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In this paper I have shown that squeezed modified quantum vacua have an effect on the background geometry by solving the semi-classical Einstein Field Equations in modified vacuum. The resultant geometry is similar to (anti) de Sitter spacetime. This geometry could explain the change of causal structure—speed of light—in such vacua without violating diffeomorphism covariance or causality. The superluminal propagation of photons in Casimir vacuum is deduced from the effective electromagnetic action in the resultant curved geometry. Singling between different vacua is shown not to violate causality as well when the geometric effect on the null rays is considered, causing a refraction of those rays when traveling between unbounded and modified vacua.

Since the 50’s light propagation in modified quantum vacua has been studied extensively. Several factors can affect the vacuum, including boundary conditions―Casimir vacuum―[

Consider an arbitrary scalar field

This field satisfies the Green’s function relations:

where

However, if the field lies in a bounded vacuum, like Casimir vacuum, the modes ought to have frequencies

larger than the cut-off frequency ω_{0}. Since the particle creation is limited to frequencies larger than

Surely this formulation can be generalized to massive fields and fields with arbitrary spin, in straightforward manner. When there is no excitation of the field, considering the vacuum state. The unbounded vacuum is expected to have a non-vanishing energy term

where

Modified vacua do not just have a different zero-point energy, but also affect the propagation of photons or other massless particles within them. This effect has been studied extensively since the 1950’s [

The refractive index for a Casimir is therefore given by:

where

(where

This is only an example of modified vacuum. Heat bath is shown to have an effect on the speed of light, the formula is given by [

Here, the speed of light is reduced by the thermal bath, same effects can be calculated in the presence of external fields, for instance. In all above examples the speed of light is either less or more than 1. An elegant unified formula for the speed of light is given in terms of vacuum (fluctuation) density

here

positive―in case of thermal bath or external E or B fields, or negative―for Casimir (squeezed) vacuum. Equation (10) might appear to break the exactness of Lorentz symmetry. We shall demonstrate that a more general symmetry holds however, the diffeomorphism symmetry. Because of the geometric back-reaction.

The starting point of our analysis is to reformulate the quantum fields discussed in the previous section in curved background. Basically, the same expansion of

setting

For Casimir vacuum b is given explicitly from the expectation value of the normally-ordered stress-energy tensor of the quantum field described above:

These are EFE’s in modified vacuum having. Surely to attempt solving (11) we need to specify the boundary conditions and symmetries of the modified vacuum metric

To illustrate this result we shall take some examples of solutions of EFE in modified vacua. We begin with a non-relativistic mass density satisfying Poisson equation and

To solve Einstein field equations for this matter, using linear perturbation theory assuming that the metric can be written as:

where

where h is the contracted perturbation metric. Now we can easily find from the spherical symmetry the terms of the metric perturbation:

The term

the Killing vectors fields of the modified vacuum metric. We write the solution to this perturbation as a modified Schwarzchild solution. Knowing

Thus, vacuum modification could either reduce or increase Schwarzchild radius depending on the value of vacuum density discussed earlier as a result of changing the background geometry, as if distances are changed in this transformation―apparent from the change of Schwarzchild radius present in (17).

We clearly observe the. It will become much clearer as the propagation of photons in modified vacua be discussed. Now, attempting to solve (11) for a maximally-symmetric vacuum satisfying

where

that b is constant within the boundary, viz independent of spatial direction. We can see that this solution is clearly a conformal transformation of the Minkowski metric

tic length

The key elements for this solution are the curvature tensors and scalar, they are independent of the coordinates chosen to represent the solution. Riemann and Ricci tensors, and Ricci scalar are written (respectively) as:

Here using the variable

notice that if

This solution alone is not sufficient to specify the light rays. We need to consider quantum corrections to it to model light propagation in modified vacua. We can use the metrics (18) or (19) to calculate the magnitude of a spacelike vector

When trying to study the propagation of photons in curved spacetime, using the effective metric as it was shown by [

Hence, using the effective metric with spacetime described in (18) should result photon propagation with speed given in 10. Thus spacetime geometry change corresponds to all cases of modifying the vacuum. We start by writing the effective metric for QED in the spacetime solving Equation (11).

where

waves

However, this does not hold when the propagation of photons is derived from (22). We see that the geodesic equation becomes:

where

This formula allows us to retrieve the expression for the (group) velocity of light in modified vacua that appears in (10) easily. Also allows to generalise the results in [

Since the effect is geometric, we can always locally set the metric to be the flat metric. Hence, we still can say that light travels in null geodesics in this vacuum. We should emphasise on an important point, assuming maximally symmetric/flat space, where polarisation of propagating photons does not matter (this is evident from the product of the polarisation vectors

It comes as a fundamental question how observers in different vacua could signal each other, preserving diffeomorphism and causality. Specially as heat bath affects the state of vacuum. Without a law to let signalling occurs between different vacua. Quantum field theory is in real conflict with exactness of diffeomorphism. But the solutions come in a very simple manner: refraction. We know already that null rays travel always with a con-

stant angle (

of the geometry change. If light rays ought to travel between different vacua, it will face refraction. This is a result of wave properties of light, or naively from Snell’s law (but in spacetime expressed in terms of null coordinates). If two observers in different types of vacua with a spacelike1 separation X try to signal each other, applying Snell’s law concluding first that they will only see light rays travelling at the speed of light in their own vacuum. However, each receptor will think that the signal is sent at a different time. In other words, the two observers do not have the same proper time, this is evident from metric variation due to modifying the vacuum

the future. And vice versa when the receiver is in the unbounded vacuum, this will certainly preserve causality. Since both observers will not be able to send superluminal signals in their with respect to their reference frame. The refraction occurs as the shift between geometries is not smooth, it happens rapidly at the boundary.

The quantum vacuum puts the exactness of diffeomorphism covariance of general relativity into a threat. Introducing an IR cut-off on the quantum vacuum will have an effect on propagation of light in an apparent contraction with diffeomorphism covariance. Nevertheless, when the background back-reaction is taken into account, this contraction is resolved. We have shown that the background of non-trivial vacua admits geometry similar to (anti) de Sitter spacetime. If the non-trivial vacuum had a negative density, the background will be negatively curved (hyperbolic); hence, if immersed into a flat spacetime distances between points will decrease compared

to what is measured in flat background. This explains the apparent increased speed of light in such vacua. We may also conclude that the quantum nature of vacuum is important for stabilising the underlying geometry. Imagine what a severe perturbation on the vacuum fluctuations can do to the underlying geometry! As such perturbation will correspond to a large absolute value on b, making the spacetime extremely curved and unstable. This treatment of modified vacua is very important in the early universe as intending to show in future work.

Warm regards to L.A Al Asfar for the helpful discussions and Prof. Nabil Bennessib for his generous technical help in writing this paper, and revising it.

This research project was supported by a grant from the “Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.

Salwa AlSaleh, (2016) Geometric Backreaction of Modified Quantum Vacua and Diffeomorphrisim Covariance. Journal of Modern Physics,07,312-319. doi: 10.4236/jmp.2016.73031