#### 2.2. Theory

A definition of the kinetic energy of a test particle with gravitational mass

${m}_{0}$ within a universe with other (gravitational, inertia inducing) masses

${m}^{\alpha}$, which is consistent with all assumptions above is given by (summation convention is used for the spatial indices

$i,j$):

where

$\gamma $ is a natural constant, presumably connected with Newton’s gravitational constant,

${\overrightarrow{r}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}={\overrightarrow{x}}_{0}-\overrightarrow{x}{\phantom{\rule{0.166667em}{0ex}}}^{\alpha}$ is the distance between

${m}_{0}$ and

${m}^{\alpha}$, with

${\overrightarrow{x}}_{0}$ and

${\overrightarrow{x}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}$ the absolute positions of

${m}_{0}$ and

${m}^{\alpha}$ in space, respectively. Similarly,

${\overrightarrow{v}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}={\dot{\overrightarrow{x}}}_{0}-{\dot{\overrightarrow{x}}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}$ is the relative (not only radial!) velocity between

${m}_{0}$ and

${m}^{\alpha}$ (as will be the relative acceleration

${\dot{\overrightarrow{v}}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}={\overrightarrow{a}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}$ later on). The sum is over all masses

${m}^{\alpha}$ in the universe with the exception of

${m}_{0}$.

The corresponding components of the momentum and force of the test particle

${m}_{0}$ are then given by

If and only if we are in a coordinate system where all inducing masses

${m}^{\alpha}$ are at rest, we have

${\overrightarrow{v}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}={\overrightarrow{v}}_{0}$ and the formulae simplify to

#### 2.3. Check of Assumptions

Let us now investigate how this definitions fulfil Assumptions 1–5. For this, we do not consider Newtonian gravitation beyond the induction of inertia.

**Assumption** **1.** Since there exists only ${m}_{0}$ and not one single ${m}^{\alpha}$, the ${M}_{ij}^{\alpha}$ are all zero. There is no kinetic energy and no momentum and also no force.

**Assumption** **2.** For translations at a constant distance, ${\overrightarrow{v}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}=0$. For (possibly non-uniform) rotations of a mass ${m}_{0}$ with constant radius R around a single point mass ${m}^{\alpha}$ (or vice versa), we assume ${\overrightarrow{x}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}=(0,0,0)\Rightarrow {\overrightarrow{r}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}={\overrightarrow{x}}_{0}-\overrightarrow{0}=(Rcos\omega t,Rsin\omega t,0)$, with ω possibly time dependent. Thus we have With our definitions above one sees easily that $\overrightarrow{p}=\overrightarrow{0},\phantom{\rule{4pt}{0ex}}E=0$ and again there is no force (this is also true if ${m}^{\alpha}$ and ${m}_{0}$ rotate synchronously around some other rotational center, e.g., both being fixed onto one single watch hand). Especially, there are no centripetal or centrifugal forces. (Therefore, the gravitational attraction between the two masses will eventually lead to a collision, regardless if there is a rotation or not.)

**Assumption** **3.** Anisotropy of inertia is realized in our definition. As an illustration, let us consider a simple experiment involving three masses, see Figure 1. ${m}_{0}$ is at rest in $(0,0,0)$. We have two masses ${m}^{\alpha}$, with mass ${m}^{1}=M$ at $(0,-R,0)$ and mass ${m}^{2}=2M$ at $(-R,0,0$). After a short calculation, we find (with

${\overrightarrow{p}}_{0}=0$)

If a force $\overrightarrow{F}=(2,2,2)$ is applied, this will lead to an acceleration of $\overrightarrow{a}=(1,2,\ast )$, which is clearly not the same direction as $\overrightarrow{F}$. The * signals the not defined acceleration in the here inertia-free z-direction.

If ${\overrightarrow{p}}_{0}\ne \overrightarrow{0}$ at the beginning, things become more complicated, since the change of ${\overrightarrow{r}}^{\phantom{\rule{0.166667em}{0ex}}1}$ and ${\overrightarrow{r}}^{\phantom{\rule{0.166667em}{0ex}}2}$ (the position of ${m}_{0}$ relative to the two other masses) gives extra terms leading to $\overrightarrow{a}\nparallel (1,2,\ast )$.

**Assumption** **4.** One sees easily that the contribution of each mass ${m}^{\alpha}$ to $\overleftrightarrow{M}$ is proportional (besides the dependence on direction) to $1/|{\overrightarrow{r}}^{\phantom{\rule{0.166667em}{0ex}}\alpha}|$.

All these considerations can easily be generalized to systems with more masses ${m}^{\alpha}$, where one also has e.g., the absence of kinetic energy and momentum if all masses move uniformly or the absence of centrifugal forces if all masses rotate uniformely without change of relative distances. This corresponds perfectly to Mach’s famous bucket-gedankenexperiment.

**Assumption** **5.** It is perhaps the most crucial and definitely the most complicated to prove the following. In Appendix A, we will give the detailed calculation (A2). The result for the inert mass tensor of a test particle ${m}_{0}$ (resting or moving with constant velocity relative to the sphere) within a thin sphere of radius R and constant area mass density σ (and hence $4\pi {R}^{2}\sigma $ its overall mass) is: This corresponds to a constant, isotropic inertia independent of the spatial position (as long as within the sphere) and would, therefore, be the physical setting for the emergence of usual Newtonian inertia. The calculation in

Appendix A shows that the contribution of a mass element

$dm$ to the inertia of a particle

${m}_{0}$ at rest (experiencing an acceleration) is proportional to

${cos}^{2}\alpha $, where

$\alpha $ is the angle between the line connecting

${m}_{0}$ and

$dm$ and the direction of the acceleration. So if the angle is

${90}^{\circ}$, there is no contribution of

$dm$ to the inertia, whereas the contribution is maximal if both directions are in parallel. Thus, apsidal precession for planets circling around a central star is not a consequence of this theory (leaving room for additional effects due to general relativity), but it avoids the prediction of an apsidal precession with wrong sign as is done by Mach-like theories with isotropic inert mass (proportional e.g., to the local gravitational potential), [

8,

9].