Coincident gauge for static spherical field configurations in symmetric teleparallel gravity

A generic connection $\tilde{\Gamma}{}^{\lambda}_{\phantom{\alpha}\mu\nu}$ with 64 independent components can be decomposed into three parts, [33, 34]

$$\tilde{\Gamma}{}^{\lambda}_{\phantom{\alpha}\mu\nu}=\accentset{\circ}{\Gamma}^{\lambda}{}_{\mu\nu}+K^{\lambda}_{\phantom{\alpha}\mu\nu}+L^{\lambda}_{\phantom{\alpha}\mu\nu}\,,$$

(1)

namely the Levi-Civita connection of the metric $g_{\mu\nu}$,

$$\accentset{\circ}{\Gamma}^{\lambda}{}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}\left(\partial_{\mu}g_{\beta\nu}+\partial_{\nu}g_{\beta\mu}-\partial_{\beta}g_{\mu\nu}\right)\,,$$

(2)

contortion tensor

$$K^{\lambda}{}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}\left(-T_{\mu\beta\nu}-T_{\nu\beta\mu}+T_{\beta\mu\nu}\right)=-\,K_{\mu}{}^{\lambda}{}_{\nu}\,,$$

(3)

and disformation tensor

$$L^{\lambda}{}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}\left(-Q_{\mu\beta\nu}-Q_{\nu\beta\mu}+Q_{\beta\mu\nu}\right)=L^{\lambda}{}_{\nu\mu}\,.$$

(4)

Here contortion is built from torsion tensors

$$T^{\lambda}{}_{\mu\nu}\equiv\tilde{\Gamma}{}^{\lambda}{}_{\mu\nu}-\tilde{\Gamma}{}^{\lambda}{}_{\nu\mu}\,,$$

(5)

while disformation is constructed from nonmetricity tensors

$$Q_{\rho\mu\nu}\equiv\nabla_{\rho}g_{\mu\nu}=\partial_{\rho}g_{\mu\nu}-\tilde{\Gamma}{}^{\beta}{}_{\mu\rho}g_{\beta\nu}-\tilde{\Gamma}{}^{\beta}{}_{\nu\rho}g_{\mu\beta}\,.$$

(6)

The latter two along with curvature tensor

$\displaystyle R^{\sigma}\,_{\rho\mu\nu}$

$\displaystyle\equiv$

$\displaystyle\partial_{\mu}\tilde{\Gamma}{}^{\sigma}\,_{\nu\rho}-\partial_{\nu}\tilde{\Gamma}{}^{\sigma}\,_{\mu\rho}+\tilde{\Gamma}{}^{\sigma}\,_{\mu\lambda}\tilde{\Gamma}{}^{\lambda}\,_{\nu\rho}-\tilde{\Gamma}{}^{\sigma}\,_{\mu\lambda}\tilde{\Gamma}{}^{\lambda}\,_{\nu\rho}\,$

(7)

are the three key properties that characterize a connection. When curvature is zero, the orientation of vectors does not change under parallel transport along a curve. When torsion is zero, the connection is symmetric in the lower indices. Hence the imposition of vanishing curvature and torsion warrants the name “symmetric teleparallel”, and we denote it by ${\Gamma}{}^{\lambda}_{\phantom{\alpha}\mu\nu}$.

It is an interesting property of the connection ${\Gamma}{}^{\lambda}_{\phantom{\alpha}\mu\nu}$ that the scalar curvature of the Levi-Civita part of the connection, $\accentset{\circ}{R}$, can be expressed as the sum of a nonmetricity scalar and a Levi-Civita divergence acting over the two independent traces of the nonmetricity tensor,

$\displaystyle\accentset{\circ}{R}=Q+\accentset{\circ}{\nabla}_{\mu}(\hat{Q}^{\mu}-Q^{\mu})\,,$

(8)

where the nonmetricity scalar and traces are defined as

	$\displaystyle Q$	$\displaystyle\equiv-\,\frac{1}{4}\,Q_{\lambda\mu\nu}Q^{\lambda\mu\nu}+\frac{1}{2}\,Q_{\lambda\mu\nu}Q^{\mu\nu\lambda}+\frac{1}{4}\,Q_{\mu}Q^{\mu}-\frac{1}{2}\,Q_{\mu}\hat{Q}^{\mu}\,,$		(9)
	$\displaystyle Q_{\mu}$	$\displaystyle\equiv Q_{\mu\nu}\,^{\nu}\,,\qquad\qquad\hat{Q}_{\mu}\equiv Q_{\nu\mu}\,^{\nu}\,.$		(10)

It is also significant that the total divergence part in (8),

$\displaystyle B_{Q}$

$\displaystyle=\accentset{\circ}{\nabla}_{\mu}(\hat{Q}^{\mu}-Q^{\mu})$

(11)

becomes a boundary term under a spacetime integral.

As the Einstein-Hilbert action of GR is given by the Levi-Civita curvature scalar $\accentset{\circ}{R}$, we can rewrite that action using the nonmetricity scalar $Q$ instead, and expect to keep the same dynamical content since the boundary term does not affect the field equations. This is the idea behind symmetric teleparallel equivalent of general relatvity. Various extensions of the symmetric teleparallel theory like substituting $Q$ in the action by $f(Q)$ or introducing a nonminimal coupling between $Q$ and a scalar field $\Phi$, however, lead to theories that are different from their counterparts $f(\accentset{\circ}{R})$ and scalar-tensor gravity originally formulated in the Riemannian geometry.

A relatively simple, but still versatile extended symmetric teleparallel action can be written as [15]

$$S=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{-g}\left(\mathcal{A}(\Phi)Q-\mathcal{B}(\Phi)g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-2{\mathcal{V}}(\Phi)\right)+S_{\mathrm{m}}\,,$$

(12)

where $\kappa^{2}=8\pi G$ and $S_{\mathrm{m}}$ denotes the action of matter (which we assume to be the same as in GR, i.e. depending on the metric alone). Like in the usual Riemannian scalar-tensor theory the nonminimal coupling function $\mathcal{A}$ sets the strength of the effective gravitational constant, $\mathcal{B}$ is the kinetic coupling function, and $\mathcal{V}$ is the scalar potential. If the nonminimal coupling function is fixed to unity, $\mathcal{A}(\Phi)\equiv 1$, and the kinetic and potential terms of the scalar field vanish, $\mathcal{B}(\Phi)\equiv\mathcal{V}(\Phi)\equiv 0$, the theory is reduced to a STEGR. If the nonminimal coupling function is unity, but and the kinetic term of the scalar field is nontrivial, then we have a theory that is equivalent to a minimally coupled scalar field in GR. In fact, also the better known $f(Q)$ gravity can be viewed as a special subcase of the scalar-tensor action (12), as it can be obtained by setting [15]

$\displaystyle\mathcal{A}=f_{Q}\,,\qquad\mathcal{B}=0\,,\qquad\mathcal{V}=\tfrac{1}{2}\left(Qf_{Q}-f\right)\,,\qquad\Phi=Q\,,$

(13)

where $f_{Q}=df/dQ$.

With the help of introducing the so-called superpotential (or conjugate) tensor

$\displaystyle P^{\alpha}{}_{\mu\nu}=-\,\frac{1}{4}Q^{\alpha}{}_{\mu\nu}+\frac{1}{2}Q_{(\mu}{}^{\alpha}{}_{\nu)}+\frac{1}{4}g_{\mu\nu}Q^{\alpha}-\frac{1}{4}(g_{\mu\nu}\hat{Q}^{\alpha}+\delta^{\alpha}{}_{(\mu}Q_{\nu)})\,$

(14)

as well as some geometric indentities the field equations arising from the variation of the action (12) with respect to the metric, symmetric teleparallel connection, and scalar field are [15, 35]

	$\displaystyle\mathcal{A}(\Phi)\accentset{\circ}{G}_{\mu\nu}+2\frac{{\rm d}\mathcal{A}}{{\rm d}\Phi}P^{\lambda}{}_{\mu\nu}\partial_{\lambda}\Phi+\frac{1}{2}g_{\mu\nu}\left(\mathcal{B}(\Phi)g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi+2{\mathcal{V}}(\Phi)\right)-\mathcal{B}(\Phi)\partial_{\mu}\Phi\partial_{\nu}\Phi$	$\displaystyle=$	$\displaystyle\kappa^{2}\mathcal{T}_{\mu\nu}\,,$		(15a)
	$\displaystyle\left(\frac{1}{2}Q_{\beta}+\nabla_{\beta}\right)\left[\partial_{\alpha}\mathcal{A}\left(\frac{1}{2}Q_{\mu}g^{\alpha\beta}-\frac{1}{2}\delta^{\alpha}_{\mu}Q^{\beta}-Q_{\mu}\,^{\alpha\beta}+\delta^{\alpha}_{\mu}Q_{\gamma}\,^{\gamma\beta}\right)\right]$	$\displaystyle=$	$\displaystyle 0\,,$		(15b)
	$\displaystyle 2\mathcal{B}\accentset{\circ}{\square}\Phi+\frac{{\rm d}\mathcal{B}}{{\rm d}\Phi}g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi+\frac{{\rm d}\mathcal{A}}{{\rm d}\Phi}Q-2\frac{{\rm d}\mathcal{V}}{{\rm d}\Phi}$	$\displaystyle=$	$\displaystyle 0\,.$		(15c)

Here $\accentset{\circ}{G}_{\mu\nu}$ is the Einstein tensor and $\accentset{\circ}{\square}$ the d’Alembert operator computed of the Levi-Civita part of the connection, while $\mathcal{T}_{\mu\nu}$ is the usual matter energy-momentum tensor. When the scalar field is globally constant, the Eq. (15a) reduces to the Einstein’s equation in GR with the value of the potential playing the role of the cosmological constant, while (15b) and (15c) immediately vanish. Therefore, the solutions of GR are trivially also the solutions of these scalar-tensor theories with the scalar field being constant.

One can also notice that if the nonminimal coupling function $\mathcal{A}(\Phi)$ is constant, the connection field equation (15b) is identically satisfied and all contributions of the nonmetricity drop out in the remaining equations as well. Thus in the case of STEGR, the non-Riemannian part of the symmetric teleparallel connection is left completely undetermined by the field equations. This feature can be traced back to the identity (8), which implies that the difference between GR and STEGR is at most in the boundary term only. The boundary term does not concern the field equations, and thus the STEGR field equations can not contain anything extra to the GR field equations. As the GR field equations know only about the Riemannian part of the connection, the non-Riemannian part of the symmetric teleparallel connection can not emerge in the STEGR field equations either. However, it might still be possible to entertain some heuristic arguments that could be used in fixing the non-Riemannian part of the connection nevertheless, relevant in modified theories of gravity or in physical situations when the value of the action plays a role (such as in black hole entropy [31]).

The action (12) and the ensuing field equations are manifestly covariant, in the sense that under a general coordinate transformation from $\xi^{\lambda}$ into $x^{\lambda}$ the metric and connection transform as usual

	$\displaystyle{g}_{\mu\nu}(x^{\lambda})$	$\displaystyle=$	$\displaystyle\frac{\partial\xi^{\alpha}}{\partial x^{\mu}}\frac{\partial\xi^{\beta}}{\partial x^{\nu}}g_{\alpha\beta}(\xi^{\lambda})\,,$		(16a)
	$\displaystyle\Gamma^{\rho}{}_{\mu\nu}(x^{\lambda})$	$\displaystyle=$	$\displaystyle\frac{\partial x^{\rho}}{\partial\xi^{\gamma}}\frac{\partial\xi^{\alpha}}{\partial x^{\mu}}\frac{\partial\xi^{\beta}}{\partial x^{\nu}}\Gamma^{\gamma}{}_{\alpha\beta}(\xi^{\lambda})+\frac{\partial^{2}\xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}}\frac{\partial x^{\rho}}{\partial\xi^{\alpha}}\,.$		(16b)

In particular, the quantities $\accentset{\circ}{R}$, $Q$, and $B_{Q}$ transform as scalars and retain their relationship (8).

A very interesting mathematical fact in the case of symmetric teleparallel geometry is that there exists a coordinate system where the connection vanishes globally on the manifold [17]. The authors of Ref. [4] have dubbed this system of coordinates a “coincident gauge”, since the vanishing of the connection makes the covariant and partial derivatives to coincide. Also then the nonmetricity tensor (6) reduces to the simple form of partial derivatives of the metric only. When the coordinate system corresponding to the coincident gauge is known, the transformation (16b) of the connection into any other coordinate system $x^{\alpha}$ reduces to

$$\Gamma^{\rho}{}_{\mu\nu}(x^{\lambda})=\left(\frac{\partial\xi^{\alpha}}{\partial x^{\rho}}\right)^{-1}\partial_{\mu}\left(\frac{\partial\xi^{\alpha}}{\partial x^{\nu}}\right)\,,$$

(17)

where $\xi^{\lambda}$ are the coordinates of the coincident gauge with $\Gamma^{\gamma}{}_{\alpha\beta}(\xi^{\lambda})=0$. Similarly, by an inverse transformation of (16) it is possible from any other coordinate system to transform back into the coincident gauge (provided the determinant of the Jacobian is not zero).

The combined property that the non-Riemannian part of the connection is not determined by the STEGR field equations and that there exists a system of coordinates where the teleparallel connection vanishes may tempt one to entertain a noncovariant approach to STEGR by just imposing vanishing teleparallel connection in association with any metric. In this approach under the coordinate transformations one would only transform the metric (16a) while instead of (16b) the connection is kept always zero. (In analogy with the metric teleparallel case [36] one may speak of a “pure metric” approach to symmetric teleparallel gravity and call this a coordinate transformation of the 2nd kind). Transforming only the metric is equivalent to making a combined transformation (16) and on top of that only the connection transformation (inverse of (17)) to make the connection to vanish. While such transformations are consistent with the STEGR field equations, it is rather questionable whether they provide a true physical symmetry of the theory which relates physically equivalent configurations (as the usual coordinate transformations (16) do). Under the pure metric transformation (16a) the nonmetricity tensor would transform noncovariantly (since $\partial_{\rho}g_{\mu\nu}$ is not a tensor), and the nonmetricity scalar would not remain at the same value but pick up an extra contribution in the form of an additional boundary term. Thus the relation (8) would be altered in the sense that $\accentset{\circ}{R}$ and $Q$ would be related to each other by a different boundary term before and after the transformation. The field equations would still assume the same covariant form, but any quantity that relies on the boundary term (like the black hole entropy) would report altered physics.

In our view Eq. (17) is rather a generating rule of a set of all possible symmetric teleparallel connections, where the connection corresponding to a particular physical configuration is one member. Indeed, for a given metric $g_{\mu\nu}(x^{\lambda})$ picking a set of all possible four functions $\xi^{\alpha}(x^{\lambda})$ generates a set of all connections that have the property of being curvature and torsion-free. Connection components which are solutions of symmetric teleparallel field equations necessarily belong to this set. In STEGR this whole set is consistent with the field equations since the STEGR field equations leave the non-Riemannian part of the connection undetermined, while for the extended theories like scalar-tensor the field equations will in general fix the connection or at least constrain it to a particular subset of the curvature and torsion-free set. An obvious question remains how to fix the connection for a physical configuration when the field equations are not able to fully settle it? In the next section we are are going to see how the demand that the connection obeys the same symmetry as the metric which constrains the set a bit further. This provides the stage for a transformation into the coincident gauge, where different heuristic considerations can be tried out.

The set 1 of static and spherically symmetric curvature and torsion-free connections is given by [32]

$\displaystyle\Gamma^{\rho}{}_{\mu\nu}$

$\displaystyle=\left[\begin{matrix}\left[\begin{matrix}c&\Gamma^{\phi}{}_{r\phi}&0&0\\ \Gamma^{\phi}{}_{r\phi}&\Gamma^{t}{}_{rr}&0&0\\ 0&0&-\frac{1}{c}&0\\ 0&0&0&-\frac{\sin^{2}{\theta}}{c}\end{matrix}\right]&\left[\begin{matrix}0&0&0&0\\ 0&\Gamma^{r}{}_{rr}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{matrix}\right]&\left[\begin{matrix}0&0&c&0\\ 0&0&\Gamma^{\phi}{}_{r\phi}&0\\ c&\Gamma^{\phi}{}_{r\phi}&0&0\\ 0&0&0&-\sin{\theta}\cos{\theta}\end{matrix}\right]\end{matrix}\begin{matrix}\left[\begin{matrix}0&0&0&c\\ 0&0&0&\Gamma^{\phi}{}_{r\phi}\\ 0&0&0&\cot{\theta}\\ c&\Gamma^{\phi}{}_{r\phi}&\cot{\theta}&0\end{matrix}\right]\end{matrix}\right]$

(22)

where the functions $\Gamma^{\phi}{}_{r\phi}$, $\Gamma^{t}{}_{rr}$, $\Gamma^{r}{}_{rr}$ depend on $r$, while $c$ is a nonzero constant. The functions have to obey the relation

$\displaystyle\frac{d}{dr}\Gamma^{\phi}{}_{r\phi}$

$\displaystyle=c\Gamma^{t}{}_{rr}-\Gamma^{\phi}{}_{r\phi}(\Gamma^{\phi}{}_{r\phi}+\Gamma^{r}{}_{rr})\,.$

(23)

to satisfy vanishing curvature and torsion, hence overall there are two functional degrees of freedom.

The set 2 of static and spherically symmetric curvature and torsion-free connections is given by [32]

	$\displaystyle\Gamma^{\rho}{}_{\mu\nu}=$	$\displaystyle\left[\left[\begin{matrix}-c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}-c+k&\frac{\left(2c-k\right)\left(c\Gamma^{t}{}_{\theta\theta}+1\right)\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&0&0\\ \frac{\left(2c-k\right)\left(c\Gamma^{t}{}_{\theta\theta}+1\right)\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&\Gamma^{t}{}_{rr}&0&0\\ 0&0&\Gamma^{t}{}_{\theta\theta}&0\\ 0&0&0&\Gamma^{t}{}_{\theta\theta}\sin^{2}\theta\end{matrix}\right]\right.$
		$\displaystyle\qquad\left[\begin{matrix}-c\left(2c-k\right)\Gamma^{r}{}_{\theta\theta}&c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}+c&0&0\\ c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}+c&\Gamma^{r}{}_{rr}&0&0\\ 0&0&\Gamma^{r}{}_{\theta\theta}&0\\ 0&0&0&\Gamma^{r}{}_{\theta\theta}\sin^{2}\theta\end{matrix}\right]$
		$\displaystyle\qquad\left.\left[\begin{matrix}0&0&c&0\\ 0&0&\frac{-c\Gamma^{t}{}_{\theta\theta}-1}{\Gamma^{r}{}_{\theta\theta}}&0\\ c&\frac{-c\Gamma^{t}{}_{\theta\theta}-1}{\Gamma^{r}{}_{\theta\theta}}&0&0\\ 0&0&0&-\sin\theta\cos\theta\end{matrix}\right]\quad\left[\begin{matrix}0&0&0&c\\ 0&0&0&\frac{-c\Gamma^{t}{}_{\theta\theta}-1}{\Gamma^{r}{}_{\theta\theta}}\\ 0&0&0&\cot\theta\\ c&\frac{-c\Gamma^{t}{}_{\theta\theta}-1}{\Gamma^{r}{}_{\theta\theta}}&\cot\theta&0\end{matrix}\right]\right]$		(24)

where the functions $\Gamma^{t}{}_{\theta\theta}$, $\Gamma^{t}{}_{rr}$, $\Gamma^{r}{}_{\theta\theta}$, $\Gamma^{r}{}_{rr}$ also depend on $r$, while $c$ and $k$ are some constant parameters. The functions have to obey the relations

	$\displaystyle\frac{d}{dr}\Gamma^{t}{}_{\theta\theta}$	$\displaystyle=-\frac{\left(\left(c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}+3c-k\right)\Gamma^{t}{}_{\theta\theta}+1\right)\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}-\Gamma^{r}{}_{\theta\theta}\Gamma^{t}{}_{rr}\,,$		(25a)
	$\displaystyle\frac{d}{dr}\Gamma^{r}{}_{\theta\theta}$	$\displaystyle=-c\left(\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}+2\right)\Gamma^{t}{}_{\theta\theta}-\Gamma^{r}{}_{\theta\theta}\Gamma^{r}{}_{rr}-1$		(25b)

for vanishing curvature and torsion, hence again there are two functional degrees of freedom remaining in the game.

For the connection set 1 given by (22), the nontrivial equations (26) turn out to be

	$\displaystyle-c\partial_{t}\xi^{\lambda}+\partial_{t}\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,,$		(27a)
	$\displaystyle-\Gamma^{\phi}{}_{r\phi}\,\partial_{t}\xi^{\lambda}+\partial_{t}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(27b)
	$\displaystyle-c\,\partial_{\theta}\xi^{\lambda}+\partial_{\theta}\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,,$		(27c)
	$\displaystyle-c\,\partial_{\phi}\xi^{\lambda}+\partial_{t}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(27d)
	$\displaystyle-\Gamma^{r}{}_{rr}\,\partial_{r}\xi^{\lambda}-\Gamma^{t}{}_{rr}\,\partial_{t}\xi^{\lambda}+\partial_{r}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(27e)
	$\displaystyle-\Gamma^{\phi}{}_{r\phi}\,\partial_{\theta}\xi^{\lambda}+\partial_{\theta}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(27f)
	$\displaystyle-\Gamma^{\phi}{}_{r\phi}\,\partial_{\phi}\xi^{\lambda}+\partial_{r}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(27g)
	$\displaystyle\partial_{\theta}^{2}\xi^{\lambda}+\frac{1}{c}\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,,$		(27h)
	$\displaystyle-\cot{\theta}\partial_{\phi}\xi^{\lambda}+\partial_{\theta}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(27i)
	$\displaystyle\sin{\theta}\cos{\theta}\,\partial_{\theta}\xi^{\lambda}+\partial_{\phi}\partial_{\phi}\xi^{\lambda}+\frac{1}{c}\sin^{2}{\theta}\,\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,.$		(27j)

They are identical for each $\xi^{\lambda}(t,r,\theta,\phi)$. The general solution of this system of partial differential equations can be written as

$\displaystyle\xi^{\lambda}$

$\displaystyle=e^{ct+\int\Gamma^{\phi}{}_{r\phi}\,dr}\left[\left(\alpha^{\lambda}\cos\phi+\beta^{\lambda}\sin\phi\right)\sin\theta+\gamma^{\lambda}\cos\theta\right]+\sigma^{\lambda}\int e^{\int\Gamma^{r}{}_{rr}\,dr}\,dr\,,$

(28)

where $\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda},\sigma^{\lambda}$ are the integration constants. We have kept the indefinite integrals explicitly in the expression (28), so it is straightforward to take the derivatives and check that Eq. (26) is indeed satisfied for the affine connection components (22) and taking into account the relation (23). The solution (28) contains only two functions $\Gamma^{\phi}{}_{r\phi}$, $\Gamma^{r}{}_{rr}$, but the third function $\Gamma^{t}{}_{rr}$ would be generated into (22) via the condition (23).

The determinant of the Jacobian matrix is given by

$\displaystyle\det\left(\frac{\partial\xi^{\lambda}}{\partial x^{\alpha}}\right)$

$\displaystyle=c\,e^{3ct+\int\Gamma^{\phi}{}_{r\phi}\,dr+\int\Gamma^{r}{}_{rr}\,dr}\sin\theta\,\det\left(||\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda},\sigma^{\lambda}||\right)\,,$

(29)

where $||\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda},\sigma^{\lambda}||$ stands for a matrix composed of the columns of the integration constants. We see that the coincident gauge coordinates $\xi^{\lambda}$ are not determined uniquely, but any combination of the integration constants that does not make the Jacobian determinant zero or singular is a possible solution. All these possibilities are of course equivalent to each other in the sense that they are related to each other by some coordinate transformations, as they are all related to the spherical coordinates. Another interesting point is that the coincident coordinates in the static spherical symmetry case depend on the two independent affine connection components allowed by the assumption of zero curvature and torsion. These components remain completely undetermined in STEGR where the connection field equations vanish identically, but could get fixed or partially fixed by the nontrivial connection field equations in extended symmetric teleparallel theories. However the connection contributions appear as exponentiated integrals and are thus guaranteed not to alter the overall sign of the Jacobian determinant and reverse the orientation of the system of coordinates.

A minimal, and reasonable looking choice of integration constants could be $\sigma^{0}=\alpha^{1}=\beta^{2}=\gamma^{3}=1$ and all the other zero, giving

	$\displaystyle\xi^{0}$	$\displaystyle=\xi_{w}=\int e^{\int\Gamma^{r}{}_{rr}\,dr}\,dr\,,$		(30a)
	$\displaystyle\xi^{1}$	$\displaystyle=\xi_{x}=e^{ct+\int\Gamma^{\phi}{}_{r\phi}\,dr}\sin{\theta}\cos{\phi}\,,$		(30b)
	$\displaystyle\xi^{2}$	$\displaystyle=\xi_{y}=e^{ct+\int\Gamma^{\phi}{}_{r\phi}\,dr}\sin{\theta}\sin{\phi}\,,$		(30c)
	$\displaystyle\xi^{3}$	$\displaystyle=\xi_{z}=e^{ct+\int\Gamma^{\phi}{}_{r\phi}\,dr}\cos{\theta}\,.$		(30d)

We may immediately notice that the angular part of this coordinate transformation reminds the relationship between the Cartesian and spherical coordinates, while the time and radial coordinates do not exactly map to their expected Cartesian counterparts. To find that the coincident gauge coordinates resemble the Cartesian system should perhaps not come as a surprise, since in order to make the affine connection to vanish certain “straightening out” of the curvilinear nature of the original spherical coordinate system is probably unavoidable. We can invert the expressions (30) to get

	$\displaystyle t$	$\displaystyle=\frac{1}{2c}\log\Big{[}R^{2}e^{-2\int\Gamma^{\phi}{}_{r\phi}dr}\Big{]}\,,$		(31a)
	$\displaystyle r$	$\displaystyle=r(\xi_{w})\,,$		(31b)
	$\displaystyle\theta$	$\displaystyle=\arccos\left(\frac{\xi_{z}}{R}\right)\,,$		(31c)
	$\displaystyle\phi$	$\displaystyle=\arctan\left(\frac{\xi_{y}}{\xi_{x}}\right)\,,$		(31d)

where we have denoted

$\displaystyle R^{2}$

$\displaystyle=\xi_{x}^{2}+\xi_{y}^{2}+\xi_{z}^{2}\,.$

(32)

The function $r(\xi_{w})$ is the inverse of Eq. (30a) and thus inherits the properties like

$\displaystyle r^{\prime}$

$\displaystyle=\frac{dr}{d\xi_{w}}=e^{\int\Gamma^{r}{}_{rr}\,dr}\,,$

$\displaystyle r^{\prime\prime}$

$\displaystyle=\frac{d^{2}r}{d\xi_{w}^{2}}=-\Gamma^{r}{}_{rr}e^{-2\int\Gamma^{r}{}_{rr}\,dr}\,,$

(33)

etc., which must be remembered in the calculations. It is interesting that the time direction $t$ in the spherical coordinates gets mixed into all four coordinates of the coincident gauge, since (31a) depends not only on $\xi_{w}$ via $\Gamma^{\phi}{}_{r\phi}(r(\xi_{w}))$, but also on the “spatial” $\xi_{x},\xi_{y},\xi_{z}$ via $R$.

To transform the metric and other tensors into the coincident gauge coordinates it is convenient to have the Jacobian matrices written out in these coordinates. Taking the derivatives, keeping in mind the properties (23) and (33), and inverting the matrix, we get

	$\displaystyle\left(\frac{\partial\xi^{\lambda}}{\partial x^{\alpha}}\right)$	$\displaystyle=\begin{pmatrix}0&\frac{1}{r^{\prime}}&0&0\\ c\xi_{x}&\xi_{x}\Gamma^{\phi}{}_{r\phi}&\frac{\xi_{x}\xi_{z}}{\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}&-\xi_{y}\\ c\xi_{y}&\xi_{y}\Gamma^{\phi}{}_{r\phi}&\frac{\xi_{y}\xi_{z}}{\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}&\xi_{x}\\ c\xi_{z}&\xi_{z}\Gamma^{\phi}{}_{r\phi}&-\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}&0\end{pmatrix}\,,$		(34)
	$\displaystyle\left(\frac{\partial x^{\alpha}}{\partial\xi^{\lambda}}\right)$	$\displaystyle=\begin{pmatrix}\vspace{0.1cm}-\frac{r^{\prime}\Gamma^{\phi}{}_{r\phi}}{c}&\frac{\xi_{x}}{cR^{2}}&\frac{\xi_{y}}{cR^{2}}&\frac{\xi_{z}}{cR^{2}}\\ r^{\prime}&0&0&0\\ 0&\frac{\xi_{x}\xi_{z}}{R^{2}\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}&\frac{\xi_{y}\xi_{z}}{R^{2}\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}&-\frac{\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}{R^{2}}\\ 0&-\frac{\xi_{y}}{\xi_{x}^{2}+\xi_{y}^{2}}&\frac{\xi_{x}}{\xi_{x}^{2}+\xi_{y}^{2}}&0\end{pmatrix}\,,$		(35)

where $\Gamma^{\phi}{}_{r\phi}(r(\xi_{w}))$, $r=r(\xi_{w})$ and $r^{\prime}=\frac{dr}{d\xi_{w}}$. With these expressions it is easy to transform (16) the metric into the coincident gauge, resulting in

$\displaystyle g_{\mu\nu}$

$\displaystyle=\begin{pmatrix}\vspace{0.1cm}\frac{\left(c^{2}{g_{rr}}-{g_{tt}}\left(\Gamma^{\phi}{}_{r\phi}\right)^{2}\right)r^{\prime 2}}{c^{2}}&\frac{\xi_{x}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}&\frac{\xi_{y}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}&\frac{\xi_{z}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}\\ \vspace{0.1cm}\frac{\xi_{x}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}&\frac{c^{2}\left(\xi_{y}^{2}+\xi_{z}^{2}\right)r^{2}-\xi_{x}^{2}{g_{tt}}}{c^{2}R^{4}}&-\frac{\xi_{x}\xi_{y}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}&-\frac{\xi_{x}\xi_{z}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}\\ \vspace{0.1cm}\frac{\xi_{y}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}&-\frac{\xi_{x}\xi_{y}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}&\frac{c^{2}\left(\xi_{x}^{2}+\xi_{z}^{2}\right)r^{2}-\xi_{y}^{2}{g_{tt}}}{c^{2}R^{4}}&-\frac{\xi_{y}\xi_{z}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}\\ \vspace{0.1cm}\frac{\xi_{z}r^{\prime}{g_{tt}}\Gamma^{\phi}{}_{r\phi}}{c^{2}R^{2}}&-\frac{\xi_{x}\xi_{z}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}&-\frac{\xi_{y}\xi_{z}\left(c^{2}r^{2}+{g_{tt}}\right)}{c^{2}R^{4}}&\frac{c^{2}\left(\xi_{x}^{2}+\xi_{y}^{2}\right)r^{2}-\xi_{z}^{2}{g_{tt}}}{c^{2}R^{4}}\end{pmatrix}\,,$

(36)

Similarly, the affine connection components (22) can be transformed (16) into the coincident gauge, giving the expected result

$\displaystyle\Gamma^{\rho}{}_{\mu\nu}=0\,.$

(37)

Note that the transformed metric in the coincident gauge contains now four independent functions, namely $g_{tt}(r(\xi_{w}))$ and $g_{rr}(r(\xi_{w}))$ which were in the metric before, as well as $\Gamma^{\phi}{}_{r\phi}(r(\xi_{w}))$ and $r(\xi_{w})$ which previously characterised the connection (the latter via the inverse of (30a)). Thus although the coordinate transformation into the coincident gauge cleared the affine connection components of all content, the same information got packed into the components of the metric tensor which took a much more complicated form as the result.

Although vanishing connection makes the covariant derivatives to reduce to the partial derivatives and certain simplifications take place, there seems to be no overall economic effect in using the coincident gauge expressions for practical computations. Actually the situation is quite the opposite. Despite zero connection, the nonmetricity tensor is full of nonvanishing components, and thus computing the nonmetricity trances and any further quantities present in the field equations becomes more involved. The reason is not that the all the free functions got packed into the metric, but that the main price of making the connection to vanish was to switch away from the coordinates that were adapted to the symmetry of the configuration, i.e. from the spherical coordinates to a sort of deformed Cartesian coordinates.

The nature of the coincident coordinates (30) can be illuminated further by transforming the Killing vectors (18) into the coincident gauge, which yields

	$\displaystyle K^{\mu}$	$\displaystyle=\begin{pmatrix}0&c\xi_{x}&c\xi_{y}&c\xi_{z}\end{pmatrix}\,,$		(38a)
	$\displaystyle R^{\mu}$	$\displaystyle=\begin{pmatrix}0&-\xi_{y}&\xi_{x}&0\end{pmatrix}\,,$		(38b)
	$\displaystyle S^{\mu}$	$\displaystyle=\begin{pmatrix}0&\xi_{z}&0&-\xi_{x}\end{pmatrix}\,,$		(38c)
	$\displaystyle T^{\mu}$	$\displaystyle=\begin{pmatrix}0&0&-\xi_{z}&\xi_{y}\end{pmatrix}\,.$		(38d)

In the form of the vectors $R^{\mu}$, $S^{\mu}$, $T^{\mu}$ we can exactly recognize the Killing vectors of spherical symmetry as expressed in the Cartesian coordinates, but the vector $K^{\mu}$ representing static time translation symmetry is quite unexpectedly orthogonal to the $\xi_{w}$ direction. Therefore we can not entertain $\xi_{w}$ as a coincident gauge analogue of time, rather the time is “smeared” among all the coordinates. Hence one should be alert that in general the physical interpretation of solutions in the coincident gauge can be tricky and problematic. However, one can still check explicitly that the the Lie derivatives (19) of the metric (36) in the directions specified by the Killing vectors (38) vanish, hence the transformed metric (36) possesses the same symmetry as the original metric (20) and connection (22). In a similar vein, the Lie derivatives of the transformed connection components (37) in the directions specified by the Killing vectors (38) vanish as well.

Taking the Minkowski limit of the original metric by letting $g_{tt}\to 1$ and $g_{rr}\to 1$ simplifies the form of of the coincident gauge metric (36). However, by closer inspection we may notice that no possible choice of the remaining functions $\Gamma^{\phi}{}_{r\phi}$ and $\Gamma^{r}{}_{rr}$ (the latter determines $r$) can take the Minkowski limit of (36) into the familiar form in the Cartesian coordinates $\eta_{\mu\nu}=\textrm{diag}(-1,1,1,1)$. At this point it is not obvious that the Minkowski limit of the coincident gauge metric should reduce to the Cartesian coordinates, but on the other hand the Minkowski spacetime in the Cartesian coordinates is already trivially in the coincident gauge, and it feels unsettling that the set 1 connection does not possess a direct link to that.

For the connection set 2 expressed by (24), the nontrivial equations (26) turn out to be

	$\displaystyle c\left(2c-k\right)\Gamma^{r}{}_{\theta\theta}\,\partial_{r}\xi^{\lambda}-\left(-c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}\,-c+k\right)\partial_{t}\xi^{\lambda}+\partial_{t}\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,,$		(39a)
	$\displaystyle-\frac{\left(2c-k\right)\left(c\Gamma^{t}{}_{\theta\theta}\,+1\right)\Gamma^{t}{}_{\theta\theta}\,\partial_{t}\xi^{\lambda}}{\Gamma^{r}{}_{\theta\theta}\,}-\left(c\left(2c-k\right)\Gamma^{t}{}_{\theta\theta}\,+c\right)\partial_{r}\xi^{\lambda}+\partial_{t}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(39b)
	$\displaystyle-c\,\partial_{\theta}\xi^{\lambda}+\partial_{\theta}\partial_{t}\xi^{\lambda}$	$\displaystyle=0\,,$		(39c)
	$\displaystyle-c\,\partial_{\phi}\xi^{\lambda}+\partial_{t}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(39d)
	$\displaystyle-\Gamma^{r}{}_{rr}\,\partial_{r}\xi^{\lambda}-\Gamma^{t}{}_{rr}\,\partial_{t}\xi^{\lambda}+\partial_{r}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(39e)
	$\displaystyle-\frac{\left(-c\Gamma^{t}{}_{\theta\theta}\,-1\right)\partial_{\theta}\xi^{\lambda}}{\Gamma^{r}{}_{\theta\theta}\,}+\partial_{\theta}\partial_{r}\xi^{\lambda}$	$\displaystyle=0\,,$		(39f)
	$\displaystyle-\frac{\left(-c\Gamma^{t}{}_{\theta\theta}\,-1\right)\partial_{\phi}\xi^{\lambda}}{\Gamma^{r}{}_{\theta\theta}\,}+\partial_{r}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(39g)
	$\displaystyle-\Gamma^{r}{}_{\theta\theta}\,\partial_{r}\xi^{\lambda}-\Gamma^{t}{}_{\theta\theta}\,\partial_{t}\xi^{\lambda}+\partial_{\theta}\partial_{\theta}\xi^{\lambda}$	$\displaystyle=0\,,$		(39h)
	$\displaystyle-\cot{\theta}\,\partial_{\phi}\xi^{\lambda}+\partial_{\theta}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(39i)
	$\displaystyle-\Gamma^{r}{}_{\theta\theta}\,\sin^{2}{\theta}\,\partial_{r}\xi^{\lambda}-\Gamma^{t}{}_{\theta\theta}\,\sin^{2}{\theta}\,\partial_{t}\xi^{\lambda}+\sin{\theta}\cos{\theta}\,\partial_{\theta}\xi^{\lambda}+\partial_{\phi}\partial_{\phi}\xi^{\lambda}$	$\displaystyle=0\,,$		(39j)

where again, the remaining equations are identical for each $\xi^{\lambda}(t,r,\theta,\phi)$. Assuming $c\neq k\neq 0$, the general solution of the system is

$\displaystyle\xi^{\lambda}$

$\displaystyle=e^{ct-\int\frac{c\Gamma^{t}{}_{\theta\theta}+1}{\Gamma^{r}{}_{\theta\theta}}\,dr}\left(\left(\alpha^{\lambda}\cos\phi+\beta^{\lambda}\sin\phi\right)\sin\theta+\gamma^{\lambda}\cos\theta\right)+\sigma^{\lambda}e^{(k-c)(t-\int\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\,dr)}\,.$

(40)

Thus the coordinate system of the coincident gauge can take various forms depending on the choice of the integration constants $\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda},\sigma^{\lambda}$. All these possibilities are permissible, provided that the Jacobian determinant

$\displaystyle\det\left(\frac{\partial\xi^{\lambda}}{\partial x^{\alpha}}\right)$

$\displaystyle=\frac{(c-k)e^{3\left(ct-\int\frac{c\Gamma^{t}{}_{\theta\theta}+1}{\Gamma^{r}{}_{\theta\theta}}\,dr\right)-(k-c)\left(t-b\int\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\,dr\right)}\sin\theta}{\Gamma^{r}{}_{\theta\theta}}\det\left(||\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda},\sigma^{\lambda}||\right)\,,$

(41)

is regular. The expression (40) contains only the functions $\Gamma^{t}{}_{\theta\theta}$ and $\Gamma^{r}{}_{\theta\theta}$ while the other two functions $\Gamma^{t}{}_{rr}$ and $\Gamma^{r}{}_{rr}$ in the connection (24) are generated by the conditions (25).