Re: Why atheists believe in a universe capable of Judgment and

Ok much better thanks

----- Original Message -----
From: "D. F. Siemens, Jr." <dfsiemensjr@juno.com>
To: <drsyme@cablespeed.com>
Cc: <glennmorton@entouch.net>; <asa@calvin.edu>
Sent: Saturday, June 25, 2005 3:13 PM
Subject: Re: Why atheists believe in a universe capable of Judgment and

>
> On Fri, 1 Jan 1988 14:34:32 -0500 "jack syme" <drsyme@cablespeed.com>
> writes:
>> Did anyone else get this email with formatting errors? Unfortunately
>> I cant
>> read it becuase of this. If you got this email with no formatting
>> problems
>> could you forward me a copy? I would like to read this.
>>
>>
>> ----- Original Message -----
>> From: "Glenn Morton" <glennmorton@entouch.net>
>> To: <asa@calvin.edu>
>> Sent: Thursday, June 23, 2005 11:18 PM
>
> Had the same problem. Have tried to edit Glenn's text in Notepad. Won't
> guarantee I got it all right, but it is better, though without italics
> and other niceties.
> Dave
>
> Prophecy
> Date: Thu, 23 Jun 2005 17:42:10 -0500
> Sender: asa-owner@lists.calvin.edu
> Precedence: bulk
>
> This didn't go through yesterday. I return to Beijing tomorrow
> so won't be able to discuss until Saturday. Thanks to everyone
> for their prayers and concern, we don't know what my wife has
> but it doesn't appear to be cancer.
>
> Why Atheists Believe in Eternal Judgment and Prophecy and
> Don't Even Know It.
>
> By Glenn R. Morton June, 2005
>
> 'I do not mean to ask you to accept anything without reasonable ground
> for it. You will soon admit as much as I need from you. You know of
> course that a mathematical line, a line of thickness nil, has no real
> existence. They taught you that? Neither has a mathematical plane.
> These things are mere abstractions.'
>
> 'That is all right,' said the Psychologist.
>
> 'Nor, having only length, breadth, and thickness, can a cube have a
> real existence.'
>
> 'There I object,' said Filby. 'Of course a solid body may exist. All
> real things - '
>
> 'So most people think. But wait a moment. Can an instantaneous cube
> exist?'
>
> 'Don't follow you,' said Filby.
>
> 'Can a cube that does not last for any time at all, have a real
> existence?'
>
> (Wells, 1895, chapter 1)
>
> Thus starts H. G. Wells' the Time Machine. The question he asks
> gets to the heart of existence. Existence means or implies, a temporal
> duration. Without that, one can=E2=80=99t actually exist, or so it
> seems. As discussed in the last post, the existence of the universe
> is a permanent gap in our understanding. In principle, the
> universe's existence will never really be explained apart from
> positing god-like properties and/or powers to the universe or some
> constituent of the universe.
>
> In 1949, Kurt Godel presented a novel solution to Einstein's
> gravitational equations. The solution he presented is now known as
> Godel's universe and it has profound implications to the science/theology
> debate. Godel's paper, presented in a book in
> honor of Einstein, argued that time does not exist. He argued that
> time is an ideal, not real. Time is an illusion. His paper was met with
> silence, some say because he was an outsider to relativity and an
> outsider to philosophy. One will find occasional references to his
> work in the relativistic literature but only to say that in rotating
> universes there are closed time-like paths(i.e. time travel is
> possible). Whatever the reason for the previous silence, his paper
> has drawn much more interest in the past 10 years, due mainly to the
> efforts of Palle Yourgrau, a philosopher at Brandeis University.
> Godel's argument has much to say about the nature of time and is worth
> hearing. I will bring out some theological implications from this work.
>
> The main point in writing this article is to show that if one follows
> a reductionist path, one is forced into believing that the universe is
> one which would allow things like prophecy and judgment, things which the
> secular world would rather omit from consideration.
>
> The technical details of Godel's bizarre universe is outlined by
> Deser and Jackiw(1992) and can be found below the references. The
> main item we need to pay attention to is that with a particular
> arrangement of matter in the universe, time travel is possible.
> Technically, there are closed time-like curves.
>
> What are the implications of a closed time-like curves? Well, it means
> that a person can travel into his past in such a universe. Clearly
> this implies all sorts of acausal paradoxes=E2=80=94things like killing
> your grandmother before your mother was born. If you succeed, then you
> are not born and could not travel back into the past to kill her so she
> didn't die and you were born. Most people who believe time travel is
> physically possible think that one can only do that which is logically
> possible to do thus ruling out killing granny. Most physicists have
> rejected Godel's universe because the mass distribution he
> assumed is not observed in our universe. Thus, most have ignored
> Godel's Universe and its implications. More on this below, but
> suffice it to say that Godel's argument includes arguments for
> the applicability to our world.
>
> I will use Dorato's reconstruction of the argument, which in outline is:
>
> (0) Time is real only if change is real.
> (1) Change is real only if there exists an objective lapse of time.
> change becomes possible only through the lapse of time=C2=BB
> (1949a, p. 558/1990, p. 202)
> (2) Time is real only if there exists an objective lapse of time [from
> (0) and (1)]
> (3) =C2=ABThe existence of an objective lapse of time means or at
> least is equivalent to the fact, that reality consists of an infinity
> of layers of "now" which come into existence
> successively (1949a, p. 558/1990, p. 202).
> (4) Reality consist of an infinity of layers of "now" which come into
> existence successively only if spacetime admits of a global time
> function (cosmic time).
> (5) Time is real only if spacetime admits of a global time function
> [from (2), (3) (4)]
> (6) G6del's rotating-model M, qua solution to Einstein's field
> equations, is a physically possible model, and
> despite the presence of closed timelike curves (circular time) and
> looming grandfather paradoxes, cannot be ruled out a priori.
> (7) Since for every x in M, x chronologically precedes itself, M does
> not possess a global time function.
> (8) In the physically possible world M, time is ideal [from (5) (6) =
> (7)].
> (9) The main, contingent, non-lawlike difference between M and our
> universe is given by the (probable) absence of a net rotation of
> matter, which implies the existence of cosmic time in our world.
> (Dorato, 2001)
>
> [the references 1949a are to Godel 1949 in this paper]
>
> Let's look at each step in the argument closely.
>
> (0) Time is real only if change is real.
>
> (1) Change is real only if there exists an objective lapse of time.
> change becomes possible only through the lapse of time
> (1949a, p. 558/1990, p. 202)
>
> The objections to these two assumptions is that it implies absolute
> change. It has also been objected that this appears to be the
> constantly moving "now" concept and this is a view which
> philosophers seem to have rejected. The way to interpret these two
> assumptions within Godel's argument is that at time t certain
> events exist and at time t' later than t, other events exist
> mind-independently. In other words, this implies that there is an
> objectiveness to the passage of time.
>
> (2) Time is real only if there exists an objective lapse of time [from
> (0) and (1)] (Dorato, 2001).
>
> This conclusion is logically deduced from the first two.
>
> Readers will object that special relativity shows that there is no
> objective or global time but we still intuitively feel the passage of
> time, i.e. the present flowage of time into the existing present then
> into past. Godel is ready for that objection.
>
> (3) The existence of an objective lapse of time means or at least is
> equivalent to the fact, that reality consists of an infinity of
> layers of "now" which come into existence successively (1949a, p.
> 558/1990, p. 202).
>
> (4) Reality consist of an infinity of layers of "now"
> which come into existence successively only if spacetime admits of a
> global time function (cosmic time).
>
> (5) Time is real only if spacetime admits of a global time function
> [from (2), (3) (4)](Dorato 2001)
>
>
> Step 3 is an acknowledgement of McTaggart's A-series. One can go
> to J. M. E. McTaggart's 1908 article in which he argues for the
> ideality of time to see why time can't be real if the world is
> governed by special relativity. McTaggart defined the A-series, the
> B-series. The B-series is a formal, geometricized time scale. 1776
> will forever be after 2005, but 2005 will always be before 2006. The
> A-series is the ever present now, that dynamic present moment. Points
> come into existence during the now and go out of existence into the
> past. The point in step 4 concerns the need for each moment to
> /come into existence/. This is the intuitive
> understanding of time in which the future does not yet exist.
>
> McTaggart, who notes that Kant, Spinoza, Hegel and Schopenaur believed
> in an idealized time, argued for an idealized time based upon the
> incompatibility of the A-series and the B-series.
>
> It would, I suppose, be universally admitted that time involves
> change. A particular thing, indeed, may exist unchanged through
> any amount of time. But when we ask what we mean by saying that there
> were different moments of time, or a certain duration of time, through
> which the thing was the same, we find that we mean that it remained the
> same while other things were changing. A universe in which nothing
> whatever changed (including the thoughts of the conscious beings in it)
> would be a timeless universe.
> If, then, a B series without an A series can constitute
> time, change must be possible without an A series. Let us suppose that
> the distinction of past, present and future does not apply to reality.
> Can change apply to reality? What is it that changes?
> Could we say that, in a time which formed a B series but
> not an A series, the change consisted in the fact that an event ceased
> to be an event, while another event began to be an event? If this were
> the case, we should certainly have got a change.
> But this is impossible. An event can never cease to be an
> event. It can never get out of any time series in which it once
> is. (McTaggart, 1908)
>
> He then says:
> Neither can the change be looked for in the numerically different
> moments of absolute time, supposing such moments to exist. For the same
> arguments will apply here. Each such moment would have its own place in
> the B series, since each would be earlier or later than each of the
> others. And as the B series indicate permanent relations, no moment
> could ever cease to be, nor could it become another moment.
> (McTaggart, 1908)
>
> Once again, McTaggart's line of reasoning leads him to believe
> that moments can not cease to be.
>
> The deduction that there is need for a global time function would rule
> out the compatibility of intuitive time with special relativity (step 5)
> as Godel argues. He says that the A-series, the intuitive flow of
> time, can not be real if special relativity is true.
>
> The opening move concerns the more limited special theory of
> relativity. Given that the A-series contains the flux of
> "now", the absence of an objective, worldwide "now" in special relativity
> rules out its ex=C2=ADistence. But absent the
> A-series there is no intuitive time. What remains, formal time as
> represented by the little "t" of Einstein-Minkowski space-time, cannot
> be identified with the intuitive time of everyday ex=C2=ADperience.
> The conclusion, for Godel, is inescapable: if relativity theory is valid,
> intuitive time disappears.=E2=80=9D (Yourgrau, 2005, p. 128-129)
>
> To re-iterate, if everyone has a different flow of time (something
> true in special relativity) and there is no formal, global time, time
> becomes entirely subjective and ideal. Another way of looking at this is
> that the A-series views future events as not yet existing, and past
> events and no longer existing. Only the present exists. Future events
> come into existence and then disappear into the past. This is
> incompatible with special relativity and mind-independent objective time
> since two observers will have different perceptions of what exists and
> what doesn't. And this is where Godel brings down the hammer. He
> says, "The concept of existence (...) cannot be relativized without
> destroying its meaning completely." (Godel,1949, p. 559)
>
> A moment of reality can not exist and not exist at the same time, and
> still retain the concept of existence. This is true as long as one
> accepts the concept that A and Not A are a contradictory. This is not
> the same as A plus Not A, which is a quantum mechanical statement of
> superposition.
>
> Godel then reminds the reader that Special Relativity is just
> that, special. It does not include accelerations. In General Relativity,
> a type of global time is allowed. It is the time of the clocks which are
> co-moving with the average matter content of the universe(Peebles, 1993,
> p. 112 note). This "global time" is a good candidate for an objective
> time that Godel is looking for. However, Godel has a surprise in store.
>
> (6) G6del's rotating-model M, qua solution to Einstein's field
> equations, is a physically possible model, and despite the presence of
> closed timelike curves (circular time) and looming grandfather
> paradoxes, cannot be ruled out a priori.
>
> (7) Since for every x in M, x chronologically precedes itself, M does
> not possess a global time function.
>
> (8) In the physically possible world M, time is ideal [from (5) (6)
> (7)] (Dorato, 2001).
>
>
> Einstein presented a view of the world in which space-time is fully
> geometricized=E2=80=94the block universe, in which all events appear as
> points in such a universe. Duration appears as lines moving through
> the block. The paths through the block are subject to the mass
> distribution in the universe. And this is where Godel pulls out his
> surprise. Godel showed that if you had a rotating universe, that there
> would be closed time-like curves. If you are on one of these closed
> timelike paths, you can travel forward in time but meet yourself in your
> past. In Godel's universe, he showed how a rocket with a certain
> acceleration could enable one to travel back to the past and meet himself
> at an earlier time. Such a pathological universe was generally greeted
> by silence on the part of philosophers and physicists.
>
> Godel then argued that if one can visit himself in the past, the past
> has not really disappeared. After all, if you can visit New Jersey,
> then New Jersey must exist. And this brings into question the reality
> of the A-series, in McTaggart=E2=80=99s terminology. We think the past
> no longer exists and the future doesn=E2=80=99t exist yet. But if you
> appear from the future to tell yourself some trivial piece of
> information (like what stocks to buy) it also means that the future
> actually exists. It isn=E2=80=99t that great undetermined thing that
> we normally think it is. Yourgrau writes:
>
> "But if it is possible in such worlds, Godel argues, to return
> to one's past, then what was past never passed at all. But a time that
> never truly passes cannot pass for real, intuitive time. The reality
> of time travel in the Godel universe signals the unreality of time. Once
> again, time disappears." (Yourgrau, 2005, p. 129-130)
>
> The physicists rejected the universe as being "unphysical". Our universe
> does not seem to be rotating. Godel's distribution of matter also
> required that the galaxies stay rigidly separated, neither expanding or
> contracting towards each other.
>
> Godel would counter argue that the physical reality of his mass
> distribution is not of importance in determining the truth or falsity
> of what he says about time:
>
> (9) The main, contingent, non-lawlike difference between M and our
> universe is given by the (probable) absence of a net rotation of
> matter, which implies the existence of cosmic time in our world.
> (Dorato, 2001)
>
> Afterall, he used the very same natural laws which govern our
> universe. The only difference between Godel=E2=80=99s universe and ours
> is a contingency in the distribution of matter, e.g. an accident in
> how matter is arranged. I would suggest it would be like saying that
> just because all the roads to New Jersey have been destroyed
> doesn't mean that New Jersey doesn't exist. You may not be able to visit
> it, but that is merely an accident due to the fact that
> no roads are left intact. Similarly, the fact that we can't
> visit the past in our universe is merely an accident of the distribution
> of matter. However, as we shall see, we very well might be able to visit
> New Jersey.
>
> Monday, I read a marvelous book by Richard Gott, called Time Travel
> in Einstein's Universe. He outlines several methods of time
> travel in our universe assuming that there are cosmic strings and there
> is some evidence for their existence.
>
> "The mysterious gamma rays that emanate from the central bulge
> of our galaxy could arise from a seething tangle of 'cosmic
> strings'." (anonymous, 2005, p. 16)
>
> While that is a possible tangle of strings, if you have two strings
> moving past each other at rapid velocities Deser and Jackiw (1992)
> state:
> "The reason for current interest in time travel ideas derives
> from the recent realization that infinitely long and arbitrarily thin
> cosmic strings can support closed time-like curves."
>
> Gott states.
> "To allow time travel to the past, cosmic strings with a
> mass-per-unit length of about 10 million billion tons per centimeter
> must each move in opposite directions at speeds of at least
> 99.999999996 percent the speed of light. We have observed high-energy
> protons in the universe moving at least this fast, so such speeds are
> possible." (Gott, 2001, p. 104)
>
> And Tipler has a solution which allows time travel in our universe:
>
> "CFrank Tipler, now at Tulane University, found that if you have an
> infinitely tall cylinder rotating at nearly the speed of light on its
> surface, you could go back in time by flying around the cylinder. This
> solution is reminiscent of mine, with the two infinite cosmic strings
> passing each other." (Gott, 2001, p. 117)
>
> There is one more interesting counter argument to Godel's
> possible universe. If the multiverse is true, then Godel's
> universe is not merely a possible universe, it is a real universe
> somewhere in the multiverse. There are only 2^(10^118) (Tegmark, 2003,
> p. 42) different ways to arrange matter in a universe the size of our
> observable universe. If the multiverse consists of all possible
> universes (as Hawking's wave equation of the universe would
> include all possible universes(Kaku, 1994, p. 254), and that should
> include Godel's universe, making it a real item.
>
> Today we know of many arrangements of matter which will allow time
> travel in our universe with our distribution of matter, and this has
> theological implications. We CAN visit New Jersy. These time machine
> solutions mean that the past isn=E2=80=99t non-existent. It also means
> that the future isn=E2=80=99t non-existent. It means that our intuitive
> time, the A-series, in which past and future are non-existent is not
> real as McTaggart argued in 1908. And that has implications to whether
> or not there is real change in the universe as opposed to perceived
> change. And as McTaggart says:
>
> "Neither can the change be looked for in the numerically
> different moments of absolute time, supposing such moments to exist.
> For the same arguments will apply here. Each such moment would have its
> own place in the B series, since each would be earlier or later than each
> of the others. And as the B series indicate permanent relations, no
> moment could ever cease to be, nor could it become another moment."
> (McTaggart, 1908)
>
> And
> "Without the A series then, there would be no change, and
> consequently the B series by itself is not sufficient for time, since
> time involves change.=E2=80=9D (McTaggart 1908)
>
> Consider the implications of this for chance. A physicist travels back
> into the past to visit his aging father again before he died 10 years
> ago. Ten years ago he had set up an experiment to observe virtual
> particles via the Lamb-Retherford shift. Before going to see his
> father, he stops by the office to check the experiment. If he is
> actually in the past, the jitters of the electron circling the hydrogen
> atom should be the same, the very same in this past as it was 10 years
> ago when he performed the experiment. The only difference is that this
> time he feels he is the guy walking in the door to see himself standing
> over the equipment instead of being the guy who was standing over the
> equipment seeing himself walk in. To visit the actual past means that
> virtual particles, those paragons of random chance, can't
> possibly be random at all but are fixed into the block universe of the
> B-series--the 4D manifold of General Relativity. Time travel
> destroys chance being anything other than the illusion of chance like
> that we get from most random number generators in our computers. Like
> the output from the old GWBasic language where one could get the same
> "random" sequence time and time again by specifying the
> same seed number, the universe would yield the same pattern of virtual
> particles every time you visited your aging father in the past.
>
> Well if chance dies in the block universe, then determinism reins, only
> we don=E2=80=99t know what is determined. We live in the A-series and
> can=E2=80=99t look at the B-series. But God can. That means that
> prophecy is allowed because God=E2=80=99s laws, the laws which allow
> time travel, allow Him to know what the future holds. And more than
> that, such a block universe would absolutely mean that judgement can
> occur. If you stand before the divine being who is looking at your life
> and you object that you were really a good guy and didn't murder
> 487 people, God can give you a glimpse of your timeline, effectively
> giving a trial by replay for your benefit.
>
> What does this do to free will? Clearly this is where everyone will
> complain. Indeed, I don't like this aspect of Godel's argument. Free
> will is a form of intentionality and physics has little to say about how
> that arises:
>
> "In the hierarchy of complexity, each level links to the one
> above: chemistry links to biochemistry, to cell biology, physiology,
> psychology, to sociology, economics and politics. Particle physics is
> the foundational subject underlying=E2=80=94and so in some sense
> explaining=E2=80=94all the others. In a reductionist world view, physics
> is all there is. The Cartesian picture of man as a machine seems to be
> vindicated.
> "But this view omits important aspects of the world that
> physics has yet to come to terms with. Our environment is dominated by
> objects that embody the outcomes of intentional design (buildings,
> books computers, teaspoons). Today's physics has nothing to say about
> the intentionality that has resulted in the existence of such objects,
> even though this intentionality is clearly causally effective."
> (Ellis, 2005, p. 743.)
>
> But, this is not what one usually means by the term intentionality.
> In the block view of the universe one could give a reductionist view of
> intentionality. The very electrons in our brains would be determined
> and (yes this is physicalism) thus our thoughts. For those who
> don't think that our thoughts are determined by our brains, look
> what happens to someone whose brain is scrambled by some horrible
> accident. Their thoughts get scrambled as well.
>
> How are we held accountable? I don't know. Godel's argument for the
> existence of time travel seems to preclude free will.
> I can think of two ways this might not be. First if, as Hawking has
> suggested, quantum considerations rule out time travel. This is often
> called the Chronology protection conjecture. Hawking claimed that the
> vacuum would always blow up creating a singularity as you approached an
> area of the space time which would allow time travel. The biggest
> complaint against this is that it seems to be ad hoc.
>
> Secondly, one might be able to avoid this conundrum when the full
> quantum mechanical gravitation is finally developed.
>
> Thirdly, Dorato (2001) believes that Godel failed in proving the
> ideality of time. That being said, I find his objections to be weak. He
> denies the postulate that one can't relativize existence. But,
> if he is right, then one can get out of Godel's argument, but maybe not
> out of the fact that both future and past must still exist if
> time travel is to be considered a physical possibility given
> Einstein's equations.
>
> Now, before people misunderstand the reason I am writing this (I am
> sure people will write responses before getting to this place in the
> text) I am writing primarily to suggest to the atheist that if they do
> engage in reductionism, they end up with a universe which is quite
> capable of things like prophecy and eternal judgment, qualities which
> they deny. My personal reasoning was that I finally got tired of being
> looked upon as the village idiot by atheists, who think they are oh so
> intellectual, and yet they don't realize that they must have something as
> the creator of the universe and that if they are logically consistent in
> believing science, they end up with a universe capable of all the things
> the theist says.
>
> Do I believe in a totally reductionist universe? No, of course not.
> Do I believe that all my actions are determined? No, but I believe this
> in spite of the line of logic above. My belief isn't science; my =
> belief is faith. I will defend the line of logic, but I will not
> defend the view and I would appreciate the responders remembering this
> distinction. The views are not something that I am advocating but
> something I am trying to say that one must hold if one is a radical
> reductionist. In general, atheism holds to a radical reductionism.
>
> References.
>
>
> Anonymous, 2005. Mystery Rays Could Be Sign of Cosmic Strings, New
> Scientist, June 4, 2005
>
> Deser, S. and Jackiw, R., 1992. TIME TRAVEL? Extended version of talk
> presented at =E2=80=9846 LNS 46 Cambridge, MA,May 1992=20
>
> http://arxiv.org/PS_cache/hep-th/pdf/9206/9206094.pdf
>
> Dorato, Mauro 2001 ON BECOMING, COSMIC TIME AND ROTATING UNIVERSES,
> Forthcoming in C. Callender (ed.), Time, Reality
> and Experience (provisional title), Royal Institute of Philosophy Series,
> Cambridge University Press, 2001)
> http://philsci-archive.pitt.edu/archive/00000150/00/becoming.pdf
>
> Ellis, George F. R., 2005. Physics, Complexity and Causality, Nature,
> 435:(June 9):743
>
> Godel, Kurt, 1949. A Remark about the Relationshiip Between
> Relativity Theory and Idealistic Philosophy, in P. A. Schilp, ed.
> Albert Einstein: Philosopher-Scientist (La Salle, Il: Open Court). p.
> 557-562
>
> Gott, Richard, 2001. Time Travel in Einstein's Universe, (New
> York: Houghton Mifflin Co.)
>
> Kaku, Michio, 1994, Hyperspace, (New York: Anchor Books).
>
> McTaggart, John Ellis, 1908. The Unreality of Time, Mind: A
> Quarterly Review of Psychology and Philosophy, 17:456-473.
> http://www.ditext.com/mctaggart/time.html
>
> Peebles, P. J. E. 1993. Principles of Physical Cosmology, (Princeton: =
> Princeton University Press)
>
> Max Tegmark, 2003 Parallel Universes, Scientific American, May.
>
> H. G. Wells, The Time Machine, 1895=20
>
> Yourgrau, Palle, 2005. A World Without Time, (New York: Perseus Books).
>
> Godel's Universe
>
> Godel took T[mu,nu] to be a space-time constant, not vanishing
> only in its time-time component (energy density),
>
> T[infinity] =3Dc^4/8=CF=80G > 0 . (5)
>
> The metric tensor that then solves Einstein's equations leads to
> the space-time interval
>
> ds^2 =3D g[mu,nu] dx^[mu] dx^[nu]=20
>
> =3D {cdt =E2=88=92 =E2=88=9A2/[Lambda] (cosh=E2=88=9A[Lambda]r =
> =E2=88=92 1)d[theta]}^2 =E2=88=92 dr^2 =E2=88=92 1/[Lambda] sinh^2 =
> =E2=88=9A[Lambda]r d[theta]^2 =E2=88=92 dz^2 , (6)
>
> where r, [theta] are planar circular coordinates, with [theta] =3D 0
> and 2[pi] identified, and there is no interesting structure in the
> z-direction. A curve x^[mu](=CF=84) is closed and time-like if both
> x=C2=B5(0) =3D x^[mu](1) (closed) and (ds/d=CF=84)^2 =3D g=C2=B5=CE=BD
> (dx^[mu]/d=CF=84) (dx^[nu]/d=CF=84) > 0 (time-like). It is therefore
> clear that a circular path in the Godel universe for which t, r and z
> remain constant, while [theta] varies from 0 to 2[pi], is closed and
> time-like provided cosh=E2=88=9A[Lambda]r > 3, i.e., r >
> 2/=E2=88=9A[Lambda] ln(1 + =E2=88=9A2 ).=E2=80=9D (Deser & Jackiw,1992)
>
> If you hold t,r, and z constant, dt, dr and dz go to zero and all you =
> have left is the d[theta] terms. Setting ds^2 =3D 0 and solving for r =
> yields the answer in the above citation.
>
Received on Sat Jun 25 19:28:42 2005