Bernhard Riemann (1826 Bernhard Riemann (1826-1866) Bernhard ...

► Riemann considered an appropriate new basis and
compared the two forms associated to the “particle of
substance” b ” at times t and d t’. ’ The h difference dff between b the h
two forms is given by
2 2 2 2 2 2 2
δ ( ds) = ds'
−ds
= ( G −1)
ds + ( G −1)
ds + ( G −1)
ds
where
ds
1
, ds
2
, ds
3
is
1
the
new
1
basis.
His result can be interpreted in terms of the classical
theory y of elasticity. y Since ether is an elastic, , homogeneous g
and isotropic substance, then one can consider an
infinitesimal displacement u and deduce that the difference
δ(ds) depends on the strain tensor due to the
displacement. Riemann supposed that this variation δ(ds)
produced a force able to modify the particle in such a way
that the same particle, by opposing this deformation, would
propagate p p g the pphysical y forces through g space. p
2
2
3
2
3