Geodesic deviation at higher orders via covariant bitensors

Abstract

We review a simple but instructive application of the formalism of covariant bitensors, to use a deviation vector field along a fiducial geodesic to describe a neighboring worldline, in an exact and manifestly covariant manner, via the exponential map. Requiring the neighboring worldline to be a geodesic leads to the usual linear geodesic deviation equation for the deviation vector, plus corrections at higher order in the deviation and relative velocity. We show how these corrections can be efficiently computed to arbitrary orders via covariant bitensor expansions, deriving a form of the geodesic deviation equation valid to all orders, and producing its explicit expanded form through fourth order. We also discuss the generalized Jacobi equation, action principles for the higher-order geodesic deviation equations, results useful for describing accelerated neighboring worldlines, and the formal general solution to the geodesic deviation equation through second order.

Appendices

Appendix 1: Generic correspondences/parametrizations and the normal correspondence $(u\cdot \xi =0)$

The main text above restricted attention to the isochronous correspondence, in which both the fiducial geodesic $y(\tau )$ and the neighboring worldline $z(\tau )$, related to the deviation vector field by $\xi ^\alpha (\tau )=-\sigma ^\alpha \big (y(\tau ),z(\tau )\big )$, are affinely parametrized. Relaxing this restriction leads to different forms for the GDE and its action principle.

The key results relating the deviation vector $\xi (\tau )$ and the worldline $z(\tau )$ which are still valid for arbitrary correspondences/parametrizations, and which are valid to all orders in the deviation and relative velocity, are Eqs. (41) and (42), which express the tangent $v^\mu$ to $z(\tau )$ and its covariant $\tau$-derivative:

$$\begin{aligned} v^\mu&={K^\mu }_\beta u^\beta + {H^\mu }_\beta \dot{\xi }^\beta ,\end{aligned}$$

(52)

$$\begin{aligned} \dot{v}^\mu&={H^\mu }_\alpha \ddot{\xi }^\alpha +{L^\mu }_{\beta \gamma }u^\beta u^\gamma +2{J^\mu }_{\beta \gamma }u^\beta \dot{\xi }^\gamma +{I^\mu }_{\beta \gamma }\dot{\xi }^\beta \dot{\xi }^\gamma . \end{aligned}$$

(53)

The forms of the GDE and its action which apply to arbitrary correspondences can be found quite simply by replacing the geodesic equation (45) for $z(\tau )$ and its action (46), which applied when $z(\tau )$ was affinely parametrized, with their reparametrization-invariant versions (assuming $v^\mu$ is not null):

$$\begin{aligned} 0&=a^\mu =\frac{1}{\sqrt{|v^2|}}\frac{D}{d\tau }\left( \frac{v^\mu }{\sqrt{|v^2|}}\right) , \end{aligned}$$

(54)

$$\begin{aligned} S[\xi ]&=-\int d\tau \sqrt{|v^2|}. \end{aligned}$$

(55)

Though we will only consider a few cases explicitly, Eqs. (54, 55) can be expanded to any order in the $O(\xi ,\dot{\xi })^n$ or $O(\xi ^n)$ expansion scheme, via Eqs. (52, 53) and the results of Appendix 3.

1.1 The generic case to $O(\xi ,\dot{\xi })$

Considering for simplicity the timelike case with $u^2=-1$, expanding the action (55) to $O(\xi ,\dot{\xi })^2$ yields

$$\begin{aligned} S[\xi ]&=\int d\tau \Big [-1+u\cdot \dot{\xi } +\tfrac{1}{2}\dot{\xi }^2+\tfrac{1}{2}(u\cdot \dot{\xi })^2-\tfrac{1}{2}R_{\xi u\xi u}+O(\xi ,\dot{\xi })^3 \Big ], \end{aligned}$$

(56)

which, ignoring the constant and the total derivative, differs from the $S_\mathrm {isoc.}$ of Eq. (22) by the addition of the $\tfrac{1}{2}(u\cdot \dot{\xi })^2$ term. This term modifies the equation of motion from the usual linear GDE (23) to

$$\begin{aligned} {P^\alpha }_\beta \ddot{\xi }^\beta +{R^\alpha }_{u\xi u}=O(\xi ,\dot{\xi })^2. \end{aligned}$$

(57)

where ${P^\alpha }_\beta ={\delta ^\alpha }_\beta +u^\alpha u_\beta$ is the tensor which projects orthogonal to $u$. Thus, this generic GDE (57) constrains only the components of $\xi$ orthogonal to $u$, leaving the component along $u$ completely unconstrained.

Note that, in the analysis of Eqs. (56, 57), we have assumed $\ddot{\xi }=O(\xi ,\dot{\xi })$, which is true when $z(\tau )$ is a geodesic, as implied by the GDE (57). When $a^\mu \ne 0$, however, this is not true. The proper generalization of Eq. (25), giving the acceleration vector to $O(\xi ,\dot{\xi })$, to the generic case, is found by expanding Eq. (56) while assuming $\ddot{\xi }=O(\xi ,\dot{\xi })^0$, yielding

$$\begin{aligned} a^\mu= & {} {g^\mu }_\alpha \Big [(1+2u\cdot \dot{\xi }){P^\alpha }_\beta \ddot{\xi }^\beta +{R^\alpha }_{u\xi u} +(u\cdot \ddot{\xi }){P^\alpha }_\beta \dot{\xi }^\beta \nonumber \\&+\Big \{\dot{\xi }\cdot \ddot{\xi }+(u\cdot \dot{\xi })(u\cdot \ddot{\xi })\Big \}u^\alpha +O(\xi ,\dot{\xi })^2\Big ]. \end{aligned}$$

(58)

Note that setting $a^\mu =0$ still yields the generic GDE (57).

1.2 The normal correspondence and Fermi normal coordinates

Consider the case where the deviation vector $\xi$ is constrained to be orthogonal to $u$, satisfying $u\cdot \xi =0$, which defines the ‘normal correspondence’. In this case, if $\{u^\alpha ,e^\alpha _i\}$ with $i=1,2,3$ is an orthonormal tetrad which is parallel-transported along the fiducial geodesic, then

$$\begin{aligned} \xi ^\alpha = x^i e^\alpha _i, \end{aligned}$$

(59)

and the components $x^i$ (along with $\tau$) correspond to Fermi normal coordinates [11] based on the fiducial geodesic. We will use the notation exemplified by ${R^i}_{0j0}=e^i_\alpha u^\beta e^\gamma _j u^\delta {R^\alpha }_{\beta \gamma \delta },$ for the frame components of the Riemann tensor, where $\{-u_\alpha ,e^i_\alpha \}$ is the dual tetrad. Spatial frame indices will be raised and lowered with the Euclidean 3-metric $\delta _{ij}$.

1.2.1 The usual linear geodesic deviation equation

This case can be treated by simply taking the results of Sect. 1.1 for the generic correspondence and replacing $\xi ^\alpha$ with Eq. (59). The action (56) [dropping the first line] and GDE (57) become

$$\begin{aligned} S_\mathrm {norm.}[x]&=\int d\tau \Big [\tfrac{1}{2}\dot{x}^i\dot{x}_i-\tfrac{1}{2}R_{i0j0}x^i x^j+O(x,\dot{x})^3\Big ]\end{aligned}$$

(60)

$$\begin{aligned}&\Rightarrow \quad \quad \ddot{x}^i+{R^i}_{0j0} x^j=O(x,\dot{x})^2, \end{aligned}$$

(61)

which are straight-forward adaptations of their isochronous counterparts. As mentioned above, the normal correspondence is a special case of the isochronous correspondence, to $O(\xi ,\dot{\xi })$, when $z(\tau )$ is a geodesic. A significant difference with the isochronous case occurs for accelerated worldlines $z(\tau )$; from Eq. (58), the acceleration vector of $z(\tau )$ in the normal correspondence is given by

$$\begin{aligned} a_\mathrm {norm.}^\mu ={g^\mu }_\alpha \Big [\left( \ddot{x}^i+{R^i}_{0j0}x^j\right) e^\alpha _i+\dot{x}^i\ddot{x}_iu^\alpha +O(x,\dot{x})^2\Big ], \end{aligned}$$

(62)

which differs from the $a^\mu _\mathrm {isoc.}$ of Eq. (25) by the addition of the final term; this term term does not affect the GDE, as one can verify that $a^\mu _\mathrm {norm.}=0$ still leads to Eq. (61).

1.2.2 Bażański’s equation

From Eqs. (52–55, 59, 47), Bażański’s equation and its Lagrangian in the normal correspondence are

(63)

which are straight-froward adaptations of Eqs. (48, 49). The normal correspondence is still a special case of the isochronous correspondence to this order, but one can verify that this fails to be true at the next order and higher.

1.2.3 The generalized Jacobi equation

From Eqs. (52-55, 59, 47), the GJE and its Lagrangian in the normal correspondence are

$$\begin{aligned} \mathcal {L}&=-\sqrt{1-\dot{x}^2}\bigg [1+\frac{1}{2(1-\dot{x}^2)}\Big (R_{i0j0}+\tfrac{4}{3}R_{ikj0}\dot{x}^k +R_{ikjl}\dot{x}^k\dot{x}^l\Big )x^i x^j+O(x^3)\bigg ], \nonumber \\&\Rightarrow \quad \ddot{x}^i+{R^i}_{0j0}x^j+2{R^i}_{jk0}\dot{x}^j x^k+\tfrac{2}{3}{R^i}_{jkl}\dot{x}^j x^k \dot{x}^l \nonumber \\&\quad \quad \quad \quad +\left( 2{R}_{0jk0}\dot{x}^j x^k+\tfrac{2}{3}{R}_{0jkl}\dot{x}^j x^k \dot{x}^l\right) \dot{x}^i=O(x^2). \end{aligned}$$

(64)

Appendix 2: The general solution to the second-order geodesic deviation equation

To derive the general solution to the second-order GDE, Bażański’s equation (48), in the isochronous correspondence, the setup is precisely the same as in Sect. 4, where we derived the solution to the linear GDE; see Fig. 4. The equations which define the geometrical relations summarized by Fig. 4 to all orders are Eqs. (28, 26, 27) and the first equality of Eq. (29),

$$\begin{aligned} \xi ^\alpha&=-\sigma ^\alpha (y,z), \nonumber \\ \tau u^\alpha&=-\sigma ^\alpha (y,y'), \nonumber \\ \xi ^{\alpha '}&=-\sigma ^{\alpha '}(y',z'), \nonumber \\ \tau v^\mu&=-\sigma ^\mu (z,z'), \end{aligned}$$

(65)

and the valid-to-all-orders generalization of the second equality of Eq. (29), which relates $v^\mu$ to $\dot{\xi }^\alpha$, is Eq. (41b),

$$\begin{aligned} v^\mu&={K^\mu }_\alpha u^\alpha +{H^\mu }_\alpha \dot{\xi }^\alpha ={g^\mu }_\alpha \left[ u^\alpha +\dot{\xi }^\alpha -\tfrac{1}{2}{R^\alpha }_{\xi u\xi }+O(\xi ,\dot{\xi })^3\right] , \end{aligned}$$

(66)

where the second line has expanded to second order, matching Eq. (20).

Fig. 4

The deviation vector $\xi ^\alpha$ and its derivative $\dot{\xi }^\alpha$ at the initial point $y$ on the fiducial geodesic determine the solution for the deviation vector $\xi ^{\alpha '}$ at the second point $y'$

Next, just as in Eqs. (30, 31) but now to second order, we expand the function $\sigma ^\mu (z,z')$ as $z\rightarrow y$ with $z'$ fixed, in powers of $\xi ^\alpha =-\sigma ^\alpha (y,z)$,

$$\begin{aligned} \sigma ^\mu (z,z')&={g^\mu }_\alpha \Big [\sigma ^\alpha (y,z') +{\sigma ^\alpha }_\beta (y,z')\,\xi ^\beta +\tfrac{1}{2}{\sigma ^\alpha }_{\beta \gamma }(y,z')\,\xi ^\beta \xi ^\gamma +O(\xi ^3)\Big ], \end{aligned}$$

(67)

and then re-expand the resultant $(y,z')$ bitensors as $z'\rightarrow y'$ with $y$ fixed, in powers of $\xi ^{\alpha '}=-\sigma ^{\alpha '}(y',z')$,

$$\begin{aligned} \sigma ^\alpha (y,z')&=\sigma ^\alpha (y,y')+{\sigma ^\alpha }_{\beta '}(y,y')\,\xi ^{\beta '} +\tfrac{1}{2}{\sigma ^\alpha }_{\beta '\gamma '}(y,y')\,\xi ^{\beta '}\xi ^{\gamma '}+O(\xi ^3), \nonumber \\ {\sigma ^\alpha }_\beta (y,z')&={\sigma ^\alpha }_\beta (y,y')+{\sigma ^\alpha }_{\beta \gamma '}(y,y')\,\xi ^{\gamma '}+O(\xi ^2),\end{aligned}$$

(68)

$$\begin{aligned} {\sigma ^\alpha }_{\beta \gamma }(y,z')&={\sigma ^\alpha }_{\beta \gamma }(y,y')+O(\xi ). \end{aligned}$$

(69)

Combining Eqs. (65–69) yields the key relation

$$\begin{aligned} -\tau v^\mu&=-\tau {g^\mu }_\alpha \left[ u^\alpha +\dot{\xi }^\alpha -\tfrac{1}{2}{R^\alpha }_{\xi u\xi }+O(\xi ,\dot{\xi })^3\right] \nonumber \\&=\sigma ^\mu (z,z')={g^\mu }_\alpha \Big [-\tau u^\alpha +{\sigma ^\alpha }_\beta \xi ^\beta +{\sigma ^\alpha }_{\beta '}\xi ^{\beta '} \nonumber \\&\quad +\tfrac{1}{2}{\sigma ^\alpha }_{\beta \gamma }\xi ^\beta \xi ^\gamma +{\sigma ^\alpha }_{\beta \gamma '}\xi ^\beta \xi ^{\gamma '}+\tfrac{1}{2}{\sigma ^\alpha }_{\beta '\gamma '}\xi ^{\beta '}\xi ^{\gamma '}+O(\xi ,\dot{\xi })^3\Big ], \end{aligned}$$

(70)

where the arguments of all the second and third derivatives of the world function are $(y,y')$. This equation can be perturbatively solved for $\xi ^{\alpha '}$, using the first-order solution $\xi ^{\alpha '}={K^{\alpha '}}_\beta \xi ^\beta +\tau {H^{\alpha '}}_\beta \dot{\xi }^\beta +O(\xi ,\dot{\xi })^2$ from Eqs. (32–34) in the last two terms. The result, the solution to the second order GDE, is

$$\begin{aligned} \xi ^{\alpha '}&={K^{\alpha '}}_\beta \xi ^\beta +\tau {H^{\alpha '}}_\beta \dot{\xi }^\beta \nonumber \\&+\tfrac{1}{2}\left( {L^{\alpha '}}_{\beta \gamma }+\tau {H^{\alpha '}}_\delta {R^\delta }_{\beta \gamma u}\right) \xi ^\beta \xi ^\gamma \nonumber \\&+\tau {J^{\alpha '}}_{\beta \gamma }\xi ^\beta \dot{\xi }^\gamma +\tfrac{1}{2} \tau ^2 {I^{\alpha '}}_{\beta \gamma } \dot{\xi }^\beta \dot{\xi }^\gamma +O(\xi ,\dot{\xi })^3, \end{aligned}$$

(71)

where the expressions for the bitensors $L$, $J$, and $I$ resulting from this derivation are as in Eq. (43a) [with the $\mu$-type indices replaced by $\alpha '$-type indices].

Appendix 3: Covariant expansions via the semi-recursive/transport-equation method

We summarize here recursion relations which allow one to efficiently generate high-order covariant expansions of fundamental bitensors. We import several results from Ottewill and Wardell (OW) [15, 16], who built on the work of Avramidi [12, 13] and Décanini and Folacci [14], deriving the recursion relations from transport equations obeyed by the bitensors. We have implemented the recursion relations in a Mathematica notebook employing the xTensor package [44].

Given any bitensor ${T^\mu }_{\beta \ldots }(y,z)$ [using the index conventions of Sect. 2], we can define a bitensor ${T^\alpha }_{\beta \ldots }(y,z)$ with all the indices at $z$ parallel-transported to indices at $y$,

$$\begin{aligned} {T^\mu }_{\beta \ldots }={g^\mu }_\alpha \ldots {T^\alpha }_{\beta \ldots }, \end{aligned}$$

(72)

and ${T^\alpha }_{\beta \ldots }$ can be expanded in powers of the deviation vector $\xi ^\alpha \equiv -\sigma ^\alpha (y,z)$ as

$$\begin{aligned} {T^\alpha }_{\beta \ldots }=\sum _{n=0}^\infty \frac{1}{n!}{T^\alpha }_{\beta \ldots (n)}, \end{aligned}$$

(73)

where ${T^\alpha }_{\beta \ldots (n)}$ is some tensor at $y$ contracted with $n$ factors of $\xi ^\alpha$:

$$\begin{aligned} {T^\alpha }_{\beta \ldots (n)}(y,z)={t^\alpha }_{\beta \ldots \delta _1\ldots \delta _n}(y)\,\xi ^{\delta _1}\ldots \xi ^{\delta _n}. \end{aligned}$$

(74)

The following subsections present recursion relations for the summands ${T^\alpha }_{\beta \ldots (n)}$ for several key bitensors, with many taken directly from OW and others constructed from results of Ref. [30]. Later recursion relations involve the results of earlier ones. Note that our summands ${T^\alpha }_{\beta \ldots (n)}$ are $(-1)^n$ times those of OW [c.f. their Eq. (2.39) and our Eqs. (72–74) and $\xi ^\alpha =-\sigma ^\alpha$], though these signs cancel out in all of the recursion relations, and we have renamed several symbols as noted below.

1.1 World-function second derivatives and Jacobi propagators

As in Eq. (72), we can write the three world-function second derivatives and the two Jacobi propagators (33, 34) in terms of bitensors with indices only at $y$ as

$$\begin{aligned} {\sigma ^\mu }_\nu= & {} {g^\mu }_\alpha {g^\beta }_\nu {C^\alpha }_\beta , \\ {\sigma ^\alpha }_\mu= & {} -{g^\beta }_\mu {D^\alpha }_\beta , \\ {\sigma ^\alpha }_\beta= & {} {E^\alpha }_\beta , \\ {H^\mu }_\beta= & {} {g^\mu }_\alpha {H^\alpha }_\beta , \\ {K^\mu }_\beta= & {} {g^\mu }_\alpha {K^\alpha }_\beta . \end{aligned}$$

We have renamed symbols, OW $\rightarrow$ here, as $\xi \rightarrow C$, $\eta \rightarrow -D$, $\lambda \rightarrow E$, $\gamma \rightarrow -H$, and $K$ is not discussed by OW.

Defining the quantities

$$\begin{aligned} {\kappa ^\alpha }_{\beta (n)}={R^\alpha }_{\delta _1\beta \delta _2;\delta _3\ldots \delta _n}\xi ^{\delta _1}\ldots \xi ^{\delta _n} \end{aligned}$$

(75)

($\mathcal {K}\rightarrow \kappa$), the recursion relations for $H$, $D$, and $C$, which are OW’s Eqs. (4.7, 4.9, 4.10), and which follow from the transport equation OW (3.15) and the relations OW (4.8, 3.14), are given by

$$\begin{aligned} {H^\alpha }_{\beta (0)}&={D^\alpha }_{\beta (0)}={C^\alpha }_{\beta (0)}={\delta ^\alpha }_\beta , \nonumber \\ {H^\alpha }_{\beta (1)}&={D^\alpha }_{\beta (1)}={C^\alpha }_{\beta (1)}=0, \nonumber \\ {H^\alpha }_{\beta (n)}&=-\frac{n-1}{n+1}\sum _{k=0}^{n-2}{n-2 \atopwithdelims ()k}{\kappa ^\alpha }_{\gamma (n-k)}{H^\gamma }_{\beta (k)}, \nonumber \\ {D^\alpha }_{\beta (n)}&=\sum _{k=2}^{n}{n \atopwithdelims ()k}{H^\alpha }_{\gamma (k)}{D^\gamma }_{\beta (n-k)}, \nonumber \\ {C^\alpha }_{\beta (n)}&=-n\,{D^\alpha }_{\beta (n)}-\sum _{k=2}^{n-2}{n \atopwithdelims ()k}{H^\alpha }_{\gamma (n-k)}{D^\gamma }_{\beta (k)}. \end{aligned}$$

(76)

A recursion relation for our $E$ (their $\lambda$) is given by OW (4.11, 4.12), following from the transport equation OW (3.17). We have found an alternate route to $E$, by first finding $K$ from a ‘transport equation’ involving horizontal and vertical covariant derivatives,

$$\begin{aligned} {K^\mu }_\alpha =\left( \xi ^\beta \nabla _{*\beta }-\xi ^\beta \nabla _{\beta *}+1\right) {H^\mu }_\alpha , \end{aligned}$$

as detailed in Ref. [30] (see also Footnote 5), and then inverting Eq. (33). Introducing the auxiliary quantities $dH$, the resultant recursion relations are

$$\begin{aligned} {dH^\alpha }_{\beta (n)}&=-\frac{n-2}{n}\sum _{k=0}^{n-3}{n-3 \atopwithdelims ()k}\Big [{\kappa ^\alpha }_{\gamma (n-k)}{H^\gamma }_{\beta (k)} +{\kappa ^\alpha }_{\gamma (n-k-1)}{dH^\gamma }_{\beta (k+1)}\Big ], \nonumber \\ {K^\alpha }_{\beta (n)}&=(n+1){H^\alpha }_{\beta (n)}-n\;{dH^\alpha }_{\beta (n)}. \nonumber \\ {E^\alpha }_{\beta (n)}&=\sum _{k=0}^{n}{n \atopwithdelims ()k}{D^\alpha }_{\gamma (n-k)}{K^\gamma }_{\beta (k)}. \end{aligned}$$

(77)

1.2 Parallel-propagator first derivatives and ‘second-order Jacobi propagators’

We can write the two first derivatives of the parallel propagator and the three bitensors of Eq. (43), which one might call the ‘second-order Jacobi propagators,’ as

$$\begin{aligned} g_{\mu \alpha ;\nu }&={g^\beta }_\mu {g^\gamma }_\nu A_{\alpha \beta \gamma }, \nonumber \\ g_{\mu \alpha ;\gamma }&={g^\beta }_\mu B_{\alpha \beta \gamma }, \nonumber \\ {I^\mu }_{\beta \gamma }&={g^\mu }_\alpha {I^\alpha }_{\beta \gamma }, \nonumber \\ {J^\mu }_{\beta \gamma }&={g^\mu }_\alpha {J^\alpha }_{\beta \gamma }, \nonumber \\ {L^\mu }_{\beta \gamma }&={g^\mu }_\alpha {L^\alpha }_{\beta \gamma }. \end{aligned}$$

(78)

The bitensors $A$ and $B$ coincide with those of OW, and they do not discuss $I$, $J$, and $L$.

Defining the quantities

$$\begin{aligned} {\mathcal {R}}_{\alpha \beta \gamma (n)}=R_{\alpha \beta \delta _1\gamma ;\delta _2\ldots \delta _n}\xi ^{\delta _1}\ldots \xi ^{\delta _n}, \end{aligned}$$

the recursion relations for $A$ and $B$ are

$$\begin{aligned} A_{\alpha \beta \gamma (n)}&=-\frac{n}{n+1}\mathcal {R}_{\alpha \beta \gamma (n)} -\frac{1}{n+1}\sum _{k=0}^{n-2}{n \atopwithdelims ()k}A_{\alpha \beta \delta (k)}{C^\delta }_{\gamma (n-k)}, \nonumber \\ B_{\alpha \beta \gamma (n)}&=\frac{1}{n}\sum _{k=0}^{n}{n \atopwithdelims ()k}A_{\alpha \beta \delta (k)}{D^\delta }_{\gamma (n-k)}. \end{aligned}$$

(79)

Though recursion relations for $I$, $J$, and $L$ could be constructed via Eq. (43a) using those for the world-function third derivatives (some of which are given by OW), or via Eq. (43b), we have found it most efficient to use the relations

$$\begin{aligned} {I^\mu }_{\beta \gamma }= & {} \nabla _{*\gamma }{H^\mu }_\beta ,\nonumber \\ {J^\mu }_{\beta \gamma }= & {} \nabla _{\beta *}{H^\mu }_\gamma ,\nonumber \\ {L^\mu }_{\beta \gamma }= & {} \nabla _{\gamma *}{K^\mu }_\beta \end{aligned}$$

(80)

(see Footnotes 5 and 6 and Ref. [30]). The resultant recursion relations, introducing several auxiliary quantities along the way, are

$$\begin{aligned} {\delta \kappa ^\alpha }_{\beta \gamma (n)}= & {} {R^\alpha }_{(\gamma |\beta |\delta _1;\delta _2\ldots \delta _n)}\xi ^{\delta _1}\ldots \xi ^{\delta _n},\nonumber \\ {\Delta \kappa ^\alpha }_{\beta \gamma (n)}= & {} {R^\alpha }_{\delta _1\beta \delta _2;\delta _3\ldots \delta _n\gamma }\,\xi ^{\delta _1}\ldots \xi ^{\delta _n}, \nonumber \\ \delta g_{\alpha \beta \gamma (n)}= & {} \sum _{k=0}^{n}{n\atopwithdelims ()k}A_{\alpha \beta \delta (n-k)}{H^\delta }_{\gamma (k)}, \nonumber \\ \Delta g_{\alpha \beta \gamma (n)}= & {} B_{\alpha \beta \gamma (n)}+\sum _{k=0}^{n}{n\atopwithdelims ()k}A_{\alpha \beta \delta (n-k)}{K^\delta }_{\gamma (k)}, \end{aligned}$$

(81)

(82)

The explicit results needed to write out the $O(\xi ,\dot{\xi })^4$ GDE [from Eq. (45)] are

$$\begin{aligned} {K^\mu }_\beta= & {} \;{g^\mu }_\alpha \Bigg [{\delta ^\alpha }_\beta -\tfrac{1}{2}{R^\alpha }_{\xi \beta \xi }-\tfrac{1}{6}{R^\alpha }_{\xi \beta \xi ;\xi } +\tfrac{1}{24}\Big (-{R^\alpha }_{\xi \beta \xi ;\xi \xi }+{R^\alpha }_{\xi \gamma \xi }{R^\gamma }_{\xi \beta \xi }\bigg ) \nonumber \\&+\tfrac{1}{120}\Big (-{R^\alpha }_{\xi \beta \xi ;\xi \xi \xi }+3{R^\alpha }_{\xi \gamma \xi ;\xi }{R^\gamma }_{\xi \beta \xi }+{R^\alpha }_{\xi \gamma \xi }{R^\gamma }_{\xi \beta \xi ;\xi }\bigg )+O(\xi ^6)\Bigg ], \nonumber \\ {H^\mu }_\beta= & {} \;{g^\mu }_\alpha \Bigg [{\delta ^\alpha }_\beta {-}\tfrac{1}{6}{R^\alpha }_{\xi \beta \xi }{-}\tfrac{1}{12}{R^\alpha }_{\xi \beta \xi ;\xi } {+}\tfrac{1}{120}\Big ({-}3{R^\alpha }_{\xi \beta \xi ;\xi \xi }{+}{R^\alpha }_{\xi \gamma \xi }{R^\gamma }_{\xi \beta \xi }\bigg )+O(\xi ^5)\Bigg ],\nonumber \\ {L^\mu }_{\beta \gamma }= & {} \;{g^\mu }_\alpha \Bigg [{R^\alpha }_{\beta \xi \gamma }+\tfrac{1}{2}\bigg ({R^\alpha }_{\beta \xi \gamma ;\xi }-{R^\alpha }_{\xi \beta \xi ;\gamma }\bigg ) \nonumber \\&+\tfrac{1}{6}\bigg ({R^\alpha }_{\beta \xi \gamma ;\xi \xi }-{R^\alpha }_{\xi \beta \xi ;\xi \gamma } -{R^\alpha }_{\beta \xi \delta }{R^\delta }_{\xi \gamma \xi }-3{R^\alpha }_{\delta \xi \gamma }{R^\delta }_{\xi \beta \xi }\bigg ) \nonumber \\&+\tfrac{1}{24}\bigg ({R^\alpha }_{\beta \xi \gamma ;\xi \xi \xi }-{R^\alpha }_{\xi \beta \xi ;\xi \xi \gamma }+\{6{R^\alpha }_{\delta \xi \gamma ;\xi }-{R^\alpha }_{\xi \xi \delta ;\gamma }\}{R^\delta }_{\xi \beta \xi } \nonumber \\&-3{R^\alpha }_{\beta \xi \delta ;\xi }{R^\delta }_{\xi \gamma \xi }{-}{R^\alpha }_{\xi \xi \delta }{R^\delta }_{\xi \beta \xi ;\gamma } {-}{R^\alpha }_{\beta \xi \delta }{R^\delta }_{\xi \gamma \xi ;\xi }\!+\!{R^\alpha }_{\delta \xi \gamma }{R^\delta }_{\xi \beta \xi ;\xi }\bigg )\!+\!O(\xi ^5)\Bigg ], \nonumber \\ {J^\mu }_{\beta \gamma }= & {} \;{g^\mu }_\alpha \Bigg [{R^\alpha }_{\gamma \xi \beta }+\tfrac{1}{6}\bigg (3{R^\alpha }_{\gamma \xi \beta ;\xi }-{R^\alpha }_{\xi \gamma \xi ;\beta }\bigg ) \nonumber \\&+\tfrac{1}{12}\bigg (2{R^\alpha }_{\gamma \xi \beta ;\xi \xi }-{R^\alpha }_{\xi \gamma \xi ;\xi \beta } +2{R^\alpha }_{\gamma \xi \delta }{R^\delta }_{\xi \beta \xi }+2{R^\alpha }_{\delta \xi \beta }{R^\delta }_{\xi \gamma \xi }\bigg ) +O(\xi ^4)\Bigg ],\nonumber \\ {I^\mu }_{\beta \gamma }= & {} \;{g^\mu }_\alpha \Bigg [-\tfrac{2}{3}{R^\alpha }_{(\beta \gamma )\xi }+\tfrac{1}{12}\bigg ({R^\alpha }_{\xi \xi (\beta ;\gamma )}-5{R^\alpha }_{(\beta \gamma )\xi ;\xi }\bigg ) +O(\xi ^3)\Bigg ]. \end{aligned}$$

(83)