1 Correction to: Gen Relativ Gravit https://doi.org/10.1007/s10714-020-02693-z

2 Error warning for sections 3 and 4

Equ. (23) is wrong and thus the contribution of \(L_{D^2\phi }\) to the energy tensor of the scalar field. This flaw is fatal for the derivations in the Milgrom regime. The derivations in sections 3 and 4 can no longer be upheld.

3 Corrigenda of typos and intermediate calculations

For readers interested in the methodology of the approach the following local corrections may be useful. In several quadratic derivative terms the field symbol after the first derivative is omitted, which gives erroneously the impression of a second order derivative. In addition some signs are to be corrected. Some entries in the Einstein equation (17) change accordingly accordingly:

Replace the right hand side (rhs) of eq. (13) by

$$\begin{aligned} \underset{Eg}{\doteq }- \frac{\gamma }{2} (\xi \phi _0)^2 (_g{}\nabla _{\nu }\partial ^{\nu } \sigma +\partial _{\nu }\sigma \partial ^{\nu } \sigma ) \end{aligned}$$

Replace the rhs the formula preceding eq. (19) by

$$\begin{aligned} = 2 \phi ^{-2} \left( D_{\mu }\phi D_{\nu }\phi + \phi D_{(\mu }D_{\nu )}\phi - (\phi D_{\lambda }D^{\lambda }\phi + D_{\lambda }\phi \, D^{\lambda }\phi )\, g_{\mu \nu } \right) \end{aligned}$$

Replace the rhs of eq. (21) by

$$\begin{aligned} \underset{Eg}{\doteq } \alpha ( \partial _{\mu } \sigma \partial _{\nu } \sigma - \frac{1}{2}\partial _{\lambda }\sigma \partial ^{\lambda } \sigma \, g_{\mu \nu } ) \end{aligned}$$

Replace eq. (25) by

$$\begin{aligned} - 2 L_H - tr\, T^{(bar)} + (-\gamma +6)\xi ^2 \phi D_{\lambda }D^{\lambda }\phi + (\alpha +6)\xi ^2 D_{\lambda }\phi D^{\lambda }\phi - L_{D\phi ^3} - 4 L_V = 0 \end{aligned}$$

Replace eq. (27) by

$$\begin{aligned} 2 L_H + (\alpha + \gamma )\, \xi ^2 \phi \, D_{\lambda } D^{\lambda }\phi + 4 L_{D\phi ^3} + 2 \beta \xi ^3 \phi ^{-1}\, D_{\lambda }\left( |D \phi | D^{\lambda }\phi \right) + 4 L_V = 0 \end{aligned}$$

Replace eq. (57) by

$$\begin{aligned} \sigma (r) = \sqrt{a_0 M}\, \ln r \end{aligned}$$

Replace eq. (81) by

$$\begin{aligned} {}_{\varphi }R = - (n-1)(n-2)\, \varphi _{\lambda }\varphi ^{\lambda }- 2(n-1 )\,_g\nabla _{\lambda }\varphi ^{\lambda } \end{aligned}$$

Moreover, the following coefficient errors are to be corrected:

Replace the rhs of (19) by

$$\begin{aligned} \underset{Eg}{\doteq }-2 \, _g{}{\nabla }_{(\mu } \partial _{\nu )} \sigma + 2\, _g \square \, \sigma \, g_{\mu \nu } \end{aligned}$$

Replace the rhs of eq. (23) by

$$\begin{aligned} \underset{Eg}{\doteq }\gamma \left( _g \nabla _{\mu }\partial _{\nu } \sigma + \partial _{\mu } \sigma \partial _{\nu } \sigma \right) - \frac{\gamma }{2} \left( \, _g\nabla _{\lambda } \sigma ^{\lambda } + \partial _{\lambda } \sigma \, \partial ^{\lambda } \sigma \right) \, g_{\mu \nu } \end{aligned}$$

Replace the rhs of the formula preceding eq. (85) by

$$\begin{aligned} \underset{Rg}{\doteq }- \tilde{\phi }\; _g\nabla _{\mu }\partial ^{\mu } \sigma \end{aligned}$$