Unnatural Causes
by Jerry Fausz, Ph.D.
[EDITOR’S NOTE: A.P. auxillary staff scientist Dr. Fausz holds a Ph.D. in Aerospace Engineering from Georgia Tech.]
The most fundamental axiom of science is Causality: the belief that every material effect observed must have a sufficient cause preceding or simultaneous with it (Miller, 2011). The observed motion of a cart is difficult to explain, for example, unless we assume the existence of an appropriately placed horse (figuratively speaking, of course). Scientific philosopher Sir Karl Popper wrote that the “rule” of causality “guides the scientific investigator in his work” (1968, p. 61). Actually, Popper refused to accept causality as a scientific “principle,” per se, but instead stated as a “methodological rule” that we should never “give up our attempts to explain causally any kind of event we can describe” (p. 61). I will not be so picky, as I believe that causality has been so thoroughly and consistently demonstrated by observation to be readily considered axiomatic, i.e., accepted without proof. Popper clearly indicates by his comments his belief that causality is heavily embedded in scientific thought and method. Nobel Laureate Erwin Schrödinger, upon defining causality, commented:
This postulate is sometimes called the “principle of causality.” Our belief in it has been steadily confirmed again and again by the progressive discovery of causes that specially condition each event (1957, p. 135).
The thoughts expressed by Popper and Schrödinger certainly support the idea that without an assumed “cause,” the scientific search would be quite aimless, if, indeed, possible at all.
The scientific importance of causality can be further illustrated with an amusing story related by noted theoretical physicist Stephen Hawking:
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!” (1988, p. 1).
Hawking’s anecdote invites several immediate observations regarding causality:
Causality is fundamental to scientific reasoning. Note the scientist’s question in response to the lady’s statement. Why assume the tortoise is standing on anything, if not for the sake of causality?
The principle requires effects to be natural or material (thus, observable to us). This is a somewhat trivial inference, for should an effect be unobservable in nature, then we would certainly not be concerned with knowing its cause. The observation implied in the story is nothing less than the existence of the Earth and its perceived place in the Universe.
The statement of causality does not of necessity require a natural or material cause. Causes need only be sufficient and antecedent to/simultaneous with the effect to satisfy causality. For example, as a direct consequence of causality, the lady’s hypothesis in the story (a giant tortoise holding up the Earth) resulted, according to her explanation, in an infinite tower of turtles. While this conclusion is not natural by any means (and, arguably, not logical), it is admissible from the standpoint of causality. (For further interesting discussion of the “tower of turtles” analogy, see Davies, 1992, pp. 223-226).
The third statement, while last, is certainly not least important. Note that even miraculous events recorded in the Bible produced effects that were observable or measurable (e.g., water becoming grape juice, the Red Sea visibly parting, people who were observably dead becoming alive again, etc.). Otherwise, miraculous causes would probably be of little interest to us.
It turns out, though, that the admissibility of non-natural causes is the only way to avoid a serious dilemma in the causality assumption. Consider the following: if a cause is, or is assumed to be, material (observable) then, like its effect, it is also contingent—the cause itself must be the effect of yet another cause according to the scientific axiom of Causality. Consequently, a predetermination to assume only material causes will, necessarily, lead to an infinite sequence of them. For example, strict material causality dictates that the existence of life at its present state of complexity may only be explained by some form of evolution (special creation, while an admissible cause, is not material). In order for life to evolve to its present state of complexity, life had to develop from non-living matter; non-living matter had to somehow organize in a very specific way to provide the necessary constituents for life to generate (assuming, of course, this is even possible); in order to so organize, this matter had to have existed under certain special conditions; and so on.
The dilemma in this reasoning is clearly illustrated by the turtle analogy. If one assumes that the Earth rests on the back of a giant turtle, then the turtle needs to be supported by something. And, since the lady in the story was inclined to make the turtle assumption to begin with, why not just assume that it is standing on the back of another turtle, which is exactly what she did. But, what is the second turtle standing on? As long as one has predetermined to accept the turtle hypothesis, like material causality, the turtles will continue to pile up, like material causes, until we have an infinite number of them. To avoid this dilemma, an assumption must be made at some point in the chain that does not involve a turtle, or by analogy, does not conform to strict materialism.
A related principle has been demonstrated within the most logically consistent subject of study known to mankind—mathematics. In 1931, Kurt Gödel, an Austrian mathematician, proved a theorem holding that “for any consistent mathematical system there exists within the system a well formed statement that is not provable under the rules of the system” (Overman, 1997, p. 27). Commonly known as Gödel’s Incompleteness Theorem, this result simply implies that to make progress within a mathematical system, certain facts need to be accepted axiomatically (that is, they cannot be proven from other facts in the system).
This principal also holds in other fields of reasoning. In fact, Paul Davies makes use of this idea when he proclaims:
I would rather not believe in supernatural events personally. Although I obviously can’t prove that they never happen, I see no reason to suppose that they do. My inclination is to assume that the laws of nature are obeyed at all times (1992, p. 15).
Here Davies assumes something to be true that, he admits, cannot be proven within his system of reasoning: the assumption that supernatural events do not occur. A tacit assumption in Davies’ statement, however, is the existence of time. While the natural laws may hold “at all times” (and supernatural events may not occur), we have already seen that assuming this to be true back to the beginning of time will lead to the equivalent of an infinite tower of turtles (an infinite causal sequence).
Speaking of the beginning of time, the most widely accepted theory for this event is the so-called “Big Bang.” This was hypothesized subsequent to the discovery that the Universe is expanding (Hawking, 1988, p. 38). Specifically, what astronomers discovered was that the light spectra of most of the stars in other galaxies were “red-shifted” (reducing in frequency), by which they assumed this indicates that those stars were moving away from us with increasing velocity. If the assumed expansion is extrapolated backwards in time, then one might suppose that there is a point in time at which all of the matter in the Universe was collocated (existed at the same point). This, of course, requires the highly non-trivial assumption that time actually extrapolates back to that point. Note that the expansion assumption does not necessarily follow since one might hypothesize other possible causes for the red-shift of the stars, such as nonlinear changes in the “elasticity” of space-time itself, perhaps as if the Universe was “stretched out” at its beginning (cf. “stretched out” [Isaiah 42:5; Isaiah 45:12; Jeremiah 10:12], or “spread out” [Job 9:8; Isaiah 40:22]).
For now, though, consider what science thinks it knows about this assumed beginning point, called the Big Bang singularity. The term singularity in this context denotes a point at which some fundamental property ceases to exist or certain processes are undefined. For example, in mathematics dividing by zero creates a singularity because division by zero is undefined. To state this more precisely mathematically, the mathematical operation of division is undefined at zero, so the origin (zero) is called a singularity point with respect to the operation of division. In the case of the Big Bang singularity, the assumed reverse time extrapolation creates a point at which all of the matter in the Universe is collocated, or has “zero size” (Hawking, p. 117). This would necessarily imply infinite mass density which, according to general relativity, implies infinite curvature of space-time (Einstein, 1920). Physicists Stephen Hawking and Roger Penrose have studied the properties of singularities predicted by general relativity quite extensively. Hawking observed that at such a point, “the laws of science and our ability to predict the future would break down” (p. 88). Since the laws of science as we understand them are undefined at this point, the point is called a singularity or, in this case, the Big Bang singularity.
So it seems, perhaps not unsurprisingly, that if the chain of causality could be followed back to a beginning of time as defined by the Big Bang, it would take us to a point in which natural laws do not apply. Hawking goes on to say: “[T]his means that one might as well cut the big bang, and any events before it, out of the theory, because they can have no effect on what we observe” (p. 122). However, cutting the singularity out of the theory just serves to bog the theory down once again in the dilemma of material causality. When the ultimate conclusion of causality turns out to be an unnatural cause, and we cut it out of our theory simply because we desire to stick with material causality, then our reasoning can only lead us, again, to an infinite tower of turtles. Note that Davies points out that it doesn’t make sense to talk about “before” with reference to the Big Bang singularity since time presumably had its beginning there (p. 50).
All of our imagination and reasoning, as well as our experience and observation, point to a necessary break-down in strict material causality. Theories of an infinite Universe, mathematical incompleteness, and space-time singularities testify to a physical reality that cannot be completely deduced by the rules of the system. The logical conclusion is a cosmological model that admits unnatural cause.
Furthermore, if we are forced to assume an unnatural cause for the beginning of time, as the evidence suggests that, indeed, we are, then why should we necessarily presume, specifically, that in order to pinpoint the beginning of time, we should extrapolate back to a point in which all the mass in the Universe is collocated? As we have seen, Hawking suggested discarding the Big Bang singularity because its effects cannot be predicted by science. However, Hawking also points out that the singularity theory implies that space-time had a beginning and a boundary, prompting him to ask “What were the ‘boundary conditions’ at the beginning of time?” He then comments: “One possible answer is that God chose the initial configuration of the universe” (p. 122).
Such boundary conditions as these could specify any of an infinite number of initial configurations and states for space-time and the matter that it contains. Indeed, boundary conditions are, by definition, these types of specifications. I see no particular reason to assume that these boundary conditions start with a point of infinite mass density and equally infinite space-time curvature, other than a predetermined desire to push material causality to its unnatural limit. These boundary conditions may, with equal probability, specify a Universe at the beginning of time that is not much different than what we now observe, thus implying that time has not progressed as far as some have conjectured. Perhaps that is why Hawking, using a mathematical contrivance he calls “imaginary time,” has more recently endeavored to create a consistent model of space-time that is finite but does not have a boundary, therefore would presumably not require any boundary conditions. Hawking, however, points out that this idea is “just a proposal: it cannot be deduced from some other principle” (p. 136).
The biblical Old Testament records that God said to Job: “Where were you when I laid the foundations of the Earth? Tell me, if you have understanding” (Job 38:4). Job understood well the rhetorical nature of God’s question, for he had already proclaimed to his companions:
But where can wisdom be found? And where is the place of understanding? Man does not know its value, nor is it found in the land of the living. The deep says, “It is not in me”; and the sea says, “It is not with me.” It cannot be purchased for gold, nor can silver be weighed for its price (Job 28:12-15).
Clearly, Job observed that complete understanding does not lie in materialism or nature and, if he did not know it before, God pointed out to him that neither does the knowledge of how “the foundations of the Earth” were established. Even what understanding we do have contains immutable evidence of its own hard-coded limitations in explaining the observable Universe.
Our observations and reasoning tell us that the foundations of the Universe could not possibly have been laid through strictly material or natural causality. Theories of cosmology, physics, and even mathematics point to the necessity of a Cause that operates independent of the rules of the system. These theories, however, while they can point to the necessity of unnatural causes, are impotent when it comes to explaining those causes. The Bible closes this gap in our theories by telling us of an omnipotent and omniscient Creator who is capable of operating outside of nature to make everything we observe out of what we cannot observe. The divinely guided Hebrews writer articulated this very concept in his striking statement against strict material causality: “By faith we understand that the worlds were framed by the word of God, so that the things which are seen were not made of things which are visible” (11:3).
If we pay attention to what all of our observation and reasoning are telling us, then our mind’s eye will “see” the foundations of the Universe being laid, not by material or natural causality, but by the omnipotent Creator, God, working outside of nature to set the boundary conditions, bringing natural laws on-line to shape and direct His creation. And our reasoning will, in turn, no longer require an infinite tower of turtles.
REFERENCES
Davies, Paul (1992), The Mind of God: The Scientific Basis for a Rational World (New York: Simon & Schuster).
Einstein, Albert (1920), Relativity: The Special and the General Theory (New York: Barnes & Noble).
Hawking, Stephen (1988), A Brief History of Time: From the Big Bang to Black Holes (New York: Bantam Books).
Miller, Jeff (2011), “God and the Laws of Science: The Law of Causality,” Apologetics Press, http://www.apologeticspress.org/APContent.aspx?category=9&article=3716.
Overman, Dean (1997), A Case Against Accident and Self-Organization (Lanham, MD: Rowman & Littlefield).
Popper Karl (1968), The Logic of Scientific Discovery (New York: Harper & Row).
Schrödinger, Ernst (1957), Science Theory and Man (New York: Dover Publications).https://apologeticspress.org/APContent.aspx?category=9&article=3831
再生核研究所声明262 (2015.12.09) 宇宙回帰説 ― ゼロ除算の拓いた世界観
最近展開しているゼロ除算が、新しい世界観を示しているのは 大変興味深い。直線とは一体どうなっているだろうか．空間とはどのようになっているだろうか。これについて、現代人は、双方向にどこまでも どこまでも 続いている直線を想像するであろう。限りなく広がった平面や空間である。ところが 立体射影によって 平面全体を球面上に１対１に写せば、全平面は 球面から北極を除いた球面上に1対１にきちんと写るから、無限に広がる 全平面の全貌が捉えられる。ところが平面上には存在しない想像上の点 それはあらゆる方向に限りなく遠くに存在する無限遠点の導入によって、その点を球面の欠けた1点北極に対応させれば、無限遠点を含めた平面全体は 球面全体と1対１にきちんと対応する。
このような対応で 平面上の円や直線全体は 球面上では共に円に対応するという美しい対応になり、平面上の直線は 球面上では、北極（無限遠点）を通る円に写ると、直線と円の区別は 球面上では不要になる。また、平面上の平行線とは 無限遠点で 角度ゼロで交わっている（接している）と平面上の構造がよく見えて、無限遠点を含めての平面の全構造が 捉えられる。このように、考えると、直線とは、球面上では北極を通る円、平面上では無限遠点を通る直線となる。この構造は、直線を1方向にどこまでも, どこまでも進めば、無限遠点を 通って、逆方向から戻ってくるという、永劫回帰の思想をちょうど実現している。それは、球面上では、 円を繰り返し回ることを意味する。 その様は 何もかも すっかり良く見える。
これが、従来100年以上も続いた世界観で、関数y=x やW=zは 無限遠点に近づけば、それらの像も無限遠点に近づいていると考えるだろう。 関数y=x の値は正方向にどんどん行けば、どんどん大きくなると考えるだろう。
しかるに、ゼロ除算1/0=0は、それらの関数は無限遠点にいくらでも近づくと 無限遠点にいくらでも近づくが、無限遠点自身では、突然ゼロになっていることが 幾何学的にも確認された。上記、北極は 実は原点ゼロに一致しているという。
話しを簡単にするために、 関数y=x を考えよう。右に行けば、プラス無限に、負の方向左に行けば 負の無限に限りなく近づくは 従来通りである。ところが、ゼロ除算では いずれの方向でも上記無限遠点では 値ゼロをきちんと取っているという。ゼロ除算の数学では、どんどん、増加した先、突然、ゼロ、原点に戻っているという。また、円でも球面でも半径Rをどんどん大きくすると、当然、円の面積や球の体積はどんどん限りなく大きくなるが、半径が無限のとき、突然、それらはゼロになるという。それらの理由も数学ばかりではなく、幾何学的にも明確に見えている。
この数学的な事実は、我々の世界、宇宙がどんどん拡大して行くと突然、ゼロに帰するということを暗示させている。 ― これは 宇宙回帰説を意味しているようである。
これは、ユニバースの普遍的な現象、どんどん進んだ先が、元に突然戻る原理を示しているようである。
そもそも人生とは如何なるものか。― よくは分からないが、事実として、生まれて、どんどん物心がついて、人間として精神活動が活発化して、多くは本能原理によって生かされて、そして、突然元に戻ることを意味しているようである。このことを深く捉えられれば、世界がよりよく観え、悟りの境地に達する大きなヒントを得ることができるだろう。
ここでは ゼロ除算の帰結として、宇宙回帰説、ユニバースの回帰説を唱えたい。この考えでは、どんどん進めば、突然元に戻るという原理を述べている。珠算における 御破算で願いましては で 再び始めることを想起させる。これは、また、reset と同様であると考えられる。
以 上
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
割り算のできる人には、どんなことも難しくない
世の中には多くのむずかしいものがあるが、加減乗除の四則演算ほどむずかしいものはほかにない。
ベーダ・ヴェネラビリス
数学名言集：ヴィルチェンコ編：松野武 山崎昇 訳大竹出版1989年
Reality of the Division by Zero z/0 = 0
http://www.ijapm.org/show-63-504-1.html
http://okmr.yamatoblog.net/
再生核研究所声明 264 (2015.12.23)： 永遠とは何か ― 永遠から
現代人は 空間とは 座標軸で表される数の組の集合 で表させるものと発想しているだろう。 基礎である直線は 実数を直線上に並べたもの、逆に直線とは 実は 実数全体の表現と考えられる。 すなわち、直線とは 基準点である原点ゼロから、正方向と負方向に正の実数と負の実数が大小関係で順序づけられ無限に双方向に伸びていると考えられる。
そこで、永遠とは 直線に時間を対応させ、限りなく正方向に進んだ先のことを 想像している。どこまでも どこまでも 先に行けばどうなるだろうか。直線上でも、平面上でも である。 砂漠の伝統を有する欧米文化の背景、キリスト教などの背後には、 永遠とは限りなく 果てしなく先にあると発想しているという。 どこまでも、どこまでも きりのない世界である。 ユークリッド幾何学が そのような空間を考えていることは確かである。
ところが四季に恵まれたアジアの民は、限りなく広がる世界に、不安や淋しさを直感して、 正の先と、負の先が一致していて、直線は円で どこまでも どこまでも行くと反対方向から、現在に至り、永遠は繰り返しであると、四季の繰り返し、天空の繰り返し、円運動のように発想して 仄かな安心感を覚えているという。永劫回帰、輪廻の思想を深く懐いている。実に面白いことには 美しい複素解析学では、立体射影の考えによって、直線を球面上の円と表現し、無限遠点の導入によって、 これらの思想を 数学的に厳格に実現させ、全ユークリッド平面の全貌を捉え、無限の彼方さえ捉えることが出来た。 その時 永遠を 確かに捉え、掴むことさえ出来たと言える。立体射影による球面上の北極に 確かに存在すると言える。素晴しい、数学を手に入れていた。この美しい数学は 100年以上もリーマン球面として、複素解析学の基本となってきている。
ところが２０１４．２．２偶然に発見されたゼロ除算の結果は、この無限遠点が 実は原点に一致していた という衝撃的な事実を述べていた。 永遠、無限の彼方と想像していたら、それが 実は原点に戻っていたという事実である。 それが我々の数学であり、ユークリッド空間の実相である。幾何学の性質や物理的な法則をきちんと説明している、我々の世界の数学である。
それで、永遠や無限遠点、我々の空間の 十分先の考え方、発想を考える必要がある。
無限の先が原点に一致している事実、それを如何に理解すべきであろうか。
それについて、 次のように解説してきた：
再生核研究所声明232（2015.5.26）無限大とは何か、無限遠点とは何か。― 驚嘆すべきゼロ除算の結果
再生核研究所声明257 (2015.11.05) 無限大とは何か、 無限遠点とは何か ー 新しい視点
再生核研究所声明262 (2015.12.09) 宇宙回帰説 ― ゼロ除算の拓いた世界観
新しい世界観は 始まりから始まり 最後には 突然戻るということを述べている。 しからば、始めとは何で 終りとは何だろうか。 これについて、 始めも終わりも、質的な変化であると定義できるのではないだろうか。 簡単な数学で万物、universe の現象を説明するのは難しい状況は確かにあるだろう．しかし、ゼロ除算の思想は、新羅万象が絶えず変化して 繰り返している様を表現しているように感じられる。
大事な人生の視点は 今日は 明日のためや遠い未来のためにあるのではなく、 現在、現在における在るべき適切な在りようが大事だと言っているようである。もちろん、現在は、未来と過去に関係する存在であり、それらは関係付けられ、繋がっているが 焦点はもちろん、 現在にあるということである。
ビッグバンの宇宙論は 適切に理解され、始めとは 大きな変化で 現状の元が始まり、
やがて突然、元に戻って 終わることを暗示しているようである。人生とは 要するに 内なる自分と環境に調和するように在れ と ゼロ除算は言っているようである。
ゼロ除算は 仏教の偉大なる思想 を暗示させているように感じられる。
以 上
Reality of the Division by Zero z/0 = 0
http://www.ijapm.org/show-63-504-1.html
http://okmr.yamatoblog.net/
Impact of 'Division by Zero' in Einstein's Static Universe and ...
gsjournal.net/Science-Journals/.../Download/2084
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Impact of 'Division by Zero' in Einstein's Static Universe and Newton's Equations in Classical Mechanics. Ajay Sharma physicsajay@yahoo.com. Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India.
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
Impact of ‘Division by Zero’ in Einstein’s Static Universe and Newton’s Equations in Classical Mechanics.
Ajay Sharma physicsajay@yahoo.com
Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India
Key Words Aristotle, Universe, Einstein, Newton
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
再生核研究所声明 271(2016.01.04)： 永遠は、無限は確かに見えるが、不思議な現象
直線を どこまでも どこまでも行ったら、どうなるだろうか。立体射影の考えで、全直線は 球面上 北極、無限遠点を通る無限遠点を除く円にちょうど写るから、我々は、無限も、永遠も明確に見える、捉えることができると言える。 数学的な解説などは下記を参照：
再生核研究所声明 264 (2015.12.23)：永遠とは何か―永遠から
再生核研究所声明257(2015.11.05): 無限大とは何か、無限遠点とは何か―新しい視点
再生核研究所声明232(2015.5.26): 無限大とは何か、無限遠点とは何か。―驚嘆すべきゼロ除算の結果
再生核研究所声明262(2015.12.09): 宇宙回帰説―ゼロ除算の拓いた世界観
とにかく、全直線が まるまる見える、立体射影の考えは、実に楽しく、面白いと言える。この考えは、美しい複素解析学を支える100年以上の伝統を持つ、私たちの空間に対する認識であった。これは永劫回帰の思想を裏付ける世界観を 楽しく表現していると考えて来た。
ところが、２０１４．２．２．に発見されたゼロ除算は、何とその無限遠点が、実は原点に一致しているという、事実を示している。それが、我々の数学であり、我々の世界を表現しているという。数学的にも、物理的にもいろいろ それらを保証する事実が明らかにされた。これは世界観を変える、世界史的な事件と考えられる：
地球平面説→地球球体説
天動説→地動説
1/0=∞若しくは未定義 →1/0=0
現在、まるで、宗教論争のような状態と言えるが、問題は、無限の彼方、無限遠点がどうして、突然、原点に戻っているかという、強力な不連続性の現象である。複数のEUの数学者に直接意見を伺ったところ、アリストテレスの世界観、世は連続であるに背馳して、そのような世界観、数学は受け入れられないと まるで、魔物でも見るかのように表情を歪めたものである。新しい数学は いろいろ証拠的な現象が沢山発見されたものの、まるで、マインドコントロールにでもかかったかのように 新しい数学を避けているように感じられる。数学的な内容は せいぜい高校生レベルの内容であるにも関わらず、考え方、予断、思い込み、発想の違いの為に、受けいれられない状況がある。
発見されてから あと1ヶ月で丸2年目を迎え、いろいろな実証に当たる現象が見つかったので、本年は世界的に 受けいれられることを期待している。
ゼロ除算の発見の遅れは、争いが絶えない世界史と同様に、人類の知能の乏しさの証拠であり、世界史の恥であると考えられる。できないことを、いろいろ考えて出来るようにしてきたのが、数学の偉大なる歴史であったにも関わらず、ゼロでは割れない、割れないとインドで６２８年ゼロの発見時から問題にされながら１３００年以上も 繰り返してきた。余りにも基本的なことであるから、特に、数学者の歴史的な汚点になるものと考える。そのために数学ばかりではなく、物理学や哲学の発展の遅れを招いてきたのは、歴然である。
以 上
Impact of 'Division by Zero' in Einstein's Static Universe and ...
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Impact of 'Division by Zero' in Einstein's Static Universe and Newton's Equations in Classical Mechanics. Ajay Sharma physicsajay@yahoo.com. Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India.
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
by Jerry Fausz, Ph.D.
[EDITOR’S NOTE: A.P. auxillary staff scientist Dr. Fausz holds a Ph.D. in Aerospace Engineering from Georgia Tech.]
The most fundamental axiom of science is Causality: the belief that every material effect observed must have a sufficient cause preceding or simultaneous with it (Miller, 2011). The observed motion of a cart is difficult to explain, for example, unless we assume the existence of an appropriately placed horse (figuratively speaking, of course). Scientific philosopher Sir Karl Popper wrote that the “rule” of causality “guides the scientific investigator in his work” (1968, p. 61). Actually, Popper refused to accept causality as a scientific “principle,” per se, but instead stated as a “methodological rule” that we should never “give up our attempts to explain causally any kind of event we can describe” (p. 61). I will not be so picky, as I believe that causality has been so thoroughly and consistently demonstrated by observation to be readily considered axiomatic, i.e., accepted without proof. Popper clearly indicates by his comments his belief that causality is heavily embedded in scientific thought and method. Nobel Laureate Erwin Schrödinger, upon defining causality, commented:
This postulate is sometimes called the “principle of causality.” Our belief in it has been steadily confirmed again and again by the progressive discovery of causes that specially condition each event (1957, p. 135).
The thoughts expressed by Popper and Schrödinger certainly support the idea that without an assumed “cause,” the scientific search would be quite aimless, if, indeed, possible at all.
The scientific importance of causality can be further illustrated with an amusing story related by noted theoretical physicist Stephen Hawking:
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!” (1988, p. 1).
Hawking’s anecdote invites several immediate observations regarding causality:
Causality is fundamental to scientific reasoning. Note the scientist’s question in response to the lady’s statement. Why assume the tortoise is standing on anything, if not for the sake of causality?
The principle requires effects to be natural or material (thus, observable to us). This is a somewhat trivial inference, for should an effect be unobservable in nature, then we would certainly not be concerned with knowing its cause. The observation implied in the story is nothing less than the existence of the Earth and its perceived place in the Universe.
The statement of causality does not of necessity require a natural or material cause. Causes need only be sufficient and antecedent to/simultaneous with the effect to satisfy causality. For example, as a direct consequence of causality, the lady’s hypothesis in the story (a giant tortoise holding up the Earth) resulted, according to her explanation, in an infinite tower of turtles. While this conclusion is not natural by any means (and, arguably, not logical), it is admissible from the standpoint of causality. (For further interesting discussion of the “tower of turtles” analogy, see Davies, 1992, pp. 223-226).
The third statement, while last, is certainly not least important. Note that even miraculous events recorded in the Bible produced effects that were observable or measurable (e.g., water becoming grape juice, the Red Sea visibly parting, people who were observably dead becoming alive again, etc.). Otherwise, miraculous causes would probably be of little interest to us.
It turns out, though, that the admissibility of non-natural causes is the only way to avoid a serious dilemma in the causality assumption. Consider the following: if a cause is, or is assumed to be, material (observable) then, like its effect, it is also contingent—the cause itself must be the effect of yet another cause according to the scientific axiom of Causality. Consequently, a predetermination to assume only material causes will, necessarily, lead to an infinite sequence of them. For example, strict material causality dictates that the existence of life at its present state of complexity may only be explained by some form of evolution (special creation, while an admissible cause, is not material). In order for life to evolve to its present state of complexity, life had to develop from non-living matter; non-living matter had to somehow organize in a very specific way to provide the necessary constituents for life to generate (assuming, of course, this is even possible); in order to so organize, this matter had to have existed under certain special conditions; and so on.
The dilemma in this reasoning is clearly illustrated by the turtle analogy. If one assumes that the Earth rests on the back of a giant turtle, then the turtle needs to be supported by something. And, since the lady in the story was inclined to make the turtle assumption to begin with, why not just assume that it is standing on the back of another turtle, which is exactly what she did. But, what is the second turtle standing on? As long as one has predetermined to accept the turtle hypothesis, like material causality, the turtles will continue to pile up, like material causes, until we have an infinite number of them. To avoid this dilemma, an assumption must be made at some point in the chain that does not involve a turtle, or by analogy, does not conform to strict materialism.
A related principle has been demonstrated within the most logically consistent subject of study known to mankind—mathematics. In 1931, Kurt Gödel, an Austrian mathematician, proved a theorem holding that “for any consistent mathematical system there exists within the system a well formed statement that is not provable under the rules of the system” (Overman, 1997, p. 27). Commonly known as Gödel’s Incompleteness Theorem, this result simply implies that to make progress within a mathematical system, certain facts need to be accepted axiomatically (that is, they cannot be proven from other facts in the system).
This principal also holds in other fields of reasoning. In fact, Paul Davies makes use of this idea when he proclaims:
I would rather not believe in supernatural events personally. Although I obviously can’t prove that they never happen, I see no reason to suppose that they do. My inclination is to assume that the laws of nature are obeyed at all times (1992, p. 15).
Here Davies assumes something to be true that, he admits, cannot be proven within his system of reasoning: the assumption that supernatural events do not occur. A tacit assumption in Davies’ statement, however, is the existence of time. While the natural laws may hold “at all times” (and supernatural events may not occur), we have already seen that assuming this to be true back to the beginning of time will lead to the equivalent of an infinite tower of turtles (an infinite causal sequence).
Speaking of the beginning of time, the most widely accepted theory for this event is the so-called “Big Bang.” This was hypothesized subsequent to the discovery that the Universe is expanding (Hawking, 1988, p. 38). Specifically, what astronomers discovered was that the light spectra of most of the stars in other galaxies were “red-shifted” (reducing in frequency), by which they assumed this indicates that those stars were moving away from us with increasing velocity. If the assumed expansion is extrapolated backwards in time, then one might suppose that there is a point in time at which all of the matter in the Universe was collocated (existed at the same point). This, of course, requires the highly non-trivial assumption that time actually extrapolates back to that point. Note that the expansion assumption does not necessarily follow since one might hypothesize other possible causes for the red-shift of the stars, such as nonlinear changes in the “elasticity” of space-time itself, perhaps as if the Universe was “stretched out” at its beginning (cf. “stretched out” [Isaiah 42:5; Isaiah 45:12; Jeremiah 10:12], or “spread out” [Job 9:8; Isaiah 40:22]).
For now, though, consider what science thinks it knows about this assumed beginning point, called the Big Bang singularity. The term singularity in this context denotes a point at which some fundamental property ceases to exist or certain processes are undefined. For example, in mathematics dividing by zero creates a singularity because division by zero is undefined. To state this more precisely mathematically, the mathematical operation of division is undefined at zero, so the origin (zero) is called a singularity point with respect to the operation of division. In the case of the Big Bang singularity, the assumed reverse time extrapolation creates a point at which all of the matter in the Universe is collocated, or has “zero size” (Hawking, p. 117). This would necessarily imply infinite mass density which, according to general relativity, implies infinite curvature of space-time (Einstein, 1920). Physicists Stephen Hawking and Roger Penrose have studied the properties of singularities predicted by general relativity quite extensively. Hawking observed that at such a point, “the laws of science and our ability to predict the future would break down” (p. 88). Since the laws of science as we understand them are undefined at this point, the point is called a singularity or, in this case, the Big Bang singularity.
So it seems, perhaps not unsurprisingly, that if the chain of causality could be followed back to a beginning of time as defined by the Big Bang, it would take us to a point in which natural laws do not apply. Hawking goes on to say: “[T]his means that one might as well cut the big bang, and any events before it, out of the theory, because they can have no effect on what we observe” (p. 122). However, cutting the singularity out of the theory just serves to bog the theory down once again in the dilemma of material causality. When the ultimate conclusion of causality turns out to be an unnatural cause, and we cut it out of our theory simply because we desire to stick with material causality, then our reasoning can only lead us, again, to an infinite tower of turtles. Note that Davies points out that it doesn’t make sense to talk about “before” with reference to the Big Bang singularity since time presumably had its beginning there (p. 50).
All of our imagination and reasoning, as well as our experience and observation, point to a necessary break-down in strict material causality. Theories of an infinite Universe, mathematical incompleteness, and space-time singularities testify to a physical reality that cannot be completely deduced by the rules of the system. The logical conclusion is a cosmological model that admits unnatural cause.
Furthermore, if we are forced to assume an unnatural cause for the beginning of time, as the evidence suggests that, indeed, we are, then why should we necessarily presume, specifically, that in order to pinpoint the beginning of time, we should extrapolate back to a point in which all the mass in the Universe is collocated? As we have seen, Hawking suggested discarding the Big Bang singularity because its effects cannot be predicted by science. However, Hawking also points out that the singularity theory implies that space-time had a beginning and a boundary, prompting him to ask “What were the ‘boundary conditions’ at the beginning of time?” He then comments: “One possible answer is that God chose the initial configuration of the universe” (p. 122).
Such boundary conditions as these could specify any of an infinite number of initial configurations and states for space-time and the matter that it contains. Indeed, boundary conditions are, by definition, these types of specifications. I see no particular reason to assume that these boundary conditions start with a point of infinite mass density and equally infinite space-time curvature, other than a predetermined desire to push material causality to its unnatural limit. These boundary conditions may, with equal probability, specify a Universe at the beginning of time that is not much different than what we now observe, thus implying that time has not progressed as far as some have conjectured. Perhaps that is why Hawking, using a mathematical contrivance he calls “imaginary time,” has more recently endeavored to create a consistent model of space-time that is finite but does not have a boundary, therefore would presumably not require any boundary conditions. Hawking, however, points out that this idea is “just a proposal: it cannot be deduced from some other principle” (p. 136).
The biblical Old Testament records that God said to Job: “Where were you when I laid the foundations of the Earth? Tell me, if you have understanding” (Job 38:4). Job understood well the rhetorical nature of God’s question, for he had already proclaimed to his companions:
But where can wisdom be found? And where is the place of understanding? Man does not know its value, nor is it found in the land of the living. The deep says, “It is not in me”; and the sea says, “It is not with me.” It cannot be purchased for gold, nor can silver be weighed for its price (Job 28:12-15).
Clearly, Job observed that complete understanding does not lie in materialism or nature and, if he did not know it before, God pointed out to him that neither does the knowledge of how “the foundations of the Earth” were established. Even what understanding we do have contains immutable evidence of its own hard-coded limitations in explaining the observable Universe.
Our observations and reasoning tell us that the foundations of the Universe could not possibly have been laid through strictly material or natural causality. Theories of cosmology, physics, and even mathematics point to the necessity of a Cause that operates independent of the rules of the system. These theories, however, while they can point to the necessity of unnatural causes, are impotent when it comes to explaining those causes. The Bible closes this gap in our theories by telling us of an omnipotent and omniscient Creator who is capable of operating outside of nature to make everything we observe out of what we cannot observe. The divinely guided Hebrews writer articulated this very concept in his striking statement against strict material causality: “By faith we understand that the worlds were framed by the word of God, so that the things which are seen were not made of things which are visible” (11:3).
If we pay attention to what all of our observation and reasoning are telling us, then our mind’s eye will “see” the foundations of the Universe being laid, not by material or natural causality, but by the omnipotent Creator, God, working outside of nature to set the boundary conditions, bringing natural laws on-line to shape and direct His creation. And our reasoning will, in turn, no longer require an infinite tower of turtles.
REFERENCES
Davies, Paul (1992), The Mind of God: The Scientific Basis for a Rational World (New York: Simon & Schuster).
Einstein, Albert (1920), Relativity: The Special and the General Theory (New York: Barnes & Noble).
Hawking, Stephen (1988), A Brief History of Time: From the Big Bang to Black Holes (New York: Bantam Books).
Miller, Jeff (2011), “God and the Laws of Science: The Law of Causality,” Apologetics Press, http://www.apologeticspress.org/APContent.aspx?category=9&article=3716.
Overman, Dean (1997), A Case Against Accident and Self-Organization (Lanham, MD: Rowman & Littlefield).
Popper Karl (1968), The Logic of Scientific Discovery (New York: Harper & Row).
Schrödinger, Ernst (1957), Science Theory and Man (New York: Dover Publications).https://apologeticspress.org/APContent.aspx?category=9&article=3831
再生核研究所声明262 (2015.12.09) 宇宙回帰説 ― ゼロ除算の拓いた世界観
最近展開しているゼロ除算が、新しい世界観を示しているのは 大変興味深い。直線とは一体どうなっているだろうか．空間とはどのようになっているだろうか。これについて、現代人は、双方向にどこまでも どこまでも 続いている直線を想像するであろう。限りなく広がった平面や空間である。ところが 立体射影によって 平面全体を球面上に１対１に写せば、全平面は 球面から北極を除いた球面上に1対１にきちんと写るから、無限に広がる 全平面の全貌が捉えられる。ところが平面上には存在しない想像上の点 それはあらゆる方向に限りなく遠くに存在する無限遠点の導入によって、その点を球面の欠けた1点北極に対応させれば、無限遠点を含めた平面全体は 球面全体と1対１にきちんと対応する。
このような対応で 平面上の円や直線全体は 球面上では共に円に対応するという美しい対応になり、平面上の直線は 球面上では、北極（無限遠点）を通る円に写ると、直線と円の区別は 球面上では不要になる。また、平面上の平行線とは 無限遠点で 角度ゼロで交わっている（接している）と平面上の構造がよく見えて、無限遠点を含めての平面の全構造が 捉えられる。このように、考えると、直線とは、球面上では北極を通る円、平面上では無限遠点を通る直線となる。この構造は、直線を1方向にどこまでも, どこまでも進めば、無限遠点を 通って、逆方向から戻ってくるという、永劫回帰の思想をちょうど実現している。それは、球面上では、 円を繰り返し回ることを意味する。 その様は 何もかも すっかり良く見える。
これが、従来100年以上も続いた世界観で、関数y=x やW=zは 無限遠点に近づけば、それらの像も無限遠点に近づいていると考えるだろう。 関数y=x の値は正方向にどんどん行けば、どんどん大きくなると考えるだろう。
しかるに、ゼロ除算1/0=0は、それらの関数は無限遠点にいくらでも近づくと 無限遠点にいくらでも近づくが、無限遠点自身では、突然ゼロになっていることが 幾何学的にも確認された。上記、北極は 実は原点ゼロに一致しているという。
話しを簡単にするために、 関数y=x を考えよう。右に行けば、プラス無限に、負の方向左に行けば 負の無限に限りなく近づくは 従来通りである。ところが、ゼロ除算では いずれの方向でも上記無限遠点では 値ゼロをきちんと取っているという。ゼロ除算の数学では、どんどん、増加した先、突然、ゼロ、原点に戻っているという。また、円でも球面でも半径Rをどんどん大きくすると、当然、円の面積や球の体積はどんどん限りなく大きくなるが、半径が無限のとき、突然、それらはゼロになるという。それらの理由も数学ばかりではなく、幾何学的にも明確に見えている。
この数学的な事実は、我々の世界、宇宙がどんどん拡大して行くと突然、ゼロに帰するということを暗示させている。 ― これは 宇宙回帰説を意味しているようである。
これは、ユニバースの普遍的な現象、どんどん進んだ先が、元に突然戻る原理を示しているようである。
そもそも人生とは如何なるものか。― よくは分からないが、事実として、生まれて、どんどん物心がついて、人間として精神活動が活発化して、多くは本能原理によって生かされて、そして、突然元に戻ることを意味しているようである。このことを深く捉えられれば、世界がよりよく観え、悟りの境地に達する大きなヒントを得ることができるだろう。
ここでは ゼロ除算の帰結として、宇宙回帰説、ユニバースの回帰説を唱えたい。この考えでは、どんどん進めば、突然元に戻るという原理を述べている。珠算における 御破算で願いましては で 再び始めることを想起させる。これは、また、reset と同様であると考えられる。
以 上
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
割り算のできる人には、どんなことも難しくない
世の中には多くのむずかしいものがあるが、加減乗除の四則演算ほどむずかしいものはほかにない。
ベーダ・ヴェネラビリス
数学名言集：ヴィルチェンコ編：松野武 山崎昇 訳大竹出版1989年
Reality of the Division by Zero z/0 = 0
http://www.ijapm.org/show-63-504-1.html
http://okmr.yamatoblog.net/
再生核研究所声明 264 (2015.12.23)： 永遠とは何か ― 永遠から
現代人は 空間とは 座標軸で表される数の組の集合 で表させるものと発想しているだろう。 基礎である直線は 実数を直線上に並べたもの、逆に直線とは 実は 実数全体の表現と考えられる。 すなわち、直線とは 基準点である原点ゼロから、正方向と負方向に正の実数と負の実数が大小関係で順序づけられ無限に双方向に伸びていると考えられる。
そこで、永遠とは 直線に時間を対応させ、限りなく正方向に進んだ先のことを 想像している。どこまでも どこまでも 先に行けばどうなるだろうか。直線上でも、平面上でも である。 砂漠の伝統を有する欧米文化の背景、キリスト教などの背後には、 永遠とは限りなく 果てしなく先にあると発想しているという。 どこまでも、どこまでも きりのない世界である。 ユークリッド幾何学が そのような空間を考えていることは確かである。
ところが四季に恵まれたアジアの民は、限りなく広がる世界に、不安や淋しさを直感して、 正の先と、負の先が一致していて、直線は円で どこまでも どこまでも行くと反対方向から、現在に至り、永遠は繰り返しであると、四季の繰り返し、天空の繰り返し、円運動のように発想して 仄かな安心感を覚えているという。永劫回帰、輪廻の思想を深く懐いている。実に面白いことには 美しい複素解析学では、立体射影の考えによって、直線を球面上の円と表現し、無限遠点の導入によって、 これらの思想を 数学的に厳格に実現させ、全ユークリッド平面の全貌を捉え、無限の彼方さえ捉えることが出来た。 その時 永遠を 確かに捉え、掴むことさえ出来たと言える。立体射影による球面上の北極に 確かに存在すると言える。素晴しい、数学を手に入れていた。この美しい数学は 100年以上もリーマン球面として、複素解析学の基本となってきている。
ところが２０１４．２．２偶然に発見されたゼロ除算の結果は、この無限遠点が 実は原点に一致していた という衝撃的な事実を述べていた。 永遠、無限の彼方と想像していたら、それが 実は原点に戻っていたという事実である。 それが我々の数学であり、ユークリッド空間の実相である。幾何学の性質や物理的な法則をきちんと説明している、我々の世界の数学である。
それで、永遠や無限遠点、我々の空間の 十分先の考え方、発想を考える必要がある。
無限の先が原点に一致している事実、それを如何に理解すべきであろうか。
それについて、 次のように解説してきた：
再生核研究所声明232（2015.5.26）無限大とは何か、無限遠点とは何か。― 驚嘆すべきゼロ除算の結果
再生核研究所声明257 (2015.11.05) 無限大とは何か、 無限遠点とは何か ー 新しい視点
再生核研究所声明262 (2015.12.09) 宇宙回帰説 ― ゼロ除算の拓いた世界観
新しい世界観は 始まりから始まり 最後には 突然戻るということを述べている。 しからば、始めとは何で 終りとは何だろうか。 これについて、 始めも終わりも、質的な変化であると定義できるのではないだろうか。 簡単な数学で万物、universe の現象を説明するのは難しい状況は確かにあるだろう．しかし、ゼロ除算の思想は、新羅万象が絶えず変化して 繰り返している様を表現しているように感じられる。
大事な人生の視点は 今日は 明日のためや遠い未来のためにあるのではなく、 現在、現在における在るべき適切な在りようが大事だと言っているようである。もちろん、現在は、未来と過去に関係する存在であり、それらは関係付けられ、繋がっているが 焦点はもちろん、 現在にあるということである。
ビッグバンの宇宙論は 適切に理解され、始めとは 大きな変化で 現状の元が始まり、
やがて突然、元に戻って 終わることを暗示しているようである。人生とは 要するに 内なる自分と環境に調和するように在れ と ゼロ除算は言っているようである。
ゼロ除算は 仏教の偉大なる思想 を暗示させているように感じられる。
以 上
Reality of the Division by Zero z/0 = 0
http://www.ijapm.org/show-63-504-1.html
http://okmr.yamatoblog.net/
Impact of 'Division by Zero' in Einstein's Static Universe and ...
gsjournal.net/Science-Journals/.../Download/2084
このページを訳す
Impact of 'Division by Zero' in Einstein's Static Universe and Newton's Equations in Classical Mechanics. Ajay Sharma physicsajay@yahoo.com. Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India.
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
Impact of ‘Division by Zero’ in Einstein’s Static Universe and Newton’s Equations in Classical Mechanics.
Ajay Sharma physicsajay@yahoo.com
Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India
Key Words Aristotle, Universe, Einstein, Newton
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
再生核研究所声明 271(2016.01.04)： 永遠は、無限は確かに見えるが、不思議な現象
直線を どこまでも どこまでも行ったら、どうなるだろうか。立体射影の考えで、全直線は 球面上 北極、無限遠点を通る無限遠点を除く円にちょうど写るから、我々は、無限も、永遠も明確に見える、捉えることができると言える。 数学的な解説などは下記を参照：
再生核研究所声明 264 (2015.12.23)：永遠とは何か―永遠から
再生核研究所声明257(2015.11.05): 無限大とは何か、無限遠点とは何か―新しい視点
再生核研究所声明232(2015.5.26): 無限大とは何か、無限遠点とは何か。―驚嘆すべきゼロ除算の結果
再生核研究所声明262(2015.12.09): 宇宙回帰説―ゼロ除算の拓いた世界観
とにかく、全直線が まるまる見える、立体射影の考えは、実に楽しく、面白いと言える。この考えは、美しい複素解析学を支える100年以上の伝統を持つ、私たちの空間に対する認識であった。これは永劫回帰の思想を裏付ける世界観を 楽しく表現していると考えて来た。
ところが、２０１４．２．２．に発見されたゼロ除算は、何とその無限遠点が、実は原点に一致しているという、事実を示している。それが、我々の数学であり、我々の世界を表現しているという。数学的にも、物理的にもいろいろ それらを保証する事実が明らかにされた。これは世界観を変える、世界史的な事件と考えられる：
地球平面説→地球球体説
天動説→地動説
1/0=∞若しくは未定義 →1/0=0
現在、まるで、宗教論争のような状態と言えるが、問題は、無限の彼方、無限遠点がどうして、突然、原点に戻っているかという、強力な不連続性の現象である。複数のEUの数学者に直接意見を伺ったところ、アリストテレスの世界観、世は連続であるに背馳して、そのような世界観、数学は受け入れられないと まるで、魔物でも見るかのように表情を歪めたものである。新しい数学は いろいろ証拠的な現象が沢山発見されたものの、まるで、マインドコントロールにでもかかったかのように 新しい数学を避けているように感じられる。数学的な内容は せいぜい高校生レベルの内容であるにも関わらず、考え方、予断、思い込み、発想の違いの為に、受けいれられない状況がある。
発見されてから あと1ヶ月で丸2年目を迎え、いろいろな実証に当たる現象が見つかったので、本年は世界的に 受けいれられることを期待している。
ゼロ除算の発見の遅れは、争いが絶えない世界史と同様に、人類の知能の乏しさの証拠であり、世界史の恥であると考えられる。できないことを、いろいろ考えて出来るようにしてきたのが、数学の偉大なる歴史であったにも関わらず、ゼロでは割れない、割れないとインドで６２８年ゼロの発見時から問題にされながら１３００年以上も 繰り返してきた。余りにも基本的なことであるから、特に、数学者の歴史的な汚点になるものと考える。そのために数学ばかりではなく、物理学や哲学の発展の遅れを招いてきたのは、歴然である。
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Impact of 'Division by Zero' in Einstein's Static Universe and ...
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Impact of 'Division by Zero' in Einstein's Static Universe and Newton's Equations in Classical Mechanics. Ajay Sharma physicsajay@yahoo.com. Community Science Centre. Post Box 107 Directorate of Education Shimla 171001 India.
http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2084
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