Frame of reference (physics)

For the general term "frame of reference", see Frame of reference.

A frame of reference in physics most usually emphasizes the dependence of the description of physical events upon an observer's state of motion, a usage emphasized by the term observational reference frame. However, frame of reference frequently is used to refer to a coordinate system or, even more simply, a set of axes, within which to measure the position, orientation, and other properties of objects. More generally, a frame of reference may include three elements: an observational reference frame, an attached coordinate system, and a measurement apparatus for making observations, as a combined unit.

Different aspects

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 a frame is related to a family of frames 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. In contrast, 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 various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below:

• 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. Some coordinate systems may be a better choice for some observations than are others. 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.

• Choice of what to measure and with what observational apparatus is a matter logically separate from the observer's state of motion and choice of coordinate system. The observational apparatus is not necessarily localized to a single observer, but may involve an observer team relaying observations to a central data collector.

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

“	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

and this on the utility of separating the notions of ${\mathfrak {R}}$ and [R, R' , etc.]:

“	As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified.	”
—L. Brillouin in Relativity Reexamined (quoted by Patrick Cornille)

and this, also on the distinction between ${\mathfrak {R}}$ and [R, R' , etc.]:

“	The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems.	”
—Graham Nerlich: What Spacetime Explains

and from J. D. Norton:

“	In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.	”
—John D. Norton: General Covariance and the Foundations of General Relativity

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.

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 and Lasenby. 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 Schwarzschild 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 inertial 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 Earth's axis, which introduces an inertial force known as the Coriolis force (among others).

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). Coordinates, coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system. If the basis vectors are mutually perpendicular at every point, the coordinate system is an orthogonal coordinate system. These components are discussed further below.

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.

To introduce coordinate surfaces, for simplicity, let us restrict consideration to differentiable manifolds based upon real numbers. Suppose the 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 coordinate lines are the intersections of the coordinate surfaces.

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 background see tangent space.

An important aspect of a coordinate system is its metric tensor 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 be used to describe motion by interpreting one coordinate as time. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.

Lest one underestimate the subtlety of coordinate systems, Ehrenfest's paradox may be mentioned. The paradox applies to a rigid rotating circular cylinder as examined in special relativity. Each element of the perimeter is aligned with its direction of motion due to the rotation and so suffers Lorentz contraction, but the radius to each element on the perimeter from the center of rotation is perpendicular to its motion and so does not contract. Thus, the circumference as measured around the perimeter differs from the calculated perimeter 2πR using the radius R measured from the axis of rotation, and Euclidean geometry apparently does not apply. This paradox has been discussed for decades, even quite recently. Proposed resolutions of the paradox involve the synchronizing of clocks among non-inertial frames of reference located at the perimeter.

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.