Omega Point (Tipler)

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The Omega Point is a term used by Tulane University professor of physics and mathematics Frank J. Tipler to describe what he maintains is a physically-necessary cosmological state in the far future of the universe. According to his Omega Point Theory, as the universe comes to an end at a singularity in a particular form of the Big Crunch, the computational capacity of the universe (in terms of both its processor speed and memory storage) increases unlimitedly with a hyperbolic growth rate as the radius of the universe goes to zero, allowing an infinite number of bits to be processed and stored before the end of spacetime. Via this supertask, a simulation run on this universal computer can thereby continue forever in its own terms (i.e., in "experiential time"), even though the universe lasts only a finite amount of proper time. Tipler states this theory requires that the current known laws of physics are true descriptions of reality, which he says implies that there be intelligent civilizations in existence at the appropriate time in order to force the collapse of the universe and then manipulate its collapse so that the computational capacity of the universe can diverge to infinity.
Tipler identifies this final singularity and its state of infinite informational capacity with God. The implication of this theory for present-day humans is that Tipler maintains this ultimate cosmic computer will be able to run computer emulations which are perfectly accurate down to the quantum level of every physically-possible universe, and any life contained in them, from the start of the Big Bang (which Tipler states starts at zero informational capacity and diverges to infinite informational capacity as the universe progresses in time, thereby allowing sufficiently later states of the universe to perfectly render earlier states). According to Tipler, from the perspective of the recreated inhabitants, the states near the Omega Point would represent their resurrection in an infinite-duration afterlife, which could take any imaginable form due to its virtual nature.
Assuming that achieving the Omega Point is physically possible, Tipler says this would be accomplished by "downloaded" human consciousness on quantum computers in tiny starships that could exponentially explore space, many times faster than biological human beings. Tipler argues that the incredible expense of keeping humans alive in space implies that flesh-and-blood humans will never personally travel to other stars. Instead, highly efficient uploads of human minds ("mind children" as Tipler calls them, they being the mental uploads of our descendants, or of ourselves[1]) and artificial intelligences will spread civilization throughout space. According to Tipler, this should likely start before 2100. Small spaceships under heavy acceleration up to relativistic speeds could then reach nearby stars in less than a decade. In one million years, these intelligent von Neumann probes would have completely colonized the Milky Way galaxy. In 100 million years, the Virgo Supercluster would be colonized. From that point on, the entire visible universe would be engulfed by these "mind children" as it approaches the point of maximum expansion. Per this cosmological model, the final singularity of the Omega Point itself will be reached between 1018 and 1019 years.[2]

History

Tipler has published his Omega Point Theory in a number of peer-reviewed scientific journals since 1986.[3][4][5][6][7][8][9][10][11][12][13] The first book wherein the Omega Point Theory was described was 1986's The Anthropic Cosmological Principle, written by astrophysicist John D. Barrow and Tipler, wherein they concluded the book by writing that[14]
if life evolves in all of the many universes in a quantum cosmology, and if life continues to exist in all of these universes, then all of these universes, which include all possible histories among them, will approach the Omega Point. At the instant the Omega Point is reached, life will have gained control of all matter and forces not only in a single universe, but in all universes whose existence is logically possible; life will have spread into all spatial regions in all universes which could logically exist, and will have stored an infinite amount of information, including all bits of knowledge which it is logically possible to know. And this is the end.
In an endnote to the above paragraph, Barrow and Tipler added that "A modern-day theologian might wish to say that the totality of life at the Omega Point is omnipotent, omnipresent, and omniscient!"[15] The first book solely concentrating on the Omega Point Theory was Tipler's The Physics of Immortality in 1994.[16]
Physicist David Deutsch in his 1997 book The Fabric of Reality defends the physics of Tipler's Omega Point Theory in Chapter 14: "The Ends of the Universe" (of which chapter concentrates mainly on the Omega Point Theory):[17]
I believe that the omega-point theory deserves to become the prevailing theory of the future of spacetime until and unless it is experimentally (or otherwise) refuted. (Experimental refutation is possible because the existence of an omega point in our future places certain constraints on the condition of the universe today.)
Deutsch later comments within a concluding paragraph of the same chapter regarding the synthesis of his "four strands" of fundamental reality, which includes the strengthened version of mathematician Alan Turing's theory of universal computation in the form of the Omega Point Theory:
It seems to me that at the current state of our scientific knowledge, this is the 'natural' view to hold. It is the conservative view, the one that does not propose any startling change in our best fundamental explanations. Therefore it ought to be the prevailing view, the one against which proposed innovations are judged. That is the role I am advocating for it. I am not hoping to create a new orthodoxy; far from it. As I have said, I think it is time to move on. But we can move to better theories only if we take our best existing theories seriously, as explanations of the world.
In 2007 Tipler's book The Physics of Christianity was published, which analyzes the Omega Point Theory's pertinence to Christian theology.[18] In the book Tipler identifies the Omega Point as being the Judeo-Christian God, particularly as described by Christian theological tradition, e.g., that the Omega Point cosmology when formulated in multiversal terms (of which multiverse conception isn't necessary for the physics upon which the Omega Point itself is based) is fundamentally triune in its structure: the Final Singularity (i.e., the Omega Point), the All-Presents Singularity (which Tipler states exists at all times at the edge of the multiverse), and the Initial Singularity (i.e., the beginning of the Big Bang), which Tipler identifies with the Father, the Son and the Holy Spirit, respectively (successively, the First, Second and Third Persons of the Trinity).
In this book Tipler also analyzes how Jesus Christ could have performed the miracles attributed to him in the New Testament without violating any known laws of physics, even if one were to assume that we currently don't exist on a level of implementation in a computer simulation (in the case that we did, then according to Tipler such miracles would be trivially easy to perform for the society which was running the simulation, even though it would seem amazing from our perspective). This proposed process uses baryon annihilation by way of electroweak quantum tunneling, and the inverse of this process, caused via the principle of least action by the requirement of the existence of the final Omega Point cosmological singularity. Tipler also proposes that the virgin birth of Jesus by Mary could be possible via Jesus being a special type of XX male who obtained all of his genetic material from Mary (i.e., an instance of parthenogenesis). If the Incarnation of Jesus Christ and the miracles attributed to him in the New Testament were necessary in order to lead to the formation of the Omega Point—and if the Omega Point is a physical necessity—then according to Tipler the probability of these events occuring is certain. Furthermore, Tipler proposes tests on particular relics associated with Jesus which, if the relics are genuine, could verify whether in fact said miracles took place via the aforementioned mechanisms. Tipler writes in this book that miracles, if they indeed exist, do not violate physical law, but instead are events which are so improbable that they would only be likely to occur within human history via the least-action principle if the universe is required to evolve into the Omega Point.
The Physics of Christianity shows a change from Tipler's earlier position within The Physics of Immortality regarding theism and Christianity. In the opening paragraph of Chapter XII: "The Omega Point and Christianity" of The Physics of Immortality, Tipler wrote the following:
To emphasize the scientific nature of the Omega Point Theory, let me state here that I am at present forced to consider myself an atheist, in the literal sense that I am not a theist. (A-theist means "not theist.") I do not yet even believe in the Omega Point. The Omega Point Theory is a viable scientific theory of the future of the physical universe, but the only evidence in its favor at the moment is theoretical beauty, for there is as yet no confirming experimental evidence for it. Thus scientifically one is not compelled to accept it at the time of my writing these words. So I do not. Flew, among others, has in my opinion made a convincing case for the presumption of atheism. If the Omega Point Theory and all possible variations of it are disconfirmed, then I think atheism in the sense of Flew, Hume, Russell, and the other self-described atheists is the only rational alternative. But of course I also think the Omega Point Theory has a very good chance of being right, otherwise I would never have troubled to write this book. If the Omega Point Theory is confirmed, I shall then consider myself a theist.
Tipler now regards himself as a theist due to what he states have been advancements in his Omega Point Theory which occurred after the publication of The Physics of Immortality.[19][20] Namely, Tipler now says the known laws of physics—specifically, quantum mechanics, general relativity, the second law of thermodynamics, and the Standard Model of particle physics—require the existence of the Omega Point singularity in order to avoid their violation;[10][11][12] whereas in The Physics of Immortality Tipler investigated what would be necessary from the postulate that life continues forever while still keeping the analysis confined to the known laws of physics. Tipler states that these physical laws have been repeatedly confirmed by every experiment to date. According to Tipler, this constitutes a massive body of empirical evidence for the Omega Point Theory's correctness. And as indicated above, Tipler also now considers himself a Christian due to his identification of the Omega Point with the God of Christian theological tradition.
A diagram of the multiverse formulation of the Omega Point.

Physics

According to Tipler from a 2005 paper[12] in the journal Reports on Progress in Physics, he outlines the following reasons why he maintains the universe must end in the Omega Point in order for the known laws of physics (i.e., the second law of thermodynamics, general relativity, and quantum mechanics) to be mutually consistent at all times:
Astrophysical black holes almost certainly exist, but Hawking[q 1] and Wald[q 2] have shown that if black holes are allowed to exist for unlimited proper time, then they will completely evaporate, and unitarity will be violated. Thus, unitarity requires that the universe must cease to exist after finite proper time, which implies that the universe has spatial topology S3.[q 3] The Second Law of Thermodynamics says the amount of entropy in the universe cannot decrease, but Ellis and Coule[q 4] and I[q 5] have shown that the amount of entropy already in the CMBR will eventually contradict the Bekenstein Bound near the final singularity unless there are no event horizons, since in the presence of horizons the Bekenstein Bound implies the universal entropy S ≤ constant × R2, where R is the radius of the universe, and general relativity requires R → 0 at the final singularity. If there are no horizons then the (shear) energy density can grow as R−6 which means that the total available energy grows as (R−6 ) R3 ~ R−3, and so the Bekenstein Bound yields E R ~ (R−3)R ~ R−2 which diverges as R−2 as R → 0 at the final singularity.[q 6][q 5] The absence of event horizons by definition means that the universe's future c-boundary is a single point,[q 7] call it the Omega Point. MacCallum[q 8] has shown that an S3 closed universe with a single point future c-boundary is of measure zero in initial data space. Barrow,[q 9][q 10] Cornish and Levin[q 11] and Motter[q 12] have shown that the evolution of an S3 closed universe into its final singularity is chaotic. Yorke et al.[q 13][q 14] have shown that a chaotic physical system is likely to evolve into a measure zero state if and only if its control parameters are intelligently manipulated. Thus life (≡intelligent computers) almost certainly must be present arbitrarily close to the final singularity in order for the known laws of physics to be mutually consistent at all times. Misner[q 15][q 16][q 17] has shown in effect that event horizon elimination requires an infinite number of distinct manipulations, so an infinite amount of information must be processed between now and the final singularity. The amount of information stored at any time diverges to infinity as the Omega Point is approached, since S → +∞ there, implying divergence of the complexity of the system that must be understood to be controlled.
Some have pointed out that the current acceleration of the universe's expansion due to the positive cosmological constant would appear to obviate the Omega Point.[21] On this matter Tipler states[22] (see also references [12] and [11]) that baryon annihilation—which he says would be the ideal form of energy resource and rocket propulsion during the colonization of the universe—will force the Higgs field to its absolute vacuum state, resulting in the universe's collapse:
The SM provides such a mechanism, which I actually discussed in the last section of the Appendix for Scientists in ([q 18], p. 515). This mechanism is the creation/destruction of baryon number by electroweak quantum tunneling. (Baryons are the heavy particles made up of quarks. Examples are neutrons and protons.) In my book, I pointed out that this mechanism would be ideal for propelling interstellar spacecraft, but I did not discuss its implications for the Higgs vacuum, a serious oversight on my part. (An oversight which invalidates the second part of my Fifth Prediction on page 149 of [q 18].) If the SM is true—ALL experiments conducted to date indicate that it is (e.g.[q 19] and [q 20], last full paragraph on p. 35)—then the net baryon number observed in the universe must have been created in the early universe by this mechanism of electroweak quantum tunneling. If the baryons were so created, then this process necessarily forces the Higgs field to be in a vacuum state that is not its absolute vacuum. But if the baryons in the universe were to be annihilated by this process, say by the action of intelligent life, then this would force the Higgs field toward its absolute vacuum, canceling the positive cosmological constant, stopping the acceleration, and allowing the universe to collapse into the Omega Point. Conversely, if enough baryons are not annihilated by this process, the positive cosmological constant will never be canceled, the universe will expand forever, unitarity will be violated, and the Omega Point will never come into existence. Only if life makes use of this process to annihilate baryons will the Omega Point come into existence.

The Omega Point and the quantum gravity Theory of Everything

In his 2005 paper[12] in the journal Reports on Progress in Physics, Tipler maintains that the correct quantum gravity theory has existed since 1962, first discovered by Richard Feynman in that year,[23] and independently discovered by Steven Weinberg and Bryce DeWitt, among others. But, according to Tipler, because these physicists were looking for equations with a finite number of terms (i.e., derivatives no higher than second order), they abandoned this qualitatively unique quantum gravity theory since in order for it to be consistent it requires an arbitrarily higher number of terms.[24] "They also did not realize that the correct quantum gravity theory is consistent only if a certain set of boundary conditions are imposed ...", writes Tipler (which includes the initial Big Bang, and the final Omega Point, cosmological singularities).[12] Tipler says that the equations for this theory of quantum gravity are term-by-term finite, but the same mechanism that forces each term in the series to be finite also forces the entire series to be infinite (i.e., infinities that would otherwise occur in spacetime, consequently destabilizing it, are transferred to the cosmological singularities, thereby preventing the universe from immediately collapsing into nonexistence[25]). Tipler writes that "It is a fundamental mathematical fact that this [infinite series] is the best that we can do. ... This is somewhat analogous to Liouville's theorem in complex analysis, which says that all analytic functions other than constants have singularities either a finite distance from the origin of coordinates or at infinity."[26]
From the aforesaid Reports on Progress in Physics paper,[12] Tipler elaborates on the mathematics and physics of this issue, in part explained below:
So basic quantum field theory quickly forces upon us the general invariant action
S = \int d^4x\, \sqrt{-g}(\Lambda + \frac{1}{8\pi G}R + c^2_1R^2 +c^3_1R^3 \ldots + c^2_2R_{\mu\nu}R^{\mu\nu} + \ldots + c^3_1R_{\mu\nu;\alpha}R^{\mu\nu;\alpha} + \ldots)
(3)
This is the qualitatively unique gravitational Lagrangian picked out by quantum mechanics. Physicists do not like it because (1) it has an infinite number of (renormalizable) constants c^i_j, all of which must be determined by experiment and (2) it will not yield second order differential equations which all physicists know and love. But the countable number of constants are in effect axioms of the theory, and I pointed out in an earlier section that the Löwenheim–Skolem theorem suggests there is no real difference between a theory with a countable number of axioms and a theory with a finite number of axioms. The finite case is just easier for humans to deal with, provided the 'finite' number is a small number. Further, as Weinberg[q 21] has emphasized, this Lagrangian generates a quantum theory of gravity that is just as renormalizable as QED and the SM.
Since quantum field theory itself is forcing the Lagrangian (3) on us, I propose that we accept the judgement of quantum mechanics and accept (3) (and the countable number of additional terms involving the non-gravitational fields interacting with the h_{\mu\nu}) as the actual Lagrangian of reality.
Donoghue[q 22] and Donoghue and Torma[q 23] have shown that Lagrangian (3) will not contradict experiment provided the (renormalized) values of the infinite number of new coupling constants are sufficiently small. ...
One consequence of the above Lagrangian being the true description of quantum gravity, explains Tipler, would be that so long as one is within spacetime, then one can never obtain a complete description of quantum gravity and hence of physics: there will always be infinitely more to learn and discover in the field of physics, including by requiring the use of experiment.[24] He says that physics will be able to become ever-more refined, knowledgeable and precise, but never complete (i.e., within spacetime). Only at the final singularity of the Omega Point (which is not in spacetime[27]) will the full description of physics be obtained, states Tipler.
In the same aforestated journal article, Tipler combines the above theory of quantum gravity with an extended Standard Model in order to form what he maintains is the correct Theory of Everything (TOE) describing and unifying all the forces in physics.[12][18]
Out of 50 articles, Tipler's said paper[12] was selected as one of 12 for the "Highlights of 2005" accolade as "the very best articles published in Reports on Progress in Physics in 2005 [Vol. 68]. Articles were selected by the Editorial Board for their outstanding reviews of the field. They all received the highest praise from our international referees and a high number of downloads from the journal Website."[28] Reports on Progress in Physics is the leading journal of the Institute of Physics (based on its impact factor, according to Journal Citation Reports[29][30]), Britain's main professional body for physicists.

Implications from string theory

If string theory is valid, it would seem to contradict the Omega Point Theory, since the Omega Point Theory requires the existence of a cosmological singularity at the end of time. Whereas, according to string theory, singularities do not actually exist because no material object can be compressed below the Planck length.[31]
Stephen Hawking has proposed a solution to the black hole information issue in order to preserve unitarity but without the universe collapsing which is dependent on the conjectured string theory-based anti-de Sitter space/conformal field theory correspondence (AdS/CFT correspondence).[32]
Tipler himself argues against the validity of string theory.[12]

Criticisms

To date the only peer-reviewed paper in a physics journal that has criticized Tipler's Omega Point Theory has been in 1994 by physicists George Ellis and David Coule in the journal General Relativity and Gravitation.[33] In the paper, Ellis and Coule gave an argument that the Bekenstein bound violates the second law of thermodynamics if the universe collapses without having event horizons eliminated. Tipler argues that in order to bring about the Omega Point that event horizons must be eliminated, and Tipler cites this paper in favor of his contention that the known laws of physics require the Omega Point to exist.[12]
There have also been a number of non-refereed book reviews appearing in science journals and popular science magazines which have been critical of Tipler's Omega Point Theory. Writing in the "Book Reviews" section of the journal Nature, Ellis described Tipler's book The Physics of Immortality as "a masterpiece of pseudoscience. ... the product of a fertile and creative imagination unhampered by the normal constraints of scientific or philosophical discipline."[34] In the magazine New Scientist, physicist Lawrence M. Krauss referred to Tipler's book The Physics of Christianity as "a collection of half-truths and exaggerations, I am tempted to describe Tipler's new book as nonsense—but that would be unfair to the concept of nonsense."[35]

See also

Physics books dealing with the Omega Point Theory

References

  1. Jump up ^ The term is perhaps from a 1988 book of the same name by Hans Moravec, according to Rebecca Sato, "'Mind Children': Transhumanism & the Search For Genetic Perfection", The Daily Galaxy: Great Discoveries Channel, August 2, 2007
  2. Jump up ^ F. J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode: 2005RPPh...68..897T, pp. 915-916. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", arXiv:0704.3276, April 24, 2007, pp. 28-29.
  3. Jump up ^ Frank J. Tipler, "Cosmological Limits on Computation", International Journal of Theoretical Physics, Vol. 25, No. 6 (June 1986), pp. 617-661, doi:10.1007/BF00670475, Bibcode: 1986IJTP...25..617T. (First paper on the Omega Point Theory.)
  4. Jump up ^ Frank J. Tipler, "Achieved spacetime infinity", Nature, Vol. 325, No. 6101 (January 15, 1987), pp. 201-202, doi:10.1038/325201c0, Bibcode: 1987Natur.325..201T.
  5. Jump up ^ Frank J. Tipler, "The Anthropic Principle: A Primer for Philosophers", PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1988, Volume Two: Symposia and Invited Papers (1988), pp. 27-48; published by University of Chicago Press on behalf of the Philosophy of Science Association.
  6. Jump up ^ Frank J. Tipler, "The Omega Point as Eschaton: Answers to Pannenberg's Questions for Scientists", Zygon: Journal of Religion & Science, Vol. 24, Issue 2 (June 1989), pp. 217-253, doi:10.1111/j.1467-9744.1989.tb01112.x. Mirror link. Republished as Chapter 7: "The Omega Point as Eschaton: Answers to Pannenberg's Questions to Scientists" in Carol Rausch Albright and Joel Haugen (editors), Beginning with the End: God, Science, and Wolfhart Pannenberg (Chicago, Ill.: Open Court Publishing Company, 1997), pp. 156-194, ISBN 0812693256, LCCN 97-000114.
  7. Jump up ^ Frank J. Tipler, "The ultimate fate of life in universes which undergo inflation", Physics Letters B, Vol. 286, Issues 1-2 (July 23, 1992), pp. 36-43, doi:10.1016/0370-2693(92)90155-W, Bibcode: 1992PhLB..286...36T.
  8. Jump up ^ Frank J. Tipler, "A New Condition Implying the Existence of a Constant Mean Curvature Foliation", Bibcode: 1993dgr2.conf..306T, in B. L. Hu and T. A. Jacobson (editors), Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland, Volume 2: Papers in Honor of Dieter Brill (Cambridge: Cambridge University Press, 1993), pp. 306-315, ISBN 0521452678, Bibcode: 1993dgr2.conf.....H. Mirror link.
  9. Jump up ^ Frank J. Tipler, "Ultrarelativistic Rockets and the Ultimate Future of the Universe", NASA Breakthrough Propulsion Physics Workshop Proceedings, National Aeronautics and Space Administration, January 1999, pp. 111-119 (mirror link); an invited paper in the proceedings of a conference held at and sponsored by NASA Lewis Research Center, Cleveland, Ohio, August 12–14, 1998;Error: Bad DOI specified!. Document ID: 19990023204. Report Number: E-11429; NAS 1.55:208694; NASA/CP-1999-208694. Mirror link.
  10. ^ Jump up to: a b Frank J. Tipler, "The Ultimate Future of the Universe, Black Hole Event Horizon Topologies, Holography, and the Value of the Cosmological Constant", arXiv:astro-ph/0104011, April 1, 2001. Published in J. Craig Wheeler and Hugo Martel (editors), Relativistic Astrophysics: 20th Texas Symposium, Austin, TX, 10-15 December 2000 (Melville, N.Y.: American Institute of Physics, 2001), pp. 769-772, ISBN 0735400261, LCCN 2001-094694, which is AIP Conference Proceedings, Vol. 586 (October 15, 2001), doi:10.1063/1.1419654, Bibcode: 2001AIPC..586.....W.
  11. ^ Jump up to: a b c Frank J. Tipler, "Intelligent life in cosmology", International Journal of Astrobiology, Vol. 2, Issue 2 (April 2003), pp. 141-148, doi:10.1017/S1473550403001526, Bibcode: 2003IJAsB...2..141T. Mirror links here and here; also available here. Also at arXiv:0704.0058, March 31, 2007.
  12. ^ Jump up to: a b c d e f g h i j k F. J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode: 2005RPPh...68..897T. Mirror link. (Note: citation formatting in the above-quoted passages have been modified for clarity. Typographical errors in the third quoted passage have been corrected, again for clarity.) Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", arXiv:0704.3276, April 24, 2007.
  13. Jump up ^ Frank J. Tipler, Jessica Graber, Matthew McGinley, Joshua Nichols-Barrer and Christopher Staecker, "Closed Universes With Black Holes But No Event Horizons As a Solution to the Black Hole Information Problem", arXiv:gr-qc/0003082, March 20, 2000. Published in Monthly Notices of the Royal Astronomical Society, Vol. 379, Issue 2 (August 2007), pp. 629-640, doi:10.1111/j.1365-2966.2007.11895.x, Bibcode: 2007MNRAS.379..629T.
  14. Jump up ^ John D. Barrow and Frank J. Tipler, "Foreword" by John A. Wheeler, The Anthropic Cosmological Principle (Oxford: Oxford University Press, 1986), pp. 676-677, ISBN 0198519494, LCCN 85-004824, Bibcode: 1986acp..book.....B. Excerpt from Chapter 1.
  15. Jump up ^ John D. Barrow and Frank J. Tipler, "Foreword" by John A. Wheeler, The Anthropic Cosmological Principle (Oxford: Oxford University Press, 1986), p. 682, ISBN 0198519494, LCCN 85-004824, Bibcode: 1986acp..book.....B. Excerpt from Chapter 1.
  16. Jump up ^ Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0385467982, LCCN 93-045046, Bibcode: 1994pimc.book.....T.
  17. Jump up ^ David Deutsch, The Fabric of Reality: The Science of Parallel Universes—and Its Implications (London: Allen Lane The Penguin Press, 1997), ISBN 0713990619, LCCN 97-006171. Extracts from Chapter 14: "The Ends of the Universe", with additional comments by Tipler; also available here, here and here.
  18. ^ Jump up to: a b Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, LCCN 2006-039028. Chapter I and excerpt from Chapter II. Chapter I also available here.
  19. Jump up ^ Sam Vincent Meddis, "Computers of the distant future", USA Today, four parts, August 3–31, 1998. Part 1: "Machines evolve" (August 3), Part 2: "A quantum leap" (August 10), Part 3: "Universal truths" (August 17), and Part 4: "Web of thought" (August 31). See Part 1 concerning Tipler no longer being an atheist.
  20. Jump up ^ Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, LCCN 2006-039028, Chapter III: "Life and the Ultimate Future of the Universe", p. 62.
  21. Jump up ^ Although see Lawrence M. Krauss and Michael S. Turner, "Geometry and Destiny", General Relativity and Gravitation, Vol. 31, No. 10 (October 1999), pp. 1453-1459, doi:10.1023/A:1026757718530, Bibcode: 1999GReGr..31.1453K. Also at arXiv:astro-ph/9904020, April 1, 1999.
  22. Jump up ^ Frank Tipler, "The Omega Point and Christianity", Gamma, Vol. 10, No. 2 (April 2003), pp. 14-23 (mirror link); note that the foregoing version corrects character formatting errors of the versions available here, here and here. (Note: citation formatting in the above-quoted passage has been modified for clarity.) For the version in Dutch, see "Het Punt Omega en het christendom", Gamma, Jrg. 10, Nr. 2 (April 2003), pp. 14-23; also available here and here.
  23. Jump up ^ Richard P. Feynman, notes taken by Fernando B. Morinigo and William G. Wagner, edited by Brian Hatfield, "Foreword" by John Preskill and Kip S. Thorne, Feynman Lectures on Gravitation (Reading, Mass.: Addison-Wesley Publishing Company, 1995), ISBN 0201627345, LCCN 95-011076, Bibcode: 1995flg..book.....F. "Foreword" mirror link; also available here.
  24. ^ Jump up to: a b Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, LCCN 2006-039028, pp. 34-35. Chapter I and excerpt from Chapter II. Chapter I also available here.
  25. Jump up ^ On this matter, in addition to Tipler's 2005 Reports on Progress in Physics paper and his 2007 book, see John D. Barrow and Frank J. Tipler, "Action principles in nature", Nature, Vol. 331, No. 6151 (January 7, 1988), pp. 31-34, doi:10.1038/331031a0, Bibcode: 1988Natur.331...31B. Also released as "The Finite Action Principle; or, Singularities without Singularities" in the Gravity Research Foundation's 1987 essay competition. Mirror link.
  26. Jump up ^ Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, LCCN 2006-039028, pp. 49 and 279. Chapter I and excerpt from Chapter II. Chapter I also available here.
  27. Jump up ^ S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (London: Cambridge University Press, 1973), ISBN 0521200164, LCCN 72-093671, Bibcode: 1973lsss.book.....H, pp. 217-221.
  28. Jump up ^ Richard Palmer, Publisher, "Highlights of 2005", Reports on Progress in Physics. Mirror link. See also "Editorial board", Reports on Progress in Physics. Mirror link.
  29. Jump up ^ "Journal Citation Reports (JCR) Year 2006—Science Edition", September 2007. Mirror link.
  30. Jump up ^ "Journals Catalogue 2008", IOP Publishing (Institute of Physics), Section: "IOP Impact Factors", p. 37. Mirror link.
  31. Jump up ^ Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (New York, N.Y.: W. W. Norton, 1999), ISBN 0393046885, LCCN 98-025695, Bibcode: 1999eush.conf.....G, pp. 252-253.
  32. Jump up ^ S. W. Hawking, "Information loss in black holes", Physical Review D, Vol. 72, No. 8 (October 2005), Art. No. 084013, 4 pages, doi:10.1103/PhysRevD.72.084013, Bibcode: 2005PhRvD..72h4013H. Mirror link. Also at arXiv:hep-th/0507171, July 18, 2005.
  33. Jump up ^ G. F. R. Ellis and D. H. Coule, "Life at the end of the universe?", General Relativity and Gravitation, Vol. 26, No. 7 (July 1994), pp. 731-739, doi:10.1007/BF02116959, Bibcode: 1994GReGr..26..731E.
  34. Jump up ^ George Ellis, "Piety in the sky", Nature, Vol. 371, No. 6493 (September 8, 1994), p. 115, doi:10.1038/371115a0. Mirror link. A review of Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0385467982, LCCN 93-045046, Bibcode: 1994pimc.book.....T.
  35. Jump up ^ Lawrence Krauss, "More dangerous than nonsense", New Scientist, Vol. 194, Issue 2603 (May 12, 2007), p. 53, doi:10.1016/S0262-4079(07)61199-3. Mirror link. A review of Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, LCCN 2006-039028.

References originally in quoted passages

  1. Jump up ^ S. W. Hawking, "Breakdown of predictability in gravitational collapse", Physical Review D, Vol. 14, Issue 10 (November 1976), pp. 2460-2473, doi:10.1103/PhysRevD.14.2460, Bibcode: 1976PhRvD..14.2460H. Mirror link.
  2. Jump up ^ Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago: University of Chicago Press, 1994), ISBN 0226870251, LCCN 94-011065, Section 7.3, pp. 182-185.
  3. Jump up ^ John D. Barrow, Gregory J. Galloway and Frank J. Tipler, "The closed-universe recollapse conjecture", Monthly Notices of the Royal Astronomical Society, Vol. 223, No. 4 (December 1986), pp. 835-844, Bibcode: 1986MNRAS.223..835B, CAT.INIST No. 8251334. On p. 926 of the same 2005 Reports on Progress in Physics paper, Tipler writes that "A dynamical proof for S3 can be found in Barrow (1986)", with "Barrow (1986)" being this reference.
  4. Jump up ^ G. F. R. Ellis and D. H. Coule, "Life at the end of the universe?", General Relativity and Gravitation, Vol. 26, No. 7 (July 1994), pp. 731-739, doi:10.1007/BF02116959, Bibcode: 1994GReGr..26..731E.
  5. ^ Jump up to: a b Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0385467982, LCCN 93-045046, Bibcode: 1994pimc.book.....T, Appendix C: "The Bekenstein Bound", p. 410. Said Appendix is reproduced in Frank J. Tipler, "Genesis: How the Universe Began According to Standard Model Particle Physics", arXiv:astro-ph/0111520, November 28, 2001, Section 2: "Apparent Inconsistences in the Physical Laws in the Early Universe", Subsection a: "Bekenstein Bound Inconsistent with Second Law of Thermodynamics".
  6. Jump up ^ Frank J. Tipler, "Intelligent life in cosmology", International Journal of Astrobiology, Vol. 2, Issue 2 (April 2003), pp. 141-148, doi:10.1017/S1473550403001526, Bibcode: 2003IJAsB...2..141T. Mirror links here and here; also available here. Also at arXiv:0704.0058, March 31, 2007.
  7. Jump up ^ S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (London: Cambridge University Press, 1973), ISBN 0521200164, LCCN 72-093671, Bibcode: 1973lsss.book.....H, pp. 217-221.
  8. Jump up ^ Malcolm A. H. MacCallum, "Mixmaster universe problem", Nature—Physical Science, Vol. 230 (March 1971), pp. 112-113, Bibcode: 1971Natur.230..112M, OSTI 4048469. See also here.
  9. Jump up ^ John D. Barrow, "Chaotic behaviour in general relativity", Physics Reports, Vol. 85, Issue 1 (May 1982), pp. 1-49, doi:10.1016/0370-1573(82)90171-5, Bibcode: 1982PhR....85....1B.
  10. Jump up ^ John D. Barrow and Janna Levin, "Chaos in the Einstein-Yang-Mills Equations", Physical Review Letters, Vol. 80, Issue 4 (January 1998), pp. 656-659, doi:10.1103/PhysRevLett.80.656, Bibcode: 1998PhRvL..80..656B. Also at arXiv:gr-qc/9706065, June 20, 1997.
  11. Jump up ^ Neil J. Cornish and Janna J. Levin, "Mixmaster universe: A chaotic Farey tale", Physical Review D, Vol. 55, Issue 12 (June 1997), pp. 7489-7510, doi:10.1103/PhysRevD.55.7489, Bibcode: 1997PhRvD..55.7489C. Also at arXiv:gr-qc/9612066, December 30, 1996.
  12. Jump up ^ Adilson E. Motter, "Relativistic Chaos is Coordinate Invariant", Physical Review Letters, Vol. 91, Issue 23 (December 2003), Art. No. 231101, 4 pages, doi:10.1103/PhysRevLett.91.231101, Bibcode: 2003PhRvL..91w1101M. Mirror link. Also at arXiv:gr-qc/0305020, May 5, 2003.
  13. Jump up ^ Troy Shinbrot, Edward Ott, Celso Grebogi and James A. Yorke, "Using chaos to direct trajectories to targets", Physical Review Letters, Vol. 65, Issue 26 (December 1990), pp. 3215-3218, doi:10.1103/PhysRevLett.65.3215, Bibcode: 1990PhRvL..65.3215S.
  14. Jump up ^ Troy Shinbrot, William Ditto, Celso Grebogi, Edward Ott, Mark Spano and James A. Yorke, "Using the sensitive dependence of chaos (the 'butterfly effect') to direct trajectories in an experimental chaotic system", Physical Review Letters, Vol. 68, Issue 19 (May 1992), pp. 2863-2866, doi:10.1103/PhysRevLett.68.2863, Bibcode: 1992PhRvL..68.2863S.
  15. Jump up ^ Charles W. Misner, "The Isotropy of the Universe", Astrophysical Journal, Vol. 151 (February 1968), pp. 431-457, doi:10.1086/149448, Bibcode: 1968ApJ...151..431M.
  16. Jump up ^ Charles W. Misner, "Quantum Cosmology. I", Physical Review, Vol. 186, Issue 5 (October 1969), pp. 1319-1327, doi:10.1103/PhysRev.186.1319, Bibcode: 1969PhRv..186.1319M.
  17. Jump up ^ Charles W. Misner, "Mixmaster Universe", Physical Review Letters, Vol. 22, Issue 20 (May 1969), pp. 1071-1074, doi:10.1103/PhysRevLett.22.1071, Bibcode: 1969PhRvL..22.1071M. Mirror link. Also available as an entry in the Gravity Research Foundation's 1969 essay competition. Mirror link.
  18. ^ Jump up to: a b Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0385467982, LCCN 93-045046, Bibcode: 1994pimc.book.....T.
  19. Jump up ^ Frank Wilczek, "Scaling Mount Planck III: Is That All There Is?", Physics Today, Vol. 55, Issue 8 (August 2002), pp. 10-11, doi:10.1063/1.1510264, Bibcode: 2002PhT....55h..10W. Mirror link; also available here.
  20. Jump up ^ Helen R. Quinn, "The Asymmetry Between Matter and Antimatter", Physics Today, Vol. 56, Issue 2 (February 2003), pp. 30-35, doi:10.1063/1.1564346, Bibcode: 2003PhT....56b..30Q. Mirror link; also available here.
  21. Jump up ^ Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations (Cambridge: Cambridge University Press, 1995), ISBN 0521550017, LCCN 95-002782 Bibcode: 1995qtf..book.....W, pp. 499 and 518-519.
  22. Jump up ^ John F. Donoghue, "General relativity as an effective field theory: The leading quantum corrections", Physical Review D, Vol. 50, Issue 6 (September 1994), pp. 3874-3888, doi:10.1103/PhysRevD.50.3874, Bibcode: 1994PhRvD..50.3874D. Also at arXiv:gr-qc/9405057, May 25, 1994.
  23. Jump up ^ John F. Donoghue and Tibor Torma, "Power counting of loop diagrams in general relativity", Physical Review D, Vol. 54, Issue 8 (October 1996), pp. 4963-4972, doi:10.1103/PhysRevD.54.4963, Bibcode: 1996PhRvD..54.4963D. Also released as "On the power counting of loop diagrams in general relativity", arXiv:hep-th/9602121, February 22, 1996.

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