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Quantum fluctuations in space, science, exploration and other cosmic fields... served up regularly by MSNBC.com science editor Alan Boyle since 2002.

Alan Boyle covers the physical sciences, anthropology, technological innovation and space science and exploration for MSNBC.com. He is a winner of the AAAS Science Journalism Award, the NASW Science-in-Society Award and other honors; a contributor to "A Field Guide for Science Writers"; and a member of the board of the Council for the Advancement of Science Writing.

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Time after time travel

Posted: Friday, November 24, 2006 6:05 PM by Alan Boyle

Moviegoers received a double dose of time-travel fiction this week, with the present-day(s) thriller "Deja Vu" on one hand and "The Fountain," a time-trippy love story, on the other. As we discussed earlier this week, "Deja Vu" reflects a bit more of the current scientific thinking about what time travel into the past, a.k.a. retro-causality, might look like if it were possible. But if you ask the physicists to list their favorite time-travel tales, the ones they mention are golden oldies going back to the days of classic "Star Trek."

Of course, the Starship Enterprise's crew was ready to travel back in time at the drop of a hat (preferably one that covered Spock's pointy ears). But for string theorist Brian Greene, a Columbia University physicist who was a consultant for "Deja Vu," the episode that stands out is "City on the Edge of Forever," in which Dr. McCoy changes mid-20th-century history and it's up to Kirk and the gang to change it back.

As history is rewritten and re-rewritten, items (such as the Enterprise) disappear and reappear to reflect the universe's changing timeline. That follows the same time-travel conventions put in place for the "Back to the Future" time-travel films (and "Star Trek IV: The Voyage Home," for that matter), but does not follow the two main scientific possibilities that Greene and others have laid out:

  • The flow of past events is unchangeable, because any effect time travelers had is already reflected in the universe (Stephen Hawking's "chronology protection conjecture").
  • The flow of past events can be changed - for example, by sending a signal or a person through a wormhole. But that merely creates a new branching in what is a multitude of universes. The universe from which the time traveler was sent back is unchanged (the "many-worlds" interpretation of quantum mechanics).

Thankfully, as a work of fiction, "City by the Edge of Forever" doesn't have to pass strictly scientific muster, and that's why Greene felt confident giving the episode an endorsement. "Put me down for that one," he told me.

Other physicists who have written about time travel (such as the University of New Hampshire's Paul Nahin and Princeton's J. Richard Gott III) have given thumbs-up to an even more unlikely entrant in the time-travel oeuvre: "Bill and Ted's Excellent Adventure," with a pre-"Matrix" Keanu Reeves as one of the slacker stars.

Of course, Greene is partial to the time-travel films in which he's been personally involved, including "Deja Vu" as well as "Frequency," where he played a time-warped cameo role. And Greene said he's involved in yet another time-travel movie project titled "Mimzy," based on the short story "Mimsy Were the Borogoves" and slated for release next year.

The physicist said he doesn't want to give the impression that time travel is a major focus of research. "Many of us see it as a lingering question in the back of our minds," Greene said. He and his colleagues are even more intrigued by the trippy concepts spawned by string theory - indicating that the universe could follow any of 10500 possible courses, and that our course seems to be going down just the right path to allow for the development of stars, galaxies and life.

For some physicists, this multiplicity of potential universes - a seemingly unprovable claim - is a scientific scandal. The criticism is voiced loud and clear in books such as "Not Even Wrong" and "The Trouble With Physics." But for Greene, the idea poses a challenge well worth accepting.

"If this is a feature of the theory, then it's one we have to better understand," he said. "If it is true, it's telling us something mind-bogglingly remarkable about the universe."

The deeper implications of physics will surely set the tone for the start-up of the Large Hadron Collider, scheduled for next year - as well as for the World Science Festival, a science-plus-art celebration scheduled to make its debut in New York in 2008. "I'm one of the festival's co-founders," Greene said.

Here are some of your own thoughts about time travel:

Stuart Greene: "I have always wondered myself about time travel. I was fascinated as a kid. My take on time travel is very simple.

"Let's assume in the future that there is a way to travel back in time. This would imply that time travel would have been infinitely possible, given that someone from the future has already traveled back to the past. And this person who traveled from the future would have transferred this and other technologies and knowledge to the past. From the fact that we know very little about the future, we could conclude that either the traveler from the future is either very selfish (giving his past self scores of the World Series), or he thinks that our knowledge of the future would be detrimental, or there is no future, or time travel is impossible. ...

"My opinion is that time is linear, and the only time travel possible would be to see images of  the past by traveling faster than the speed of light. Kind of sad, but way more interesting not to know the future."

Darrell A. Jones: "If you get a chance, go back and research the Philadelphia Experiment - not the movie, but the actual military research. ... It revealed that the time-travel experiment started out as a failure, then did succeed, yet was still a failure for they had to [go] back because of the hole that had developed and was left open. The scientist involved left a message and documents behind, so that the government did not get the complete files. ..."

Harold Estep, Clarksville, Tenn.: "I once read a novel which dealt with time travel.  It was titled 'Time and Again,' by Jack Finney.  Over the years I have read it three times because the author's premise of how to travel back in time, seemed so believable.  If you like this genre of fiction, I highly recommend you obtain a copy."

That sounds like this month's selection for the Cosmic Log Used Book Club, which highlights books with cosmic themes that you should be able to find at your local library or used-book shop. Before he passed away, Finney wrote a sequel to the novel titled "From Time to Time."

Do you have additional recommendations for time-travel tales - or observations on the facts and fictions surrounding the flow of time and causality? Feel free to add your comments below.

 

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Sir, you might be interested in a photographic time travel. Check 99 multiple exposure, phusion images created only within the Canon A-1 - no editing see wwwphotomysticart.com ENJOY Best Regards Jim McCrea
Some other great books dealing with time travel are several by Robert Heinlein- he also deals with the 'multiple universes' theory in "Number of the Beast", and in fact makes most of his other stories cross by the end of the book by simply making all the separate universes accessible. I think some of the books he wrote after that one feature a group that tries to deal with the various universes and some consequenses of lines being crossed. Definitely not books for children, as there tend to be a lot of sexual themes, but for adults that can handle innuendos and a little unconventionality, they're wonderful, funny, well-written books.
Kate and Leopold; that's the time-travel movie you want to see. Cool and charming, it's the perfect date movie.
Recommendations: Robert Heinlein's short story "All You Zombies" The first, best and tightest intersecting time travel plot. Kage Baker's series "The Company." Wild romance, horrible villians and and incredibly complex plot revealed like a 500,000 year strip tease. Audrey Niffenegger's "The Time Traveler's Wife." A love story, not a science fiction novel. I thought I had read every conceivable use of time travel as a plot device, but this one rises to a new level of inventiveness.
Robert Forward wrote a very cerebral hard-SF novel called "Timemaster" that dealt with the chronology protection scenario. In the novel, the mechanism for protecting the timeline is that increasingly improbable events become probable to prevent paradox--in other words, paradox would be asymptotic and exert a "repulsive" force on events. One of the bizarre concepts Forward explores is what could be called "looping self-correction," where someone or something in danger of causing paradox is knocked through the time portal into the immediate past by its immediate future self, and then goes on to repeat the same event by knocking the recent-past version of itself through the portal. As mind-bending as it is, the offending object or person would witness the same events twice within seconds, but from two alternate perspectives. A film that must be mentioned is "Primer"--quite possibly the hardest, most rigorous, most realistic, and most intellectually stimulating movie about time travel yet created. We see a realistic process of discovery as two engineers stumble on a strange phenomenon, feel their excitement and awe as the implications begin to dawn on them, their wonder as they step into the (recent) past, and share their confusion and fear as the layers of causality begin to pile up. Two central premises are that (a)you can only backwards, and (b)you can only go back as far as the time machine has been in operation. It also seems to say that the machine has to be in continuous operation at all points on the included timeline. Then it examines (quite obliquely) what happens if you put a time machine within a time machine. This is one of the few films that dares to present a universe more complex than its audience is prepared to immediately understand, and for that it's a classic.
Hi Alan Ok. The ultimate time-travel novel is "The End of Eternity" (1955) in which Isaac Asimov describes a trans-Time organisation, Eternity, which modifies the flow of history from the 28th Century to the 70,000th Century. Initially it was begun in the 30th century by some scientists anxious to avoid a nuclear war by travelling back and causing a key Senator to have a car-crash. But the urge to control for the sake of security gives birth to an organisation existing outside of Time which adjusts history "selflessly" to achieve maximum happiness for all. But by doing so they distort social evolution, suppress technology that's "too dangerous" (too hard for them to control) and stifle humanity's Outward Urge. The people of the 120,000th century place a time-block on Eternity at the 70,000 Century so they can't change human history directly after that time. In those "Hidden centuries" humans develop hyperdrive and discover the Galaxy has been totally colonised by other species. Humanity retreats to Earth in despair and dies out by the 150,000th century. But another faction find that 'Eternity' is an anomaly in all the threads of possible history, a closed time loop which generates itself by sending the 'inventor' of time-travel back to the 24th Century to bring time-travel and 'Eternity' into being. Thus 'Eternity' can be destroyed in all possible worlds by undoing the loop. Literally the time-travel story to end all time-travel stories.
The golden age of science fiction is whatever was popular when you were a kid.
what about prophsey? Future events and people seen hundreds or thousands of years before it happens.
I don't think you can discuss great works that involve time travel without mentioning "Contact" by the late Carl Sagan. It is a very fascinating story because it could concievably happen tomorrow.
I made a comment on this yesterday, before the article was published.
Rosswell's observation of time: "Time emanates from the center" Careful study reveals that time flows from the center of matter, outward, evenly, in all directions (not from left to right or twords a gravity well). It's the opposite of gravity, so we have named the force "Levity". It is a dimension, not a particle. It accelerates from zero to the speed of time in no time flat. Phillip Jose Farmer was correct when he said (Time's Last Gift) "Everything that can be done in the past HAS been done in the past", because we can measure and visually confirm the uniformly onward and outward passage of time. Stephen Hawking was correct when he incorporated time as a dimension in String Theory, and Albert Einstein was correct when he said "It's like an old man with a teaspoon of sand, you don't know WHAT he's got!"
One of the problems with time travel of more than a few milliseconds is the fact that our planet, solar system, and galaxy are in constant motion. If it were possible to travel in time, a traveler on the planet earth would also have to travel in absolute 3 dimensional space by the same amount that the planet has moved during the so called time displacement. For example, if you make the assumption that our sun is stationary in absolute space (which it is not), then a time traveler who goes back in time by six months must also travel the diameter of the earth's solar orbit to the location of the earth six months ago--on the opposite side of the sun! When the motion of the sun around the center of the galaxy is considered along with the motion of the galaxy within the universe--well you get the idea.
I DO BEALEVE IN A DEJA VU AS YOU SAY, IN THE BOOK JACK FINNEY, SOMEWERE IN TIME IT CANT HAPEN, YES, ALWAYS, LIKE THE THE SIMPHONY OF RACHMANINOFF, RHAPSODIA ON A THEME OF PAGANINI.
Of interest to this discussion is Larry Niven's short story/essay, "Theory and Practise of Time Travel," in a 1970's collection "All the Myriad Ways." In it, he theorizes that if it's possible to go back in time and change the past, sooner or later the past and future will stabilize on a timeline in which no time travel machine is ever invented and in which no other way of changing the past is possible, because that's the only possible stable configuration.
One of my favorite time-travel cartoons (by Dan Piraro) has a confused-looking guy in a fancy sports car pulled over in front of the saloon in an old Wild West town. The sheriff is standing next to him, saying "Son, do you have any idea how fast you must have been going?"
"time" is the measurment of the relative distance between two moving objects. thus, the "measurement" of "time". it is more like "inches" then any unexplainable "dimentions". And it doesn't "go" anywhere, especially backwards.
Although it is not exactly "travel", it is apparent that "motion in time" is akin to "motion in space". It is certainly something that is accessible to physicists, as there are plenty of ordinary examples. See http://www.geocities.com/thirdgenerationphysics/ for a short, student paper on the topic.
I have a hypothesis on the subject of Time. I just spent the past week hashing through it on google group's minds-eye group. The topic is called "What is Time" by Hadoz. Check it out!
Let's add "Thrice Upon A Time" by James P. Hogan. Changing time by sending messages to a past version of yourself.
One of the stories that I find my mind returning to over the years is "The Man Who Folded Himself." I guess it's considered a classic though I never realized it. I see it's now back in print too. http://www.chtorr.com/books-folded/folded.htm
Good Time Travel stories- check out Dr. Who. btw- you do realize that Deja Vue is a remake of the old UPN TV show "7 days".
I like the idea of combining the themes from "Back to the Future" where if you change something in the past, it creates an alternate timeline askew of the one you were in and "The One", where there are a set number of Universes that directly relate to your own. I actually think that there are INFINITE universes, but only so many that share a likeness to our own, as in "The One" where Jet Li kills himself in over 100 other universes.
Bill is on the mark with his comment about the three-dimensional compensation required for time travel. A "time machine" will need to be capable of operating not only in time, but also in all the other dimensions and even outside them. If you can crack time, you can go anywhere, anywhen, and you would, because you could. (But I don't think for a moment that this is a the ultimate possible technology.) Time machines are not a technology you'd use to go pick lotto numbers or pick up groceries, or perform a hit, and it's not something you'd use to go save Kennedy or Lincoln. Why do those things if you have the ability to cross to the other side of the galaxy? Why meddle in such affairs? It is important to get away from the "history change paradox" element of time travel because it's far too limiting. Oh yes, there could be local paradoxes. But none that effect more than this earth. No paradox that matter, in other words. The universe is just too huge -and that's before adding in the multiverse. Yes, time and paradoxes works on the small scale, but "time" doesn't care as much as you think about any of it. What happens here, whether A man lives or dies differently, simply doesn't matter at all. If a time traveler managed to obliterate the Earth before it ever formed, the universe would take no notice. So what WOULD it take to create and operate such a vehicle? What would it look like? It's not likely to be a police box or a DeLorean. Would we know it if we saw it? I'm about to mention two words that will get me labeled as a fruitcake by some of you, which is ironic because you're happily reading about time travel and that's not TOO exotic, but what I am about to say will be. Flying saucers. They're not just visitors from space: they're visitors from time or from other dimensions. What the heck does that have to do with time travel? The machines they use are evidence that it can be done. Are we smart enough to figure it out? Will we be allowed to? Interesting that Asimov apparently wrote this story. I will seek that book.
May I humbly suggest the A.E van Vogt's Weapon Shops of Isher get a nod? Starts out with one hell (and I mean that literally!) of a time trip. If you are going to tootle around in 4 dimensions (plus a few to contain them!), you must be able to locate yourself or re-locate if you prefer, to account for the new or old location. Just figure the movement of any point on Earth and factor in rotation, Earth-Moon orbit, Sol-centered orbit, Solar galactic orbit, galaxy movement, universal expansion, etc., etc., etc. Best compute verry carefully before you jump!
An outstanding fiction book to read is, "The Light of Other Days" written by Arthur C Clarke and Stephen Baxter. ***The Light of Other Days follows a soulless tech billionaire (sort of an older, more crotchety Bill Gates), a soulful muckraking journalist, and the billionaire's two (separated since birth) sons. It's 2035, and all four hold ringside seats at the birth of a new paradigm-destroying technology, a system of "WormCams," harnessing the power of wormholes to see absolutely anyone or anything, anywhere, at any distance (even light years away). As if that weren't enough, the sons eventually figure out how to exploit a time-dilation effect, allowing them to use the holes to peer back in time** The ending is shall we say Out of the world or time!
What? No one remembers "A Wrinkle in Time"? Where the author describes time as a ribbon. Much like Frank Herbert's "Dune." Traveling without moving. Loved all your comments - thought I'd add my own.
H. Beam Piper has an excellent parallel earth book called "Paratime". He doesn't discuss the theory at length, but gives a good feel for what travelling across the different iterations of the earth would be like.
You know, the thought that there are infinite possible timelines has been around for as long as there has been science fiction. It makes sense, if you think about it. Our universe is just one string of probabilities that has been followed. Every universe represents another string of probabilities that has also occurred but that we (as yet) have no access to. Thus, our universe is perfect for us because it occurred along the lines of probability that allowed it to happen. It is not that we were extremely lucky; it is that this particular version of the universe exists this way because the probability existed that it could be this way.
Good Grief! Someone pulled out Heinlein's "All You Zombies" but not one of you came up with "Doctor Who"? Now I agree that "...Zombies" is easily the best short story time trip, but Doctor Who is by far the most fun on TV.
Time travel has always been fascinating and frightful to me. Who wouldn't like to go back and change some things? A book about time travel that I read as a youth was Door Into Summer by the aforementioned Robert Heinlein - nothing major, just a lot of fun. Of course, can't forget H.G. Wells, can we? Andre Norton wrote a series starting with The Time Traders, again, more for adolescents than adults but still fanciful and fun to read. Interesting topic.
If talking time-travel, you must include Doctor Who and his TARDIS. And continuity is not a big problem with the Doctor's stories, one big,happy mess.

- Einstein tethered us to linear time because he linked relativity to the speed of light (velocity). You cannot have a negative velocity so, according to Einstein, time is ever moving forward in a linear way.

- "Time", as we know it, is a contrivance of Man (stuff happens and we devised a means of measuring durations, like 'quarters' in a football game). Time, as we measure it, is merely increments of eternity.  

Time travel is a realistic possibility, however, time travel does not include a 'device' (eg. time machine). It (time travel) is a function of the soul (or spirit) which exists both within and outside of time simultaneously.

- Our 'ego/physicalbody' is a time-based entity which, when transcended, liberates us to navigate to any imaginable space-time context. In brief, time travel is an inherent characteristic (ability) of our 'true self'.

- The 'reality of time' is that is is always "now" and the human condition seems to be that our minds and physical bodies are not (usually) in sync.  Our bodies are always 'present in the now' and our minds are always (usually) in either the past or future. When the mind and body merge into the instant then magic happens and eternity reveals herself.

"Master of Space and Time". This is a funny good novel about how you should be carefull what you wish for when your exploring quantum phyisics and time travel.
The idea of multiple universes splitting off was covered in a short story I read some years ago. Once it was 'discovered' that mutliple universes existed and they could be travelled to people went nuts and did all manner of crazy things knowing that another branch of the universe had the same individual doing the not so crazy stuff. The story not withstanding, multiple universes seems plausible; if we accept that the universe (multiverse)system is always headed towards a state of greater entropy then the sprouting of multiple universes in a completely random fashion as time expands fits with the entropy idea. But this also implies that there is a (or a multiple of) steady universe(s), contracting universe(s), ever expanding universe(s) which brings me to causality: is the very act of our observation of space-time in our universe affecting the possible outcome(s) of our universe as time expands? Are we in effect willing a steady state universe to exist or an expanding one with each new super collider experiment (time branching as we do so?)....ponder on great minds...
Another interesting time-travel book is Pastwatch: The Redemption of Christopher Columbus by Orson Scott Card, where people in the future of an Earth destroyed by ecological damage have discovered a way to view the past, and then to change the past to attempt to stave off ecological destruction of the world, even while knowing that by changing the past they will destroy the "world" or universe in which they live.
My favorite time travel movie hasn't been mentioned yet: The Final Countdown, with Kirk Douglas and Martin Sheen. The premise isn't exactly subtle: the nuclear carrier USS Nimitz is somehow thrown back to 1941 right before Pearl Harbor. What's great about the movie is that you give the screenwriters one giant Tooth Fairy (what natural phenomenon could possibly cause an aircraft carrier to travel in time?) and from there everyone reacts like real people would react: confusion, disbelief, arguments over "what do we do now?", etc. [Warning: Spoilers Ahead!!] Also, Hawkings' chronology protection conjecture is strictly enforced.
"Timescape" is an excellent book which addressed the problem of earth's movement: tachyons were beamed at a portion of sky where earth had been located in the past. The effort was successful in warning the past about what the future had become: a polluted and destroyed world. The book's ending is enigmatic - in all, a very satisfying read!

How about this...there is really no such thing as "time". All reality exists simultaneously. It is only our limited perspective that causes us to divide events into digestible past, present and future pieces. If we had the "God view," we would see all things at once. It's a perspective thing.

Considering "time" as the fourth dimention, when I was a kid, I read a book about dragons. Turns out they can slide uphill because they are able to keep the tip of their tail in the fourth dimension. Wish I could find that one again.

"Time" as we perceive it is not really a dimension or measurement; it is only the way our brains interpret movement, which is itself described in terms of time and space, which is described in terms of tim and movement.They are all expressions of the basic idea of existence,which is the material of existence itself.
Another interesting Niven short story is: "Rotating Cylinders and Global Causality Violation" whose title is taken from that of a paper by theorist Frank Tipler where he shows how a long neutron-star dense rotating (at about half lightspeed) cylinder would create a region near its middle where travel into the future or past should be possible (limit: You can't go back to a time prior to the existence of the cylinder, or forward beyond the time the cylinder is shut off or destroyed). Roberr Forward also describs this device in some of his non-fiction works. http://en.wikipedia.org/wiki/Tipler_Cylinder The Niven story, however, suggests that nature 'conspires' to preserve causalty and prevent paradoxes, even where a workable time travel device is possible, and how one side in an interstellar conflict tries to exploit that tendency, rather than the machine itself...
If you want to look at other views of time, Clifford Simak's "Ring around the sun" explores the many worlds theory with a different take and James Blish's "Quincunx of Time"(originally:Beep) explores the idea of FTL communications with a twist. Poul Anderson played with Time Travel in his "Time Patrol" storied, but his most clever take was "There will be time". And a side-note on Asimov's "End of Eternity" that gets neglected is that it is actually Chapter Zero of his Robots/Galactic Empire timeline. TV-wise, I'm surprised nobody has mentioned "The time tunnel", inspired by a story of the same title by the great Murray Leinster, who also did "Sideways in time". Also worth checking out is the surprisingly good pilot for the proposed remake of the show a couple years back (its in the Time Tunnel DVD set) where an experiment in nuclear fussion creates a time vortex that cuts across time and alters history irrevocably. (Hint: Boston Yankees...) I suspect that as fun as these speculations all are, reality will, if ever figure it out, will turn out to be even more mind boggling. :-)
comment on a comment, kate and leopold is not the movie you want to see. ever. do not watch this movie. warning! spoiler! it sucks...
All I did was scan the entries, but is NO ONE going to point out that the Philadelphia Experiment is about as real as 9/11 conspiracies and unicorns? The US Navy ripped a hole in space-time...I'll believe that as soon as someone shows me these "holographic cruise missiles".
Heinlein's "All you Zombies" has been mentioned, but he also has another story "By His Bootstraps" involving multiple time travel trips with overlapping appearances by the main character. At one point in time, three versions of the same character (each from a different point in his time-line) overlap in a college dorm room. Turns out his future self sent an earlier version back to recruit himself to step into the future. It works, and in the future he starts the chain up again. We experience the one scene three times from the viewpoint of each version of the same character. At no point does he change time, but does travel back and forth between a specific day in our era and a ten year period far in the future.
I have to put my vote in for the lovely 1980 movie starring Christopher Reeve and Jane Seymour. I believe it's called "Time After Time." Come to think of it, wasn't there another movie with the same title that starred Malcolm McDowell in the mid-1980s? McDowell's character was chasing Jack the Ripper.
I believe time travel is indicated in the Bible, along with other alien technologies. How does one know what will happen in the future unless one has either been there, or informed about future events?
these discussions remind me of deja vu all over again.
Don't forget The Final Countdown as a good movie about a modern aircraft carrier arriving near pearl harbor hours before the Japanese attack, only to find they can't actually change the past.

Wow.  I am impressed with all of the comments, recommendations and diverse thought in this string. Some pretty "heady" stuff that definitely causes my cranium to expand a bit as I am sitting here trying to get a mental handle on all the posts.

Being an avid sci-fi reader and viewer I am familiar with many of the books, films, etc. and will make a determined effort to begin hunting for those I am not.

If you are into this kind of material I would not hesitate to recommend a very lengthy but equally worthwhile book called "Cities In Flight".  It is by James Blish, whom many of you might know as a writer of many of the Star Trek (after the original series ended). The book deals with many issues relevant to today's and also encompasses the challenge of travel through time.  In fact the ending (without being a spoiler) deals with the paradox of time actually being a finite process.    

I don't believe time travel would be possible.  If we take the universe as it is as the collapse of the sum of all wave functions (observation) that could have resulted in the universe as it is then one would have to "uncollapse" the wave functions, which would take an extremely large, if not infinite, amount of power.  Furthermore, the entropic state of the universe must also be taken into account.  The traveler would be going from a less ordered state to a more highly ordered state (from higher entropy to lower entropy), which takes energy and, if one considers the entropic state of the universe, a vastly large amount of energy.

So, if time travel is possible then it would be as an observer only, or it is not possible.  It may be argued that the traveler has already impacted the past and therefore the wave functions contain that information, but, as stated before, one must consider difference between the entropic states.



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