Index notation from hell –
and how to make it more fun

written by Splines
published

I’m currently taking a course in General Relativity. One major, quite technical part of it is index notation, which can quickly get out of hand. In this harmless example, let’s consider how the connection coefficients Γμλν\Gamma^\nu_{\mu\lambda} transform under a smooth change of coordinates xμ→x′μ(x)x^\mu \rightarrow x'^\mu(x):

Γμλ′ν=∂xρ∂x′μ∂xγ∂x′λ∂x′ν∂xσΓργσ−∂xρ∂x′μ∂xγ∂x′λ∂2x′ν∂xρ∂xγ \Gamma'^\nu_{\mu\lambda} = \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\gamma}{\partial x'^\lambda} \frac{\partial x'^\nu}{\partial x^\sigma} \Gamma^\sigma_{\rho\gamma} - \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\gamma}{\partial x'^\lambda} \frac{\partial^2 x'^\nu}{\partial x^\rho \partial x^\gamma}

We use greek letters to indicate the four dimensions (00,11,22,33) we sum over, in contrast to latin symbols, where one would just sum over the spatial dimensions (11,22,33). The beauty of Einstein’s summation notation is that we can leave out the summation symbols ∑\sum: whenever we have a repeated index in one summand, we implicitly know that we have to sum over it. In the example above, such summation indices are ρ\rho, γ\gamma and σ\sigma.

Fresh indices

However, with all these greek letters and their scrollwork, I personally find it hard to keep track of the overall structure of the formula and to identify where exactly one variable occurs again in another part. So let’s make this better. I was motivated by Bret Victor’s thought-provoking talk “The Humane Representation of Thought” where he states:

Leibniz was the UI designer of the 17th century. He was obsessed with notation, always trying out different notations, always talking with his friends about notation. Because he realized that a lot of the power in an idea lies in the form in which it’s expressed, because that’s what allows people to think it. ~ Bret Victor

Our “new” notation will look as follows, using the covariant derivative as an example.

Only greek letters
With new symbols
∇μVν=∂μVν+ΓμανVα \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\alpha} V^\alpha
∇∙V∘=∂∙V∘+Γ∙∼∘V∼ \nabla_\bullet V^\circ = \partial_\bullet V^\circ + \Gamma^\circ_{\bullet \sim} V^\sim

The constraints for the new notation are:

To not clash with existing notation, we should avoid using symbols like ∇\nabla, Δ\Delta, the prime ′', parentheses of any kind, ∗*, †\dag, ∧\wedge, ⊗\otimes, ∂\partial (and a lot more). We could theoretically use Emojis like 🎈, but these are hard to draw by hand, so the symbols shouldn’t be too detailed. Here is a proposal for indices that hopefully satisfy my constraints:

∘  ∙  □  ■  ∼  α  :  ⌞  ⌜  ⌝  ⌟  ∩  ∪  ⊓  ⊔  ∨  ∧ \circ \:\: \bullet \:\: \squaree \:\: \blacksquaree \:\: \sim \:\: \alpha \:\: : \:\: \llcorner \:\: \ulcorner \:\: \urcorner \:\: \lrcorner \:\: \cap \:\: \cup \:\: \sqcap \:\: \sqcup \:\: \vee \:\: \wedge

The α\alpha should be understood as a placeholder for any greek latter. I don’t want to ban them, just enrich the palette we can use. In the above formula, with the new notation, I find it easier to identify where the given indices ∙\bullet and ∘\circ occur on the right side and where we just have a summation over ∼\sim. And it’s just a lot more fun to draw these shapes as indices instead of greek letters.

Example: Covariant Derivative

Let’s see both notations in action to derive the expression of the covariant derivative. On wider screens, the usual notation sits on the left and the new symbols on the right. We make use of the Leibniz rule and the definition of the connection coefficients Γμνα\Gamma^\alpha_{\mu \nu} to expand ∇μV\nabla_\mu V in terms of partial derivatives and connection coefficients.

Only greek letters
With new symbols
(V=Vν∂ν)⇒∇μV=∇μ(Vν∂ν)=(∂μVν)∂ν+Vν∂μ∂ν(∂μ∂ν=Γμνα∂α)⇒∇μV=(∂μVν)∂ν+VνΓμνα∂α=(∂μVβ+ΓμνβVν)∂β \begin{align*} \bigl(V = V^\nu \partial_\nu\bigr) \Rightarrow \nabla_\mu V &= \nabla_\mu(V^\nu \partial_\nu)\\ &= (\partial_\mu V^\nu) \partial_\nu + V^\nu \partial_\mu \partial_\nu\\ \bigl(\partial_\mu \partial_\nu = \Gamma^\alpha_{\mu \nu} \partial_\alpha\bigr) \Rightarrow \nabla_\mu V &= (\partial_\mu V^\nu) \partial_\nu + V^\nu \Gamma^\alpha_{\mu \nu} \partial_\alpha\\ &= (\partial_\mu V^\beta + \Gamma^\beta_{\mu \nu} V^\nu) \partial_\beta \end{align*}
(V=V∘∂∘)⇒∇∙V=∇∙(V∘∂∘)=(∂∙V∘)∂∘+V∘∂∙∂∘(∂∙∂∘=Γ∙∘∼∂∼)⇒∇∙V=(∂∙V∘)∂∘+V∘Γ∙∘∼∂∼=(∂∙V:+Γ∙∘:V∘)∂: \begin{align*} \bigl(V = V^\circ \partial_\circ\bigr) \Rightarrow \nabla_\bullet V &= \nabla_\bullet(V^\circ \partial_\circ)\\ &= (\partial_\bullet V^\circ) \partial_\circ + V^\circ \partial_\bullet \partial_\circ\\ \bigl(\partial_\bullet \partial_\circ = \Gamma^\sim_{\bullet \circ} \partial_\sim\bigr) \Rightarrow \nabla_\bullet V &= (\partial_\bullet V^\circ) \partial_\circ + V^\circ \Gamma^\sim_{\bullet \circ} \partial_\sim\\ &= (\partial_\bullet V^: + \Gamma^:_{\bullet \circ} V^\circ) \partial_: \end{align*}
Then, take the component and rename the dummy index to arrive at the formula.
(∇μV)β=∇μVβ=∂μVβ+ΓμνβVν⇒∇μVν=∂μVν+ΓμανVα \begin{align*} (\nabla_\mu V)^\beta = \nabla_\mu V^\beta &= \partial_\mu V^\beta + \Gamma^\beta_{\mu \nu} V^\nu\\ \Rightarrow \nabla_\mu V^\nu &= \partial_\mu V^\nu + \Gamma^\nu_{\mu \alpha} V^\alpha \end{align*}
(∇∙V):=∇∙V:=∂∙V:+Γ∙∘:V∘⇒∇∙V∘=∂∙V∘+Γ∙∼∘V∼ \begin{align*} (\nabla_\bullet V)^: = \nabla_\bullet V^: &= \partial_\bullet V^: + \Gamma^:_{\bullet \circ} V^\circ\\ \Rightarrow \nabla_\bullet V^\circ &= \partial_\bullet V^\circ + \Gamma^\circ_{\bullet \sim} V^\sim \end{align*}

Partial derivatives

Additionally, I wanted to reduce the effort of writing many partial derivatives in index notation, while also improving readability and clarity. Here is my suggestion. It should be clear from the context that this is not a regular fraction since we use indices (with special symbols) as quantities for the nominator and denominator.

With new symbols
With new symbols and shorthand for derivative
∂x′∙∂x∘∂2x′∙∂x∘∂x□ \frac{\partial x'^\bullet}{\partial x^\circ} \qquad \frac{\partial^2 x'^\bullet}{\partial x^\circ\partial x^\square}
∙′∘∙′∘□ \frac{\bullet^\prime}{\circ} \qquad \frac{\bullet^\prime}{\circ \square}

Now let’s derive how the connection coefficients transform under a smooth change of coordinates x∘→x′∘(x)x^\circ \rightarrow x'^\circ(x). We use the fact that ∇∙V∘\nabla_\bullet V^\circ is a (1,1)(1,1)-tensor and thus transforms like one.

Only greek letters
With new symbols and shorthand for derivative
∇μ′V′ν=∂μ′V′ν+Γμλ′νV′λ=∂xρ∂x′μ∂ρ ⁣(∂x′ν∂xαVα)+Γμλ′ν∂x′λ∂xαVα=∂xρ∂x′μ∂2x′ν∂xρ∂xαVα+∂xρ∂x′μ∂x′ν∂xα∂ρVα+Γμλ′ν∂x′λ∂xαVα \begin{align*} \nabla'_\mu V'^\nu &= \partial'_\mu V'^\nu + \Gamma'^\nu_{\mu\lambda}V'^\lambda\\ &= \frac{\partial x^\rho}{\partial x'^\mu} \partial_\rho\!\left(\frac{\partial x'^\nu}{\partial x^\alpha}V^\alpha\right) + \Gamma'^\nu_{\mu\lambda}\frac{\partial x'^\lambda}{\partial x^\alpha}V^\alpha\\ &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial^2 x'^\nu}{\partial x^\rho\partial x^\alpha}V^\alpha + \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\alpha}\partial_\rho V^\alpha\\ &\quad + \Gamma'^\nu_{\mu\lambda}\frac{\partial x'^\lambda}{\partial x^\alpha}V^\alpha \end{align*}
∇∙′V′∘=∂∙′V′∘+Γ∙□′∘V′□=∼∙′∂∼ ⁣(∘′:V:)+Γ∙□′∘□′:V:=∼∙′∘′∼:V:+∼∙′∘′:∂∼V:+Γ∙□′∘□′:V: \begin{align*} \nabla'_\bullet V'^\circ &= \partial'_\bullet V'^\circ + \Gamma'^\circ_{\bullet\square}V'^\square\\ &= \frac{\sim}{\bullet^\prime}\partial_\sim\!\left(\frac{\circ^\prime}{:}V^:\right) + \Gamma'^\circ_{\bullet\square}\frac{\square^\prime}{:}V^:\\ &= \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{\sim :}V^: + \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{:}\partial_\sim V^:\\ &\quad +\Gamma'^\circ_{\bullet\square}\frac{\square^\prime}{:}V^: \end{align*}
Now enforce the tensor transformation law for ∇V\nabla V and expand ∇∼V:\nabla_\sim V^: once more.
∇μ′V′ν=∂xρ∂x′μ∂x′ν∂xα∇ρVα=∂xρ∂x′μ∂x′ν∂xα(∂ρVα+ΓργαVγ)=∂xρ∂x′μ∂x′ν∂xα∂ρVα+∂xρ∂x′μ∂x′ν∂xσΓρασVα \begin{align*} \nabla'_\mu V'^\nu &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\alpha}\nabla_\rho V^\alpha\\ &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\alpha} \left(\partial_\rho V^\alpha + \Gamma^\alpha_{\rho\gamma}V^\gamma\right)\\ &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\alpha}\partial_\rho V^\alpha\\ &\quad + \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\sigma}\Gamma^\sigma_{\rho\alpha}V^\alpha \end{align*}
∇∙′V′∘=∼∙′∘′:∇∼V:=∼∙′∘′:(∂∼V:+Γ∼α:Vα)=∼∙′∘′:∂∼V:+∼∙′∘′αΓ∼:αV: \begin{align*} \nabla'_\bullet V'^\circ &= \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{:}\nabla_\sim V^:\\ &= \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{:} \left(\partial_\sim V^: + \Gamma^:_{\sim\alpha}V^\alpha\right)\\ &= \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{:}\partial_\sim V^: + \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{\alpha}\Gamma^\alpha_{\sim:}V^: \end{align*}
The terms ∼∙′∘′:∂∼V:\frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{:}\partial_\sim V^: are exactly the same in both expansions. Cancel them, use that this holds for arbitrary VV, then multiply by the inverse Jacobian.
Γμλ′ν∂x′λ∂xα=∂xρ∂x′μ∂x′ν∂xσΓρασ−∂xρ∂x′μ∂2x′ν∂xρ∂xα⇒Γμλ′ν=∂xρ∂x′μ∂xγ∂x′λ∂x′ν∂xσΓργσ−∂xρ∂x′μ∂xγ∂x′λ∂2x′ν∂xρ∂xγ \begin{align*} \Gamma'^\nu_{\mu\lambda}\frac{\partial x'^\lambda}{\partial x^\alpha} &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x'^\nu}{\partial x^\sigma}\Gamma^\sigma_{\rho\alpha} - \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial^2 x'^\nu}{\partial x^\rho\partial x^\alpha}\\ \Rightarrow\qquad \Gamma'^\nu_{\mu\lambda} &= \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\gamma}{\partial x'^\lambda} \frac{\partial x'^\nu}{\partial x^\sigma}\Gamma^\sigma_{\rho\gamma}\\ &\quad - \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\gamma}{\partial x'^\lambda} \frac{\partial^2 x'^\nu}{\partial x^\rho\partial x^\gamma} \end{align*}
Γ∙□′∘□′:=∼∙′∘′αΓ∼:α−∼∙′∘′∼: \begin{align*} \Gamma'^\circ_{\bullet\square}\frac{\square^\prime}{:} &= \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{\alpha}\Gamma^\alpha_{\sim:} - \frac{\sim}{\bullet^\prime}\frac{\circ^\prime}{\sim:}\\ \end{align*} Γ∙□′∘=∼∙′:□′∘′αΓ∼:α−∼∙′:□′∘′∼: \begin{aligned} \Gamma'^\circ_{\bullet\square} &= \frac{\sim}{\bullet^\prime}\frac{:}{\square^\prime}\frac{\circ^\prime}{\alpha}\Gamma^\alpha_{\sim:} - \frac{\sim}{\bullet^\prime}\frac{:}{\square^\prime}\frac{\circ^\prime}{\sim:} \end{aligned}

Note how the second derivative is the reason why the connection coefficients do not transform like a tensor.

I’m curious to know what you think about this “enhancement” of an existing notation. Some of my fellow students were quite skeptical and didn’t like, saying it wasn’t an improvement. But I think it’s a fun way to make the notation more intuitive and easier to read as my spatial recognition skills are better for shapes than for letters. I fully agree that it might take some time getting used to.

Even with the new notation, index calculations are still technical and tedious, but at least they are a bit more enjoyable. I’m also happy to learn about edge cases where this notation might not work well in practice. If you have any suggestions for improvements, please let me know!

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