ASA - June 2005: Why atheists believe in a universe capable of J

From: Glenn Morton <glennmorton@entouch.net>
Date: Fri Jun 24 2005 - 00:18:08 EDT

Prophecy
Date: Thu, 23 Jun 2005 17:42:10 -0500
Sender: asa-owner@lists.calvin.edu
Precedence: bulk

This didn=E2=80=99t go through yesterday. I return to Beijing
tomorrow =
so won=E2=80=99t be able to discuss until Saturday. Thanks to
everyone =
for their prayers and concern, we don=E2=80=99t know what my wife has =
but it doesn=E2=80=99t appear to be cancer. =20

=20

Why Atheists Believe in Eternal Judgment and Prophecy and
Don=E2=80=99t =
Even Know It. =20

By Glenn R. Morton June, 2005

=20

'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?'=E2=80=9D=20

(Wells, 1895, chapter 1)

=20

Thus starts H. G. Well=E2=80=99s 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=E2=80=99s existence will never really be explained apart from =
positing god-like properties and/or powers to the universe or some =
constituent of the universe. =20

=20

In 1949, Kurt Godel presented a novel solution to Einstein=E2=80=99s =
gravitational equations. The solution he presented is now known as =
Godel=E2=80=99s universe and it has profound implications to the =
science/theology debate. Godel=E2=80=99s 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=E2=80=99s =
argument has much to say about the nature of time and is worth
hearing. =
I will bring out some theological implications from this work.

=20

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. =20

=20

The technical details of Godel=E2=80=99s 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.

=20

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=E2=80=99t 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=E2=80=99s universe because the mass distribution
he =
assumed is not observed in our universe. Thus, most have ignored =
Godel=E2=80=99s Universe and its implications. More on this below,
but =
suffice it to say that Godel=E2=80=99s argument includes arguments for =
the applicability to our world.

=20

I will use Dorato=E2=80=99s reconstruction of the argument, which in =
outline is:

=20

=E2=80=9C(0) Time is real only if change is real.

(1) Change is real only if there exists an objective lapse of time. =
=C2=ABchange 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 =E2=80=9Cnow=E2=80=9D which come into existence =
successively=C2=BB (1949a, p. 558/1990, p. 202).

(4) Reality consist of an infinity of layers of =E2=80=9Cnow=E2=80=9D =
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) G=C3=B6del=E2=80=99s rotating-model M, qua solution to =
Einstein=E2=80=99s 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)]=E2=80=9D

(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.=E2=80=9D =
(Dorato, 2001)

=20

[the references 1949a are to Godel 1949 in this paper]

=20

Let=E2=80=99s look at each step in the argument closely.

=20

(0) Time is real only if change is real.

=20

(1) Change is real only if there exists an objective lapse of time. =
=C2=ABchange becomes possible only through the lapse of time=C2=BB =
(1949a, p. 558/1990, p. 202)

=20

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 =E2=80=98now=E2=80=99 concept and this is a view
which =
philosophers seem to have rejected. The way to interpret these two =
assumptions within Godel=E2=80=99s argument is that at time t certain =
events exist and at time t=E2=80=99 later than t, other events exist =
mind-independently. In other words, this implies that there is an =
objectiveness to the passage of time.

=20

(2) Time is real only if there exists an objective lapse of time [from =
(0) and (1)] (Dorato, 2001).

=20

This conclusion is logically deduced from the first two. =20

=20

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.

=20

(4) Reality consist of an infinity of layers of =E2=80=9Cnow=E2=80=9D =
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)

=20

Step 3 is an acknowledgement of McTaggart=E2=80=99s A-series. One can
go =
to J. M. E. McTaggart=E2=80=99s 1908 article in which he argues for
the =
ideality of time to see why time can=E2=80=99t 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 =
=E2=80=98come into existence=E2=80=99. This is the intuitive =
understanding of time in which the future does not yet exist. =20

=20

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.

=20

      =E2=80=9CIt 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.=E2=80=9D=20
       =E2=80=9CIf, 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?=E2=80=9D
       =E2=80=9CCould 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. =E2=80=9C
       =E2=80=9CBut 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.=E2=80=9D (McTaggart, 1908)

He then says:

=E2=80=9CNeither 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.=E2=80=9D =
(McTaggart, 1908)

Once again, McTaggart=E2=80=99s 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.=20

=20

=E2=80=9CThe opening move concerns the more limited special theory of =
relativity. Given that the A-series contains the flux of =
=E2=80=98now,=E2=80=99 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)

=20

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=E2=80=99t. And this is where Godel brings down the hammer. He =
says, =E2=80=9CThe concept of existence (...) cannot be relativized =
without destroying its meaning completely.=E2=80=9D (Godel,1949, p. =
559)

=20

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.

=20

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 =E2=80=98global time=E2=80=99 is a good candidate =
for an objective time that Godel is looking for. However, Godel has a =
surprise in store.

=20

(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)]=E2=80=9D (Dorato, 2001)

=20

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 Godels =
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. =20

=20

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:

=20

=E2=80=9CBut 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.=E2=80=9D (Yourgrau, 2005, p. 129-130)

=20

The physicists rejected the universe as being =
=E2=80=98unphysical=E2=80=99. Our universe does not seem to be
rotating. =
Godel=E2=80=99s distribution of matter also required that the
galaxies =
stay rigidly separated, neither expanding or contracting towards each =
other.=20

=20

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:

=20

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=E2=80=94e.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=E2=80=99t mean that New Jersey doesn=E2=80=99t exist. You may
not =
be able to visit it, but that is merely an accident due to the fact
that =
no roads are left in tact. Similarly, the fact that we can=E2=80=99t =
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.

=20

Monday, I read a marvelous book by Richard Gott, called Time Travel
in =
Einstein=E2=80=99s 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.

=20

=E2=80=9CThe mysterious gamma rays that emanate from the central bulge =
of our galaxy could arise from a seething tangle of =E2=80=98cosmic =
strings=E2=80=99.=E2=80=9D (anonymous, 2005, p. 16)=20

=20

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:

=20

=E2=80=9CThe 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.=E2=80=9D

=20

Gott states,

=20

=E2=80=9CTo 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.=E2=80=9D (Gott, 2001, p. 104)

=20

And Tipler has a solution which allows time travel in our universe:

=20

=E2=80=9CFrank 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.=E2=80=9D(Gott, 2001, p. 117)

=20

There is one more interesting counter argument to Godel=E2=80=99s =
possible universe. If the multiverse is true, then Godel=E2=80=99s =
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=E2=80=99s wave equation of the universe would =
include all possible universes(Kaku, 1994, p. 254), and that should =
include Godel=E2=80=99s universe, making it a real item.=20

=20

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:

=20

And

=20

=E2=80=9CWithout 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)

=20

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=E2=80=99t =
possibly be random at all but are fixed into the block universe of the =
B-series=E2=80=94the 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 =
=E2=80=98random=E2=80=99 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.

=20

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=E2=80=99t
murder =
487 people, God can give you a glimpse of your timeline, effectively =
giving a trial by replay for your benefit.

=20

What does this do to free will? Clearly this is where everyone will =
complain. Indeed, I don=E2=80=99t like this aspect of Godel=E2=80=99s =
argument. Free will is a form of intentionality and physics has
little =
to say about how that arises:

=20

=E2=80=9CIn 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.
=E2=80=9CBut 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=E2=80=99s 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.=E2=80=9D =
(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=E2=80=99t 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=E2=80=99t know. Godel=E2=80=99s =
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=E2=80=99t relativize existence.
But, =
if he is right, then one can get out of Godel=E2=80=99s 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=E2=80=99s 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=E2=80=99t 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.=20

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=E2=80=99t 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.

=20

References.=20

=20

Anonymous, 2005. =E2=80=9CMystery Rays Could Be Sign of Cosmic =
Strings,=E2=80=9D New Scientist, June 4, 2005

=20

Deser, S. and Jackiw, R., 1992. =E2=80=9CTIME TRAVEL?=E2=80=9D
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

=20

Dorato, Mauro 2001 =E2=80=9CON BECOMING, COSMIC TIME AND ROTATING =
UNIVERSES,=E2=80=9D 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

=20

Ellis, George F. R., 2005. =E2=80=9CPhysics, Complexity and =
Causality,=E2=80=9D Nature, 435:(June 9):743

=20

Godel, Kurt, 1949. =E2=80=9CA 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

=20

Gott, Richard, 2001. Time Travel in Einstein=E2=80=99s Universe, (New =
York: Houghton Mifflin Co.)

=20

Kaku, Michio, 1994, Hyperspace, (New York: Anchor Books).

=20

McTaggart, John Ellis, 1908. =E2=80=9CThe Unreality of Time, Mind: A =
Quarterly Review of Psychology and Philosophy, 17:456-473. =
http://www.ditext.com/mctaggart/time.html

=20

Peebles, P. J. E. 1993. Principles of Physical Cosmology, (Princeton: =
Princeton University Press)

Max Tegmark, 2003 =E2=80=9CParallel Universes,=E2=80=9D Scientific =
American, May.

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H. G. Wells, The Time Machine, 1895=20

=20

Yourgrau, Palle, 2005. A World Without Time, (New York: Perseus Books).

=20

Godel=E2=80=99s Universe

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=E2=80=9CGodel took T[mu,nu] to be a space-time constant, not
vanishing =
only in its time-time component (energy density),

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T[infinity] =3Dc^4/8=CF=80G > 0 . (5)

=20

The metric tensor that then solves Einstein=E2=80=99s equations leads
to =
the space-time interval

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ds^2 =3D g[mu,nu] dx^[mu] dx^[nu]=20

=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)

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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)

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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.

=20

=20
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Why atheists believe in a universe capable of Judgment and