光の片道速度

光速の語を使う時、片道速度と往復速度を区別する必要がある場合がある。ある光源から検出器までの片道速度は、光源と検出器それぞれの地点での時刻をどのように同期させるかについての規約（時計の同期方法に関する恣意的な決定）と独立に測定することができず、実際に実験的に測定されているのは光源から検出器までの往復速度である。アインシュタインは光の片道速度が往復速度と等しくなるような時計の同期法を選んだ（en:アインシュタインの同期法）。任意の慣性系において光の片道速度が一定であることが彼の特殊相対性理論の基礎となってはいるが、その理論の予言のうち実験的に検証可能であったものは全て、この同期法の選び方には依存していない。（つまり特殊相対性理論は証明されているが、その時計の同期法については任意性が残っている）

同期法に依存せず、直接に光の片道速度を測定しようと試みた実験がいくつかあるが、どれも成功には至っていない。 それらの実験は直接的には遅い時計輸送による同期法（slow clock-transport）がアインシュタインの同期法と等価であることを確立しており、これは特殊相対性理論の重要な特徴となっている。これらの実験は遅い時計輸送と等価であることが示されているため、光の片道速度の等方性を直接的に確立できていないが、ニュートン力学や慣性系の定義の仕方自体についても片道速度が等方的であることを仮定しているため、恣意性に関してはどれも同じ問題を抱えている。 一般に、これらの実験は、光の往復速度が等方的であり、かつ片道速度が非等方的である場合と整合可能であることが示されている。

この記事における「光速」とは全ての電磁波の真空中の速度のことをいう。

往復速度

マイケルソン干渉計における光の往復路

光の往復速度とは、ある点（光源）から鏡まで行って帰ってきた光の、平均速度である。光は同一地点から発し、また達するため、合計時間の測定には一つの時計しか必要としない。それゆえ、この速度は、いかなる時計の同期方法からも独立に、実験的に測定することができる。直線を往復する場合だけでなく、光が閉路（closed path）をつくるいかなる測定も、往復速度の測定をしていることになる。

マイケルソン・モーリーの実験やKennedy–Thorndikeの実験といった多くの特殊相対性理論の検証実験は、ある慣性系における光の往復速度は、等方的であり、閉路の取り方と独立である、ということを極めて高い精度で示してきた。マイケルソン・モーリー型の等方性に関する実験は、マイケルソン干渉計の全て腕が特定の周期を持っており、全体で相対的な方角の依存性を検証することができる光時計とみなせることから、「時計同期実験」とも呼ばれることがある。

1983年以来、光が真空中で1秒あたりに進む距離の1⁄299,792,458として1メートルが定義されてきた。これはつまり、光速はもはや国際単位系では実験的に測定できず、メートルの長さを他の長さの基準に対して比較することが必要であることを意味する。

片道速度

光行差を考える際の片道の光路

往復路にわたる平均速度は測定可能であるが、ある方向への片道速度については、二つの別々の地点での「同時」とは何であるかを定義しない限り、未定義のままである。光がある場所から別の場所へと進むのにかかった時間を測定するには、出発時刻と到着時刻を同じ時間尺度で測定する必要がある。これを実現するためには、二つの同期した時計を出発地点と到着地点それぞれに置くか、出発地点から到着地点まで瞬時に何らかの手段で出発時刻に関する信号を送るかのどちらかが必要となる。ある情報を、瞬間的に移送する手段は知られていない（光速を超える情報伝搬の手段自体、一般的には存在しないと考えられている）。それゆえ、片道の平均速度の測定値はいつも出発地点と到着地点の時計の同期に用いられた方法に依存しており、人間の側で恣意的に定義を決める話となっているのである。ローレンツ変換は、光の片道速度が慣性系の選び方と独立に測定されるように定義されている。

MansouriやSexl (1977)またClifford Will (1992)は、ある特定の（エーテル）座標系Σに対する相対的な方向依存性の変化を考えるなどすれば、この問題は光の片道速度の等方性の測定に影響しないと主張した。彼らの分析はRMS検証理論の、光の片道路を測る実験や遅い時計輸送の実験との関係における特定の解釈に基づいている。Willは同期法なしに光の光行時間を用いて二つの時計の間の片道速度を測定することは不可能であるとしているが、「...伝搬路の方向がΣ系に対して変化するときの、同じ二つの時計の間の速度の等方性の検証は、それらがどのように同期されたかによらないはずである」と主張している。彼はアドホックな仮説を紹介することによって、エーテル理論だけが相対性と整合をとることができると加えている。 また最近の論文(2005, 2006)でWillはそれらの実験を「片道伝搬を用いた光速の等方性」を測定するものと呼んでいる。

しかし他の、Zhang (1995, 1997) やAnderson et al. (1998)などは、この解釈が誤りであると示している。例えば、Anderson et al.は、ある特定の座標系を選ぶ時点で既に同時性についての恣意的決定がなされており、その座標系における光の片道速度や他の速度の等方性に関する全ての仮定もまた恣意的決定であることを指摘している。それゆえ、RMS理論はローレンツ不変性と光の往復速度について分析するのに有用な検証理論に留まっており、光の片道速度についてはそうではない。彼らは「...光の片道速度の等方性については、同一の実験内で、原理的には少なくとも片道速度の数値を導出しなくては検証する望みがないが、それは同期法に関する恣意性と矛盾することになる」と結論付けている。 ローレンツ変換の、片道速度に関する非等方性を考慮した一般化を用いて、ZhangとAndersonはローレンツ変換と光の片道速度の等方性に整合する全ての事象と実験結果が、光の往復速度の一定性と等方性を保ったまま、片道速度の非等方性を許すものとも整合することを指摘した。

同期法の規約

The way in which distant clocks are synchronized can have an effect on all time-related measurements over distance, such as speed or acceleration measurements. In isotropy experiments, simultaneity conventions are often not explicitly stated but are implicitly present in the way coordinates are defined or in the laws of physics employed.

Einstein convention

→「Einstein synchronization」も参照

This method synchronizes distant clocks in such a way that the one-way speed of light becomes equal to the two-way speed of light. If a signal sent from A at time $t_{1}$ is arriving at B at time $t_{2}$ and coming back to A at time $t_{3}$, then the following convention applies:

$t_{2}=t_{1}+{\tfrac {1}{2}}\left(t_{3}-t_{1}\right)$.

The details of this method, and the conditions that assure its consistency are discussed in Einstein synchronization.

Slow clock-transport

It is easily demonstrated that if two clocks are brought together and synchronized, then one clock is moved rapidly away and back again, the two clocks will no longer be synchronized due to time dilation. This was measured in a variety of tests and is related to the twin paradox.

However, if one clock is moved away slowly in frame S and returned the two clocks will be very nearly synchronized when they are back together again. The clocks can remain synchronized to an arbitrary accuracy by moving them sufficiently slowly. If it is taken that, if moved slowly, the clocks remain synchronized at all times, even when separated, this method can be used to synchronize two spatially separated clocks. In the limit as the speed of transport tends to zero, this method is experimentally and theoretically equivalent to the Einstein convention. Though the effect of time dilation on those clocks cannot be neglected anymore when analyzed in another relatively moving frame S'. This explains why the clocks remain synchronized in S, whereas they are not synchronized anymore from the viewpoint of S', establishing relativity of simultaneity in agreement with Einstein synchronization. Therefore, testing the equivalence between these clock synchronization schemes is important for special relativity, and some experiments in which light follows a unidirectional path have proven this equivalence to high precision.

Non-standard synchronizations

As demonstrated by Hans Reichenbach and Adolf Grünbaum, Einstein synchronization is only a special case of a more broader synchronization scheme, which leaves the two-way speed of light invariant, but allows for different one-way speeds. The formula for Einstein synchronization is modified by replacing ½ with ε:

$t_{2}=t_{1}+\varepsilon \left(t_{3}-t_{1}\right).$

ε can have values between 0 and 1. It was shown that this scheme can be used for observationally equivalent reformulations of the Lorentz transformation, see Generalizations of Lorentz transformations with anisotropic one-way speeds.

As required by the experimentally proven equivalence between Einstein synchronization and slow clock-transport synchronization, which requires knowledge of time dilation of moving clocks, the same non-standard synchronisations must also affect time dilation. It was indeed pointed out that time dilation of moving clocks depends on the convention for the one-way velocities used in its formula. That is, time dilation can be measured by synchronizing two stationary clocks A and B, and then the readings of a moving clock C are compared with them. Changing the convention of synchronization for A and B makes the value for time dilation (like the one-way speed of light) directional dependent. The same conventionality also applies to the influence of time dilation on the Doppler effect. Only when time dilation is measured on closed paths, it is not conventional and can unequivocally be measured like the two-way speed of light. Time dilation on closed paths was measured in the Hafele–Keating experiment and in experiments on the Time dilation of moving particles such as Bailey et al. (1977). Thus the so-called twin paradox occurs in all transformations preserving the constancy of the two-way speed of light.

Inertial frames and dynamics

It was argued against the conventionality of the one-way speed of light that this concept is closely related to dynamics, the laws of motion and inertial reference frames. Salmon described some variations of this argument using momentum conservation, from which it follows that two equal bodies at the same place which are equally accelerated in opposite directions, should move with the same one-way velocity. Similarly, Ohanian argued that inertial reference frames are defined so that Newton's laws of motion hold in first approximation. Therefore, since the laws of motion predict isotropic one-way speeds of moving bodies with equal acceleration, and because of the experiments demonstrating the equivalence between Einstein synchronization and slow clock-transport synchronization, it appears to be required and directly measured that the one-way speed of light is isotropic in inertial frames. Otherwise, both the concept of inertial reference frames and the laws of motion must be replaced by much more complicated ones involving anisotropic coordinates.

However, it was shown by others that this is principally not in contradiction with the conventionality of the one-way speed of light. Salmon argued that momentum conservation in its standard form assumes isotropic one-way speed of moving bodies from the outset. So it involves practically the same convention as in the case of isotropic one-way speed of light, thus using this as an argument against light speed conventionality would be circular. And in response to Ohanian, both Macdonald and Martinez argued that even though the laws of physics become more complicated with non-standard synchrony, they still are a consistent way to describe the phenomena. They also argued that it's not necessary to define inertial frames in terms of Newton's laws of motion, because other methods are possible as well. In addition, Iyer and Prabhu distinguished between "isotropic inertial frames" with standard synchrony and "anisotropic inertial frames" with non-standard synchrony.

Experiments which appear to measure the one-way speed of light

Experiments which claimed to use a one-way light signal

The Greaves, Rodriguez and Ruiz-Camacho experiment

In the October 2009 issue of the American Journal of Physics Greaves, Rodriguez and Ruiz-Camacho reported a new method of measurement of the one-way speed of light. In the June 2013 issue of the American Journal of Physics Hankins, Rackson and Kim repeated the Greaves et al. experiment obtaining with greater accuracy the one way speed of light. This experiment proves with greater accuracy that the signal return path to the measuring device has a constant delay, independent of the end point of the light flight path, allowing measurement of the time of flight in a single direction.

J. Finkelstein claimed that the Greaves et al. experiment actually measures the round trip (two-way) speed of light.

In the November issue of the Indian Journal of Physics, Ahmed et al. published a comprehensive review of One-Way and Two-Way Experiments to test the isotropy of the speed of light.

Experiments in which light follows a unidirectional path

List of eclipses of the Jupiter moon Io that formed the basis of Ole Rømer's discovery of the finite speed of light.

Many experiments intended to measure the one-way speed of light, or its variation with direction, have been (and occasionally still are) performed in which light follows a unidirectional path. Claims have been made that those experiments have measured the one-way speed of light independently of any clock synchronisation convention, but they have all been shown to actually measure the two-way speed, because they are consistent with generalized Lorentz transformations including synchronizations with different one-way speeds on the basis of isotropic two-way speed of light (see sections the one-way speed and generalized Lorentz transformations).

These experiments also confirm agreement between clock synchronization by slow transport and Einstein synchronization. Even though some authors argued that this is sufficient to demonstrate the isotropy of the one-way speed of light, it has been shown that such experiments cannot, in any meaningful way, measure the (an)isotropy of the one way speed of light unless inertial frames and coordinates are defined from the outset so that space and time coordinates as well as slow clock-transport are described isotropically (see sections inertial frames and dynamics and the one-way speed). Regardless of those different interpretations, the observed agreement between those synchronization schemes is an important prediction of special relativity, because this requires that transported clocks undergo time dilation (which itself is synchronization dependent) when viewed from another frame (see sections Slow clock-transport and Non-standard synchronizations).

The JPL experiment

This experiment, carried out in 1990 by the NASA Jet Propulsion Laboratory, measured the time of flight of light signals through a fibre optic link between two hydrogen maser clocks. In 1992 the experimental results were analysed by Clifford Will who concluded that the experiment did actually measure the one-way speed of light.

In 1997 the experiment was re-analysed by Zhang who showed that, in fact, only the two-way speed had been measured.

Rømer's measurement

The first experimental determination of the speed of light was made by Ole Christensen Rømer. It may seem that this experiment measures the time for light to traverse part of the Earth's orbit and thus determines its one-way speed, however, this experiment was carefully re-analysed by Zhang, who showed that the measurement does not measure the speed independently of a clock synchronization scheme but actually used the Jupiter system as a slowly-transported clock to measure the light transit times.

The Australian physicist Karlov also showed that Rømer actually measured the speed of light by implicitly making the assumption of the equality of the speeds of light back and forth.

Other experiments comparing Einstein synchronization with slow clock-transport synchronization

Experiments	Year		$\Delta c/c$
Moessbauer rotor experiments	1960s	Gamma radiation was sent from the rear of a rotating disc into its center. It was expected that anisotropy of the speed of light would lead to Doppler shifts.	$<3\times 10^{-8}$
Vessot et al.	1980	Comparing the times-of-flight of the uplink- and downlink signal of Gravity Probe A.	$\sim 10^{-8}\!$
Riis et al.	1988	Comparing the frequency of two-photon absorption in a fast particle beam, whose direction was changed relative to the fixed stars, with the frequency of a resting absorber.	$<3\times 10^{-9}$
Nelson et al.	1992	Comparing the frequencies of a hydrogen maser clock and laser light pulses. The path length was 26 km.	$<1.5\times 10^{-6}$
Wolf & Petit	1997	Clock comparisons between hydrogen maser clocks on the ground and cesium and rubidium clocks on board 25 GPS satellites.	$<5\times 10^{-9}$

Experiments that can be done on the one-way speed of light

Artist's illustration of a bright Gamma-ray burst. Measurements on light from such objects were used to show that the one-way speed of light does not vary with frequency.

Although experiments cannot be done in which the one-way speed of light is measured independently of any clock synchronization scheme, it is possible to carry out experiments that measure a change in the one-way speed of light due, for example, to the motion of the source. Such experiments are the De Sitter double star experiment (1913), conclusively repeated in the x-ray spectrum by K. Brecher in 1977; or the terrestrial experiment by Alväger, et al. (1963); they show that, when measured in an inertial frame, the one-way speed of light is independent of the motion of the source within the limits of experimental accuracy. In such experiments the clocks may be synchronized in any convenient way, since it is only a change of speed that is being measured.

Observations of the arrival of radiation from distant astronomical events have shown that the one-way speed of light does not vary with frequency, that is, there is no vacuum dispersion of light. Similarly, differences in the one-way propagation between left- and right-handed photons, leading to vacuum birefringence, were excluded by observation of the simultaneous arrival of distant star light. For current limits on both effects, often analyzed with the Standard-Model Extension, see Vacuum dispersion and Vacuum birefringence.

Experiments on two-way and one-way speeds using the Standard-Model Extension

→詳細は「Modern searches for Lorentz violation」を参照

While the experiments above were analyzed using generalized Lorentz transformations as in the Robertson–Mansouri–Sexl test theory, many modern tests are based on the Standard-Model Extension (SME). This test theory includes all possible Lorentz violations not only of special relativity, but of the Standard Model and General relativity as well. Regarding the isotropy of the speed of light, both two-way and one-way limits are described using coefficients (3x3 matrices):

• ${\tilde {\kappa }}_{e-}$ representing anisotropic shifts in the two-way speed of light,
• ${\tilde {\kappa }}_{o+}$ representing anisotropic differences in the one-way speed of counterpropagating beams along an axis,
• ${\tilde {\kappa }}_{tr}$ representing isotropic (orientation independent) shifts in the one-way phase velocity of light.

A series of experiments have been (and still are) performed since 2002 testing all of those coefficients using, for instance, symmetric and asymmetric optical resonators. No Lorentz violations have been observed as of 2013, providing current upper limits for Lorentz violations: ${\tilde {\kappa }}_{e-}=\scriptstyle (-0.31\pm 0.73)\times 10^{-17}$, ${\tilde {\kappa }}_{o+}=\scriptstyle 0.7\pm 1\times 10^{-14}$, and ${\tilde {\kappa }}_{tr}=\scriptstyle -0.4\pm 0.9\times 10^{-10}$. For details and sources see Modern searches for Lorentz violation#Speed of light.

However, the partially conventional character of those quantities was demonstrated by Kostelecky et al, pointing out that such variations in the speed of light can be removed by suitable coordinate transformations and field redefinitions. Though this doesn't remove the Lorentz violation per se, since such a redefinition only transfers the Lorentz violation form the photon sector to the matter sector of SME, thus those experiments remain valid tests of Lorentz invariance violation. There are one-way coefficients of the SME that cannot be redefined into other sectors, since different light rays from the same distance location are directly compared with each other, see the previous section.

Theories in which the one-way speed of light is not equal to the two-way speed

Theories equivalent to special relativity

Lorentz ether theory

→「Lorentz ether theory」も参照

Hendrik Antoon Lorentz

In 1904 and 1905, Hendrik Lorentz and Henri Poincaré proposed a theory which explained this result as being due the effect of motion through the aether on the lengths of physical objects and the speed at which clocks ran. Due to motion through the aether objects would shrink along the direction of motion and clocks would slow down. Thus, in this theory, slowly transported clocks do not, in general, remain synchronized although this effect cannot be observed. The equations describing this theory are known as the Lorentz transformations. In 1905 these transformations became the basic equations of Einstein's special theory of relativity which proposed the same results without reference to an aether.

In the theory, the one-way speed of light is principally only equal to the two-way speed in the aether frame, though not in other frames due to the motion of the observer through the aether. However, the difference between the one-way and two-way speeds of light can never be observed due to the action of the aether on the clocks and lengths. Therefore, the Poincaré-Einstein convention is also employed in this model, making the one-way speed of light isotropic in all frames of reference.

Even though this theory is experimentally indistinguishable from special relativity, Lorentz's theory is no longer used for reasons of philosophical preference and because of the development of general relativity.

Generalizations of Lorentz transformations with anisotropic one-way speeds

A sychronisation scheme proposed by Reichenbach and Grünbaum, which they called ε-synchronization, was further developed by authors such as Edwards (1963), Winnie (1970), Anderson and Stedman (1977), who reformulated the Lorentz transformation without changing its physical predictions. For instance, Edwards replaced Einstein's postulate that the one-way speed of light is constant when measured in an inertial frame with the postulate:

The two way speed of light in a vacuum as measured in two (inertial) coordinate systems moving with constant relative velocity is the same regardless of any assumptions regarding the one-way speed.

So the average speed for the round trip remains the experimentally verifiable two-way speed, whereas the one-way speed of light is allowed to take the form in opposite directions:

$c_{\pm }={\frac {c}{1\pm \kappa }}.$

κ can have values between 0 and 1. In the extreme as κ approaches 1, light might propagate in one direction instantaneously, provided it takes the entire round-trip time to travel in the opposite direction. Following Edwards and Winnie, Anderson et al. formulated generalized Lorentz transformations for arbitrary boosts of the form:

${\begin{aligned}d{\tilde {t}}'=&{\tilde {\gamma }}\left[1+\kappa \cdot {\tilde {\mathbf {v} }}/c-\kappa '\cdot {\tilde {\mathbf {v} }}'/c\right]d{\tilde {t}}-\left(\kappa '+{\tilde {\gamma }}{\tilde {\mathbf {v} }}'\right)\cdot d{\tilde {\mathbf {x} }}/c\\&-\left[{\tilde {\gamma }}\left(1+\kappa \cdot {\tilde {\mathbf {v} }}/c\right)-1\right]{\frac {\kappa '\cdot {\tilde {\mathbf {v} }}}{{\tilde {\mathbf {v} }}^{2}c}}{\tilde {\mathbf {v} }}\cdot d{\tilde {\mathbf {x} }}+{\tilde {\gamma }}\kappa \cdot {\tilde {\mathbf {v} }}\left(\kappa \cdot d{\tilde {\mathbf {x} }}\right)/c,\\d{\tilde {\mathbf {x} }}'=&-{\tilde {\gamma }}{\tilde {\mathbf {v} }}d{\tilde {t}}+d{\tilde {\mathbf {x} }}+\left[{\tilde {\gamma }}\left(1+\kappa \cdot {\tilde {\mathbf {v} }}/c\right)-1\right]{\frac {{\tilde {\mathbf {v} }}\cdot d\mathbf {x} }{{\tilde {\mathbf {v} }}^{2}}}{\tilde {\mathbf {v} }}-{\tilde {\gamma }}{\tilde {\mathbf {v} }}\left(\kappa \cdot d{\tilde {\mathbf {x} }}\right)/c,\\{\tilde {\gamma }}=&\gamma \left(1-\kappa \cdot \mathbf {v} /c\right),\\{\tilde {\mathbf {v} }}=&{\frac {\mathbf {v} }{1-\kappa \cdot \mathbf {v} /c}},\end{aligned}}$

(with κ and κ' being the synchrony vectors in frames S and S', respectively). This transformation indicates the one-way speed of light is conventional in all frames, leaving the two-way speed invariant. κ=0 means Einstein synchronization which results in the standard Lorentz transformation. As shown by Edwards, Winnie and Mansouri-Sexl, by suitable rearrangement of the synchrony parameters even some sort of "absolute simultaneity" can be achieved, in order to simulate the basic assumption of Lorentz ether theory. That is, in one frame the one-way speed of light is chosen to be isotropic, while all other frames take over the values of this "preferred" frame by "external synchronization".

All predictions derived from such a transformation are experimentally indistinguishable from those of the standard Lorentz transformation; the difference is only that the defined clock time varies from Einstein's according to the distance in a specific direction.

Theories not equivalent to special relativity

Test theories

→「Test theory」および「Test theories of special relativity」も参照

A number of theories have been developed to allow assessment of the degree to which experimental results differ from the predictions of relativity. These are known as test theories and include the Robertson and Mansouri-Sexl (RMS) theories. To date, all experimental results agree with special relativity within the experimental uncertainty.

Another test theory is the Standard-Model Extension (SME). It employs a broad variety of coefficients indicating Lorentz symmetry violations in special relativity, general relativity, and the Standard Model. Some of those parameters indicate anisotropies of the two-way and one-way speed of light. However, it was pointed out that such variations in the speed of light can be removed by suitable redefinitions of the coordinates and fields employed. Though this doesn't remove Lorentz violations per se, it only shifts their appearance from the photon sector into the matter sector of SME (see above Experiments on two-way and one-way speeds using the Standard-Model Extension.

Aether theories

→「Luminiferous aether」も参照

Before 1887 it was generally believed that light travelled as a wave at a constant speed relative to the hypothesised medium of the aether. For an observer in motion with respect to the aether, this would result in slightly different two-way speeds of light in different directions. In 1887, the Michelson–Morley experiment showed that the two-way speed of light was constant regardless of direction or motion through the aether. At the time, the obvious explanation for this effect was that objects in motion through the aether experience the combined effects of time dilation and length contraction in the direction of motion.

Preferred reference frame

A preferred reference frame is a reference frame in which the laws of physics take on a special form. The ability to make measurements which show the one-way speed of light to be different from its two-way speed would, in principle, enable a preferred reference frame to be determined. This would be the reference frame in which the two-way speed of light was equal to the one-way speed.

In Einstein's special theory of relativity, all inertial frames of reference are equivalent and there is no preferred frame. There are theories, such as Lorentz ether theory that are experimentally and mathematically equivalent to special relativity but have a preferred reference frame. In order for these theories to be compatible with experimental results the preferred frame must be undetectable. In other words, it is a preferred frame in principle only, in practice all inertial frames must be equivalent, as in special relativity.

Further reading

• Janis, Allen (2010). "Conventionality of Simultaneity". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (英語).
• Mathpages: Conventional Wisdom, Round Trips and One-Way Speeds, Teaching Special Relativity
• Rizzi, Guido; Ruggiero, Matteo Luca; Serafini, Alessio (2004). “Synchronization Gauges and the Principles of Special Relativity”. Foundations of Physics 34 (12): 1835–1887. arXiv:gr-qc/0409105. Bibcode: 2004FoPh...34.1835R. doi:10.1007/s10701-004-1624-3. 
• Sonego, Sebastiano; Pin, Massimo (2008). “Foundations of anisotropic relativistic mechanics”. Journal of Mathematical Physics 50 (4): 042902-1–042902-28. arXiv:0812.1294. Bibcode: 2009JMP....50d2902S. doi:10.1063/1.3104065.