On the Metric Lorentz Invariant Newtonian Cosmology

Abstract

We review a metric theory of gravitation that combines Newtonian gravity with Lorentz invariance. Beginning with a conformastatic metric justified by the Weak Equivalence Principle. We describe, within the Newtonian approximation, the spacetime geometry generated by a static distribution of dust matter. To extend this description to moving sources, we apply a Lorentz transformation to the static metric. This procedure yields, again within the Newtonian approximation, the metric associated with moving bodies. In doing so, we construct a gravitational framework that captures key relativistic features—such as covariance under Lorentz transformations—while remaining rooted in Newtonian dynamics. This approach offers an alternative route to describing weak-field gravitational interactions, without relying directly on Einstein’s field equations.

1. Introduction

One of the most profound conceptual consequences of the gauge invariance inherent to General Relativity (GR) is the challenge it poses to the substantivalist interpretation of spacetime. Because GR is invariant under active diffeomorphisms—smooth transformations that relocate points on the spacetime manifold without altering physical observables—it becomes difficult to assign physical meaning to individual spacetime points independent of the dynamical fields that inhabit them. This has led many to interpret GR as favoring a relational view of spacetime, in which geometry has no ontological status apart from the matter and fields it structures. Nevertheless, the relational perspective is not the only possible framework for understanding gravity. In this essay, we explore an alternative approach that seeks to preserve a form of spacetime substantivalism—at least within the Newtonian limit—where gravitational fields are weak and velocities are small. In this regime, it is possible to define a meaningful geometric structure directly in terms of intrinsic gravitational potentials, without relying on the full machinery of Einstein’s field equations. Rather than viewing spacetime curvature as something that must be derived from the dynamics of matter and energy through the field equations, we propose that the geometry can emerge from physical principles such as the Weak Equivalence Principle and Lorentz symmetry, leading to a consistent description of gravity in weak-field scenarios.

This approach, while more limited in scope than the full general relativistic treatment, opens the door to alternative interpretations of gravitation that are compatible with a substantivalist view of spacetime—offering a fresh perspective on classical gravitational physics and its foundational implications.

The first step in pursuing our objective involves modeling a static mass distribution using a conformastatic metric. This choice naturally leads to a form that, in linearized General Relativity, emerges within the de Donder (or Lorenz) gauge framework [1,2,3,4]. However, in our approach, this metric structure is justified independently of the field equations, relying instead on the Principle of Equivalence as a guiding principle (see [5] for a critical examination of this principle and [6] for its standard formulation). The second key idea is to extend the construction to configurations involving moving bodies. This is accomplished by applying a general infinitesimal Lorentz transformation to the static metric, allowing us to account for the relativistic effects induced by motion without resorting to the Einstein field equations. The resulting metric, when retardation effects in the potentials are neglected, coincides with that of linearized GR and also aligns with the so-called Parametrized Post-Newtonian (PPN) approximation [7,8,9,10]. Accordingly, it reproduces the classical tests of General Relativity with equal success.

To the best of our knowledge, the idea of applying a Lorentz transformation to obtain the metric of a moving mass was first outlined by Kenneth Nordtvedt Jr. in [11], in the context of the Schwarzschild metric, and more recently, an application of this methodology to the metric generated by a point mass is presented in a controversial unpublished manuscript [12], whose conclusions remain subject to scrutiny due to the lack of peer-reviewed validation. Furthermore, the topic was merely touched upon in a brief footnote on page 46 of the book [13], without further elaboration or detailed discussion, suggesting that its significance may have been overlooked or assumed to be already understood by the reader. That is, little progress has been made in developing this approach further.

In contrast, the present essay develops this approach further by deriving the metric generated by moving bodies within the weak-field approximation, without invoking Einstein’s field equations. This framework enables us to reproduce results consistent with linearized General Relativity solely on the basis of kinematic arguments and the weak Principle of Equivalence (the equivalence of the inertial and gravitational mass).

As an application, we will consider the static case, where we compute the line element generated by a point particle at rest. Using this metric, we then analyze the motion of a planet in orbit and calculate the precession of its perihelion. Remarkably, the result coincides with that obtained using the Schwarzschild solution in General Relativity, thereby illustrating the consistency of the approach and its ability to reproduce well-established relativistic effects within a simplified framework.

The essay is structured as follows: We begin by reviewing the geometric foundations necessary to construct a covariant metric theory of gravity on a Lorentzian manifold—that is, a theory built exclusively from intrinsic elements of the manifold itself. The paradigmatic example, due to its elegance and simplicity, is the General Theory of Relativity: a theory formulated in terms of the manifold’s curvature, in particular the Ricci curvature, which remains invariant under active diffeomorphisms. As a consequence, the Einstein field equations are invariant under a specific class of these transformations—namely, hole diffeomorphisms (see, for instance, [14,15,16,17]). This invariance leads to a striking feature: the non-uniqueness of solutions to the field equations, even given fixed initial or boundary data. This lack of uniqueness has deep philosophical implications, as it challenges the Newtonian notion of substantival space and time, and instead supports a relational view of spacetime—where points in spacetime have no identity independent of the fields and relations defined upon them.

We then proceed to the main objective of the essay: the development of a Lorentz invariant, a metric-based formulation of Newtonian cosmology. This approach extends the classical Newtonian framework by incorporating Lorentz symmetry, thereby enabling a consistent treatment of gravitational phenomena in the weak-field and low-velocity regime—such as that characterizing our solar system. In particular, we derive the gravitational potentials associated with moving bodies by applying a Lorentz transformation to a static weak-field metric. This method reveals how motion modifies the gravitational field, providing a bridge between Newtonian gravity and relativistic corrections while remaining within a conceptually familiar and computationally tractable framework.Finally, we derive the geodesic equations within this approximation and apply them specifically to the metric generated by a rotating, spherically symmetric homogeneous body. This allows us to explore the motion of test particles in the presence of both gravitational and frame-dragging effects, highlighting how rotation influences spacetime geometry and particle trajectories—even in a weak-field regime.

2. Pseudo-Riemann Geometry and Covariance

The goal of this section is to briefly review the foundational concepts of differential geometry in the context of a Lorentzian manifold. This mathematical framework is essential for formulating physical theories that rely solely on intrinsic geometric structures—namely, covariant theories that remain valid under general coordinate transformations. As a paradigmatic example, we will examine General Relativity, not only as a theory of gravitation but also in terms of its broader philosophical implications, particularly regarding the nature of space, time, and the role of geometry in fundamental physics.

Let me start with an $1+3$-dimensional pseudo-Riemann, in this particular case, a Lorentz manifold represented by the pair $(\mathcal{M},\mathfrak{g})$, where it is possible to select a frame field ${\left\{{\mathbf{w}}_{\mu }\right\}}_{\mu =0,1,2,3}$ in $\mathfrak{X}\left(\mathcal{M}\right)$ (the set of vector—contravariant vector in physical language—fields in $\mathcal{M}$) that satisfies the following:

$$\begin{array}{c}\hfill \mathfrak{g}({\mathbf{w}}_0,{\mathbf{w}}_0)=1,\mathfrak{g}({\mathbf{w}}_1,{\mathbf{w}}_1)=\mathfrak{g}({\mathbf{w}}_2,{\mathbf{w}}_2)=\mathfrak{g}({\mathbf{w}}_3,{\mathbf{w}}_3)=-1,\end{array}$$

(1)

and $\mathfrak{g}({\mathbf{w}}_{\mu },{\mathbf{w}}_{\nu })=0$ when $\mu \ne \nu$.

Let ${\left\{{\mathfrak{w}}^{\mu }\right\}}_{\mu =0,1,2,3}$ represent its dual form (covariant vector or simply covector), i.e., ${\mathfrak{w}}^{\mu }\left({\mathbf{w}}_{\nu }\right)={\delta }_{\nu }^{\mu }$. Then, the metric $\mathfrak{g}$, which is a 2-form (2-covariant tensor), can be written as $\mathfrak{g}={\mathfrak{w}}^0\otimes {\mathfrak{w}}^0-{\sum }_{i=1}^3{\mathfrak{w}}^i\otimes {\mathfrak{w}}^i$.

We can also introduce the so-called Musical Operators, which are defined as follows:

• Given the contravariant vector $\mathbf{w}$, we have (application of the flat operator) the covariant vector ${\mathbf{w}}^{♭}$ defined as ${\mathbf{w}}^{♭}\left(\mathbf{v}\right)=\mathfrak{g}(\mathbf{w},\mathbf{v}).$
• Given a covariant vector $\mathfrak{w}$, we have (application of the sharp operator) the contravariant vector ${\mathfrak{w}}^{♯}$ defined as $\mathfrak{g}({\mathfrak{w}}^{♯},\mathbf{v})=\mathfrak{w}\left(\mathbf{v}\right)$.

To make calculations we need a coordinate system: a way to assign a set of four numbers $\left\{x^{\mu }\right\}$ (coordinates) to each point on the differentiable manifold, allowing the description of geometric objects and functions locally using tools from calculus. More specifically, we need a local chart $(\mathcal{V},\psi )$ satisfying the following: $\mathcal{V}$ is an open subset of $\mathcal{M}$, and $\psi$ is a homeomorphism (a continuous bijection with a continuous inverse) from $\mathcal{V}$ to an open subset of ${\mathbb{R}}^4$. To each point P of $\mathcal{M}$, we have $\psi \left(P\right)=(x^0\left(P\right),x^1\left(P\right),x^2\left(P\right),x^3\left(P\right))$.

In ${\mathbb{R}}^4$ a vector field can be identified to a directional derivative. Thus, we can define the vector basis associate to the coordinate system as ${\left\{{\partial }_{\mu }\right\}}_{\mu =0,1,2,3}$, and, using this basis, the vector field $\mathbf{w}$ can be written as $\mathbf{w}=w^{\mu }{\partial }_{\mu }$.

On the other hand, any vector field $\mathbf{w}$ on ${\mathbb{R}}^4$ can be sent to the manifold via the push-forward of ${\psi }^{-1}$, denoted by ${\psi }_{\ast }^{-1}$; that is, ${\psi }_{\ast }^{-1}\mathbf{w}$ is a vector field on $\mathfrak{X}\left(\mathcal{M}\right)$. Denoting the push-forward of the coordinate vector basis by ${\mathbf{e}}_{\mu }={\psi }_{\ast }^{-1}{\partial }_{\mu }$, the inner product of two vector fields ${\mathbf{v}}_1=v_1^{\mu }{\mathbf{e}}_{\mu }$ and ${\mathbf{v}}_2=v_2^{\mu }{\mathbf{e}}_{\mu }$ in $\mathfrak{X}\left(\mathcal{M}\right)$ is $\mathfrak{g}({\mathbf{v}}_1,{\mathbf{v}}_2)=g_{\mu \nu }{v}_1^{\mu }{v}_2^{\nu }$, with $g_{\mu \nu }=\mathfrak{g}({\mathbf{e}}_{\mu },{\mathbf{e}}_{\nu })$. In the same way considering a 2-form (covantiant tensor) $\mathfrak{h}=h_{\mu \nu }{\mathfrak{w}}^{\mu }\otimes {\mathfrak{w}}^{\nu }$, the the pull-back of ${\psi }^{-1}$, denoted by ${{\psi }^{-1}}^{\ast }$, sends the tensor $\mathfrak{h}$ to the tensor ${{\psi }^{-1}}^{\ast }\mathfrak{h}$ in ${\mathbb{R}}^4$, defined as

$$\begin{array}{c}\hfill {{\psi }^{-1}}^{\ast }\mathfrak{h}({\partial }_{\mu },{\partial }_{\nu })=\mathfrak{h}({\mathbf{e}}_{\mu },{\mathbf{e}}_{\nu })\equiv h_{\mu \nu }.\end{array}$$

(2)

Next, we choose two points in ${\mathbb{R}}^4$, namely $A=(A^0,A^1,A^2,A^3)$ and $A+dx=(A^0+d{x}^0,A^1+d{x}^1,A^2+d{x}^2,A^3+d{x}^3)$, and we consider the vector form A to $A+dx$, namely $d\mathbf{x}=d{x}^{\mu }{\partial }_{\mu }$. Once we have this vector, we can define the square of the line element, namely $d{s}^2$, as $d{s}^2\equiv {{\psi }^{-1}}^{\ast }\mathfrak{g}(d\mathbf{x},d\mathbf{x})=\mathfrak{g}({\psi }_{\ast }^{-1}d\mathbf{x},{\psi }_{\ast }^{-1}d\mathbf{x})=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, which is the key element that allows us to make calculations.

The definition is independent of the coordinate system. In another coordinate system with the basis ${\left\{{\partial }_{\mu }^{\prime }\right\}}_{\mu =0,1,2,3}$, we will have $d{s}^2=g_{\mu \nu }^{\prime }d{x^{\prime }}^{\mu }d{x^{\prime }}^{\nu }$, with $g_{\mu \nu }^{\prime }d{x^{\prime }}^{\mu }=\mathfrak{g}({\mathbf{e}}_{\mu }^{\prime },{\mathbf{e}}_{\nu }^{\prime })$, being ${\mathbf{e}}_{\mu }^{\prime }={{\psi }^{\prime }}_{\ast }^{-1}{\partial }_{\mu }^{\prime }$.

Once we have introduced this notation, we can construct a frame-independent (independent of the coordinate system) physical theory by working directly in the Lorentz manifold $\mathcal{M}$, that is, imposing some relations between intrinsic elements, such as vectors or tensors, of the manifold. This is what one understands as a covariant theory.

For example, given two 2-forms $\mathfrak{h}$ and $\mathfrak{t}$, we can impose the equation $\mathfrak{h}=\mathfrak{t}$, which, in coordinates, will have the form $h_{\mu \nu }=t_{\mu \nu }$, and, in another coordinate frame, the form $h_{\mu \nu }^{\prime }=t_{\mu \nu }^{\prime }$.

The relation between both equations comes form the change in coordinates (in modern language a passive diffeomorphism):

$$\begin{array}{c}\hfill {\partial }_{\mu }={\displaystyle \frac{\partial {x^{\prime }}^{\alpha }}{\partial x^{\mu }}}{\partial }_{\alpha }^{\prime }⟺{\mathbf{e}}_{\mu }={\displaystyle \frac{\partial {x^{\prime }}^{\alpha }}{\partial x^{\mu }}}{\mathbf{e}}_{\alpha }^{\prime }.\end{array}$$

(3)

Then, one finds the following:

$$\begin{array}{c}\hfill h_{\mu \nu }={{\psi }^{-1}}^{\ast }\mathfrak{h}({\partial }_{\mu },{\partial }_{\nu })=\mathfrak{h}({\mathbf{e}}_{\mu },{\mathbf{e}}_{\nu })={\displaystyle \frac{\partial {x^{\prime }}^{\alpha }}{\partial x^{\mu }}}{\displaystyle \frac{\partial {x^{\prime }}^{\beta }}{\partial x^{\nu }}}\mathfrak{h}({\mathbf{e}}_{\alpha }^{\prime },{\mathbf{e}}_{\beta }^{\prime })={\displaystyle \frac{\partial {x^{\prime }}^{\alpha }}{\partial x^{\mu }}}{\displaystyle \frac{\partial {x^{\prime }}^{\beta }}{\partial x^{\nu }}}{{{\psi }^{\prime }}^{-1}}^{\ast }\mathfrak{h}({\partial }_{\alpha }^{\prime },{\partial }_{\beta }^{\prime })={\displaystyle \frac{\partial {x^{\prime }}^{\alpha }}{\partial x^{\mu }}}{\displaystyle \frac{\partial {x^{\prime }}^{\beta }}{\partial x^{\nu }}}{h}_{\alpha \beta }^{\prime }.\end{array}$$

(4)

Example: General Relativity

The General Theory of Relativity constitutes a paradigmatic example of a covariant field theory, in which the geometric structure of the spacetime manifold $\mathcal{M}$ is dynamically determined by its matter and energy content. This relationship is encoded in the Einstein field equations, which establish a direct link between the curvature of spacetime and the stress–energy tensor via the following relation:

$$\begin{array}{c}\hfill {\mathfrak{Ric}}_{\mathfrak{g}}=8\pi G{\overline{\mathfrak{T}}}_{\mathfrak{g}},\end{array}$$

(5)

where ${\mathfrak{Ric}}_{\mathfrak{g}}$ is the Ricci tensor and ${\overline{\mathfrak{T}}}_{\mathfrak{g}}={\mathfrak{T}}_{\mathfrak{g}}-{\displaystyle \frac{1}{2}}\mathfrak{g}T$, with ${\mathfrak{T}}_{\mathfrak{g}}$ being the energy–stress tensor and T its trace. Specifically

$$\begin{array}{c}\hfill \overline{\mathfrak{T}}=\mathfrak{T}-{\displaystyle \frac{1}{2}}\mathfrak{g}T=(\rho +p)\mathfrak{u}\otimes \mathfrak{u}-{\displaystyle \frac{1}{2}}(\rho -p)\mathfrak{g},\end{array}$$

(6)

where $\mathfrak{u}$ is the dual of the four-velocity $\mathbf{u}=({\displaystyle \frac{dt}{ds}},\mathbf{v})$, with $\mathbf{v}={\displaystyle \frac{d\mathbf{x}}{ds}}$.

Remark 1.

It is worth noting that we have written the field equations in their original form, as presented by Einstein in his lectures at the Prussian Academy of Sciences in November 1915 [18]. We have deliberately avoided using the more familiar modern expression:

$$\begin{array}{c}\hfill {\mathfrak{Ric}}_{\mathfrak{g}}-{\displaystyle \frac{1}{2}}\mathfrak{g}R=8\pi G{\mathfrak{T}}_{\mathfrak{g}},\end{array}$$

(7)

where R is the Ricci scalar. Although this form is now standard, particularly because the Bianchi identities ensure that the left-hand side is divergence-free—implying the local conservation of the energy–momentum tensor—it can be misleading from a historical and conceptual standpoint.

In fact, the conservation of the stress–energy tensor is not merely a consequence of the field equations; rather, it played a foundational role in their formulation. As Einstein’s November 1915 lectures clearly show, the requirement of energy–momentum conservation—originally derived by Einstein himself in the case of a pressureless dust fluid—served as a crucial heuristic principle in his search for the correct gravitational field equations. Rather than being a mere consequence of the final equations, the conservation condition acted as a guiding constraint, shaping the form that the field equations had to take. This insight underscores the deep interplay between physical intuition and mathematical structure in the development of General Relativity, where the demand for consistency with known conservation laws was instrumental in identifying the geometrical foundations of gravity.

Next, it is important to point out the following important property of the Ricci tensor: Consider an active diffeomorphism, that is, a diffeomorphism $\varphi :\mathcal{M}\to \mathcal{M}$ different from the identity. The push-forward of $\varphi$ sends vector fields to vector fields

$$\begin{array}{c}\hfill {\varphi }_{\ast }:\mathfrak{X}\left(\mathcal{M}\right)\to \mathfrak{X}\left(\mathcal{M}\right);\mathbf{w}\to {\varphi }_{\ast }\mathbf{w},\end{array}$$

(8)

and the pull-back of $\varphi$ applies to the forms. For example, for a 2-form $\mathfrak{h}$, one has ${\varphi }^{\ast }\mathfrak{h}$ defined as

$$\begin{array}{c}\hfill {\varphi }^{\ast }\mathfrak{h}(\mathbf{w},\mathbf{v})=\mathfrak{h}({\varphi }_{\ast }\mathbf{w},{\varphi }_{\ast }\mathbf{v}).\end{array}$$

(9)

Then it follows that the Ricci tensor is invariant under active diffeomorphisms, in the sense that ${\varphi }^{\ast }{\mathfrak{Ric}}_{\mathfrak{g}}={\mathfrak{Ric}}_{{\varphi }^{\ast }\mathfrak{g}}$.

This means that, when we apply an active diffeomorphism to Einstein’s field equations, we obtain the following:

$$\begin{array}{c}\hfill {\mathfrak{Ric}}_{{\varphi }^{\ast }\mathfrak{g}}=8\pi G{\varphi }^{\ast }{\overline{\mathfrak{T}}}_{\mathfrak{g}}.\end{array}$$

(10)

This has a profound implication. Consider a hole diffeomorphism ${\varphi }_h$, i.e., a diffeomorphism that acts non-trivially outside the domain of the energy–momentum tensor but leaves it unchanged where it does not vanish. In other words, it modifies the metric in “empty” regions without affecting the physically meaningful regions containing matter or energy. In this case, we have ${\varphi }_h^{\ast }{\overline{\mathfrak{T}}}_{\mathfrak{g}}={\overline{\mathfrak{T}}}_{{\varphi }_h^{\ast }\mathfrak{g}}$.

This expresses the fact that the same matter distribution can correspond to different metric configurations related by hole diffeomorphisms—yet all of them satisfy Einstein’s field equations. This leads directly to the hole argument and to the conclusion that the metric field cannot be uniquely determined by the matter distribution alone—unless one accepts that only diffeomorphism equivalence classes of solutions represent physical reality, or in physical terms, General Relativity is a gauge theory.

However, this shift carries significant philosophical implications. The substantivalist conception of spacetime—central to Newtonian mechanics—assumes that space and time exist independently of any material content. Space is conceived as an absolute, immobile stage—a kind of container—within which physical objects move and interact, regardless of their presence. Time, in this view, flows uniformly and independently of events. A similar substantivalist stance is present in the Special Theory of Relativity, where Minkowski spacetime functions as a four-dimensional geometric arena in which events take place. Although there is no absolute simultaneity or preferred reference frame, Minkowski spacetime remains a fixed, non-dynamical background, unaffected by the distribution of matter or energy.

This framework, however, undergoes a profound transformation in the context of General Relativity, which marks a decisive break with substantivalism and instead embraces a relational view of space and time. Inspired by philosophical ideas developed by thinkers such as Descartes and Leibniz, this relationalist perspective holds that space and time do not exist independently of the physical world, but arise from the network of relations among material entities. Spatial and temporal structures, in this view, are not fundamental substances but abstractions derived from how physical systems coexist and interact.

Einstein’s mature formulation of General Relativity gives this relational intuition a precise physical expression through the concept of “point-coincidences”: the idea that only the intersections of worldlines—i.e., localized interactions between physical systems—constitute physically meaningful events. Spacetime points, in themselves, lack intrinsic significance unless they are associated with such coincidences. Thus, the physical content of the theory resides not in a fixed geometric backdrop, but in the evolving web of dynamical relations among observable entities.

Nonetheless, one must be careful with a common interpretation in physics that describes spacetime in General Relativity as “dynamic”—meaning that it interacts with matter by acquiring a metric structure. This interpretation might suggest that spacetime events have an individuality akin to that of Newtonian absolute space. However, such a view is misleading within General Relativity. In this theory, spacetime cannot be separated from the matter and other fields it contains; rather, it is intrinsically connected to them. Events do not exist in isolation—they gain meaning only through their relationship with fields.

In this light, one may argue that positions are defined relative to fields, through the coincidence of world lines or geodesics whose trajectories depend on the metric. Since the metric itself is determined by the distribution of fields, it follows that spatial and temporal locations are not absolute but emerge from the interaction between matter and geometry.

At this point, one might ask how to construct a theory of gravity that aligns with a substantivalist view of spacetime—one in which spacetime has an existence independent of the physical fields it contains. Such a theory would need to be formulated using intrinsic elements of the manifold in a way that preserves general covariance, but crucially, without exhibiting invariance under active diffeomorphisms. The challenge here is profound: as far as we know, geometric objects that can be used to build such a theory—most notably curvature tensors—are inherently invariant under active diffeomorphisms. This makes it exceedingly difficult to reconcile the substantivalist stance with the mathematical structure of General Relativity, where the invariance under diffeomorphisms is not merely a technical property, but a reflection of the relational nature of spacetime itself.

For the present author, constructing a gravitational theory that satisfies both substantivalist intuitions and mathematical consistency seems highly nontrivial. Nonetheless, rather than abandoning the project entirely, we may consider a more modest goal: to restrict our attention to the Newtonian regime, which in the context of a homogeneous expanding universe has been successfully investigated in several works [19,20], and investigate whether it is possible to formulate a metric theory of gravity—at least approximately—that does not rely on curvature tensors, but instead on Newtonian potentials. Such an approach would not be fully general or relativistic, but it might provide a coherent description of gravity at solar system scales, where relativistic effects are small and the Newtonian approximation remains valid.

In this framework, the geometry of spacetime would emerge not from curvature tensors but from the structure of the Newtonian potentials, allowing us to preserve a notion of spacetime as a fixed stage upon which matter evolves. While such a construction would likely lack the elegance and generality of General Relativity, it could offer valuable insights into the conceptual foundations of gravitational theory and serve as a bridge between Newtonian mechanics and relativistic physics. This is the idea we aim to explore in the next section.

3. Newtonian Gravity Plus Lorentz Transformation

In this section, we develop a framework to recover the spacetime metric predicted by linearized General Relativity, without relying explicitly on Einstein’s field equations. Our approach focuses on the weak-field, low-velocity regime relevant to the solar system, where relativistic effects are small but measurable. Given that, at present, the cosmological expansion is negligible on solar system scales, it is justified to adopt a static approximation for the background spacetime. Consequently, we consider a non-expanding universe and limit our analysis to gravitational fields generated by slowly moving bodies—such as planets and stars—within our local astrophysical environment. This approximation allows us to isolate the essential relativistic corrections to Newtonian gravity while working within a conceptually transparent and computationally tractable framework.

Remark 2.

Although the present essay does not directly address the expansion of the universe, it is important to highlight that several significant works have analyzed this phenomenon using Newtonian hydrodinamics. For instance, in [21], by applying the classical conservation, Euler, and Poisson equations and decomposing the velocity field into its three components—shear, rotation, and expansion—the authors derive the Heckmann–Schücking equations [22,23]. These equations allow for a dynamical interpretation of cosmic evolution in terms of Helmholtz’s vortex theorems and the non-homogeneous expansion of the scale factor.

Moreover, in the homogeneous case, these equations generalize the homogeneous Friedmann equations, taking into account the rotation of matter [21,24], whose effect is, at early times, to prevent the Big Bang singularity [21] and, at late times, to enhance the acceleration of cosmic expansion [25]. Nonetheless, some form of dark energy is still required to fully account for the observational data.

Our starting point is a conformastatic metric, which describes a static mass distribution in a given inertial frame K. This choice is motivated not only by simplicity but also by the fact that such a metric naturally arises in the weak-field limit of General Relativity formulated in the de Donder (or Lorenz) gauge [3]. However, in our approach, the adoption of this metric is justified solely by symmetry arguments and the Principle of Equivalence, independently of the field equations.

To develop our argument we start with what is named the Weak Principle of Equivalence, which states that gravitational mass and inertial mass are equivalent, implying that all bodies fall with the same acceleration to the Earth’s surface (where one assumes that the potential is nearly homogeneous), regardless of their composition or internal structure. This last observation, inherited from Galileo and systematized by Newton—and understood in terms of force balance by D’Alembert—finds in Einstein a radical extension: if, in a state of free fall, there is no sensation of gravity, then there is no way to distinguish, via local physical experiments, between a uniform gravitational field and a uniformly accelerated reference frame.

Taking into account the identity between gravitational and inertial mass, Newton’s second law can be reformulated from D’Alembert’s perspective as ${\mathbf{F}}_{\mathrm{i}}+{\mathbf{F}}_{\mathrm{g}}=0$ [26], where ${\mathbf{F}}_{\mathrm{i}}=-m\ddot{\mathbf{x}}$ represents the inertial force—the vis insita [27] described by Newton as the innate force of a body to resist any change in its state of motion—and ${\mathbf{F}}_g=-m\nabla {\Phi }_{\mathrm{N}}$ the gravitational force. This formulation expresses the fact that the inertial reaction force exactly cancels the gravitational attraction—bodies do not feel their weight—and when the inertial and gravitational masses are equal, all bodies near the Earth’s surface fall with the same acceleration. This establishes an equivalence between a uniformly accelerated frame and a frame at rest in a uniform gravitational field, as stated by the Weak Equivalence Principle. This implies that acceleration is not absolute. In Einstein’s words [28],

This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system.

In other words, acceleration cannot be defined in absolute terms if this physical equivalence with a gravitational field holds. From this, it follows that “gravity and acceleration are physical manifestations of the same phenomenon.” Thus, extending the Principle of Relativity to encompass uniformly accelerated frames naturally leads to the inclusion of gravitational fields, revealing the true core of Einstein’s interpretation of the Equivalence Principle.

Then, exploiting the invariance of the line element $d{s}^2$ and assuming spatial flatness, Einstein derived the coordinate transformation between an inertial system and another one uniformly accelerated along, for instance, the $OZ$ direction with constant acceleration a. From this, he obtained the corresponding line element [29]:

$$\begin{array}{c}\hfill d{s}^2={(1+az)}^{2}d{t}^2-(d{x}^2+d{y}^2+d{z}^2)\cong (1+2az)d{t}^2-(d{x}^2+d{y}^2+d{z}^2),\end{array}$$

(11)

which is valid in the weak-field approximation.

To incorporate gravitation into the relativistic framework, guided by the Equivalence Principle and the form of the line element in (11), Einstein proposed (for a non-rotating static field) a spacetime metric of the form $d{s}^2=c^2(x,y,z)d{t}^2-(d{x}^2+d{y}^2+d{z}^2)$, where the function $c(x,y,z)$ determines the gravitational field and satisfies a Poisson-like equation:

$$\begin{array}{c}\hfill \Delta c=kc\rho ,\end{array}$$

(12)

with k being a constant related to Newton’s gravitational constant and $\rho$ the mass density.

It is important to note that, at this stage (in 1912), Einstein still assumed that space remained flat: spatial distances were measured as in Euclidean geometry, and gravity affected only the rate at which time flows [29]. As we will see, this assumption eventually became a major obstacle in his search for the correct field equations, delaying the final formulation of General Relativity.

Rather than fully adopting Einstein’s original viewpoint—which posits a complete equivalence between a uniformly accelerated frame and a static frame in a uniform gravitational field—we will pursue an alternative approach grounded in D’Alembert’s principle. To derive the line element, we also incorporate gravity into the metric within the framework of Special Relativity by modifying the Minkowski interval to include the gravitational potential. But crucially, we impose the condition that, at least near the Earth’s surface, the inertial force must exactly cancel the gravitational force. This implies that, under such conditions, a freely falling body does not feel its own weight. Moreover, due to the equality of inertial and gravitational mass, all freely falling bodies experience the same acceleration.

To gain an intuition about which metric to choose, we consider a uniform gravitational field, such as $\Phi =gz$, and try to reproduce Newton’s second law exactly in D’Alembert’s form:

$$\begin{array}{c}\hfill {\mathbf{F}}_{\mathrm{i}}+{\mathbf{F}}_{\mathrm{g}}=0,\end{array}$$

(13)

where, as we have seen, ${\mathbf{F}}_{\mathrm{i}}=-m{\mathbf{a}}_{\mathrm{p}}$ is the inertial force and ${\mathbf{F}}_{\mathrm{g}}=-m(0,0,g)$ is the gravitational force. Here ${\mathbf{a}}_{\mathrm{p}}$ denotes the acceleration (which we will see must be the proper acceleration used in Special Relativity) of the body along the $OZ$ direction.

To arrive at the corresponding dynamical equation, consider the following line element:

$$\begin{array}{c}\hfill d{s}^2=A\left(z\right)d{t}^2-B\left(z\right)(d{x}^2+d{y}^2+d{z}^2),\end{array}$$

(14)

where we have assumed spacial isotropy.

From the minimization of the action $\delta S=-m\delta \int ds=0$, we obtain

$$\begin{array}{c}\hfill m{\displaystyle \frac{d^{2}z}{d{s}^2}}=-{\displaystyle \frac{m}{2A\left(z\right)B\left(z\right)}}\left[{\partial }_{z}A+{\partial }_z\left(AB\right){\left({\displaystyle \frac{dz}{ds}}\right)}^2\right].\end{array}$$

(15)

Note that to obtain Newton’s second law, it is necessary to impose that the velocity-dependent term on the right-hand side vanish. This condition holds when $A\left(z\right)B\left(z\right)=C,$ where C is a constant. Without loss of generality, this constant can be taken as one by appropriately rescaling coordinates. Thus, we obtain

$$\begin{array}{c}\hfill -m{\displaystyle \frac{d^{2}z}{d{s}^2}}-{\displaystyle \frac{m}{2}}{\partial }_{z}A=0,\end{array}$$

(16)

and comparing with (13) leads to

$$\begin{array}{c}\hfill {\mathbf{a}}_{\mathrm{p}}={\displaystyle \frac{d^{2}z}{d{s}^2}}\mathrm{and}A\left(z\right)=b+2\Phi \left(z\right),B\left(z\right)={\displaystyle \frac{1}{b+2\Phi \left(z\right)}},\end{array}$$

(17)

where b is a constant which we must set equal to 1 to recover the Minkowski metric when the potential vanishes. We conclude, therefore, that for a uniform gravitational field described by $\Phi \left(z\right)=gz$, a freely falling object obeys Newton’s second law when the acceleration ${\mathbf{a}}_{\mathrm{p}}$ is understood as the proper acceleration. Moreover, we may say that the object does not feel its own weight, since the proper inertial force exactly cancels the gravitational force—just as described in Newtonian mechanics.

Given the significance of this result, we will henceforth use this conformastatic metric as the starting point for constructing the metric associated with moving masses. This will be conducted by applying local Lorentz transformations and accumulating their effects in a way that is consistent with the relativistic structure of spacetime.

Therefore, in accordance with our geometric framework and in light of the result obtained for a homogeneous field, we begin with

$$\begin{array}{c}\hfill \mathfrak{g}=a_{\mathrm{N}}^{-2}{\mathfrak{e}}^0\otimes {\mathfrak{e}}^0-a_{\mathrm{N}}^2\mathfrak{E},\end{array}$$

(18)

where $\mathfrak{E}={\delta }_{ij}{\mathfrak{e}}^i\otimes {\mathfrak{e}}^j$ is the Euclidean metric. Then, the equation of geodesics is

$$\begin{array}{c}\hfill \ddot{\mathbf{x}}=-{\mathbf{H}}_{\mathrm{N}}-2a_{\mathrm{N}}^2{\left|\dot{\mathbf{x}}\right|}^2{\mathbf{H}}_{\mathrm{N},\perp },\end{array}$$

(19)

where we have introduced the notation ${\mathbf{H}}_{\mathrm{N}}=-{\displaystyle \frac{1}{a_{\mathrm{N}}^2}}\nabla a_{\mathrm{N}}$ and its orthogonal rejection onto $\dot{\mathbf{x}}$, namely ${\mathbf{H}}_{\mathrm{N},\perp }={\mathbf{H}}_{\mathrm{N}}-{\displaystyle \frac{\dot{\mathbf{x}}·{\mathbf{H}}_{\mathrm{N}}}{|\dot{\mathbf{x}}{|}^2}}\dot{\mathbf{x}}$.

We can introduce the Newtonian potential, namely ${\Phi }_{\mathrm{N}}$, as $1+2{\Phi }_{\mathrm{N}}=a_{\mathrm{N}}^{-2}⟹{\mathbf{H}}_{\mathrm{N}}=\nabla {\Phi }_{\mathrm{N}}$, and thus, the geodesic equation becomes

$$\begin{array}{c}\hfill \ddot{\mathbf{x}}=-\nabla {\Phi }_{\mathrm{N}}-2{\displaystyle \frac{|\dot{\mathbf{x}}{|}^2}{1+2{\Phi }_{\mathrm{N}}}}{\nabla }^{\perp }{\Phi }_{\mathrm{N}},\end{array}$$

(20)

where ${\nabla }^{\perp }{\Phi }_{\mathrm{N}}=\nabla {\Phi }_{\mathrm{N}}-{\displaystyle \frac{\dot{\mathbf{x}}·\nabla {\Phi }_{\mathrm{N}}}{|\dot{\mathbf{x}}{|}^2}}\dot{\mathbf{x}}$ is the orthogonal rejection of $\nabla {\Phi }_{\mathrm{N}}$ onto $\dot{\mathbf{x}}$.

Note that when the velocity is parallel to the gradient of ${\Phi }_{\mathrm{N}}$—as in the case of a radially falling object in a spherically symmetric gravitational field—one recovers the exact Newtonian equation of motion $\ddot{\mathbf{x}}=-\nabla {\Phi }_{\mathrm{N}}$. This result illustrates the Weak Equivalence Principle, understood here as the exact cancellation between the gravitational and inertial forces. Specifically, if we interpret the inertial force acting on a particle of mass m as the proper force $-m\ddot{\mathbf{x}}$ (note that one must use the proper acceleration), then the equation of motion becomes a balance of forces:

$$\begin{array}{c}\hfill -m\ddot{\mathbf{x}}-m\nabla {\Phi }_{\mathrm{N}}=0,\end{array}$$

(21)

where we have assumed the identity of the inertial and gravitational mass.

In particular, consider a freely falling body moving along a purely radial trajectory. That is, its velocity vector is aligned with the position vector $\dot{\mathbf{x}}\Vert \mathbf{x}$, implying a trajectory of the form $\mathbf{x}\left(t\right)\equiv x\left(t\right)\mathbf{x}$. In this case, the Newtonian gravitational potential is given by ${\Phi }_{\mathrm{N}}\left(\mathbf{x}\right)=-{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}}$, so its gradient is $\nabla {\Phi }_{\mathrm{N}}={\displaystyle \frac{MG}{{\left|\mathbf{x}\right|}^3}}\mathbf{x}$. Then, the exact equation of motion becomes $\ddot{x}=-{\displaystyle \frac{MG}{x^2}}$.

Near the Earth’s surface, this corresponds numerically to an acceleration of $g\cong$ 9.8 m/s² in SI units. This result exemplifies the equivalence of gravitational and inertial effects: the inertial force, which corresponds to the proper force, exactly cancels the gravitational one. Thus, the body does not feel its own weight—a manifestation of Einstein’s early formulation of the Equivalence Principle.

Furthermore, the conformastat metric leads to the conservation of the energy. Specifically, by multiplying, using the inner product, Equation (20) by the momentum $m\dot{\mathbf{x}}$, one obtains the energy conservation ${\displaystyle \frac{m|\dot{\mathbf{x}}{|}^2}{2}}+m{\Phi }_{\mathrm{N}}\left(\mathbf{x}\right)=E$.

Another compelling argument in favor of our chosen metric is that it yields the same prediction for the deflection of light as the Schwarzschild solution, namely $\delta \varphi ={\displaystyle \frac{4MG}{b}}$, where b denotes the impact parameter.

Therefore, once we have physically argued the reasons of the choice of the static metric in the weak-field approximation, we will have $a_{\mathrm{N}}^2={\displaystyle \frac{1}{1+2{\Phi }_{\mathrm{N}}}}\cong 1-2{\Phi }_{\mathrm{N}}+4{\Phi }_{\mathrm{N}}^2$, and the metric will be $\mathfrak{g}=(1+2{\Phi }_{\mathrm{N}}){\mathfrak{e}}^0\otimes {\mathfrak{e}}^0-(1-2{\Phi }_{\mathrm{N}}+4{\Phi }_{\mathrm{N}}^2)\mathfrak{E}$.

Remark 3.

As we have already discussed, recall that, at the outset of his investigation into gravity, Einstein introduced uniformly accelerated frames within the framework of Special Relativity. By combining this idea with the Principle of Equivalence, he derived a spatially flat metric of the form $\mathfrak{g}=c^2{\mathfrak{e}}^0\otimes {\mathfrak{e}}^0-\mathfrak{E},$ where c is a scalar function encoding the gravitational field. And the use of this metric could be one of the reasons why he rejected, in his first attempt to find the fields equations of GR, the field equation:

$$\begin{array}{c}\hfill {\mathfrak{Ric}}_{\mathfrak{g}}=8\pi G{\mathfrak{T}}_{\mathfrak{g}}.\end{array}$$

(22)

Due to his spatially flat metric, in the weak-field approximation, one cannot recover the classical Poisson [30]. In addition, this metric predicts only half of the correct light deflection, which constitutes another argument against its physical validity.

In fact, Einstein went even further and, for a period of time, gave up general covariance in order to obtain equations that, in the weak-field approximation, would align with the classical Poisson equation.

Remark 4.

It is important to emphasize what Einstein understood as the Equivalence Principle during the early stages of his work on gravitation. At that time, he believed in the exact equivalence between a uniformly accelerated frame and a frame at rest in a uniform gravitational field—without invoking any notion of locality. This interpretation is clearly reflected in his early papers, particularly in [29,31].

However, this conceptual equivalence, taken in a global and exact sense, is not strictly valid and can lead to misconceptions—especially regarding the nature of spacetime curvature. A uniformly accelerated frame in flat Minkowski spacetime and a frame at rest in a gravitational field may exhibit similar physical phenomena (such as the universality of free fall), but they are not globally equivalent from a geometric standpoint. The former exists in flat spacetime, while the latter typically corresponds to a curved spacetime geometry.

This misinterpretation was later clarified, notably by Synge, who criticized the overly literal use of the Equivalence Principle. As he wrote in [2],

“I have never been able to understand this Principle of Equivalence. Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer’s acceleration? If so, it is false. In Einstein’s theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observer’s world-line. Space-time is either flat or curved, and in several places in the book I have been at considerable pains to separate truly gravitational effects due to curvature of space-time from those due to curvature of the observer’s world-line… The Principle of Equivalence performed the essential office of midwife at the birth of General Relativity… I suggest that the midwife be now buried with appropriate honours and the facts of absolute space-time faced.”

Synge’s remark underscores the idea that while the Equivalence Principle served as a powerful heuristic tool in the development of General Relativity, it should not be taken as a foundational principle in its early, non-local, and global form. Instead, the local version—asserting the equivalence between a freely falling frame and an inertial frame only locally—is what survives in modern formulations of the theory.

Next, following Nordtvedt’s idea presented in [11], we consider a frame $K^{\prime }$ moving with three-dimensional velocity $-{\mathbf{v}}^{\prime }$ with respect to another frame K, which is equipped with the metric in (18). Equivalently, K moves with velocity ${\mathbf{v}}^{\prime }$ with respect $K^{\prime }$.

Let $\gamma$ be the Lorentz factor, where $\gamma ={\displaystyle \frac{1}{\sqrt{1-|{\mathbf{v}}^{\prime }{|}^2}}}$. Then the infinitesimal Lorentz transformation is

$$\begin{array}{c}\hfill d\mathbf{x}=\gamma \left[{\displaystyle \frac{1}{\gamma }}d{\mathbf{x}}^{\prime }+{\displaystyle \frac{\gamma }{1+\gamma }}({\mathbf{v}}^{\prime }·d{\mathbf{x}}^{\prime }){\mathbf{v}}^{\prime }-{\mathbf{v}}^{\prime }d{t}^{\prime }\right],dt=\gamma (d{t}^{\prime }-{\mathbf{v}}^{\prime }·d{\mathbf{x}}^{\prime }).\end{array}$$

(23)

Up to second-order terms in the velocity and after dropping the primes for simplicity, the metric transforms into

$$\begin{array}{c}\hfill \mathfrak{g}=(1+2{\Phi }_{\mathrm{N}}+4{{\rm Y}}_{\mathrm{N}}){\mathfrak{e}}^0\otimes {\mathfrak{e}}^0+({\mathfrak{N}}_{\mathrm{N}}\otimes {\mathfrak{e}}^0+{\mathfrak{e}}^0\otimes {\mathfrak{N}}_{\mathrm{N}})-(1-2{\Phi }_{\mathrm{N}}+4{\Phi }_{\mathrm{N}}^2)\mathfrak{E}+{\mathfrak{h}}_{\mathrm{N}},\end{array}$$

(24)

where we have introduced the scalar ${{\rm Y}}_{\mathrm{N}}={\Phi }_{\mathrm{N}}{\left|\mathbf{v}\right|}^2$ and the covector ${\mathfrak{N}}_{\mathrm{N}}^{\prime }\equiv 4{\Phi }_{\mathrm{N}}{{\mathbf{v}}^{\prime }}^{♭}$, that is, in components ${\mathfrak{N}}_{\mathrm{N}}=N_i{\mathfrak{e}}^i$, with $N_i\equiv -4{\Phi }_{\mathrm{N}}{v}^i\cong +4{\Phi }_{\mathrm{N}}{v}_i$, and ${\mathfrak{h}}_{\mathrm{N}}=4{\Phi }_{\mathrm{N}}{v}^i{v}^k{\mathfrak{e}}^i\otimes {\mathfrak{e}}^k\cong 4{\Phi }_{\mathrm{N}}{v}_i{v}_k{\mathfrak{e}}^i\otimes {\mathfrak{e}}^k$ is related with the tensor perturbations.

To derive the gravitational potentials, when we deal with dust matter, i.e., in the case where $p\ll \rho$, it is essential to recognize that their effects are cumulative: every infinitesimal element of the moving mass distribution contributes independently to the overall metric. Since each of these elements may possess a different velocity, a distinct Lorentz transformation must be applied to each one according to its local motion. The total potential is then obtained by summing—or, more precisely, integrating—all these individual contributions over the entire distribution of matter. This approach ensures that the resulting metric properly reflects the influence of the full configuration of moving sources.

Therefore, the Newtonian potential is given by:

$$\begin{array}{c}\hfill {\Phi }_{\mathrm{N}}(t,\mathbf{x})=-G\int {\displaystyle \frac{\rho (t,\overline{\mathbf{x}})}{|\mathbf{x}-\overline{\mathbf{x}}|}}d\overline{V}⟹\Delta {\Phi }_{\mathrm{N}}=4\pi G\rho .\end{array}$$

(25)

In the same way,

$$\begin{array}{c}\hfill {{\rm Y}}_{\mathrm{N}}(t,\mathbf{x})=-G\int {\displaystyle \frac{\rho (t,\overline{\mathbf{x}})}{|\mathbf{x}-\overline{\mathbf{x}}|}}{\left|\mathbf{v}(t,\overline{\mathbf{x}})\right|}^{2}d\overline{V},\end{array}$$

(26)

so the expression of ${\mathfrak{N}}_{\mathrm{N}}$ has to be

$$\begin{array}{c}\hfill {\mathfrak{N}}_{\mathrm{N}}(t,\mathbf{x})=-4G\int {\displaystyle \frac{\rho (t,\overline{\mathbf{x}}){\mathbf{v}}^{♭}(t,\overline{\mathbf{x}})}{|\mathbf{x}-\overline{\mathbf{x}}|}}d\overline{V},\end{array}$$

(27)

and the one of $\mathfrak{h}$ is

$$\begin{array}{c}\hfill {\mathfrak{h}}_{\mathrm{N}}(t,\mathbf{x})=-4G\int {\displaystyle \frac{\rho (t,\overline{\mathbf{x}})}{|\mathbf{x}-\overline{\mathbf{x}}|}}{\mathbf{v}}^{♭}(t,\overline{\mathbf{x}})\otimes {\mathbf{v}}^{♭}(t,\overline{\mathbf{x}})d\overline{V}.\end{array}$$

(28)

Together with the conservation and Euler equations, the following is true:

$$\begin{array}{c}\hfill {\partial }_t\rho +\nabla ·\left(\rho \mathbf{v}\right)=0,\rho {\partial }_t\mathbf{v}+\rho (\mathbf{v}·\nabla )\mathbf{v}+\nabla p+\rho \nabla {\Phi }_{\mathrm{N}}=0.\end{array}$$

(29)

Specifically, the line element takes the following form:

$$\begin{array}{c}\hfill d{s}^2=(1+2{\Phi }_{\mathrm{N}}+4{{\rm Y}}_{\mathrm{N}})d{t}^2+2N_{i}d{x}^{i}dt-(1-2{\Phi }_{\mathrm{N}}+4{\Phi }_{\mathrm{N}}^2){\delta }_{ij}d{x}^{i}d{x}^j+h_{ij}d{x}^{i}d{x}^j,\end{array}$$

(30)

with the equations

$$\begin{array}{c}\hfill \Delta {\Phi }_{\mathrm{N}}=4\pi G\rho ,\Delta {{\rm Y}}_{\mathrm{N}}=4\pi G\rho {\left|\mathbf{v}\right|}^2,\Delta N_i=16\pi G\rho v_i,\mathrm{and}\Delta h_{ij}=16\pi G\rho v_i{v}_j.\end{array}$$

(31)

A final remark is in order: if one seeks a theory closer to linearized General Relativity in the de Donder (or Lorenz) gauge, one can replace the Newtonian potentials with their retarded counterparts, that is, to replace the Laplacian Equation with the D’Alambertian Equation in (31), which account for the finite propagation speed of interactions. In this more relativistic setting, the classical continuity and Euler equations can be replaced by the conservation of the stress–energy tensor:

$$\begin{array}{c}\hfill \mathrm{div}\left(\mathfrak{T}\right)=0⟹{\displaystyle \frac{D\rho }{ds}}=-(\rho +p)\mathrm{div}\left(\mathbf{u}\right)\mathrm{and}(\rho +p){\displaystyle \frac{D\mathbf{u}}{ds}}=\mathrm{grad}\left(p\right)-{\displaystyle \frac{Dp}{ds}}\mathbf{u},\end{array}$$

(32)

where $\mathbf{u}=({\displaystyle \frac{dt}{ds}},\mathbf{v})$, with $\mathbf{v}={\displaystyle \frac{d\mathbf{x}}{ds}}$ being is the four-velocity, and D denotes the covariant derivative.

This shift not only aligns the theory more closely with General Relativity but also naturally incorporates the causal structure and dynamical behavior expected in a relativistic framework.

The Static Case: Field Produced by a Point Mass

We consider, in a static universe, the Poisson equation looking for the stationary spherical symmetric solution produced by a point particle of mass M situated at the origin of $\mathbf{x}$ coordinates. In vacuum, we have to solve

$$\begin{array}{c}\hfill \Delta {\Phi }_{\mathrm{N}}=0.\end{array}$$

(33)

The well-known solution of this equation is

$$\begin{array}{c}\hfill {\Phi }_{\mathrm{N}}\left(\mathbf{x}\right)=-{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}}.\end{array}$$

(34)

Next, to find the relationship with the spherical coordinates, that is, between $\left|\mathbf{x}\right|$ and the Euclidean distance r, we see that our line element is

$$\begin{array}{c}\hfill d{s}^2=\left(1-{\displaystyle \frac{2MG}{\left|\mathbf{x}\right|}}\right)d{t}^2-\left(1+{\displaystyle \frac{2MG}{\left|\mathbf{x}\right|}}+{\displaystyle \frac{4M^2{G}^2}{{\left|\mathbf{x}\right|}^2}}\right){\left(d\right|\mathbf{x}|}^2+{\left|\mathbf{x}\right|}^{2}d\Omega ),\end{array}$$

(35)

with $d\Omega =d{\theta }^2+sin\theta d{\varphi }^2$. Since, as was pointed out by Hilbert in [32], the angular term in spherical coordinates is $r^{2}d\Omega$ (see also Equation (8.3.4) of [33]), one has

$$\begin{array}{c}\hfill r^2={\left|\mathbf{x}\right|}^2+2MG\left|\mathbf{x}\right|+4M^2{G}^2⟹\left|\mathbf{x}\right|=-MG+\sqrt{r^2-3M^2{G}^2}\\ \hfill ⟹{\Phi }_{\mathrm{N}}\left(r\right)=-{\displaystyle \frac{MG}{\sqrt{r^2-3M^2{G}^2}-MG}},\end{array}$$

(36)

and, in spherical coordinates, the metric becomes

$$\begin{array}{c}\hfill d{s}^2=\left(1+2{\Phi }_{\mathrm{N}}\left(r\right)\right)d{t}^2-\left(1-2{\Phi }_{\mathrm{N}}\left(r\right)+4{\Phi }_{\mathrm{N}}^2\left(r\right)\right){\left(1-{\displaystyle \frac{3M^2{G}^2}{r^2}}\right)}^{-1}d{r}^2-r^{2}d\Omega \\ \hfill ⟹d{s}^2\cong \left(1+2\left(-{\displaystyle \frac{MG}{r}}+{\displaystyle \frac{M^2{G}^2}{r^2}}\right)\right)d{t}^2-\left(1+{\displaystyle \frac{2MG}{r}}+{\displaystyle \frac{9M^2{G}^2}{r^2}}\right)d{r}^2-r^{2}d\Omega ,\end{array}$$

(37)

which, as General Relativity, is singular at the Schwarzschild radius $r=2MG$ (note that the Newtonian potential in polar coordinates ${\Phi }_{\mathrm{N}}\left(r\right)$ diverges at the Schwarzschild radius). Note also that the last approximation of (37) coincides with the post-Newtonian approximation obtained in Formula (9.4.25) of [33].

On the other hand, in the approximation $MG/r\ll 1$, the metric becomes

$$\begin{array}{c}\hfill d{s}^2\cong \left(1+2{\Phi }_{\mathrm{N}}\left(r\right)\right)d{t}^2-\left(1-2{\Phi }_{\mathrm{N}}\left(r\right)\right)d{r}^2-r^{2}d\Omega \mathrm{with}{\Phi }_{\mathrm{N}}\left(r\right)\cong -{\displaystyle \frac{MG}{r}},\end{array}$$

(38)

which, in this approximation, that is, when $|{\Phi }_{\mathrm{N}}|\ll 1$, coincides with the Schwarzschild line element, where

$$\begin{array}{c}\hfill d{s}^2=\left(1+2{\Phi }_{\mathrm{N}}\left(r\right)\right)d{t}^2-{\left(1+2{\Phi }_{\mathrm{N}}\left(r\right)\right)}^{-1}d{r}^2-r^{2}d\Omega \mathrm{with}{\Phi }_{\mathrm{N}}\left(r\right)=-{\displaystyle \frac{MG}{r}}.\end{array}$$

(39)

We will conclude by highlighting two important points that are often misunderstood or misrepresented in the literature:

• It is important to emphasize that the line element commonly known as the Schwarzschild metric was not exclusively derived by Karl Schwarzschild. In fact, similar solutions emerged independently from the work by Johannes Droste [34] and David Hilbert [32]. Droste, working under Hendrik Lorentz, arrived at the same class of static, spherically symmetric solutions in 1916, deriving them from Einstein’s earlier Entwurf field equations of 1914 [35], without any knowledge of Schwarzschild’s original results. Shortly thereafter, Hilbert also derived an equivalent solution, which was about a year after Schwarzschild had first introduced his own form of the metric.
  These metrics describe the gravitational field surrounding a non-rotating, spherically symmetric mass situated at the origin. Among the possible solutions, Schwarzschild identified the one that displayed a singularity only at the spatial origin, avoiding coordinate singularities elsewhere. When re-expressed in the appropriate coordinate system—namely, spherical polar coordinates—his result takes the form now universally recognized as the Schwarzschild solution [36].
  Although Schwarzschild’s name became attached to the solution, the mathematical structure was part of a broader effort, with multiple contributors reaching the same result independently, each from slightly different conceptual frameworks.
  Finally, there is also an ontological lesson embedded in this history. The Schwarzschild metric, as a solution to Einstein’s equations, does not exist merely as a mathematical artifact—it embodies a specific conception of spacetime as something dynamic shaped by mass and energy. Yet the fact that it can be derived in different coordinate systems, from different formulations of Einstein’s equations, and even under differing philosophical attitudes toward the nature of gravity reveals the relational and formal character of spacetime in General Relativity.
• The Schwarzschild line element can be written in isotropic coordinates as follows [37]:

  $$\begin{array}{c}\hfill d{s}^2={\left({\displaystyle \frac{2\left|\mathbf{x}\right|-MG}{2\left|\mathbf{x}\right|+MG}}\right)}^{2}d{t}^2-{\left(1+{\displaystyle \frac{MG}{2\left|\mathbf{x}\right|}}\right)}^4{\left(d\right|\mathbf{x}|}^2+{\left|\mathbf{x}\right|}^{2}d\Omega ),\end{array}$$

  (40)

  where the relation with the polar coordinate r is $r=\left|\mathbf{x}\right|{\left(1+{\displaystyle \frac{MG}{2\left|\mathbf{x}\right|}}\right)}^2$. Disregarding quadratic terms on $MG/\left|\mathbf{x}\right|$, we find

  $$\begin{array}{c}\hfill d{s}^2=\left(1-{\displaystyle \frac{2MG}{\left|\mathbf{x}\right|}}\right)d{t}^2-\left(1+{\displaystyle \frac{2MG}{\left|\mathbf{x}\right|}}\right){\left(d\right|\mathbf{x}|}^2+{\left|\mathbf{x}\right|}^{2}d\Omega ).\end{array}$$

  (41)
  At this stage, it is crucial to clarify a common misconception: the coordinate $\left|\mathbf{x}\right|$ should not be naively equated with the radial distance in standard spherical coordinates, even though such an identification is sometimes made, for instance, in [37]. If one proceeds with this assumption, the resulting analysis fails to yield the correct value for Mercury’s perihelion precession.
  To avoid this pitfall, one must consider the precise relation between the polar coordinate r and the Euclidean norm $\left|\mathbf{x}\right|$. By neglecting terms of second order or higher in the gravitational potential, one finds the approximation $r=\left|\mathbf{x}\right|+MG$. When this adjustment is taken into account and polar coordinates are defined appropriately, the metric naturally reduces to the form given in Equation (38), which—as will be shown—produces the correct classical predictions.
  This highlights a deeper insight: the physical interpretation of coordinate systems is not fixed a priori, but emerges from the structure of the metric itself. Coordinates acquire meaning only in relation to the geometry described by the solution. This is especially evident in the case of the Schwarzschild metric, where the coordinate r has a specific operational definition tied to physical measurements (see [38] for a detailed discussion). In short, it is the dynamics and geometry of spacetime that give coordinates their physical significance—not the other way around.

Precession of the Perihelion

To calculate the precession of a planet’s perihelion, we begin by analyzing its motion confined to a single plane $\theta =\pi /2$. This simplification is justified by the conservation of angular momentum, which ensures that the orbit remains planar in a central gravitational field. By focusing on the dynamics within this plane, we can derive an effective radial equation of motion and identify relativistic corrections to the classical Keplerian orbit. These corrections ultimately lead to a small but measurable shift in the point of closest approach—i.e., the perihelion—after each revolution.

The line element in (38) is sufficient for our purposes. Accordingly, the corresponding Lagrangian is

$$\begin{array}{c}\hfill \mathcal{L}=\left(1+2{\Phi }_{\mathrm{N}}\right){\dot{t}}^2-\left(1-2{\Phi }_{\mathrm{N}}\right){\dot{r}}^2-r^2{\dot{\varphi }}^2,\mathrm{with}\mathcal{L}=1.\end{array}$$

(42)

Since the Lagrangian is independent on t and $\varphi$, we have two conserved quantities:

$$\begin{array}{c}\hfill \left(1+2{\Phi }_{\mathrm{N}}\right)\dot{t}=\sqrt{2E+1}\mathrm{and}{r}^2\dot{\varphi }=L,\end{array}$$

(43)

and together with the condition $\mathcal{L}=1$, this leads to

$$\begin{array}{c}\hfill {\dot{r}}^2\cong 2E-2{\Phi }_{\mathrm{N}}\left(r\right)-(1+2{\Phi }_{\mathrm{N}}\left(r\right)){\displaystyle \frac{L^2}{r^2}}.\end{array}$$

(44)

That is

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{dr}{r^{2}d\varphi }}\right)}^2\cong {\displaystyle \frac{2E}{L^2}}+{\displaystyle \frac{2MG}{L^{2}r}}-\left(1-{\displaystyle \frac{2MG}{r}}\right){\displaystyle \frac{1}{r^2}},\end{array}$$

(45)

and by identifying the change in variable $u=1/r$, we find

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{du}{d\varphi }}\right)}^2\cong P\left(u\right),\end{array}$$

(46)

where

$$\begin{array}{c}\hfill P\left(u\right)=2MG{u}^3-u^2+{\displaystyle \frac{2MG}{L^2}}u+{\displaystyle \frac{2E}{L^2}}.\end{array}$$

(47)

We now turn our attention to the points of the closest and farthest approaches in the planetary orbit, denoted by $r_{-}$ (perihelion) and $r_{+}$ (aphelion), respectively. To facilitate the analysis, we define $u_{\pm }=1/r_{\pm }$, corresponding to the extremal values of the inverse radial coordinate $u\left(\varphi \right)$ along the orbit. Since the radial equation can be expressed as a differential equation involving a cubic polynomial $P\left(u\right)$, the turning points of the motion occur at the values of u for which ${\displaystyle \frac{du}{d\varphi }}=0$, that is, the roots of $P\left(u\right)$.

Hence, $u_{-}$ and $u_{+}$ are two of the three real roots of this polynomial. According to the fundamental theorem of algebra, a cubic equation has exactly three roots (real or complex), and in our case—representing bound orbits—we expect all three to be real. We denote the third root by $u_0$, which typically lies between $u_{-}$ and $u_{+}$, depending on the specific energy and angular momentum of the orbit. Thus, we can write the following:

$$\begin{array}{c}\hfill P\left(u\right)=2MG(u-u_{+})(u-u_{-})(u-u_0),\end{array}$$

(48)

and by expanding it and comparing it with the quadratic term of (47), we find $2MG(u_{+}+u_{-}+u_0)=1$; that is, $u_0={\displaystyle \frac{1}{2MG}}-(u_{+}+u_{-})$. And we can write the following:

$$\begin{array}{c}\hfill P\left(u\right)=(u-u_{+})(u_{-}-u)(1-2MG(u+u_{+}+u_{-}))⟹{\displaystyle \frac{1}{P\left(u\right)}}\cong {\displaystyle \frac{1+2MG(u+u_{+}+u_{-})}{(u-u_{+})(u_{-}-u)}}.\end{array}$$

(49)

This leads us to the central differential equation governing the orbital motion in terms of the inverse radial coordinate u:

$$\begin{array}{c}\hfill {\displaystyle \frac{d\varphi }{du}}\cong {\displaystyle \frac{1+MG(u+u_{+}+u_{-})}{\sqrt{(u-u_{+})(u_{-}-u)}}}.\end{array}$$

(50)

Notably, this expression is identical to the one derived from the Schwarzschild metric in the weak-field limit (see [39]). It encapsulates the key relativistic correction responsible for the perihelion advance and shows that the same physics emerges from our alternative formulation, confirming its consistency with General Relativity in this approximation.

4. Geodesics

We focus our attention on the present state of the universe, where—on the scale of the solar system—the effects of cosmic expansion can be safely neglected. This is justified by the fact that gravitational binding within such systems overwhelmingly dominates over the influence of the Hubble flow.

Under this approximation and by neglecting terms that are quadratic in either the velocity or the Newtonian potential, the spacetime geometry can be described by the following simplified metric:

$$\begin{array}{c}\hfill \mathfrak{g}=(1+2{\Phi }_{\mathrm{N}}){\mathfrak{e}}^0\otimes {\mathfrak{e}}^0+({\mathfrak{N}}_{\mathrm{N}}\otimes {\mathfrak{e}}^0+{\mathfrak{e}}^0\otimes {\mathfrak{N}}_{\mathrm{N}})-(1-2{\Phi }_{\mathrm{N}})\mathfrak{E}.\end{array}$$

(51)

The line element is given by $d{s}^2=(1+2{\Phi }_{\mathrm{N}})d{t}^2-2{\mathbf{N}}_{\mathrm{N}}·d\mathbf{x}dt-(1-2{\Phi }_{\mathrm{N}}){\left|d\mathbf{x}\right|}^2$, where we have used the notation ${\mathbf{N}}_{\mathrm{N}}={\mathfrak{N}}_{\mathrm{N}}^{♯}$ and the standard notation $\mathfrak{E}(\mathbf{v},\mathbf{w})\equiv \mathbf{v}·\mathbf{w}.$

The Lagrangian is given by

$$\begin{array}{c}\hfill \mathcal{L}=(1+2{\Phi }_{\mathrm{N}}){\dot{t}}^2-2\dot{t}{\mathbf{N}}_{\mathrm{N}}·\dot{\mathbf{x}}-(1-2{\Phi }_{\mathrm{N}}){\left|\dot{\mathbf{x}}\right|}^2,\mathrm{with} \mathrm{the} \mathrm{constraint}\mathcal{L}=1.\end{array}$$

(52)

We have

$$\begin{array}{c}\hfill {\partial }_{\dot{t}}\mathcal{L}=2\left((1+2{\Phi }_{\mathrm{N}})\dot{t}-{\mathbf{N}}_{\mathrm{N}}·\dot{\mathbf{x}}\right).\end{array}$$

(53)

In addition,

$$\begin{array}{c}\hfill {\partial }_{\dot{\mathbf{x}}}\mathcal{L}=-2(\dot{t}{\mathbf{N}}_{\mathrm{N}}+(1-2{\Phi }_{\mathrm{N}})\dot{\mathbf{x}}).\end{array}$$

(54)

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{ds}}{\partial }_{\dot{\mathbf{x}}}\mathcal{L}=-2\left(\ddot{t}{\mathbf{N}}_{\mathrm{N}}+{\dot{t}}^2{\partial }_t{\mathbf{N}}_{\mathrm{N}}+\dot{t}(\dot{\mathbf{x}}·\nabla ){\mathbf{N}}_{\mathrm{N}}-2{\partial }_t{\Phi }_{\mathrm{N}}\dot{t}\dot{\mathbf{x}}-2(\dot{\mathbf{x}}·\nabla {\Phi }_{\mathrm{N}})\dot{\mathbf{x}}+(1-2{\Phi }_{\mathrm{N}})\ddot{\mathbf{x}}\right).\end{array}$$

(55)

$$\begin{array}{c}\hfill {\partial }_{\mathbf{x}}\mathcal{L}=2\nabla {\Phi }_{\mathrm{N}}{\dot{t}}^2-2\dot{t}[(\dot{\mathbf{x}}·\nabla ){\mathbf{N}}_{\mathrm{N}}+\dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}}]+2\nabla {\Phi }_{\mathrm{N}}{\left|\dot{\mathbf{x}}\right|}^2.\end{array}$$

(56)

Taking into account the constraint where $\mathcal{L}=1$, one has

$$\begin{array}{c}\hfill \dot{t}={\displaystyle \frac{1}{1+2{\Phi }_{\mathrm{N}}}}\left({\mathbf{N}}_{\mathrm{N}}·\dot{\mathbf{x}}+\sqrt{1+2\Phi +{({\mathbf{N}}_{\mathrm{N}}·\dot{\mathbf{x}})}^2+(1-4{\Phi }_{\mathrm{N}}^2){\left|\dot{\mathbf{x}}\right|}^2}\right)\cong 1-{\Phi }_{\mathrm{N}}+{\displaystyle \frac{|\dot{\mathbf{x}}{|}^2}{2}}.\end{array}$$

(57)

To have an expression at the first order of perturbations, we have to recall that ${\Phi }_{\mathrm{N}}$ is of the order ${\displaystyle \frac{MG}{L}}\ll 1$, where L is the characteristic length of the movement. Since $|\dot{\mathbf{x}}{|}^2$ is of the order of ${\Phi }_{\mathrm{N}}$ (recall the conservation of the energy in Newtonian physics), we can assume that the velocity of the objects, i.e., $\mathbf{v}$, is of the order $\sqrt{{\displaystyle \frac{MG}{L}}}$, and thus ${\mathbf{N}}_{\mathrm{N}}\sim {\Phi }_{\mathrm{N}}\mathbf{v}$ is of the order ${\left({\displaystyle \frac{MG}{L}}\right)}^{3/2}$. In addition ∇ is of the order ${\displaystyle \frac{1}{L}}$, and ${\partial }_t\sim \dot{\mathbf{x}}·\nabla$ is of the order ${\displaystyle \frac{1}{L}}\sqrt{{\displaystyle \frac{MG}{L}}}$. Finally, $\ddot{\mathbf{x}}$ is of order ${\displaystyle \frac{MG}{L^2}}$, and thus, $\ddot{t}$ is of the order ${\left({\displaystyle \frac{MG}{L}}\right)}^{3/2}{\displaystyle \frac{1}{L}}$.

Then, up to order ${\displaystyle \frac{M^2{G}^2}{L^3}}$, one obtains

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{ds}}{\partial }_{\dot{\mathbf{x}}}\mathcal{L}=-2\left({\partial }_t{\mathbf{N}}_{\mathrm{N}}+(\dot{\mathbf{x}}·\nabla ){\mathbf{N}}_{\mathrm{N}}-2({\partial }_t{\Phi }_{\mathrm{N}}+\dot{\mathbf{x}}·\nabla {\Phi }_{\mathrm{N}})\dot{\mathbf{x}}+(1-2{\Phi }_{\mathrm{N}})\ddot{\mathbf{x}}\right).\end{array}$$

(58)

$$\begin{array}{c}\hfill {\partial }_{\mathbf{x}}\mathcal{L}=2\nabla {\Phi }_{\mathrm{N}}-2[\dot{\mathbf{x}}·\nabla {\mathbf{N}}_{\mathrm{N}}+\dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}}]+2\nabla {\Phi }_{\mathrm{N}}{\left|\dot{\mathbf{x}}\right|}^2.\end{array}$$

(59)

Therefore, up to order ${\displaystyle \frac{M^2{G}^2}{L^3}}$, one has

$$\begin{array}{c}\hfill \ddot{\mathbf{x}}=-\nabla {\Phi }_{\mathrm{N}}-{\partial }_t{\mathbf{N}}_{\mathrm{N}}+\dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}}+2{\partial }_t{\Phi }_{\mathrm{N}}\dot{\mathbf{x}}-2{\left|\dot{\mathbf{x}}\right|}^2{\nabla }^{\perp }{\Phi }_{\mathrm{N}},\end{array}$$

(60)

where ${\nabla }^{\perp }{\Phi }_{\mathrm{N}}\equiv \nabla {\Phi }_{\mathrm{N}}-{\displaystyle \frac{\dot{\mathbf{x}}·\nabla {\Phi }_{\mathrm{N}}}{|\dot{\mathbf{x}}{|}^2}}\dot{\mathbf{x}}$ is the orthogonal rejection of ${\nabla }_{\mathbf{q}}{\Phi }_{\mathrm{N}}$ onto $\dot{\mathbf{x}}$.

Remark 5.

In GR and using the linear approximation where ${\mathbf{N}}_{\mathrm{N}}$ is assumed to be of the same order than ${\Phi }_{\mathrm{N}}$, the velocity of the matter is of order 1. Thus, at the order ${\displaystyle \frac{1}{L}}{\left({\displaystyle \frac{MG}{q}}\right)}^{3/2}$, one has

$$\begin{array}{c}\hfill \ddot{\mathbf{x}}=-\nabla {\Phi }_{\mathrm{N}}-{\partial }_t{\mathbf{N}}_{\mathrm{N}}+\dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}},\end{array}$$

(61)

and, in this approximation, taking into account that $ds\cong (1+{\Phi }_{\mathrm{N}})dt$, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left((1-{\Phi }_{\mathrm{N}}){\displaystyle \frac{d\mathbf{x}}{dt}}\right)=-\nabla {\Phi }_{\mathrm{N}}-{\partial }_t{\mathbf{N}}_{\mathrm{N}}+{\displaystyle \frac{d\mathbf{x}}{dt}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}},\end{array}$$

(62)

where we have disregarded the quadratic term ${\Phi }_{\mathrm{N}}\nabla {\Phi }_{\mathrm{N}}$, which coincides with Einstein’s result (see page 122 of [1]).

5. Perturbations

For values of $\left|\mathbf{x}\right|$ larger than the typical dimensions of the bodies, we can make the following approximation:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{|\mathbf{x}-\overline{\mathbf{x}}|}}\cong {\displaystyle \frac{1}{\left|\mathbf{x}\right|}}+{\displaystyle \frac{\mathbf{x}·\overline{\mathbf{x}}}{{\left|\mathbf{x}\right|}^3}}.\end{array}$$

(63)

In this approximation we have

$$\begin{array}{c}\hfill {\Phi }_{\mathrm{N}}(t,\mathbf{x})\cong -{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}}\left(1+{\displaystyle \frac{\mathbf{x}·{\mathbf{x}}_{\mathrm{CM}}}{{\left|\mathbf{x}\right|}^2}}\right),\end{array}$$

(64)

where ${\mathbf{x}}_{\mathrm{CM}}$ is the center of mass of the body. Note that due to the conservation equation ${\partial }_t\rho +\nabla ·\left(\rho \mathbf{v}\right)=0$, the total mass of the body is conserved.

$$\begin{array}{c}\hfill {\mathbf{N}}_{\mathrm{N}}\cong -{\displaystyle \frac{4G}{\left|\mathbf{x}\right|}}\mathbf{P}-{\displaystyle \frac{4G}{{\left|\mathbf{x}\right|}^3}}\int (\mathbf{x}·\overline{\mathbf{x}})\rho (t,\overline{\mathbf{x}})\mathbf{v}(t,\overline{\mathbf{x}})d\overline{V},\end{array}$$

(65)

where $\mathbf{P}=\int \rho (t,\overline{\mathbf{x}})\mathbf{v}(t,\overline{\mathbf{x}})d\overline{V}$ is the momentum of the body.

Note that the calculation simplifies when one chooses, as an origin of coordinates, the center of mass; in that case ${\mathbf{x}}_{\mathrm{CM}}=\mathbf{P}=\mathbf{0}$, and we have

$$\begin{array}{c}\hfill {\Phi }_{\mathrm{N}}(t,\mathbf{x})\cong -{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}}\mathrm{and}{\mathbf{N}}_{\mathrm{N}}\cong -{\displaystyle \frac{4G}{{\left|\mathbf{x}\right|}^3}}\int (\mathbf{x}·\overline{\mathbf{x}})\rho \mathbf{v}d\overline{V}.\end{array}$$

(66)

Next, we are interested in the term $\nabla \wedge {\mathbf{N}}_{\mathrm{N}}$. To perform the calculation, we use the vector identity

$$\begin{array}{c}\hfill (\mathbf{x}·\overline{\mathbf{x}})\mathbf{v}=(\mathbf{x}·\mathbf{v})\overline{\mathbf{x}}-\mathbf{x}\wedge (\overline{\mathbf{x}}\wedge \mathbf{v}),\end{array}$$

(67)

to write

$$\begin{array}{c}\hfill {\mathbf{N}}_{\mathrm{N}}\cong -{\displaystyle \frac{4G}{{\left|\mathbf{x}\right|}^3}}\int (\mathbf{x}·\mathbf{v})\rho \overline{\mathbf{x}}d\overline{V}+{\displaystyle \frac{4G}{{\left|\mathbf{x}\right|}^3}}\mathbf{x}\wedge \mathbf{S},\end{array}$$

(68)

where $\mathbf{S}\equiv \int \overline{\mathbf{x}}\wedge \mathbf{v}\rho d\overline{V}$ is the angular momentum of the body.

Now, we calculate

$$\begin{array}{c}\hfill \nabla \wedge \left({\displaystyle \frac{(\mathbf{x}·\mathbf{v})\overline{\mathbf{x}}}{{\left|\mathbf{x}\right|}^3}}\right)=-{\displaystyle \frac{\overline{\mathbf{x}}\wedge \mathbf{v}}{{\left|\mathbf{x}\right|}^3}}-{\displaystyle \frac{3(\mathbf{x}·\mathbf{v})}{{\left|\mathbf{x}\right|}^5}}(\mathbf{x}\wedge \overline{\mathbf{x}})\mathrm{and}\\ \hfill \nabla \wedge \left({\displaystyle \frac{\mathbf{x}\wedge (\overline{\mathbf{x}}\wedge \mathbf{v})}{{\left|\mathbf{x}\right|}^3}}\right)={\displaystyle \frac{1}{{\left|\mathbf{x}\right|}^5}}\left[{\left|\mathbf{x}\right|}^2(\overline{\mathbf{x}}\wedge \mathbf{v})-3(\mathbf{x}·(\overline{\mathbf{x}}\wedge \mathbf{v}))\mathbf{x}\right],\end{array}$$

(69)

which leads to

$$\begin{array}{c}\hfill \nabla \wedge {\mathbf{N}}_{\mathrm{N}}={\displaystyle \frac{8G}{{\left|\mathbf{x}\right|}^3}}\mathbf{S}-{\displaystyle \frac{12G}{{\left|\mathbf{x}\right|}^5}}\left((\mathbf{x}·\mathbf{S})\mathbf{x}-\int (\mathbf{x}·\mathbf{v})(\mathbf{x}\wedge \overline{\mathbf{x}})\rho d\overline{V}\right).\end{array}$$

(70)

As an application we will consider a homogeneous rotating sphere.

Spherical Homogeneous Bodies

We choose, as an origin of coordinates, the center of a rotating homogeneous sphere of radius R, that is, its center of mass.

We assume that the velocity is given by $\mathbf{v}=\mathbf{w}\wedge \overline{\mathbf{x}}$, where the angular velocity $\mathbf{w}$ is constant.

The Newtonian potential is shown in the usual form ${\Phi }_{\mathrm{N}}=-{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}}$, and the shift vector is given by

$$\begin{array}{c}\hfill {\mathbf{N}}_{\mathrm{N}}=-{\displaystyle \frac{2G}{{\left|\mathbf{x}\right|}^3}}\mathbf{S}\wedge \mathbf{x},\end{array}$$

(71)

where for a rotating homogeneous sphere, the angular momentum is $\mathbf{S}={\displaystyle \frac{2M{R}^2}{5}}\mathbf{w}$.

We are interested in the term $\dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}}$. To perform this calculation, we will use, once again, the following vector identity:

$$\begin{array}{c}\hfill \nabla \wedge \left({\displaystyle \frac{\mathbf{x}\wedge \mathbf{S}}{{\left|\mathbf{x}\right|}^3}}\right)={\displaystyle \frac{1}{{\left|\mathbf{x}\right|}^5}}\left({\left|\mathbf{x}\right|}^2\mathbf{S}-3(\mathbf{x}·\mathbf{S})\mathbf{x}\right).\end{array}$$

(72)

Then, introducing the angular velocity of the precession

$$\begin{array}{c}\hfill {\Omega }_{\mathrm{N}}\equiv {\displaystyle \frac{1}{2}}\nabla \wedge {\mathbf{N}}_{\mathrm{N}}={\displaystyle \frac{G}{{\left|\mathbf{x}\right|}^5}}\left({\left|\mathbf{x}\right|}^2\mathbf{S}-3(\mathbf{x}·\mathbf{S})\mathbf{x}\right),\end{array}$$

(73)

we have

$$\begin{array}{c}\hfill \dot{\mathbf{x}}\wedge \nabla \wedge {\mathbf{N}}_{\mathrm{N}}=-2{\Omega }_{\mathrm{N}}\wedge \dot{\mathbf{x}},\end{array}$$

(74)

which is a Coriolis force.

Taking that into account, since $\mathbf{w}$ is constant, the potentials are constant, and thus, the geodesic equation is

$$\begin{array}{c}\hfill \ddot{\mathbf{x}}=-{\displaystyle \frac{MG}{{\left|\mathbf{x}\right|}^3}}\mathbf{x}-2{\Omega }_{\mathrm{N}}\wedge \dot{\mathbf{x}}-2{\displaystyle \frac{MG|\dot{\mathbf{x}}{|}^2}{{\left|\mathbf{x}\right|}^3}}{\mathbf{x}}_{\perp },\end{array}$$

(75)

where ${\mathbf{x}}_{\perp }$ is the orthogonal rejection of $\mathbf{x}$ onto $\dot{\mathbf{x}}$.

Second-Order Perturbations

Finally, we want to obtain the effects of the second order on the velocity of the body. To do it, we need the metric in Equation (22) to calculate the line element ${{\rm Y}}_{\mathrm{N}}$ and $h_{ij}$.

We have

$$\begin{array}{c}\hfill {\int |\mathbf{v}|}^2\rho d\overline{V}={\displaystyle \frac{2M{R}^2}{5}}{\left|\mathbf{w}\right|}^2,\mathrm{and}\int (\mathbf{x}·\overline{\mathbf{x}}){\left|\mathbf{v}\right|}^2\rho d\overline{V}=0,\end{array}$$

(76)

and

$$\begin{array}{c}\hfill \int v^i{v}^j\rho d\overline{V}={\displaystyle \frac{M{R}^2}{5}}\left[\left(\right|\mathbf{w}{|}^2{\delta }_{ij}-w^i{w}^j\right],\mathrm{and}\int (\mathbf{x}·\overline{\mathbf{x}})v^i{v}^j\rho d\overline{V}=0.\end{array}$$

(77)

Thus

$$\begin{array}{c}\hfill {{\rm Y}}_{\mathrm{N}}=-{\displaystyle \frac{IG}{\left|\mathbf{x}\right|}}{\left|\mathbf{w}\right|}^2\mathrm{and}{h}_{ij}=-{\displaystyle \frac{2IG}{\left|\mathbf{x}\right|}}\left[{\left|\mathbf{w}\right|}^2{\delta }_{ij}-w^i{w}^j\right],\end{array}$$

(78)

where $I={\displaystyle \frac{2M{R}^2}{5}}$ is the inertial momentum of the homogeneous sphere.

In summary, in the second-order approximation, the potentials are given by:

$$\begin{array}{c}\hfill {\Phi }_{\mathrm{N}}=-{\displaystyle \frac{MG}{\left|\mathbf{x}\right|}},{\mathbf{N}}_{\mathrm{N}}=-{\displaystyle \frac{2IG}{{\left|\mathbf{x}\right|}^3}}\mathbf{w}\wedge \mathbf{x},{{\rm Y}}_{\mathrm{N}}=-{\displaystyle \frac{IG}{\left|\mathbf{x}\right|}}{\left|\mathbf{w}\right|}^2\mathrm{and}{h}_{ij}=-{\displaystyle \frac{2IG}{\left|\mathbf{x}\right|}}\left[{\left|\mathbf{w}\right|}^2{\delta }_{ij}-w^i{w}^j\right],\end{array}$$

(79)

with $I={\displaystyle \frac{2M{R}^2}{5}}$.

And the expression of the Lagrangian is given by

$$\begin{array}{c}\hfill \mathcal{L}=\left(1-{\displaystyle \frac{2G}{\left|\mathbf{x}\right|}}(M+2I{\left|\mathbf{w}\right|}^2)\right){\dot{t}}^2+{\displaystyle \frac{4IG}{{\left|\mathbf{x}\right|}^3}}((\mathbf{w}\wedge \mathbf{x})·\dot{\mathbf{x}})\dot{t}-\left(1+{\displaystyle \frac{2G}{\left|\mathbf{x}\right|}}{(M+I|\mathbf{w}|}^2)+{\displaystyle \frac{4M^2{G}^2}{{\left|\mathbf{x}\right|}^2}}\right){\left|\dot{\mathbf{x}}\right|}^2+{\displaystyle \frac{2IG}{\left|\mathbf{x}\right|}}{(\mathbf{w}·\dot{\mathbf{x}})}^2.\end{array}$$

(80)

6. Conclusions and Conceptual Implications

In this essay, we have sought to preserve a substantivalist conception of spacetime, akin to Newton’s absolute space or the flat Minkowski spacetime of Special Relativity. Our approach is restricted to the weak-field regime, where it is still meaningful to consider spacetime as a physical entity with its own ontological status, rather than as a mere relational construct dependent on the distribution of matter and energy.

Rather than assuming from the outset the full geometrization of gravity as in General Relativity, we have explored the possibility of constructing a spacetime metric using only kinematical considerations, symmetry arguments, and the Principle of Equivalence. This allows us to maintain a fixed spacetime background—at least approximately—without resorting to the curvature tensors of a Lorentzian manifold, which are what lead to the relational interpretation. By doing so, we aim to reassess the foundational assumptions of gravitational theory and highlight that the success of Einstein’s theory does not necessarily force a relationalist reading of spacetime, at least not in all regimes of applicability.

It is important to emphasizes that we do not claim that this picture can be extended to the full, non-linear regime of General Relativity, where the curvature of the manifold becomes essential. Nonetheless, the existence of a consistent, physically motivated approximation that retains substantival spacetime lends support to the idea that relationalism is not forced upon us by physics, but emerges only under specific conditions.

Concerning the mathematical aspects of this essay, the core idea of our approach is to begin with a metric that describes a weak gravitational field—derived uniquely from the Weak Equivalence Principle—and to apply a Lorentz transformation to it. This yields, in the weak-field approximation, the metric generated by moving bodies.

Surprisingly, at least for the present author, the resulting metric matches the one obtained via the linearized field equations of General Relativity when working in the de Donder (harmonic) gauge. This result is striking because it shows that essential relativistic features of gravity can be derived through a conceptually straightforward path that is grounded in symmetry principles and the Weak Equivalence Principle, without recourse to the full machinery of Einstein’s field equations.

This leads to a natural historical question: why did Einstein not follow such a route in his early attempts to unify gravity with the framework of Special Relativity? As pointed out by scholars such as John Stachel and Jürgen Renn, Einstein’s early view of gravity was shaped by the belief that a static gravitational field would not curve space but merely affect the passage of time. This perspective is clearly reflected in his 1912 paper “Speed of Light and the Static Gravitational Field”, where Einstein proposes a position-dependent speed of light but retains a spatially Euclidean geometry [29]. As a result, in his early formulations, Einstein modified only the time component of the metric (the $g_{00}$ term) while leaving the spatial components flat. Naturally, a Lorentz transformation applied to such an incomplete metric does not produce the correct spacetime structure around moving bodies, which may partly explain why this approach was not fruitful at the time.

Moreover, this limited view of spacetime geometry contributed to one of the key difficulties in the so-called Entwurf theory [35], which Einstein co-developed with Marcel Grossmann. In that theory, Einstein initially rejected generally covariant field equations and the use of the Ricci tensor, largely because of his doubts about whether the resulting theory would satisfy energy–momentum conservation and reduce Newtonian gravity in the appropriate limit. Historians such as Norton and Renn have suggested that Einstein’s adherence to a Euclidean spatial geometry and the assumption that gravity only modified the rate of clocks obstructed his path toward a fully geometric and covariant theory of gravity [30,40,41,42]. For an English translation of Einstein’s original 1907 paper, in which he made his first attempt to reconcile gravity with the principles of relativity, see [43]. In this seminal work, often regarded as the true starting point of General Relativity, Einstein introduced the idea that an observer in free fall does not feel the effects of gravity—a realization that would later be formalized as the Equivalence Principle. This insight led him to propose that the laws of physics in a uniformly accelerated reference frame are indistinguishable from those in a uniform gravitational field, thus laying the conceptual foundation for a relativistic theory of gravitation. The 1907 paper stands as a milestone in the transition from Special to General Relativity, reflecting Einstein’s early efforts to extend the principle of relativity beyond inertial frames.

It was only in late 1915, after reconsidering the implications of generally covariant conservation laws and correctly calculating the precession of the perihelion of Mercury, that Einstein returned to the Ricci tensor and formulated the field equations in their now-familiar form. Ironically, as this essay has illustrated, much of the weak-field behavior that Einstein had been seeking can already be recovered through a more consistent application of the Equivalence Principle, Newtonian gravity, and Lorentz symmetry—tools that were already available to him during the earliest stages of his investigation.

In the author’s view, the core difficulty that Einstein initially faced lay in his attempt to incorporate both uniformly accelerated frames and gravitation into the framework of Special Relativity. This approach led Einstein to propose a line element whose spatial part remained Euclidean—a result that stemmed from extending Special Relativity to include accelerated observers and from an early, somewhat controversial application of the Principle of Equivalence.

At this preliminary stage, Einstein equated uniform acceleration with the presence of a gravitational field. His reasoning was based on the observation that a freely falling body near the Earth’s surface does not feel its own weight—which, from our viewpoint, suggests, in line with D’Alembert’s Principle, that the inertial force experienced in an accelerated frame cancels the gravitational force. This insight captures the core idea of the Equivalence Principle: the physical indistinguishability, at least locally, between acceleration and gravitation.

However, Einstein’s method in his early attempts—directly incorporating gravity into the Minkowski line element in the same way as acceleration—does not lead to a fully consistent description of gravitational phenomena. In particular, this approach preserves flat spatial geometry and yields geodesic equations that fail to fully reflect the equivalence between inertial and gravitational effects; in other words, the inertial and gravitational forces do not exactly cancel in general.

As discussed in Section 3, to accurately capture the physical content of the Equivalence Principle—especially the condition that gravitational and inertial forces exactly cancel in the appropriate limit—it is necessary to modify not only the temporal component of the metric but also its spatial part. Only then do the resulting geodesics—particularly those that follow trajectories aligned with the gradient of the gravitational potential, such as radial motion in a spherically symmetric field—correctly describe the behavior of freely falling bodies and genuinely express the equivalence between uniform acceleration and a uniform gravitational field in a dynamically consistent way.