# Time as a Dimension or Projection?

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1. Aug 27, 2010

### inflector

I see lots of references to time being the fourth dimension as well as there being 3 + 1 dimensions to spacetime as we know it, etc. I also see that time has to be treated differently in some of the constructs of physics. So it seems that time seems to be both similar and dissimilar to the other more traditional physical dimensions.

It seems clear that everyone proposes that there are more than 3 dimensions since there must be some mechanism to account for the changes that occur in systems which we all observe and interact with all the time.

Has there been any work that considers time as changes in a projection of 4D space onto 3D space? Or is this a traditional perspective?

In case, what I'm asking isn't clear, by "projection of 4D space onto 3D space," I mean something analogous to a 2D projection of 3D space. For example, a slice of a 3D object is a 2D projection of sorts. A slice of a human body like you might see at a science museum being one example, or the various 2D plans for a building an architect might use.

This seems like a subject that should be well covered, so I'm looking for pointers to the relevant discussions, if possible, papers, sites, etc.

Curtis

2. Aug 27, 2010

### marcus

Except as a provisional computation strategy, I don't think it is a good idea of geometrize time (as if the crystal of eternity really existed and we were snail trails in it) as if the present moment was just a psychological illusion.

In 2008 there was an essay contest on "what is time" with some rather distinguished entries and two sets of judges (so two first prizes, two seconds etc.)

The Fundamental Questions Institute "Fqxi", which gives out research grants, held the contest.
The prize-winning essays are online.

As I recall, the two first prize winners were Julian Barbour and Carlo Rovelli.

Neither of them treated time as a geometrized "fourth dimension". I don't think any of the top four or five essays did that. There are logical problems with that when you take both quantum theory and general relativity together, so people tend to avoid it.

But certainly as a practical computational device we are always provisionally geometrizing or "spatializing" time.

BTW in the unimodular version of GR, time is the volume of the past, or comes out to be uncannily like unto that. That didn't come out in any of the Fqxi prize essays but it is something to know about anyway.

I will get some links, at least to the Barbour and Rovelli essays. You can google for "fqxi essay contest"

3. Aug 27, 2010

### inflector

Thanks marcus, I'll look the essays over.

I suppose: "What is time?," is really is one of the BIG questions.

- Curtis

Last edited: Aug 27, 2010
4. Aug 27, 2010

### marcus

for some reason that caused uncontrollable silent chortling
probably good for the stomach muscles, like doing sit-ups.

Yes I suppose that too.

I went off to get a snack in the kitchen and forgot to fetch links. Just now remembered.

Here is recent offshoot (2 years after the essay contest)
http://arxiv.org/abs/1005.2985
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
4 pages
Carlo Rovelli, Matteo Smerlak
(Submitted on 17 May 2010)
"The thermal time hypothesis has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we use the notion of thermal time to study thermal equilibrium on stationary spacetimes. Notably, we show that the Tolman-Ehrenfest effect (the variation of temperature in space so that $$T\sqrt{g_{00}}$$ remains constant) can be reappraised as a manifestation of this fact: at thermal equilibrium, temperature is locally the rate of flow of thermal time with respect to proper time - pictorially, "the speed of (thermal) time". Our derivation of the Tolman-Ehrenfest effect makes no reference to the physical mechanisms underlying thermalization, thus illustrating the import of the notion of thermal time."

The essay submitted to Fqxi in autumn 2008 and then later put on Arxiv:
http://arxiv.org/abs/0903.3832
"Forget time"
Carlo Rovelli
(Submitted on 23 Mar 2009)
"Following a line of research that I have developed for several years, I argue that the best strategy for understanding quantum gravity is to build a picture of the physical world where the notion of time plays no role. I summarize here this point of view, explaining why I think that in a fundamental description of nature we must "forget time", and how this can be done in the classical and in the quantum theory. The idea is to develop a formalism that treats dependent and independent variables on the same footing. In short, I propose to interpret mechanics as a theory of relations between variables, rather than the theory of the evolution of variables in time."

Last edited: Aug 28, 2010
5. Aug 27, 2010

### atyy

They can get emergent space, but apparently not yet emergent time.
http://arxiv.org/abs/hep-th/0601234
Emergent Spacetime
Nathan Seiberg

But an interesting proposal is in section 4.1 of
http://arxiv.org/abs/0711.1656
The Arrow Of Time In The Landscape
Brett McInnes

Another interesting proposal for emergent time is
http://arxiv.org/abs/gr-qc/0702030
An Introduction to Loop Quantum Gravity Through Cosmology
Abhay Ashtekar

I like
http://arxiv.org/abs/0909.1861
Space does not exist, so time can
Fotini Markopoulou
"Background independence means to make no distinction between geometry and matter and to describe the dynamics of the universe as seen by observers inside it. Diffmorphisms are not about timelessness but about being inside a dynamical universe, affecting it and being affected by it, constituting it."

Last edited: Aug 27, 2010
6. Aug 28, 2010

The space of projections from 4D to 3D is, roughly speaking, locally 4x3=12 dimensional. There are many ways in which a projection can change. You would like to split it somehow into 11+1?

7. Aug 28, 2010

### marcus

Who is meant here when you say "they", Atyy? Doesn't it appear that the proposal of thermal time is completely emergent? Somewhat like temperature---temperature is an emergent property expressing something about a large collection of molecules. It only appears at large scale, statistically so to speak. This is how the proposal of Alain Connes and Carlo Rovelli works for time.
This year's paper (Rovelli Smerlak) just carries the idea further.

So who is it you mean when you say "not yet [get] emergent time"?

Last edited: Aug 28, 2010
8. Aug 28, 2010

### atyy

Oh, I just meant whatever Seiberg was reporting on - the sentences were meant to describe only the references directly below them.

The Ashtekar LQC reference I gave does have some sort of emergent time.

9. Aug 28, 2010

### atyy

BTW, isn't thermal time just time given by the second law, which has always been emergent anyway?

Isn't Rovelli still using proper time in that paper, and that time is not emergent?

10. Aug 28, 2010

### marcus

I wouldn't think so. The second law gives a direction but that is a long way from giving a clock or a time coordinate. Entropy can increase at variable rate.

The main paper to look at here (unless you go back to the one with Alain Connes in the 1994) is "Forget Time", Rovelli's 2008 essay.

In that paper he argues forcefully that in nature time is emergent. A macroscopic feature of large systems. It is a pretty hard idea to grasp and the only thing I can suggest to do is read the essay. You will see how he proposes to derive time (a measure of time) using something called the Tomita flow (which I don't understand) and you will see that it is not based on entropy (though it certainly relates to it.)

Then there is the followup paper Rovelli Smerlak. I will take another look. Most likely he uses coordinate time for some purpose at some point but that would not be the whole story. There is a fascinating idea in that paper that the gravitational field itself has a temperature and different places different temperatures. That was discovered by Ehrenfest, and independently by Tolman. Now Rovelli Smerlak are saying yes indeed the gravitational field has a temperature at whatever designated location, and that temperature corresponds to how rapidly time flows at that location.

I don't think that could refer to the proper time of an arbitrary particle because the particle might not stay at the given location. There are probably several measures being used here, related in various ways. Not all that simple.

Probably not quite that simple. But I can't parse the Rovelli Smerlak paper for us right now. Will try to take a look later.

========================

Yes, I took a look. The Rovelli Smerlak paper is not about proper time. It is explicitly about the TTH (thermal time hypothesis) which it starts talking about in the very first sentence. The thermal time is only defined (with the Tomita flow) on page 4 in the section called
Thermal time hypothesis for general relativistic quantum systems .

How they describe it on page 1 is simplified (classical and not relativistic) just to give intuition---it conveys the unrigorous approximate idea---how to think of it. The real thing is based on a couple of Connes papers which they cite: [24] and [25]. I haven't tried to get those or to understand the real version of the TTH.

Last edited: Aug 28, 2010
11. Aug 28, 2010

### atyy

Doesn't an emergent time just has to increase monotonically - like Ashtekar's emergent time in LQC?

But the thermal time involves the proper time: "The core of the TTH hypothesis comes into play in the identification of the temperature as the ratio between the local proper time and thermal time flows."

12. Aug 28, 2010

### marcus

Well that is the hypothesis, isn't it. And it may be right and you may be too!

But then that depends on our living in a gravitational field of which we know the temperature. What is the temperature of the gravitational field where you are?
You know that this is not the ordinary air temperature. What is it?

Tolman and Ehrenfest knew what the temperature of gravity was. Atyy this is a hair-raisingly obscure subject. We are used to Unruh telling us that an accelerating observer experiences a temperature of space (of the gravitational field he is in, I guess you could say)

We are used to the black hole horizon having a temperature. Again a case of the gravitational field having a temperature.

But are you used to just any old location in any old gravitational field having a temperature?

Maybe this is simpler than I suppose, and perhaps you will explain.

Last edited: Aug 28, 2010
13. Aug 28, 2010

### atyy

Yes, it's obscure to me. But it seems that he is talking about classical thermodynamics in curved classical spacetime - so the temperature is due to matter.

"The first was proposed in the original work of Tolman and Ehrenfest, using the idea of a “radiation thermometer” – electromagnetic radiation whose pressure p measures the local temperature T"

"Consider a macroscopic system, say a gas, in a stationary spacetime."

14. Aug 29, 2010

### Fra

For the sake of discussion, this direction of thought is similar to how I personally understand time, but there are some issues with this. This is what I think of the issues and some possible ideas for resolutions.

(1) there is the problem of deciding what entropy measure to use. As we know entropy is just a mesaure of missing information about microstates, but such a measure can be constructed in different ways. So which is it? We also have the closely related ambigousness of choice of prior. These are "issues" of all "entropic methods".

(2) Also, unless we're talking about differential entropy, usually entropy with a fixed prior and measure are "global in nature" as they assign a global flow on the microstructure. That may suggest that there are fundamental fixed degrees of freedom in nature, on which to apply this. I think this is far from clear.

If we use a global entropy then that would correspond to an equiblirium, but it's probably valid for exploring some simple ideas, but I think eventually the full treatment needs to understand also the nature of equilibrium shifts, corresponding to that the global flow defined by the entropy is also evolving. Otherwise it's like an analogy considering gravity as curved spacetime, but forgetting that spacetime itself also evolves. (except what is evolving here is the entropy measure and the microstructure).

A version of this is that instead use a differential entropy measure to define a local flow each each point, that rather corresponds to dissipation not relative to a global flow, but relative to the current state which defines a local flow as a form of perturbation in information space. The same logic is used, except that the entropic flow is not globally valid, it's only valid in the differential sense (ie in order to evaluate the next step).

Two observer can expect different flows, but the measurable implications; when they compare their results by interacting, is rather that they will "see" different interactions - which in the ideal case are related by transformations, like in current models.

To get clock time or a pace from the entropic flow, I think we can just compare ratios of transition events in the overall state vs the state in any subsystem we can arbitrary label clock. I don't see that as a major problem. But again, there can not be EXCEPT in certain forms of equilibrium cases any global times. But conceptually I think this is good, because it really does not make conceptual sense. Global time doesn't belong in a sound reasoning IMO.

It was some time since I read rovelli's view, but as far as I think I remember my own conclusion, rovelli's view corresponds to a certain equilibrium case. This doesn't mean that there is no local times, it means that he hold a form of structural realism of the global system. This is typical rovelli as far as I've analysed his other reasoning too. These assumptions makes things more decidable! and it's probably why he adopts them, but the question is if the decisions are right. I'm suggesting that we don't need to make all decisions. All the decision we need to make is the next step. I think it's how nature also works. Some decisions simply can't be made until in the future. Inconsistencies in reasoning appear if we try to do it prematurely.

The relation to a more abstract notion of temperature is interesting. I looked at my own notes when I've tried to compare some of my expressions with simple stat mech models and I've come to associate the "temperature" with "evidence counts per distinguishable statespace". IT's somehow a measure of how "massive" a "probability distribution is", and when one considers transiton probabilities combinatorically it's clear that this in related to a form of inertia of the distributions. Ie. their resistance to change. So the connection to rate of the entropic flow here is clear.

Given that this is extremely immature i think this kind of thinking and the approaches that are related to this, is great, and it's research in the right direction. We need to go back to the foundations of statistics and probability, and understand what that means, in terms of new logic, such as quanutm logic, and to see why the classical logic is only a special case.

All these abstract ideas are great because they are independent of specific programs such as string theory or spin networks. They can be phrased in more general terms, and connections with insights can probably be made to several existing programs.

/Fredrik

15. Sep 2, 2010

### Bobby Weardal

Perhaps a more interesting projection is 3+1 to 4. I say this because the natural geometry of space-time is 3+1 but through our measuring and representation systems we visualise the world as 4. A space time diagram represents measurements of time and distance on a Euclidean reference grid. Since the true geometry and the represented geometry of the world are different then the interval between any two events as graphically represented on the space-time diagram deviates from the proper interval that separates them. In the extreme case when the distance element of the metric is equal to the temporal element then the deviation is equal to the full magnitude of the interval represented on the space-time diagram. As a consequence of this an event in Minkowski (or Schwarzschild etc) space-time cannot be uniquely represented on a space-time diagram. There is not a one to one correspondence between any event in Minkowski space-time and its representation on our reference grids. Consequently an event in Minkowski space time must be primarily projected on to our reference grids as a light cone. However each point on the light cone itself cannot be uniquely represented therefore each point on the light cone must itself be projected as a secondary light cone. Thus a proper event in Minkowski space-time is projected onto our reference grids as an infinite succession of light cones. The laws of Physics can be reformulated using this property (The relationship between the locations of events in proper space-time and the way they must be projected onto our reference grids.) A quantum event (a moment in the history of a quantum system) must be projected onto our reference grids as a light cone. This event projection will contain all the information about the quantum state of the system during the event, including how the system can absorb and donate energy. For instance the wave function associated with a photon is formed of the future light cones projected from successive events of the donor system. This allows the wave-function to be established in the vicinity of an absorber system for an indefinite period before an interaction occurs. (The future projections of the donor system will become super positioned with the past projections of potential absorber systems allowing the photon to be absorbed by the most attuned absorber system.)

Understanding what is happening in proper space-time and how this behaviour projects onto our reference grids is perhaps the key to understanding quantum behaviour and developing a system of physics free of contradictions between quantum mechanics and special relativity.

16. Sep 2, 2010

### granpa

say what?

What the heck do you mean by 'the interval between any two events as graphically represented on the space-time diagram'? are you talking about naively measuring with a ruler the line drawn on the diagram between 2 events in space-time. That isnt how space-time diagrams are meant to be used.

In a space-time diagram a given frame is a 3D 'slice' of the 4D space-time not a projection.

and the position of any given event is uniquely defined not only for one given frame in that space-time diagram but for all possible frames in that diagram and they all agree completely. I would even go so far as to say that that is the whole point of a space-time diagram.

Last edited: Sep 2, 2010
17. Sep 3, 2010

### Bobby Weardal

say what?

Yes, part of my example was to naively measure the interval between two events on a space-time diagram. This is compared with the proper interval between the events as obtained from the metric. The difference between the two measures demonstrates the error in the space-time diagram representation of the interval. In the extreme case all events on a light cone are separated from its apex by proper intervals of zero magnitude. This shows that mathematically the apex of the light cone is inseparable from all other events on the light cone. Therefore events in the natural world cannot be uniquely represented by four coordinates on our reference grids but must be projected on to them as light cones. This is not a theory but a theorem and it is necessary to apply the theorem appropriately to avoid the development of contradictions in physical theory. For example, contradictions between quantum mechanics and relativity!

More often a 2D slice.

However I’m referring to the projection of events onto our reference grids that are generated by the limitations of our measuring and representational systems.

The nature of our reference grids depends intrinsically on how we measure time and distance. The methodology will therefore impose limitations on the scope of its application. In the case of inertial reference grids it is only large observable bodies that can be attributed unique sets of coordinates. Even so, the locality possessed by large bodies is a pseudo locality, since such bodies are composed of countless quantum entities and it is their group behaviour as observed by the observer that gives rise to a unique locality in the world. Beneath this the events in the histories of individual quantum entities will form lights cones relative to our reference grids.

From this we can conclude the your statement cannot be universally true but can be applied to large observable bodies relative to which the methods of classical mechanics and relativity directly apply. For quantum objects we must recognise that events cannot be assigned unique set of coordinates and we must modify our descriptions and methodology accordingly.

18. Sep 17, 2010

### Rebound

19. Sep 17, 2010