Corrigendum: Area deficits and the Bel–Robinson tensor (2018 Class. Quantum Grav.35 085005)

We have noticed a computational mistake in the paper. Though the main conclusions and results do not change, there are some sentences and explanations that must be corrected as they could be misleading.

The following is the list of needed changes given in order of appearance:

(i) In p 3, the sentence 'This turns out to be equivalent to demanding that the null expansions are constant over the surface of the ball', towards the middle of the first paragraph, must be removed.

(ii) In p 13, the last sentence in the paragraph after (60) must be removed.

(iii) The index in the summation (61) must be s instead of i.

(iv) In p 11, paragraph after formula (47), 'if the radius is held constant' should be replaced by 'if the spherical average X of the second order contribution to the radius variation is zero'.

(v) At the end of the same paragraph, 'there must be a radius variation' should be replaced by 'X must be nonzero'.

(vi) Formula (65) should read

$$\begin{equation*}1{{{\Gamma}}_{AB}^{r}}=\frac{2}{3}{r}^{3}{n}^{k}{n}^{l}\frac{\partial {n}^{i}}{\partial {\theta }^{A}}\frac{\partial {n}^{j}}{\partial {\theta }^{B}}{D}_{\mathit{\text{ikjl}}}-~{\delta }{\ell }_{1}{{\Omega}}_{AB}.\end{equation*}$$

This is the main error as it propagates into several other formulae.

(vii) The first summation in (67) should have the factor (s + 1) instead of s.

(viii) Equations (68) and (69) should be, respectively,

$$\begin{align*}K& =\frac{d-2}{\ell }+\frac{1}{{\ell }^{2}}\sum\limits _{s=1}^{\infty }(d-2+s)(s-1){Y}_{{i}_{1}\dots {i}_{s}}{n}^{{i}_{1}}\dots {n}^{{i}_{s}}-\frac{\ell }{3}{E}_{ij}{n}^{i}{n}^{j},\\ K& =\frac{d-2}{\ell }+\frac{1}{{\ell }^{2}}\sum\limits _{s\ne 2}(d-2+s)(s-1){Y}_{{i}_{1}\dots {i}_{s}}{n}^{{i}_{1}}\dots {n}^{{i}_{s}}\\ & \quad +{n}^{i}{n}^{j}\left(\frac{d}{{\ell }^{2}}{Y}_{ij}-\frac{\ell }{3}{E}_{ij}\right).\end{align*}$$

(ix) Relation (70) should finally be

$$\begin{equation*}{Y}_{ij}=\frac{{\ell }^{3}}{3d}{E}_{ij}\end{equation*}$$

and the first sentence just following (70) should be ignored.

(x) In the last-but-one sentence of section 5 the value γ = 1/6 must be corrected to γ = 1/(3d).

(xi) Formula (119) contains an erratum and a missing term. Its correct expression reads

$$\begin{align*}K& =\frac{d-2}{\ell }-\frac{\ell }{3}{E}_{ij}{n}^{i}{n}^{j}+\frac{7}{12}{\ell }^{2}{n}^{k}{n}^{l}{n}^{m}{\nabla }_{k}{R}_{0l0m}\\ & \quad -\frac{{\ell }^{3}}{45}{n}^{i}{n}^{j}{n}^{k}{n}^{l}({D}_{i}^{\enspace \enspace m}{}_{ j}{}^{\enspace \enspace n}D_{kmln}+4{H}_{i}^{\enspace \enspace m}{}_{j}H_{kml})\\ & \quad -\frac{1}{10}{\ell }^{3}{n}^{k}{n}^{l}{n}^{m}{n}^{n}{\nabla }_{k}{\nabla }_{l}{R}_{0m0n.}\end{align*}$$

(xii) In p 24, 2nd paragraph, the sentence 'or, equivalently, keeping its null expansions constant over the surface' must be deleted.

Acknowledgments

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