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<title>Frame of reference</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Frame of reference</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Frame_of_reference_(disambiguation)" class="mw-disambig" title="Frame of reference (disambiguation)">Frame of reference (disambiguation)</a>.</div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks cm-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle cm-sidebar-title"><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation>

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</math></span><img src="./c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}" loading="lazy"></span><div class="sidebar-caption"><a href="Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading cm-sidebar-above">

<ul><li><a href="History_of_classical_mechanics" title="History of classical mechanics">History</a></li>

<li><a href="Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li>

<li><a href="List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul>

</div></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Applied_mechanics" title="Applied mechanics">Applied</a></li>

<li><a href="Celestial_mechanics" title="Celestial mechanics">Celestial</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum</a></li>

<li><a href="Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</a></li>

<li><a href="Classical_field_theory" title="Classical field theory">Field theory</a></li>

<li><a href="Kinematics" title="Kinematics">Kinematics</a></li>

<li><a href="Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li>

<li><a href="Statics" title="Statics">Statics</a></li>

<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul>

</div></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Acceleration" title="Acceleration">Acceleration</a></li>

<li><a href="Angular_momentum" title="Angular momentum">Angular momentum</a></li>

<li><a href="Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li>

<li><a href="D'Alembert's_principle" title="D'Alembert's principle">D'Alembert's principle</a></li>

<li><a href="Energy" title="Energy">Energy</a>

<ul><li><a href="Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li>

<li><a href="Potential_energy" title="Potential energy">potential</a></li></ul></li>

<li><a href="Force" title="Force">Force</a></li>

<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>

<li><a href="Impulse_(physics)" title="Impulse (physics)">Impulse</a></li>

<li><span class="nowrap"><a href="Inertia" title="Inertia">Inertia</a> / <a href="Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li>

<li><br><a href="Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li>

<li><a href="Work_(physics)" title="Work (physics)">Mechanical work</a></li>

<li><br><a href="Moment_(physics)" title="Moment (physics)">Moment</a></li>

<li><a href="Momentum" title="Momentum">Momentum</a></li>

<li><a href="Space" title="Space">Space</a></li>

<li><a href="Speed" title="Speed">Speed</a></li>

<li><a href="Torque" title="Torque">Torque</a></li>

<li><a href="Velocity" title="Velocity">Velocity</a></li>

<li><a href="Virtual_work" title="Virtual work">Virtual work</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li>

<li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="Appell's_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Core topics</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Damping" title="Damping">Damping</a></li>

<li><a href="Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li>

<li><a href="Equations_of_motion" title="Equations of motion">Equations of motion</a></li>

<li><a href="Euler's_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li>

<li><a href="Fictitious_force" title="Fictitious force">Fictitious force</a></li>

<li><a href="Friction" title="Friction">Friction</a></li>

<li><a href="Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul>

</div>

<ul><li><span class="nowrap"><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul>

<ul><li><a href="Motion" title="Motion">Motion</a> (<a href="Linear_motion" title="Linear motion">linear</a>)</li>

<li><a href="Newton's_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li>

<li><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li>

<li><a href="Relative_velocity" title="Relative velocity">Relative velocity</a></li>

<li><a href="Rigid_body" title="Rigid body">Rigid body</a>

<ul><li><a href="Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li>

<li><a href="Euler's_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li>

<li><a href="Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li>

<li><a href="Vibration" title="Vibration">Vibration</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Circular_motion" title="Circular motion">Circular motion</a></li>

<li><a href="Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li>

<li><a href="Centripetal_force" title="Centripetal force">Centripetal force</a></li>

<li><a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a>

<ul><li><a href="Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>

<li><a href="Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li>

<li><a href="Tangential_speed" title="Tangential speed">Tangential speed</a></li>

<li><a href="Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul>

</div>

<ul><li><a href="Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="Angular_displacement" title="Angular displacement">displacement</a> / <a href="Angular_frequency" title="Angular frequency">frequency</a> / <a href="Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Johannes_Kepler" title="Johannes Kepler">Kepler</a></li>

<li><a href="Galileo_Galilei" title="Galileo Galilei">Galileo</a></li>

<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Newton</a></li>

<li><a href="Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li>

<li><a href="Edmond_Halley" title="Edmond Halley">Halley</a></li>

<li><a href="Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li>

<li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li>

<li><a href="Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>

<li><a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li>

<li><a href="Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li>

<li><a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li>

<li><a href="Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li>

<li><a href="Edward_Routh" title="Edward Routh">Routh</a></li>

<li><a href="Joseph_Liouville" title="Joseph Liouville">Liouville</a></li>

<li><a href="Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li>

<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>

<li><a href="Bernard_Koopman" title="Bernard Koopman">Koopman</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul>

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<p>In <a href="Physics" title="Physics">physics</a> and <a href="Astronomy" title="Astronomy">astronomy</a>, a <b>frame of reference</b> (or <b>reference frame</b>) is an abstract <a href="Coordinate_system" title="Coordinate system">coordinate system</a>, whose <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a>, <a href="Orientation_(geometry)" title="Orientation (geometry)">orientation</a>, and <a href="Scale_(geometry)" class="mw-redirect" title="Scale (geometry)">scale</a> have been specified in <a href="Physical_space" class="mw-redirect" title="Physical space">physical space</a>. It is based on a set of <b>reference points</b>, defined as <a href="Point_(geometry)" title="Point (geometry)">geometric points</a> whose <a href="Position_(geometry)" title="Position (geometry)">position</a> is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).<sup id="cite_ref-Kovalevsky_Mueller_1989_pp._1–12_1-0" class="reference"><a href="#cite_note-Kovalevsky_Mueller_1989_pp._1–12-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

An important special case is that of <i><a href="Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">inertial reference frames</a></i>, a stationary or uniformly moving frame.

</p><p>For <i>n</i> dimensions, <span class="nowrap"><i>n</i> + 1</span> reference points are sufficient to fully define a reference frame. Using <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">rectangular Cartesian coordinates</a>, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the <i>n</i> coordinate <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">axes</a>.

</p><p>In <a href="Theory_of_relativity" title="Theory of relativity">Einsteinian relativity</a>, reference frames are used to specify the relationship between a moving <a href="Observer_(special_relativity)" title="Observer (special relativity)">observer</a> and the phenomenon under observation. In this context, the term often becomes <b>observational frame of reference</b> (or <b>observational reference frame</b>), which implies that the observer is at rest in the frame, although not necessarily located at its <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a>. A relativistic reference frame includes (or implies) the <a href="Coordinate_time" title="Coordinate time">coordinate time</a>, which does not equate across different reference frames <a href="Relative_motion" class="mw-redirect" title="Relative motion">moving relatively</a> to each other. The situation thus differs from <a href="Galilean_invariance" title="Galilean invariance">Galilean relativity</a>, in which all possible coordinate times are essentially equivalent.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>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 <i>Cartesian frame of reference</i>. Sometimes the state of motion is emphasized, as in <i><a href="Rotating_reference_frame" title="Rotating reference frame">rotating frame of reference</a></i>. Sometimes the way it transforms to frames considered as related is emphasized as in <i><a href="Galilean_frame_of_reference" class="mw-redirect" title="Galilean frame of reference">Galilean frame of reference</a></i>. Sometimes frames are distinguished by the scale of their observations, as in <i>macroscopic</i> and <i>microscopic frames of reference</i>.<sup id="cite_ref-macroscopic_2-0" class="reference"><a href="#cite_note-macroscopic-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>In this article, the term <i>observational frame of reference</i> is used when emphasis is upon the <i>state of motion</i> 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 <i>coordinate system</i> 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 employs <i><a href="Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a></i>, <i><a href="Normal_modes" class="mw-redirect" title="Normal modes">normal modes</a></i> or <i><a href="Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a></i>, 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:

</p>

<ul><li>An observational frame (such as an <a href="Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a> or <a href="Non-inertial_frame_of_reference" class="mw-redirect" title="Non-inertial frame of reference">non-inertial frame of reference</a>) is a physical concept related to state of motion.</li>

<li>A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.<sup id="cite_ref-Pontriagin_3-0" class="reference"><a href="#cite_note-Pontriagin-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> 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 <i>observational</i> frame of reference. This viewpoint can be found elsewhere as well.<sup id="cite_ref-Johansson_4-0" class="reference"><a href="#cite_note-Johansson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.</li></ul>

<ul><li>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.</li></ul>

</p>

<div class="mw-heading mw-heading2"><h2 id="Coordinate_systems">Coordinate systems</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Coordinate_systems" class="mw-redirect" title="Coordinate systems">Coordinate systems</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Generalized_coordinates" title="Generalized coordinates">Generalized coordinates</a> and <a href="Axes_conventions" title="Axes conventions">Axes conventions</a></div>

<p>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.

</p><p>A coordinate system in mathematics is a facet of <a href="Geometry" title="Geometry">geometry</a> or of <a href="Algebra" title="Algebra">algebra</a>,<sup id="cite_ref-Barker_10-0" class="reference"><a href="#cite_note-Barker-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ramsay_11-0" class="reference"><a href="#cite_note-Ramsay-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in particular, a property of <a href="Manifold" title="Manifold">manifolds</a> (for example, in physics, <a href="Configuration_space_(physics)" title="Configuration space (physics)">configuration spaces</a> or <a href="Phase_space" title="Phase space">phase spaces</a>).<sup id="cite_ref-Hawking_12-0" class="reference"><a href="#cite_note-Hawking-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Morita_13-0" class="reference"><a href="#cite_note-Morita-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">coordinates</a> of a point <b>r</b> in an <i>n</i>-dimensional space are simply an ordered set of <i>n</i> numbers:<sup id="cite_ref-Korn_14-0" class="reference"><a href="#cite_note-Korn-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-encarta_15-0" class="reference"><a href="#cite_note-encarta-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

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</math></span><img src="./e7dd8270c4f5fa053e35aeab704c2d09b59e3275.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.798ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} =[x^{1},\ x^{2},\ \dots ,\ x^{n}].}" loading="lazy"></span></dd></dl>

<p>In a general <a href="Banach_space" title="Banach space">Banach space</a>, these numbers could be (for example) coefficients in a functional expansion like a <a href="Fourier_series" title="Fourier series">Fourier series</a>. In a physical problem, they could be <a href="Spacetime" title="Spacetime">spacetime</a> coordinates or <a href="Normal_mode" title="Normal mode">normal mode</a> amplitudes. In a <a href="Robotics" title="Robotics">robot design</a>, they could be angles of relative rotations, linear displacements, or deformations of <a href="Linkage_(mechanical)" title="Linkage (mechanical)">joints</a>.<sup id="cite_ref-Yamane_16-0" class="reference"><a href="#cite_note-Yamane-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Here we will suppose these coordinates can be related to a <a href="Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinate</a> system by a set of functions:

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<annotation encoding="application/x-tex">{\displaystyle x^{j}=x^{j}(x,\ y,\ z,\ \dots ),\quad j=1,\ \dots ,\ n,}</annotation>

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</math></span><img src="./70cffae3c981a2746c07924eeca0014f72cef764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.258ex; height:3.176ex;" alt="{\displaystyle x^{j}=x^{j}(x,\ y,\ z,\ \dots ),\quad j=1,\ \dots ,\ n,}" loading="lazy"></span></dd></dl>

<p>where <i>x</i>, <i>y</i>, <i>z</i>, <i>etc.</i> are the <i>n</i> Cartesian coordinates of the point. Given these functions, <b>coordinate surfaces</b> are defined by the relations:

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</math></span><img src="./5c16299c230de0f2759d72960643cae6289f35f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.557ex; height:3.176ex;" alt="{\displaystyle x^{j}(x,y,z,\dots )=\mathrm {constant} ,\quad j=1,\ \dots ,\ n.}" loading="lazy"></span></dd></dl>

<p>The intersection of these surfaces define <b>coordinate lines</b>. At any selected point, tangents to the intersecting coordinate lines at that point define a set of <b>basis vectors</b> {<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, ..., <b>e</b><sub>n</sub>} at that point. That is:<sup id="cite_ref-Papapetrou_17-0" class="reference"><a href="#cite_note-Papapetrou-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \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 }},\quad i=1,\ \dots ,\ n,}</annotation>

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</math></span><img src="./1d4fd676934b280fde6f899304144693ddc3c114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:84.568ex; height:6.176ex;" alt="{\displaystyle \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 }},\quad i=1,\ \dots ,\ n,}" loading="lazy"></span></dd></dl>

<p>which can be normalized to be of unit length. For more detail see <a href="Curvilinear_coordinates#Covariant_basis" title="Curvilinear coordinates">curvilinear coordinates</a>.

</p><p>Coordinate surfaces, coordinate lines, and <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis vectors</a> are components of a <b>coordinate system</b>.<sup id="cite_ref-Zdunkowski_18-0" class="reference"><a href="#cite_note-Zdunkowski-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> If the basis vectors are orthogonal at every point, the coordinate system is an <a href="Orthogonal_coordinates" title="Orthogonal coordinates">orthogonal coordinate system</a>.

</p><p>An important aspect of a coordinate system is its <a href="Metric_tensor" title="Metric tensor">metric tensor</a> <i>g<sub>ik</sub></i>, which determines the <a href="Arc_length" title="Arc length">arc length</a> <i>ds</i> in the coordinate system in terms of its coordinates:<sup id="cite_ref-Borisenko_19-0" class="reference"><a href="#cite_note-Borisenko-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

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<p>where repeated indices are summed over.

</p><p>As is apparent from these remarks, a coordinate system is a <a href="Model_theory" title="Model theory">mathematical construct</a>, part of an <a href="Axiomatic_system" title="Axiomatic system">axiomatic system</a>. 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, <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> and <a href="Galilean_transformation" title="Galilean transformation">Galilean transformations</a> may be viewed as <a href="Coordinate_system#Transformations" title="Coordinate system">coordinate transformations</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Observational_frame_of_reference">Observational frame of reference</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></div>

<p>An <b>observational frame of reference</b>, often referred to as a <i>physical frame of reference</i>, a <i>frame of reference</i>, or simply a <i>frame</i>, 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 <i>only by its state of motion</i>.<sup id="cite_ref-Kubar_20-0" class="reference"><a href="#cite_note-Kubar-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an <i>observer</i> and a <i>frame</i>. According to this view, a <i>frame</i> is an <i>observer</i> plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.<sup id="cite_ref-Doran_21-0" class="reference"><a href="#cite_note-Doran-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This restricted view is not used here, and is not universally adopted even in discussions of relativity.<sup id="cite_ref-Moller_22-0" class="reference"><a href="#cite_note-Moller-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lightman_23-0" class="reference"><a href="#cite_note-Lightman-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In <a href="General_relativity" title="General relativity">general relativity</a> the use of general coordinate systems is common (see, for example, the <a href="Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a> solution for the gravitational field outside an isolated sphere<sup id="cite_ref-Faber_24-0" class="reference"><a href="#cite_note-Faber-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>).

</p><p>There are two types of observational reference frame: <a href="Inertial_frame_of_reference" title="Inertial frame of reference">inertial</a> and <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial</a>. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In <a href="Special_relativity" title="Special relativity">special relativity</a> these frames are related by <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, which are parametrized by <a href="Rapidity" title="Rapidity">rapidity</a>. In Newtonian mechanics, a more restricted definition requires only that <a href="Newton's_first_law" class="mw-redirect" title="Newton's first law">Newton's first law</a> holds true; that is, a Newtonian inertial frame is one in which a <a href="Free_particle" title="Free particle">free particle</a> travels in a <a href="Straight_line" class="mw-redirect" title="Straight line">straight line</a> at constant <a href="Speed" title="Speed">speed</a>, or is at rest. These frames are related by <a href="Galilean_transformation" title="Galilean transformation">Galilean transformations</a>. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of <a href="Representation_theory" title="Representation theory">representations</a> of the <a href="Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Poincaré group</a> and of the <a href="Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Galilean group</a>.

</p><p>In contrast to the inertial frame, a non-inertial frame of reference is one in which <a href="Fictitious_force" title="Fictitious force">fictitious forces</a> 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 the <a href="Coriolis_force" title="Coriolis force">Coriolis force</a>, <a href="Centrifugal_force" title="Centrifugal force">centrifugal force</a>, and <a href="Gravitational_force" class="mw-redirect" title="Gravitational force">gravitational force</a>. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)

</p>

<div class="mw-heading mw-heading2"><h2 id="Measurement_apparatus">Measurement apparatus</h2></div>

<p>A further aspect of a frame of reference is the role of the <a href="Metrology" title="Metrology">measurement apparatus</a> (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 <a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">quantum mechanics</a>, where the relation between observer and measurement is still under discussion (see <a href="Measurement_problem" title="Measurement problem">measurement problem</a>).

</p><p>In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the <a href="Laboratory_frame" class="mw-redirect" title="Laboratory frame">laboratory frame</a> 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 the <a href="Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a> "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.

</p><p>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 <a href="Metrology" title="Metrology">metrology</a> that is connected to the nature of the <a href="Vacuum" title="Vacuum">vacuum</a>, and uses <a href="Atomic_clocks" class="mw-redirect" title="Atomic clocks">atomic clocks</a> that operate according to the <a href="Standard_model" class="mw-redirect" title="Standard model">standard model</a> and that must be corrected for <a href="Gravitational_time_dilation" title="Gravitational time dilation">gravitational time dilation</a>.<sup id="cite_ref-Wolfson_25-0" class="reference"><a href="#cite_note-Wolfson-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> (See <a href="Second" title="Second">second</a>, <a href="Meter" class="mw-redirect" title="Meter">meter</a> and <a href="Kilogram" title="Kilogram">kilogram</a>).

</p><p>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.<sup id="cite_ref-Rizzi_26-0" class="reference"><a href="#cite_note-Rizzi-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2></div>

<p>The discussion is taken beyond simple space-time coordinate systems by <a href="Katherine_Brading" title="Katherine Brading">Brading</a> and Castellani.<sup id="cite_ref-Brading_27-0" class="reference"><a href="#cite_note-Brading-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Extension to coordinate systems using generalized coordinates underlies the <a href="Hamilton's_principle" title="Hamilton's principle">Hamiltonian</a> and <a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> formulations<sup id="cite_ref-Johns_28-0" class="reference"><a href="#cite_note-Johns-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, <a href="Classical_mechanics" title="Classical mechanics">classical relativistic mechanics</a>, and <a href="Quantum_gravity" title="Quantum gravity">quantum gravity</a>.<sup id="cite_ref-Greenwood_29-0" class="reference"><a href="#cite_note-Greenwood-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Trump_30-0" class="reference"><a href="#cite_note-Trump-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kompaneyets_31-0" class="reference"><a href="#cite_note-Kompaneyets-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Srednicki_32-0" class="reference"><a href="#cite_note-Srednicki-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Rovelli_33-0" class="reference"><a href="#cite_note-Rovelli-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Instances">Instances</h2></div>

<ul><li><a href="International_Terrestrial_Reference_Frame" class="mw-redirect" title="International Terrestrial Reference Frame">International Terrestrial Reference Frame</a></li>

<li><a href="International_Celestial_Reference_Frame" class="mw-redirect" title="International Celestial Reference Frame">International Celestial Reference Frame</a></li>

<li>In fluid mechanics, <a href="Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Lagrangian and Eulerian specification of the flow field</a></li></ul>

<dl><dt>Other frames</dt></dl>

<ul><li><a href="Frame_fields_in_general_relativity" title="Frame fields in general relativity">Frame fields in general relativity</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame in mathematics</a></li></ul>

<ul><li><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></li>

<li><a href="Applied_mechanics" title="Applied mechanics">Applied mechanics</a></li>

<li><a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></li>

<li><a href="Center-of-momentum_frame" title="Center-of-momentum frame">Center-of-momentum frame</a></li>

<li><a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a></li>

<li><a href="Centripetal_force" title="Centripetal force">Centripetal force</a></li>

<li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>

<li><a href="Curvilinear_coordinates" title="Curvilinear coordinates">Curvilinear coordinates</a></li>

<li><a href="Datum_reference" title="Datum reference">Datum reference</a></li>

<li><a href="Dynamics_(physics)" class="mw-redirect" title="Dynamics (physics)">Dynamics (physics)</a></li>

<li><a href="Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a></li>

<li><a href="Galilean_invariance" title="Galilean invariance">Galilean invariance</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Generalized_coordinates" title="Generalized coordinates">Generalized coordinates</a></li>

<li><a href="Generalized_forces" title="Generalized forces">Generalized forces</a></li>

<li><a href="Geodetic_reference_frame" class="mw-redirect" title="Geodetic reference frame">Geodetic reference frame</a></li>

<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>

<li><a href="Local_coordinates" title="Local coordinates">Local coordinates</a></li>

<li><a href="Material_frame-indifference" class="mw-redirect" title="Material frame-indifference">Material frame-indifference</a></li>

<li><a href="Rod_and_frame_test" title="Rod and frame test">Rod and frame test</a></li>

<li><a href="Kinematics" title="Kinematics">Kinematics</a></li>

<li><a href="Laboratory_frame_of_reference" class="mw-redirect" title="Laboratory frame of reference">Laboratory frame of reference</a></li>

<li><a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li>

<li><a href="Mach's_principle" title="Mach's principle">Mach's principle</a></li>

<li><a href="Orthogonal_coordinates" title="Orthogonal coordinates">Orthogonal coordinates</a></li>

<li><a href="Principle_of_relativity" title="Principle of relativity">Principle of relativity</a></li>

<li><a href="Quantum_reference_frame" title="Quantum reference frame">Quantum reference frame</a></li></ul>

</div>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Relativity254" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><div id="Relativity254" style="font-size:114%;margin:0 4em"><a href="Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Special_relativity" title="Special relativity">Special<br>relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li>

<li><a href="Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li>

<li><a href="Special_relativity" title="Special relativity">Special relativity</a></li>

<li><a href="Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br>concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Speed_of_light" title="Speed of light">Speed of light</a></li>

<li><a href="Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li>

<li><a href="Rapidity" title="Rapidity">Rapidity</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Proper_length" title="Proper length">Proper length</a></li>

<li><a href="Proper_time" title="Proper time">Proper time</a></li>

<li><a href="Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li>

<li><a href="Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li>

<li><a href="List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Time_dilation" title="Time dilation">Time dilation</a></li>

<li><a href="Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li>

<li><a href="Length_contraction" title="Length contraction">Length contraction</a></li>

<li><a href="Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li>

<li><a href="Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li>

<li><a href="Thomas_precession" title="Thomas precession">Thomas precession</a></li>

<li><a href="Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li>

<li><a href="Twin_paradox" title="Twin paradox">Twin paradox</a></li>

<li><a href="Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Light_cone" title="Light cone">Light cone</a></li>

<li><a href="World_line" title="World line">World line</a></li>

<li><a href="Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li>

<li><a href="Biquaternion" title="Biquaternion">Biquaternions</a></li>

<li><a href="Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="General_relativity" title="General relativity">General<br>relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li>

<li><a href="Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br>concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li>

<li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li>

<li><a href="Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li>

<li><a href="Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li>

<li><a href="Mach's_principle" title="Mach's principle">Mach's principle</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="ADM_formalism" title="ADM formalism">ADM formalism</a></li>

<li><a href="BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li>

<li><a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li>

<li><a href="Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li>

<li><a href="Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li>

<li><a href="Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li>

<li><a href="Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li>

<li><a href="Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Black_hole" title="Black hole">Black hole</a></li>

<li><a href="Event_horizon" title="Event horizon">Event horizon</a></li>

<li><a href="Gravitational_singularity" title="Gravitational singularity">Singularity</a></li>

<li><a href="Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul>

<ul><li><a href="Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li>

<li><a href="Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="LIGO" title="LIGO">LIGO</a> and <a href="LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li>

<li><a href="Virgo_interferometer" title="Virgo interferometer">Virgo</a></li>

<li><a href="LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li>

<li><a href="Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul>

<ul><li><a href="Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li>

<li><a href="Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li>

<li><a href="Gravitational_redshift" title="Gravitational redshift">redshift</a></li>

<li><a href="Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li>

<li><a href="Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li>

<li><a href="Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br>theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li>Cosmological: <a href="Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li>

<li><a href="Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li>

<li><a href="Kasner_metric" title="Kasner metric">Kasner</a></li>

<li><a href="BKL_singularity" title="BKL singularity">BKL singularity</a></li>

<li><a href="G%C3%B6del_metric" title="Gödel metric">Gödel</a></li>

<li><a href="Milne_model" title="Milne model">Milne</a></li></ul>

<ul><li>Spherical: <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li>

<li><a href="Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li>

<li><a href="Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul>

<ul><li>Axisymmetric: <a href="Kerr_metric" title="Kerr metric">Kerr</a> (<a href="Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li>

<li><a href="Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li>

<li><a href="Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li>

<li><a href="Relativistic_disk" title="Relativistic disk">discs</a></li></ul>

<ul><li>Others: <a href="Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li>

<li><a href="Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li>

<li><a href="Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li>

<li><a href="Ellis_wormhole" title="Ellis wormhole">Ellis</a></li></ul>

<ul><li>In computational physics: <a href="Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>

<li><a href="Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li>

<li><a href="Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li>

<li><a href="Arthur_Eddington" title="Arthur Eddington">Eddington</a></li>

<li><a href="Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li>

<li><a href="Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li>

<li><a href="Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li>

<li><a href="Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li>

<li><a href="Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li>

<li><a href="Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li>

<li><a href="John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li>

<li><a href="Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li>

<li><a href="Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li>

<li><a href="Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li>

<li><a href="J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li>

<li><a href="Robert_Geroch" title="Robert Geroch">Geroch</a></li>

<li><a href="Roger_Penrose" title="Roger Penrose">Penrose</a></li>

<li><a href="Stephen_Hawking" title="Stephen Hawking">Hawking</a></li>

<li><a href="Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li>

<li><a href="Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li>

<li><a href="Hermann_Bondi" title="Hermann Bondi">Bondi</a></li>

<li><a href="Charles_W._Misner" title="Charles W. Misner">Misner</a></li>

<li><a href="Kip_Thorne" title="Kip Thorne">Thorne</a></li>

<li><a href="Rainer_Weiss" title="Rainer Weiss">Weiss</a></li>

<li><a href="List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox390" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Authority control databases: National </th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4126032-6">Germany</a></span></li></ul></div></td></tr></tbody></table></div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Here is a quotation applicable to moving observational frames <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span> and various associated Euclidean three-space coordinate systems [<i>R</i>, <i>R′</i>, <i>etc.</i>]:<sup id="cite_ref-Lyle_5-0" class="reference"><a href="#cite_note-Lyle-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

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</style><blockquote class="templatequote"><p>We first introduce the notion of <i>reference frame</i>, itself related to the idea of <i>observer</i>: 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span>, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span> by establishing a <i>coordinate system</i> <i>R</i> with origin <i>O</i>. The corresponding set of axes, sharing the rigid body motion of the frame <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span>, can be considered to give a physical realization of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span>. In a frame <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span>, coordinates are changed from <i>R</i> to <i>R′</i> by carrying out, at each instant of time, the same coordinate transformation on the components of <i>intrinsic</i> objects (vectors and tensors) introduced to represent physical quantities <i>in this frame</i>.</p></blockquote>

<p>and this on the utility of separating the notions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span> and [<i>R</i>, <i>R′</i>, <i>etc.</i>]:<sup id="cite_ref-Lakhtakia_6-0" class="reference"><a href="#cite_note-Lakhtakia-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<blockquote class="templatequote"><p>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.</p></blockquote>

<p>and this, also on the distinction between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>

</semantics>

</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span> and [<i>R</i>, <i>R′</i>, <i>etc.</i>]:<sup id="cite_ref-Nerlich_7-0" class="reference"><a href="#cite_note-Nerlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<blockquote class="templatequote"><p>The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different <i>spaces</i> (sets of <i>rest</i> 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 <i>classes</i> of coordinate systems.</p></blockquote>

<p>and from J. D. Norton:<sup id="cite_ref-Norton_8-0" class="reference"><a href="#cite_note-Norton-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<blockquote class="templatequote"><p>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.</p></blockquote></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFKovalevskyMueller1989" class="citation book cs1"><a href="Jean_Kovalevsky" title="Jean Kovalevsky">Kovalevsky, J.</a>; <a href="Ivan_I._Mueller" title="Ivan I. Mueller">Mueller, Ivan I.</a> (1989). "Introduction". <i>Reference Frames</i>. Astrophysics and Space Science Library. Vol. 154. Dordrecht: Springer Netherlands. pp. <span class="nowrap">1–</span>12. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-009-0933-5_1">10.1007/978-94-009-0933-5_1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-010-6909-0</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0067-0057">0067-0057</a>.</cite></span>

</li>

<li id="cite_note-macroscopic-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-macroscopic_2-0">^</a></b></span> <span class="reference-text">The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where <a href="Constitutive_equation" title="Constitutive equation">constitutive relations</a> of various time and length scales are used to determine the current and charge densities entering <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a>. See, for example, <cite id="CITEREFKurt_Edmund_Oughstun2006" class="citation book cs1">Kurt Edmund Oughstun (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=behRnNRiueAC&q=macroscopic+frame++electromagnetism&pg=PA165"><i>Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media</i></a>. Springer. p. 165. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-34599-X</bdi>.</cite>. These distinctions also appear in thermodynamics. See <cite id="CITEREFPaul_McEvoy2002" class="citation book cs1">Paul McEvoy (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dj0wFIxn-PoC&q=macroscopic+frame&pg=PA206"><i>Classical Theory</i></a>. MicroAnalytix. p. 205. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-930832-02-8</bdi>.</cite>.</span>

</li>

In very general terms, a coordinate system is a set of arcs <i>x</i><sup>i</sup> = <i>x</i><sup>i</sup> (<i>t</i>) in a complex <a href="Lie_group" title="Lie group">Lie group</a>; see <cite id="CITEREFLev_Semenovich_Pontri͡agin1986" class="citation book cs1">Lev Semenovich Pontri͡agin (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JU0DT_wXu2oC&q=algebra+%22coordinate+system%22&pg=PA429"><i>L.S. Pontryagin: Selected Works Vol. 2: Topological Groups</i></a> (3rd ed.). Gordon and Breach. p. 429. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>2-88124-133-6</bdi>.</cite>. Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>,... <b>e</b><sub>n</sub>}; see <cite id="CITEREFEdoardo_SernesiJ._Montaldi1993" class="citation book cs1">Edoardo Sernesi; J. Montaldi (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1dZOuFo1QYMC&q=algebra+%22coordinate+system%22&pg=PA95"><i>Linear Algebra: A Geometric Approach</i></a>. CRC Press. p. 95. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-412-40680-2</bdi>.</cite> As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.</span>

</li>

<cite id="CITEREFJ_X_Zheng-JohanssonPer-Ivar_Johansson2006" class="citation book cs1">J X Zheng-Johansson; Per-Ivar Johansson (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I1FU37uru6QC&q=frame+coordinate+johansson&pg=PA13"><i>Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces</i></a>. Nova Publishers. p. 13. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-59454-260-0</bdi>.</cite></span>

</li>

<li id="cite_note-Lyle-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lyle_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFJean_SalençonStephen_Lyle2001" class="citation book cs1">Jean Salençon; Stephen Lyle (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=H3xIED8ctfUC&q=physical+%22frame+of+reference%22&pg=PA9"><i>Handbook of Continuum Mechanics: General Concepts, Thermoelasticity</i></a>. Springer. p. 9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-41443-6</bdi>.</cite></span>

</li>

<li id="cite_note-Lakhtakia-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lakhtakia_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFPatrick_Cornille_(Akhlesh_Lakhtakia,_editor)1993" class="citation book cs1">Patrick Cornille (Akhlesh Lakhtakia, editor) (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qsOBhKVM1qYC&q=coordinate+system+%22reference+frame%22&pg=PA149"><i>Essays on the Formal Aspects of Electromagnetic Theory</i></a>. World Scientific. p. 149. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-02-0854-5</bdi>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|author=</code> has generic name (help)</span></span>

</li>

<li id="cite_note-Nerlich-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nerlich_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFNerlich1994" class="citation book cs1"><a href="Graham_Nerlich" title="Graham Nerlich">Nerlich, Graham</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fKK7rKOpc7AC&q=%22idea+of+a+reference+frame%22&pg=PA64"><i>What Spacetime Explains: Metaphysical essays on space and time</i></a>. Cambridge University Press. p. 64. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-45261-9</bdi>.</cite></span>

</li>

<li id="cite_note-Norton-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Norton_8-0">^</a></b></span> <span class="reference-text">John D. Norton (1993). <a rel="nofollow" class="external text" href="http://www.pitt.edu/~jdnorton/papers/decades.pdf"><i>General covariance and the foundations of general relativity: eight decades of dispute</i></a>, <i>Rep. Prog. Phys.</i>, <b>56</b>, pp. 835-7.</span>

</li>

<li id="cite_note-Barker-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Barker_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilliam_BarkerRoger_Howe2008" class="citation book cs1">William Barker; Roger Howe (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NIxExnr2EjYC&q=geometry++axiom+%22coordinate+system%22&pg=PA17"><i>Continuous symmetry: from Euclid to Klein</i></a>. American Mathematical Society. p. 18 ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3900-3</bdi>.</cite></span>

</li>

<li id="cite_note-Ramsay-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ramsay_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFArlan_RamsayRobert_D._Richtmyer1995" class="citation book cs1">Arlan Ramsay; Robert D. Richtmyer (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams"><i>Introduction to Hyperbolic Geometry</i></a></span>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams/page/11">11</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94339-0</bdi>. <q>geometry axiom coordinate system.</q></cite></span>

</li>

<li id="cite_note-Hawking-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hawking_12-0">^</a></b></span> <span class="reference-text">According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." <cite id="CITEREFStephen_W._HawkingGeorge_Francis_Rayner_Ellis1973" class="citation book cs1">Stephen W. Hawking; George Francis Rayner Ellis (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QagG_KI7Ll8C&q=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&pg=PA59"><i>The Large Scale Structure of Space-Time</i></a>. Cambridge University Press. p. 11. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-09906-4</bdi>.</cite> A mathematical definition is: <i>A connected <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a> </i>M<i> is called an </i>n<i>-dimensional manifold if each point of </i>M<i> is contained in an open set that is homeomorphic to an open set in Euclidean </i>n<i>-dimensional space.</i></span>

</li>

<li id="cite_note-Morita-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Morita_13-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFShigeyuki_MoritaTeruko_NagaseKatsumi_Nomizu2001" class="citation book cs1">Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometryofdiffer00mori"><i>Geometry of Differential Forms</i></a></span>. American Mathematical Society Bookstore. p. <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofdiffer00mori/page/12">12</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1045-6</bdi>. <q>geometry axiom coordinate system.</q></cite></span>

</li>

<li id="cite_note-Korn-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Korn_14-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGranino_Arthur_KornTheresa_M._Korn2000" class="citation book cs1">Granino Arthur Korn; <a href="Theresa_M._Korn" title="Theresa M. Korn">Theresa M. Korn</a> (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xHNd5zCXt-EC&q=curvilinear+%22coordinate+system%22&pg=PA169"><i>Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review</i></a>. Courier Dover Publications. p. 169. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-41147-8</bdi>.</cite></span>

</li>

<li id="cite_note-encarta-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-encarta_15-0">^</a></b></span> <span class="reference-text">See <a rel="nofollow" class="external text" href="https://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html">Encarta definition</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091030054251/http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html">Archived</a> 2009-10-31.</span>

</li>

<li id="cite_note-Yamane-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yamane_16-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKatsu_Yamane2004" class="citation book cs1">Katsu Yamane (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tNrMiIx3fToC&q=generalized+coordinates+%22kinematic+chain%22&pg=PA12"><i>Simulating and Generating Motions of Human Figures</i></a>. Springer. pp. <span class="nowrap">12–</span>13. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-20317-6</bdi>.</cite></span>

</li>

<li id="cite_note-Papapetrou-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Papapetrou_17-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAchilleus_Papapetrou1974" class="citation book cs1">Achilleus Papapetrou (1974). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SWeOggyp1ZsC&q=relativistic++%22general+coordinates%22&pg=PA3"><i>Lectures on General Relativity</i></a>. Springer. p. 5. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-277-0540-2</bdi>.</cite></span>

</li>

<li id="cite_note-Zdunkowski-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zdunkowski_18-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilford_ZdunkowskiAndreas_Bott2003" class="citation book cs1">Wilford Zdunkowski; Andreas Bott (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GuYvC21v3g8C&q=%22curvilinear+coordinate+system%22&pg=RA1-PA84"><i>Dynamics of the Atmosphere</i></a>. Cambridge University Press. p. 84. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-00666-X</bdi>.</cite></span>

</li>

<li id="cite_note-Borisenko-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Borisenko_19-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFA._I._BorisenkoI._E._TarapovRichard_A._Silverman1979" class="citation book cs1">A. I. Borisenko; I. E. Tarapov; Richard A. Silverman (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CRIjIx2ac6AC&q=coordinate+metric&pg=PA86"><i>Vector and Tensor Analysis with Applications</i></a>. Courier Dover Publications. p. 86. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-63833-2</bdi>.</cite></span>

</li>

<li id="cite_note-Kubar-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kubar_20-0">^</a></b></span> <span class="reference-text">See <cite id="CITEREFArvind_KumarShrish_Barve2003" class="citation book cs1">Arvind Kumar; Shrish Barve (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=czlUPz38MOQC&q=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar&pg=PA115"><i>How and Why in Basic Mechanics</i></a>. Orient Longman. p. 115. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-7371-420-7</bdi>.</cite></span>

</li>

<li id="cite_note-Doran-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Doran_21-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFChris_DoranAnthony_Lasenby2003" class="citation book cs1">Chris Doran; Anthony Lasenby (2003). <a rel="nofollow" class="external text" href="http://www.worldcat.org/search?q=9780521715959&qt=owc_search"><i>Geometric Algebra for Physicists</i></a>. Cambridge University Press. p. §5.2.2, p. 133. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-71595-9</bdi>.</cite>.</span>

</li>

<li id="cite_note-Moller-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Moller_22-0">^</a></b></span> <span class="reference-text">For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates <i>x</i><sup>i</sup> in four-space…." <cite id="CITEREFC._Møller1952" class="citation book cs1">C. Møller (1952). <i>The Theory of Relativity</i>. Oxford University Press. p. 222 and p. 233.</cite></span>

</li>

<li id="cite_note-Lightman-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lightman_23-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFA._P._LightmanW._H._PressR._H._PriceS._A._Teukolsky1975" class="citation book cs1">A. P. Lightman; W. H. Press; R. H. Price; S. A. Teukolsky (1975). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/problembookinrel00ligh"><i>Problem Book in Relativity and Gravitation</i></a></span>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/problembookinrel00ligh/page/15">15</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-08162-X</bdi>. <q>relativistic general coordinates.</q></cite></span>

</li>

<li id="cite_note-Faber-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Faber_24-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRichard_L_Faber1983" class="citation book cs1">Richard L Faber (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ctM3_afLuVEC&q=relativistic++%22general+coordinates%22&pg=PA149"><i>Differential Geometry and Relativity Theory: an introduction</i></a>. CRC Press. p. 211. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-1749-X</bdi>.</cite></span>

</li>

<li id="cite_note-Wolfson-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wolfson_25-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRichard_Wolfson2003" class="citation book cs1">Richard Wolfson (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OUJWKdlFKeQC&q=%22gravitational+time+dilation+%22&pg=PA216"><i>Simply Einstein</i></a>. W W Norton & Co. p. 216. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-05154-4</bdi>.</cite></span>

</li>

<li id="cite_note-Rizzi-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rizzi_26-0">^</a></b></span> <span class="reference-text">See <cite id="CITEREFGuido_RizziMatteo_Luca_Ruggiero2003" class="citation book cs1">Guido Rizzi; Matteo Luca Ruggiero (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_PGrlCLkkIgC&q=centrifugal+%22+%22+relativity+OR+relativistic&pg=PA226"><i>Relativity in rotating frames</i></a>. Springer. p. 33. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-1805-3</bdi>.</cite>.</span>

</li>

<li id="cite_note-Brading-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brading_27-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKatherine_BradingElena_Castellani2003" class="citation book cs1"><a href="Katherine_Brading" title="Katherine Brading">Katherine Brading</a>; Elena Castellani (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SnmBN64cAdYC&q=%22idea+of+a+reference+frame%22&pg=PA417"><i>Symmetries in Physics: Philosophical Reflections</i></a>. Cambridge University Press. p. 417. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-82137-1</bdi>.</cite></span>

</li>

<li id="cite_note-Johns-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Johns_28-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFOliver_Davis_Johns2005" class="citation book cs1">Oliver Davis Johns (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PNuM9YDN8CIC&q=coordinate+observer&pg=PA318"><i>Analytical Mechanics for Relativity and Quantum Mechanics</i></a>. Oxford University Press. Chapter 16. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-856726-X</bdi>.</cite></span>

</li>

<li id="cite_note-Greenwood-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greenwood_29-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDonald_T_Greenwood1997" class="citation book cs1">Donald T Greenwood (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x7rj83I98yMC&q=%22relativistic+%22+Lagrangian+OR+Hamiltonian&pg=RA2-PA314"><i>Classical dynamics</i></a> (Reprint of 1977 edition by Prentice-Hall ed.). Courier Dover Publications. p. 313. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-69690-1</bdi>.</cite></span>

</li>

<li id="cite_note-Trump-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Trump_30-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFMatthew_A._TrumpW._C._Schieve1999" class="citation book cs1">Matthew A. Trump; W. C. Schieve (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g2yfLOp0IzwC&q=relativity+%22generalized+coordinates%22&pg=PA99"><i>Classical Relativistic Many-Body Dynamics</i></a>. Springer. p. 99. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-5737-X</bdi>.</cite></span>

</li>

<li id="cite_note-Kompaneyets-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kompaneyets_31-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAlexander_Solomonovich_Kompaneyets2003" class="citation book cs1">Alexander Solomonovich Kompaneyets (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CQ2gBrL5T4YC&q=relativity+%22generalized+coordinates%22&pg=PA118"><i>Theoretical Physics</i></a> (Reprint of the 1962 2nd ed.). Courier Dover Publications. p. 118. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-49532-9</bdi>.</cite></span>

</li>

<li id="cite_note-Srednicki-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Srednicki_32-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFM_Srednicki2007" class="citation book cs1">M Srednicki (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5OepxIG42B4C&pg=PA266"><i>Quantum Field Theory</i></a>. Cambridge University Press. Chapter 4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-86449-7</bdi>.</cite></span>

</li>

<li id="cite_note-Rovelli-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rovelli_33-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCarlo_Rovelli2004" class="citation book cs1">Carlo Rovelli (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HrAzTmXdssQC&q=%22relativistic+%22+Lagrangian+OR+Hamiltonian&pg=PA179"><i>Quantum Gravity</i></a>. Cambridge University Press. p. 98 ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-83733-2</bdi>.</cite></span>

</li>

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