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∓b ∆c± = c. (4) 1 ∓ b

For b ≪ 1 the change in the speed of light due to transformation (2) becomes:

∆c± = ∓bc. (5)

Whereas transformation (1) introduces spatial anisotropy into the propagation of light, transformation (2) does not. According to the conventionality of simultaneity, the one-way velocity of light cannot be measured. Therefore, by combining thesis of the conventionality of simultaneity with the Hugyens-Fresnel-Miller model of wave propagation, we are led to the following fundamental invariance principle: The laws of physics are invariant under transformation (2) with b constant, namely, the laws of physics are invariant under spherically symmetric transformations of the one-way speed of light at each point in space. Note that r is not a global variable, but is defined only locally in the local coordinate systems that define the propagation from each spherical point source. As is well known, the Einstein-Cartan-Kibble-Sciama (ECKS) theory is a gauge theory of the Poincar´e group [11] (see also [12]). The traditional theory of is a subset of ECKS that follows naturally by gauging only the translation group. The additional spin coefficients emerge in ECKS because of the invariance of physical laws under local Lorentz rotations of the vierbein fields. Similarly, the thesis of the conventionality of simultaneity in the context of the Huygens-Fresnel-Miller model of wave propagation forces us to introduce a new gauge field, Bµ, that preserves the local synchrony transformations when b becomes an arbitrary function of the space-time variables. We consider a Lagrangian that is a function of a set of field variables, χ(xµ), and the coordinates xµ:

µ µ L ≡ L {χ(x ),χ,µ, x } , (6)

where χ,µ ≡ ∂µχ. The variations of the coordinates and field variables under an infinitesimal transformation are:

xµ → x′µ = xµ + δxµ χ(xµ) → χ′(x′µ)= χ(xµ)+ δχ(xµ). (7)

We consider infinitesimal synchrony transformations:

′µ µ δx = δ0 br δχ = bW χ, (8)

where b represents a real infinitesimal parameter and W is the generator of the spherically-symmetric syn- chrony group. According to the gauge prescription one assumes the action is invariant under a transformation group for constant parameters and then covariant derivatives are introduced to retain invariance when the parameters of the group become arbitrary functions of the coordinates. To preserve invariance of the action under

2 generalized synchrony transformations, we must replace the derivative χ,µ with a covariant derivative, χ;µ, according to: χ;µ ≡ χ,µ + BµW χ, (9) This new field transforms under synchrony transformations as:

δBµ = −b,µ (10) to retain invariance of the action when the parameters of the synchrony group become arbitrary functions of the coordinates. We write the Lagrangian for the free fields as 1 L0 = − F0, (11) 4 µν where F0 = Fµν F and Fµν = Bµ,ν − Bν,µ is calculated from the commutator of b-covariant derivatives. This produces the following field equations: µν µ F ;ν = J , (12) µ where J ≡−∂L/∂Bµ and L is the Lagrangian that is a function of a set of fields. We identify Bµ as being proportional to the electromagnetic potentials, φµ:

Bµ = αφµ, (13)

where α is a constant. Whereas Kibble demonstrated the fundamental relationship between the gravitational field and the in- variance properties of the Lagrangian under the 10-parameter Poincar´egroup, we see that the electromag- netic field follows from the 5-parameter group that includes space-time translations and local, spherically- symmetric synchrony transformations. In the same way that Einstein’s serves as the physical guide that yields torsion in the Einstein-Cartan-Kibble theory, the Huygen’s-Fresnel-Miller principal along with the conventionality of simultaneity serves as the physical guide that produces the electromagnetic field in a space-time theory that is invariant under local translations. Note that this prescription successfully combines the gravitational field with the electromagnetic field, but does not incorporate the spin coefficients since Lorentz transformations do not form a group with synchrony transformations. We conclude that the electromagnetic field modifies the one-way speed of light. Whereas the one-way speed of light remains unobservable, the relative change in the one-way speed of light that results from the presence of non-zero electromagnetic potentials can be observed. For example, the presence of non-zero electromagnetic potentials can account for many of the observed phenomena that variable speed of light (VSL) theories have attempted to explain. Indeed, a non-zero electric potential in the early universe can account for a wide range of observed phenomena consistent with VSL theories. Let us consider a simple experiment in which light propagates along a length L from an initial point P to a final point Q. We are free to choose a synchrony convention (Einstein’s convention) such that the one-way speed of light from P to Q is equal to the two-way speed of light. Once this synchronization convention is established we introduce an electromagnetic field, φµ. The change in the speed of light due to the introduction of the electromagnetic field is given by:

µ αφµ∆x ∆c = µ c. (14) 1 − αφµ∆x

For example, consider the introduction of a constant electric potential, φ0, in a region of space for which Einstein synchronization has been established. Whereas the one-way speed of light is unobservable and defined only by the adopted convention, the relative change in the one-way speed of light when a non-zero constant electric potential φ0 is introduced can be measured: αφ ∆t ∆c = 0 c. (15) 1 − αφ0∆t

For a weak field, ∆c = αφ0∆tc. The measurement of the relative change in the speed of light due to the introduction of an electromagnetic field in a region of space for which a synchronization convention has been

3 established can serve to verify the proposed relationship between the electromagnetic potentials and local spherically symmetric synchrony transformations as well as to determine the value of the proportionality constant α in Equation (13). Let us also consider the case where two separate light beams propagate in vacuum and combine after traveling equal distances. We further assume that along the first path a constant and weak electric potential φ0 is introduced. Whereas the associated electric field vanishes, φ0 will manifest itself by producing an evolving interference pattern due to the changing relative speed of the wavefront along the path subject to φ0, according to: ∆c = αcφ0. (16) ∆t References

[1] Allen Janis. Conventionality of simultaneity. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Fall 2014 edition, 2014. [2] M. Reichenbach and J. Freund. The Philosophy of Space and Time. Dover Publications, New York, 1958. [3] H. Grunbaum. Philosophical Problems of Space and Time, volume 12 of Boston Studies in the Philosophy of Science. Dordrecht, Boston, 2nd enlarged edition, 1973. [4] Y. Zhang. Special Relativity and Its Experimental Foundations. World Scientific, Singapore, 1997. [5] Ronald Anderson, E. Stedman, Geoffrey, and I. Vetharaniam. Conventionality of synchronisation, gauge dependence and test theories of relativity. Phys. Rept., 295:93–180, 1998. [6] W. Salmon. The philosophical significance of the one-way speed of light. Noˆus, 11:253–292, 1977. [7] D. Malament. Causal theories of time and the conventionality of simultaniety. Noˆus, 11:293–300, 1977. [8] H. Ohanian. The role of dynamics in the synchronization problem. American Journal of Physics, 72:141–148, 2004. [9] A. Einstein. On the electrodynamics of moving bodies. Annalen der Physik, 17, 1905. [10] David A. B. Miller. Huygens’s wave propagation principle corrected. Opt. Lett., 16(18):1370–1372, Sep 1991. [11] T. W. B. Kibble. Lorentz invariance and the gravitational field. Journal of Mathematical Physics, 2(2):212–221, February 1961. [12] R. Utiyama. Invariant theoretical interpretation of interaction. Physical Review, 101(5):1597–1607, march 1956.

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