Question
Asked 12 December 2014

Did the Big Bang create a parallel universe where time goes backwards?

I recently read an interesting article with topic as this query in "The Telegraph".
 (By Gregory Walton, 10 Dec 2014)
Some Excerpts:
"A scientific breakthrough could redefine how the universe is understood.
Radical new research led by a British scientist has suggested that there may be a second universe where time runs backwards.
The theoretical claims put forward in the Physical Review Letters journal could revolutionise the field of research into the origin and future of the universe.
In the paper titled 'Identification of a Gravitational Arrow of Time', an international team of world renowned scientists led by Oxfordshire-based Dr Julian Barbour challenge assumptions about the so called 'arrow of time'.
The 'arrow of time' is the theory that time is symmetric and therefore time moves forward. They contend that there is no scientific reason that a mirror universe could not have been created where time moved in an distinct way from our own.
But in a quirk of science it is thought that if a parallel universe did exist where time moved backward, any sentient beings there would consider that time in our universe in fact moved backward.."

Most recent answer

Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
Dear Rajat,
well, but again - – the coordinate time isn’t “fictitious”, that is the “time rule/possibility” which is real, as the true time rule is (though it is “more possibility, then rule”, comparing with the true time); and, besides, – it is (as the time interval) just “the time is what clocks show” – as that is established just in the SR.
All clocks show their own proper times in the absolute 4D Euclidian spacetime (and so show their position on the real coordinate time axis in the spacetime – but what is usually unknown for an observer since he doesn’t know his absolute speed), so a clock ticks more frequently, if it has lesser spatial (and so larger temporal) speed in the spacetime.
But in the SR, as the consequence of rather evidently logically self-inconsistent postulate that all [inertial] reference frames are totally equivalent, any SR-observer believes that he is at rest (at all, always and everywhere,  since “there is no absolute spacetime and corresponding absolute RF”) an his RF’s clocks tick faster then any other, if other clock moves relating to him. What is, of course can be not correct, in reality only one condition determines the clock rate – see above (that’s, besides, the solution of the Dingle problem; moreover, in the SR the time coordinate is, in fact, imaginary; etc.)
Nonetheless, just the SR- observer uses the clocks’ showings as clock’s coordinates in the Minkowski spacetime, at that – since he doesn’t know his real speed in the real spacetime – he calibrate the clock practically arbitrary.
But in reality any observer, for example a human, can measure his absolute speed (see “To measure the absolute speed is possible?” http://viXra.org/abs/1311.0190) if has money $30-$50 *106; further - calculate, using the measured speed value and established preliminary clock rate, corresponding tick rate for the clock at [spatial] rest; and, further, re-calibrate his clock so, that the clock showing would be as the clock is at [spatial] rest in the spacetime and so the clock will show true time till will not be accelerated…
Cheers
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All Answers (26)

Sundarapandian Vaidyanathan
Vel Tech - Technical University
A related article in "The Independent" (UK):
'Mirror universe' suggests Big Bang created place where time goes backwards (Andrew Griffin, Dec 11, 2014)
Some excerpts:
"The current theory suggests that entropy — the force of the universe that means that it tends to get more disordered over time — also drives time forwards. Since the universe began as an ordered thing, as it gets more disorganised, so does time.
But that depends on an assuming that the universe was exceptionally ordered at its beginning, according to Steven Carlip, of the University of California at Davis. And while that is many scientists’ working theory, it is impossible to prove."
"But the new theory suggests that time doesn’t have to flow in just one direction, which would settle the problem entirely. When the universe began, it could have created another one, flowing in the other direction, write Julian Barbour, Tim Koslowski, and Flavio Mercati in the Physical Review Letters."
“Any internal observer must be in one half of the solution and will only be aware of the records of one branch and deduce a unique past and future direction from inspection of the available records,” they write. People think that time moves in one direction because they can only see one half of the universe, in other words.
4 Recommendations
Rajat Kumar Pradhan
Utkal University
dear Sundar,
It did create another. In fact creation by big bang and in general any process would be impossible unless accompanied by the time-reversed process.
take a look at my "Psychophysical Interpretation of Quantum theory".
My finding is that the backward time universe exists, not in physical space but in the mind of the observer.
The universe is a psychophysical Reality.
Regards,
Rajat
4 Recommendations
Sundarapandian Vaidyanathan
Vel Tech - Technical University
Dear Rajat Pradhan -jee... Namaste.. Thanks for your kind reply.
After your reply, I checked and found that your research paper titled, "Psychophysical Interpretation of Quantum Theory" got published in NeuroQuantology, December 2012,  vol. 10, no. 4, pp. 629-645. I have enclosed a soft copy of your paper for the benefit of Research-Gate friends, who follow this query & replies. Thank you!
 
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Abedallah M Rababah
United Arab Emirates University
Dear Sundar, Thank you for the interesting paper.  The Big Bang is an inverse process that created a parallel universe; in this model many things are inverted including  time that runs  backward.
4 Recommendations
Douglas R. Daugherty
University of New Mexico
Hello Rajat and thank you Sundarapandian for this interesting topic,
Sundarapandian & Rajat, forgive me but the physical sciences are not my area of expertise and having only given Rajat's article a once through and will give a good read latter. In my once through however is appears from your writing "that formalism of quantum theory naturally incorporats psychophysical parallelism" that this other universe could be viewd as ~A. Meaning our universe is A and the parallel universe would be ~A this a fair read? If we think of time then again (A &~A)?
Yet many mathematical papers as well as cosmological arguements are presented that time is an illusion, only "exsits" in mathematics, or has no "flow"in one direction or another. If so, how then how can time have a "gravitational arrow"? Would that not counter the arguement of time existing only in mathematics?
Thnak you,
Douglas
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H.G. Callaway
Temple University
Philadelphia, PA
This strikes me as an interesting report. Let's take a look. See in particular the link to the short piece by Steven Carlip, from October 29, 2014. (DOI: 10.1.103Physics.7.111) He views the paper by Barbour, Koslowski and Mercati, in Physical Review Letters, as suggestive but not conclusive. Their paper is a piece of mathematical physics, based on Newtonian gravity. One question is whether similar results could be obtained with a physically more realistic study of the mathematics of GR. 
In a sense, the proposal seems to exchange one mystery for another. To answer the question about the arrow of time we observe, or about the initial condition of low entropy in the big bang, say, we have to postulate another universe, with a contrasting arrow of time, but which remains unobserved--and perhaps unobservable? There have been many similar proposals in the past. What appears new here is just the simplicity of the Newtonian mathematical physics. 
H.G. Callaway
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Douglas R. Daugherty
University of New Mexico
Dear Marcel,
Given a timeless universe? I don't know but I do know, as of yet, but I do know how I would begin to theorize about it.
Let us suppose you and I were sitting having lunch over this philosophical question. From our perspective would this scenario ever end? From an observers perspective? Would I even here your words, as well could you here mine? How would we build an argument?
Having no past or future only "place" and "physicality", what would be language, knowledge, all which in our current state require a reference point. A marker from then, now, and unknown without which communication in our current state is made impossible.
So I guess what I am asking is even if it is theoretically possible how would we know from previous (X) to now (Y)?
Your thoughts please.
Douglas
 
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Dr Julian Barber announced that "‘If you look at a simple model with a swarm of bees in the middle of the Big Bang but breaking up in either direction, then you would say there are two arrows of time, pointing in opposite directions from the swarm. One arrow would be forwards and one backwards.’
2 Recommendations
Douglas R. Daugherty
University of New Mexico
Dear Kamal,
If I understand the Big Bang Theory is it not conceptualized as an expanding baloon? Would this then mean if time exists all the arrows would be going outward in all directions from that singularity? The concept of time being only being two directional seems off to me. Could you add to J. Barber's thoughts? 
Does he describe what would be the limiting foactors or influences that limit time to these two outcomes instead of being one direction outward from all directions?
Thank you
Douglas
2 Recommendations
Dear Douglas,
Thank you so much for your kind remarks.
Time for me is a mystery!! Don't forget that I am a surgeon not a physicist!! but I am trying to learn more by following this interesting discussion!!
Peace upon you
Kamal
4 Recommendations
Dear ALL
LET US go BACKWARD ! (otherwise discussion may deviate)
The discovery of the “Big-Bang” theory highlights following basic facts (although our/cosmologists/scientists gaze and body cannot travel faster than the speed of the light but ‘the experiment of THOUGHT travelled’ !)
(a) Initially Universe had POINT-MASS that had zero volume
(b) It had “ Zero Volume” because of its immense gravitational force
(c) If a and b is true then ZERO = means NOTHING
(d) IF C is true then Universe had NO (thing) volume.
(e) Therefore, science/scientists/ COSMOLOGISTS/ (what you have mentioned) have arrived at “NOTHING”. The question of PARALELL I THINK HAS BEEN BORROWED FROM THE IDEA OF HOMOPLACY IN BIOLOGY).
(f) The biggest scientific UNDISCOVERY so far is that “UNIVERSE/MATTER came out of NOTHING”
(g) If (a) to (f) is true then what next “Does/did the MATTER/universe have/had intelligence/EXTRAORDINARY PRECISION (expansion & formations) of self-formation with so much of complexity and diversity? Does matter has any self intelligence for self-formation? CAN’T we think !
(i) If (i) and h is true then then cant we raise a question who created heavens and the Earth out of Nothing. 
3 Recommendations
Douglas R. Daugherty
University of New Mexico
Hello All,
Following dear  Mr. Alam's lead lets do go back. Let us go back to the question. Did the Big Bang create two universes on moving forward and one maching backwards? In thinking and discussing on the matter I brought up "that formalism of quantum theory naturally incorporats psychophysical parallelism" that this other universe could be viewd as ~A. Meaning our universe is A and the parallel universe would be ~A this a fair read? If we think of time then again (A &~A)? 
Continuing to follow Mr. Alam's led the connection between light and a "gravitional arrow" was raised as to how this could be possible. If as many do time only exists in mathematics. A  gravitional arrow would imply (I would think) some mass. I don't think "nothingness" has gravity. One more issue I am having a hard time conceptualizing is from the moment of ignition like a balloon the expantion of the universe in all directions energy, gravity, and light. This point is where I am having a problem conceptualizing. If at that moment of ignition and time does exist outside of mathematics, the ~A. universe was? Time was moving backward's to when? The universe is a nanosecond old and time is running backwords to when?
Can someone help please?
Peace and Joy
Douglas 
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Krishnan Umachandran
WRW Services LLP
An experiment to recreate the beginning of our universe suggests that as the Big Bang happened, it sent off another one, the mirror image of ours, with the 'arrow of time' running the other way. The experiment solves a key problem in theoretical physics: that all of its fundamental laws would work just as well if time is going the other way.
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Sundarapandian Vaidyanathan
Vel Tech - Technical University
Dear Professor Krishnan - Many Indian newspapers have given coverage to this debate. The source paper was written by Julian Barbour, Tim Koslowski, and Flavio Mercati in the Physical Review Letters (2014).
The Abstract of this celebrated paper is reproduced here:
"It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behavior of any kind that defines an “arrow of time.” We show that this is not so. The simplest nontrivial time-symmetric law that can be used to model a dynamically closed universe is the Newtonian N-body problem with vanishing total energy and angular momentum. Because of special properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a uniquely defined point into two halves. In each, a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Structures that store dynamical information are created as the complexity grows and act as “records.” Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and will only be aware of the records of one branch and deduce a unique past and future direction from inspection of the available records."
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Rajat Kumar Pradhan
Utkal University
Dear kamal,
Take heart- Time is a mystery even for a physicist ! You are in good company. Time is a psychophysical phenomenon. Without psyche, no time can be there, since nothing could be perceived without psyche. No ordering of events, no time, since no events are there in the first place.
In deep sleep, we kill time literally. We come back unscathed !
Time is succession of perceived events. No events, no time.
Regards,
Rajat
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Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
(1) To answer on this question is necessary firstly to understand – what is time? Some corresponding comments one can find in the thread “Is the flow of time an illusion?”, SS post https://www.researchgate.net/post/Is_the_flow_of_time_an_illusion/12
(…see “Space and Time” http://arxiv.org/abs/1110.0003 ..., etc.)
(2) To apply to this question, one can pay more attention on that in Matter of our Universe two rules/ possibilities “time” act – “true time” and “coordinate time”.
The true time rule establishes that for any change of any object – including the change of a position in the space - is necessary to spend “true time interval”. True time is not a dimension in the spacetime.
The coordinate time – which is 4-th dimension in our absolute (3+1) Euclidian spacetime – establishes that a coordinate time interval is spent if a material object changes only its internal state.
Since after “Big Bang” our Matter was created as “realization” of a huge portion of energy any material object is continuously changing – what is observed as continuous motion of all objects in the spacetime with identical speeds that are equal to the speed of light.
This evolution of Matter – which is eventually an evolution of the constant portion of energy energy (more correctly here, though, – the increase of concomitant to this process true time interval) – a human calls “the time flow”, or “time arrow” .
(3) In Matter there are two times,  because of indeed, every material particle (practically, if one remember about CP symmetry weakly broken for some exotic particles) is some reversible algorithm, so there are pairs “particle - antiparticle”; when it was proven, that the information can be changed without energy dissipation only if the change is reversible.
So for Matter’s evolution, at least in first approximation,  there is no necessity to spend some additional energy – it seems that Matter’s Creator was rather thrifty.
Since the algorithms command orders for particles and antiparticles are opposite, they move in the coordinate time in opposite directions, or “have opposite coordinate time flows” . But every particle moves in the true time only in one direction. So a particle and antiparticle can be in very different time points in the spacetime, but if they meet in the space they annihilate – all interactions happen only in the space and true time.
(4) So at a Big Bang indeed could two Matters appear – Matter and AntiMatter, and in the coordinate time they also move in opposite directions. But from that by no means follow that their true time flows would be different. Though every elementary particles in both Matters “lives” in opposite directions, both Matters evolutions would be practically identical, in certain sense: Big Bang – particles – atoms – stars – producing of elements – galaxies, etc.
(5) In some sense there could be something as “reverse evolution”, if, e.g., the start energy is small enough, so after Matters expansion will be, because of the gravity force, some compression – which will end in a corresponding collapse. But, nonetheless, all that will be within the same true time flow…
Cheers
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Demetris Christopoulos
National and Kapodistrian University of Athens
Tempest in skull:
  1. time is not such an extreme concept as is presented to be, see https://www.researchgate.net/publication/263104630
  2. our universe is just a local one, see: https://www.researchgate.net/publication/264537258
The above mentioned journal article is a good example of what we can read in a respectable journal if somebody is a big name...
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Rajat Kumar Pradhan
Utkal University
Dear All,
Since time is never seen to move backwards in the physical world, even its forward movement is only inferred from succession of perceived events in an ordered fashion, there can be no way but to assume that backward time is possible only in imagination, i.e. it can be possible mentally. In the psychic world we do have the ability to move backwards in time "DOWN MEMORY LANE" as they say. Thus Memory as a function of the psyche shows that backward time movement is possible in the psychic world, and not in the physical world.
The closest that we come to backward time travel is in the Feynman-Stuckelberg interpretation of negative energy solutions of field equations in QFT. Even in this case the backward time movement is only an assumed movement with negative energy (again in the mental world!).
All said and done we cannot escape the conclusion sooner or later that there is indeed a backward time arrow and a backward moving universe, not in the pahysical world but in the psychic world and it is an exact image of the physical world.
This will be dealt with in my forthcoming article.
Regards,
Rajat
2 Recommendations
Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
Dear Rajat,
the introducing of some “psychic world” doesn’t mean that this world “is an exact image of the physical world” – since there is no “backward time movement” as “an assumed movement with negative energy” in Matter.
Because of there is no “negative energy” in Matter’s reality. The “ Feynman-Stuckelberg interpretation of negative energy” is only an interpretation, which  work well in the QFT
– as, for example, the “time dilation” interpretation for the physical fact of slowing down of clocks’ tick rate if a clock moves in the space in the absolute Matter’s 4D Euclidian spacetime. This SR interpretation works well in a very lot of cases if a system of material objects, including rules and clocks, constitutes a rigid system; and, since Earth’s gravity supports corresponding rigidity in practically any usual now [physical] situation, there exist so many “successful experimental verifications” of the SR.
Though, of course, in the reality the time is absolute and is so independent on anything material or mental object and by no means can be “dilated”. But, again – in many technical cases the interpretation works well.
Analogously the Feynman-Stuckelberg interpretation works well, though in reality it is incorrect –“backward time” isn’t an effect of some “negative energy”. The antiparticles indeed move in the coordinate time in the negative direction; simply their internal algorithms are reverse relating to the particles. But, because of in Matter both time rules act, in the true time, antiparticles - as well as everything else – move only in one – positive by definition – direction.
The true time works universally on whole “Information” Set. To what about the true time is in the SS posts here (and in RG elsewhere), there is also some other definition of the time as an universal rule that was done by (if I remember the name correctly) Raymond Cummings: “time is what keeps everything from happening at once". And the “happening not at once” means that if something changes, then a parallel “true time no-zero interval” obligatorily exists. And it exists objectively, without any dependence on – there is or not some consciousness, the consciousness sleeps or not.
So, since any consciousness is some informational dynamical system/program – some element of the Set, any psychic process spends always corresponding true time interval, when it seems as for processes in the consciousness – in contrast to Matter - the coordinate time doesn’t work.
So “…a backward moving universe, the psychic world… [is as] … an exact image of the physical world..” relates to the objective reality only till the QFD works correctly. But the QFD – as well as the cosmology are created basing on the postulate that real 4D spacetime is pseudoEuclidian (pseudoRiemannian), with partially imaginary metrics, what is rather evidently wrong…
Cheers
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Rajat Kumar Pradhan
Utkal University
Dear Sergey,
You are right in regard to Stuckelberg-Feynman interpretation. Please wait for my future publication which will extend it to the psychophysical realm. The world of ours is a representation before us of "something we know not" through the avenues of the senses and through the perceptive apparatus of the psyche, which processes the signals received through the senses. There is no way one can escape this in science.
"Particles and antiparticles all move in positive time-direction" ---This is alright in QFT.
But negative energy particles are not found and they must be accommodated somewhere. The antiparticles serve the purpose well.
Regards,
Rajat.
2 Recommendations
Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
Dear Rajat,
- well, though I should make some remark relating to “Particles and antiparticles all move in positive time-direction" ---This is alright in QFT.
The point is that in the SR – what is a base for the QFT – there exist two times also – “proper time” and “simple time” – the SR in a number of aspect relates to the reality; including in that the spacetime is 4D there, but it is "pceudo Euclidian". Thus both SR’s times are/contain some mixture of the real times that indeed work in Matter, e.g., - of “true time” and “coordinate time” .
An example - in the Feynman diagrams quite correctly ,say, a positron is marked as moving back in time. But in this theory such a designation is considered as something “virtual”, while in the QFT-reality the particle moves in the positive time direction only.
But in indeed reality antiparticle indeed moves in the coordinate time in negative coordinate time direction, while it – and everything in Matter – moves in the positive true time (i.e. outside the 4D spacetime) direction only.
Cheers
2 Recommendations
Rajat Kumar Pradhan
Utkal University
Dear Sergey,
What you say is right.
But, I am talking of "cosmic time" (true and simple) in the sense of Big bang occurring at t = 0 and not the fictitious "coordinate time".  This time that originates ( or assumed to originate) with bigbang is the time for us and the psychic recordings all move backwards as this time evolves forwards. this is my position.
Regards,
Rajat
1 Recommendation
Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
Dear Rajat,
Well, but I should make some terminological remarks again. The “true time” is not “simple time”; the “coordinate time” isn’t “fictitious time”, and any time doesn’t “evolve”.
The true and the coordinate times both are real and both are some real logical rules/ possibilities that are necessary for every material object and for Matter (all are objectively real) to evolve as they evolve and as we observe this evolution.
“Simple time” – that’s only my concrete term here to define the time in the SR – may be it is not too felicitous, but here – let it be. And though the “simple time” in the SR is, in certain sense, the proper time of clocks of given reference frame in the absolute spacetime (absolute reference frame), Matter is built so, that for an observer in his RF this time is seen in many experiments as “true time”. Though that isn’t correct principally – therefore the SR is self-contradictory (first of all – the Dingle problem), since this time – as “true time” doesn’t logically allow reversible operations, etc.
And – a next terminological point – as some universal (“coordinate time” – universal for Matter) rules/ possibilities, both times exist forever, independently on - originated or not something, including, say, the Big Bang; the Big Bang (if that was) – and following Matter’s evolution - happened (happens) in accordance with these times. So the times cannot “evolve”, but one of the rules’ applications, i.e. “the time interval” indeed accompanies Matter’s evolution, and, indeed, originated as “t=0” at possible Beginning; observers – and physicists – use this interval’s value as the variable “time”.
But this interval doesn’t evolve, it bluntly – and ultimately uniformly - growths.
And, again, – the coordinate time isn’t “fictitious” – it is (as the time interval) just “the time is what clocks show”. But if somebody makes a pair of clocks – one material and other antimaterial, the clocks’ pointers will turn in opposite directions; but the ways, which the pointers’ ends pass – and corresponding true time intervals – will be identical…
Cheers
3 Recommendations
Rajat Kumar Pradhan
Utkal University
Thank you Sergey for your interesting remarks on my comments.
The coordinate time is fictitious since it is relative to the frame used.
Of course everything is relative to a frame. But this coordinate time can be transformed in many ways and no definite value can be assigned to it.  But what you say is right. After the bigbang it is  continuously growing interval that we are taking in SR as the time coordinate. I like your insightful remark.
Regards,
Rajat
2 Recommendations
Sergey Shevchenko
Institute of Physics of the National Academy of Science of Ukraine
Dear Rajat,
well, but again - – the coordinate time isn’t “fictitious”, that is the “time rule/possibility” which is real, as the true time rule is (though it is “more possibility, then rule”, comparing with the true time); and, besides, – it is (as the time interval) just “the time is what clocks show” – as that is established just in the SR.
All clocks show their own proper times in the absolute 4D Euclidian spacetime (and so show their position on the real coordinate time axis in the spacetime – but what is usually unknown for an observer since he doesn’t know his absolute speed), so a clock ticks more frequently, if it has lesser spatial (and so larger temporal) speed in the spacetime.
But in the SR, as the consequence of rather evidently logically self-inconsistent postulate that all [inertial] reference frames are totally equivalent, any SR-observer believes that he is at rest (at all, always and everywhere,  since “there is no absolute spacetime and corresponding absolute RF”) an his RF’s clocks tick faster then any other, if other clock moves relating to him. What is, of course can be not correct, in reality only one condition determines the clock rate – see above (that’s, besides, the solution of the Dingle problem; moreover, in the SR the time coordinate is, in fact, imaginary; etc.)
Nonetheless, just the SR- observer uses the clocks’ showings as clock’s coordinates in the Minkowski spacetime, at that – since he doesn’t know his real speed in the real spacetime – he calibrate the clock practically arbitrary.
But in reality any observer, for example a human, can measure his absolute speed (see “To measure the absolute speed is possible?” http://viXra.org/abs/1311.0190) if has money $30-$50 *106; further - calculate, using the measured speed value and established preliminary clock rate, corresponding tick rate for the clock at [spatial] rest; and, further, re-calibrate his clock so, that the clock showing would be as the clock is at [spatial] rest in the spacetime and so the clock will show true time till will not be accelerated…
Cheers
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Our questions are:
1) Space-time is interconnected, confined by the speed of light c =Δx /Δt; the factor a(t) that determines space-time is of a kinetic nature; what makes it relevant only to time (it affects all of space in the same way as time passes) [4] and not to space?
2) Can the Einstein field equations essentially be written as two separate equations, the bending effect equation and the expansion effect equation?
3) How does Space-Time know to distinguish between energy and dark energy if Space-Time Curvature and Expansion are both different properties?
4) Can local Space-Time Curvature geometrically affect expansion if it appears to be strongly curved?
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Notes
** How the View of Space-Time is Unified (4) - Is Space-Time Expansion a Space-Time Creation?
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Refererncs
[1] Grøn, Ø., & Hervik, S. (2007). Einstein's Field Equations. In Einstein's General Theory of Relativity: With Modern Applications in Cosmology (pp. 179-194). Springer New York. https://doi.org/10.1007/978-0-387-69200-5_8
[2] Weinberg, S. (2008). COSMOLOGY (Chinese ed.). Oxford University Press.
[3] Peebles, P. J. E., & Ratra, B. (2003). The cosmological constant and dark energy. Reviews of Modern Physics, 75(2), 559.

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