Wednesday, December 31, 2008

The Counterintuitive Time: 4. Time and General Relativity

This is a series of posts about the counterintuitive nature of time in Physics. This post presents some difficult aspects of time in General Relativity, such as the notion of curved spacetime, the looping time, and the idea of the beginning of time.

After the Special Relativity, Einstein tried to express various physical laws in this formalism. Basically, their mathematical form is required to be invariant at Lorentz transformations. The main difficulty he encountered was to express Newton’s gravitation field in this way. After many years of research, Einstein obtained the movement of bodies under the effect of gravity as simply an inertial movement in a curved spacetime. The inertia and the gravity become unified. The spacetime itself is curved by the masses, and the universal attraction was just an effect of this curvature. General Relativity was born. The experimental consequences eventually confirmed the theory, which become widely accepted. Among its surprising features is that that the time flow changes in the presence of massive bodies.

The main equation of General Relativity, Einstein’s equation, relates the curvature to the distribution of energy. One very important difference between this equation, and the previously known equations in mathematical Physics, is the following: finding the solution, means also finding the background (meaning the spacetime itself). In the Newtonian and special relativistic cases, the spacetime was fixed, but in General Relativity, it is part of the solution itself. Perhaps, this is the most striking difference.

The main counterintuitive aspect of the curved spacetime is caused by our tendency to consider it as a subspace of a space with more dimensions. Many persons, when learn for the first time that the spacetime is curved, tend to interpret this as being curved in a fifth dimension. As a simpler but historic example, when we think at a curved surface, we tend to consider it a subspace of the Euclidean space. Gauss realized that the intrinsic geometry of every surface can be expressed independently on the Euclidean space in which this is embedded. The main ingredient is the metric tensor, which provides a point-dependant measure of the lengths of the curves embedded in the surface. Riemann generalized the surfaces to curved spaces with any number of dimensions. Their work helps understanding that the curved spaces in Riemannian geometry do not rely on a Euclidean space in which they may be embedded. Einstein found the four-dimensional Riemannian geometry as the ideal tool for General Relativity, provided that we replace the Euclidean metric tensor with the Lorentz metric.

The Einstein’s equation may have solutions that contain closed timelike curves. Spacetime may be curved in such a manner, that the future of an event becomes also its past. This looping time highly contradicts our intuition. Yet, unlike the other counterintuitive aspects of time, this one may not even exist, as Hawking’s Chronology Protection Conjecture states.

Another hard to grasp aspect of time is the beginning. Our experience teaches us to consider the time as being linear, infinitely continued in past and future. Why do we have this intuition, considering that our lives are finite? Perhaps because the daily events succeed linearly, at our scales, and because the History of our countries, and of our planet, and solar systems, appear to be linear. But when we hear about the Big Bang, two questions we may find natural is “what was before the Big Bang?”, and “when happened the Big Bang?”. We find difficult to accept that even the time may have a beginning.

Other difficult aspects of time in General Relativity are related to special situations, like the time in the presence of a Black Hole, in a Worm Hole, time traveling using Worm Holes, Hawking’s imaginary time, the time near/at the initial singularity. I will not detail these problems.

The Counterintuitive Time: 3. The Time's Arrows

This is a series of posts about the counterintuitive nature of time in Physics. In this post it is analyzed the difference we perceive between past and future, as it appears in irreversible phenomena.

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.

The Counterintuitive Time: 2. The Geometric Time

This is a series of posts about the counterintuitive nature of time in Physics. This post tries to identify the problem in accepting the geometric nature of time implied by Special Relativity, as well as the differences between space and time in relativistic spacetime.

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 PoincarĂ© 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.

Tuesday, December 30, 2008

The Counterintuitive Time: 1. Time and Determinism

This is a series of posts about the counterintuitive nature of time in Physics. This first post tries to identify the reasons why people may not agree with the deterministic view, and with the block spacetime view, in our sense of free-will and of flowing time.

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.

Sunday, October 12, 2008

The Spinning Dancer's Mistery

I explain why the "spinning dancer" is perceived by the most as spinning clockwise. I will then show why the laws of perspective indicate that in fact she spins counter-clockwise.

What is "The Spinning Dancer"?

The spinning dancer is an interesting optical illusion created by Nobuyuki Kayahara. The ambivalence of the image makes some observers seeing that the dancer is spinning clockwise, while others have the impression that she is spinning counter-clockwise. This is an example of bistable optical illusion.

Some tests presented at Cognitive Daily show that the ones perceiving the dancer spinning clockwise are twice as many as the others.

Why two people out of three see the dancer spinning clockwise?

I will explain now why. The answer is based on our experience. We are accustomed to see other people from one specific angle - from about the level of their eyes. For example, in the case of the dancer, the rotating leg’s toe describes a (almost) horizontal circle. Well, at least the circle would be horizontal, if the dancer would not go up and down. Our brains tend to make the supposition that we are looking at that circle from the above, that the circle is at a lower level than our eyes. The figure below shows this:

This is also the reason why, when you look at the dancer’s reflection, you switch the to counter-clockwise mode. The reflected circle, seen from above, corresponds to the counter-clockwise spin.

Why the correct answer is that the dancer spins counter-clockwise?

Well, although the illusion is usually presented as lacking any hint for the depth, I will show you that this is not the case. If we look at the dancer’s reflection in the floor, we can see that, assuming that the circle is more or less horizontal, we obtain a clue. If we assume that the circle described by the dancer’s toe is horizontal, then its plane should be parallel to the plane containing the circle described by the reflection of the toe in the floor. But the laws of the perspective tell us that a closer object look larger than a distant one. Accordingly, the distance between the toe and its reflection should look larger when the toe is closer to the observer. In our case, this happens if the dancer is spinning counter-clockwise. Of course, this means that the observer’s point of view is somewhere near the floor, because the circle described by the toe is now perceived from below.

To give a chance to the clockwise interpretation, we can assume that the floor is very inclined, so that the circle is far from being parallel to its reflection. Even in this case, the point of view should be under the floor’s level, because the closest side of the reflected circle appears to be above the other one. But in this case, the reflection makes no sense. If it is shadow and not reflection, it should not appear upside down, when we see it from below the floor. Therefore, the only interpretation compatible with the laws of perspective is the counter-clockwise one.

It is therefore safe to affirm that the original 3d animation (perhaps created with a 3d computer graphics software like 3ds Max) from which Nobuyuki Kayahara exported the animation rotates indeed counter-clockwise.

Breaking the illusion

How many of our perceptions, how many of our beliefs, are illusions? Is there a possibility to understand that we are living in an illusory world? I not necessarily refer to the objective world as being an illusion. By definition, the illusions are subjective. Between us and the objective world, is a thick layer of more or less illusory beliefs.

To realize whether a belief is based on illusion, we can use the logic. We can check the consistency; this is the best reality test. If the laws of perspective tell us that the reflection of the dancer is not consistent with her spinning clockwise, then we can try the opposite interpretation. We can try the opposite interpretation anyway, as a mean to wonder whether our point of view is the only possible, or as a way to understand other perspectives.

The opinion of the majority cannot be taken as a reality test, or as a proof of objectiveness. Their opinion matters in politics, it matters in their personal lives, or in marketing. The opinion of the others matter for us, because we care about them. But we should not consider the opinion of the majority as our reference frame. The majority believes that the dancer is spinning clockwise, but the logic and the geometry showed us that she is spinning counter-clockwise.

Even if you are among the few that saw the dancer spinning counter-clockwise, this may be an illusion too. You can dream that you are in your bed, and when you wake up you find that you were right, but it was only an illusion. You can consider that you were not fooled by the illusion only when you not only were right, but you understood, or experienced, the reason why you are right.

An example makes the things clearer. If you simplify in the fraction 16/64 the figure 6, you obtain the correct answer 1/4. But you obtained it by pure chance; the same solution will not apply to other fractions. The way to solve the problem was completely wrong, but this does not guarantee you that the answer will be wrong too.

Let me summarize: don’t let you be fooled by the illusions, question every preconception you are tempted to adopt. Whether the preconception belongs to the majority, to a minority you trust, to a person you cherish… Even better: question every preconception of yourself. A preconception is not necessarily false, but it is based on false arguments. The logical consistency should be your first guide out of the illusion.

Thursday, September 25, 2008

The Illusion of Center

There really exists a center, or it is an illusion?

The O’Reilly’s rotating grid is an example of an illusion of center, as I will explain. Then, some math skills will allow us to find a grid rotating around two exact centers. The next part will be about the self and the illusion of center.

O'Reilly's rotating grid

Let's take a look at an interesting effect, discovered by David O’Reilly:

As its discoverer noticed, it is an effect of temporal aliasing, but I will add an ingredient - the illusion of center. The temporal aliasing is an effect which usually appears when the moving images are presented as a sequence of frames. Presenting the image by frames creates the illusion of motion, but sometimes the information contained in the frames is ambiguous. As we can see below, if we highlight the parts that seem to rotate, the region delimited by the yellow square is rotating around a central point. The regions delimited by the red squares, although may appear to have their own centers, at a second examination are found not to have a center.

If we look at a real rotating grid, the effect cannot appear. It happens only because the frames present us only some lines. Two rotating orthogonal lines pass through the center. Together with them, other parallel lines rotate around the center. Near every point on the image there are lines, which are parallel to the lines passing through the center. The "approximate centers" are near more lines than the others, but it is only the center that contains lines for all possible directions. We can see this by overlapping all the frames:

Rotation around two centers

If we rotate the grid at each step with an angle of 360/n, the center is the only point containing all the time vertices of the grid (except, of course, the case when n=1, 2 or 4, when no rotation is viewed). But if we are good enough at math, we can modify the animation such that we obtain more than one fixed point (please click to see the full image):

The animation above is based on some properties of the number 65. This number plays the role of the hypotenuse in eight Pythagorean triples:


If we allow the hypotenuse not necessarily be integer (for example to square to 325), then the obtained image can be smaller:

and a tile version:

Here is a checker board pattern:

and you can see some dots:

The self and the illusion of center

Why are our minds tricking us into believing that there is a center where in fact it is not? Perhaps this happens because, in a perpetually moving world, we need to believe that there is a fixed point of the Universe. We need to believe that we can see where the phenomena converge, so that we can understand the laws of Nature, so that we can survive and have benefits. In order to want to survive and evolve, we need to believe that we are important, that we have a central self.

At different stages in our lives, we may be so different, that we hardly can say that we are the same person. Our interests and ideals as children may differ very much from those we have as adults. Our priorities and even values at the office or school may be very different than the ones we manifest in family, or with our friends. If we are changing that much during our lifetimes, is there a convergence point of all our facets? Do we really have a self? Or this is just another trick of our mind?

In everything happens to us, sometimes we may feel the need to believe that there is a reason, other than blind chance. We may be tempted to hope that there is a reason behind everything, a central point towards everything converges. Is there such a Center of the World, like the one marked by cross? If there is, how are we, the other beings, like the areas marked with red squares, with no real center? Or are we all centers in the same time, like the vertices marked with red dots? Or there is nothing else but the emptiness of the grid?