Flowing with a Frozen River by Cristinel Stoica

If you would be a rose, would you believe that lilies and tulips are weeds?

Dear Dr. Crowell,

It happens that I also liked that paper. I think that GR's ideas can be applied to a better understanding, so that the dark energy is explained without extra hypotheses. Your comments and the connections you make around that article are interesting.

Best wishes,

Cristi Stoica

The equivalence principle dictates that local inertial frames are indistinguishable from Minkowski spacetime. Yet I think there is a general extension of this to conformal spacetimes, where not GR is formulated according to a conformal or cosmological equivalence priniple.

Comsologies resist having a Killing vector field which defines conservation principles. Yet in this setting a conformal Killing vector field can be defined. In this setting the conformal energy is conserved, but energy defined according to coordinate time is not. However, that is not really a problem for coordinate time in general relativity is not a physical time.

This is an aspect I think of dark energy and how the exponential expansion of the universe can "perform work" by pulling apart galaxy clusters.

With general relativity the Lagrangian for the motion of a particle is mc^2 int ds, where ds is the invarant interval. The action, would in general be Lds = pdq - Hdt, if we think of there being potentials and the like. This is a curious thing, for the action of GR motion including potentials, or mc^2 = k^2 - E^2 etc with DG equations, then we have an intertwiner between the proper time s and the coordinate time t. A Hamiltonian requires the imposition of a spatial surface (a part of a contact manifold) with a coordinate time. Yet in relativity this is a matter of a coordinate condition, similar to a gauge, and this defines a different notion of time entirely.

The Hilbert-Palatini action int d^4x(-g)^{-1/2}R has that determinant of the metric which will give an overall conformal term to the action for g_{ab} ---> Q^2g_{ab}. Hence the concept of energy conservation only holds at best according to tensor densities, and if certain symmetries can be applied to the system. The Ricci curvature scalar there is also

R = pi^{ij}dg_{ij} - NH - N^iH_i boundary terms,

and includes no apriori notion of time at all. Without the notion of time the concept of a Hamiltonian as the generator of time translations is lost. All this H does above is to define a soldering form between spatial surfaces.

For an Einstein space g_{ab} ~ R_{ab} (such as deSitter spacetime) and the Ricci scalar is then proportional to a proper time. So the H-P action reduces to a similar Lagrangian as for particle motion. So the action defines some proper time or inteval, which we regard as measurable in GR, but now for gravitational variables the Hamiltonian has no coordinate notion of time at all!

Ultimately we have an problem of two time variables, a proper interval which is tied to GR variables, and a coordinate time which is gives quantum wave dynamics, but is not physically relevant in GR.

Lawrence B. Crowell

Issues are raised about the basic nature of all physical phenomena being random in nature, i.e., no one can predict when the next event shall take place exactly. On the other hand , the whole process of the evolution of the Universe shows a logical design, formation of first nucli, atoms, molecules, birth of first star, galaxies,.. and finally on earth we had stone age, plants, animals and then humans.. How can everything be so random or by chance. Where is the reality in all these. Can the 'design'signal remains cleverly hidden without detection in the 'disorderly' nature of individual events of a process? There is a chi-square test in the theory of probabiloity that detects any regular signal taht may get mixed with random events, or in a competed Experimental set-up measuring a rare phenomena, one can detect instrumental dyfunctioning! Are we missing that 'guiding logic signal' somewhere in this strange mixture of measurement limitations!

Varaition in h, c and fine -structure constant have been claimed in more than one or two experiments over the lifetime of the Universe. What do you feel, the Physics we have build up in the past few hundred years may not be sufficient to study or explain phenomena over the space /time expanse of the Universe, as we confine measurements bound to the earth!

Stoica,

in the above post , there are a few spelling mistakes that come inadvertently in typing the text direct from the mind!For example,it is 'complicated' and not 'competed' Experimental set-up, 'taht' is actually 'that', etc.\ nn

Dear Dr. Narendra Nath,

Your questions deserve careful consideration. Please see the attached document.

Best regards,

Cristi StoicaAttachment #1: global_convergence.pdf

seen, life has many interpretations and your is certainly is one possibility. let us respect the individual faiths / beliefs of one another and continue to lead a life that is not only useful for the self but of all others associated with us.

Science is a profession, a noble one, but humanity and life are far bigger in perspectives. It is humanity that has permitted us to become scientists , not just for our sake but for the good of society and the world we live and belong to.

Dear Dr. Narendra Nath,

Nice words, I agree with what you said.

Best regards,

Cristi Stoica

I have through another venue been in a debate over whether constants change, in particlar the speed of light, but also hbar. I will make this brief, but I argue then do not change at all. The reason is that if you consider Planck units, such as L_p = sqrt{G hbar/c^3} any change in c is not apparent. The fine structure constant is unchanged as well, since in Planck units e = sqrt{4pi eps c}, and e^2/hbar-c reamins the same.

The speed of light is involved with the projective system called light cones. These define the psl(2,C) group, or the projective Lorentz group. Similarly the unit of action hbar is a measure for the projective map in geometric quantization. In particular the phase phi = sqrt{(H^2) - (H)^2}t/hbar, here I use parantheses for the bra-ket terms. So in keeping with the nature of projective geometry, where a point is a line modulo length, any rescaling of c and hbar leaves things invariant.

Lawrence B. Crowell

Dear Dr. Crowell,

Although in principle may be possible, I see no reason for admitting a time or space dependence of the fundamental constants.

Best regards,

Cristi Stoica

Regarding Crowell comments on 'c' and 'h' constancy, experiments though few already seem to confirm that the value of 'c' was higher for cosmic objects 12 billion years away. The same also hold for the value of alpha. Thus maintaining 'e' constant, 'h' may also change correspondingly. In fact i feel that closer one may go to the birth of the Universe, say within 1 billion years, the changes in these magnitudes may even be even more significant. Time and space concepts can still hold, but the nature of space in early universe may come out to be different from what it is presently. i have indicated in another post to an essay that curved space is already a fact from Theory of General Relativity. Being concepts generated by mind to understand the physical phenomena, space and time are not directly verifyable experimentally except through the consequences. The same stands established thus far. But one can still play with curvature of space and time to explain/ understand what still remains as a 'mystery'.

When it comes to the variability of the two constants hbar and c, or measurements of the fine structure constant e^2/hbar-c have from my understanding not revealed any change. There are occassional reports of such changes, usually attempts to detect hyperfine splittings of very distant atoms, but these have so far under close examination been found to be unrepeatable results.

When it comes to the variability of the speed of light, it must realized that this constant is really a way of telling us how to measure one coordinate variable with a certain ruler and how to convert those measurements to another coordinate measured by another ruler (clocks and rods). So in that sense it is a conversion factor, which in naturalized units is dimensionless. The speed of light is similar to a rule that tells us how to convert measurements of depth and length into consistent units.

It takes a bit of work, but if you consider the Planck units of lengths and adjust the speed of light you find that what changes occur are consistently cancelled out. Such as the Bohr radius of an atom might change with the expansion of the Planck length, but since everything else also scales accordingly there is no detectable change. This is similar to what Poincare indicated over 100 years ago with an invariance of length scalings.

Hbar and c are measures on a projective blow up of a point. In the case of relativity this projective blow up in spacetime are light cones, and in QM the projective blow up of a classical point in phase space gives units of fundamental action.

Lawrence B. Crowell

To Lawrence B. Cromwell,

Your explanations are apparently valid toaccoount for variation in the value of the physical constants as seen recently in a few experiments. However, i have doubts from another angle that the measurements may well be correct. The situation in early stages of the universe were much more volatile relative to the near universe. All our present day Physics accounts for currently observed phenomena. The early universe conditions appear to be far too divergent . There is no theoretical reason that the physical constants and for that matter the relative strengths of the four force/fields should remain constant for the universe at birth or closer to it!In fact, it is well nigh possible that gravity was repulsive in nature around the Big Bang when entropy was very low and mass distribution homogenous. Present conditions are just on the other side. It is quite possible that early universe atoms were mesonic rather than electronic. It is the experiments done carefully for that period that will indicate the truth. We develop Physics to conform the nature's behavior and not that the nature must follow what we have developed in Physics, unless we can create a tiny bit of universe ourselves!

This essay contest managed to bring together a wide diversity of people who seriously thought about time: from well-known physicists to less known but interesting thinkers; from scientist, to philosophers, from professionals, to amateurs. I really enjoyed this diversity of opinions orbiting the nature of time. I had the opportunity to meet very interesting persons, and to understand and appreciate viewpoints that are different or even opposed to mine. I am glad for this.

Cristi

Narendra,

The speed of light is a measure of the projectivizaton of Lorentzian coordinates, the light cone. The Planck unit of action is a measure of the blow-up of a point (projectivization) in phase space. The scale of projectivization is an independent parameter,

PR^n = R^n/(x ---> cx,c=!0).

Further, if one where to consider Planck units of action, any rescaling of the speed of light leaves things invariant. Remember that even though units such as the Bohr radius changes with L_p, so does everything else. The fine structure constant is invariant as well for while alpha = e^2/hbar-c, the electric charge is e = sqrt(4pi eps c), and this remains the same under any rescaling of c. The gauge parameter of interest is then really the dielectric constant eps.

Other constants of nature scale with energy in renormalization group systems of running parameters. This is a more subtle issue, for the value of these constants depends upon an energy scaling. The point of my essay on #370 is to show that this scaling is determined by a correspondence between the conformal fields and the conformal boundary of AdS.

Lawrence B. Crowell

Dear Lawrence,

May i take it that the concepts evolved and Maths. used in Physics thus far controls the Physical processes that are no longer taking place in the present universe. As indicated by me and others who observed the very early universe, experimentally measured that the light signal coming from an object 12 billion away from us , indicated that 'c' was higher than what one has measured for our present universe. You need to change the concepts involved along with the Mathematics neded to explain such a behaviour. i prefer to believe the published results reported only in the past 2 years in this regard. The explanations are still to come in that sense of understanding such results. Alpha measurements from similarly placed distant objects also show the variation.

i fail to understand energy or time scaling chsnged only in the early universe and then it stopped changing for most of its subsequent evolution?Even the value of h may well have changed, but the same has or could not be measured for a distantly placed cosmic object!

THE COUNTERINTUITIVE TIME - PART 1

TIME AND DETERMINISM

The paradigmatic example of a world governed by determinism is provided by a system of differential equations (DE), or, more generally, of partial differential equations (PDE). In Physics, the PDE are required to have as solutions functions of position and time. In Newtonian Mechanics, the position is a point in the 3-dimensional Euclidean space, and the time is a real number. The positions and the instants form a 4-dimensional real vector space. The solutions of the PDE are functions f(x, t) defined on this space. The state of the system at a moment of time t is given by f_t(x) = f(x, t).

The PDE systems appearing in mathematical physics have the nice property that, by knowing the state at a moment of time t_0, and the values of some additional partial derivatives of f (in general the first order ones are enough), we can determine uniquely the states for another time t. For the solution to exist at t, the initial states and the derivatives appearing in the initial conditions are required to be well-posed, but I will not detail here. What is important is that we can extend the state at a time t to contain not only f_t, but also the partial derivatives involved in the initial condition. This way, all the information about the system and its time evolution is contained in the extended state at each instant.

The fundamental equations of Physics are PDE, and they satisfy these conditions. One important exception seems to be provided by the Quantum Mechanics, where the indeterminism seems to be fundamental, but for the moment I will concentrate on the deterministic situation.

In such a deterministic world, the extended states contain all the information about the system. There is no physical property which is not contained in the extended state. Is the world we live in, of this type? It may be, or it may be not. If the world is like this, then it is a block world. The solution of the PDE is defined on the spacetime, and together they form a timeless, frozen entity, the function f(x, t).

Many biologists and neurobiologists believe that, at least in principle, life and consciousness can be explained by making use of the deterministic properties of atoms and molecules, and perhaps more complex systems only, and not appealing to the indeterminism. Many persons understand what a deterministic world is, and even believe that our world may be of this type. Yet, they hardly accept the block world. A deterministic world contains all the information regarding all moments of time, at the extended state at each instant. There is no need to "play" this world, like playing a pick-up disk. If our world is deterministic, and if the minds are reducible to configurations of matter, then the extended state contains also the mind state of a possible observer. Are the observers just states depending on a real number (which is interpreted as time)? If it would be so, then there will be no change, in the sense that, at any instant, the 3d-observer at that instant will contain in its state the impression that he or she perceive a time flow, and a dynamic evolution of his/hers state. There will be only timeless 3d observers containing in their states the impression of time evolution. For a person, there will be one such 3d timeless observer, associated to each instant (which is just a real number). Considering all the instants making a "lifetime", there will be an infinity of such 3d timeless observers, connected.

It is easy to understand why such view is rejected by many. Determinism leads to a block world view. Some may accept determinism, and reject block view. When they understand the relation between them, they may continue rejecting the block view, and therefore reject also the determinism. The main problem seems to be the block view. If our world is such, then we are also reduced to parts of a set of timeless states.

Perhaps the main parts of our intuition contradicted by this view are the following. First, the feeling of subjectivity, the sense of "I". Second, the feeling that we have free-will. A block view seems to make everything frozen, predetermined.

To be continued...

Cristi

THE COUNTERINTUITIVE TIME - PART 2

THE GEOMETRIC TIME

In Newtonian Mechanics, the world evolves deterministically. The time is a parameter similar to the space coordinates. The PDE describing the evolution respect symmetries at orthogonal transformations of space, and at time translation and time reversal. Another symmetry is related to the speed of the inertial reference frames: the laws do not depend on the speed of an inertial frame.

A challenge of the Galilean relativity is provided by the Maxwell's equations. The Electrodynamics suggested another group of symmetries, the Poincaré group, and its Lorentz subgroup, which are associated to the Special Relativity.

The introduction of the Lorentz transformations shed a new light on the nature of time. The time is no longer a parameter, but it gains a geometric meaning, which brings new counterintuitive aspects. The geometric meaning of time comes from the Lorentz invariance. The Lorentz transforms can "mix" space and time dimensions, like a spatial rotation can mix two directions of space. The relativity of simultaneity, which is a central point of Einstein's theory, provides a physical interpretation of this character. This challenges our intuition, because it suggests that spacetime is a single geometric and timeless entity. Each direction in the Minkowski spacetime corresponds to a speed. The relative speed between two such directions can be obtained by applying the hyperbolic arctangent to the angle between them. This shows that two inertial frames moving with a relative speed, have different time direction in spacetime.

When somebody hears about the Minkowski spacetime as a symmetric space, may think why we couldn't move through time like we are moving through space. The usual answer involves the idea of lightcone, but this explanation is not enough. But let us first discuss the lightcone and the causal structure of Relativity.

The lightcone is the set of all spacetime directions which corresponds to light speed. The 4-vectors from inside the cone, represents time directions, and the ones from outside, spatial directions. The squared norm of a spacelike vector has opposite sign than the squared norm of a timelike vector, and the lightlike vectors have zero norm, being also named null vectors. The Lorentz transformations preserve the norms, therefore they cannot be used to turn a timelike vector into a spacelike vector.

It seems impossible for an object having a velocity smaller than the speed of light to change smoothly the direction in spacetime and go back in time. The main reason is that its velocity will need to become the speed of light, and then larger (to go out of the light cone). But what is the problem with a body being accelerated to the speed of light? The answer is that we would need an infinite amount of energy for doing this. When the body increases its speed, its mass also increases, and the energy required for increasing further its speed becomes larger. For going to the speed of light, we will need to give it an infinite amount of energy.

Although we understand that the Relativity explains well our limitations in moving through time like we are moving through space, this difference between space and time are still so deep rooted in our intuition, that we find very difficult to accept the geometric nature of time.

A second counterintuitive aspect is the difference between the spacelike and the timelike vectors. If they can be rotated one into another by Lorentz transforms, this rotation is only partial, because we cannot transform a spacelike vector into a timelike vector. This asymmetry is not that annoying from mathematical viewpoint, and, as we saw from the previous argument, it is even useful. But many find it disturbing, and they feel like there is a need to replace the Lorentz metric with a Euclidean one (by some mathematical trickery). In general, these attempts ended up by complicating the things, and the mainstream physicists remained with the Lorentz metric. But, I cannot deny that there may be persons who consider simpler the Euclidean approach, because the price of accepting an indefinite metric seems too high for them. Maybe it is a matter of taste.

To be continued...

Cristi

THE COUNTERINTUITIVE TIME - PART 3

THE TIME'S ARROWS

Seeing that the equations are symmetric at time reversal, we may legitimately wonder why the time has a direction. Boltzmann answered this question long time ago, when he explained the entropy, but since then, many felt that the things are not clear yet.

If at microscopic level the laws are symmetric to time reversal, why are they irreversible at larger scales? At larger scales, two systems which differ at small scale, may look identical. For example, to spheres made of the same material, and of the same radius, having the same density, may be considered identical, although their microscopic structures are far from being identical. Two glass balloons of identical shapes, filled with the same quantity of the same gas, will look identical at macroscopic level, but very different at atomic scale. Boltzmann defined the entropy of a macroscopic state of a system as minus the logarithm of the number of distinct microscopic states that macroscopically look identical to the macroscopic state. This definition fit well the entropy as it was known in Physics, and also has an analog in Shannon's information theory, which led to an informational interpretation of the entropy. For our discussion, we will deal with its probabilistic meaning. A system tends to evolve to a more probable state, and a state with larger entropy is more probable. This is the key to understanding phenomena which are thermodynamically irreversible, like boiling an egg or breaking a cup.

The entropy will increase only to a maximum value corresponding to the most probable state, after that it will just fluctuate around that value. But then, it seems to follow that the present state is most likely to be one of the most probable, with the maximum of entropy, therefore we should not observe an increase of entropy, and no special arrow of time. The answer is that our present state is one of the most improbable, and therefore the entropy has enough room to increase. Moreover, it appears that the entropy increases since the Big Bang, and at that initial moment the entropy was very low. Very low entropy means very improbable, so the matter distribution at the Big Bang was very improbable. The permanent increase of entropy is explained not by a universal law of Physics, like the fundamental laws, but by a special property of the initial conditions. It is a "historical law", and not a "universal law".

The Big Bang itself seems to provide initial conditions improbable enough to activate the Second Law of Thermodynamics, by the simple fact that the matter was all concentrated in a very small region, most probably a singularity. But not all scientists consider this concentration enough. For example, Roger Penrose proposed an explanation of the thermodynamic arrow of time based on the condition that the Weyl tensor canceled. The tensor describing the curvature of the spacetime in General Relativity contains a part corresponding to the energy-momentum tensor, the other part is the Weyl tensor. But the Weyl tensor can be viewed, by the mean of the Bianchi identity, as corresponding to the gravitational field generated by the energy-momentum tensor of the matter. I interpret Penrose's condition Weyl=0 as simply stating that in gravitation, only the "retarded gravitational potential" should be considered (similar to the retarded potential in Electrodynamics). Therefore, it seems that Penrose's condition refers to a "radiative arrow of time". It seems that the Big Bang, the cosmological arrow, is tied with the thermodynamic and radiative arrows.

The psychological arrow of time, corresponding to our minds remembering only the past, is perhaps the most difficult to grasp. It is habitually to be explained by comparing the brain with a computer who, in order to use its memory, needs to heat the environment, increasing the entropy.

I believe that the explanations of the arrows of time are very counterintuitive, and one reason is that they are based on symmetry breaking. The PDE expressing the fundamental physical laws are time-symmetric, but the solutions are not necessarily so. The time asymmetry is related very well with the existence of a special time, of minimum entropy, and that time is, naturally, the origin of time's arrows. Because of the difficulty in accepting the arrow of time in a world governed by time-symmetric fundamental laws, some physicists try to find fundamental laws which exhibit time-asymmetry. In most cases, the asymmetry is searched in quantum phenomena, especially in the measurement process. But many consider the time arrows explained well enough, not requiring supplemental assumptions.

Yet, if one of the time's arrows is less understood, I think that this is a psychological one, not necessarily restrained to memory, but to the whole psychological meaning of the words "time flows". Perhaps the central point of the flow of time is the subject experiencing it, the "I" of each one of us.

To be continued...

Cristi