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Loop quantum gravity predicts a variable speed of light. What is the range of such a variation in its speed?

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I'll paraphrase my answer to your twin question, saying mainly that breaking Lorentz invariance doesn't necessarily break the constancy of c

  • loop quantum gravity theories discuss of the Lorentz invariance but most of them consider c with a constant value.

in the wikipedia page of the Lorentz invariance in loop quantum gravity , you can read :

Lorentz invariance is a measure of universal features in hypothetical loop quantum gravity universes. The various hypothetical multiverse loop quantum gravity universe design models, could have various Lorentz invariance results.

Why physicists try to keep c constant when they build speculative theories ? they play with a set of variables and mathematical tools. The latter were developped with c constant. Generally, theorists are in favor changing anything but not c , letting them a contact with the hub. Even if modification of the constancy of c would likely imply to evolve its exact meaning.

a little further in the same page :

These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Lee Smolin and Joao Magueijo then went on to study doubly special relativity, in which not only there is a constant velocity c but also a constant distance l.

Notice than in General relativity, c is postulated locally constant. Potential pitfalls at another scales are handled by the flexibility of the spacetime properties in differrent models.

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