344 Mr Milne, Tensor Form of the Equations of Viscous Motion 



The Tensor Form of the Equations of Viscous Motion. By 

 E. A. Milne, B.A., Trinity College. 



[Received 5 January, Read 7 February 1921.] j 



In the general theory of relativity the Principle of Equivalence 

 asserts that all laws relating to phenomena in a geometrical field 

 of force which depend on the g's. and their first derivatives only will 

 also hold in a permanent gravitational field. Eddington comments 

 on this* that "it would be quite consistent with the general idea of 

 jelativity if the true expression of such laws involved the Riemann- 

 Christofiel tensor, which vanishes in the artificial field and would 

 have to be replaced before the equations were apphed to the 

 gravitational field. But were we to admit that, the principle of 

 equivalence would become absolutely useless." The following ex- 

 ample from three dimensions illustrates the significance of this point 

 by analogy. 



The equations of motion of a viscous fluid in terms of the 

 velocities may be obtained in tensor form either by generahsing 

 the corresponding Cartesian equations, or by first generahsing the 

 equations of motion involving the pressures and then substituting 

 for the pressures in terms of the velocities. The two forms are 

 found to differ by a term involving G^,, which is of course zero in 

 Gahlean space, so that the two forms are in fact equivalent. The 

 exphcit emergence of G^, in such a simple case is however inter- 

 esting; although there would in any case be no field for the apphca- 

 tion of an analogue of the principle of equivalence since the second 

 derivatives of the g'& are elsewhere involved in both forms. 



The stress system (p^^, ^^xv, ■•■) in rectangular three-dimen- 

 sional co-ordmates is a symmetrical tensor of the second rank. The 

 precise generahsed definition of j)^^, etc., in general co-ordinates 

 is to some extent arbitrary; let us assume they are defined so as to 

 constitute a contravariant tensor Pi^\ It is to be noted that in 

 this case the contravariant vector expressing the force across the 

 element of surface dS is ] -pu.v riQo^. /i x 



here the element of surface is represented by the antisymmetrical 

 contravariant tensor dS^^ (such that dS^^ = when p = a and 

 dS^'' = - dS^^ when p =t= a), and e,^^ denotes the covariant tensorf 



* "ReportontheRelativityTheoryofGravitation,"P%s.5oc.io/irf.(1918) p 43 

 ,.J ^°f *^® properties of the e-tensors see J. E. Wright, "Invariants of quadratic 

 differential forms, Camb. Math. Tracts, No. 9, p. 21. They may be used for convertino- 

 a tensor of rank ]} into one of rank \n -p\,?i being the number of dimensions; for 

 example, by the use of them it is easy to show how it is that antisvmmetrical tensors 

 of the second rank m three dimensions (such as the vector product) deoenerate into 

 vectors. ^ 



