General Relativity in Light of the Necessary Logic

1. Introduction

What is a gravitational force? Is it from real or apparent origin? Within the elevator and relative to it, all happens as if the gravitational force were non-existent. Each free particle moves there with an uniform velocity and in straight line. From its side is sure: there is an inertial system of reference. On the other side is also sure: the earth is in motion with the reversed acceleration relative to this system in the elevator: can thus the free fall of the particles be regarded as a purely kinematic effect? The physical fact is by all means indisputable: the gravitational force vanishes in a falling elevator. Is this fact a consequence of its compensation (=the stand of classical physics) or its disappearence (=the stand of Einstein)? Einstein preferred to assume that it really vanishes from a deep reason: it betrays (=just like a force of inertia) a relationship between different frames of reference. Is it at all a real force? Or an apparent force? That is the question!

Yes! For both questions holds (=either „true“ or „false“) from the standpoint of the two-valued Aristotelian Logic! A tautology in this logic is an expression of the two possibilities in the freedom of contingency: it is true regardless of the truth value of the proposition within it. Thus „A or not A“ is a tautology just by virtue of this kind of logic. Such freedom of possibilities is rejected by the Necessary Logic with the only one-valued proposition (=neither „true“ nor „false“ but „something else“). Its transcendental power emerges from the condition of knowledge (=sameness of mental thinking and ontological occurring): it „knows“ nothing about freedom. The necessary knowledge transcends so the contingent knowledge from a simple reason: we do not know how we know. This fact was confirmed by Gödel in the framework of mathematics. It arises also from our impossibility to answer: is our knowledge at all a result of cognition or recognition? The necessary logic remains from that reason an ideal. But! Its real existence is indisputable: we should know nothing without it. The absence of the condition of knowledge requires simply its reality. Its immediate trace is just „we know something“ in contingence. And we are able to understand: both concepts (=„true“ and „false“) are meaningless from the standpoint of the concept „truth“. So we assert with certainty: the demand

– neither „true“ nor „false“ but „necessary“– 

emerges from the Necessary Logic with the concept „truth“.

Only such kind of one-way reasoning brings us a hope to understand the current occurrence in terms of the following proposition:

– neither „compensation“ nor „disappearence“ but „something else“.

The Necessary Logic does not permit any game with the nonsense of „non-existence“. On the other hand, a possibility „compensation“ is against the spirit of Einstein's relativity. For that reason it remains:

– neither „compensation“ nor „disappearence“ but „conversion“.

And indeed! The concepts „force“ and „motion“ require compilation:

– neither „force“ nor „motion“ but „force-and-motion“– 

in the sense

– „force“ contains in itself a potential „motion“– 

and vice versa

– „motion“ comprises in itself its condition „force“.

Just like a relationship of mutation (=an inner conversion) between mass and energy:

– „energy“ is the form of „mass“– 

and vice versa

– „mass“ is the essence of „energy“.

This thesis is easily demonstrable in description of the free fall via the formalism in the framework of general relativity.

2. Relativistic interpretation of the free fall

The four coordinate functions of an object moving to a particular frame of reference are defined by

\(\displaystyle \vec{x}=x^{\mu}(\tau)=\begin{pmatrix} x^0(\tau) \\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \end{pmatrix} = \begin{pmatrix} ct \\ x^1(t) \\ x^2(t) \\ x^3(t) \end{pmatrix} \)       (2.1)

They represent its four-velocity which is the tangent four-vector of a world line defined as

\( \displaystyle\vec{U}=\frac{d\vec{x}}{d\tau} \)       (2.2)

The relation between the coordinate time and the proper time appears as a consequence of the time dilatation

\( \displaystyle t=\gamma\tau, x^0=ct=c\gamma\tau \)       (2.3)


\( \displaystyle\gamma = (1 - \frac{v^2}{c^2})^{-1/2} \)       (2.4)

is the Lorentz factor, c is the speed of light. We are now able to express the components of the four-velocity vector

\( \displaystyle U^0 = \frac{dx^0}{d\tau} = c\gamma \)


\( \displaystyle U^i = \frac{dx^i}{d\tau} = \frac{dx^i}{dx^0} \frac{dx^0}{d\tau} = \frac{dx^i}{dx^0}c\gamma = \frac{dx^i}{d(ct)}c\gamma = \frac{1}{c} \frac{dx^i}{dt}c\gamma = \gamma \frac{dx^i}{dt} = \gamma v^i \)       


\( \displaystyle\vec{U} = \gamma (c, \vec{v}) \)       (2.6)


\( \displaystyle v^i = \frac{dx^i}{dt} \)       (2.7)

are the components of velocity from classical theory.

The four-momentum for the particle's invariant mass is further given by

\( \displaystyle\vec{P} = m_0\vec{U} \)       (2.8)

and the four-force becomes

\( \displaystyle\vec{F} = \frac{d\vec{P}}{d\tau} = m_0\vec{A} = (\gamma \frac{\vec{f}\cdot\vec{v}}{c}, \gamma \vec{f}) \)       (2.9)

where we have (v, p, f are 3-vectors and dE/dt rate of energy)

\( \displaystyle\vec{f} = \frac{d}{dt}(\gamma m_0\vec{v}) = \frac{d\vec{p}}{dt} \)


\( \displaystyle\vec{f}\cdot\vec{v} = \frac{d}{dt}(\gamma m_0 c^2) = \frac{dE}{dt} \)

It is still to emphasize: the elements of the four-force are also related to the elements of the four-momentum through a covariant derivative with respect to proper time

\( \displaystyle F^{\lambda} : = \frac{DP^{\lambda}}{d\tau} = \frac{DP^{\lambda}}{d\tau} + \begin{Bmatrix} \lambda \\ \mu \nu \end{Bmatrix} U^{\mu} P^{\nu} \)       (2.11)

The non-zero Christoffel symbols of the second kind in the case of radial motion (=for the free fall) are

\(\displaystyle\begin{Bmatrix} \lambda \\ \mu \mu \end{Bmatrix} = - \frac{1}{2}\frac{1}{g_{\mu\mu}}\frac{\partial g_{\mu\mu}}{\partial x^{\lambda}}, \begin{Bmatrix} \lambda \\ \lambda \mu \end{Bmatrix} = \frac{1}{2} \frac{\partial \log g_{\lambda\lambda}}{\partial x^{\mu}}, \begin{Bmatrix} \lambda \\ \lambda \lambda \end{Bmatrix} = \frac{1}{2} \frac{\partial\log g_{\lambda\lambda}}{\partial x^{\lambda}}\)       (2.12)

and the equations of the geodesics

\( \displaystyle\frac{d}{ds}\left(g_{\sigma\nu}\frac{dx^{\nu}}{ds}\right)-\frac{1}{2} \frac{\partial g_{\lambda\nu}}{\partial x^{\sigma}}\frac{dx^{\lambda}}{ds}\frac{dx^{\nu}}{ds}=0\)       (2.13)


\(\displaystyle \frac{d}{ds}\left(g_{\lambda\lambda}\frac{dx^{\lambda}}{ds}\right) -\frac{1}{2} \sum_{\nu=0}^{3}\frac{\partial g_{\nu\nu}}{\partial x^{\lambda}}\left(\frac{dx^{\nu}}{ds}\right)^2=0, (\lambda =0,1,2,3)\)       (2.14)

We need still the Einstein equations for the free-space field

\( \displaystyle R_{\mu\nu}=\frac{\partial^2\log\sqrt{-g}}{\partial x^{\nu}\partial x^{\mu}}- \frac{\partial}{\partial x^{\sigma}}\begin{Bmatrix} \sigma \\ \mu \nu \end{Bmatrix} + \begin{Bmatrix} \tau \\ \mu \sigma \end{Bmatrix}\begin{Bmatrix} \sigma \\ \nu \tau \end{Bmatrix} - \begin{Bmatrix} \tau \\ \mu \nu \end{Bmatrix}\frac{\partial\log\sqrt{-g}}{\partial x^{\tau}}=0\)       (2.15)

and their solution

\(\displaystyle e^{\nu}=e^{-\mu}=1-\frac{\alpha}{r} \)       (2.16)

for the Schwarzschild space-time

\(\displaystyle d\tau^2 = e^{\nu}dt^2 - \frac{1}{c^2}\left\{ e^{\mu}dr^2 + r^2\left(d\theta^2 + \sin^2\theta d\varphi^2\right)\right\} \)       (2.17)

with the result: the gravitational field, as a cause of the free fall, is reducible through the geodesic equations to the form

\(\displaystyle \frac{d^2r}{dt^2} = - \frac{1}{2}\frac{\alpha c^2}{r^2} \)       (2.18)

which coincides with the Newtonian equation of (radial) motion for a „link“

\(\displaystyle \alpha = \frac{2GM}{c^2} \)       (2.19)

of this constant with the gravitational constant. Consequently the gravitational „force“ appears to be the metric effect only (=it does vanish as a force!)

Is it so indeed? That is the question!

3. Relativistic reinterpretation of the free fall

The gravitational force is perfectly real: it can be detected and measured. Thus is wrong to say „it is no longer an absolute“. Yes! It is relative and it can be annulled like a force of inertia, since its value varies with the choice of a frame of reference. But a notion „relativization“ loses a sense in the case of its „disappearence“, not in the case of its „absence“. What is then a sense of such absence? Is it in some exposition of „something else“? Yes! An intervention of necessity is unavoidable: „something else“ is nothing but conversion or mutation of the reality! This thesis must be proven.

The usual logic of generalization in the framework of general relativity could accept a strange thesis about variability of the mass of a body in the proper time during its motion in the gravitational field through space. We cannot avoid the idea about an invariant mass. Its role is however indispensable only throgh an attempt of physicist to describe a reality with ever changing generalized mass of a body. Description of a phenomenon is in no way its explanation. The presence of the generalized mass in the Newtonian law of gravitation is conditio sine qua non:

\(\displaystyle F^1 = -G \frac{Mm}{r^2} \)       (3.1)

If so, then the radial component of the four-force from (2.10) requires also the generalized form

\(\displaystyle F^1 = \frac{d}{d\tau}\left(m\frac{dr}{d\tau}\right) = m\frac{d^2r}{d\tau^2} +\left(\frac{dr^2}{d\tau}\right)^2\frac{dm}{dr} \)       (3.2)

where the proper velocity can be expressed by means of the Schwarzschild metric (2.17)

\(\displaystyle \left(\frac{dr}{d\tau}\right)^2 = c^2\frac{\alpha}{r} \)       (3.3)

The rate of energy from (2.10) becomes in this case

\(\displaystyle F^1\frac{dr}{d\tau} = \frac{dmc^2}{d\tau} = -G\frac{Mm}{r^2}\frac{dr}{d\tau} \)       (3.4)

and its integration gives the generalized mass

\(\displaystyle m=m_0 e^{\frac{\alpha}{2r}} \)       (3.5)

which together with the Schwarzschild solution enables to express the four-force (3.2) in the form

\(\displaystyle F^1 = m\frac{d^2r}{d\tau^2} + \left(1-e^{\nu}\right)\left(-\frac{\alpha c^2}{2r^2}\right)m \)       (3.6)

whose content contains the two equations: neither the first

\(\displaystyle F^1=-\frac{1}{2}\frac{\alpha c^2}{r^2} m \)       (3.7)

represents the presence of the gravitational force (3.1) (=since its consequence is also there) nor the second

\(\displaystyle m\frac{d^2r}{d\tau^2} + \frac{1}{2}\frac{\alpha c^2}{r^2} m\left(1-\frac{\alpha}{r}\right) =0 \)       (3.8)

represents the presence of the geodesic motion

\(\displaystyle \frac{d^2r}{d\tau^2} + G\frac{M}{r^2} =0, \left( \alpha \ll \gamma \right) \)       (3.9)

(=since its cause is also there). In such a way only the two negations lead to the compilation of both phenomena. The metric effect (=the geodesic motion) and the physical effect (=the gravitational force) are two faces of reality from the two demonstrations in complete harmony up to the boundary of existence at the Schwarzschild radius. Only there is an explanation of our impossibility to conquer the total knowledge.

Nothing strange happens at the Schwarzschild boundary? It seems to be a funny question! Why?There is the last shore of physics! Each attempt of human mind (=with the apparatus of physics) to reconstruct the structure of occurrences behind the curtain (=without the apparatus of necessity) is nothing but a thing of our fancy.

Let us to conclude!

The force

\(\displaystyle F^1 = -G\frac{Mm}{r^2} = -GMm_0 \frac{e^{\frac{\alpha}{2r}}}{r^2} \)       (3.10)

does increase up to infinity after a change of sign in acceleration at the boundary

\(\displaystyle m\frac{d^2r}{d\tau^2} +G\frac{Mm}{r^2}\left(1-\frac{\alpha}{r}\right)=0 \)       (3.11)

Why? Neither force nor motion but „something else“ from the Necessary Logic governs with the state of affairs in the free fall. Hence we can conclude: the thesis

– either the presence of the gravitational force (= the extract 1 from „something else“) or the presence of the geodesic motion (=the extract 2 from „somethng else“)– 

is wrong. It must be replaced by the thesis:

– neither appearence nor reality but necessity comprises the ultimate truth about physical laws in the contingency.

So far as the phenomena are concerned, the precise knowledge is impossible indeed. Neither subject nor object but a single whole possesses an understanding of the last shore of physics:

– neither the particle-picture nor the wave-picture but the necessity-picture– 

may explain the origin of physics from the background of our world. There is a pure mystery. It is in tension with the ethics of science. We are therfore cautioned by Sir Edmund Whittaker:

„We have no right to postulate the existence of entities which lie beyond the knowledge actually obtainable by observation, and which have no part in the prediction of future effects. Thus the classical concept of a particle must be discarded: in its stead there has been introduced a new fundamental element in the description of the external world, which is called a state …“


But we have right to postulate the existence of conditions (=the work of mind) for those entities which lie beyond the knowledge and participate in the prediction of future effects. If not, then the classical concept of a particle could not be discarded. A new fundamental element „state“ confirms such statement. It enables us to survive in the fog of the mystery called „knowledge“.

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