coriolis force.

Measurement apparatus

A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relation between observer and measurement is still under discussion (see measurement problem).

In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation. (See second, meter and kilogram).

In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.

Inertial frames of reference

The remainder of this article focuses upon the two types of observational frame of reference outlined above. The first is the inertial frame of reference characterized by the simplicity and universality of physical laws in these frames. They are related by Lorentz transformations in special relativity and by Galilean transformations in Newtonian mechanics.

Simple example

Consider a situation common in everyday life. Two cars travel along a road, both moving at a constant velocity. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars are accelerating, we can determine their positions by the following formulas, where $x_1(t)$ is the position in meters of car one after time t seconds and $x_2(t)$ is the position of car two after time t.

x_1(t)= d + v_1 t = 200 + 22t ; quad x_2(t)= v_2 t = 30t

Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which $x_1=x_2$ . Therefore we set $x_1=x_2$ and solve for $t$ , that is:

200 + 22 t = 30t quad

8t = 200 quad

t = 25 quad mathrm{seconds}

Alternatively, we could choose a frame of reference S' situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m / s. In order to catch up to the first car, it will take a time of d /( v₂ − v₁) = 200 / 8 s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s.

It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Additional examples

For a simple example, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.

For a more complex 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.

Remarks

It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display the exact same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (Lorentz transformations).

It is also important to note that the definition of inertial reference frame can be extended beyond three dimensional Euclidean space. Newton's assumed a Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, let us consider the non-Euclidean geometry of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a geodesic path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now believed that we exist in a four dimensional geometry known as spacetime. It is believed that the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity.

Non-inertial frames

Here we consider the relation between inertial and non-inertial observational frames of reference. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

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

mathbf r = mathbf R + mathbf r'

Taking the first and second derivatives of this, we obtain

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, we can now write Newton's second law as

mathbf F = mmathbf a = mmathbf A + mmathbf 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 tangential direction, the Coriolis effect). Actually 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.

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{boldsymbolomega} times mathbf r' + 2boldsymbolomega times mathbf v' + boldsymbolomega times (boldsymbolomega times mathbf r') + mathbf A_0

or, to solve for the acceleration in the accelerated frame,

mathbf a' = mathbf a - dot{boldsymbolomega} times mathbf r' - 2boldsymbolomega times mathbf v' - boldsymbolomega times (boldsymbolomega 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} - mmathbf A_0

where

mathbf F'_mathrm{Euler} = -mdot{boldsymbolomega} times mathbf r'

(Euler force)

mathbf F'_mathrm{Coriolis} = -2mboldsymbolomega times mathbf v'

(Coriolis force)

mathbf F'_mathrm{centrifugal} = -mboldsymbolomega times (boldsymbolomega times mathbf r')=m(omega^2 mathbf r'- (boldsymbolomega cdot mathbf r')boldsymbolomega)

(centrifugal force)

Particular frames of reference in common use

International Terrestrial Reference Frame
International Celestial Reference Frame
In fluid mechanics, Eulerian Reference Frame and Lagrangian Reference Frame

Other frames

Footnotes

Distortions can vary from place to place, with gravity appearing to be the common cause. In fact, General relativity predicts a frame-dragging effect (aka Lense-Thirring effect).