One-way speed of light

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 ${\displaystyle t_{1}}$ is arriving at B at time ${\displaystyle t_{2}}$ and coming back to A at time ${\displaystyle t_{3}}$, then the following convention applies:

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

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

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. This effect is due to time dilation. This was measured in a variety of tests and is related to the twin paradox.[14][15]

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.[4] 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.[16] 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.

As demonstrated by Hans Reichenbach and Adolf Grünbaum, Einstein synchronization is only a special case of a 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:[4]

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

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.[17] 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.[18] 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).[19] Thus the so-called twin paradox occurs in all transformations preserving the constancy of the two-way speed of light.

In the October 2009 issue of the American Journal of Physics Greaves, Rodriguez and Ruiz-Camacho proposed a new method of measurement of the one-way speed of light.[26] In the June 2013 issue of the American Journal of Physics Hankins, Rackson and Kim repeated the Greaves et al. experiment intending to obtain with greater accuracy the one way speed of light.[27] This experiment assumed 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 showed that the Greaves et al. experiment actually measures the round trip (two-way) speed of light.[28]

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.[29] 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).[1]

These experiments also confirm agreement between clock synchronization by slow transport and Einstein synchronization.[2] Even though some authors argued that this is sufficient to demonstrate the isotropy of the one-way speed of light,[10][11] 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[2] (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).

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.[30] In 1992 the experimental results were analysed by Clifford Will who concluded that the experiment did actually measure the one-way speed of light.[11]

In 1997 the experiment was re-analysed by Zhang who showed that, in fact, only the two-way speed had been measured.[31]

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.[32]

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.[33][34]

Hendrik Antoon Lorentz

In 1904 and 1905, Hendrik Lorentz and Henri Poincaré proposed a theory which explained this result as being due to 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.

A sychronisation scheme proposed by Reichenbach and Grünbaum, which they called ε-synchronization, was further developed by authors such as Edwards (1963),[47] Winnie (1970),[17] Anderson and Stedman (1977), who reformulated the Lorentz transformation without changing its physical predictions.[1][2] 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.[47]

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:

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

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:[2]

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

(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".[9]

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.[48]

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[9] (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.[43]