From the Readers

As far as I understand Hitoshi's theory of time, what he says is that there is no universal time which is governing the whole cosmos; in other words, there is no beginning nor end to the universe. Time should be recognized only locally. When we pick up some particular phenomena going on under certain circumstances involving certain participants (no matter what they are or who they are), only then we feel we can measure time. Hitoshi claims that time as a scale does not "exist" as we look into the inner cosmos which is in ourselves. Nobody is conscious of time usually. When we pay special attention to "change" of something, we try to explain it by referring to time. Time, however, is most intricate creation of "modern" human beings. Would that make any sense to you? Or do you think such a statement is a total nonsense? What is time for you? Let us know that.
From a reader of Hitoshi's works

Time, what is it? Where is it? We made it up so long ago. What if time where space, and space where time? What if our lives which we beleive are to be say 70 to 80 years, was just a fraction of a second our time. Im so confused, only if we could travel the universe, then could we start research. But untill then time to me is everything I am and do. Untill next TIME over and out.
Jeffery Bowen

Suppose that we have two identical and synchronized light clocks and we send one of them on a round trip around the world, and after its return, we observe that the two clocks show different times. If we then ask an intelligent layman about his judgement of these clocks, we are likely to hear that these clocks are no good clocks. They do not produce a valid measurement of what common people, in agreement with Galilei and Newton, consider as "time". Einstein considered it impossible to measure this common sense time in a physically objective way, but now we can do it, using a "spacetime odometer" in addition to a light clock. I hope this will make it possible for physicists to switch back to common sense.
Hartmut Traunm ler URL:http://www.ling.su.se/staff/hartmut/uhr.htm

I feel that there must be some "real" time but I agree that it should only be defined locally. Experiments that confirmed special relativistic "time dilation" using radioactive clocks suggest to me that nature does indeed behave in such a way that seems consistent with a notion of time.
I think that the nonuniformity of time in special relativity also leads to some misleading statements. People often say time is running slower. What does it mean for time to have a velocity? This is ridiculous of course.
Jason Sharples
 Is the speed of light slowing down? And if it is, as computations taken over the past 150 or so years seems to indicate, does this mean that our measure of time is skewed, too? Just a thought to brighten our days.
Michael Kitada
 Aren't time dilations conjugate to space (length) dilations ? Can't you look at these either way ?
In this sense, isn't relativity an uncertainty relation like Heisenberg's ?
Robert Fung
 Could an ultrametric be defined among the local systems?
cf.
http://mathworld.wolfram.com/Stephen Paul King URL:http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html
 Could you ruminate a bit on Bell's Theorem? Preferably on Lance's TIME list  I'll be sure to catch it there. Thanks.
Lloyd Doering
 thanks for privileges of visiting your home page and reading your article on time. No comment as yet but will.
stanley Chan
 Hi, the photo of you on the train in Denmark looks to me yours is a very plesant and relaxing trip. If so, I echo to that which was my experience only fortnight ago.
My interest in time is in a clinical sense, seeking a linkage,the existence of which is in my mind beyond a doubt, between the conceptual application of the theoories of time to one world view of life, particularly in a terminal stage of one's earthly existence. Your theory of local time as can be implied in a dying patient's inner time is worty of further reflection. will be in touch.thanks. Stanley
stanley chan
 Read an engineer's approach to the same "time" problem:
http://webpages.charter.net/stephenk1/Outlaw/Outlaw.htmlJerry Malsam
 Thought, we travel the world over, to find the time,
we must carry it with us,
or we find it not.
excellent site!!
Keijo Kettunen
 IV. Defining the Local Clock: an alternative notion of time
I've come up with an idea and some data that would seem to require QM to use a notion of time that appears to be very similar to what is expresed in the above section.
The basic idea is than Minkowski space prevails as reallity and that Riemann is how our mind interperts the reality. This idea appears to be testable and the discarded data in the MM class expeiments would seem to support it.
Perhaps you might find this interesting.
Steve Jensen
 Will time travel ever be possible?
lauren
 To whomever it may concern,
I am currently a debate team member at my high school. One of the ideas my team came up with is to travel back in time and blow up the earth. therefore, all the harms and bad things on the planet will have never existed. Now here is the question. Do you think that time travel is possible, and that our idea is feasible? Also, we are not only limited to just blowing up the planet earth, we could also stop the universe from ever happening, ie stop the "bigbang". Now do you think that this is possible? Thank you very much and please send any answers to X Spawn777@aol.com
Thanks again
Nate
 A compreensão do tempo Ealgo intimamente vinculado Ecompreensão das condições fúicas locais de cada ponto do Universo. Se realmente Eassim, então o tempo Eenergia dependente do espaço e poderEser explicado por meio da interação das forças de gravidade, eletromagnética, forte e fraca, quando essa interação estiver resolvida.
italo ramos
 To experience the passage of time is to be alive !
 to be made of carbon and to have memories.
Oh to be wandering the universe as a photon
instant and in pepetuity.
DNA is coded for time (see superstrings )
JOhn Valle

Time cannot possibly exist by itself, as a thing in itself. Time is a symptom of awareness, but then it has
to exist if there is any reality to the universe.
So what came first  Time or Space ?
John Valle
 My thoughts are taking shape in this private area. Thank you for providing an appropriate forum for such offerings.
Chris Cowsley
Dr C W Cowsley
 You mention Julian Barbour on your site. Does anyone know WHEN his longannounced new book "END OF TIME" will be published??
Regards G Coldham
G Coldham
 I am not well studied in these matters, but what are the chances of time being a fundamental force of nature, much as gravity, electromagnetic, strong and weak neuclear force, instead of being considered a dimentional element? It would seem that time could be observered as a wave, a particle, etc., and have both a positive and a negative element (i.e. time, antitime). Forgive me for being ignorant, I have had this question on my mind for some time.
Paul J. Heidt
 I think that the progression of time is due to the cosmological scale expansion. Please check my web page http://www.wolfenet.com/~ritech/ and my upcoming paper in Astrophysics and Space Science later this year.
C. Johan Masreliez
C. Johan Masreliez
 How can one define borders to each local system?
Altamirando L. Leal Jr.
 This is a response to the following inquiry:
>Name: Altamirando L. Leal Jr. (email: allj_leo@hotmail.com )
>
>How can one define borders to each local system?Local systems have no boundaries in either spatial or temporal sense. A local system is just a collection (set) of plural particles. It can extend to any place far from its center of mass. This extension does not require time, for in a local system, no time is necessary to reach any point of the system. It is an insideworld as our mind.
Hitoshi Kitada
 do you think gravity=spacetime^2 meaning the universe would simply be a huge or relativistically small inverse square law?
CD
Craig Day
 very interesting. Time. Everybody thinks they
know what is but does anyone really? To a child,
next week is a lifetime away, to the grandfather
who's promised to take him to the ballgame(or
wherever) then it seems like an instant. The
extent of our experience of it seems to control
our perception of the speed of it's passage. The
child may be 5yrs., the grandfather 75, the
universe ???. BUT! Perception depends upon
consciousness. Is the universe conscious? If it
is, given its age, does the passage of time even
mean anything to it? Can it? or is everything
always? Are time and consciousness two sides of
the same coin? They seem to be dependent on each
other.
len soukup
 A very interesting set of problems addressed with the sort of intelligent enthusiasm that eventuates in exciting theory. A couple of exegetical suggestions that may improve communications:
 A clearer presentation of exactly what "local" time is essential if we are to understand exactly what you are talking about. What is "nonlocal" time, if there is such an idea.
 The general contours of the relation of quantum theory and relativity theory should be made more explicite: I think this will strengthen the position.
 The statement of the "problem of time" needs sharpening. Some folks working say entirely in topology may not be apprised of a lot of the physics. These people face analogous problems in making their theories clear: of course we all experience space but how to we convey in sensory terms the notion of a metric space or a strictly Hausdorffian space? It is not "undoable" but the interface of intuition and formalism must be discrete.
 There is a lot going on here. It is places such as this little web site that will most likely pioneer the new idesas that drive science, a pursuit which like philosophy is threatened by institutional ossification. Good luck.
Steve Bayne

Your views on Time are refreshing to those of us who have been offering the same conception for years. Good luck.
James Rockefeller

Perhaps you may be interested in something I wrote for the "Long Now Foundation" on the web:
No discussion of a clock should proceed without a precise conception of what is being discussed.
Days don't make clocks, we know that the earth's rotation is not constant. In fact no celestrial motions are. Even atomic clocks are effected to some degree by the exact evolving force field structure in which they are situated. So what is the basis of the concept of a clock? What definition of a clock can outlast our current best guess about the form of the true theory of everything? How could we even attempt to compare alternative views of the world if theories can not agree on a conception of a clock? How are we supposed to lay one time period next to a subsequent one and tell if they are of the same magnitude? The best definition available is:
The ideal clock is the recurrent process that makes the most other recurrent processes periodic.
This concept reaches out into the heart of nature and takes the measure of its heartbeat. This is a concept that will be viable to a future that might conceptualize the rest of the world very differently than we do today. If the present idea of time survives then so will this conceptualization of its measurement. This can at least position the concept being pursued by clockmakers.
Now it can be discussed which clocks best fulfill their mission.
Andre Mirabelli
 I AM AMAZED! AS AN UNDERGRAD MY PHYSICS DEPT TOLD ME IT WAS "USELESS" TO EXPLORE A CONCEPT OF TIME VERY SIMILAR TO YOURS, OR WHITEHEAD'S RELATIVITY, OR BOHM'S MECHANICS...NOW THAT I'M PUSHING FIFTY I WAS TAKING ALL OF THAT UP AGAIN WHEN I FOUND YOUR WEBSITE...MAY I ASK QUESTIONS AND CORRESPOND?
Mark "Radar" BrowneMiddleton

My comments is: it can explain the exist of the new dimension, as the membranes of the strings theory?
Alejandra Hernández Altaba
 A Comment to the above:
It has a relation with the new dimension of the string theories. Refer to the discussions in time group at time archive, especially the issues at the end of 2000, e.g., 1493 On RBT, Internal Time, and LT Hitoshi Kitada Thu 12/21/2000 (and 1497 On Life (was: On RBT, Internal Time, and LT) Hitoshi Kitada Fri 12/22/2000)Hitoshi Kitada

In general relativity, we have no problem explaining time. Time and space are different aspects of one thing called spacetime. Any 4vector has the form [t,x,y,z]. We are probably making assimptions about the nature of spacetime that we don't even realize we are making. In classical physics, they made assumptions about time that they didn't even realize were assumptions.
Jeffery Winkler

Hello
Thank you very much for your very thoughtful work on time. I have included some thoughts here that I have had in my attempt to understand time.It is common for people (west) to speak of temporal time and enternal time. Perhaps there are two distinct times.
Time can be compared to space:
 The "present" is like three dimensional space. That is movement and change can take place. Also there can be two way communication.
 The past is like two dimensional plane. That is like a picture. Something flat that cannot be changed and impossible to communicate with.
 The future is like a one dimensional point. It contains the least amount of information because the future is dependent upon the present and therefor always to some extent undetermined.
"Temporal Time" can be thought of as the continuous movement from one dimension to three dimensions to two dimensions.
"Eternal Time" is this continuous movement but in which the future is two dimensional and the past is one dimensional. That is, in "Eternal Time" the future would be as knowable as the past is to us today. It would be a time will a clear future and a dim past.Bill Givens

You claim that the following ``theorem'' of calculus ``includes a selfreference'':
Any nonempty set A of real numbers has an upper bound denoted sup A, if A is bounded from above.
My first comment is that in modern analysis, this is not a theorem, but an axiom for the real number system. My second comment is that you have misstated the axiom. It should read as follows:
``Let A be a nonempty set of real numbers that is bounded above. Then A has a least upper bound.''
My third comment is that the apparent selfreference can be removed from the definition of supremum. Here is how this is done:
1. Definition: Let A be a set of real numbers. If there is a real number r such that for every member a of A, a<r or a=r, then we say that A is *bounded above*, and such a real number r is called an *upper bound* for the set A. (Note that the empty set is bounded above.)
2. Definition: Let A be a set of real numbers. We denote by U(A) the set of all upper bounds of A:
U(A)={r\in\R  a\in A implies that a=r or a<r}.
3. Definition: Let X be a set of real numbers. If there is a number m in X such that for every number x in X, either x=m or m<x, then m is a *minimal element* of X.
4. Axiom: Let a and b be real numbers. Then a<b or a=b or b<a.
5. Axiom: Let a, b and c be real numbers. If a<b and b<c, then a<c.
6. Axiom: Let a and b be real numbers. If a<b, then a is not equal to b and b is not less than a.
7. Theorem: Let X be a set of real numbers. Then X has at most one minimal element. Therefore, if X has a minimal element, then X has a unique minimal element.
Proof:
Let m and n be minimal elements of X. Since m is minimal and n is in X, either m=n or m<n. Since n is minimal and m is in X, either n=m or n<m. Thus If m<n, then m is not equal to n, so n<m. This violates one of the above axioms, and so is false. Thus m=n.
8. Definition: Let X be a set of real numbers that has a minimal element m. Then m is called *the least element* of X.
9. Axiom: Let A be a nonempty set of real numbers that is bounded above. Then U(A) has a minimal element.
10. Definition: Let A be a set of real numbers that is bounded above. A minimal element of U(A) is called a *supremum* of A.
11. Theorem: Let A be a set of real numbers that is bounded above. Then A has a unique supremum.
12. Definition: Let A be a set of real numbers that is bounded above. Then the supremum of A is denoted by
sup(A).
13. Definition: Let B be a set of real numbers. If there is a real number r such that for every number b in B, either r=b or r<b, then we say that B is *bounded below* and we call the number r a *lower bound* for B. The set of all lower bounds for the set B is denoted by L(B):
L(B)={r\in\R  if b\in B, then either b=r or r<b}.
14. Definition: Let X be a set of real numbers. If there is a number M in X such that for every number x in X, either x=M or x<M, then we call the number M a *maximal element* of X.
15. Theorem: Let X be a set of real numbers. Then X has at most one maximal element.
16. Definition: Let X be a set of real numbers that has a maximal element M. Then M is called *the greatest element* of X.
17. Theorem: Let B be a nonempty set of real numbers that is bounded below. Then L(B) has a maximal element.
Proof:
Let A=L(B), and let b be any number in B. If a is in A, then a=b or a<b, since a is a lower bound for B. Thus A is bounded above. Since B is bounded below, A is not empty. Since A is a nonempty set of real numbers that is bounded above, A has a supremum, sup(A). Since a supremum of A is a maximal element of A and A=L(B), it follows that
L(B) has a maximal element, as claimed.
18. Theorem: Let B be a set of real numbers that is bounded below. Then B has a greatest lower bound.
19. Definition: Let B be a set of real numbers that has a greatest lower bound. Then we denote sup(L(B)) by inf(B). This number is called *the infimum* of B.
20. Theorem: Let A be a set of real numbers that is bounded above. Then sup(A)=inf(U(A))=sup(L(U(A)).Thus, in this last theorem, an ``appearance'' of selfreference is found, in the sense that the supremum of a boundedabove set A of real numbers can be calculated in terms of a supremum of some other set of real numbers, but this is only one of many ways to calculate the supremum of a set of real numbers that is bounded above. It is not the only way to calculate a supremum, as was shown above.
Matt Insall
Matt Insall
 A comment to the above message from Matt Insall:
Thanks, Matt, for your comment and correction to my mistyping. When I wrote it I noticed the lack of the word "least" but did not find time to supplement it.On other points, my answer is as follows:
 > My first comment is that in modern analysis,
> this is not a theorem, but an axiom
> for the real number system.Of course it can be assumed as an additional axiom to set theory as done in usual courses on calculus for freshmen insofar as we agree that such an exposition only treats real numbers. I took the lectured form of exposition of real number theory as the issue of the lecture was science that is thought formalizable as a single theory. Then necessarily I need to explain a construction of numbers, mathematics, and physics based solely on the set theory, not assuming axioms other than those of set theory.
 > My third comment is that the apparent selfreference
> can be removed from the definition of supremum. Here
> is how this is done: ...
...
> Thus, in this last theorem, an ``appearance'' of
> selfreference is found, in the sense that the
> supremum of a boundedabove set A of real numbers
> can be calculated in terms of a supremum of some
> other set of real numbers, but this is only one of
> many ways to calculate the supremum of a set of
> real numbers that is bounded above. It is not
> the only way to calculate a supremum,
> as was shown above.I disagree with that "selfreference" is removed in your argument from step 1 to step 20. You just assume, as an a priori axiom, the existence of supremum of a set A of real numbers under the assumption that A is a nonempty, boundedabove set of real numbers. This argument of yours or "modern analysis" in your sense does give no way of calculating the supremum, but just assumes its existence. This conceals the procedure of construction of the least upper bound behind the axiom you referred to, thus does not give any solution to the problem of "selfreference."
 Further, your argument in your third point seems incomplete. You state in item 11 as follows:
> 11. Theorem: Let A be a set of real numbers
> that is bounded above. Then A has a unique supremum.Reading the later part 17, the proof of the theorem asserted in 17 looks assuming that "A has a supremum, sup(A)" implies that "A includes sup(A) as an element." I.e. in the part: "Since A is a nonempty set of real numbers that is bounded above, A has a supremum, sup(A). Since a supremum of A is a maximal element of A and A=L(B), it follows that L(B) has a maximal element, as claimed," you seem to "deduce" that sup(A) belongs to the set A from Theorem 11. But this is incomprehensible. In fact consider an open interval (a, b) (with  infinity < a < b < infinity) of real numbers. It is nonempty and bounded above, but does not include b = sup(A).
Hitoshi Kitada
 > My first comment is that in modern analysis,

I was astonished that your definition of a "quantum clock" is very similar
to mine (quantph/9906130 and 9912021):
exp( 2pi itL/(hm )) .
In my case, an "internaltime" is introduced in a similar form:
exp( 4pi i\int dtL/ h )
but the crucial difference of multipliers (2 or 4). According to a Bohm's book, de Broglie originated such a localclock interpretation of a wavefunction.
I agree with you on that time should be relative to the object observed or considered. In addition, such time should be discretized as a process of phenomena (due to Whitehead). In my opinion, the difference between subject and object creates time not only as such a series of instants (Bachelard) but also as duration (Bergson).
On the other hand, I am interested in how you can take account of true singularities of the universe into the most global definition of time over all the fourdimensional universe (Hawking&Ellis). I have not understood the detail of your solution on this matter, yet.
At least, I think that any quantum theories can not produce any clearcut definitions of time. The opposite is true: the basic nature of time would create quantum mechanical feature of the world.
Toshihiko Ono

I wonder about the mystics "experience" of seeing time endon. This cannot be an experience because it is outside of time, formally similar (it seems) to the difference between universal time and local time. Only mystics seem to be able to apprehend the true infinite time which is not sempiternal; the inability to speak of it is known as the "via negativa." I have an idea of how to speak of it. The second law of thermodynamics, related to the phenomenological experience of time as explained by science, can be restated to say nature abhors differences, i.e., gradients. Pressure, electron potential, and temperature differences represent previous improbabilities that are creatively destroyed by selflike inanimate systems as well as biological selves. In relativity theory, however, space and time are not separate. Thus it occurs to me that the difference between known, "frozen" past and unknown "open" future also represents a gradienta temporal !
one. Breaking this one down is the "direction" of the via negativa.
Dorion Sagan URL:http://www.xsnrg.com/renaissance
 I am reminded of Kant's definition of time as "that which permits two objects to occupy the same space" and of space as "that which permits two objects to exist at the same time." With due apologies to that obscure German, I have never been able to see much more than cleverness in his definitions. But your theory, it seems to me (a nonphysicist) is more compatible with Spinoza's concepts. But I see that Lance Fletcher has already pointed that out.
Frank Dixon
 Dear Professor Hitoshi Kitada,
QM and Einstein's SR together, will may resolve the problem of time. In SR, "I" (as human observer) is the center of the observation. "I" (mind) represents an immeasurable and absolute "nowpoint." The relativity becomes into the woldpicture, because there are many nowpoints (cordinate systems) at the same time (I and you etc.) In order to resolve the measurement problem of QM, the "now" must be taken seriously.
Sincerely
Atso Eerikainen
Th.DAtso Eerikainen