Inphysics andastronomy, aframe of reference (orreference frame) is an abstractcoordinate system, whoseorigin,orientation, andscale have been specified inphysical space. It is based on a set ofreference points, defined asgeometric points whoseposition is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).^[1]An important special case is that ofinertial reference frames, a stationary or uniformly moving frame.

Forn dimensions,n + 1 reference points are sufficient to fully define a reference frame. Usingrectangular Cartesian coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of then coordinateaxes.^{[citation needed]}

InEinsteinian relativity, reference frames are used to specify the relationship between a movingobserver and the phenomenon under observation. In this context, the term often becomesobservational frame of reference (orobservational reference frame), which implies that the observer is at rest in the frame, although not necessarily located at itsorigin. A relativistic reference frame includes (or implies) thecoordinate time, which does not equate across different reference framesmoving relatively to each other. The situation thus differs fromGalilean relativity, in which all possible coordinate times are essentially equivalent.^{[citation needed]}

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 inCartesian frame of reference. Sometimes the state of motion is emphasized, as inrotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as inGalilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as inmacroscopic andmicroscopic frames of reference.^[2]

In this article, the termobservational frame of reference is used when emphasis is upon thestate 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, acoordinate 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, the formulation of many problems in physics employsgeneralized coordinates,normal modes oreigenvectors, 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 aninertial frame ornon-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.^[3] 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'sobservational frame of reference. This viewpoint can be found elsewhere as well.^[4] Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.

• 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.

^[a]

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 ofgeometry or ofalgebra,^[9][10] in particular, a property ofmanifolds (for example, in physics,configuration spaces orphase spaces).^[11][12] Thecoordinates of a pointr in ann-dimensional space are simply an ordered set ofn numbers:^[13][14]

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

In a generalBanach space, these numbers could be (for example) coefficients in a functional expansion like aFourier series. In a physical problem, they could bespacetime coordinates ornormal mode amplitudes. In arobot design, they could be angles of relative rotations, linear displacements, or deformations ofjoints.^[15] Here we will suppose these coordinates can be related to aCartesian coordinate system by a set of functions:

${\displaystyle x^j=x^j(x,y,z,\dots ),j=1,\dots ,n,}$ 

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

${\displaystyle x^j(x,y,z,\dots )=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t},j=1,\dots ,n.}$ 

The intersection of these surfaces definecoordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set ofbasis vectors {e₁,e₂, ...,e_n} at that point. That is:^[16]

${\displaystyle {\mathbf{e}}_i(\mathbf{r})=\underset{ϵ\to 0}{lim}\frac{\mathbf{r}\left(x^1,\dots ,x^i+ϵ,\dots ,x^n\right)-\mathbf{r}\left(x^1,\dots ,x^i,\dots ,x^n\right)}{ϵ},i=1,\dots ,n,}$ 

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

Coordinate surfaces, coordinate lines, andbasis vectors are components of acoordinate system.^[17] If the basis vectors are orthogonal at every point, the coordinate system is anorthogonal coordinate system.

An important aspect of a coordinate system is itsmetric tensorg_ik, which determines thearc lengthds in the coordinate system in terms of its coordinates:^[18]

${\displaystyle (ds{)}^2=g_{ik}d{x}^{i}d{x}^k,}$ 

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is amathematical construct, part of anaxiomatic 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 andGalilean transformations may be viewed ascoordinate transformations.

Anobservational frame of reference, often referred to as aphysical frame of reference, aframe of reference, or simply aframe, 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 characterizedonly by its state of motion.^[19] However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between anobserver and aframe. According to this view, aframe is anobserver plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.^[20] This restricted view is not used here, and is not universally adopted even in discussions of relativity.^[21][22] Ingeneral relativity the use of general coordinate systems is common (see, for example, theSchwarzschild solution for the gravitational field outside an isolated sphere^[23]).

There are two types of observational reference frame:inertial andnon-inertial. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. Inspecial relativity these frames are related byLorentz transformations, which are parametrized byrapidity. In Newtonian mechanics, a more restricted definition requires only thatNewton's first law holds true; that is, a Newtonian inertial frame is one in which afree particle travels in astraight line at constantspeed, or is at rest. These frames are related byGalilean transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms ofrepresentations of thePoincaré group and of theGalilean group.

In contrast to the inertial frame, a non-inertial frame of reference is one in whichfictitious 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 the fictitious forces known as theCoriolis force,centrifugal force, andgravitational force. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)

A further aspect of a frame of reference is the role of themeasurement 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 inquantum mechanics, where the relation between observer and measurement is still under discussion (seemeasurement problem).

In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as thelaboratory frame or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to thecenter of momentum frame "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles.

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 indirectmetrology that is connected to the nature of thevacuum, and usesatomic clocks that operate according to thestandard model and that must be corrected forgravitational time dilation.^[24] (Seesecond,meter andkilogram).

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.^[25]

The discussion is taken beyond simple space-time coordinate systems byBrading and Castellani.^[26] Extension to coordinate systems using generalized coordinates underlies theHamiltonian andLagrangian formulations^[27] ofquantum field theory,classical relativistic mechanics, andquantum gravity.^{[28][29][30][31][32]}

• International Terrestrial Reference Frame
• International Celestial Reference Frame
• In fluid mechanics,Lagrangian and Eulerian specification of the flow field

Other frames

• Frame fields in general relativity
• Moving frame in mathematics