Tag Archives: Astrophysics

What else other than the Big Bang Singularity?

16 May

What else other than the Big Bang Singularity?

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Since my childhood I have been affascinato with astronomy and cosmology. I lived in the relative darkness at night in suburban Illinois and the night sky was a display of planets, stars and even satellites. Early on, seeing my enthusiasm, my parents bought me a small telescope with which I could see even clearer and, also, more distant celestial objects, and we went to the nearby observatory and planetariums to learn more about the universe. Of course, as with all science, over the years with the advancement of more sophisticated technology, the hypothesis of the origin of the universe has changed and will continue to do so. This short blog is written as a personal reflection on the latest, more reasonable hypotheses on the topic of the beginning of our universe.

Those familiar with cosmology or the scientific study of the origin and structure of the universe are aware of the competition of two primary hypotheses about the nature of the universe in which we live. These two are the Steady State and Big Bang. In brief, the steady-state theory, claims that the density of matter in the expanding universe remains unchanged due to a continuous creation of matter, thus adhering to the perfect cosmological principle, a principle that asserts that the observable universe is basically the same at any time as well as at any place. While the steady state model enjoyed some popularity in the mid-20th century, the Steady State Theory is now no longer accepted by most cosmologists. Today the majority of astronomers consider the Big Bang theory to be the best description of the origin of the universe as the observational evidence points to a hot Big Bang cosmology with a finite age of the universe, which the Steady State model does not predict.

It seems that the field of cosmology, therefore, yields good evidence that there was an initial beginning to our universe. According to the Big Bang theory, our universe began as an infinitesimally small, infinitely hot, and infinitely dense something – a singularity. The universe began to exist as a hot, dense phase, which can be considered the “birth” of our universe in which was contained all of the mass and spacetime of the Universe before it rapidly expanded with subsequent inflation, creating the present-day Universe. Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. The initial singularity is part of what is called the Plank  Epoch , or the earliest period of time in the history of the universe. So according to the standard theory, based on measurements of the expansion using Type IA supernova and measurements of temperature fluctuations in the cosmic microwave background, our universe came into existence as a singularity of an estimated age of 13.799 ± 0.021 billion years ago.

Well, as an ‘affascinato’ of cosmology, this has always been pretty convincing through the science that supports the Big Bang until one arrives at the logical question of “How did all that mass come into existence from nothing?”. One thinks that perhaps not being a professional in the subject, one has missed a subtle and important link in the argument for the Big Bang model. This self-doubt ends rather quickly when the usual cosmologist at the end of the Big Bang lecture admits, “Where did the initial singularity come from? We don’t know. Why did it appear? Well, to be honest, we don’t know. This is a question that stretches physics to its limit simply because before the initial singularity there was no space and no time. Therefore, the word ‘before’ becomes meaningless.” In fact, the Big Bang singularity can explain only what happened immediately after—not at or before—the singularity. Also, this implies that the universe was born with a tendency to expand, which overcame the tendency of matter to collapse. Why it initially chose to do so is also still a mystery… the Big Bang model has numerous theoretical difficulties to it.

So while the Big Bang is the most accepted model, there are many holes (not black ones) in this proposition. But now there is a new perspective which gives a more parsimonious answer to the questionable issues associated with that singularity. The perspective is quantum physics. In quantum physics, particularly the transactional interpretation (TI) of quantum mechanics, as discussed by Ruth Kastner, explains that the macroscopic world of mass, space and time is based on the foundation or scaffold of the quantum interactions found in ‘quantumland’. This interpretation purports that there is more to known reality than ‘spacetime’, and that quantum theory describes that subtler, unseen reality. In this hypothesis, quantum processes take place in a realm scaffolding the ‘spacetime’ realm. Quanta are not contained in our spacetime world but in the realm of possibilities outside spacetime. Kastner explains that according to the transaction interpretation of quantum systems, e.g. electrons, travel by a physical entity called an offer wave, which is offered from a source called an emitter, to a destination called an absorber. The microscopic emitters and absorbers are quantum objects and not in spacetime. When there is absorption of the offer, this process gives rise to a confirmation wave that travels back to the emitter. This process of an offer responded to by a confirmation is the basic ‘handshake’. The confirmation is also like a mirror image of the offer representing an incipient transaction whose essence is merely possible energy rather than real energy. The process of the creation of new particles can only be treated by relativistic quantum mechanics.

Once there is a matching confirmation, then the property is defined as actualized, brought into spacetime, and is a classical property. The incipient transaction is actualized and becomes an observable event in the macroworld or ‘our’ world of mass in space and time. A macroscopic object begins at the point at which a confirmation has been generated. Real energy is only conveyed in the actualized transaction, in fact; only through an actualized transaction can real energy be radiated or transferred from one object to another. So indeed, a reliable macroscopic object is a consistent absorber and can be defined as a system of many actualized transactions. Kastner uses the example of a geiger counter to illustrate the difference of the two ‘worlds’. A geiger counter exists as an object in the macroscopic world being a conglomerate of actualized transactions. But it also maintains its roots in the quantumland domain of possibilities because it is comprised of atoms, which can act as emitters or absorbers. Measurement occurs both whenever an absorber is accessible to an emitter and when confirmations are generated.

In actuality, it is the interaction of subatomic material that brings forth the material world as we know it and as it exists. So, in terms of the beginning of our universe, using the TI model, the speculation that would make sense is that at a point about 13 billion years ago there was a quantum fluctuation that created the macroscopic elements which ‘broke through’ and created our realm of existence. While, of course, the why, how and what are still a mystery for this as is still much of our comprehension of ‘quantumland’, we are no longer faced with the impossibility of explaining the ‘before’ the singularity event of the Big Band using the infinite macroscopic mass/space/time model of the Big Bang but instead the more heuristic, efficient quantum model which bypasses the impossible.

Fascinating and Mysterious Life

11 Aug

This small project of mine began a few winters ago after I had time to begin to search for some further trends in thinking from some of my previous experiences and research in the area of Buddhism and science. It has turned into a fun and interesting search of ideas which are presented here as quotes. Life ideas are like reading a good mystery novel where the plot twists and turns as you contunue the story and you are never really sure what is the “truth”. While no “truth” is always forthcoming, ‘trends’, patterns, “evidence” starts to appear which create important hypothesis or propositions about the nature of life, reality, knowledge, etc, as well as show the mystery, complexity and strangeness of life. I share those now on my blog(s). I hope you enjoy the thread!                                                                                                                                                                                             —————————————————————————————————————————-

Mystery is not something negative that has to be eliminated. On the contrary, it is one of the constitutive elements of being. B.D’Espagnat  

Not only is the universe stranger than we imagine, it is stranger than we can imagine. A. Eddington

The Big Bang Theory is just a model and is not accepted by all astronomers – although the vast majority of the do accept it. This idea proposes that 14 billion years ago the energy of the Universe was suddenly created, with all the energy required to produce it concentrated into what was essentially a point, a point with no volume referred to by scientists as a singularity. From that point the Universe expanded outwards and 14 billion years later has evolved into what we observe today. The implication of this model is that, at the instant the Universe began, space and time did not exist. One cannot refer to a time before the Big Bang because there was no such thing as time before the Universe began. Again, one cannot ask what the Universe expanded into because the only space that exists is within the bounds of the expanding Universe. … On the basis of the Big Bang theory what was the Universe like at the instant it began? It was an unimaginable concentration of just pure energy – no matter could exist. Then it began to expand and, once this happened, then time and space came into existence. … From the beginning to 10(-12) seconds, there would be no clear distinction between energy and matter and the forces that operated would be of an unfamilar kind. During this period there occurred a rapid expansion … at a speed greater than light. Michael M. Woolfson

Steven Hawking, George Ellis, and Roger Penrose. According to their calculations, time and space had a finite beginning that corresponded to the origin of matter and energy.”3 The singularity didn’t appear in space; rather, space began inside of the singularity. Prior to the singularity, nothing existed, not space, time, matter, or energy – nothing. So where and in what did the singularity appear if not in space? We don’t know. We don’t know where it came from, why it’s here, or even where it is. All we really know is that we are inside of it and at one time it didn’t exist and neither did we. There was no explosion; there was (and continues to be) an expansion. Rather than imagining a balloon popping and releasing its contents, imagine a balloon expanding: an infinitesimally small balloon expanding to the size of our current universe. anonymous

The cyclic universe model *space and time exist forever *the big bang is not the beginning of time; rather, it is a bridge to a pre-existing contracting era *the Universe undergoes an endless sequence of cycles in which it contracts in a big crunch and re-emerges in an expanding big bang, with trillions of years of evolution in between *the temperature and density of the universe do not become infinite at any point in the cycle; indeed, they never exceed a finite bound (about a trillion trillion degrees) *no inflation has taken place since the big bang; the current homogeneity and flatness were created by events that occurred before the most recent big bang *the seeds for galaxy formation were created by instabilities arising as the Universe was collapsing towards a big crunch, prior to our big bang. In the new paradigm, each cycle proceeds through a period of radiation and matter domination consistent with standard cosmology, producing the observed primordial abundance of elements, the cosmic microwave background, the expansion of galaxies, etc. For the next trillion years or more, the Universe undergoes a period of slow cosmic acceleration (as detected in recent observations1), which ultimately empties the Universe of all of the entropy and black holes produced in the preceding cycle and triggers the events that lead to contraction and a big crunch. Note that dark energy is not simply added on — it plays an essential role. The transition from big crunch to big bang automatically replenishes the Universe by creating new matter and radiation. Gravity and the transition from big crunch to big bang keep the cycles going forever.Paul J. Steinhardt

Eventually the universe will become a cold, dead wasteland with a temperature approaching what scientists term “absolute zero”. Professor Priyamvada Natarajan (which contradicts the Cyclic Model)

And so without even quoting about string theory or M theory, the obvious question that goes beyond our present ability of understanding is the question “how did it all begin” or “where did the massive energy that appeared to become our known universe orginate?”, etc.. These questions can also become the fodder for religious thought, ie “God created the Universe” and it can be left at that for at this point science just can’t answer that question. Neither “answer” can be “correct” because science admits it is unanswerable and religion takes it on faith. So, as T. Caryle says, the world is an inscrutable and magical place. Rodger

Wonder is retained by wise pondering.” –Ravi Zacharias

Xuan(Tao) is the dark mystery beyond all mysteries.Han Zhongli

The more I learn, the more I understand that life is amazing, sensitive, responsive, mysterious  and totally interconnected . -Rodger  

It remains admitted that to this day we don’t have a convincing notion of why there is something, why not merely nothing: What “spiritus rectot breathes fire into the equations and makes the universe for them to describe?”- raising the question, what do the laws of nature permit beyond what actually exists? H. Genz  

In the realm of particle collisions and quantum processes, antimatter is produced as often as ordinary matter. In fact, the big bang should have produced equal amounts of both—not a good thing, because each piece of antimatter would destroy an equal amount of matter. The big bang should thus have created universe of only light and energy, free of any solids, liquids or gases. So, some sort of asymmetry occurred that skewed the universe’s evolution toward matter. It would not have taken much—just one extra matter particle for every billion particle-antiparticle pairs. Researchers have discovered an asymmetry between the behavior of matter and of antimatter, called charge–parity violation, which could have skewed things to our side of the material world. But for this subtle bias to translate into an excess of matter, the primordial universe would have had to go through a wrenching period of imbalanced conditions, and so far no one knows how that might have happened.Philip Yam      

We are mounds of quarks in trios, we are proton-and-electron families. …There is but a single family on this planet, just one life-form stretching out its tendrils, testing possibilities as dust and stars did once upon a time. Face it, we are all in this together, microbes, seaweed, starfish, salamanders, humans, every strange extrusion of nucleic acid chains. We are the kin of yeast, the brothers of cockroaches, the sisters of sugar beets, and the cousins of maize. We share a common birthright born of ancient gene-and-membrane teams. All of us are children in the clan of DNA. Howard Bloom

The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed. –Albert Einstein  

The religion of future will be a cosmic religion. It should transcend a personal God and avoid dogmas and theology. Covering both the natural and the spiritual, it should be based on a religious sense arising from the experience of all things, natural and spiritual as a meaningful unity. Buddhism answers this description… If there is any religion that would cope with modern scientific needs, it would be Buddhism. perhaps Albert Einstein

A human being is a part of the whole, called by us the “Universe,”a part limited in time and space. He experiences himself, his thoughts and feelings as something seperate from the rest – a kind of optical illusion of his consciouness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from the prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security. Einstein

We know from science that nothing in the universe exists as an isolated or independent entity. M.Wheatley

The most important scientific revolutions all include, as their only common feature, the dethronement of human arrogance from one pedestal after another of previous convictions about our centrality in the cosmos.S.J.Gould

This world, after all our science and sciences, is still a miracle; wonderful, inscrutable, magical and more, to whosoever will think of it. T. Carlye

What is the mind of ours? Last week’s potatoes!…The atoms come into my brain, dance and dance and then go out-there are always new atoms but always doing the same dance, remembering what the dance was yesterday.  RP Feynman  

“… all those who apprehend the single significant whole, or experience cosmic religious feeling, with or without the awareness of the existence of the principle of cosmic order, are engaged in similar acts of communion with the Whole. Yet any translation into conscious content of that experience , in scientific or religious thought, invokes reductionism where it cannot be applied. …all knowledge in the conscious content is a differentiated system that cannot by definition articulate the universal principle of order. Just as there can be no one-to-one correspondence between physical theory and physical reality, there can be no such correspondence between religious descriptions of beings and Being itself.” and ” … conceiving of a human being, as Einstein put it, as “part of the whole” is the leap of perspective that will prove most critical. It is only in making this leap that we can begin, as he suggests, to free ourselves of the ‘optical illusions’ of our present conception of self as a “part limited in time and space”, and to widen “our circle of compassion to embrace all living creatures and the whole of nature in its beauty”.  and ” The central problem… has been trying to “prove” the existence of Being when it can never be proven because of its inherent undivided wholeness. Being neither requires or permits “proof”. It merely is, and accepting this abundantly obvious fact can provide a “foundation”, as Einstein put it, ‘for our inner security.’ …the description of the parts cannot disclose the existence or nature of the Whole. Yet one cannot, of course, merely reason or argue oneself into an acceptance of this proposition. One must have the capacity for what Einstein termed ‘cosmic religious feeling.’  Hopefully many of those who have the capacity will also communicate their awareness to others in metaphoric representations in ordinary language with enormous emotional appeal. …As described by Jonas Salk: ‘…By using the processes of Nature as metaphor, to describe the forces of the Cosmos by which it operates upon and within Man, we come as close to describing ‘reality’ as we can within the limits of our comprehension. Men will be very uneven in their capacity for such understanding, which, naturally, differs for different ages and cultures, and develops and changes over the course of time. For these reasons it will always be necessary to use metaphor and myth to provide ‘comprehensible’ guides to living. In this way, Man’s imagination and intellect play vital roles in his survival and evolution’.” from The Conscious Universe  

“The whole is something else than the sum of its parts” K. Koffka

The usual conception of the world is that matter is “embedded in” space and time. …this cannot be the case. There are many factors which indicate that reality is “projected onto” space and time. … The picture and also its frame, space-time, are located in the mind of the observer….the fact that reality is not embedded in space-time but is rather projected onto space-time. …space-time is not installed in the brain as a definate system but it is only “inserted” if there is actually something to be portrayed or represented, i.e., when our sense organs register objects and processes from the reality outside. W. Schommers  

I am therefore inclinded to think that “the Real” – alais human-independent reality- is not embedded in space-time. And, indeed, I go so far as speculating that, quite on the contrary, the nature of space-time is… not “noumenal” but “phenomenal”, that space-time is a “reality-for-us.” B.d’Espagnat  

Time is a dimension in the domain of descriptions, not a feature of the ambience. H.Maturana

According to classical physics, the universe consists of bodies in space. We are tempted to assume, therefore, that we live in a physical world consisting of bodies in space and that what we percieve consists of objects in space. But this is very dubious. J.J.Gibson

Most of us still think like Newton, regarding space as sort of a vast container that has no walls. But our notion of space is false. Like time, space is neither physical nor fundamentally real in our view. Rather, it is a mode of interpretation and understanding. It is part of an animal’s mental software that molds sensations into multidimensional objects. time does not exist independently of the life that notices it. Robert Lanza and Bob Berman  

The concept of time cannot actually be understood. We are accustomed to thinking that time is something which can be found or which one has. But it actually has no existence. The physicist says time is something that can be measured in one way or another by a clock. But what does the clock measure. Nothing but time!E. Dammann   

In recent years many planets have been found around nearby main-sequence stars, all of the Population 1 variety, and it is generally accepted that the material of these planets will have come from the same source as that which formed the parent star. This is how the material that formed our world came into being – its atmosphere and solid substance, the biosphere and everything within it, including us. An eminent American astronomer, Carl Sagen (1934-96), a great expositor of popular science, once described humankind as being evolved from star stuff. How right he was! Look around you- look in a mirror. All the matter that you see, living and non-living, has been through the violence of several supernovae, and may do so many times more in the distant future. Michael M Woolfson  

Unknowingly, we plow the dust of stars, blown about us by the wind, and drink the universe in a glass of rain.–Ihab Hassan  

What is life? It is the flash of a firefly in the night. It is the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset. Crowfoot, Blackfoot warrior and orator 1830 – 1890 

So, I don’t really know how to justify my love for the electron. Perhaps it is because an electron has no location. Electrons interact via the electromagnetic field, aka the photon. All the electrons in the universe and all the photons in the universe are talking to each other all the time. They are all connected, no matter how far apart, by the electromagnetic field, which has infinite range.Jon Butterworth 

TIME IS A MYSTERY

26 Jul
  • Time is a mystery. Some thoughts by thinkers. ——————————————————————-  

The illusion of the passage of time arises from the confusing of the given with the real. Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality.” Kurt Gödel

“Gödel concluded that time travel is indeed theoretically possible, rendering time, as we know it, meaningless. Time, that mysterious and seemingly self-contradictory being, as Gödel put it, which, on the other hand, seems to form the basis of the world and our own existence, turned out in the end to be the world’s greatest illusion. For Gödel, time was the philosophical question” Palle Yourgrau, ‘A world without time’

“Ultimately all moments are really one. Therefore, now is eternity.” David Bohm
“…for us physicists believe the separation between past, present, and future is only an illusion, although a convincing one.” Albert Einstein

“I am therefore inclined to think that “the Real”- alais human independent reality- is not embedded in space-time. And indeed, I go as far as speculating that, quite the contrary, the nature of space-time is… not ‘noumenal’ but ‘phenomenal’, that space-time is a ‘reality-for-us.’” Bernard d’Espagnat  

” …As a result, all change must be gauged by correlations. Ultimately, everything must be correlated with the size of the Universe. Any vestige of a moving present has faded completely…Remember that Einstein spoke of an illusion…Might it be that our strong sense of the flow of time is an illusion akin to giddiness, perhaps connected with the way our memories operate?” Paul Davies   

“The world just does not happen, it simply is…” Hermann Weyl 

“All these Nows,  just exist.  They do not appear and then vanish; they just are.   From a global perspective, there is no answer to the question, “What time is it?”  There are just different experiences at different Nows. From any given vantage point, you look back, and remember other times – so that the question, “Why is it this time right now, rather than some other time?” seems to make sense.  But there is no answer. When I came to this understanding, I forgot the meaning that Time had once held for me. Time has dissolved for me, has been reduced to something simpler that is not itself timeful. I can no longer conceive that there might really be a universal time, which is somehow “moving” from the past to the future.  This now seems like nonsense.” Eliezer Yudkowsky 

“One cannot though conclude how the variations are taking place, over what timescales they are taking place or even how old the universe is. The universe could be 10×10 years old or 5 x 10-44 sec (the Planck time)old, or any time in between. Time is strictly a parameter that can be introduced in the scale-invariant relationships. It has no meaning by itself. The universe appears to be evolving as the number of particles and ratios are varying.” Menas Kafatos, Sisir Roy and Richard Amoroso   

“…the present moments of reality are now replacing one another, where if one had direct awareness of reality one would know reality as transcendental efficient moments” Fyodor Ippolitovich Stcherbatsky

“From non-nirvanic awareness, one experiences imaginary causal connections between moments and thus fabricates the illusion of persistence and duration of objects. But when one experiences reality, one only experiences the efficient moments, rapidly replacing one another, like the flames of a fire. Buddha: the whole world is burning. The non-nirvanic observer, unaware of the fiery, boiling nature of reality, is unaware that they will not exist beyond this present instant, and that any volition or hope is unneeded and illusory, and is based on the misery of unreality.” J. Grupp  

The Buddha said, “Even so was my past existence at that time real, but unreal the future and present existence; and my future existence will be at one time real but unreal the past and present existence; and my present existence is now real, but unreal the past and future existence. All these are merely popular designations and expressions, mere conventional term of speaking, mere popular notions” (Digha-nikaya). Hence the Buddha has only one time, that is “Eternal Present”.” Chen Hsiongcai

“I have realized that the past and future are real illusions, that they exist in the present, which is what there is and all there is.” Alan Watts

“Time, as self-determination of the eternal “Now”, is essentially contained in this Now. There where time is, contained and extinquished, personality appears, as content of eternity.” Nishida

“The Universe is a dynamic system in a permanent dynamic equilibrium, there is no beginning and no end of the universe.” A.S.Sorli

“So in summary, the universe we see is just a fragment nested in a timeless (everything) whole, rather than a single material world magically arisen above some primordial nothing. All universes exist without beginning or end in the ultimate arena of time, and each moment we experience exists forever.” Gevin Giorbran  

“This means that because, in reality, there is no coming or going in time, when we cross the river or climb the mountain we exist in the eternal present of time; this time includes all past and present time.” Dogen  

“Dogen writes, ‘Entire being, the entire world, exists in the time of each and every now.’ Thus the mind, body, being, world, and time form a unity. Not only are entities time, and not only is time in me, but activities are time. ‘As the time now is all there is, each being-time is without exception entire time.’ Dogen emphasizes the now moment because there is never a time that has not been or a time that is coming. ‘…all is the immediate presencing here and now of being-time.’ Thus time is a continuous occurrence of ‘nows’. The Buddha-nature is a present actuality.” Carl Olson

“Dogen frees himself from the representational mode of thinking without subverting time and remaining convinced that each moment is complete in itself and cannot be undermined.” Carl Olson    

“The universe would be completely self contained and not affected by anything outside itself. It would neither be created nor destroyed. It would just BE.”  Stephen Hawking

“Whether or not reality has one universe or many, it had no beginning and was not created. It neither was nor will be. It just is.” Victor J. Stenger

“What suggests itself is that psychologically – and perhaps eventually for the deepest level physically – we can’t use time as the essence. Rather, the moment NOW is the essence, because all the past and the future that we ever will know are in this moment. The past and the future are now – namely, in so far as it has left any impression, whatever has happened is now. And our experiences are now. Thus we could say that now may be the starting point. One picture you could make of an electron would be that it sort of flashes into and out of existence so fast that when picked up in the usual equipment it looks continuously existent. It might have a certain regularity, so that it appears to obey an order of necessity. But it might be that it is basically creative; the creative act may create this order of necessity.” David Bohm

“We are always moving (inflationary cosmology)(arrow of time) but also always remain in the now. The ‘phenomenal’ now is relative, experienced as eternal but intuitively we grasp and psychologically represent a moving existence by remembering a past and projecting into a future.” Rodger Ricketts

“Time is like a ‘field’ – there is no fixed, locatable present. It always vanishes upon close examination. The moment you try to pinpoint it, it has already passed and a future moment is now present.” Trinh Thuan

In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least, possible. Within the framework of Einstein’s general theory of relativity Gödel produced cosmological solutions to Einstein’s field equations which contain closed time-like curves, that is, curves in spacetime which, despite being closed, still represent possible paths of bodies. An object moving along such a path would travel back into its own past, to the very moment at which it ‘began’ the journey. More generally, Gödel showed that, in his ‘universe’, for any two points P and Q on a body’s track through spacetime (its world line), such that P temporally precedes Q, there is a time-like curve linking P and Q on which Q temporally precedes P. This means that, in principle at least, one could board a ‘time machine’ and travel to any point of the past. Gödel inferred, in consonance (as he observes) with the views of Parmenides, Kant and the modern idealists, that under these circumstances there could be no such thing as an objective lapse of time, that time or, more generally, change, is an illusion arising from our special mode of perception.” John L. Bell