General relativity question Rule Violations This must b be complete in parenthe
ID: 3281007 • Letter: G
Question
General relativity question
Rule Violations This must b be complete in parenthe component is zero wh nonzero, when a BOX 4.5 Consider the following equations: (4 25 (4.25 (425 4.250 (4.25e Exercise 4.5.1. Which of these equations violate Rule 1 about free indices? c) dt= 0for a free particle Exerc ing li dp Write "OK"or "violates" next to each equation, as appropriate Consider the following examples of renaming an index: a) /1#2 A' renamed to _Aa_ A' (4.260) (4.26b) (4.26c) (4.26d) 4.26e) (4.26 We net c) ap =0 renamed to up-=0 dt dt 8pvpw = renamed to 8" vpwv-pW Exercise 4.5.2, Which of these equations violate Rule 2 about re ces? Write "OK" or "violates" next to each equation, as appropri renaming indi-Explanation / Answer
4.5.1 The rule about free indices is that they must agree on both sides of the equation.
a) Free index on LHS is mu but on RHS alpha, beta and mu. Therefore, violated.
b) Both RHS and LHS no free index. OK.
c) LHS has mu and RHS is no free index. Violated. (Sometimes this type of notation is used informally to indicate that all components of a vector are 0 but it is not correct because LHS and RHS have different ranks).
d) Both RHS and LHS have free indices alpha and beta. OK.
e) LHS has alpha but RHS has alpha, mu and nu. Violated
4.5.2: Rule about renaming indices is that indices may be renamed assuming that other rules are not violated
a)Free indices do not agree in the renamed equation. Violated.
b) Same index appears more than twice in the product. Violated
c) All rules followed. OK
d) All rules followed. OK
e) All rules followed. OK. (Remember that the rule about same index not appearing more than twice is only for a single term, ie products. They may appear in different terms without issue.)
f)Same index appearing more than twice in a single term. Violated.
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