Describing the deformation of a medium as a diffeomorphism

Question

In this paper online, the author models the deformation of a medium as a diffeomorphism $ mathbb{R}^3 rightarrow mathbb{R}^3$ as given by:
$$ y^i mapsto x^i(y)=y^i + u^i(x) $$
as given by equation (1). The diffeomorphism induces a transformation of the metric
$$ g_{ij}(x) = frac{partial y^k}{partial x^i} frac{partial y^l}{partial x^j} delta_{ij}$$
which is just the push forward of $delta_{ij}$ under the diffeomorphism, as shown in equation (5).
It is stated that after the diffeomorphism, which transforms the metric from $delta_{ij}$ to $g_{ij}$, the geodesics of the material will become curved because the metric $g_{ij}$ is non-trivial. Therefore, sound waves through the medium will now take curved paths as they are postulated to follow geodesics. However, this seems completely bizarre to me. A diffeomorphism is equivalent to a change of coordinates, so the geodesics of $g_{ij}$ will be the same as the geodesics of $delta_{ij}$, which are straight lines, not curved. It is just now the geodesic equation will look a bit more complicated because we are working in a general curvilinear coordinate system. In fact, both metrics are flat because the curvature is invariant under diffeomorphisms, so I assume this is another reason to argue that the geodesics will be straight lines too?
My question
How can one describe a deformation of a material, something which physically affects the density of the material and the paths sound waves travel, as a diffeomorphism, something which does not change the manifold structure and can be viewed as a change of coordinates so should be unphysical?

A.V.S. · Accepted Answer

… as a diffeomorphism, something which does not change the manifold structure and can be viewed as a change of coordinates so should be unphysical …

Diffeomorphism does not have to be the change of coordinates. It can also have nontrivial physical meaning, as it has here. This is because the Euclidean space under consideration here has an elastic material occupying it. And when Euclidean space is mapped onto itself this is also accompanied by the displacement and deformation of that material. So when we write:
$$
y^i mapsto x^i(y)=y^i + u^i(x)
$$
the physical interpretation is that the physically small element of the material that initially occupied the neighborhood around position $y$ is now occupying the neighborhood around $x(y)$. The displacement of this material element is given by $u$ and its deformation is described by the tensor of small deformations $epsilon$. Note, that $y^i$ and $x^i$ are the Cartesian coordinates of the initial and final placements of the elastic substance, and so the straight lines in initial state generally would not be mapped to the straight lines in the final placement.

Describing the deformation of a medium as a diffeomorphism

One Answer

Add your own answers!

Ask a Question