Mathematical geophysics

A. Tarantola

Space of measurable physical quality

Even though the physics is based, usually, on the notion of physical quantity, the more basic notion of measurable physical quality is fundamental. A simple example: behind the notions of temperature quantities $ T $ , thermodynamic parameter $ \beta = 1/kT $ , logarithmic temperature $ t = \log T/T_0 $ or cubic temperature $ \tau = T^3 $ , a qualitative notion exists, the notion of hot-cold. The different quantities $ T $ , $ \beta $ , $ t $ or $ \tau $ are possible coordinates in the (1-D) space that represent the quality hot-cold. A less trivial example is the quality ideal elastic medium, corresponding to a space of dimension 21: the 21 components of the rigigidity tensor $c_{ijk\ell}$ , or the 21 components of the compliance tensor (inverse of the rigigidity tensor) $ d_{ijk\ell} $ , or the six eigenvalues and the 15 eigen angles of one or the other, etc. Like that, this remark is quite trivial, and corresponds to Aristote's point of view (who explicitely discuss of the quality hot-cold). It's only after Galilee that physicists start to favour the notion of quantity. A non trivial result, prooved in our work, is that for each physical quality an intrinsic notion of distance exists, and is unique (under certain natural invariance conditions). For example, two points in the hot-cold space can be characterized by their temperatures $ T_1 $ and $ T_2 $ , their inverse temperatures $ \beta_1 $ et $ \beta_2 $ , etc. The distance between two points est $ D = \vert \log T_1/T_2 \vert = \vert \log \beta_1/\beta_2 \vert $ . On can note that this distance is not linked in a simple way to the difference of temperatures. Another example, the distance between two ideal elastic solids, characterized, for example, by their rigidity tensors $ {\bf c}_1 $ and $ {\bf c}_2 $ is the logarithmic norm (tensorial) of $ {\bf c}_2^{-1} \cdot {\bf c}_1^{\phantom{1}} $.

Geotensorial Space

When the notion of distance is introduced, in an abstract space, several questions arise. For example: is the space flat ? Many spaces, so-called tensorial, have been shown not to be flat (they have a curvature which can be computed). For example, in the theory of the deformation of a continuous medium, the stress $ \sigma_{ij} $ and the strain $ \varepsilon_{ij} $  are generally introduced. While the stress tensor belongs to a linear space (a tensor in the usual sense) the strain ``tensor'' belongs to a space with curvature. It is only for small deformations that the usual tensorial space is retrieved. This has important implications in physics, mentionned later. These ``tensors'', which are not tensors, are called here geodesic tensors or geotensors. The theory of linear spaces (vectorial or tensorial) has to be redeveloped to include these new objects. From a mathematical point of view, the geotensorial spaces appear quite similar to the usual tensorial spaces, except that the sum of two geotensors is not in general commutative (beside when the geotensors are ``small''). This is not just a mathematical abstraction, it is a formalisation of properties that are otherwise well known (for example, the ``composition'' of two strains $ \mbox{\boldmath$\varepsilon$}_1 $ and $ \mbox{\boldmath$\varepsilon$}_2 $ does not commute unless when the two deformations are small [or when they have the same principle axis]). The theory of the ``geotensorial'' space is under construction. Usual notions like that of a basis, a norm, correspond to new concepts. This spaces are not only spaces with curvature, they do have also a torsion, but we will not elaborate here on this point. We propose to call a physical theory intrinsic when it can be formulated not in terms of particular physical quantities, but in terms of measurable physical qualities with the corresponding distances. Some of the usual physical theories are, from this point of view, intrinsic. Others are not (deformation theory, Fourier's transport theory of heati, etc). These theories are then considered as mathematically non consistant, and as to be reformulated.

Declinative

A basic mathematical tool is the derivative of a physical quantity with respect to another quantity. It is defined, following Newton and Leibniz, as the limit of a quotient of differences. But the difference between two values of a physical quantity is of little interest, when considering that the notion of distance exists, and that it does not generally correspond to a difference. Therefore, we propose to replace the notion of derivative by that of the declinative, defined as the limit of a quotient of distances between points (and not as differences between values of quantities). This open the field of ``distancial'' calculus (by opposition to ``differential''). The notion of tangent linear applications, of divergence, etc. are replaced by intrinsic notions. For example, lets take a solid body in rotation around a point. At each instant $ t $ , the orientation of the body in space can be described by the vector of rotation $ {\bf r}(t) $ (with respect to an arbitrary origine $ t_0 $ ). Let $ \mbox{\boldmath$\omega$}(t) $ be the vector instantaneous rate of rotation. The books explain well that $ \mbox{\boldmath$\omega$}(t) $ is not the time derivative of $ {\bf r}(t) $ , and give an ad-hoc definition of $ \mbox{\boldmath$\omega$}(t) $ . In our theory, $ \mbox{\boldmath$\omega$}(t) $ is the declinative and not the time derivative of $ {\bf r}(t) $ . This shows the overuse of a definition (the derivative) that has, somehow, little interest. One of the objectives of our work is to rewrite the theory of differential calculus, so as to convert it into a theory of ``distancial'' calculus. Most of the usual properties break down (for example, the declinative of a sum is not always that of the sum of the declinatives). Linked to this new theory of distancial calculus, there is of course, a theory of ``integral'' calculus where integrals are no more considered as limits of usual sums but as limits of sums that are not always commutative. The intrinsic physical theories - mentionned above - must be theories including declinatives, and not derivatives (beside the rare cases where both notions coincide). There is a fundamental difference between derivatives and declinatives: while a quantity is derived with respect to another quantity, it is a quality which is declinated with respect to another quality. So, the temperature is not declinated with respect to a coordinate, but it is the quality ``hot-cold'' which is declinated with respect to a quality `` spatial position''. To actually do the calculus, one can choose to represent the quality ``hot-cold'' by the temperature, or the inverse temperature, etc., however the result of the declinative is independant of that choice. In that sense, one can formulate the equations of the physics in an intrinsic manner. Then, if tensorial calculus was developed to make the equations free from the choice of spatial coordinates (or spatio-temporal coordinates), the theory developped here aim to make the equations independant of the choice of the physical quantities which can be used to represent a measurable physical quality (i.e.. a new storey for the notion of intrinsic calculus).

External collaboration:

Bartolomé Coll, Observatoire de Paris;
Klaus Mosegaard, Copenhagen University.