## Saturday, September 03, 2011

### Ten new things modern physics has learned about time

Note that it may take half a minute for your browser to display various formulae in $$\LaTeX$$ that are included in the text below.

Here is a somewhat balanced, completed, and corrected version of a Cosmic Variance (CV) summary:
1. Time is just another, imaginary dimension of space: this is the basic insight of Albert Einstein's theory of relativity. Just like the distances $$ds$$ in 3D space are given by the Pythagorean theorem involving coordinate differences, $ds^2 = dx^2+dy^2+dz^2,$ the squared invariant separation in the spacetime - a space enhanced by time - is given by the same Pythagorean theorem where time enters just like space, is converted via $$c$$, the speed of light, and appears with the opposite sign: $ds^2 = -c^2 dt^2+dx^2+dy^2+dz^2.$ Note that $$-dt^2$$ may be written as $$+(i\cdot dt)^2$$, i.e. the minus sign may be converted to a plus sign if we use a new imaginary spatial coordinate $$x_4 = ic\,dt$$.

Simultaneity of two events depends on the observer (differently moving observers slice the spacetime in different directions); the moving clocks may be shown to tick slower relatively to a static observer by the factor of $$\sqrt{1-v^2/c^2}$$ (twin paradox etc.); according to general relativity, spacetime gets curved so the speed of clocks' ticking depends on the gravitational potential. A theory that can't explain that space and time are close cousins (i.e. that can't reproduce the Lorentz symmetry) is incompatible with the fundamental knowledge we have about Nature. CV largely ignored relativity in its description of time; the exception is CV #3, "everyone observes time differently".
2. The question about the existence of the past depends on our definition of existence: the discussions whether "the past exists" (or, more precisely, for experts, "the past light cone exists") depend on what we mean by the verb "exist" and whether the verb "exist" is allowed to have a grammatically unusual tense that unifies the present tense and the past tense, i.e. the past and the present. So one shouldn't make a big deal out of the "eternalism vs presentism" debates: they're just battles about what terminology should be used. CV #2 discusses this point but incorrectly includes the future which is addressed in the following point.
3. The future doesn't exist at the present regardless of any definitions: the information about the events that will occur in the future is strictly non-existent at the present; according to the free-will theorem by Conway and Kochen, building on general postulates of quantum mechanics, the quantum mechanically random outcomes of the experiments/events are only decided at the "right time" and can't be pre-determined; as long as we admit that the experimeters have a free will, all other quantum objects have to have a free will as well so their outcomes can't be functions of the data (hypothetical "hidden variables") that exist in the past light cone. CV #2 is invalid because it claims that the future exists at the present - a proposition that violates the rules of quantum mechanics: we may speculate about the future but all currently unknown facts about the future will only be decided in the future.
4. Some units of time are better than others: in technical terms, there is no symmetry of Nature that would allow you to slow processes down by a factor $$k$$ and expect that all these processes will continue unchanged. For example, the duration of one period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom is always equal to 1/9,192,631,770 of a second and one can't construct any "mutation of the caesium 133 atom" in which the denominator would be different. This important observation - contradicting a widespread myth on a scale invariance of the laws of Nature and underlying a key concept of modern quantum field theory developed in the 1970s by Ken Wilson et al. and known as the "Renormalization Group" (different time scales are associated with different "effective" laws of physics) - is being ignored by CV. The same comment applies to space as well, as guaranteed by TRF #1: the size of the atoms can't be "inflated" or "shrunk", either. So situations morally resembling the picture below are forbidden by the laws of Nature:

5. Among conserved quantities, time has a special relationship with energy: again, this key insight of physics of time - which holds both in classical physics and quantum physics - is being ignored by CV. Emmy Noether has shown that each conservation law may be attributed to a symmetry of Nature and vice versa. The time-translational symmetry of the laws of Nature (saying that the phenomena proceed identically at time $$t$$ and at a later time $$t+\Delta t$$) implies the conservation of energy and vice versa. Noether's deep time-energy relationship may be also used to show that in quantum mechanics, the evolution in time is "generated" by a particular operator called the Hamiltonian - which is just a fancy technical name for the total energy of the system (or, especially, the linear operator in quantum mechanics that represents it). In particular, systems with a totally well-defined energy have their wave function depending on time as $$\exp(Et/i\hbar)$$. Because energy is related to time in this intimate way, temperature (the average energy per one degree of freedom, with some proper refinements and in some units) is related to time as well. In quantum physics (especially when it's studied by Richard Feynman's path integral), absolute temperature $$T$$ may be visualized as a periodic imaginary time $$\tau=it$$ with a period obeying $$k_{{\rm Boltzmann}}T = \hbar / \Delta \tau$$.
6. All physical systems with many degrees of freedom inevitably possess a future-time asymmetry, the so-called logical arrow of time: in any physical description of anything as well as the reality itself, one must always be able to say that among two causally related events A, B, one of them is the consequence of the other or that it has evolved from the other. If the influence existed in both directions, one would produce closed time-like curves (including the paradox in which one may kill his granddad before he met his grandmother). Without a loss of generality, B has evolved out of A. In other words, we say that B belongs to the future light cone of A; A is an event in the past and B is an event in the future. The relation of A, B is completely asymmetric and implies very different ways how to derive probabilities of various properties of A and B from one another. In particular, direct probabilistic formulae for properties of B exist as functions of properties of A - these calculations are known as predictions; on the other hand, one needs to choose a priori arbitrary "prior probabilities" and perform a "retrodiction" - a form of Bayesian inference - if he wants to derive properties of A out of the properties of B. Such a retrodiction never works too well because macroscopic phenomena (i.e. laws of their evolution) are irreversible. This insight implies that CV #2 and CV #9 are wrong.
7. The logical arrow of time implies the asymmetry of all macroscopic processes, i.e. the thermodynamic arrow of time and other arrows of time: one may mathematically prove the so-called H-theorem which is a rigorous quantitative version of the "second law of thermodynamics" which says that "the entropy (the quantity describing the amount of disorder of a system) never decreases by a macroscopic amount as time increases". As a result, the proof of the H-theorem - first presented by Ludwig Boltzmann - identifies the direction of the "logical arrow of time" mentioned in TRF #6 with the direction of time in which the entropy increases. People are aging and the heat always goes from a warmer body to a cooler one; these are two well-known and typical examples of the second law of thermodynamics. There are millions of others: eggs break but don't unbreak, and so on. The second law applies not only globally but it also constrains arbitrary small regions of spacetime as long as the number of degrees of freedom is large in this region and as long as the interactions between them are fast enough; this fact invalidates most of the propositions in CV #9. The laws of physics allow ingenious surgeries that will make one feel younger, or cloning which may start a life from the beginning; but they do not allow to run a macroscopic process backwards so that one would be getting gradually younger in the same (mirror) way as he is getting older. Many things should be said about the second law of thermodynamics and the thermodynamic arrow of time. For example, it doesn't contradict the emergence of more organized and more symmetric structures (higher forms of life etc.) or the existence of fridges: both fridges and organisms are optimized to reduce their own entropy accompanied by the pumping of an even larger entropy to their environment. The ability to reduce one's own entropy is indeed a typical sign of life, in an almost direct contradiction with CV #8 that says that "purpose of life" is to increase the entropy. "A purpose of life" is to locally decrease the entropy which is pretty unusual among the natural phenomena.
8. An exact time-reversing symmetry which holds in Nature is called CPT and only applies to the microscopic laws: the weak nuclear force implies that the phenomena which are just reverted in time (future exchanged with the past: the T operation) do not proceed identically as the original ones. However, one may prove and Wolfgang Pauli and others have proved in the 1950s that if you combine the T reversal with parity P (the map $$(x,y,z)\to(-x,-y,-z)$$ and with the charge conjugation C (the replacement of particles by their antiparticles and vice versa: this C comes pretty naturally together with T because particles moving backwards in time - with the opposite sign of energy - become antiparticles), we get a symmetry called CPT that is the exact symmetry of the laws of Nature because it may be identified with a kind of rotation of the (Euclideanized) spacetime by 180 degrees - which is guaranteed to be a symmetry by TRF #1. However, this symmetry only has a "straightforward interpretation" in the case of microscopic processes in which all the information about the future and the past is determined so the map between the past and the future is one-to-one; whenever there is some incomplete information about the past or about the future (or both), and there is always some incomplete information when the number of degrees of freedom is large (much greater than one: and this claim is true both in classical and quantum physics), the logical arrow of time from TRF #6 implies that the probabilities in the future and in the past have to be treated differently (one averages, with some subjective prior probabilities, the probabilities over different initial states; but one sums over all final states and this summing has no subjective component). Much like in TRF #7, this guarantees that the macroscopic processes proceed so that the entropy increases; however, the CPT-reversed processes in which the entropy decreases by $$-\Delta S < 0$$ are less likely than the original processes by the factor of $$\exp(-\Delta S / k_{{\rm Boltzmann}})$$, as explained many times on this blog. For a finite entropy difference and assuming the $$k_{{\rm Boltzmann}}\to 0$$ limit, the entropy-decreasing transitions quickly become impossible.
9. To properly understand the psychological perception of time, one needs to analyze the functions of the brain: this doesn't really belong to "physics proper" but the brain or the mind isn't the exact copy of the spacetime; it is a device that collects impulses from the past and produces an image. It takes almost 0.1 second for us to realize that we have seen or otherwise witnessed something: see CV #4. Brains may be fooled to have a flawed idea about the past (known as the "memory"): see CV #5. Also, I don't quite understand what's the point of CV #6 but it would clearly belong to this item, too. Sean Carroll has dedicated way too much space to these psychological speculations because he has returned from an interdisciplinary conference in which many neuroscientists as well as vacuous philosophical babblers apparently tried to hype their occasionally unphysical musings. I think that one point is enough to summarize them - and I chose the summary that says that one must be careful about the inner workings of the brain if he wants to make any valid statements about time that depend on human (or other) psychology.
10. When times become as short as the Planck time, $$10^{-43}$$ seconds, time develops some wholly unfamiliar properties: quantum gravity allows one to calculate (and Max Planck was able to compute as early as 100 years ago) that there is something special going on at durations that are as short as the Planck time, $$t_{\rm Planck} = \sqrt{\hbar G/c^5}$$, which is a function of the most universal constants of physics only. The geometry of spacetime can no longer be imagined to be a simple pseudo-Euclidean geometry when the separations in time become this short. The Planck time is the shortest time at which our intuition about time (which is continuous, causal, in principle accurately determined, and locally isomorphic to $${\mathbb R}$$) remains marginally valid. "Shorter times" don't really exist in a conventional sense; alternatively, totally new phenomena allowing us to connect the past and future etc. in novel ways (quantum foam and other classes of unusual phenomena, including processes involving strings, branes, and topology change) take over when you try to access shorter times than the Planck time. Also, the Planck time is the (very ephemeral) lifetime after which the smallest black hole (marginally) worth its name evaporates by the Hawking radiation. Quantum gravity i.e. string/M-theory has to be properly understood (by you or by the whole mankind) to say anything sensible yet nontrivial about these extremely short periods of time (especially about the first moments after the Big Bang; the last moments of an evaporating black hole; or the vicinity of the singularities of larger black holes: unluckily enough, these 3 examples are among the least understood short-distance questions in string theory). This point - which is the only point of "truly contemporary physics" in this list - is ignored by CV, too. Time may be "emergent", as hinted by CV #1, and the Planck time regime may make such an emergent description of time more important. Emergent and fundamental descriptions may also be equivalent to each other. However, the equivalence of time and space as described in TRF #1 has to be valid at all times and all descriptions which is a huge constraint on possible candidate theories of physics - and candidate theories of time. This constraint immediately invalidates, among many other hypotheses, all hypotheses in which time is universally discrete or quantized or in which the spacetime curvature requires many degrees of freedom to be excited ("entropic gravity").
Let me mention that I don't believe that CV #10 ("each life lasts 1 billion heartbeats") has a broader range of validity that goes well beyond the organisms we know on Earth or that would hold very accurately. It's an OK observation that larger animals live longer but have slower heartbeats: one hearbeat brings you about 1 ppb closer to the death. However, pretty ordinary intelligent structures such as computers live for much longer a time than 1 billion cycles of the microprocessor (which would be less than a second). ;-)

I am sure that hypothetical extraterrestrial life forms (assuming that they exist) may be "hybrids" of our biological organisms or computers, or even more accurately and flawless computers than ours, so they may have a very different number of heatbeats per lifetime.

If you look at the list of points at CV, several points are valid (#3, #4, #5, #6, #7), one point is almost completely vacuous or "undecided" (#1), three points are wrong or at least their major propositions are wrong (#2, #8, #9), and one of them is just a very inaccurate phenomenological observation about a limited number of life forms on Earth (#10). Moreover, most of the points that are valid (#3, #4, #5, #6, #7) are morally redundant and overexpose a very small subset of scientific questions that have very little to do with physics.

Most of the points about time that I found totally paramount for the description of time as understood by contemporary physics (TRF #1, #4, #5, #6, #8, #10) were completely or almost completely omitted by the Cosmic Variance. This much for the scientific consensus. It's kind of remarkable that a famous cosmology blogger picks less than one-half of the important insights that physics has made about time; and in those that he did pick, about one-half is fundamentally invalid or at least vacuous.

1. Lubos:I am sure that hypothetical extraterrestrial life forms (assuming that they exist) may be "hybrids" of our biological organisms or computers, or even more accurately and flawless computers than ours, so they may have a very different number of heatbeats per lifetime.

We do not have to go to far from what is known, to conclude that the life span of any of the species on earth varies according to those heart beats measured in time:)

I appreciate your explanations on time and maybe you can talk more on what Sean Carroll states there about Einstein. They are quite selective there about comments, so I have been blocked.

Sean Carroll:As Einstein put it, “It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.”

Getting introduced by Grossman's to views on Riemann were liberating for Einstein as well as moving all of us to non euclidean views of reality?

Would you agree?

Best,

2. In 1952, in his book Relativity, in discussing Minkowski's Space World interpretation of his theory of relativity, Einstein writes:

Since there exist in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.

Einstein writes a letter to Besso's family in regards to Bessopreceding him in death ........for us physicists believe the separation between past, present, and future is only an illusion, although a convincing one."

Best,

3. The future doesn't exist at the present regardless of any definitions...

Isn't it true that given any event X in spacetime, any other event outside the future lightcone is, in some observer's frame, either simultaneous with X or in the past of X? Therefore, given a (infinite) hypersurface of simultaneity to some observer, what is unambiguously the future lies only in the intersection of all the future lightcones emanating from that surface, which is the empty set?

4. No, Arun, what lies unambiguously in the future of a given event/point in spacetime is this point's future light cone, not an empty set. (Yes, the intersection of all your bizarre collection of light cones is an empty set but it has no relevance for the original question - and by the way, your question is unrelated to my original statement, too.)

5. The future doesn't exist at the present regardless of any definitions...

Luboš - I'll try asking in another way. If events A and B are simultaneous as per some observer, and events B and C are simultaneous as per some other observer, and if C is in the future of A, then in what sense does C not exist when A occurs?

Either I have to take into account all the simultaneity reports of all observers, or of none of them.

That would be the solipsist approach that the only things which have existed or exist are those that are in my past light cone. The laws of physics don't stop working at my horizon, though. I need a way of talking about whether things that have wandered out of my past light cone still are, were, will be.

6. Let me try it another way, tell me if you agree.

Talking of future, present, past is misleading. I should talk only about knowable things and predictions, where I mean by knowable, everything inside my past light cone; and predictions everything outside my past light cone.

Anything I say about events outside my past light cone is a prediction, even though in my global coordinate system, it might lie in the past. In particular, the existence of a good part of the rulers and clocks that make up my global coordinate system is a prediction. That the far-away "now" event X will eventually enter my past light cone at which event I will find X to be coeval with the event 12:00 PM GMT, 2011-09-06, is a prediction (or a fundamental postulate about the nature of Lorentzian space).

I don't like the word "knowable" much, perhaps there is a better term for the knowable region - which is the region from which the results of a measurement could have reached my worldline.

7. Dear Arun,

what you try to paint as "solipsism" is exactly the right answer. Only the past light cone "exists now".

The events that are spacelike-separated from the event "I, now" can't have influenced me yet which is why I can't even know whether these spacelike separated regions exist or not. The vacuum could have decayed over there, a fact that will only arrive here later.

But even if you extended the definition of existence so that all events with time "t" smaller than your time, according to your coordinate system, exist "now", there would still be no contradiction because you can't incoherently mix different reference systems when defining "now" or the "future".

Your zigzag way of getting from A to C just has no meaning. Of course that you can get any interval in spacetime - timelike or spacelike - as a sum of purely spacelike intervals; or as a sum of purely timelike intervals. And what? There's no importance about this point.

I think that you haven't even begun to start to understand the depth of my comment that "future doesn't exist now" which is based on the free-will theorem. This statement means that in the quantum world, one cannot consider the future to be a function in the past. The future events aren't decided by any hidden variables that could exist in the past light cone.

Cheers
LM

8. Of course it is experimentally important to try and answer how any universe came into being. Is time a emergent product then?

Cycle of Birth, Life, and Death-Origin, Indentity, and Destiny by Gabriele Veneziano

In one form or another, the issue of the ultimate beginning has engaged philosophers and theologians in nearly every culture. It is entwined with a grand set of concerns, one famously encapsulated in an 1897 painting by Paul Gauguin: D'ou venons-nous? Que sommes-nous? Ou allons-nous? "Where do we come from? What are we? Where are we going?"

How would information be transferred from the previous universe, and is this an important question in terms of the now, as too, any universe as containing the past and future??

"Where," would this be possible in our universe? String theorists have push back our perspective on time. How far can they go?