# Variable speed of light in cosmology

In this paper, D. H. Coule argues that warp drive metrics, like the one proposed by Alcubierre, require the exotic matter to be laid beforehand on the travel path by conventional travel. At section 5 of this paper "Alteration of the light-cone structure" he basically goes for the same trick that Alcubierre did: write some metric up from his sleeve with the desired properties, and see what properties the energy-stress tensor needs to have in order to satisfy Einstein equations. The metric he writes describes a spacetime with a bigger effective speed of light in the $x$ axis. He explains that this demands a $T_{\mu \nu}$ that violates the dominant energy condition.

Now, if you head to the variable speed of light wikipedia page and head to the cosmology section, none of the papers by Magueijo or Albrecht rely on a mechanism of this sort to obtain variable speed of light; They go instead by the arguably harder route of proposing alternative lagrangians or theories. I'm wondering if either there is something so blatantly wrong the Coule metric alteration that the VLS authors didn't dignify it with a rebuttal, or if they simply ignore this simple mechanism to stretch light cones?

We know that for explaining cosmic inflation, we need a scalar field that violates the dominant energy condition, but if this scalar had important rotational anisotropies, such a model (if correct) would imply that far-away parts of the primordial universe might have been causally connected after all (at least in some directions that would change from place to place) which might explain a lot of the long scale uniformity we are seeing? isn't this a better start for a simpler explanation that uses well-known gravitational theory?

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The exclamation mark in the abstract looks suspicious ... –  Dilaton Jul 15 '12 at 22:24
@Dilaton: I thought so too, but there's a "Clas. Quant. Gravity." at the Top - Left Corner . –  Dimensio1n0 Aug 7 '13 at 14:57

The metric in the paper where the speed of light in the x-direction is stretched out violates the weak energy condition and is certainly unphysical, at least if the violation is localized in y and z. This is not the type of thing you expect from any physical solution of GR.

There is an area theorem in GR which only uses the weak energy condition. If you look at a lightfront crossing a region from left to right, such that it consists of those rays which are furthest possible to the right, area must always increase when the lightfront reaches positive infinity. This is proved here: Second Law of Black Hole Thermodynamics . The proof is for the black hole case, but the light front version is just the infinite black hole limit.

If you consider a hypothetical region where the light going through the region is sped up relative to parallel light at infinity, if you trace back the boundary of communication with a far away point at right infinity, you find a plane that gets dimpled when crosses the region where light is sped up, then the dimple spreads, so that the area increase when you look backwards. This means that the area of the light-front in this case is decreasing, not increasing, and this requires a violation of the weak energy condition, and this is linked to the implied violation of causality. This is discussed more here: Does a Weak Energy Condition Violation Typically Lead to Causality Violation? and in the linked question and answers.

Regarding the early universe, the violation of the strong energy condition in the early universe is not the same as the violation of the weak energy condition. The causal structure of spacetime is not affected by the strong condition violating cosmological constant, all that happens is that the stable situation is a deSitter space, with a finite area horizon. Violating the weak energy condition on the other hand is tantamount to violating the laws of thermodynamics, since you can use a local violation of this condition to make a classical black hole shrink.

The literature you quote is not very good. The "variable speed of light" ideas are phrased in the least enlightening way possible. The speed of light is a dimensional constant, and should be set to 1. The notion that speeding up light in the early universe somehow brings it into causal contact is false, since speeding up light doesn't do anything in itself.

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"violating the weak energy condition is tantamount to violating the laws of thermodynamics" not necessarily true. It depends on how much entropy we have to add to the universe to create a certain amount of energy-condition-violating matter/energy –  diffeomorphism Jul 17 '12 at 2:14