Non-Relativistic Regime and Topology: Topological Term in the Einstein Equation

Abstract

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of the Einstein equation is only possible in Euclidean topologies, i.e., for which the covering space is $\mathbb {E}^3$. We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a universe with a non-Euclidean topology and admitting a non-relativistic limit.

Appendices

Appendix A The NEN Theory

In this section, we present the properties of the non-relativistic theory in non-Euclidean topologies, i.e., the NEN theory, developed in [15] and [36, section 5.6 ].

1.1 A.1 Spacetime Equations

The theory is defined on a 4-manifold ${\mathcal {M}}= \mathbb {R}\times \Sigma$ where $\Sigma$ is closed. ${\mathcal {M}}$ is equipped with a Galilean structure $({\hat{h}}^{\alpha \beta }, \tau _\mu , {\hat{\nabla }}_\nu )$ as defined in Sect. 3.2. The spacetime equations of the NEN theory are

$$\begin{aligned}&{\hat{h}}^{\mu \beta }{{{\hat{R}}}^\alpha }_{\gamma \mu \sigma } - {\hat{h}}^{\mu \alpha }{{{\hat{R}}}^\beta }_{\sigma \mu \gamma } = 0, \end{aligned}$$

(69)

(70)

$$\begin{aligned} {\hat{\nabla }}_\mu T^{\mu \alpha } = 0, \end{aligned}$$

(71)

where $\rho$ is the matter density; $T^{\alpha \beta }$ is the energy-momentum tensor; $\Lambda$ is the cosmological constant; is the orthogonal projector (defined by relation (6)) to a vector ${{\tilde{G}}}^\alpha$ with ${{{\tilde{G}}}^\mu \tau _\mu {:}{=}1}$ called the Galilean vector (or Galilean observer) in ${\mathcal {M}}$. It is defined by two conditions: (i) in coordinates adapted to the $\tau$-foliation and such that $\varvec{\partial }_t = \tilde{\varvec{G}}$, the time derivative of the spatial components of the space metric (which correspond to the expansion tensor of $\tilde{\varvec{G}}$) are

$$\begin{aligned} \frac{1}{2}\partial _t {\hat{h}}^{ij} = -H(t){\hat{h}}^{ij} - \Xi ^{ij}, \end{aligned}$$

(72)

and (ii) $\tilde{\varvec{G}}$ is vorticity free with respect to the Galilean structure, i.e.,

$$\begin{aligned} {\hat{h}}^{\mu [\alpha }{\hat{\nabla }}_\mu {{\tilde{G}}}^{\beta ]} = 0. \end{aligned}$$

(73)

H(t) is the expansion rate of the finite volume of the spatial sections defined by $\varvec{\tau }$, and $\Xi ^{ij}$ is traceless-transverse and represents anisotropic expansion. Property (72) ensures that, without anisotropic expansion, there exists an adapted coordinate system (the one related to $\tilde{{\varvec{G}}}$) in the NEN theory such that the spatial metric can be separated in space and time dependence: $h_{ij}(t,x^k) = a(t)^2 {{\tilde{h}}}_{ij}(x^k)$. This coordinate system defines what we call an inertial frame in classical Newton’s theory. Therefore, property (72) ensures its existence for the non-Euclidean version of the theory.⁸

Equation (69) is the Trautmann condition, also present in Newton’s theory with the Newton-Cartan system. It ensures that no gravitomagnetism is present in the theory. In equation (70), is an additional term with respect to the Newton-Cartan system. This term was added by [15, 36] for the theory to be defined on spherical or hyperbolic topologies, as it implies the spatial Ricci tensor (from equation (70)) to have the form ${\hat{{\mathcal {R}}}}^{ij} = \pm 2/a(t)^2\, {\hat{h}}^{ij}$, where a is the scale factor of expansion.

1.2 A.2 Spatial Equations

To allow for a more intuitive understanding of this theory, we present the spatial equations resulting from the above the spacetime equations. Assuming isotropic expansion ($\Xi _{ij} = 0$), and introducing the gravitational potential $\Phi$ defined via the 4-acceleration of the Galilean vector (i.e., ${\hat{D}}^\mu \Phi {:}{=}{{\tilde{G}}}^\nu {\hat{\nabla }}_\nu {{\tilde{G}}}^\mu$), we have (see [36] for a detailed derivation)

$$\begin{aligned} \dot{v}^a&= -{\hat{D}}^a \Phi - 2v^a H + (a_{\not = \mathrm grav})^a, \end{aligned}$$

(74)

$$\begin{aligned} {\dot{\rho }}/\rho&= -3H - {\hat{D}}_c v^c, \end{aligned}$$

(75)

$$\begin{aligned} {\hat{\Delta }} \Phi&= \frac{\kappa }{2} \left( \rho - \left\langle \rho \right\rangle _{\Sigma }\right) , \end{aligned}$$

(76)

$$\begin{aligned} {\hat{{\mathcal {R}}}}_{ab}&= \frac{\pm 2}{a(t)^2}{\hat{h}}_{ab}, \end{aligned}$$

(77)

with the expansion law

$$\begin{aligned} 3\ddot{a}/a&= -\frac{\kappa }{2} \left\langle \rho \right\rangle _{\Sigma } + \Lambda , \end{aligned}$$

(78)

where the dot derivative corresponds to $\partial _t + v^c{\hat{D}}_c$ with ${\hat{D}}_i$ the covariant derivative with respect to the spatial metric; $\varvec{v}$ is the spatial fluid velocity; $(a_{\not = \mathrm grav})^a$ is the non-gravitational spatial acceleration of the fluid; $\left\langle \psi \right\rangle _{\Sigma }(t) {:}{=}\frac{1}{V_\Sigma }\int _\Sigma \psi \sqrt{\textrm{det}({\hat{h}}_{ij})}\textrm{d}^3 x$ is the average of a scalar field $\psi$ over the volume $V_\Sigma$ of $\Sigma$. This system is valid in an inertial frame, i.e., in coordinates where the spatial metric is separated is space and time dependence.

We see that the spatial equations defining the NEN theory are algebraically equivalent to the cosmological Newton equations, but with the presence of a non-zero spatial Ricci tensor related to a spherical or hyperbolic topology. This shows that the NEN theory can be seen as an adaptation of Newton’s theory (by adding a curvature) such that it is defined on non-Euclidean topologies.

Appendix B Natural Metrics on Thurston’s Topological Class

We give the natural metrics for each Thurston’s topological class and some properties of their Ricci tensor (we denote $N_{\ker {}} {:}{=}\dim [\ker ({\bar{{\mathcal {R}}}}_{ij})]$):

• For Euclidean topologies (the covering space ${\tilde{\Sigma }}$ of $\Sigma$ is $\mathbb {E}^3$):

  $$\begin{aligned} {{\bar{h}}}_{ij} = \delta _{ij}, \end{aligned}$$

  (79)

  with ${\bar{{\mathcal {R}}}}_{ij} = 0$ and $N_{\ker {}} = 3$.

• For ${\tilde{\Sigma }} = \mathbb {S}^3$ (spherical topologies):

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j =\textrm{d}r^2 + \sin ^2r \left( \textrm{d}\theta ^2 + \sin ^2\theta \textrm{d}\varphi ^2\right) , \end{aligned}$$

  (80)

  with ${\bar{{\mathcal {R}}}}_{ij} = 2{{\bar{h}}}_{ij}$ and $N_{\ker {}} = 0$.

• For ${\tilde{\Sigma }} = \mathbb {H}^3$ (hyperbolic topologies):

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j =\textrm{d}r^2 + \sinh ^2r \left( \textrm{d}\theta ^2 + \sin ^2\theta \textrm{d}\varphi ^2\right) , \end{aligned}$$

  (81)

  with ${\bar{{\mathcal {R}}}}_{ij} = - 2{{\bar{h}}}_{ij}$ and $N_{\ker {}} = 0$.

• For ${\tilde{\Sigma }} = \mathbb {R}\times \mathbb {S}^2$:

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j = \textrm{d}r^2 + \sin ^2 r \, \textrm{d}\varphi ^2 + \textrm{d}z^2, \end{aligned}$$

  (82)

  with ${\bar{{\mathcal {R}}}} = 2$, ${\bar{{\mathcal {R}}}}_{\langle ij\rangle } \not =0$ and $N_{\ker {}} = 1$.

• For ${\tilde{\Sigma }} = \mathbb {R}\times \mathbb {H}^2$:

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j = \textrm{d}r^2 + \sinh ^2 r \, \textrm{d}\varphi ^2 + \textrm{d}z^2, \end{aligned}$$

  (83)

  with ${\bar{{\mathcal {R}}}} = -2$, ${\bar{{\mathcal {R}}}}_{\langle ij\rangle } \not =0$ and $N_{\ker {}} = 1$.

• For ${\tilde{\Sigma }} = \tilde{SL2\mathbb {R}}$:

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j = \textrm{d}x^2 + \cosh ^2 x \, \textrm{d}y^2 + \left( \textrm{d}z + \sinh ^2x \, \textrm{d}y\right) ^2, \end{aligned}$$

  (84)

  with ${\bar{{\mathcal {R}}}} = -5/2$, ${\bar{{\mathcal {R}}}}_{\langle ij\rangle } \not =0$ and $N_{\ker {}} = 0$.

• For Nil topologies:

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j = \textrm{d}x^2 + \textrm{d}y^2 + \left( \textrm{d}z - x \textrm{d}y\right) ^2, \end{aligned}$$

  (85)

  with ${\bar{{\mathcal {R}}}} = -1/2$, ${\bar{{\mathcal {R}}}}_{\langle ij\rangle } \not =0$ and $N_{\ker {}} = 0$.

• For Sol topologies:

  $$\begin{aligned} {{\bar{h}}}_{ij}\textrm{d}x^i\textrm{d}x^j = e^{-2z}\textrm{d}x^2 + e^{2z}\textrm{d}y^2 + \textrm{d}z^2, \end{aligned}$$

  (86)

  with ${\bar{{\mathcal {R}}}} = -2$, ${\bar{{\mathcal {R}}}}_{\langle ij\rangle } \not =0$ and $N_{\ker {}} = 2$.

Appendix C Formulas for the NR Limit

In this section, the notion of spatial is related to the foliation defined by $\varvec{\tau }$ and to the metric $\hat{\varvec{h}}$ present in the Galilean structure.

1.1 C.1 Leading Order of the Fluid Orthogonal Projector

The projector orthogonal to a g-timelike vector is defined as

(87)

Its limit is

(88)

(89)

(90)

Equation (88) can also be written as

(91)

where is the projector orthogonal to , with respect to the Galilean structure (defined by relations (6)).

1.2 C.2 Expansion and Vorticity Tensors

The formalism of this section is presented in more details in [34].

For a Galilean structure, the expansion and vorticity tensors of a unit $\tau$-timelike vector $\varvec{U}$ are defined as

(92)

$$\begin{aligned}&= -\frac{1}{2}{\mathcal {L}}_{\varvec{U}}{\hat{h}}^{\alpha \beta }, \end{aligned}$$

(93)

(94)

These tensors are spatial. For two unit $\tau$-timelike vectors $\varvec{U}$ and $\varvec{T}$, such that $\varvec{U} - \varvec{T} = \varvec{w}$, which is spatial, we have

(95)

(96)

where ${\hat{D}}_\alpha$ is the spatial derivative induced by ${\hat{h}}^{\alpha \beta }$ on the $\tau$-foliation, with ${\hat{D}}^\alpha {:}{=}{\hat{h}}^{\alpha \mu } {\hat{D}}_\mu$ (see section II.C.4. in [34]). From the general definition of a Galilean vector $\tilde{\varvec{G}}$ given in Section 6.2.2, and using (95) and (96), for any unit $\tau$-timelike vector $\varvec{U}$ we have

(97)

(98)

where .

A symmetric tensor can be decomposed into a scalar part (i.e., proportional to the metric), a gradient part (i.e., the gradient of a vector) and a tensor part (a traceless-transverse tensor) [39], called SVT decomposition. In general, the scalar part has spatial dependence. The existence of a Galilean vector implies that the expansion tensors (in the framework of Galilean structures) of any $\tau$-timelike vector have a spatially constant scalar mode as shown by relation (97). Another consequence is the form of the vorticity (98) which is expressed as function of the vector present in the gradient part of the expansion tensor. These relations were already shown to hold in [34, 36] for Newton-Cartan and the NEN theory.

Finally, the definition (94) implies that the vector $\varvec{B}$ present in the Galilean connection in the NR limit (see equation (20)) is irrotational (). Therefore formulas (97) and (98) applied for $\varvec{B}$ give

(99)

where $\sigma$ is a scalar field and $B^\mu - {{\tilde{G}}}^\mu = {\hat{D}}^\mu \sigma$.

1.3 C.3 First Order of the Lorentzian Connection

From the relation of the difference between two connections, we have

(100)

This leads to

(101)

This formula is in agreement with the “connection perturbation” introduced by [31] for the post-Newton-Cartan approximation (Table II in [31]).

Using , which follows from (15), where $k^{\alpha \beta }$ is defined with respect to the first order in equation (17), we have

(102)

(103)

1.4 C.4 First Order of the Spacetime Curvature Tensor

Using the formula for the difference between two Riemann tensors

(104)

where , we obtain:

(105)

(106)

Then,

(107)

Using equations (102) and (103) along with as , we obtain

(108)

(109)

where we only keep spatial components as is spatial, and .

Appendix D Spatial Metric at the Limit and Existence of the Galilean Vector

1.1 D.1 A Solution to Equation (56)

While we have not yet found the general solution for ${\hat{h}}_{ij}$ to the equation ${\hat{{\mathcal {R}}}}_{ij} = {\bar{{\mathcal {R}}}}_{ij}$, we give in this section a simple non-trivial solution. It can be found by writing the physical metric with a scalar-vector-tensor (SVT) decomposition with respect to the reference spatial metric ${{\bar{h}}}_{ij}$. In general, the contravariant and covariant components of the physical metric have a different decomposition:

$$\begin{aligned} {\hat{h}}_{ij}&= A{{\bar{h}}}_{ij} + {{\bar{D}}}_{(i}A_{j)} + A_{ij}, \end{aligned}$$

(110)

$$\begin{aligned} {\hat{h}}^{ij}&= B{{\bar{h}}}^{ij} + {{\bar{D}}}^{(i}B^{j)} + B^{ij}, \end{aligned}$$

(111)

where $A_{ij}$ and $B^{ij}$ are traceless-transverse tensors with respect to ${\bar{h}}_{ij}$, and ${{\bar{D}}}_i$ is the covariant derivative with respect to that same metric, with ${{\bar{D}}}^i {:}{=}{{\bar{h}}}^{ci}{{\bar{D}}}_c$. We recall that ${\hat{h}}_{ij}$ and ${\hat{h}}^{ij}$ are by definition inverse from each-other, and the same for ${{\bar{h}}}_{ij}$ and ${{\bar{h}}}^{ij}$.

We can rewrite equation (56) as

$$\begin{aligned} 2 {{\bar{D}}}_{[c}C^c_{j]i} + 2C^c_{d[c}C^d_{j]i} = 0, \end{aligned}$$

(112)

with $C^a_{ij} {:}{=}{\hat{h}}^{ak}\left( {{\bar{D}}}_{(i}{\hat{h}}_{j)k} - \frac{1}{2}{{\bar{D}}}_k{\hat{h}}_{ij}\right)$. Then, a sufficient condition on ${\hat{h}}_{ij}$ to be solution of the above equation is

$$\begin{aligned} {{\bar{D}}}_{k}{\hat{h}}_{ij} = 0. \end{aligned}$$

(113)

This condition implies that ${\hat{h}}_{ij} - {\hat{h}}_{kl}\bar{h}^{kl}\,{{\bar{h}}}_{ij}/3$ is a traceless-transverse tensor with respect to $\bar{\varvec{h}}$. Then, from the uniqueness of the SVT decomposition, the gradient mode (i.e., ${\hat{D}}_{(i} A_{j)}$) of ${\hat{h}}_{ij}$ is zero and we have

$$\begin{aligned} {\hat{h}}_{ij}&= A{{\bar{h}}}_{ij} + A_{ij}, \end{aligned}$$

(114)

$$\begin{aligned} \bar{D}_i A&= 0 \quad ; \quad {{\bar{D}}}_k A_{ij} = 0. \end{aligned}$$

(115)

The condition (113) implies $C^a_{ij} =0$, which is the difference between the physical ${\hat{D}}_i$ and the reference ${{\bar{D}}}_i$ connections. Therefore we have ${\hat{D}}_i = {{\bar{D}}}_i$, implying ${{\bar{D}}}_{k}{\hat{h}}^{ij} = 0$. Then, the contravariant gradient mode also vanishes and we have

$$\begin{aligned} {\hat{h}}^{ij}&= B{{\bar{h}}}^{ij} + B^{ij}, \end{aligned}$$

(116)

$$\begin{aligned} {{\bar{D}}}_i B&= 0 \quad ; \quad {{\bar{D}}}_k B^{ij} = 0. \end{aligned}$$

(117)

As for the homogeneity of ${\hat{{\mathcal {R}}}}_{ij}$, we have

$$\begin{aligned} \partial _i{\hat{{\mathcal {R}}}}= \partial _i\left( B^{cd}{\bar{{\mathcal {R}}}}_{cd}\right) , \end{aligned}$$

(118)

using $\partial _i({{\bar{h}}}^{cd}{\bar{{\mathcal {R}}}}_{cd}) = 0$ and equation (56). Therefore, for spherical or hyperbolic topologies, where by definition ${\bar{{\mathcal {R}}}}_{ij} = \pm 2 \bar{h}_{ij}$, we obtain $\partial _i{\hat{{\mathcal {R}}}} = 0$. In summary, from the guess (113), for $\tilde{\Sigma } = \mathbb {S}^3$ or $\mathbb {H}^3$, the spatial scalar curvature of the Galilean structure is homogeneous. However, it is not necessarily isotropic as we have the following relation:

$$\begin{aligned} {\hat{{\mathcal {R}}}}_{ij} = \pm 2/A \left( {\hat{h}}_{ij} - A_{ij}\right) , \end{aligned}$$

(119)

with ${\hat{h}}_{ij} - \frac{1}{3} h^{cd}A_{cd} \, {\hat{h}}_{ij} \not = 0$.

1.2 D.2 Expansion Tensor of $G^\mu$

We consider a vector $\varvec{G} \in \textrm{ker}({{\bar{R}}}_{\mu \nu })$. Then, from relation (48) and the definition of $\bar{R}_{\mu \nu }$, in a coordinate system $\{\varvec{\partial }_t, \varvec{\partial }x^i\}$ adapted to the $\tau$-foliation with $\varvec{G} = \varvec{\partial }_t$, we can choose ${{\bar{h}}}_{ij}$ such that

$$\begin{aligned} \partial _{t|_G} {{\bar{h}}}_{ij} = 0. \end{aligned}$$

(120)

From that equation and relation (115), the time derivative along $\varvec{G}$ of the physical metric ${\hat{h}}_{ij}$ is

$$\begin{aligned} \partial _{t|_G} {\hat{h}}_{ij} = \frac{\partial _{t|_G} A}{A} {\hat{h}}_{ij} - \frac{\partial _{t|_G} A}{A} A_{ij} + \partial _{t|_G} A_{ij}. \end{aligned}$$

(121)

From the definition of the expansion tensor (92), this leads to . Then from (121) and the fact that ${\hat{D}}_k A = 0$, ${\hat{D}}_k A_{ij} = 0$ and ${\hat{D}}_k \partial _{t|_G} A_{ij} = 0$ (using (120) along with the property ${\hat{D}}_i = {{\bar{D}}}_i$ for the last relation), we have

(122)

That property is equivalent as having (this follows from the unicity of the SVT decomposition)

(123)

with ${\hat{D}}_k H = 0$ and $D_k\Xi _{ij} = 0$. From and the definition of the volume of $\Sigma$, i.e., $V_\Sigma {:}{=}\int _\Sigma \sqrt{\textrm{det}\, {\hat{h}}_{ij}} \, \textrm{d}x^3$, we have $H(t) = \dot{V}_\Sigma /(3V_\Sigma )$. Result (123) holds once condition (113) is fulfilled, or if we directly impose ${\hat{{\mathcal {R}}}}_{ij} = {\hat{{\mathcal {R}}}}(t)\, {\hat{h}}_{ij}/3$ in the case ${\tilde{\Sigma }} = \mathbb {S}^3$ or $\mathbb {H}^3$. Equation 123 is the first of the two conditions (57) and (58) for $\varvec{G}$ to be a Galilean vector.

From the general form of the physical spatial Ricci curvature (119) (depending on the guess (113)) and with (114), we see that imposing the physical spatial curvature to be only trace (i.e., ${\hat{{\mathcal {R}}}}_{ij} \propto {\hat{h}}_{ij}$) leads to $A_{ij} = 0$, and $\partial _{t|_G} {\hat{h}}_{ij} \propto {{\bar{h}}}_{ij} \propto {\hat{h}}_{ij}$. Therefore, we have the following property:

$$\begin{aligned} {\hat{{\mathcal {R}}}}_{ij} \propto {\hat{h}}_{ij} \Rightarrow \Xi _{ij} = 0. \end{aligned}$$

(124)

That is: an isotropic spatial curvature implies an isotropic expansion. It is however not yet clear whether or not the reverse property is true.

Remark 8

We expect that the general solution of equation (56) would rather constrain $\Xi _{ij}$ to be only harmonic, i.e., ${\hat{\Delta }} \Xi _{ij} = 0$, instead of gradient free. Indeed, being traceless-transverse and harmonic is the expected condition for a tensor representing global anisotropic expansion.

1.3 D.3 Vorticity of $G^\mu$

From the Galilean limit, the vector $\varvec{B}$ present in the first order of is irrotational with respect to the Galilean structure. Using relation (96) for the difference between two vorticity tensors, this implies that where $\varvec{w} = \varvec{G} - \varvec{B}$ is a spatial vector. Then using ${\hat{R}}_{\mu \nu }{\hat{h}}^{\mu \alpha }G^\nu = 0$, which results from (50), we obtain

$$\begin{aligned} {\hat{D}}_c{\hat{D}}^{[i}w^{c]} - 2{\hat{D}}^i H = 0. \end{aligned}$$

(125)

From the unicity of the Hodge decomposition, we have ${\hat{D}}_c{\hat{D}}^{[i}w^{c]} = 0$, which implies ${\hat{D}}^{[i}w^{j]} =0$. We just showed that .

In conclusion, the vector $\varvec{G}$ is a Galilean vector: $\varvec{G} = \tilde{\varvec{G}}$.

1.4 D.4 Uniqueness of the Galilean Vector?

While $\varvec{G}$ is a Galilean vector, it is not uniquely defined if $\text {dim}[\ker ({{\bar{R}}}_{\alpha \beta })] > 1$ (see Sect. 5.2.3). For two independent Galilean vectors $\tilde{\varvec{G}}_{(1)}$ and $\tilde{\varvec{G}}_{(2)}$, we define the spatial vector ${\varvec{w} = \tilde{\varvec{G}}_{(2)} - \tilde{\varvec{G}}_{(1)}}$. Then from the difference between two expansion and vorticity tensors (equations (95) and (96)), the vector $\varvec{v}$ is constrained by:

$$\begin{aligned} {\hat{D}}_i w^j = 0 \quad ; \quad w^k{\hat{{\mathcal {R}}}}_{ki} = 0. \end{aligned}$$

(126)

The second property is a direct consequence of the first one when calculating $D_{[i}D_{j]} w^k$. In conclusion, if there is not a unique inertial frame at the NR limit, then this frame is define up to a global translation in the direction of zero spatial curvature.

Remark 9

Even if there is not a unique inertial frame, the closedness of $\Sigma$ still implies the existence of a “preferred” one, due to topological properties [27, 32]. In other words, once we require the existence of a NR limit of either general relativity or the bi-connection theory, this implies the existence of a preferred observer in the Universe. Question: could the primordial power spectrum of the CMB be explained by the fondamental presence of such a preferred frame in the theory, and therefore be of topological origin?

Appendix E Derivation of the Dictionary

In this section, we consider isotropic expansion ($\Xi _{ij} = 0$). Using the Taylor series (11) and (12) of the spacetime metric, in a coordinate system $\{\varvec{\partial }_t, \varvec{\partial }x^i\}$ adapted to the $\tau$-foliation and where $\varvec{\partial }_t = \tilde{\varvec{G}}$, we have

(127)

and

(128)

where, as shown in Appendix C.2, $B^\mu - {{\tilde{G}}}^\mu {=}{:}{\hat{D}}^\mu \sigma$ with ${\hat{D}}_i \sigma {:}{=}{\hat{h}}_{ki}{\hat{D}}^k \sigma$. We also defined $k_{ij} {:}{=}k^{cd}{\hat{h}}_{ci} {\hat{h}}_{dj}$. We recall that $B^\mu$ and $\phi$ are defined in Sect. 3.4 where the NR limit is presented. For now, $\phi$ is just a scalar field related to the 4-acceleration of $B^\mu$, i.e., .

The goal is to find if there is a relation between $\phi$, $\sigma$, $k^{ij}$ and the gravitational field $\Phi$. A powerful property is the fact that in both Newton’s theory and the NEN theory, $\Phi$ is given by the 4-acceleration of the Galilean vector [34, 36], i.e., the 4-acceleration of inertial frames:

(129)

Using formula (60) in [34], we can express as function of and obtain (we keep only spatial indices as all the tensors involved are spatial):

(130)

Then using (99) and , we obtain ${\hat{D}}^i \left( \phi -\frac{1}{2} {\hat{D}}^k \sigma {\hat{D}}_k \sigma - \partial _{t|_G}\sigma \right) = {\hat{D}}^i\Phi$, which leads to

$$\begin{aligned} \phi -\frac{1}{2} {\hat{D}}^k \sigma {\hat{D}}_k \sigma - \partial _{t|_G}\sigma = \Phi , \end{aligned}$$

(131)

up to an integration constant. We now have a link between $\phi$, $\sigma$ and $\Phi$.

Let us consider the SVT decomposition of $k^{ij} - {\hat{D}}^i \sigma {\hat{D}}^j \sigma$, as is usually done in the weak field limit:

$$\begin{aligned} k^{ij} - {\hat{D}}^i \sigma {\hat{D}}^j \sigma {=}{:}2\psi {\hat{h}}^{ij} - 2{\hat{D}}^i{\hat{D}}^j E - 2{\hat{D}}^{(i}F^{j)} - 2f^{ij}, \end{aligned}$$

(132)

with ${\hat{D}}_kF^k {:}{=}0$ and $f^{ij}{\hat{h}}_{ij} {:}{=}0 {=}{:}{\hat{D}}_k f^{ki}$. The first order of the (modified) Einstein equation (42) is

(133)

This holds either in the Euclidean case (general relativity), or in the non-Euclidean case (bi-connection theory). Using formula (108) for , with (132) and (131) we obtain

$$\begin{aligned} \left( \frac{\kappa }{2}\rho + \Lambda \right) {\hat{h}}^{ij}&= {\hat{h}}^{ij}(\dot{H} + 3H^2) + {\hat{h}}^{ij}\left[ {\hat{\Delta }}\psi + {\hat{\Delta }}(H\sigma )\right] \nonumber \\&\qquad + {\hat{D}}^i{\hat{D}}^j\left[ -\Phi + H\sigma + \psi + 4KE\right] \nonumber \\&\qquad + 4K{\hat{D}}^{(i}F^{j)} \nonumber \\&\qquad + \left( 6K - {\hat{\Delta }}\right) f^{ij}, \end{aligned}$$

(134)

where K is the spatial Gaussian curvature: ${\hat{{\mathcal {R}}}}_{ij} = 2K/a^2\, {\hat{h}}_{ij}$. The unicity of the SVT decomposition applied on the above equation leads to

$$\begin{aligned}&\frac{\kappa }{2}\rho + \Lambda = \dot{H} + 3H^2 +{\hat{\Delta }}\psi + {\hat{\Delta }}(H\sigma ), \end{aligned}$$

(135)

$$\begin{aligned}&-\Phi + H\sigma + \psi + 4KE = 0, \end{aligned}$$

(136)

$$\begin{aligned}&4K{\hat{D}}^{(i}F^{j)} = 0, \end{aligned}$$

(137)

$$\begin{aligned}&\left( 6K - {\hat{\Delta }}\right) f^{ij} = 0. \end{aligned}$$

(138)

So for any K, we have $f^{ij} = 0$ (more precisely, $f^{ij}$ is harmonic), and using the Poisson equation (76) for $\Phi$ with relation (135) we obtain

$$\begin{aligned} \psi + H\sigma = \Phi . \end{aligned}$$

(139)

In the Euclidean case (i.e., $K=0$), which corresponds to the NR limit of Einstein’s equation and the usual Newton-to-GR dictionary, E and $F^i$ are free. This is a usual gauge freedom in post-Newtonian theory. However, we see that in the non-Euclidean case, relations (136) and (137) imply $E=0$ and $F^i=0$.

Finally, the spacetime metric can be written as follows as function of the gravitational potential:

(140)

and

(141)

where $E\not =0$ and $F^i\not =0$ only if $K = 0$.

It seems that in the non-Euclidean case, E and $F^i$ are not anymore gauge variables and are necessarily zero, i.e., the metric needs to be written with the same form as in the standard post-Newtonian gauge with Newton’s theory. In the next Appendix, we show that this peculiarity in the non-Euclidean case comes from a gauge fixing we implicitly made in Sect. 6 by assuming that ${{\bar{R}}}_{\mu \nu }$ should not depend on $\lambda$.

Appendix F Gauge Freedom in the NR limit

1.1 F.1 Gauge Freedom

Let us consider $\mathbb {R}^4$ on which we define two Minkowski metrics $\eta _{\mu \nu }$ and ${\tilde{\eta }}_{\mu \nu }$. The set of foliations which are flat with respect to $\eta _{\mu \nu }$ are not necessarily flat with respect to ${\tilde{\eta }}_{\mu \nu }$. In other words, in a coordinate system where $\eta _{\mu \nu } = \textrm{diag}(-1; 1; 1; 1)$, the relation ${\tilde{\eta }}_{\mu \nu } = \textrm{diag}(-1; 1; 1; 1)$ does not necessarily hold. This is because there are four degrees of freedom in defining ${\tilde{\eta }}_{\mu \nu }$ with respect to $\eta _{\mu \nu }$, corresponding to a diffeomorphism $\varphi$ of $\mathbb {R}^4$ in itself. The freedom in changing ${\tilde{\eta }}_{\mu \nu }$ while preserving its properties of being a Minkowski metric is related to the differential $\textrm{d}\varphi : T\mathbb {R}^4 \rightarrow T\mathbb {R}^4$ of $\varphi$:

$$\begin{aligned} {\tilde{\eta }}_{\mu \nu } \overset{\textrm{freedom}}{\longrightarrow } \textrm{d}\varphi \left( {\tilde{\eta }}_{\mu \nu }\right) . \end{aligned}$$

(142)

The same freedom applies for $R_{\mu \nu }$ and ${{\bar{R}}}_{\mu \nu }$ in the bi-connection theory. Therefore, the role of the bi-connection condition (43) is to constrain the four degrees of freedom in the definition of the physical connection with respect to the reference one (or inversely).

This freedom arising when two tensors are defined on the same manifold is also present for the family of Lorentzian metrics used in the NR limit: there is a diffeomorphism freedom in defining with respect to , with $\lambda \not = \lambda _0$ for any $\lambda _0 > 0$. However, while in the example related to the Minkowski metrics, this diffeomorphism is general, in the case of the family , because we impose , that diffeomorphism freedom is constrained to be the identity in the limit $\lambda \rightarrow \lambda _0$, and thus should also be parametrised by $\lambda$: .

A consequence is that, at first order in $(\lambda - \lambda _0)$, the freedom in (re)defining with respect to is a Lie derivative (see [20] for a detailed explanation⁹):

(143)

where $\xi ^\mu$ is a vector field on ${\mathcal {M}}$. In perturbation theory, that freedom is called a gauge freedom as it does not change physics.

1.2 F.2 Gauge Invariant Variables in the NR Limit

Under the gauge transformation describe in the previous section, and transform as

(144)

(145)

The vector $\xi ^\mu$ can be written as

$$\begin{aligned} \xi ^\mu = N {{\tilde{G}}}^\mu + {\hat{D}}^\mu {\mathcal {A}}+ {\mathcal {A}}^\mu , \end{aligned}$$

(146)

where ${\mathcal {A}}^\mu \tau _\mu {:}{=}0 {=}{:}{\hat{D}}_\mu {\mathcal {A}}^\mu$, and N and ${\mathcal {A}}$ are scalar fields. We recall that ${{\tilde{G}}}^\mu$ is a Galilean vector. Then

$$\begin{aligned} {\mathcal {L}}_{\varvec{\xi }}{\hat{h}}^{\mu \nu }&= -2NH{\hat{h}}^{\mu \nu } - 2N\Xi ^{\mu \nu } - 2{\hat{h}}^{\alpha (\mu }{{\tilde{G}}}^{\nu )}\nabla _\alpha N - 2{\hat{D}}^\mu {\hat{D}}^\nu {\mathcal {A}}- 2{\hat{D}}^{(\mu } {\mathcal {A}}^{\nu )}, \end{aligned}$$

(147)

$$\begin{aligned} {\mathcal {L}}_{\varvec{\xi }} (-\tau _\mu \tau _\nu )&= -2\tau _{(\mu }\nabla _{\nu )} N, \end{aligned}$$

(148)

where we used ${\hat{h}}^{\alpha \mu }{\hat{\nabla }}_\alpha {{\tilde{G}}}^\nu = -2\,H{\hat{h}}^{\mu \nu } -2\Xi ^{\mu \nu }$. Then using the SVT decomposition introduced in Appendix E we have

(149)

Using the definition (18) of along with formula (88) (in which we replace by $B^\mu {=}{:}{{\tilde{G}}}^\mu + {\hat{D}}^\mu \sigma$; and we replace by ), we get

(150)

Then, from (149) and (150), the gauge transformations (144) and (145) induce a transformation of the variables $\sigma$, $\psi$, E, $F^\mu$, $f^{\mu \nu }$ and $\phi$ as follows:

(151)

(152)

(153)

(154)

(155)

(156)

Therefore, the following variables are gauge invariant:

$$\begin{aligned} {\tilde{\Psi }}&{:}{=}\psi + H\sigma , \end{aligned}$$

(157)

$$\begin{aligned} {\tilde{\Phi }}&{:}{=}\phi + \frac{1}{2}{\hat{h}}^{\alpha \beta }{ \hat{\nabla }}_\alpha \sigma {\hat{\nabla }}_\beta \sigma - G^\mu {\hat{\nabla }}_\mu \sigma , \end{aligned}$$

(158)

$$\begin{aligned} {{\tilde{f}}}^{\mu \nu }&{:}{=}f^{\mu \nu } - \sigma \Xi ^{\mu \nu }. \end{aligned}$$

(159)

${\tilde{\Psi }}$ and ${\tilde{\Phi }}$ are the equivalent of the Bardeen potentials in the cosmological perturbation theory (weak field limit around a homogeneous and isotropic solution). We showed in Appendix E that ${\tilde{\Psi }} = {\tilde{\Phi }}$ either in the Euclidean case (i.e., with Newton’s theory from Einstein’s equation, as expected), or in the non-Euclidean case (from the bi-connection theory). Therefore, the gravitational slip (i.e., ${\tilde{\Psi }} - {\tilde{\Phi }}$) is still zero in the bi-connection theory. Furthermore, we have ${\tilde{\Psi }} = {\tilde{\Phi }} = \Phi$, the gravitational potential, showing that this variable is gauge-invariant, again either in the Euclidean or non-Euclidean case. Equation (152)–(153) show that $\sigma$, E and $F^\mu$ can be set to zero by a gauge transformation.

1.3 F.3 Gauge Freedom in the Reference Curvature

As mentioned above, there is a diffeomorphism freedom in defining ${{\bar{R}}}_{\mu \nu }$ with respect to $R_{\mu \nu }$. In the case of the NR limit, that freedom is present with respect to all the metrics and their curvatures. This implies that, once we consider a NR limit, the diffeomorphism freedom on defining $\bar{R}_{\mu \nu }$ with respect to the family of Lorentzian metrics is parametrised by $\lambda$, leading, at first order, to

(160)

for any vector field $X^\mu$. Under a gauge transformation we have . As the difference between ${{\bar{R}}}_{\mu \nu }$ and comes solely from diffeomorphism freedom, these two tensors have the same property, i.e., they take the form (46) in a certain (one for each) coordinate system¹⁰. In particular, as for ${{\bar{R}}}_{\mu \nu }$, there exists a vector such that .

Because until now we supposed that ${{\bar{R}}}_{\mu \nu }$ was independent on $\lambda$, this means that the calculation of Sect. 6 and Appendix E was performed in a specific gauge where $X^\mu = 0$. This is why we found in Appendix E that E and $F^i$ needed to vanish. In what follows we explain why, without the gauge choice $X^\mu = 0$, E and $F^i$ do not vanish anymore.

In a general gauge, equation (55) takes the form

(161)

with , and the first order equation (133) (used to derive the dictionary) becomes

(162)

Because has the same properties as ${{{\bar{R}}}}_{\mu \nu }$, the resolution of equation (161) is equivalent as solving equation (55) in Sect. 6.2.1. In particular, corresponds to a Galilean vector, and in Euclidean, spherical or hyperbolic topologies without anisotropic expansion we have

(163)

Then, writing $X^\mu {=}{:}M{{\tilde{G}}}^\mu + {\hat{D}}^\mu {\mathcal {X}}+ {\mathcal {X}}^\mu$ with ${\mathcal {X}}^\mu \tau _\mu {:}{=}0 {=}{:}{\hat{D}}_\mu {\mathcal {X}}^\mu$, the additional term in equation (162), compared to equation (133), takes the form

(164)

A consequence is that reperforming the calculation of Section E, we obtain

$$\begin{aligned} \left( \frac{\kappa }{2}\rho + \Lambda \right) {\hat{h}}^{ij}&= {\hat{h}}^{ij}(\dot{H} + 3H^2) + {\hat{h}}^{ij}\left[ {\hat{\Delta }}\psi + {\hat{\Delta }}(H\sigma )\right] \nonumber \\&\qquad + {\hat{D}}^i{\hat{D}}^j\left[ -\Phi + H\sigma + \psi + 4K(E + {\mathcal {X}})\right] \nonumber \\&\qquad + 4K{\hat{D}}^{(i}\left( F^{j)} + {\mathcal {X}}^{j)}\right) \nonumber \\&\qquad + \left( 6K - {\hat{\Delta }}\right) f^{ij}. \end{aligned}$$

(165)

Therefore, E and $F^i$ are not anymore imposed to be zero, instead the gauge-invariant variables ${\mathcal {C}}{:}{=}E + {\mathcal {X}}$ and ${\mathcal {C}}^i {:}{=}F^i + {\mathcal {X}}^i$ are zero.

In conclusion, from either the Einstein equation (Euclidean case) or the bi-connection theory (non-Euclidean case), the spacetime metric (respectively ) takes the form (64) [respectively (65)] as function of the gravitational potential, with $\sigma$, E and $F^i$ being free gauge fields in both cases.