The perihelion of Mercury advance and the light bending calculated in (enhanced) Newton’s theory

Abstract

We show that results of a simple dynamical gedanken experiment interpreted according to standard Newton’s gravitational theory, may reveal that three-dimensional space is curved. The experiment may be used to reconstruct the curved geometry of space, i.e. its non-Euclidean metric $^3g_{ik}$. The perihelion of Mercury advance and the light bending calculated from the Poisson equation $^3g^{ik} \nabla _i \nabla _k \varPhi = -4\pi G \rho$ and the equation of motion $F^i = ma^i$ in the curved geometry $^3g_{ik}$ have the correct (observed) values. Independently, we also show that Newtonian gravity theory may be enhanced to incorporate the curvature of three dimensional space by adding an extra equation which links the Ricci scalar $^3R$ with the density of matter $\rho$. Like in Einstein’s general relativity, matter is the source of curvature. In the spherically symmetric (vacuum) case, the metric of space $^3g_{ik}$ that follows from this extra equation agrees, to the expected accuracy, with the metric measured by the Newtonian gedanken experiment mentioned above.

1 Introduction

Newton’s theory of gravity was formulated in a flat, Euclidean 3-D space but its basic laws, i.e. the Poisson equation and the equation of motion,

$$\begin{aligned} {}^{3}\,g^{ik}\,\, \nabla _i \nabla _k \varPhi&= -4\pi G \rho , \end{aligned}$$

(1)

$$\begin{aligned} F_i&= m a_i, \end{aligned}$$

(2)

make perfect sense in a 3-D space with an arbitrary geometry ${}^{3}g_{ik}$. Indeed, the curvature of space is (potentially) present in Newton’s theory. It is easy to argue that the “centrifugal” acceleration of a particle moving with velocity $V$ on a circular orbit equals $a_{\scriptscriptstyle C} = V^2/\mathcal{R}$, and the “gravitational” acceleration in the gravitational field of a spherically symmetric body with the mass $\mathcal{M}$ equals $a_{\scriptscriptstyle G} = -G\mathcal{M}/{\tilde{r}}^2$, with $\mathcal{R}$ being the curvature radius of the circle, and ${\tilde{r}}$ being its circumferential radius. In flat, i.e. Euclidean, 3-D space these two radii are equal, $\mathcal{R} = {\tilde{r}}$, but in a space with a non-zero gaussian curvature $\mathcal{G}$, they are different, $\mathcal{R} \not = {\tilde{r}}$. Therefore, by measuring centrifugal and gravitational accelerations one may independently measure $\mathcal{R}$ and ${\tilde{r}}$, and thus experimentally find whether the space is flat (Euclidean) or it has a non-zero Gaussian curvature $\mathcal{G} \not = 0$. Based on that, Abramowicz has recently suggested in [3] that a Newtonian physicist could experimentally determine the metric ${}^{3}g_{ik}$ of the real physical 3-D space and calculate, according to (1) and (2), the perihelion of Mercury advance and the light bending effects. In this paper we follow this suggestion and calculate both effects within Newton’s theory. Surprisingly, the values of the perihelion advance and the light bending agree (to the expected order of $M/r$) with predictions of Einstein’s theory. Here $M$ is the “geometrical” mass of the spherical gravitating body expressed in the convenient “geometrical” units $G = 1 = c$. It is connected to the mass $\mathcal{M}$ expressed in the standard units by $M = G\mathcal{M}/c^2$ and has the dimension of length.

Another point discussed in this paper is based on the following two remarks: (i) Obviously, Newton’s gravity theory is a limit of Einstein’s general theory of relativity. Should the limit necessarily correspond to $\mathcal{G} = 0$? Perhaps not, because Newtonian physicists could discover within Newton’s theory that $\mathcal{G} \not = 0$. (ii) They could also discover that the curvature of space depends on the distance from the gravity center. This would suggest to them, again within the framework of Newton’s theory, that gravity and curvature are not independent, but instead they are somehow linked. Here we suggest that it is possible to establish the link within an “enhanced” version of Newton’s theory, by adding to its standard version defined by (1) and (2) an extra equation,

$$\begin{aligned} {}^{3}R = 2 k \rho , \end{aligned}$$

(3)

where ${}^{3}R$ is the Ricci scalar corresponding to ${}^{3}g_{ik}$, $\rho$ is the density of matter, and $k$ is a constant. Equations (1), (2) and (3) define our enhanced version of Newtonian gravitational theory. In the special case of a spherically symmetric, vacuum ($\rho = 0$) space, they uniquely lead to the 3-D metric of the form,

$$\begin{aligned} ds^2 = \left( 1 - \frac{r_0}{r}\right) ^{-1}dr^2 + r^2 \left( d\theta ^2 + \sin ^2 \theta d\phi ^2\right) , \end{aligned}$$

(4)

where $r_0$ is a constant. A choice $r_0 = 4\,M$ leads to correct values for both the perihelion advance and the light bending effects.¹

Several authors have discussed the idea of Newton’s gravity and dynamics in a curved 3-D space, for example most recently Naresh Dadhich in “Einstein is Newton with space curved” [1] or previously, in the context of the optical geometry, [2].

Fig. 1

For a circle placed in a curved space (here on a curved 2-D surface), its geodesic radius $r_*$, circumferential radius ${\tilde{r}}$, and curvature radius $\mathcal{R}$ are all different, $r_* \not = {\tilde{r}} \not = \mathcal{R}$

2 The three radii of a circle

Consider a two dimensional curved, axisymmetric surface. In general, it has the metric,

$$\begin{aligned} ds^2 = dr_*^2 + [{\tilde{r}(r_*)}]^2d\phi ^2. \end{aligned}$$

(5)

Consider a family of concentric circles $r_* =$ const in it. One of them is shown in Fig. 1. Obviously, $r_*$ is the geodesic radius and ${\tilde{r}}$ is the circumferential radius of these circles,

$$\begin{aligned} (\mathrm{geodesic~radius})&\equiv \int \limits _0^{r_*} ds_{\vert \phi = \mathrm{const}} = r_*, \end{aligned}$$

(6)

$$\begin{aligned} (\mathrm{circumferential~radius})&\equiv \frac{1}{2\pi }\int \limits _0^{2\pi } ds_{\vert r_* = \mathrm{const}} = {\tilde{r}}. \end{aligned}$$

(7)

Let $\tau ^i = {\tilde{r}}^{-1}\delta ^i_{\,\phi }$ be a unit vector tangent to the circle. From the Frenet formula,

$$\begin{aligned} \frac{d\tau ^i}{ds} = - \frac{1}{\mathcal{R}} \lambda ^i, ~\mathrm{where}~\lambda ^i = (\mathrm{unit~normal~to~the~circle}), \end{aligned}$$

(8)

one deduces that the curvature radius $\mathcal{R}$ may be defined by,

$$\begin{aligned} (\mathrm{curvature~radius}) \equiv \left[ \frac{d\tau ^i}{ds}\,\frac{d\tau _i}{ds}\right] ^{-1/2} = \mathcal{R}. \end{aligned}$$

(9)

Two useful formulae for the curvature of the circle, $\mathcal{K} = 1/\mathcal{R}$, and for the Gaussian curvature $\mathcal{G}$ of the surface with the metric (5) read,

$$\begin{aligned} \mathcal{K}&= +\, \frac{1}{{\tilde{r}}}\left( \frac{d{\tilde{r}}}{\,dr_*} \right) , \end{aligned}$$

(10)

$$\begin{aligned} \mathcal{G}&= -\, \frac{1}{{\tilde{r}}} \left( \frac{d^{2}{\tilde{r}}}{d r_*^2}\right) . \end{aligned}$$

(11)

Formula (10) follows from (9). For derivation of (11) see e.g. [4], Section 3.4.

3 Equations of motion

Let us consider a curve in space given by a parametric equation,

$$\begin{aligned} x^i = x^i(s), \end{aligned}$$

(12)

where $x^i$ are coordinates in space, and $s$ is the length along the curve. If a body moves along this curve, its velocity equals,

$$\begin{aligned} V^i = \frac{dx^i}{dt} = \frac{ds}{dt} \frac{dx^i}{ds} = V \tau ^i. \end{aligned}$$

(13)

Here $V =ds/dt$ is the speed of the body and $\tau ^i = dx^i/ds$ is a unit vector tangent to the curve (12), i.e. the direction of motion. The acceleration may be calculated as follow,

$$\begin{aligned} a^i = \frac{dV^i}{dt} = \frac{ds}{dt} \frac{d(V \tau ^i)}{ds} = V^2 \left( \frac{d\tau ^i}{ds}\right) + \tau ^i V\frac{dV}{ds}. \end{aligned}$$

(14)

Assuming circular motion with constant velocity, $V =$ const, and applying (9) to calculate the term in brackets, we arrive at

$$\begin{aligned} a^i = -V^2\frac{1}{\mathcal{R}} \lambda ^i, \end{aligned}$$

(15)

which is the well known formula for the centripetal acceleration. Therefore the radial component of the centrifugal acceleration is

$$\begin{aligned} a_C = V^2\frac{1}{\mathcal{R}}. \end{aligned}$$

(16)

Consider now circular motion around a spherically symmetric center of gravity. The Newtonian equation of motion, $F^i = m a^i$, takes the form,

$$\begin{aligned} -\nabla ^i \varPhi = -V^2\frac{1}{\mathcal{R}} \lambda ^i, \end{aligned}$$

(17)

where $F^i = - m \nabla ^i \varPhi$ is the gravitational force, and $\varPhi$ is the gravitational potential. Three quantities characterize motion on a particular circular orbit: the angular velocity $\varOmega$, the angular speed $V$, and the specific angular momentum $\mathcal{L}$. They are related by,

$$\begin{aligned} V&= {\tilde{r}} \varOmega , \end{aligned}$$

(18)

$$\begin{aligned} \mathcal{L}&= {\tilde{r}} V = {\tilde{r}}^2 \varOmega . \end{aligned}$$

(19)

Multiplying left hand side of the equation of motion (17) by $\lambda _i$ and using (19), we transform (17) into a form which will be convenient later,

$$\begin{aligned} \lambda _i \nabla ^i \varPhi = \frac{\mathcal{L}^2}{{\tilde{r}}^2 \mathcal{R}}. \end{aligned}$$

(20)

In this expression, $\lambda ^i$ is a unit, outside pointing, vector. Here “outside” has the absolute meaning—outside the center, in the direction towards infinity. We will calculate the left-hand side of this equation in the next Section.

4 Newton’s gravity and Kepler’s law

In an empty space, the gravitational potential $\varPhi$ obeys the Laplace equation,

$$\begin{aligned} \nabla _i (\nabla ^i \varPhi ) = 0. \end{aligned}$$

(21)

Let us integrate (21) over the volume $\mathbb {V}$ that is contained between two spheres, concentric with the gravity center, with sphere $\mathbb {S}_1$ being inside sphere $\mathbb {S}_2$. We transform the volume integral into a surface integral, using the Gauss theorem

$$\begin{aligned} 0 = \int \limits _{\mathbb {V}}\nabla _i (\nabla ^i \varPhi ) d\mathbb {V} = \int \limits _{\mathbb {S}_1}(\nabla ^i \varPhi ) N^{(1)}_i d\mathbb {S} + \int \limits _{\mathbb {S}_2}(\nabla ^i \varPhi ) N^{(2)}_i d\mathbb {S}. \end{aligned}$$

(22)

The oriented surface elements on the spherical surfaces $\mathbb {S}_1$ and $\mathbb {S}_2$ may be written, respectively, as

$$\begin{aligned} N^{(1)}_i d\mathbb {S} = - \lambda _i d\mathbb {S}, ~~N^{(2)}_i d\mathbb {S} = + \lambda _i d\mathbb {S}, \end{aligned}$$

(23)

therefore,

$$\begin{aligned} \int \limits _{\mathbb {S}_1}(\nabla ^i \varPhi ) \lambda _i d\mathbb {S} = \int \limits _{\mathbb {S}_2}(\nabla ^i \varPhi ) \lambda _i d\mathbb {S}. \end{aligned}$$

(24)

This means that the value of the integral is the same, say $S_0$, for all spheres around the gravity center. In addition, because of the spherical symmetry of the potential, the quantity $(\nabla ^i \varPhi ) \lambda _i$ is constant over the sphere of integration. Thus,

$$\begin{aligned} S_0 = (\nabla ^i \varPhi ) \lambda _i \int \limits _{\mathbb {S}} d\mathbb {S} = 4\pi {\tilde{r}}^2\, (\nabla ^i \varPhi ) \lambda _i, ~~ (\nabla ^i \varPhi ) \lambda _i = \frac{S_0}{4\pi {\tilde{r}}^2} = \frac{G\mathcal{M}}{{\tilde{r}}^2}. \end{aligned}$$

(25)

From the above expression it follows that the radial component of the gravitational acceleration equals,

$$\begin{aligned} a_G = -\frac{G\mathcal{M}}{{\tilde{r}}^2}. \end{aligned}$$

(26)

Combining (25) with (20), we may finally write,

$$\begin{aligned} \mathcal{L}^2 = G\mathcal{M}\mathcal{R}. \end{aligned}$$

(27)

This is the Kepler Third Law. Using natural units for radius and frequency,

$$\begin{aligned} R_G = \frac{G\mathcal{M}}{c^2} = M, ~~ \varOmega _G = \frac{c^3}{G\mathcal{M}} = \frac{c}{M}, \end{aligned}$$

(28)

we may write the formula for the Keplerian angular velocity as,

$$\begin{aligned} \left( \frac{{\varOmega }}{\,\varOmega _G}\right) ^2 = R_G^3 \left( \frac{\mathcal{R}}{{\tilde{r}}^4}\right) . \end{aligned}$$

(29)

5 A Newtonian experiment

Imagine a Newtonian physicist who measures the gravitational $a_G$ and the centrifugal $a_C$ accelerations for a circular motion (with the orbital velocity $V$) around the Sun. In the c.g.s. (or any other unit system he may use), $a_G$ and $a_C$ are just numbers. They are independent on the measuring setup and on the theory used. As we have already explained, in Newton’s theory these measurements are equivalent to measurements of the circumferential and curvature radii of the circular orbits, because in Newton’s theory it follows from Eqs. (16) and (26) that,

$$\begin{aligned} {\tilde{r}} = \left[ -\frac{G\mathcal{M}}{a_G}\right] ^{1/2}, \quad \mathcal{R} = \frac{V^2}{a_C}. \end{aligned}$$

(30)

In the flat Euclidean space it must be ${\tilde{r}} = \mathcal{R}$, and therefore the sufficient experimental Newtonian condition for the 3-D space to be curved is,

$$\begin{aligned} \left[ -\frac{G\mathcal{M}}{a_G}\right] ^{1/2} \not = \frac{V^2}{a_C}. \end{aligned}$$

(31)

The Newtonian physicist may form a table of pairs $[{\tilde{r}}, \mathcal{R}]$, which follow from his measurements of $[a_G, a_C]$ at many different orbits. Thus, he may experimentally establish the function,

$$\begin{aligned} \mathcal{R} = \mathcal{R}({\tilde{r}}). \end{aligned}$$

(32)

The geometrical information contained in this function is sufficient to reconstruct the metric of any spherically symmetric 3-D space (spherical symmetry with the Sun at the center would be a natural “working hypothesis” adopted by the Newtonian physicist). The Newtonian physicist may achieve this reconstruction without referring to anything out of the framework of Newton’s physics. Obviously, however, he must allow for a non-zero curvature of the 3-D space.

The Solar System dynamics is given by the Einstein geodesic motion in the Schwarzschild metric; this metric is the “physical reality” here. Thus, when we calculate results of the gedanken experiment considered now, we must use the Schwarzschild metric. We will now calculate the results of the measurements of the circumferential and curvature orbital radii, expressing them by the Schwarzschild standard radial coordinate $r$,

$$\begin{aligned} \mathcal{R} = \mathcal{R}(r), ~~~ {\tilde{r}} = {\tilde{r}}(r). \end{aligned}$$

(33)

To avoid a possible confusion here, we stress that Eq. (33) should be understood as a parametric representation of the experimentally established, invariant, function (32). This function does not refer to $r$ explicitly. No explicit knowledge of $r$ is needed for any argument of the Newtonian physicist — he is only using ${\tilde{r}}$ and $\mathcal{R}$, which he directly measures. Similarly, the final formulae for the perihelion shift and light bending depend neither on $r$, nor on any non-Newtonian concepts (spacetime, timelike Killing vectors, etc.) that we have used to calculate the results of the Newtonian gedanken experiment.

In the Schwarzschild metric, the acceleration of a particle (a “planet”) moving with the orbital velocity $v$ along a circular orbit equals,

$$\begin{aligned} a_i = \nabla _i \varPsi + V^2 \frac{\nabla _i \mathbb {R}}{\mathbb {R}}. \end{aligned}$$

(34)

Here $V = v/(1-v^2)^{1/2}$, and the scalars $\varPsi$ and $\mathbb {R}$ are expressed in terms of the time-symmetry Killing vector $\eta ^i$, and the axial-symmetry Killing vector $\xi ^i$,

$$\begin{aligned} \varPsi = -\frac{1}{2}\ln (\eta ^i\,\eta _i), ~~ \mathbb {R}^2 = - \frac{(\xi ^k\,\xi _k)}{(\eta ^i\,\eta _i)}. \end{aligned}$$

(35)

In Schwarzschild coordinates this is, at the “equatorial plane” $\theta = \pi /2$,

$$\begin{aligned} (\eta ^i\,\eta _i) = g_{tt} = 1 - \frac{2M}{r}, ~~ ~~ (\xi ^k\,\xi _k) = - r^2. \end{aligned}$$

(36)

Equation (34) allows to predict the acceleration measurement results. The acceleration measured with $V = 0$, interpreted by the Newtonian physicist as gravitational acceleration $a_G$, corresponds to the first term on the right hand side of (34). Velocity dependent part of the acceleration, on the other hand, corresponding to the second term on the right hand side of (34), would be interpreted as the centrifugal acceleration, i.e.,

$$\begin{aligned} a_G = -\frac{1}{2}\frac{d}{dr}\left[ \ln \left( 1 - 2M/r\right) \right] , ~~ a_C = \frac{1}{2}\, V^2\, \frac{d}{dr}\left[ \ln \left( \frac{r^2}{1 - 2M/r}\right) \right] . \end{aligned}$$

(37)

By comparing (30) and (37), one concludes that,

$$\begin{aligned} \tilde{r}(r) = r \left( 1-2M/r \right) ^{1/2}, ~~ \mathcal{R}(r) = r \frac{ 1 - 2M/r}{1 - 3M/r}. \end{aligned}$$

(38)

Note, that ${\tilde{r}}/r = 1 + \mathcal{O}(M/r)$ and $\mathcal{R}/r = 1 + \mathcal{O}(M/r)$. Therefore, curvature effects (${\tilde{r}} \not = \mathcal{R}$) may appear at the order $\mathcal{O}(M/r)$. We can also usefully calculate the derivative $d r_* (r) / dr$, as only the derivative, not the absolute value of $r_* (r)$ is of interest. The following equation follows from the definition of Frenet’s curvature radius $\mathcal {R}$

$$\begin{aligned} \frac{dr_*}{dr} = \frac{\mathcal {R}}{\tilde{r}}\frac{d \tilde{r}}{d r} = \frac{r-M}{r-3M} . \end{aligned}$$

(39)

The above formula allows one to write the metric of the 2-D space geometry of the equatorial plane $\theta = \pi /2$, measured in this Newtonian experiment, in the form

$$\begin{aligned} ds^2&= \left( \frac{r-M}{r-3M} \right) ^2 dr^2 + r^2\left( 1-\frac{2M}{r}\right) d \phi ^2 \end{aligned}$$

(40)

$$\begin{aligned}&= \left( \frac{\mathcal{R}({\tilde{r}})}{\tilde{r}} \right) ^2 d{\tilde{r}}^2 + {\tilde{r}}^2 d\phi ^2. \end{aligned}$$

(41)

The second line of this Eq. (41), which in an obvious tautology for any axially symmetric 2-D metric, shows nevertheless that the metric “reconstructed” from the Newtonian experiment described in this Section depends only on $\tilde{r}$ and $\mathcal{R}({\tilde{r}})$, i.e. quantities that are measured by the Newtonial physicist. It needs no reference to $r$. However, one may use $r$ as a convenient parameter, as in the first line of this Eq. (40).

6 Epicyclic oscillations, the perihelion advance

Suppose that we slightly perturb a test-body on a circular orbit. This means that its angular momentum will not correspond to the Keplerian one, $\mathcal{L}^2$, given by (27), but will be slightly different $\mathcal{L}^2 + \delta \mathcal{L}^2$. There will be also a small radial motion with velocity ${\dot{(\delta r_*)}}$ and acceleration ${\ddot{(\delta r_*)}}$. From (20) it follows that

$$\begin{aligned} \frac{GM}{{\tilde{r}}^2} - \frac{\mathcal{L}^2 + \delta \mathcal{L}^2}{{\tilde{r}}^2 \mathcal{R}} = {\ddot{(\delta r_*)}}. \end{aligned}$$

(42)

Keeping the first order term in Eq. (42), and using

$$\begin{aligned} \delta \mathcal{L}^2 = \frac{d\mathcal{L}^2}{d r_*}\,(\delta r_*), \end{aligned}$$

(43)

we arrive at the simple harmonic oscillator equation,

$$\begin{aligned} \omega ^2 (\delta r_*) + {\ddot{(\delta r_*)}} = 0, \end{aligned}$$

(44)

where $\omega$ is the radial epicyclic frequency,

$$\begin{aligned} \omega ^2 = \frac{1}{{\tilde{r}}^2 \mathcal{R}}\left( \frac{d\mathcal{L}^2}{d r_*}\right) . \end{aligned}$$

(45)

Using Eqs. (27) and (28), we may write the expression for the epicyclic frequency in the form,

$$\begin{aligned} \left( \frac{\omega }{\varOmega _G}\right) ^2 = \left( \frac{d\mathcal{R}}{\,\,d r_*}\right) \frac{R_G^3}{{\tilde{r}}^2 \,\mathcal{R}}, \end{aligned}$$

(46)

or comparing this with (29),

$$\begin{aligned} \left( \frac{\omega }{\varOmega }\right) ^2 \,=\, \left( \frac{d\mathcal{R}}{\,\,d r_*}\right) \frac{{\tilde{r}}^2}{\mathcal{R}^2} \,=\, \left( \frac{d {\tilde{r}}}{\,\,d r_*} \right) ^2 - {\tilde{r}} \left( \frac{d^2 {\tilde{r}}}{d r_*^2} \right) . \end{aligned}$$

(47)

In a flat space, $r_*\,=\,{\tilde{r}}\,=\,\mathcal{R}$, and therefore $\omega \,=\,\varOmega$, which implies that the slightly non-circular orbit is a closed curve, indeed an ellipse. In a curved space with $\mathcal{G}\,\not =\,0$, one has $r_*\,\not =\,{\tilde{r}}\,\not =\,\mathcal{R}$, and consequently $\omega \,\not =\,\varOmega$. The slightly non-circular orbit would not be a closed curve. It could be represented by a precessing ellipse, with two consecutive perihelia shifted by

$$\begin{aligned} \varDelta \phi = T(\varOmega - \omega ) = 2\pi \left( 1 - \frac{\omega }{\varOmega }\right) = 2\pi \left[ 1 - \left( \frac{ d\mathcal{R} }{d{\tilde{r}} } \frac{\tilde{r} ^3}{\mathcal { R}^3} \right) ^{1/2} \right] , \end{aligned}$$

(48)

where $T = 2\pi /\varOmega$ is the orbital period. Inserting (38) into the Newtonian perihelion advance formula (48) one gets,

$$\begin{aligned} \frac{\varDelta \phi }{2 \pi } = 1 - \sqrt{1 + \frac{-6Mr^3 + 34M^2r^2 -62M^3r + 36M^4}{r^4 - 5Mr^3 + 8M^2r^2 -4M^3r}} . \end{aligned}$$

(49)

Expanding this to the desired accuracy $\mathcal{O}^2(M/r)$, and using Eq. (38), one finally gets the same value for the perihelion advance as calculated in Einstein’s theory,

$$\begin{aligned} \varDelta \phi = 6 \pi \frac{M}{r} + \mathcal{O}^2 \left( \frac{M}{r} \right) = 6 \pi \frac{M}{\tilde{r}} + \mathcal{O}^2 \left( \frac{M}{\tilde{r}} \right) . \end{aligned}$$

(50)

7 Light bending

Knowing the space geometry, given by (40), we may calculate the effect of light bending assuming that light travels along geodesic lines in space. In Newton’s theory this assumption is equivalent to the Fermat principle, i.e. that light travels (with a constant speed) between two points $A$, $B$ in space, minimizing the time travel $T_{AB}$. The equation of motion for the $\phi$ coordinate is, in these circumstances,

$$\begin{aligned} \frac{d^2 \phi }{ds^2} + 2 \frac{r-M}{r(r-2M)} \frac{dr}{ds} \frac{d\phi }{ds} = 0 , \end{aligned}$$

(51)

from which we find $\frac{d \phi }{ds}$ to be equal to

$$\begin{aligned} \frac{d\phi }{ds} = \frac{C}{r(r-2M)} . \end{aligned}$$

(52)

The integration constant can be evaluated at the perihelion location $r = R_0$ (i.e. where ${d \phi }/{ds} = 0$), yielding

$$\begin{aligned} \frac{d\phi }{ds} = \frac{\sqrt{R_0(R_0-2M)}}{r(r-2M)} . \end{aligned}$$

(53)

Using Eqs. (40) and (53) we find also

$$\begin{aligned} \frac{dr}{ds} = \frac{r-3M}{r-M}\sqrt{1 - \frac{R_0(R_0-2M)}{r(r-2M)}}. \end{aligned}$$

(54)

After dividing (53) by (54) and substituting $x = R_0/r$ the $d \phi / d r$ equation can be integrated from $R_0$ to $\infty$ (or $x$ from 0 to 1), which will give us the half of $\pi + \delta$. Let us also define $\mu = M/R_0$, then

$$\begin{aligned} \frac{\pi + \delta }{2} = \int \limits ^1_0 \frac{1 - x \mu }{(1 - 2 x \mu )(1- 3 x \mu )} \sqrt{\frac{1 -2\mu }{1 - x^2(1-2\mu )/(1-2x \mu ) }} d x . \end{aligned}$$

(55)

This integration can be expanded in a Taylor series for $\mu$:

$$\begin{aligned} \frac{\pi + \delta }{2} = \int \limits ^1_0 \frac{dx}{\sqrt{1-x^2}} + \int \limits ^1_0 \frac{(3x^2 +3x-1)dx}{\sqrt{1-x^2}(x+1)} \mu + \mathcal{O}^2(\mu ) . \end{aligned}$$

(56)

As the first component on the right hand side is equal to $\pi /2$, we conclude that

$$\begin{aligned} \delta \approx 2\mu \int \limits ^1_0 \frac{(3x^2+3x-1)dx}{\sqrt{1-x^2}(x+1)} = 4 \frac{M}{R_0} . \end{aligned}$$

(57)

Similar calculations in the Schwarzschild spacetime geometry give the same result²

$$\begin{aligned} \delta \approx 2\mu \int \limits ^1_0 \frac{1-x^3}{(1-x^2)^{3/2}}dx = 4 \frac{M}{R_0} . \end{aligned}$$

(58)

Once again, the prediction of the Newtonian theory in the non-flat space is found to be consistent with observations (and with Einstein’s general relativity).

In the real curved space of the Solar System (described by the Schwarzschild metric) a Newtonian physicist may discover the curvature by the two specific measurements that we have described, and then correctly estimate the value of the perihelion of Mercury advance and the light bending. Of course, he may perform different measurements, for example he may directly measure the circumferential and curvature radii. If he would use light signals to measure the distance (radar defined distances leading to the optical geometry), then the result will be the same as from measuring these radii “dynamically” (see e.g. Abramowicz [5]). However, if he would measure these radii using the “rigid rods”, then he would find a different result. To get the right answers for the light bending and orbital precession, he would need to make the right choice of the measure to use. He may eventually find that the optical geometry agrees with dynamics, but the geometry based on rigid rods does not.

8 Enhanced Newtonian gravitational theory

Jürgen Ehlers pointed out in 1961 that in Einstein’s theory the curvature of the rest-space of irrotational matter is determined by its distribution and relative motion (see 1221 in his article [6]). The equations governing such 3-space curvature for arbitrary irrotational flows are given in [7]; see their Eq. (54). Consequently it makes sense to consider gravitational dynamics in the context of 3-D curved Riemannian spaces. As Newtonian theory is an approximation to General Relativity Theory, it is therefore interesting to see what happens in the case of Newtonian theory in a curved 3-D background space³.

In the case of isometric flows, $\theta = \sigma _{ab} = 0$ and there is a potential such that $\dot{u}^a = U_{,a}$ where the gravitational potential $U$ relates the Killing vector $\xi$ to the unit 4-velocity $u^a$ by $\xi ^a = e^U u^a$ (see 1234 in Ehlers [6]). Then the relevant equation becomes

$$\begin{aligned} {}^3R_{ab} = \tilde{\nabla }_a\tilde{\nabla }_b U \,+ \,\tilde{\nabla }_a U \tilde{\nabla } _b U \,+ \,\frac{2}{3} \,k\,\rho \,h_{ab} , \end{aligned}$$

(59)

where $\tilde{\nabla }_a$ is the 3-D covariant derivative, $\rho$ is the energy density of matter, and we have assumed anisotropic stress is zero ($\pi _{ab} = 0$) and a vanishing cosmological constant. Here $h_{ab} = g_{ab} - u_a u_b$ is the metric of the three-spaces orthogonal to $u^a$. This case will include static spherically symmetric spacetimes. Taking the trace of this equation gives (see equation (55) in [7])

$$\begin{aligned} {}^3R = 2\, k \,\rho , \end{aligned}$$

(60)

where the potential terms have gone because of the relation between the 3-D and 4-D covariant derivatives. Together with the Poisson equation and equation of motion it defines the Enhanced Newtonian Gravitational Theory,

$$\begin{aligned} {}^{3}g^{ik}\,\, \nabla _i \nabla _k \varPhi&= -4\pi G \rho , \nonumber \\ F_i&= m a_i, \nonumber \\ {}^3R&= 2 k \,\rho . \end{aligned}$$

(61)

For spherically symmetric spaces, the most general metric has the form,

$$\begin{aligned} ds^2 = A(r)\,dr^2 + r^2 \left( d\theta ^2 + \sin ^2 \theta d\phi ^2 \right) , \end{aligned}$$

(62)

and in the vacuum case, $\rho = 0 = {}^3R$ one has

$$\begin{aligned} {}^3R_{rr}&= \frac{A^\prime }{rA}, \end{aligned}$$

(63)

$$\begin{aligned} {}^3R_{\theta \theta }&= \left( \frac{rA^\prime }{2A^2} - \frac{1}{A} + 1 \right) \sin ^2\theta , \end{aligned}$$

(64)

$$\begin{aligned} {}^3R_{\phi \phi }&= \frac{rA^\prime }{2A^2} - \frac{1}{A} + 1, \end{aligned}$$

(65)

$$\begin{aligned} {}^3R&= \frac{2}{r^2} - \frac{2}{A r^2} + \frac{2A^\prime }{A^2 r} = 0. \end{aligned}$$

(66)

Here the prime denotes a derivative with respect to $r$. Equation (66) has a unique solution,

$$\begin{aligned} A(r) = \left( 1 - \frac{r_0}{r} \right) ^{-1}, \end{aligned}$$

(67)

with $r_0$ being an integration constant. Its value cannot be determined by Eq. (61), but instead must be chosen by correspondence with experiment⁴. Using the same procedure as in Sects. 6 and 7, one proves that the choice $r_0 = 4M$ gives the correct values for the perihelion advance and light bending (with accuracy $\mathcal{O}(r_0/r)$).

9 The two metrics

We have shown that “experimentally” established and the “theoretically” postulated Newtonian metrics of the curved 3-D space corresponding to a spherically symmetric body are, respectively,

$$\begin{aligned} ds^2&= \left( \frac{r-M}{r-3M} \right) ^2 dr^2 + r^2\left( 1-\frac{2M}{r}\right) \left( d\theta ^2 + \sin ^2 \theta d\phi ^2\right) , \end{aligned}$$

(68)

$$\begin{aligned} ds^2&= \left( 1 - \frac{4 M}{r}\right) ^{-1}dr^2 + r^2 \left( d\theta ^2 + \sin ^2 \theta d\phi ^2\right) . \end{aligned}$$

(69)

We have also shown that any spherically symmetric metric that obeys $^3 R = 0$ must be isometric with (69). The Ricci scalar for the “experimental” metric may be calculated to be

$$\begin{aligned} ({}^3 R)M^2&= M^2\frac{18(M/r)^3 - 10(M/r)^2}{r^2[ 4(M/r)^3 - 8(M/r)^2 +5(M/r) - 1]} \nonumber \\&= 10(M/r)^4 + 32(M/r)^5 + \cdots \nonumber \\&= 0 + \mathcal{O}^4(M/r). \end{aligned}$$

(70)

On the other hand a metric,

$$\begin{aligned} ds^2 =\frac{dr^2}{1 - \frac{4 M}{r} + \alpha \left( \frac{M}{r}\right) ^2 + \beta \left( \frac{M}{r}\right) ^3 + \cdots } + r^2 \left( d\theta ^2 + \sin ^2 \theta d\phi ^2\right) , \end{aligned}$$

(71)

has the Ricci tensor,

$$\begin{aligned} ({}^3 R)M^2 = 2\,\alpha \left( \frac{M}{r}\right) ^4 + 4\,\beta \left( \frac{M}{r}\right) ^5 + \cdots = 0 + \mathcal{O}^4(M/r). \end{aligned}$$

(72)

Thus, the experimental metric (68) and the theoretical metric (69) describe, with accuracy $\mathcal{O}^2(M/r)$, the same geometry of space.

10 Conclusions

We demonstrated that a Newtonian physicist may experimentally determine the geometry of the 3-D space $^3g_{ik}^E$ by measuring gravitational and centrifugal accelerations. He may then predict by calculations the perihelion advance and the light bending as effects of the curvature of space. The predicted values agree with the ones measured. We also demonstrated that one may extend Newton’s theory of gravitation by adding an equation that links Ricci curvature of space with the density of matter. We calculated the resulting theoretical metric of space $^3g_{ik}^T$ assuming spherical symmetry. In this metric, the values of perihelion advance and light bending also agree with those observed. The two metrics represent the same geometry, $^3g_{ik}^E = {}^3g_{ik}^T$ with accuracy $\mathcal{O}^2(M/r)$.

Abramowicz [3] has shown that for spaces with constant Gaussian curvature Newton’s theory predicts no perihelion advance. We speculate that this is why Gauss (and other XIX century mathematicians) who might have calculated Newtonian orbits in curved spaces, would have missed the effect of perihelion advance. Most probably, they would calculate orbits in spaces with a constant Gaussian curvature first. Gauss almost certainly made this calculation. He was a master in calculating orbits. He made himself famous at the age of 23 by calculating the orbit of Ceres, discovered in 1801 by Piazzi. He seriously considered the possibility that our space is curved. He even attempted to determine the curvature of space by measuring angles in a big triangle (69, 84, 106 km) made by the summits of Brocken, Hoher Hagen and Großer Inselsberg. Gauss was not quick in publishing his results concerning curved spaces. It is known that he discovered most of Bolyai’s results, but never published them. Gauss died in February 1885, four years before Le Verrier discovered the effect of the perihelion of Mercury advance.