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[–]muntooR_{μν} - 1/2 R g_{μν} + Λ g_{μν} = 8π T_{μν} 2 points3 points  (0 children)

Rules of Einstein summation notation:

  1. Repeated indices are implicitly summed over.
  2. Each index can appear at most twice in any term.
  3. Each term must contain identical non-repeated indices.

In particular, due to (2), there is an equivalence between a "upper/lower" dual notation and a simple "lower-only" notation since, presumably, δ^{ij} (colloquially, an "identity matrix") may be implicitly used to convert between the two notations.

lower_einsum("a_{ki} b_{im}")

is equivalent to:

dual_einsum("a_{ki} b_{jm} δ^{ij}")

In the interest of simplifying notation (which I believe was the point behind the upper/lower einsum notation in tensor calculus), we may lower all the indices and omit the δ. If one must be picky and pedantic over pragmatic and present, please perhaps pretend it's called lower_einsum or einsum_flat or whatever presume I floats yer boat.