By Verley J.-L., et al.
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Additional resources for Dictionnaire des mathematiques (algebre, analyse, geometrie) (1997)(sans p.121)
Example text
X ∂x i j If X and Y are any contravariant fields on M, then X·Y is a scalar, and ∂ ∂ X·Y = Xi i ·Yj j = gijXiYj. 3, it is a type (0, 2) tensor. We call this tensor “the metric tensor inherited from the imbedding of M in Es. ” Exercise Set 5 1. Compute the transformation rules for each of the following, and hence decide whether or not they are tensors. Sub-and superscripted quantities (other than coordinates) are understood to be tensors. 37 dXij ∂2˙ ∂xi ∂Xi (c) (d) dt ∂xi∂xj ∂xj ∂xj 2. (Rund, p.
What is wrong with the following “proof” that ∂xh ∂xk = 0 regardless of what smooth functions x–j(xh ) we use: ∂2 xj– ∂ ∂x–j = Definition of the second derivative ∂xh ∂xk ∂xh ∂xk ∂ ∂x–j ∂x–l = l k h Chain rule ∂x– ∂x ∂x ∂2 xj– ∂x–l = l k h Definition of the second derivative ∂x– ∂x ∂x ∂2 xj– ∂x–l = k l h Changing the order of differentiation ∂x ∂x– ∂x ∂ ∂x–j ∂x–l = k l h Definition of the second derivative ∂x ∂x– ∂x ∂ ∂x–l ∂x–j = k ©il h Since l = ©il ∂x ∂x ∂x– =0 Since ©il is constant!
Then (1) reduces, for a photon, to dh -q =±t dt Integrating gives t = [±(1-q)(h-h0 )] 1/(1-q) For simplicity let us take q = 1/2, so that the above curves give parabolas as shown: Here, the Big Bang is represented by the horizontal (t = 0) axis. Only points inside a particular parabola are accessible from its vertex (h0 , 0), by a signal at or below light-speed, so the points A and B in the picture are not in “causal contact” with each other. ) Exercise Set 6 1. Give an example of a Riemannian metric on E2 such that the corresponding metric tensor gij is not constant.
Dictionnaire des mathematiques (algebre, analyse, geometrie) (1997)(sans p.121) by Verley J.-L., et al.
by James
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