Unpopular Electronics The Three Polarisations In our universe, light in a vacuum is a completely transverse wave: the oscillating electric and magnetic fields that constitute the wave are always orthogonal to the direction in which the wave is travelling through space. In the Riemannian universe, as we discussed in the notes on vector waves , there is also a sense in which the equivalent kind of wave will always be transverse to its direction of propagation: the four-dimensional vector A describing the oscillating field will be orthogonal to the four-dimensional propagation vector k. For example, suppose the propagation vector k lies in the xt-plane, as in the diagram on the right. Then if A0 points along the y-axis, along the z-axis, or in the xt-plane itself but at right angles to k, those are three mutually perpendicular directions, all of which are orthogonal to k.
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Unpopular Electronics The Three Polarisations In our universe, light in a vacuum is a completely transverse wave: the oscillating electric and magnetic fields that constitute the wave are always orthogonal to the direction in which the wave is travelling through space. In the Riemannian universe, as we discussed in the notes on vector waves , there is also a sense in which the equivalent kind of wave will always be transverse to its direction of propagation: the four-dimensional vector A describing the oscillating field will be orthogonal to the four-dimensional propagation vector k.
For example, suppose the propagation vector k lies in the xt-plane, as in the diagram on the right. Then if A0 points along the y-axis, along the z-axis, or in the xt-plane itself but at right angles to k, those are three mutually perpendicular directions, all of which are orthogonal to k.
But when all these vectors are projected down to three-dimensional space, while the first two directions will remain orthogonal to the direction the wave is travelling, the third ends up parallel or anti-parallel to it. In that sense, the Riemannian equivalent of light can include a longitudinal mode, where the field oscillates in the same direction as the wave is travelling. In the diagram, we show the projections into the xy-plane. In every case, k and A0 are orthogonal to each other.
That said, there is a very real difference here from the situation in our own universe, where there are only two polarisations. Doppler Shift The Doppler shift is a well-known effect in which the motion of an observer alters the wavelength and frequency they measure for a light wave. In the diagram on the right, we show the same sequence of wavefronts passing over three different observers in the Lorentzian universe; the light is coming in from the right.
The reference observer will measure a period of OR between successive wavefronts. This is known as blue shift, because blue light is at the high-frequency end of the visible spectrum. This is known as red shift. Suppose F has coordinates xF, tF. Then the blue shift factor is two, and all light coming towards you from straight ahead will have its frequency doubled, compared to when you were at rest.
Similarly, all light you see from directly behind will have its frequency halved. In our universe, red light has a longer wavelength and a longer period than blue light. In the Riemannian universe, no kind of light has both a longer wavelength and a longer period than any other kind. The relationship is always the opposite: a longer wavelength always means a shorter period. So we have to make a choice. We see an example of this with the observer in the diagram whose world line is OG.
This observer sees the light as having infinite velocity, and the minimum possible wavelength. The propagation vector coming from the source would appear to them to be pointing back in time, so in any interaction with the light they would consider themselves to be the source, with the direction of the propagation vector reversed.
We can obtain the frequency ratio for the observer moving backwards, away from the light source, simply by putting a negative value for v in the same formula. What happens when the observer starts out-pacing the light, moving backwards even faster than the light is travelling?
Our formula for the frequency ratio gives a positive value, which is now less than the maximum, so the light should still be visible.
As well as the Doppler shift changing the frequency of light, the geometry of the situation means that an observer in motion will measure different angles between light rays than the reference observer, an effect known as aberration. Suppose the reference observer sees light of a particular colour from a multitude of sources spread out evenly across the sky.
This will have two effects: it will change the lengths of the projections of the propagation vectors, and it will change the angles between those projections. The diagram shows the result for various velocities for the observer, assuming all the incoming light has a velocity of 0.
The colours here match the translation scheme used in the novel for the visible hues, with dashed lines for ultraviolet and infrared. Light coming from the opposite direction is red-shifted, and spread out across the sky. Phrased this way with red- and blue-shift referring to wavelengths rather than frequencies, this account sounds just like the Lorentzian version. Where rays of red, yellow and green mingle with ultraviolet, the redder light is actually coming from behind according to the reference observer , but the tilt of the moving observer makes it appear to arrive from ahead, and it is superimposed on light that really is coming from ahead.
The light that is visible here is all from behind, and is crowded together into a region of the sky whose angular width is just that of the cone of incoming rays. The way Doppler shift and aberration modify these star trails is discussed in the first volume of the novel. It turns out that a whole theory of Riemannian electromagnetism can be developed, in a manner that is analogous to the Lorentzian electromagnetism of our own universe.
This includes vector fields that we can sensibly call electric and magnetic fields, and a property of matter analogous to electric charge, which results in the matter in question experiencing electromagnetic forces and having the capacity to generate electromagnetic waves.
Just as the electric potential is a quantity whose rates of change across space give us the electric field, the four-vector potential is a vector whose derivatives give us the complete electromagnetic field. What is the meaning of that matrix?
Any change in u that was parallel to u itself would alter its length, so the vector describing its rate of change must always be orthogonal to u. This means that F must belong to a special class of matrices, such that multiplying any vector by F gives a result orthogonal to the original vector. The definition of F in terms of A guarantees that it will be antisymmetric. What kind of world line will a charged particle follow, if it is moving through a constant electromagnetic field?
If b is a multiple of c, the right-hand side here becomes zero. The result, in general, will look something like the diagram on the right. With u lying neither entirely in the plane spanned by b and c nor entirely orthogonal to it, the particle will continue to move orthogonally to the plane at a constant rate, while its motion in the directions parallel to the plane will be bent around into a circle. If the plane is spanned by two directions that you consider to be directions in space in other words, the plane is orthogonal to your world line , then the circular part of the motion will look like motion in space to you: either the particle will be circling in a fixed plane, or it will be moving in a helix through space.
This is precisely the behaviour that we associate in our universe with a magnetic field. Indeed, in a somewhat old-fashioned kind of particle accelerator known as a cyclotron , a constant magnetic field is used to cause particles to circle this way. The conventions for the description of a magnetic field are that we ascribe to it a direction in space orthogonal to the plane in which the particle is circling.
When a charged particle is accelerated this way in our universe, we consider that to be the effect of an electric field. In Lorentzian physics the world line of an accelerating particle is hyperbolic rather than circular, but in all other respects the effect is the same. We ascribe to the electric field a direction in space that lies in the plane.
The plane spanned by k and A0 is neither wholly spatial nor does it contain our time axis, et, so the wave will include both electric and magnetic fields. The second part, with a plane spanned by et and ey, is an oscillating electric field that points in the y direction.
This is very similar to light in our own universe: the electric field, the magnetic field, and the direction in which the light is travelling are all at right angles to each other. The result is an oscillating, purely electric field that points in the x direction, parallel to the motion of the light itself. The Corrugated Coulomb Force The Riemannian Vector Wave equation governs the four-vector potential A for an electromagnetic field in a vacuum — that is, in the absence of any charged matter that might actually give rise to the field in the first place.
But there are two such vectors at every point on any world line, pointing in opposite directions. How do we choose the right one? In one direction the charged material will look positively charged to an observer who takes that choice of u as their time axis; in the other direction the same material will look negatively charged.
So it makes no difference which choice we make. We call the vector j the four-current. If an observer who is not at rest with respect to the charged material measures the components of j, they will see both a charge density, jt, and a current density with components jx, jy and jz, describing the motion of charge.
One of the simplest and most important solutions to this equation is known as the Coulomb potential. This is the potential of a particle with a charge q that occupies a single point, at rest in some coordinate system. Nevertheless, the solution is not too hard to find. Coulomb potential.
Quantum mechanics, antimatter and new biology; lots of diagrams, I finished The Eternal Flame by Greg Egan and it was excellent; maybe not as groundbreaking as The Clockwork rocket but a top 10 of mine kind of in my 2nd tier now at around ; there was no dominant character like Yalda and the universe is now familiar, but still lots of great stuff and I am really curious where it will go in volume 3 Mr Egan gave a hint in a comment to his blog post on FBC when asked a question by someone. While Mr. Egan has written extraordinary sf before - the novelization of General Relativity in Incandescence just the latest example before last year - The Clockwork Rocket had the added dimensions of great characters, most notably in the lead female physicist Yalda, and of an emotional ending that stayed with me for a long time. While there are exceptions and finer points eg Yalda was a single with no co of her own, though she had two co-siblings and females can extend their life span by taking an inhibitory drug, holin, usually its effect attenuates with age and the chances of spontaneous with no father for the children fission increase considerably The cast of The Eternal Flame is comparable to the one in The Clockwork Rocket, but here there are three main characters whose research, action and personal struggles are the focus of the novel: Carla, the spiritual descendant of Yalda and the leading theoretical physicist of her time, Carlo, her co and a biologist who abandoned reproduction research for trying to increase the food production only to ask his former coworkers to take him back as he deems himself a failure in agronomy and Tamara, an astronomer and pilot whose impossible dreams of being the one to guide back Peerless to the home world may find an expression in the exploration of a mysterious flying object that comes closer to the starship, but whose very traditionalist farmer co and father, Tamaro and Erminio may have other ideas At the start of The Eternal Flame, the inhabitants try to solve some major internal problems like overpopulation and manufacturing fuel or a new engine to return home, when the mysterious object assumed to consist of Orthogonal matter is spotted and an expedition to investigate is planned.
The Eternal Flame
The Clockwork Rocket