By Synge, John Lighton; Schild, Alfred
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Example text
However, difficulties occur here in the case of an indefinite metric, since any two points can be joined by an indefinite number of curves of zero length. We shall not pursue further the question of the existence of a geodesic joining two given points. When we have occasion to use this construction, we shall assume that it can be done. The importance of stationary principles, buch as that used in defining a geodesic, wzrs recognized long before the tensor calculus was invented. Such a principle is 'i,mtar'iant, in the sense that no particular coordinate system is mentioned in stating the principle.
D,s] - lst,rl :$. The cmaria,nt d,crivat'i,vcof thc mctrh Excrc'i,sc. 527. 0, a'"p : 0. For the sake of completeness we define the absolute and covariant derivatives of an invariant to be the ordinary and partial derivatives respectively. If. 528. DT - dT 'l' - tTr AT Ar" 6u d,u' An important difference between the covariant derivative and the partial derivative should be noticed. Suppose we have a tensor lrr. oTn The partial derivative can be calculated if we know the component Tn 3s a function of the coordinates.
It follows from the fundamental lemma of the calculus of variations* thattheequations fR. 499. 413. ,r,)t-o *#,@,,)t are satisfied at all points on a geodesic. These are called the "Euler equations" of the variational problem. 414. d0w 0w ldwilw Aap'- T*,- zr6rTp,' so far the parameter u has been arbitrary. 415. % : E, p':y, , zt): a^n p^p,,: et H:0. 416. iI 0w ds 0p' Aw 0x'- To obtain a more explicit form, we substitute for w. 417. o:0. 4rs. 419. 420. 421. rstorrsr, Stlmors r,nn,rtr(W+#-W). This orpression lmn, rl is called the Chrtsbffcl sytnbol of tlu first hind,.