Field equation

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In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not ordinary differential equations since a field depends on space and time, which requires at least two variables.

Whereas the "wave equation", the "diffusion equation", and the "continuity equation" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation".

The topic broadly splits into equations of classical field theory and quantum field theory. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc. They also describe the fundamental forces of nature, like electromagnetism and gravity. In quantum field theory, particles or systems of "particles" like electrons and photons are associated with fields, allowing for infinite degrees of freedom (unlike finite degrees of freedom in particle mechanics) and variable particle numbers which can be created or annihilated.

Generalities

Origin

Usually, field equations are postulated (like the Einstein field equations and the Schrödinger equation, which underlies all quantum field equations) or obtained from the results of experiments (like Maxwell's equations). The extent of their validity is their extent to correctly predict and agree with experimental results. 

From a theoretical viewpoint, field equations can be formulated in the frameworks of Lagrangian field theory, Hamiltonian field theory, and field theoretic formulations of the principle of stationary action. Given a suitable Lagrangian or Hamiltonian density, a function of the fields in a given system, as well as their derivatives, the principle of stationary action will obtain the field equation.

Symmetry

In both classical and quantum theories, field equations will satisfy the symmetry of the background physical theory. Most of the time Galilean symmetry is enough, for speeds (of propagating fields) much less than light. When particles and fields propagate at speeds close to light, Lorentz symmetry is one of the most common settings because the equation and its solutions are then consistent with special relativity. 

Another symmetry arises from gauge freedom, which is intrinsic to the field equations. Fields which correspond to interactions may be gauge fields, which means they can be derived from a potential, and certain values of potentials correspond to the same value of the field.

Classification

Field equations can be classified in many ways: classical or quantum, nonrelativistsic or relativistic, according to the spin or mass of the field, and the number of components the field has and how they change under coordinate transformations (e.g. scalar fields, vector fields, tensor fields, spinor fields, twistor fields etc.). They can also inherit the classification of differential equations, as linear or nonlinear, the order of the highest derivative, or even as fractional differential equations. Gauge fields may be classified as in group theory, as abelian or nonabelian.

Waves

Field equations underlie wave equations, because periodically changing fields generate waves. Wave equations can be thought of as field equations, in the sense they can often be derived from field equations. Alternatively, given suitable Lagrangian or Hamiltonian densities and using the principle of stationary action, the wave equations can be obtained also.

For example, Maxwell's equations can be used to derive inhomogeneous electromagnetic wave equations, and from the Einstein field equations one can derive equations for gravitational waves.

Supplementary equations to field equations

Not every partial differential equation (PDE) in physics is automatically called a "field equation", even if fields are involved. They are extra equations to provide additional constraints for a given physical system. 

"Continuity equations" and "diffusion equations" describe transport phenomena, even though they may involve fields which influence the transport processes.

If a "constitutive equation" takes the form of a PDE and involves fields, it is not usually called a field equation because it does not govern the dynamical behaviour of the fields. They relate one field to another, in a given material. Constitutive equations are used along with field equations when the effects of matter need to be taken into account.

Classical field equation

Classical field equations arise in continuum mechanics (including elastodynamics and fluid mechanics), heat transfer, electromagnetism, and gravitation. 

Fundamental classical field equations include

• Newton's Law of Universal Gravitation for nonrelativistic gravitation.
• Einstein field equations for relativistic gravitation
• Maxwell's equations for electromagnetism.

Important equations derived from fundamental laws include:

• Navier–Stokes equations for fluid flow.

As part of real-life mathematical modelling processes, classical field equations are accompanied by other equations of motion, equations of state, constitutive equations, and continuity equations.

Quantum field equation

In quantum field theory, particles are described by quantum fields which satisfy the Schrödinger equation. They are also creation and annihilation operators which satisfy commutation relations and are subject to the spin–statistics theorem. 

Particular cases of relativistic quantum field equations include

• the Klein–Gordon equation for spin-0 particles
• the Dirac equation for spin-1/2 particles
• the Bargmann–Wigner equations for particles of any spin

In quantum field equations, it is common to use momentum components of the particle instead of position coordinates of the particle's location, the fields are in momentum space and Fourier transforms relate them to the position representation.

Wheeler–DeWitt equation

From Wikipedia, the free encyclopedia

 

The Wheeler–DeWitt equation is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which is the diffeomorphism group on-shell).

Quantum gravity

All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-time) in every description, so there is a preferred definition of the Hamiltonian (with nonzero eigenvalues) to evolve states of the system forwards in time. This avoids all the need to dynamically generate a time dimension using the Wheeler–DeWitt equation. Thus, the equation has not played a role in string theory thus far.

There could exist a Wheeler–DeWitt-style manner to describe the bulk dynamics of quantum theory of gravity. Some experts believe that this equation still holds the potential for understanding quantum gravity; however, decades after the equation was published, completely different approaches, such as string theory, have brought physicists as clear results about quantum gravity.

Motivation and background

In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is ${\displaystyle \gamma _{ij}}$ and given by

${\displaystyle g_{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})\,\mathrm {d} t^{2}+2\beta _{k}\,\mathrm {d} x^{k}\,\mathrm {d} t+\gamma _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}.}$

In that equation the Latin indices run over the values 1, 2, 3 and the Greek indices run over the values 1, 2, 3, 4. The three-metric ${\displaystyle \gamma _{ij}}$ is the field, and we denote its conjugate momenta as ${\displaystyle \pi ^{kl}}$. The Hamiltonian is a constraint (characteristic of most relativistic systems)

${\displaystyle {\mathcal {H}}={\frac {1}{2{\sqrt {\gamma }}}}G_{ijkl}\pi ^{ij}\pi ^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!R=0}$

where ${\displaystyle \gamma =\det(\gamma _{ij})}$ and ${\displaystyle G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})}$ is the Wheeler–DeWitt metric. 

Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator

${\displaystyle {\widehat {\mathcal {H}}}={\frac {1}{2{\sqrt {\gamma }}}}{\widehat {G}}_{ijkl}{\widehat {\pi }}^{ij}{\widehat {\pi }}^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!{\widehat {R}}.}$

Working in "position space", these operators are

${\displaystyle {\hat {\gamma }}_{ij}(t,x^{k})\to \gamma _{ij}(t,x^{k})}$

${\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\to -i{\frac {\delta }{\delta \gamma _{ij}(t,x^{k})}}.}$

One can apply the operator to a general wave functional of the metric ${\displaystyle {\widehat {\mathcal {H}}}\Psi [\gamma ]=0}$ where:

${\displaystyle \Psi [\gamma ]=a+\int \psi (x)\gamma (x)dx^{3}+\int \int \psi (x,y)\gamma (x)\gamma (y)dx^{3}dy^{3}+...}$

which would give a set of constraints amongst the coefficients ${\displaystyle \psi (x,y,...)}$. This means the amplitudes for N gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating ${\displaystyle \omega (g)}$ as an independent field so that the wave function is ${\displaystyle \Psi [\gamma ,\omega ]}$.

Derivation from path integral

The Wheeler–DeWitt equation can be derived from a path integral using the gravitational action in the Euclidean quantum gravity paradigm:

${\displaystyle Z=\int _{C}\mathrm {e} ^{-I[g_{\mu \nu },\phi ]}{\mathcal {D}}\mathbf {g} \,{\mathcal {D}}\phi }$

where one integrates over a class of Riemannian four-metrics and matter fields matching certain boundary conditions. Because the concept of a universal time coordinate seems unphysical, and at odds with the principles of general relativity, the action is evaluated around a 3-metric which we take as the boundary of the classes of four-metrics and on which a certain configuration of matter fields exists. This latter might for example be the current configuration of matter in our universe as we observe it today. Evaluating the action so that it only depends on the 3-metric and the matter fields is sufficient to remove the need for a time coordinate as it effectively fixes a point in the evolution of the universe. 

We obtain the Hamiltonian constraint from

${\displaystyle {\frac {\delta I_{EH}}{\delta N}}=0}$

where ${\displaystyle I_{EH}}$ is the Einstein–Hilbert action, and ${\displaystyle N}$ is the lapse function, i.e. the Lagrange multiplier for the Hamiltonian constraint. The demand for this variation of our gravitational action to vanish corresponds, in fact, to the background independence in general relativity. This is purely classical so far. We can recover the Wheeler–DeWitt equation from

${\displaystyle {\frac {\delta Z}{\delta N}}=0=\int \left.{\frac {\delta I[g_{\mu \nu },\phi ]}{\delta N}}\right|_{\Sigma }\exp \left(-I[g_{\mu \nu },\phi ]\right)\,{\mathcal {D}}\mathbf {g} \,{\mathcal {D}}\phi }$

where ${\displaystyle \Sigma }$ is the three-dimensional boundary. Observe that this expression vanishes, implying that the functional derivative also vanishes, giving us the Wheeler–DeWitt equation. A similar statement may be made for the diffeomorphism constraint (take functional derivative with respect to the shift functions instead).

Mathematical formalism

The Wheeler–DeWitt equation is a functional differential equation. It is ill-defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics. The Wheeler–DeWitt equation has the form of an operator acting on a wave functional; the functional reduces to a function in cosmology. Contrary to the general case, the Wheeler–DeWitt equation is well defined in minisuperspaces like the configuration space of cosmological theories. An example of such a wave function is the Hartle–Hawking state. Bryce DeWitt first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".

Hamiltonian constraint

Simply speaking, the Wheeler–DeWitt equation says 

${\displaystyle {\hat {H}}(x)|\psi \rangle =0}$

where ${\displaystyle {\hat {H}}(x)}$ is the Hamiltonian constraint in quantized general relativity and ${\displaystyle |\psi \rangle }$ stands for the wave function of the universe. Unlike ordinary quantum field theory or quantum mechanics, the Hamiltonian is a first class constraint on physical states. We also have an independent constraint for each point in space. 

Although the symbols ${\displaystyle {\hat {H}}}$ and ${\displaystyle |\psi \rangle }$ may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics. ${\displaystyle |\psi \rangle }$ is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like surface and normalized to unity. Instead it is a functional of field configurations on all of spacetime. This wave function contains all of the information about the geometry and matter content of the universe. ${\displaystyle {\hat {H}}}$ is still an operator that acts on the Hilbert space of wave functions, but it is not the same Hilbert space as in the nonrelativistic case, and the Hamiltonian no longer determines evolution of the system, so the Schrödinger equation ${\displaystyle {\hat {H}}|\psi \rangle =i\hbar \partial /\partial t|\psi \rangle }$ no longer applies. This property is known as timelessness. The reemergence of time requires the tools of decoherence and clock operators (or the use of a scalar field).

Momentum constraint

We also need to augment the Hamiltonian constraint with momentum constraints

${\displaystyle {\vec {\mathcal {P}}}(x)\left|\psi \right\rangle =0}$

associated with spatial diffeomorphism invariance. 

In minisuperspace approximations, we only have one Hamiltonian constraint (instead of infinitely many of them). 

In fact, the principle of general covariance in general relativity implies that global evolution per se does not exist; the time ${\displaystyle t}$ is just a label we assign to one of the coordinate axes. Thus, what we think about as time evolution of any physical system is just a gauge transformation, similar to that of QED induced by U(1) local gauge transformation ${\displaystyle \psi \rightarrow e^{i\theta ({\vec {r}})}\psi }$ where ${\displaystyle \theta ({\vec {r}})}$ plays the role of local time. The role of a Hamiltonian is simply to restrict the space of the "kinematic" states of the Universe to that of "physical" states - the ones that follow gauge orbits. For this reason we call it a "Hamiltonian constraint." Upon quantization, physical states become wave functions that lie in the kernel of the Hamiltonian operator. 

In general, the Hamiltonian vanishes for a theory with general covariance or time-scaling invariance.

Michelson–Morley experiment

From Wikipedia, the free encyclopedia

 

Figure 1. Michelson and Morley's interferometric setup, mounted on a stone slab that floats in an annular trough of mercury

 

The Michelson–Morley experiment was an attempt to detect the existence of aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 1887 by Albert A. Michelson and Edward W. Morley at what is now Case Western Reserve University in Cleveland, Ohio, and published in November of the same year. It compared the speed of light in perpendicular directions, in an attempt to detect the relative motion of matter through the stationary luminiferous aether ('aether wind'). The result was negative, in that Michelson and Morley found no significant difference between the speed of light in the direction of movement through the presumed aether, and the speed at right angles. This result is generally considered to be the first strong evidence against the then-prevalent aether theory, and initiated a line of research that eventually led to special relativity, which rules out a stationary aether. Of this experiment, Einstein wrote, "If the Michelson–Morley experiment had not brought us into serious embarrassment, no one would have regarded the relativity theory as a (halfway) redemption."

Michelson–Morley type experiments have been repeated many times with steadily increasing sensitivity. These include experiments from 1902 to 1905, and a series of experiments in the 1920s. More recent optical resonator experiments confirmed the absence of any aether wind at the 10⁻¹⁷ level. Together with the Ives–Stilwell and Kennedy–Thorndike experiments, Michelson–Morley type experiments form one of the fundamental tests of special relativity theory.

Detecting the aether

Physics theories of the late 19th century assumed that just as surface water waves must have a supporting substance, i.e., a "medium", to move across (in this case water), and audible sound requires a medium to transmit its wave motions (such as air or water), so light must also require a medium, the "luminiferous aether", to transmit its wave motions. Because light can travel through a vacuum, it was assumed that even a vacuum must be filled with aether. Because the speed of light is so great, and because material bodies pass through the aether without obvious friction or drag, it was assumed to have a highly unusual combination of properties. Designing experiments to investigate these properties was a high priority of 19th century physics.

Earth orbits around the Sun at a speed of around 30 km/s (18.64 mi/s), or 108,000 km/h (67,000 mph). The Earth is in motion, so two main possibilities were considered: (1) The aether is stationary and only partially dragged by Earth (proposed by Augustin-Jean Fresnel in 1818), or (2) the aether is completely dragged by Earth and thus shares its motion at Earth's surface (proposed by Sir George Stokes, 1st Baronet in 1844). In addition, James Clerk Maxwell (1865) recognized the electromagnetic nature of light and developed what are now called Maxwell's equations, but these equations were still interpreted as describing the motion of waves through an aether, whose state of motion was unknown. Eventually, Fresnel's idea of an (almost) stationary aether was preferred because it appeared to be confirmed by the Fizeau experiment (1851) and the aberration of star light.

Figure 2. A depiction of the concept of the "aether wind"

 

According to the stationary and the partially-dragged aether hypotheses, Earth and the aether are in relative motion, implying that a so-called "aether wind" (Fig. 2) should exist. Although it would be possible, in theory, for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both the direction and the speed of the motion. At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analyzing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether. The expected relative difference in the measured speed of light was quite small, given that the velocity of the Earth in its orbit around the Sun has a magnitude of about one hundredth of one percent of the speed of light.

During the mid-19th century, measurements of aether wind effects of first order, i.e., effects proportional to v/c (v being Earth's velocity, c the speed of light) were thought to be possible, but no direct measurement of the speed of light was possible with the accuracy required. For instance, the Fizeau–Foucault apparatus could measure the speed of light to perhaps 5% accuracy, which was quite inadequate for measuring directly a first-order 0.01% change in the speed of light. A number of physicists therefore attempted to make measurements of indirect first-order effects not of the speed of light itself, but of variations in the speed of light (see First order aether-drift experiments). The Hoek experiment, for example, was intended to detect interferometric fringe shifts due to speed differences of oppositely propagating light waves through water at rest. The results of such experiments were all negative. This could be explained by using Fresnel's dragging coefficient, according to which the aether and thus light are partially dragged by moving matter. Partial aether-dragging would thwart attempts to measure any first order change in the speed of light. As pointed out by Maxwell (1878), only experimental arrangements capable of measuring second order effects would have any hope of detecting aether drift, i.e., effects proportional to v²/c². Existing experimental setups, however, were not sensitive enough to measure effects of that size.

1881 and 1887 experiments

Michelson experiment (1881)

Michelson's 1881 interferometer. Although ultimately it proved incapable of distinguishing between differing theories of aether-dragging, its construction provided important lessons for the design of Michelson and Morley's 1887 instrument.

 

Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. In 1877, while teaching at his alma mater, the United States Naval Academy in Annapolis, Michelson conducted his first known light speed experiments as a part of a classroom demonstration. In 1881, he left active U.S. Naval service while in Germany concluding his studies. In that year, Michelson used a prototype experimental device to make several more measurements.

The device he designed, later known as a Michelson interferometer, sent yellow light from a sodium flame (for alignment), or white light (for the actual observations), through a half-silvered mirror that was used to split it into two beams traveling at right angles to one another. After leaving the splitter, the beams traveled out to the ends of long arms where they were reflected back into the middle by small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference whose transverse displacement would depend on the relative time it takes light to transit the longitudinal vs. the transverse arms. If the Earth is traveling through an aether medium, a beam reflecting back and forth parallel to the flow of aether would take longer than a beam reflecting perpendicular to the aether because the time gained from traveling downwind is less than that lost traveling upwind. Michelson expected that the Earth's motion would produce a fringe shift equal to 0.04 fringes—that is, of the separation between areas of the same intensity. He did not observe the expected shift; the greatest average deviation that he measured (in the northwest direction) was only 0.018 fringes; most of his measurements were much less. His conclusion was that Fresnel's hypothesis of a stationary aether with partial aether dragging would have to be rejected, and thus he confirmed Stokes' hypothesis of complete aether dragging.

However, Alfred Potier (and later Hendrik Lorentz) pointed out to Michelson that he had made an error of calculation, and that the expected fringe shift should have been only 0.02 fringes. Michelson's apparatus was subject to experimental errors far too large to say anything conclusive about the aether wind. Definitive measurement of the aether wind would require an experiment with greater accuracy and better controls than the original. Nevertheless, the prototype was successful in demonstrating that the basic method was feasible.

Michelson–Morley experiment (1887)

Figure 5. This figure illustrates the folded light path used in the Michelson–Morley interferometer that enabled a path length of 11 m. a is the light source, an oil lamp. b is a beam splitter. c is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass (important since experiments were run with white light which has an extremely short coherence length requiring precise matching of optical path lengths for fringes to be visible; monochromatic sodium light was used only for initial alignment). d, d' and e are mirrors. e' is a fine adjustment mirror. f is a telescope.

 

In 1885, Michelson began a collaboration with Edward Morley, spending considerable time and money to confirm with higher accuracy Fizeau's 1851 experiment on Fresnel's drag coefficient, to improve on Michelson's 1881 experiment, and to establish the wavelength of light as a standard of length. At this time Michelson was professor of physics at the Case School of Applied Science, and Morley was professor of chemistry at Western Reserve University (WRU), which shared a campus with the Case School on the eastern edge of Cleveland. Michelson suffered a nervous breakdown in September 1885, from which he recovered by October 1885. Morley ascribed this breakdown to the intense work of Michelson during the preparation of the experiments. In 1886, Michelson and Morley successfully confirmed Fresnel's drag coefficient – this result was also considered as a confirmation of the stationary aether concept.

This result strengthened their hope of finding the aether wind. Michelson and Morley created an improved version of the Michelson experiment with more than enough accuracy to detect this hypothetical effect. The experiment was performed in several periods of concentrated observations between April and July 1887, in the basement of Adelbert Dormitory of WRU (later renamed Pierce Hall, demolished in 1962).

As shown in Fig. 5, the light was repeatedly reflected back and forth along the arms of the interferometer, increasing the path length to 11 m (36 ft). At this length, the drift would be about 0.4 fringes. To make that easily detectable, the apparatus was assembled in a closed room in the basement of the heavy stone dormitory, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a large block of sandstone (Fig. 1), about a foot thick and five feet square, which was then floated in a circular trough of mercury. They estimated that effects of about 0.01 fringe would be detectable.

Figure 6. Fringe pattern produced with a Michelson interferometer using white light. As configured here, the central fringe is white rather than black.

 

Michelson and Morley and other early experimentalists using interferometric techniques in an attempt to measure the properties of the luminiferous aether, used (partially) monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of environmental temperature control, experimentalists struggled with continual fringe drift even when the interferometer was set up in a basement. Because the fringes would occasionally disappear due to vibrations caused by passing horse traffic, distant thunderstorms and the like, an observer could easily "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low coherence length. As Dayton Miller wrote, "White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all readings." Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light.

The mercury trough allowed the device to turn with close to zero friction, so that once having given the sandstone block a single push it would slowly rotate through the entire range of possible angles to the "aether wind," while measurements were continuously observed by looking through the eyepiece. The hypothesis of aether drift implies that because one of the arms would inevitably turn into the direction of the wind at the same time that another arm was turning perpendicularly to the wind, an effect should be noticeable even over a period of minutes.

The expectation was that the effect would be graphable as a sine wave with two peaks and two troughs per rotation of the device. This result could have been expected because during each full rotation, each arm would be parallel to the wind twice (facing into and away from the wind giving identical readings) and perpendicular to the wind twice. Additionally, due to the Earth's rotation, the wind would be expected to show periodic changes in direction and magnitude during the course of a sidereal day. 

Because of the motion of the Earth around the Sun, the measured data were also expected to show annual variations.

Most famous "failed" experiment

Figure 7. Michelson and Morley's results. The upper solid line is the curve for their observations at noon, and the lower solid line is that for their evening observations. Note that the theoretical curves and the observed curves are not plotted at the same scale: the dotted curves, in fact, represent only one-eighth of the theoretical displacements.

 

After all this thought and preparation, the experiment became what has been called the most famous failed experiment in history. Instead of providing insight into the properties of the aether, Michelson and Morley's article in the American Journal of Science reported the measurement to be as small as one-fortieth of the expected displacement (Fig. 7), but "since the displacement is proportional to the square of the velocity" they concluded that the measured velocity was "probably less than one-sixth" of the expected velocity of the Earth's motion in orbit and "certainly less than one-fourth." Although this small "velocity" was measured, it was considered far too small to be used as evidence of speed relative to the aether, and it was understood to be within the range of an experimental error that would allow the speed to actually be zero. For instance, Michelson wrote about the "decidedly negative result" in a letter to Lord Rayleigh in August 1887:

The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe – the maximum displacement was 0.02 and the average much less than 0.01 – and then not in the right place. As displacement is proportional to squares of the relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth’s velocity.

— Albert Abraham Michelson, 1887

From the standpoint of the then current aether models, the experimental results were conflicting. The Fizeau experiment and its 1886 repetition by Michelson and Morley apparently confirmed the stationary aether with partial aether dragging, and refuted complete aether dragging. On the other hand, the much more precise Michelson–Morley experiment (1887) apparently confirmed complete aether dragging and refuted the stationary aether. In addition, the Michelson–Morley null result was further substantiated by the null results of other second-order experiments of different kind, namely the Trouton–Noble experiment (1903) and the experiments of Rayleigh and Brace (1902–1904). These problems and their solution led to the development of the Lorentz transformation and special relativity.

After the "failed" experiment Michelson and Morley ceased their aether drift measurements and started to use their newly developed technique to establish the wavelength of light as a standard of length.

Light path analysis and consequences

Observer resting in the aether

Expected differential phase shift between light traveling the longitudinal versus the transverse arms of the Michelson–Morley apparatus

 

The beam travel time in the longitudinal direction can be derived as follows: Light is sent from the source and propagates with the speed of light ${\textstyle c}$ in the aether. It passes through the half-silvered mirror at the origin at ${\textstyle T=0}$. The reflecting mirror is at that moment at distance ${\textstyle L}$ (the length of the interferometer arm) and is moving with velocity ${\textstyle v}$. The beam hits the mirror at time ${\textstyle T_{1}}$ and thus travels the distance ${\textstyle cT_{1}}$. At this time, the mirror has traveled the distance ${\textstyle vT_{1}}$. Thus ${\textstyle cT_{1}=L+vT_{1}}$ and consequently the travel time ${\textstyle T_{1}=L/(c-v)}$. The same consideration applies to the backward journey, with the sign of ${\textstyle v}$ reversed, resulting in ${\textstyle cT_{2}=L-vT_{2}}$ and ${\textstyle T_{2}=L/(c+v)}$. The total travel time ${\textstyle T_{\ell }=T_{1}+T_{2}}$ is:

${\displaystyle T_{\ell }={\frac {L}{c-v}}+{\frac {L}{c+v}}={\frac {2L}{c}}{\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}\approx {\frac {2L}{c}}\left(1+{\frac {v^{2}}{c^{2}}}\right)}$

Michelson obtained this expression correctly in 1881, however, in transverse direction he obtained the incorrect expression

${\displaystyle T_{t}={\frac {2L}{c}},}$

because he overlooked the increased path length in the rest frame of the aether. This was corrected by Alfred Potier (1882) and Lorentz (1886). The derivation in the transverse direction can be given as follows (analogous to the derivation of time dilation using a light clock): The beam is propagating at the speed of light ${\textstyle c}$ and hits the mirror at time ${\textstyle T_{3}}$, traveling the distance ${\textstyle cT_{3}}$. At the same time, the mirror has traveled the distance ${\textstyle vT_{3}}$ in the x direction. So in order to hit the mirror, the travel path of the beam is ${\textstyle L}$ in the y direction (assuming equal-length arms) and ${\textstyle vT_{3}}$ in the x direction. This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame. Therefore, the Pythagorean theorem gives the actual beam travel distance of ${\textstyle {\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}}$. Thus ${\textstyle cT_{3}={\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}}$ and consequently the travel time ${\textstyle T_{3}=L/{\sqrt {c^{2}-v^{2}}}}$, which is the same for the backward journey. The total travel time ${\textstyle T_{t}=2T_{3}}$ is:

${\displaystyle T_{t}={\frac {2L}{\sqrt {c^{2}-v^{2}}}}={\frac {2L}{c}}{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\approx {\frac {2L}{c}}\left(1+{\frac {v^{2}}{2c^{2}}}\right)}$

The time difference between T_ℓ and T_t before rotation is given by^{[A 16]}

${\displaystyle T_{\ell }-T_{t}={\frac {2}{c}}\left({\frac {L}{1-{\frac {v^{2}}{c^{2}}}}}-{\frac {L}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\right).}$

By multiplying with c, the corresponding length difference before rotation is

${\displaystyle \Delta _{1}=2\left({\frac {L}{1-{\frac {v^{2}}{c^{2}}}}}-{\frac {L}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\right),}$

and after rotation

${\displaystyle \Delta _{2}=2\left({\frac {L}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-{\frac {L}{1-{\frac {v^{2}}{c^{2}}}}}\right).}$

Dividing ${\textstyle \Delta _{1}-\Delta _{2}}$ by the wavelength λ, the fringe shift n is found:

${\displaystyle n={\frac {\Delta _{1}-\Delta _{2}}{\lambda }}\approx {\frac {2Lv^{2}}{\lambda c^{2}}}.}$

Since L ≈ 11 meters and λ≈500 nanometers, the expected fringe shift was n ≈ 0.44. So the result would be a delay in one of the light beams that could be detected when the beams were recombined through interference. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. The negative result led Michelson to the conclusion that there is no measurable aether drift.

Observer comoving with the interferometer

If the same situation is described from the view of an observer co-moving with the interferometer, then the effect of aether wind is similar to the effect experienced by a swimmer, who tries to move with velocity ${\textstyle c}$ against a river flowing with velocity ${\textstyle v}$.

In the longitudinal direction the swimmer first moves upstream, so his velocity is diminished due to the river flow to ${\textstyle c-v}$. On his way back moving downstream, his velocity is increased to ${\textstyle c+v}$. This gives the beam travel times ${\textstyle T_{1}}$ and ${\textstyle T_{2}}$ as mentioned above.

In the transverse direction, the swimmer has to compensate for the river flow by moving at a certain angle against the flow direction, in order to sustain his exact transverse direction of motion and to reach the other side of the river at the correct location. This diminishes his speed to ${\textstyle {\sqrt {c^{2}-v^{2}}}}$, and gives the beam travel time ${\textstyle T_{3}}$ as mentioned above.

Mirror reflection

The classical analysis predicted a relative phase shift between the longitudinal and transverse beams which in Michelson and Morley's apparatus should have been readily measurable. What is not often appreciated (since there was no means of measuring it), is that motion through the hypothetical aether should also have caused the two beams to diverge as they emerged from the interferometer by about 10⁻⁸ radians.

For an apparatus in motion, the classical analysis requires that the beam-splitting mirror be slightly offset from an exact 45° if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed. In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams.

Length contraction and Lorentz transformation

A first step to explaining the Michelson and Morley experiment's null result was found in the FitzGerald–Lorentz contraction hypothesis, now simply called length contraction or Lorentz contraction, first proposed by George FitzGerald (1889) and Hendrik Lorentz (1892). According to this law all objects physically contract by ${\textstyle L/\gamma }$ along the line of motion (originally thought to be relative to the aether), ${\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$ being the Lorentz factor. This hypothesis was partly motivated by Oliver Heaviside's discovery in 1888 that electrostatic fields are contracting in the line of motion. But since there was no reason at that time to assume that binding forces in matter are of electric origin, length contraction of matter in motion with respect to the aether was considered an Ad hoc hypothesis.

If length contraction of ${\textstyle L}$ is inserted into the above formula for ${\textstyle T_{\ell }}$, then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction:

${\displaystyle T_{\ell }={\frac {2L{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{c}}{\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}={\frac {2L}{c}}{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=T_{t}}$

However, length contraction is only a special case of the more general relation, according to which the transverse length is larger than the longitudinal length by the ratio ${\textstyle \gamma }$. This can be achieved in many ways. If ${\textstyle L_{1}}$ is the moving longitudinal length and ${\textstyle L_{2}}$ the moving transverse length, ${\textstyle L'_{1}=L'_{2}}$ being the rest lengths, then it is given:

${\displaystyle {\frac {L_{2}}{L_{1}}}={\frac {L'_{2}}{\varphi }}\left/{\frac {L'_{1}}{\gamma \varphi }}\right.=\gamma .}$

${\textstyle \varphi }$ can be arbitrarily chosen, so there are infinitely many combinations to explain the Michelson–Morley null result. For instance, if ${\textstyle \varphi =1}$ the relativistic value of length contraction of ${\textstyle L_{1}}$ occurs, but if ${\textstyle \varphi =1/\gamma }$ then no length contraction but an elongation of ${\textstyle L_{2}}$ occurs. This hypothesis was later extended by Joseph Larmor (1897), Lorentz (1904) and Henri Poincaré (1905), who developed the complete Lorentz transformation including time dilation in order to explain the Trouton–Noble experiment, the Experiments of Rayleigh and Brace, and Kaufmann's experiments. It has the form

${\displaystyle x'=\gamma \varphi (x-vt),\ y'=\varphi y,\ z'=\varphi z,\ t'=\gamma \varphi \left(t-{\frac {vx}{c^{2}}}\right)}$

It remained to define the value of ${\textstyle \varphi }$, which was shown by Lorentz (1904) to be unity. In general, Poincaré (1905) demonstrated that only ${\textstyle \varphi =1}$ allows this transformation to form a group, so it is the only choice compatible with the principle of relativity, i.e., making the stationary aether undetectable. Given this, length contraction and time dilation obtain their exact relativistic values.

Special relativity

Albert Einstein formulated the theory of special relativity by 1905, deriving the Lorentz transformation and thus length contraction and time dilation from the relativity postulate and the constancy of the speed of light, thus removing the ad hoc character from the contraction hypothesis. Einstein emphasized the kinematic foundation of the theory and the modification of the notion of space and time, with the stationary aether no longer playing any role in his theory. He also pointed out the group character of the transformation. Einstein was motivated by Maxwell's theory of electromagnetism (in the form as it was given by Lorentz in 1895) and the lack of evidence for the luminiferous aether.

This allows a more elegant and intuitive explanation of the Michelson–Morley null result. In a comoving frame the null result is self-evident, since the apparatus can be considered as at rest in accordance with the relativity principle, thus the beam travel times are the same. In a frame relative to which the apparatus is moving, the same reasoning applies as described above in "Length contraction and Lorentz transformation", except the word "aether" has to be replaced by "non-comoving inertial frame". Einstein wrote in 1916:

Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a "specially favoured" (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.

— Albert Einstein, 1916

The extent to which the null result of the Michelson–Morley experiment influenced Einstein is disputed. Alluding to some statements of Einstein, many historians argue that it played no significant role in his path to special relativity, while other statements of Einstein probably suggest that he was influenced by it. In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.

It was later shown by Howard Percy Robertson (1949) and others, that it is possible to derive the Lorentz transformation entirely from the combination of three experiments. First, the Michelson–Morley experiment showed that the speed of light is independent of the orientation of the apparatus, establishing the relationship between longitudinal (β) and transverse (δ) lengths. Then in 1932, Roy Kennedy and Edward Thorndike modified the Michelson–Morley experiment by making the path lengths of the split beam unequal, with one arm being very short. The Kennedy–Thorndike experiment took place for many months as the Earth moved around the sun. Their negative result showed that the speed of light is independent of the velocity of the apparatus in different inertial frames. In addition it established that besides length changes, corresponding time changes must also occur, i.e., it established the relationship between longitudinal lengths (β) and time changes (α). So both experiments do not provide the individual values of these quantities. This uncertainty corresponds to the undefined factor ${\textstyle \varphi }$ as described above. It was clear due to theoretical reasons (the group character of the Lorentz transformation as required by the relativity principle) that the individual values of length contraction and time dilation must assume their exact relativistic form. But a direct measurement of one of these quantities was still desirable to confirm the theoretical results. This was achieved by the Ives–Stilwell experiment (1938), measuring α in accordance with time dilation. Combining this value for α with the Kennedy–Thorndike null result shows that β must assume the value of relativistic length contraction. Combining β with the Michelson–Morley null result shows that δ must be zero. Therefore, the Lorentz transformation with ${\textstyle \varphi =1}$ is an unavoidable consequence of the combination of these three experiments.

Special relativity is generally considered the solution to all negative aether drift (or isotropy of the speed of light) measurements, including the Michelson–Morley null result. Many high precision measurements have been conducted as tests of special relativity and modern searches for Lorentz violation in the photon, electron, nucleon, or neutrino sector, all of them confirming relativity.

Incorrect alternatives

As mentioned above, Michelson initially believed that his experiment would confirm Stokes' theory, according to which the aether was fully dragged in the vicinity of the earth. However, complete aether drag contradicts the observed aberration of light and was contradicted by other experiments as well. In addition, Lorentz showed in 1886 that Stokes's attempt to explain aberration is contradictory.

Furthermore, the assumption that the aether is not carried in the vicinity, but only within matter, was very problematic as shown by the Hammar experiment (1935). Hammar directed one leg of his interferometer through a heavy metal pipe plugged with lead. If aether were dragged by mass, it was theorized that the mass of the sealed metal pipe would have been enough to cause a visible effect. Once again, no effect was seen, so aether-drag theories are considered to be disproven.

Walther Ritz's emission theory (or ballistic theory) was also consistent with the results of the experiment, not requiring aether. The theory postulates that light has always the same velocity in respect to the source. However de Sitter noted that emitter theory predicted several optical effects that were not seen in observations of binary stars in which the light from the two stars could be measured in a spectrometer. If emission theory were correct, the light from the stars should experience unusual fringe shifting due to the velocity of the stars being added to the speed of the light, but no such effect could be seen. It was later shown by J. G. Fox that the original de Sitter experiments were flawed due to extinction, but in 1977 Brecher observed X-rays from binary star systems with similar null results. Also terrestrial tests using particle accelerators have been made that were inconsistent with source dependence of the speed of light. In addition, Emission theory might fail the Ives–Stilwell experiment, but Fox questioned that as well.

Subsequent experiments

Figure 8. Simulation of the Kennedy/Illingworth refinement of the Michelson–Morley experiment. (a) Michelson–Morley interference pattern in monochromatic mercury light, with a dark fringe precisely centered on the screen. (b) The fringes have been shifted to the left by 1/100 of the fringe spacing. It is extremely difficult to see any difference between this figure and the one above. (c) A small step in one mirror causes two views of the same fringes to be spaced 1/20 of the fringe spacing to the left and to the right of the step. (d) A telescope has been set to view only the central dark band around the mirror step. Note the symmetrical brightening about the center line. (e) The two sets of fringes have been shifted to the left by 1/100 of the fringe spacing. An abrupt discontinuity in luminosity is visible across the step.

 

Although Michelson and Morley went on to different experiments after their first publication in 1887, both remained active in the field. Other versions of the experiment were carried out with increasing sophistication. Morley was not convinced of his own results, and went on to conduct additional experiments with Dayton Miller from 1902 to 1904. Again, the result was negative within the margins of error.

Miller worked on increasingly larger interferometers, culminating in one with a 32-meter (105 ft) (effective) arm length that he tried at various sites, including on top of a mountain at the Mount Wilson Observatory. To avoid the possibility of the aether wind being blocked by solid walls, his mountaintop observations used a special shed with thin walls, mainly of canvas. From noisy, irregular data, he consistently extracted a small positive signal that varied with each rotation of the device, with the sidereal day, and on a yearly basis. His measurements in the 1920s amounted to approximately 10 km/s (6.2 mi/s) instead of the nearly 30 km/s (18.6 mi/s) expected from the Earth's orbital motion alone. He remained convinced this was due to partial entrainment or aether dragging, though he did not attempt a detailed explanation. He ignored critiques demonstrating the inconsistency of his results and the refutation by the Hammar experiment. Miller's findings were considered important at the time, and were discussed by Michelson, Lorentz and others at a meeting reported in 1928. There was general agreement that more experimentation was needed to check Miller's results. Miller later built a non-magnetic device to eliminate magnetostriction, while Michelson built one of non-expanding Invar to eliminate any remaining thermal effects. Other experimenters from around the world increased accuracy, eliminated possible side effects, or both. So far, no one has been able to replicate Miller's results, and modern experimental accuracies have ruled them out. Roberts (2006) has pointed out that the primitive data reduction techniques used by Miller and other early experimenters, including Michelson and Morley, were capable of creating apparent periodic signals even when none existed in the actual data. After reanalyzing Miller's original data using modern techniques of quantitative error analysis, Roberts found Miller's apparent signals to be statistically insignificant.

Using a special optical arrangement involving a 1/20 wave step in one mirror, Roy J. Kennedy (1926) and K.K. Illingworth (1927) (Fig. 8) converted the task of detecting fringe shifts from the relatively insensitive one of estimating their lateral displacements to the considerably more sensitive task of adjusting the light intensity on both sides of a sharp boundary for equal luminance. If they observed unequal illumination on either side of the step, such as in Fig. 8e, they would add or remove calibrated weights from the interferometer until both sides of the step were once again evenly illuminated, as in Fig. 8d. The number of weights added or removed provided a measure of the fringe shift. Different observers could detect changes as little as 1/300 to 1/1500 of a fringe. Kennedy also carried out an experiment at Mount Wilson, finding only about 1/10 the drift measured by Miller and no seasonal effects.

In 1930, Georg Joos conducted an experiment using an automated interferometer with 21-meter-long (69 ft) arms forged from pressed quartz having very low thermal coefficient of expansion, that took continuous photographic strip recordings of the fringes through dozens of revolutions of the apparatus. Displacements of 1/1000 of a fringe could be measured on the photographic plates. No periodic fringe displacements were found, placing an upper limit to the aether wind of 1.5 km/s (0.93 mi/s).

Recent experiments

Optical tests

Optical tests of the isotropy of the speed of light became commonplace. New technologies, including the use of lasers and masers, have significantly improved measurement precision. (In the following table, only Essen (1955), Jaseja (1964), and Shamir/Fox (1969) are experiments of Michelson–Morley type, i.e., comparing two perpendicular beams. The other optical experiments employed different methods.)

Author	Year	Description	Upper bounds
Louis Essen	1955	The frequency of a rotating microwave cavity resonator is compared with that of a quartz clock	~3 km/s
Cedarholm et al.	1958	Two ammonia masers were mounted on a rotating table, and their beams were directed in opposite directions.	~30 m/s
Mössbauer rotor experiments	1960–68	In a series of experiments by different researchers, the frequencies of gamma rays were observed using the Mössbauer effect.	~2.0 cm/s
Jaseja et al.	1964	The frequencies of two He–Ne masers, mounted on a rotating table, were compared. Unlike Cedarholm et al., the masers were placed perpendicular to each other.	~30 m/s
Shamir and Fox	1969	Both arms of the interferometer were contained in a transparent solid (plexiglas). The light source was a Helium–neon laser.	~7 km/s
Trimmer et al.	1973	They searched for anisotropies of the speed of light behaving as the first and third of the Legendre polynomials. They used a triangle interferometer, with one portion of the path in glass. (In comparison, the Michelson–Morley type experiments test the second Legendre polynomial)	~2.5 cm/s

Figure 9. Michelson–Morley experiment with cryogenic optical resonators of a form such as was used by Müller et al. (2003).

Recent optical resonator experiments

Over the last several years, there has been a resurgence in interest in performing precise Michelson–Morley type experiments using lasers, masers, cryogenic optical resonators, etc. This is in large part due to predictions of quantum gravity that suggest that special relativity may be violated at scales accessible to experimental study. The first of these highly accurate experiments was conducted by Brillet & Hall (1979), in which they analyzed a laser frequency stabilized to a resonance of a rotating optical Fabry–Pérot cavity. They set a limit on the anisotropy of the speed of light resulting from the Earth's motions of Δc/c ≈ 10⁻¹⁵, where Δc is the difference between the speed of light in the x- and y-directions.

As of 2009, optical and microwave resonator experiments have improved this limit to Δc/c ≈ 10⁻¹⁷. In some of them, the devices were rotated or remained stationary, and some were combined with the Kennedy–Thorndike experiment. In particular, Earth's direction and velocity (ca. 368 km/s (229 mi/s)) relative to the CMB rest frame are ordinarily used as references in these searches for anisotropies.

Author	Year	Description	Δc/c
Wolf et al.	2003	The frequency of a stationary cryogenic microwave oscillator, consisting of sapphire crystal operating in a whispering gallery mode, is compared to a hydrogen maser whose frequency was compared to caesium and rubidium atomic fountain clocks. Changes during Earth's rotation have been searched for. Data between 2001–2002 was analyzed.	${\displaystyle \lesssim 10^{-15}}$
Müller et al.	2003	Two optical resonators constructed from crystalline sapphire, controlling the frequencies of two Nd:YAG lasers, are set at right angles within a helium cryostat. A frequency comparator measures the beat frequency of the combined outputs of the two resonators.
Wolf et al.	2004	See Wolf et al. (2003). An active temperature control was implemented. Data between 2002–2003 was analyzed.
Wolf et al.	2004	See Wolf et al. (2003). Data between 2002–2004 was analyzed.
Antonini et al.	2005	Similar to Müller et al. (2003), though the apparatus itself was set into rotation. Data between 2002–2004 was analyzed.	${\displaystyle \lesssim 10^{-16}}$
Stanwix et al.	2005	Similar to Wolf et al. (2003). The frequency of two cryogenic oscillators was compared. In addition, the apparatus was set into rotation. Data between 2004–2005 was analyzed.
Herrmann et al.	2005	Similar to Müller et al. (2003). The frequencies of two optical Fabry–Pérot resonators cavities are compared – one cavity was continuously rotating while the other one was stationary oriented north–south. Data between 2004–2005 was analyzed.
Stanwix et al.	2006	See Stanwix et al. (2005). Data between 2004–2006 was analyzed.
Müller et al.	2007	See Herrmann et al. (2005) and Stanwix et al. (2006). Data of both groups collected between 2004–2006 are combined and further analyzed. Since the experiments are located at difference continents, at Berlin and Perth respectively, the effects of both the rotation of the devices themselves and the rotation of Earth could be studied.
Eisele et al.	2009	The frequencies of a pair of orthogonal oriented optical standing wave cavities are compared. The cavities were interrogated by a Nd:YAG laser. Data between 2007–2008 was analyzed.	${\displaystyle \lesssim 10^{-17}}$
Herrmann et al.	2009	Similar to Herrmann et al. (2005). The frequencies of a pair of rotating, orthogonal optical Fabry–Pérot resonators are compared. The frequencies of two Nd:YAG lasers are stabilized to resonances of these resonators.

Other tests of Lorentz invariance

Figure 10. ⁷Li-NMR spectrum of LiCl (1M) in D₂O. The sharp, unsplit NMR line of this isotope of lithium is evidence for the isotropy of mass and space.

 

Examples of other experiments not based on the Michelson–Morley principle, i.e., non-optical isotropy tests achieving an even higher level of precision, are Clock comparison or Hughes–Drever experiments. In Drever's 1961 experiment, ⁷Li nuclei in the ground state, which has total angular momentum J = 3/2, were split into four equally spaced levels by a magnetic field. Each transition between a pair of adjacent levels should emit a photon of equal frequency, resulting in a single, sharp spectral line. However, since the nuclear wave functions for different M_J have different orientations in space relative to the magnetic field, any orientation dependence, whether from an aether wind or from a dependence on the large-scale distribution of mass in space, would perturb the energy spacings between the four levels, resulting in an anomalous broadening or splitting of the line. No such broadening was observed. Modern repeats of this kind of experiment have provided some of the most accurate confirmations of the principle of Lorentz invariance.