.:krenZ:.

Gravitational Aerodynamics.

Rotation holds a surprise for anybody who studies it carefully. Angular momentum is a quantity with a magnitude and a direction. However, it is not a vector, as any mirror shows.The angular momentum of a body circling in a plane parallel to a mirror behaves in a different way from a usual arrow: its mirror image is not reflected if it points towards the the mirror. You can easily check this for yourself. For this reason, angular momentum is called a pseudovector.The fact has no important consequences in classical physics; but we have to keep it in mind for later occasions.

The length of the year changes with time. The measured variations are of the order of a few parts in 1011 or about 1ms per year. However, knowledge of these changes and of their origins is much less detailed than for the changes in the Earth's rotation. transversal acceleration = coriolis effect.

Like all motion, fluid motion obeys energy conservation. In the case that no energy is transformed into heat, the conservation of energy is particularly simple. Motion that does not generate heat implies the lack of vortices; such fluid motion is called laminar. If the speed of the fluid does not depend on time at all positions, it is called stationary. Energy conservation implies that the lower the pressure is, the larger the speed of a fluid becomes.

Important systems which show laminar flow, vortices and turbulence at the same time are wings and sails. All wings work best in laminar mode.Thee essence of a wing is that it imparts air at a downward velocity with as little turbulence as possible. The aim to minimize tubulence is the reason that wings are curved. If the engine is very powerful, a wing at an angle also works. Strong turbulence is also of an adavntage for landing safely. The downward velocity of the trailing air leads to a centrifugal force acting on the air that passes above the wing. this leads to a lower pressure, and thus to li. (Wings thus do not rely on the Bernoulli equation, where lower pressure along the flow leads to higher air speed, as unfortunately, many books used to say. Above a wing, the higher speed is related to lower pressure across the flow.) The diferent speeds of the air above and below the wing lead to vortices at the end of every wing.These vortices are especially important for the take of any insect, bird and aeroplane. More details on wings are discussed later on.

All the richness of self-organization reveals itself in the study of plain sand. Why do sand dunes have ripples, as does the sand floor at the bottom of the sea? We can also study how avalanches occur on steep heaps of sand and how sand behaves in hourglasses, in mixers, or in vibrating containers. Oscillons can move and interact with one another. Oscillons in sand are simple example for a general effect in nature: discrete systems with nonlinear interactions can exhibit localized excitations. Thisis fascinating topic is just beginning to be researched. It might well be that it will yield results relevant to our understanding of elementary particles.

Most systems that show self-organization also show another type of motion.When the driving parameter of a self-organizing system is increased to higher and higher values, order becomes more and more irregular, and in the end one usually finds chaos. For physicists, chaotic motion is the most irregular type of motion.Chaos can be defined independently of self-organization, namely as that motion of systems for which small changes in initial conditions evolve into large changes of the motion (exponentially with time). More precisely, chaos is irregular motion characterized by a positive Lyapounov exponent.The weather is such a system, as are drippingwater-taps, the fall of dice, and many other common systems. For example, research on the mechanisms by which the heart beat is generated has shown that the heart is not an oscillator, but a chaotic system with irregular cycles. this allows the heart to be continuously ready for demands for changes in beat rate which arise once the body needs to increase or decrease its efforts.

We have already mentioned above the issue of the stability of the solar system. The long-term future of the planets is unknown. In general, the behaviour of few-body systems interacting through gravitation is still a research topic of mathematical physics. Answering the simple question of how long a given set of bodies gravitating around each other will stay together is a formidable challenge.

With this clarification, we can now briefly consider rotation in relativity.The first question is how lengths and times change in a rotating frame of reference. You may want to check that an observer in a rotating frame agrees with a non-rotating colleague on the radius of a rotating body; however, both find that the rotating body, even if it is rigid, has a circumference different from the one it had before it started rotating. Sloppily speaking, the value of p changes for rotating observers. Tt increases with rotation speed.This counter-intuitive result is often called Ehrenfest's paradox. Among other things, it shows that space-time for an observer on a rotating disc is not the Minkowski space-time of special relativity.

Rotating bodies behave strangely in many ways. For example, one gets into trouble when one tries to synchronize clocks mounted on a rotating circle. If one starts synchronizing the clock at O2 with that at O1, and so on, continuing up to clock On, one finds that the last clock is not synchronized with the first. This result reflects the change in circumference just mentioned. In fact, a careful study shows that the measurements of length and time intervals lead all observers Ok to conclude that they live in a rotating space-time. Rotating discs can thus be used as an introduction to general relativity, where this curvature and its effects form the central topic. Is angular velocity limited? Yes: the tangential speed in an inertial frame of reference cannot exceed that of light. The limit thus depends on the size of the body in question.

In summary, the results mentioned so far make it clear that mass generates curvature. The mass-energy equivalence we know from special relativity then tells us that as a consequence, space should also be curved by the presence of any type of energy-momentum. The field equations for empty space-time also show scale symmetry. This is the invariance of the equations after multiplication of all coordinates by a common numerical factor. diffeomorphism symmetry and trivial scale symmetry are the only symmetries of the vacuum field equations. Apart from diffeomorphism symmetry, full general relativity, including mass-energy, has an additional symmetry which is not yet fully elucidated. This symmetry connects the various possible initial conditions of the field equations; the symmetry is extremely complex. These fascinating investigations should give new insights into the classical description of the big bang.

Another area of application concerns gravitational waves.The full field equations show that waves are not harmonic, but nonlinear. Sinewaves exist only approximately, for small amplitudes. Even more interestingly, if two waves collide, in many cases singularities are predicted to appear.

But why is the vacuum transparent?That is a deeper question. Vacuum is transparent because it contains no electric charges and no horizons: charges or horizons are indispensable in order to absorb light. In fact, quantum theory shows that vacuum does contain so-called virtual charges. However, virtual charges have no effects on the transmission of light.

Detailed investigations show that there is no process which decreases the horizon area, and thus the irreducible mass or radius, of the black hole. People have checked this in all ways possible and imaginable. For example, when two black holes merge, the total area increases. One calls processes which keep the area and energy of the black hole constant

SOMMERFELD'S FINE STRUCTURE CONSTANT, explaining the constant is one of the toughtest challenges in physics today...the mechanism gives all objects color. essential to QED.

To sum up, the list of possible representations thus shows that rotations require the existence of spin. But why then do experiments show that all fermions have half-integer spin and that all bosons have integer spin? Why do electrons obey the Pauli exclusion principle? At frst, it is not clear what the spin has to do with the statistical properties of a particle. In fact, there are several ways to show that rotations and statistics are connected. His- torically, the first proof used the details of quantum field theory and was so complicated that its essential ingredients were hidden. It took quite some years to convince everybody that a simple observation about rotation was the central part of the proof.