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Is there acceptable notation to collapse sums in an expression where the tensor is diagonal? As a simple artificial example, consider the expression: $$ F^{\alpha\beta} =g^{\alpha\mu} g^{\beta\nu} \left( a^\kappa (\partial_\mu g_{\nu\kappa}) + g_{\nu\kappa} (\partial_\mu a^\kappa ) - a^\lambda (\partial_\nu g_{\mu\lambda} ) -g_{\mu\lambda} (\partial_\nu a^\lambda ) \right) . $$ If the metric $g^{\alpha\mu}$ is diagonal, then for all nonzero terms $\mu=\alpha=\lambda$ and $\nu=\beta=\kappa$. In fact, none of the repeated indices call for a sum over more than a single term.

Could one write: $$ F^{\mu\nu} =g^{\mu\mu} g^{\nu\nu} \left( a^\nu (\partial_\mu g_{\nu\nu}) + g_{\nu\nu} (\partial_\mu a^\nu ) - a^\mu (\partial_\nu g_{\mu\mu} ) -g_{\mu\mu} (\partial_\nu a^\nu ) \right) ? $$ For more complicated expressions of this sort, repeated indices proliferate without shedding light on the actual operations involved, and make any later evaluation clumsier.

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2 Answers 2

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The intent of the index notation is to write tensor expressions that hold in any frame. A 2-index tensor being diagonal will only generally be true in one frame. So to take advantage of a property like that is usually taking you away from the point of index notation.

Having said that, when you go to calculate what should happen in a concrete scenario, you often do pick a coordinate system and writing general abstract tensor expressions becomes less important. Then it does make sense to parameterize the the tensors you are working with and work in terms of those parameters. For example, you could work directly with the diagonal entries $\lambda_i$ ($i=0, 1, 2, 3$) of the tensor. In cosmology, in FRW coordinates, the diagonal elements of the metric are $(-1, a^2(t), a^2(t), a^2(t))$ (where $a(t)$ is the scale factor), so it's common to parameterize a general metric $g_{\mu\nu}$ with $g_{00}=-1$, $g_{0i}=0$, and $g_{ij}=a^2(t) \delta_{ij}$, do a 3+1 split on all the index sums you see, and simplify the resulting expressions as far as possible. The final results for things like the Riemann tensor are expressed in terms of $a(t)$ and its time derivatives, with tensor indices only appearing in the form of the identity matrix, $\delta_{ij}$.

Basically, once you know you are leaving the world of generally true tensor expressions, there is no one notation you must use. You are free to use whatever notation makes the most sense for your problem, so long as you define it sufficiently well that a reader (which may include yourself in six months) can follow what you are doing.

Having said all that, I don't see that you've particularly simplified the expression for $F^{\mu\nu}$ you wrote down by assuming $g_{\mu\nu}$ is diagonal. One situation where you sometimes get simplifications with a condition like a given tensor being symmetric or diagonal is when you can separate a tensor expression $F^{\mu\nu}$ into a symmetric part $\frac{1}{2}(F^{\mu\nu}+F^{\nu\mu})$ plus and antisymmetric part $\frac{1}{2}(F^{\mu\nu}-F^{\nu\mu})$, then sometimes a property like $g_{\mu\nu}$ being diagonal might imply that the antisymmetric part vanishes.

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  • $\begingroup$ Thanks for the interesting and extensive response! $\endgroup$
    – macrofish
    Aug 14, 2023 at 19:30
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The whole point of this type of index notation is to be able to construct expressions that are coordinate-independent and hence are valid spacetime tensors. But whether or not a tensor is diagonal depends on the coordinates you are using. So "simplifying" these expressions on this basis makes the expressions much less useful, since they can't immediately be generalized to other "non-diagonal" coordinate systems.

(Also a minor nitpick: your first expression is not a valid tensor equation. It should be $F^{\alpha \beta}$ on the left-hand side.)

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  • $\begingroup$ Thanks for catching the alpha/beta error! I corrected it above. $\endgroup$
    – macrofish
    Aug 14, 2023 at 19:31

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