Resolving the Singularity by Looking at the Dot and Demonstrating the Undecidability of the Continuum Hypothesis

Abstract

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The hypothesis that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately $113.148\times 10^{-32}$ meter/sec$^2$ and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”.

Appendices

Appendix 1: A Critical Commentary on Einstein’s Views Regarding “Physical Meaning of Geometrical Propositions”

In what follows, I shall offer some critical comments on a few relevant statements by Einstein from the first chapter of ref. Einstein (2010) in order to show how Einstein was not operational and rather relied on logic, which by nature is abstract and detached from immediate truths of experience as Einstein himself pointed out. So, what Bridgman called as “operational” was only a contextual characteristic of Einstein’s reasoning that founded his relativity theories i.e. while Einstein explained the concepts verbally he was operational because his explanations were directly attached to experienced facts, but while he did mathematical analysis he relied on the axioms of geometry. I analyze Einstein’s statements part by part as follows.

“Geometry sets out from certain conceptions such as “plane,” “point,” and “straight line,” with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as “true.” ”

It is manifest from Einstein’s words that geometric conceptions are associated with “more or less definite” ideas i.e. geometric conceptions are approximate and not exact or definite. Furthermore, such ideas are assumed to be universal truths and that is why we need to be “inclined to accept” such truths in order to proceed with the axiomatic framework of geometry. Such inclinations or biases are the necessities for working with such axiomatic framework irrespective of whether such assumed truths defy experience or not. Indeed Einstein pointed this out shortly.

“The question of the “truth” of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. ”

Here, Einstein pointed out that the “truth” of any axiom of geometry can not be verified by remaining within the axiomatic framework of geometry. This certainly means that if any one axiom of geometry is considered as “false”, then the basis of such consideration can not be geometry itself. The experience of seeing a dot, which is an act of being operational, is thus necessary to be taken into consideration if the limited validity of the axioms of geometry is to be explored. And this is reflected from Einstein’s own admission that I point out next.

“The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.”

This statement by Einstein clearly justifies in what sense he was not operational and it was because of his reliance on the axioms of geometry to write down theories. As Einstein emphasized, the word “true” is only meaningful while we relate our ideas to some “real” objects of experience. So, considering Bridgman’s notion of being “operational”, the axioms of geometry are not operational (or anti-operational) as those axioms are statements which do not explain our experience. Further, considering Einstein’s explication of the word “true”, a statement that expresses the visible dot of the pencil can be only considered to be “true” by virtue of our experience of vision of the dot. Therefore, I may conclude that although Einstein was operational while explaining concepts based on experience of observed phenomena, but he was not operational due to his use of the axioms of geometry to write down his theories.

Appendix 2: More Clarifications on the Statement Regarding “A Point” and “A Line”

Here I clarify certain issues with some expected objections in mind those may be raised by the logic minded reader. I enlist such expected objections and the possible refutations as follows:

• Is my statement a definition of “a point” or “a line”? I should say “it is neither” and “it is both” – the answer depends on the way the situation is analyzed. It is neither because I have not considered the term “definition” at all (except on a different occasion in the sense of symbolic abbreviation—see footnote(9)). Rather, I have adopted a practical method of demonstrating facts in simple language which depends of the mindset of the reader to be deemed as acceptable. If I consider the term “definition” as “explication”, then I have explicated both “a point” and “a line” in relation to each other. This is because I can not realize any of those concepts in isolation, but only in relation to each other. When I put a dot on the paper, it seems of negligible extension only if I have the experience of drawing a line with respect to which I write the dot’s extension to be “negligible”. Otherwise, the word “negligible” by itself is incomplete in the sense that one can always question “negligible with respect to what?”

• I can draw a line only after putting a dot. So, is my statement circular in reasoning? The answer is “yes” and “no” depending on the way one chooses to analyze the situation. The statement is circular if “a dot” (“a point”) and “a line” are considered as isolated and logical truths, as it is done in accord with the axioms of geometry. However, in that process, the truth of the experience is denied – an anti-operational process of reasoning. The statement is not circular if the experience of putting a dot and drawing a line is truthfully analyzed. Certainly the dot needs to be put first and then the line can be drawn while I draw with a pencil. However, the negligible extension of a dot can be realized only after drawing, and in relation to, a line. So, neither a dot nor a line has in itself the truth of experience in isolation. Rather, both can be realized in relation to the other. Therefore, my statement is just an expression of demonstration based on experience. I do not want to categorize only parts of the statement into logical and hence, isolated absolute truths that only leads to logical paradoxes like this objection of circular reasoning.

So, instead of putting logical analysis into the context, I consider my investigation as a series of reasonable expressions that explicate my experience. Nevertheless, I must admit that such operational process of reasoning brings in the danger of consideration of an infinite process of reasoning because one can raise a legitimate doubt regarding whether I can ever stop expressing my experience e.g. colour of the paper, colour of the pencil, the pressure I apply to put the dot and so on. So, I declare the incompleteness of my expression and my choice to cut off the reasoning where I find it suitable and sufficient for the present context.

Even after all these explanations, I expect the modern scientist, who puts his complete trust in logic and hence, his belief in complete definitions, to be unwilling to believe in the process of reasoning that I have adopted here. To germinate the seed of doubt in such a mind, I have analyzed Riemann’s definition of the concept of “line” in Appendix 4.

Appendix 3: Born’s Wish of Uncertain Numbers, Klein’s Vision of “A Small Spot” and “Fusion of Arithmetic and Geometry”: an Unfulfilled Desire

The visual experience of the dot and the indispensability of the cuts, render the explication of numbers to be inexact. Such a demonstration process is always accompanied with errors due to the mode of demonstration itself and without such demonstration the ideas remain abstract and can not be applied for practical purpose, unless it is lied about. Apart from Descartes’ doubt regarding the exactness of numbers being explicated in such a way (see footnote (10)), I find it interesting and worthy to note Born’s and Klein’s views regarding such issues so that the importance of the present work is manifested in a more convincing manner.

Born wrote the following on page no. 81 of ref. Born (1968):

“Of course, I do not intend to banish from physics the idea of a real number. It is indispensable for the application of analysis. What I mean is that a physical situation must be described by means of real numbers in such a way that the natural uncertainty in all observations is taken into account.”

I believe, Born would have agreed that demonstration of the concept of number by drawing a line and making cuts, which nevertheless begins with putting a visible dot on the paper, is both a physical act and a physical observation as well. Thus, the uncertainty ($\epsilon , \delta$ etc.), that is incorporated in my demonstration, is what Born tried to indicate (my guess). Assuming Born’s agreement, I may consider that the demonstration procedure that I have adopted and the way I have analyzed the situation, can be considered as quite believable by the modern authorities of science. However, I may go further to bring to light some more convincing historical visions along such lines of thought, which Born himself provided in the statements which followed the above quoted ones. Born’s comment concerning Klein’s vision, that followed the above quoted statement only justifies the essence of the present discussion in an even more explicit and convincing way:

“Felix Klein called for a similar step to be taken in geometry. Besides abstract, exact geometry, he desired to have a practical geometry, in which a point is replaced by a small spot, straight lines by narrow strips,etc. However, nothing much resulted from this.”

Although Born did not analyze further why Klein could not accomplish what he wanted to, it appears to me that once “a point” is replaced by “a small spot”, then the question arises that “small with respect to what”. An obvious, and the simplest (according to me), answer is that “with respect to any other extension that one may call a line” i.e. the relational aspect of the scenario (i.e. the visual cognition) needs to be necessarily taken into account and the involvement of units become a requirement because neither the dot nor the line, by itself, explicates the concept of number that is represented by numeral. Then the scenario can not be categorized as either “arithmetic” or “geometry”, but somewhere in the middle—a practical method of demonstration and the associated analysis, which Klein himself might have called “fusion of arithmetic and geometry” and not any one in isolation—see page no. 2 of ref. Klein (2004). However, any act of demonstration of this “fusion”, which being a “practical method”, must rely on a mode of demonstration. This mode of demonstration is certainly a physical act that leads to the “physical situation” (in Born’s words) involving the extension of the dot that is negligible with respect to the extension of a line. Certainly Klein highlighted the issue of inexactness associated with such an explication of the scenario. However, what was missing in Klein’s analysis was this understanding of the relational nature of the underlying situation that Born also could not get the grasp of.

Nevertheless, it is worth clarifying certain aspects of this work in tandem with some of the statements of Klein from pp. 42-43 of ref. Klein et al. (1893):

“If we now ask how we can account for this distinction between the naive and refined intuition, I must say that, in my opinion, the root of the matter lies in the fact that the naive intuition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical development from axioms considered as perfectly exact.To explain the meaning of the first half of this statement it is my opinion that, in our naive intuition, when thinking of a point we do not picture to our mind an abstract mathematical point, but substitute something concrete for it. In imagining a line, we do not picture to ourselves “length without breadth”, but a strip of a certain width.”

These are the statements, from which I have quoted the appropriate words while demonstrating the impossibility of a complete refinement of the act of counting dots in Section(3) and therefore, the importance of the condition $(N^{s_i}_d+2)\delta ^{s_i}_Is_i<s_i$ which I imposed. Added to these statements, it is important to note what Klein wrote next regarding the term “definition” while it comes to the words “point”, “line”, etc.

“Now such a strip has of course always a tangent....; i.e. we can always imagine a straight strip having a small portion (element) in common with the curved strip; similarly with respect to the osculating circle. The definitions in this case are regarded as holding only approximately, or as far as may be necessary. The “exact” mathematicians will of course say that such definitions are not definitions at all. But I maintain that in ordinary life we actually operate with such inexact definitions. Thus we speak without hesitancy of the direction and curvature of a river or a road, although the“line” in this case has certainly considerable width.”

Certainly the term “definition” is logical and has an inherent appeal of being complete or exact or perfect. Then, to be logical a complete refinement of intuition is necessary and then, in the process, directly experienced truth needs to be ignored in the case of the dot or the line. This is why, considering such a view in accord with Klein, I have used terms like “explication”, “demonstration”, etc. instead of the term “definition” in the context “point”, “line”, etc. In agreement with Klein’s view, Kant would prefer the word “exposition” instead of the word “definition”:

“Instead of the term, definition, I prefer to use the term, exposition, as being a more guarded term, which the critic can accept as being up to a certain point valid, though still entertaining doubts as to the completeness of the analysis.”

– see page no. 144, Vol. 1 of ref. Ewald (2007). For a very simple demonstration of such a logical dilemma while considering the term “definition”, one may consult Section (2.1.2) of ref. Majhi (2021a) and the relevant discussion regarding Frege’s demonstration of definition of a concept analyzed in Section (2.1) of ref. Majhi (2021b).

Appendix 4: Analyzing Riemann’s Expressions Regarding “The Hypotheses on Which Geometry is Based” from the Logician’s Perspective

In what follows I analyze some of the statements of Riemann in ref. Riemann (2004) so as to bring forth the subtleties of reasoning that gets associated if each of his verbal statements are carefully analyzed. The two issues that I intend to raise concern about are the following: (i) Riemann’s definition of “a line” is circular in reasoning (ii) Riemann’s statements regarding “infinitesimal” are incomplete.

1.1 Circular Reasoning in Riemann’s Definition of the Concept of “Line”

Riemann wrote on pp 261-262 of ref. Riemann (2004):

“Measurement requires that the measure of the entities being measured must be independent of their location, and this can be the case in more than one way.”

If I suppose, a length L being measured in terms of some length unit $\lambda _0$, then the number $n^{\lambda _0}_L$ that is yielded must be independent of the location where the measurement process takes place. I believe this is what Riemann meant in the above statement. Einstein assumed that such measurement process is exact and does not contain any error as he wrote on p 4 of ref. Einstein (2010) “that there is nothing left over, i.e. that the measurement gives a whole number.”. Therefore, $n^{\lambda _0}_L$, which represents the number of times $\lambda _0$ can be superposed on L, can only take values $1, 2, 3, \cdots$. Now, Riemann went on to write,

“The assumption which first suggests itself, and which I intend to pursue here, is that the length of lines is independent of their position, so that every line can be measured by comparing it with any other line.”

The word “location” is now replaced by “position”. Then Riemann wrote the following.

“If the determination of the position of a point in a given n-dimensional manifold is reduced to the determination of n variables $x_1, x_2, x_3, \cdots , x_n$, then a line may be defined by the statement that the quantities x are given functions of a single variable.”

Here, I raise the following question: How is the position of a point determined? Since Riemann did not explain further the word “determination”, I consider the following explanation.

• I choose some origin O and draw a line by joining O and the point P (say). The line OP is the distance d of the point P from O.

• I choose another line $\lambda _0$ such that it can be superposed multiple times on d to generate a number $n^{\lambda _0}_d$ (say). Then, I write $d=n^{\lambda _0}_d\lambda _0$.

The above steps explicate the meaning of “determination” of the position of a point. Then the quantities x (which, I believe, should be written as $x_i: i=1,2,3,\cdots , n$) should be written in terms of $\lambda _0$. Therefore, this act of “determination” is itself a measurement process. Then, according to Riemann’s own statement, the yielded number $n^{\lambda _0}_d$ should be independent of the “location” of the lines that represent the distance d and the length unit $\lambda _0$. The problem is now to give meaning to the word “location/position”. This is because the distance d itself is now a line that is getting measured for the determination process to be carried out. Then, one needs to choose another origin, say $O'$, with respect to which the “location/position” of d and $\lambda _0$ needs to be “determined” so that the previous determination holds any meaning according to Riemann. However, the new “determination” is again a measurement process and to give meaning to it, another origin, say $O^{''}$, needs to be chosen. The process goes on. Hence, the term “determination” can not be completely defined and can only be explicated with partial satisfaction as there is always a doubt retained in the process of reasoning owing to such self-inquiry (Majhi, 2021a).

Further, it must be clear from the above explanation that in order to carry out the “determination” process, the lines need to be drawn i.e. the concept of “line” needs to be used. However, instead of analyzing his own words in such a way, Riemann “defined” what a “line” is, without explication of the term “determination” for which he needed to use the concept of a “line” in the first place. Since geometry is a system of logical truths or axioms then, viewing from the logician’s perspective, such definition of “line” appears to be based on circular reasoning i.e. Riemann’s definition of “line” is not a logical definition. However, if one ignores the logical rigor of such a logical system of thoughts then Riemann’s analysis is definitely useful in an operational way because we do general relativity based on such concepts. Therefore, the foundations of Riemannian geometry, which forms the basis of general relativity, are both logical and not logical in the same process of reasoning – logical because such a definition of “line” is accepted to be true and considered as axiom; illogical because such a definition is circular in reasoning and hence, can not be considered as logical by the logician.

1.2 Riemann’s Incomplete Statement: “Infinitesimal” with Respect to What?

Riemann continued to write the following:

“The problem then is to find a mathematical expression for the length of a line, and for this purpose we need to consider the quantities x as expressible in terms of units.”

As I have explained earlier while explicating the term “determination”, Riemann did acknowledge that the quantities x ($x_i$) should be expressed in terms of units. However, Riemann’s verbal statements are not truthfully translated into his equations because the units are not written explicitly. This is important to note because of what Riemann wrote next.

“I shall handle this problem only under certain restrictions, and confine myself in the first place to lines in which the relations between the quantities dx - the associated variations of the variables x - vary in continuous fashion. We can then visualize the line as being divided up into elements, within which the ratios of the increments dx can be regarded as constant, and the problem reduces to finding a general expression for line element starting from a given point, which will involve the variables x as well as the variables dx.”

Here, there is no clarification regarding whether the elements of the line are bigger than or smaller than or equal to the length unit. This is important because of what Riemann wrote next.

“Secondly, I shall assume that the length of the line element, disregarding quantities of the second order of magnitude, remains unchanged if all its points undergo the same infinitesimal displacement.”

I believe that by the word “infinitesimal” Riemann meant “infinitesimally small”. If not, then the word “infinitesimal” needs to be clarified in a more elementary fashion. If yes, then an obvious question arises about the displacement and that is, infinitesimally small with respect to what? Is it with respect to the length unit or the quantity x? If it is with respect to the length unit, then how does the theory look like when smaller length units are chosen because in that case the displacement does not remain “infinitesimal” anymore? If it is with respect to the quantity x, then the theory should be written in such a way that the quantity x must have, as Born would write (see Appendix 3), a “natural uncertainty” much greater than dx irrespective of the role of the length unit. Neither do I find any answer to such basic questions nor do I find clarifications regarding such basic doubts anywhere in ref. Riemann (2004). Therefore, I find Riemann’s use of the word “infinitesimal” to have only an incomplete sense.

Appendix 5: A Glimpse of Non-standard Physics

In this work, I have used the concept of “mass” as an independent physical dimension so as to draw the connection with standard physics literature by writing $s_i=Gm_i/c^2$. However, the relevant equations can be written in terms of $s_i$, in relation to $\lambda _0$, only and this will result in the appearance extremely small numbers in the associated analysis which is quite akin to what one encounters in Non-Standard Analysis (NSA) (Robinson, 1966)¹⁸. Goedel asserted that “there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.” (see the Preface of ref. Robinson (1966)). Certainly, what I have discussed in this work has a priori nothing to do with NSA that is founded on mathematical logic and does not involve physical dimensions. However, it would not be a criminal offense to consider it as an “other version” of NSA. I may call this Non-Standard Physics (NSP) so as to distinguish it from standard physics literature and from NSA due to the distinctions that I mentioned before. While a detailed discussion regarding NSP is beyond the scope of this article, however, some immediate results from NSP can be showcased so as to convince the reader that it is a clear possibility. In NSP, instead of being considered as an independent physical dimension, the concept of “point mass” founded on which is classical mechanics of standard physics, is expressed as an intrinsic length of an object, considered as a whole, that is extremely small compared to the chosen conventional length unit. So, the unit of intrinsic length, if called “kilogram” and abbreviated as “kg”, then unlike standard physics, now we have “kg $\lll$ meter”, where “meter” is the chosen unit of length. Any other such intrinsic lengths are some multiples of “kg”. I provide below a comparison between the standard physics and NSP, in light of the choice of units that we make in the beginning of classical mechanics while studying the laws of motion, followed by a simple problem to explicate the situation.

1.1 Choice of Units

Standard physics

Non-standard physics

$\begin{array}{l} \hbox {Force on an object, considered as a whole and represented as}\\ \hbox {a ``point mass'', is written as}\, F\propto ma. \,\hbox {Therefore,}\\ \qquad =kma,\\ \hbox {where}\, k\, \hbox {is a proportionality constant with appropriate physical}\\ \hbox {dimension. For}\, m=1\hbox {kg,}\, a=1\hbox {meter/sec}^2,\, \hbox {we have}\\ \qquad F=k\text {kg.meter}/ \text {sec}^2. \\ \hbox {We choose} k \hbox {such that} \,1 \,\hbox {unit of mass, having} \,1\, \hbox {unit of acceleration}\\ \hbox {is equivalent to} \,1 \,\hbox {unit of force. So, we choose} \,k=1 \,\hbox {to}\\ \hbox {write}\\ \qquad 1 \text {N}= 1 \text {kg.meter/sec}^2,\\ \hbox {where}\, N \,\hbox {stands for `Newton', the unit of force. This is a}\\ \hbox {convention to write the unit of force that is derived from}\\ \hbox {the units of mass and acceleration.}\\ \end{array}$

$\begin{array}{l} \hbox {Force on an object, considered as a whole and represented as}\\ \hbox {``an intrinsic length that is extremely small compared to the}\\ \hbox {chosen conventional unit of length'', is written as}\, F\propto sa.\\ \hbox {Therefore,}\\ \qquad F=ksa,\\ \hbox {where}\, k\, \hbox {is a proportionality constant with appropriate physical}\\ \hbox {dimension. Chosen conventional unit of length is called}\\ \hbox {``meter''. So, we write}\, s\lll \,\hbox {meter. Unit of}\,s\, (\hbox {not} m) \hbox {is called} \\ \hbox {``kilogram'', abbreviated as ``kg'' such that kg}\, \lll \, \hbox {meter. Then,} \\ (\hbox {instead of}\, ``m=1\,\hbox {kg''}) \hbox {we write}\, ``s=1\,\hbox {kg}\,=1.\epsilon ^{meter}_{kg}\,\hbox {meter}\\ =\epsilon ^{meter}_{kg}\,\hbox {meter}\, : 0<\epsilon ^{meter}_{kg}\lll 1''.\, \hbox {So, for} \, s=1\, \hbox {kg}\, =\epsilon ^{meter}_{kg}\, \\ \hbox {meter},\, a=1 \hbox {meter/sec}^2, \hbox {we have}\\ F =k (1\text {kg}).(1\text {meter/sec}^2): 0<\epsilon ^{meter}_{kg}\lll 1\\ \quad = k\epsilon ^{meter}_{kg} \text {meter}^2/\text {sec}^2.\\ \hbox {We choose}\, k\, \hbox {such that}\, 1\, u\hbox {nit of intrinsic length (extremely}\\ \hbox {small compared to conventional length unit), having}\, 1\, \hbox {unit} \\ \hbox {of acceleration is equivalent to}\, 1 \, \hbox {unit of force. So, we choose}\\ k=1 \,\hbox {to write}\\ 1\text { N} = \epsilon ^{meter}_{kg}\text { meter}^2/\text {sec}^2,\\ \hbox {where}\, N \,\hbox {stands for `Newton', the unit of force.}\\ \end{array}$

1.2 A Simple Problem

A force of 5 N gives a mass $m_1$, an acceleration of 10 meter/sec$^2$ and a mass $m_2$ an acceleration of 20 meter/sec$^2$. What acceleration would it give if both the masses were tied together?

Solution in standard physics

Solution in non-standard physics

$\begin{array}{l} \hbox {We note that N}\, =\, \hbox {kg.meter/sec}^2.\, \hbox {Now, we solve the}\\ \hbox {problem as follows.}\\ \quad 5\text { N} = m_1. 10 \text {meter/sec}^2\quad \Leftrightarrow \quad m_1=\frac{1}{2} \text { kg}\\ \quad 5\text { N} = m_2. 20 \text {meter/sec}^2\quad \Leftrightarrow \quad m_2=\frac{1}{4} \text {kg}\\ \hbox {If} a \hbox {is the acceleration of the joint mass, then we can}\\ \hbox {write}\\ \quad 5 \text {N} = (m_1+m_2) a = \left( \frac{1}{2}+\frac{1}{4}\right) \text {kg}. a \\ \qquad \Leftrightarrow a=\frac{20}{3} \text {meter/sec}^2.\\ \end{array}$

$\begin{array}{l} \hbox {First we replace}\, m_1, m_2\, \hbox {by}\, s_1, s_2 \hbox {and also we note that} \\ \hbox {N}=\epsilon ^{meter}_{kg}\, \hbox {meter}^2/\hbox {sec}^2~:0<\epsilon ^{meter}_{kg} \lll 1. \hbox {Now, we solve the} \\ \hbox {problem as follows.}\\ \quad 5 \text {N}= s_1.10 \text { meter}/\text {sec}^{2}\Leftrightarrow s_1=\frac{1}{2}\epsilon ^{meter}_{kg} \text { meter},\\ \quad 5 \text {N}= s_2.20 \text { meter}/\text {sec}^{2}\Leftrightarrow s_2=\frac{1}{4}\epsilon ^{meter}_{kg} \text { meter}.\\ \hbox {If}\, a\, \hbox {is the acceleration of the tied collection of bodies,} \\ \hbox {then we can write}\\ \quad 5 \text {N} = (s_1+s_2) a = \left( \frac{1}{2}+\frac{1}{4}\right) \text {kg}. a\\ \qquad \Leftrightarrow a=\frac{20}{3} \text {meter/sec}^2. \end{array}$

In view of this, I may write that it now becomes just a matter of further effort to understand how I can write down the known “laws” of standard physics as only approximate truths from NSP.

Appendix 6: Further Motivations to Write $s_i=Gm_i/c^2$

Here are some reasons that I find compelling to motivate the relation $s_i=Gm_i/c^2$.

1.1 Gravitational Field, Potential, Back Reaction of Test Mass

Let me focus on how we define gravitational field g, due to a source mass $m_2$, by considering Newton’s law of gravitation, an experimentally verified hypothesis, as the premise. The reasoning broadly consists of the following steps.

• 1. Step 1 According to Newton’s law of gravitation, the gravitational force of interaction between a source mass $m_2$ and a test mass $m_1$ is given by

  $$\begin{aligned} F=\frac{Gm_1m_2}{r^2}. \end{aligned}$$

  (39)

  We write the acceleration of the test mass $m_1$ as

  $$\begin{aligned} \frac{F}{m_1}=\frac{Gm_2}{r^2}. \end{aligned}$$

  (40)

  Here, I have disregarded the distinction between inertial mass and gravitational mass.

• 2. Step 2 We argue that the gravitational back reaction of the test mass $m_1$ must be negligible so that the acceleration $F/m_1$ can be treated as the gravitational field due to the source mass $m_2$, symbolized as “g”. Thus, g is defined as follows:

  $$\begin{aligned} g:=\lim _{m_1\rightarrow 0}\frac{F}{m_1}=\frac{GM}{r^2}. \end{aligned}$$

  (41)

  That is, the condition “$\lim _{m_1\rightarrow 0}$” is considered as the computational step corresponding to the argument concerning the gravitational back reaction of the test mass m.

• 3. Step 3 Then, we generally introduce the gravitational potential $\Phi =Gm_2/r$ by writing

  $$\begin{aligned} g=-\frac{d\Phi }{dr}~~:~~\frac{d\Phi (r)}{dr}:=\lim _{h\rightarrow 0}\frac{\Phi (r+h)-\Phi (r)}{h}, \end{aligned}$$

  (42)

  where h is some length that represents an infinitely small change of r. Validity of this definition is verified by the following steps of calculation.

  $$\begin{aligned} g=\, & {} -\frac{d\Phi }{dr}=\,-\lim _{h\rightarrow 0}\frac{\Phi (r+h)-\Phi (r)}{h}\qquad \\=\, & {} -GM\lim _{h\rightarrow 0}\frac{1}{h}\left[ \frac{1}{(r+h)}-\frac{1}{r}\right] \qquad [\text {using}\, \Phi =\,GM/r]\\=\, & {} GM\lim _{h\rightarrow 0}\frac{1}{h}\left[ \frac{h}{r^2+rh}\right] \\=\, & {} GM\lim _{h\rightarrow 0}\left[ \frac{1}{r^2+rh}\right] \qquad [h\, \text { cancels out as}\, h\ne 0]\\=\, & {} GM\left[ \frac{1}{r^2+r.0}\right] \qquad [h=0 \text { is used to calculate the limit}]\\=\, & {} \frac{GM}{r^2}. \end{aligned}$$

Ignoring the objections regarding statements like “$m_1\rightarrow 0$”, “$h\rightarrow 0$” that I have discussed in the very beginning of this article and in refs. Majhi (2021b, 2022), I make the following observations.

In Step 1, $F/m_1$ is equal to $Gm_2/r^2$. Therefore, “$\lim _{m_1\rightarrow 0}$” should apply to both $F/m_1$ and $Gm_2/r^2$ in Step 2. However, it does not make sense to write “$\lim _{m_1\rightarrow 0}Gm_2/r^2$” because $Gm_2/r^2$ contains no $m_1$-dependent term. In case we would have had an expression like

$$\begin{aligned} \frac{F}{m_1}=\frac{Gm_2}{r^2}+m_1-{\text{dependent terms that vanish as}}\, m\_1\rightarrow 0, \end{aligned}$$

then only we could have written

$$\begin{aligned} \lim _{m_1\rightarrow 0}\frac{F}{m\_1}= & {} \lim _{m_1\rightarrow 0}\left[ \frac{Gm_2}{r^2}+m_1-{\text{dependent terms that vanish as}}\, m\_1\rightarrow 0\right] =\frac{Gm_2}{r^2}.~ \end{aligned}$$

(43)

Considering this, together with the arguments that has been discussed in section(2.2), it is suggestive of the fact that the hypothesis should be of such a form that the following is true:

$$\begin{aligned} \frac{F}{m_1}=\frac{Gm_2}{r^2}+ \text {sub-leading terms, depending} \text {on}\, m_1 \text {and}\, c, \text {such that they vanish as}\, m\_1\rightarrow 0. \end{aligned}$$

So, there are clear indications of missing terms in the hypothesis concerning gravitational two body interaction. The question remains how to look for those terms and here goes the clue.

Using expressions (41) and (42), I may write

$$\begin{aligned} \lim _{m_1\rightarrow 0}\frac{F}{m_1}=-\lim _{h\rightarrow 0}\frac{\Phi (r+h)-\Phi (r)}{h}\quad : F=Gm_2m_1/r^2, \Phi =Gm_2/r. \end{aligned}$$

(44)

It does not make sense that the limiting condition on the left hand side concerns mass, and on the right hand side, concerns length. However, it can be given sense if mass is related to length by some means. A possibility is to write $s_i=Gm_i/c^2~\forall i\in [1,2]$. So, $s_1=Gm_1/c^2$. It is evident that $s_1\rightarrow 0$ as $m_1\rightarrow 0$ and $c\rightarrow \infty$. This provides the ground for speculation that $s_1$ may be playing the role of h and is involved in the sub-leading terms of (43) in positive powers.

1.2 Some Specific Questions Regarding Schwarzschild Metric

In ref. Riemann (2004) Riemann did not specify that the line element, the displacement, etc. are infinitesimal or infinitely small with respect to which length (see section (D.2)). Such line of inquiry indeed justifies the consideration of the relation $s_i=Gm_i/c^2$ in the following way. One of the most widely used “infinitesimal line element” in the literature of general relativity, as also pointed out in the very beginning of ref. The Event Horizon Telescope Collaboration (2019), is the Schwarzschild metric (Schwarzschild, 1916):

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

So, I inquire, ds is infinitesimal or infinitesimally small with respect to which of the following: (i) $s_M=GM/c^2$ where M is the source mass (ii) r, the coordinate distance (iii) the unmentioned unit of length (iv) $s_m=Gm/c^2$ where m is the test mass that plays no role otherwise.