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The description of J. B. Barbour arXiv:0903.3489, 2009 is considered to be timeless but the term "change" appears in that text.

Essentially, a configuration space is considered: One calculates the sum of the variable "action" along each of the many trajectories linking two points in that space. One sum is absolutely the smallest. The "Variational Principle of Jacobi" says that only this is the real case, i.e. only the points lying on that trajectory obey the well-known laws of classical mechanics.

I am looking for a timeless description of the classical world. However, it is not clear to me whether this can be conceived without assuming that the components which are put together to define a point in configuration space have to be taken at one and the same instant.

I see several possibilities: (1) The variational minimum automatically yields those components of a point in conf. space that are valid at ONE instant. I would not understand HOW but that is not the problem of my present question.

(2) I could deliberately select the x and y coordinates of one object at time t1 and as the z coordinate I introduce a z-coordinate of another object, and in addition not taken at time t1 but another time t2. If I understand correctly the definition of a conf. space, only the coordinates of all objects count.

Having defined a point in conf. space with this contribution, there are 2 possibilities: (2a) the the selected collection of coordinates of all objects is indeed reached by the system at SOME (unknown) time. Then, (1) applies. No problem.

(2b) the system never reaches the selected point in conf. space because the system may not go to ALL places within that space.

My questions are: - is (1) the only relevant statement? Then, case (2b) should somehow be impossible, perhaps because the calculation of the "action" fails, or the search for the variational minimum goes awry.

or

  • Barbour's proposal is not correct, insofar as one has to know beforehand what "simultaneity" is (this would be a temporal notion).
Jürgen Krüger
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    BTW-tracking papers that cite one you are interested in (look in the sidebar on arXiv) can be enlightening. For instance the author of this paper has a later one in which he (apparently) distinguishes between 'time' and 'change'... – dmckee --- ex-moderator kitten Jun 14 '17 at 17:36

2 Answers2

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It seem that a good part of the 'timeless' formulations in Ref. 1 is directly or indirectly related to the well-known fact that (for autonomous systems) one may perform a Legendre transform $$\Delta t~:=~t_f-t_i\quad\longleftrightarrow\quad E $$ to a dual energy-variable $E$ instead of time. See my Phys.SE answers here & here for details.

References:

  1. J. Barbour, The Nature of Time, arXiv:0903.3489.
Qmechanic
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I think the issue here lies with the concept of configuration space. The language used in texts might sometimes be confusing especially when describing a configuration space for only a few individual particles. However, in the configuration space every single point represents an entirely separate state for the entire system as a whole. Even if you have a billion particles, the whole current state of the system is just represented by a single point in configuration space.

So as you trace out a trajectory in configuration space you're basically defining an order in which the system evolves. If you draw a line from point P1, to P2, to P3, then you're basically saying that the entire system evolves from state 1 to state 2 to state 3. Note that despite the fact that you're only talking about 3 points in configuration space they are indeed 3 separate full descriptions of all the billions of particles in your system.

Although we can look at it as an evolution between these states to understand the role time plays, note that there is no actual explicit reference to time. So you can think of it as "timeless" and just work out what trajectories exist in the configuration space.

ison
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  • You may have billions of coordinates forming together the point P1 in conf. space, and another such set forming P2. How do you know that you should not mix up PARTS of such sets? One might even ask: where does the idea come from that P1 and P2 differ by SOMETHING? Why are all these points P.. not ONE point in a space with still more dimensions, and if so, how does one proceed to subdivide it into the P1, P2, P3... of a space with less dimensions? – Jürgen Krüger Jun 17 '17 at 12:16
  • The dimensions decide how they are separate. A set of billions of particles is going to have a conf. space with multiple billions of dimensions. But take a more simple example: You can plot 3 particles in space where each has 3 coordinates, but an alternative way of doing it is to plot a single point in a 3x3 dimensional space. There is the exact same amount of information there. You can extend this up to more properties and more particles by adding more dimensions to your conf. space. Two adjacent points in conf. space only differ by the properties of the dimensions they are separated by. – ison Jun 17 '17 at 18:09
  • You say: "Even if you have a billion particles, the whole current state of the system is just represented by a single point in configuration space." I know this. My question is, rather, what do you mean by "current"? It seems that you refer to something occurring SIMULTANEOUSLY on some time axis. Can such a "state" be defined without such a reference to "time"? – Jürgen Krüger Sep 10 '17 at 09:57
  • The idea is that you are replacing time with a path. You don't just use a single point, but rather a curve in configuration space that shows the state of the system at all times at once. So it's like time has been sort of "encoded" into the geometry of your path, there is no time axis and you don't refer to time explicitly you just deal with the path as its own thing. Now to figure out which path physically occurs you consider all possible paths, and the one which minimizes action is the physical path. – ison Sep 10 '17 at 18:01
  • I have understood that the entire curve in conf.-space also "encodes" time. But "You don't just use a single point": does that mean that a single point on that curve cannot be conceived, and not be defined? Does each point separately "encode" a complete time course? - I asked J. Barbour. He replied: "[...] The instant and the configuration are taken to be primitive concepts. [...] I was showing how duration emerges [..] All theories must presuppose certain concepts [..]". - Thus, I still believe that there is a reference to "simultaneity" --> the story is not truly timeless – Jürgen Krüger Sep 18 '17 at 16:30
  • Curves in configuration space are SEQUENCES of complete configurations. Pretend your system is 2 coins. The conf-space is 2D with 4 points. Say (0,0)=(heads,heads) (0,1)=(heads,tails) (1,0)=(tails,heads) (1,1)=(tails,tails) Your space contains all possibilities the system CAN have right? Then a line such as: (0,1)>(0,0)>(1,0) represents a SEQUENCE of 3 different configurations at 3 different times. The line itself is "timeless" only because it represents 3 different times all together. However, usually you don't know the right line, so you consider all lines to find which minimizes action – ison Sep 19 '17 at 21:52
  • For a state "heads,heads" you have to take the 2 coins with heads up at the same time. Right? Or could a well-defined state also be "heads coin1 today, heads coin2 yesterday"? Or even worse: "heads coin1 taken at some unknown instant, heads coin2 taken at some other unknown instant". Only in the latter case I would believe that the concept is really timeless, or better, if the word "unknown" was replaced by "unspecified". – Jürgen Krüger Sep 21 '17 at 11:15
  • (Part 1/2) "heads coin1 today, heads coin2 yesterday" No, time is simply not defined on states. There is no time associated with the states at all, neither today, nor yesterday, nor "at the same time". It is simply a configuration of the elements of your system without taking time into account at all. So the SPACE itself is absolutely timeless. We can then create curves in this space which define ordered sequences of configurations, but this is still not time. – ison Sep 21 '17 at 17:08
  • (Part 2/2) It's just an ordered list, just as the numberline is ordered but that doesn't imply time. Where time might come into play is in how we USE and INTERPRET this mathematical model to associate the line with physical evolution of time. So you could only argue that the CURVE, in our specific use case (as it relates to minimizing action as a temporal sequence of events) isn't timeless if you want to, but just remember that 1) the space is definitely timeless and 2) this mathematical model might have other uses where the curves aren't interpreted as time-ordered at all. – ison Sep 21 '17 at 17:08
  • Also 1 more thing: "varying" the curve as we do to minimize action is NOT a time variation. You're just varying the geometry of it, e.g. varying the ordered list of points (or if you really want to think of it even at this stage as as time ordered list, then you're varying which configurations of the system are associated to which times of your curve, but not varying time itself) This might also be another reason people call it "timeless" even if you think of the curve as time ordered – ison Sep 21 '17 at 17:24
  • How, in a world of coins, a time concept can be reached via a variational principle? Rather, my aim is to understand "J. S. Briggs: [..]emergence of time dependence [..]. Phys. Rev. A 91, 052119 (2015)" who partly relies on Barbour. One considers an isolated universe. The variational principle is applied between two points in a high-dimensional configuration space. I do not understand how one can compose these points from a concrete physical situation, WITHOUT making use of "simultaneity". Thereafter I see no problems with the further variational steps. Can you help me with this problem? – Jürgen Krüger Sep 22 '17 at 16:58
  • The points will be interpreted as two separate times, each describing a state at a single time. I mean, time has to come into play SOMEWHERE because we live in a universe with time and "equations of motion" that you get as your result will need to depend on time. I think you're overthinking it too much. What I have done above was to try and justify to you why some people would call it "timeless", but honestly it's not that important. If you want to say time is involved then that's fine. But time doesn't explicitly play any role in the actual variation itself. It's NOT a time variation. – ison Sep 24 '17 at 00:14
  • (1/2)The essential eqs. of nature are differential eqs. in time. Such a description obscures the fact that at any instant there is only a SINGLE case of a state plus a slightly different one (in a not-yet-realised "virtual" version). For nature, this is "the real world". One may compare the situation with a human seeing himself as a "virtual human" in a mirror. No one would assume that "in a next step" the virtual human becomes real, and a new virtual one appears, and so on. In contrast, for the differential eqs. such an additional assumption is done. It leads to a time concept. – Jürgen Krüger Sep 30 '17 at 16:49
  • (2/2) However, one has to rely on the non-scientific relationship "a present physical event (namely a memory readout) SIGNIFIES a past event" which leads into a phenomenal domain. Phenomenally one gets an overview over longer time spans, a duration is attributed to the world (an "instant"), and one can execute time integrals. But all this has no effect in the world. In the real world it makes no sense to talk about a simultaneity of elements within a state.- Is that approximately correct? – Jürgen Krüger Sep 30 '17 at 16:52
  • Exploiting energy constancy one can, in an isolated mechanical system, replace an infinitesimal time step dt by ALL infinitesimal object displacements (and other variables). Suppose a keen mathematician is given the fundamtental equation of mechanics in this timeless fashion. Suppose he/she ignores the physics behind it. Is it straightforward to find the usual version WITH (ephemeris) time? Would it appear as a simplification? Can this give an idea of how "time" came into nature? – Jürgen Krüger Oct 28 '17 at 17:46