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The redshift in Hubble's constant

A generalisation to electrodynamics and Yang-Mills theory is presented that permits computation of the speed of light. The model presented herewithin indicates that the speed of light in vacuo is not a universal constant. This may be relevant to the current debate in astronomy over large values of the Hubble constant obtained through the use of the latest generation of ground and space-based telescopes. An experiment is proposed based on Compton scattering to measure deviations in the speed of light.


Introduction
Astronomers have recently reported on observations for the Hubble constant that predict an age for the universe younger than the estimated age of some globular clusters [11][4] [14]. The obvious tension contained in this result has been christened the 'age crisis'. There are three possibilities of course: the measurements made for the Hubble constant are incorrect; current models for stellar evolution are incorrect; or, there is new physics to be understood. Without further observation it is too early to view favourably recent experiments for the Hubble constant. And as noted by Sandage, if the results are supported by further experimentation, it must be understood why previous estimates for the Hubble constant based on observations of type Ia supernovae are only half as large. It may be that models for stellar evolution need to be re-evaluated as well.
Over the course of the last fifteen years, observation has uncovered deviations in the Hubble constant. The Hubble relation turns sharply upwards from linearity on a redshift vs. distance plot [8] [13]. Refinements to the distance scale achieved by [11][4] [14] cannot of course account for the non-linearity in the Hubble constant. Sandage has argued that the non-linearity is due to a bias in the choice of objects chosen by astronomers for study, and that at large distances such a bias must be filtered out in order to correctly determine the Hubble constant. Such is the current debate.
Another hypothesis is suggested by the deviations in the Hubble constant: the speed of light in vacuum at space-time points remote from Earth is less than the present, terrestial speed of light. For suppose that the speed of light in a vacuum were not an absolute constant, then the red-shift would need to be reassessed. The redshift is given by where we assume that the velocity of the emitter v ≪c, the speed of light when the photon is emitted. If the speed of light at the time of emission is smaller than the present speed of light on earth, then the observed red-shift in (1) would be greater. This would make the Hubble constant larger, and thereby make the universe appear to be younger than it actually is. A lower speed of light in the early universe might therefore be helpful in understanding the purported 'age crisis'. In addition, one notes that under these circumstances the enormous powers associated with quasars would be reduced. In the next section we present a model that permits us to derive the speed of light in a natural way. In section three we propose an experiment based on Compton scattering that might detect variations in the speed of light.

A model
Let π : P → M be a principal U (2)-bundle over an oriented, compact, connected four-manifold M , and denote by E the associated rank two adjoint vector bundle. A(P ) is the space of connections on P . Let A, B ∈ A(P ), and introduce local coordinate charts with indices µ = 0, . . . , 3 on M . The Lie algebra-valued connections or vector potentials, A µ and B µ , induce exterior covariant deriva- In this way H A and K B are two-forms on M taking values in E, that is H A , K B ∈ Λ 2 (M, E). The Lagrangian Action that forms the basis of our model is given by where H A and K B are gauge field curvatures over the four-manifold, M . I E denotes the identity transformation on the adjoint bundle, E, and the bundle inner product is represented by < >. The Killing-Cartan inner product can be adopted. The inner product is normalized so that < I 2 E >= 1. The form of the Lagrangian (2) generalises the topological gauge field theories studied some time ago by Horowitz [6]. In local space-time coordinates and using the Killing-Cartan inner product the Lagrangian Action can be written explicitly The generators of the Lie algebra are denoted by T a . For gauge groups where tr(T a ) = 0, the Lagrangian (3) reduces to the second integral-these are the topological field theories studied by Baulieu and Singer [2]. The variational field equations for the Lagrangian (2) are The field equations are clearly independent of any metric structure on M . The set of solutions to (4) is clearly not trivial, because when A = B the field equations reduce to the Bianchi identities.
Observe that when a space-time metric is placed on M (by whatever means) and used to define the Hodge star operator, * , the topological field equations (4) become the source-free Yang-Mills or electrodynamic field equations when the gauge field B is chosen so that K B = * H A . The topological field theory above therefore, is seen to contain Yang-Mills theory and electromagnetism. It is for this reason that we defined a field theory with two vector potentials, A and B, and not one.
We turn now to the Bogomol'nyi structure. By completing the square, the Lagrangian (2) can be rewritten as The Lagrangian (5) is now in Bogomol'nyi form. The Bogomol'nyi equations arising from (5) are By an index computation equations (6) imply that H A and K B are projectively flat, where F is a real-valued two form on M . Clearly solutions to the Bogomol'nyi equations (6) automatically satisfy the variational field equations (4) when F is closed.
Let E A and E B be the adjoint vector bundles equipped with D A or D B , respectively. E * is the dual bundle to E. The curvature of the tensor product bundle E A ⊗ E * B is given by [7] In view of this, the Bogomol'nyi equations in (6) are seen to be a vanishing curvature condition on the tensor product bundle E A ⊗ E * B , and implies that The second integral in (5) can also be written as a characteristic class, The second Chern character is denoted by ch 2 (E). It is clear that under a perturbation of the vector potentials both integrals in (5) are invariant, and when the Bogomol'nyi equations are satisfied the Lagrangian is proportional to the characteristic class (9). Non-singular solutions to the Bogomol'nyi equations are non-trivial and stable when (9) is non-vanishing.
We have therefore identified a class of non-singular, finite-energy, stable solutions to the variational field equations (4). However, most solutions to the Bogomol'nyi equations do not appear to be 'particle-like'. Thus within the class of Bogomol'nyi solutions we must now define the 'particle-like' solutions to the field equations: solutions to the Bogomol'nyi equations (7) will be said to be 'particle-like' if together they form a phase space manifold that is nonsingular, Hausdorff, and of finite dimension. This is a tacit assumption in the particle picture, both classically and quantum mechanically. The phase space is given by the moduli space of solutions to the Bogomol'nyi equations (7) where the operator Λ : Λ n,m → Λ n−1,m−1 is defined as the adjoint operator to L ≡ Φ ∧ ·. A vector bundle is said to be Einstein-Hermitian when for k a real constant. When k is a non-constant function on M , then the vector bundle is said to be weak Einstein-Hermitian.
One cannot help but notice the similarity of equation (10) with the projective flatness of the Bogomol'nyi equations (7). For holomorphic vector bundles F ∈ Λ 1,1 (M ).
Applying the operator Λ to both sides of (11), we see that a projectively flat holomorphic vector bundle over a Kähler surface is weak Einstein-Hermitian:  (8) are holomorphic projectively flat [9].
Thus, up to a conformal change in the Hermitian structure, a holomorphic projectively flat connection and an Einstein-Hermitian connection are equivalent.
To obtain a phase space that is topologically well-behaved we shall therefore The next order of business is to include special relativity into the theory locally by modelling M k with M massless , the phase space for massless particles.
Consider the phase space for massless particles in The phase space of the massless particle on R 3 is equivalent to the tangent bundle T S 2 c where the radius of the sphere is the speed of light. The natural metric on T S 2 c is given by where (θ, ϕ) are spherical polar coordinates on the two-sphere, and (r = d , ψ) plane polar coordinates on the tangent plane. The functions f (r) and a(r) are arbitrary, and we have multiplied c by unit time. We require that the local geometry of M k reduce to the geometry of special relativity.
For a Kähler-torus space-time the phase space manifold of topological instantons, M k , was shown by Kobayashi to be symplectic Kähler [7]. This means that the holonomy group of M k is Sp(1) ≃ SU (2), and immediately implies that where {σ i } is the dual basis for so (3). In terms of Euler angles, we define σ 1 = dϕ sin θ cos ψ − dθ sin ψ, σ 2 = dϕ sin θ sin ψ + dθ cos ψ, In Euler coordinates the Eguchi-Hanson metric becomes where γ(r) ≡ 1 − (a/r) 4 . Compare (15) with (14). is also subject to spatial variation. However, the spatial coordinate r in the tangent space is a little ambiguous, e.g., where is the origin? We shall suppose that the origin is Earth, and that r = 2c − α d where c is the present terrestrial speed of light in vacuum multiplied by unit time, α is a dimensionless constant, and d is the distance from Earth. This is our physical parameterization of For simplicity, we assume that the photon scatters off a loosely bound electron in the detector (a graphite detector, for example).
Let the incident photon have energy E and momentum E/c, wherec is the incident speed of light. The electron with mass m is assumed to be initially at rest. After interacting with the electron, the scattered photon has energy E ′ and momentum E ′ /c, and the electron has momentum P ′ . Let c =c + ∆c, define ǫ ≡ ∆c/c, and use conservation of momentum and conservation of energy to obtain where we have usedc Define ∆λ = λ ′ −λ and take the lowest-order correction to the Compton formula to obtain The second term is dependent on the wavelength, while the terrestrial Compton effect is obviously independent of the wavelength. Scattering dependence on the incident wavelength would be a clear signal for spatial variation in the speed of light.