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Although the transformation algorithm is very well established and implemented, some intriguing questions remain unanswered.
1) In what precise mathematical sense is the transformation optics algorithm ‘exact’? The invariance of Maxwell’s equations is well understood, but in what sense does the same principle not apply to acoustics (say)?
2) Even if the fields are transformed in a way that apparently mimic vacuum perfectly, it is easy to construct very simple examples where the impedance of the transformed medium is no longer isotropic and homogeneous. This would seem to imply a fundamental shortcoming in any claim that electromagnetic cloaking has been reduced to technology.
3) Transformations are known to exist that introduce a discrepancy between the Poynting vector and the wave-vector. Does this distinction carry any physical significance?
We have worked extensively on understanding a commonality between transformation theories that operates at the level of rays – being interpreted as geodesics of an appropriate manifold. At this level we now understand that the *key* problem underlying all attempts to unify the transformational approach to disparate areas of physics is how to relate the transformation of the base metric (be it Euclidean for spatial transformation optics, or Minkowskian for spacetime transformation optics) to the medium parameters of a given physical domain (e.g. constitutive parameters for electromagnetism, bulk modulus and mass density for acoustics, diffusion constant and number density for diffusion physics).
Another misconception we will seek to address is the notion of the relationship between transformation optics and curvature. Many have indicated that transformation optics evinces similarities with Einstein’s curvature of spacetime. Here we will show emphatically that transformation optics cannot induce curvature. Inducing curvature in an electromagnetic medium requires the equivalent of a gravitational source. We will propose a scheme that achieves this.
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