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I think this likely comes down to the following expression,

$$g’^{ab}e’_a e’_b = \delta ^a_b $$

Is this in agreement with the Einstein summation convention? Because even though the two indices are summed over, they still appear on the right.

This leads to issues when trying to rearrange for $g’$

In this case how would you rearrange for $g’$?

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  • $\begingroup$ What are $e_a$ meant to be? The components of a unit vector? $\endgroup$ – jacob1729 Nov 5 at 17:07
  • $\begingroup$ Yes sorry, they are unit vectors wrt some primed basis $\endgroup$ – Jake Rose Nov 5 at 17:10
  • $\begingroup$ Is this supposed to be a version of $g(e_i,e_j)=\delta_{ij}$ in co-ordinates? $\endgroup$ – jacob1729 Nov 5 at 17:16
  • $\begingroup$ Yes it is an inner product $\endgroup$ – Jake Rose Nov 5 at 17:18
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    $\begingroup$ Why did someone downvote this? This is a perfectly legitimate question. $\endgroup$ – Ben Crowell Nov 5 at 20:55
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I assume that you're using Latin indices to mean abstract indices, whereas Greek would imply concrete indices. This would be the modern convention (as opposed to in older publications where Latin vs Greek would indicate timelike vs spacelike coordinates).

Why are the primes there?

This is a notational clash between the use of subscripts for two different purposes: (1) to identify what basis vector we're talking about, and (2) as an abstract index. If you just have a list of basis vectors, in no particular order and with no connotation of being unit vectors along a certain coordinate axis, then an index to state which vector you're using from the list is neither a concrete index nor an abstract index in the sense of Einstein summation notation. Let's use $i$ and $j$ for this type of index.

Then a way to notate this without the clash would be $g^{ab}e_{i,a}e_{j,b}=\delta_{ij}$. (Since the $i$ and $j$ aren't concrete or abstract indices, it doesn't make sense to write them as superscripts.)

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  • $\begingroup$ This makes sense. Do you know any books that give a more formal treatment of these indices? Hobson doesn't seem to address this! $\endgroup$ – Jake Rose Nov 5 at 23:06

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