Time, relativity imaginary

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

Incidentally, the telegrapher s equation (17) with x = ihjlmc2 is satisfied by the Klein-Gordon (also Dirac) wavefunction for a free particle, if the factor exp [—(ijh)mc2f is split off from it. Thus, the time lag according to relativity corresponds to an imaginary relaxation time x. 

The events as seen by any particular observer happen one after another in different positions. In the s-diagram they do not happen, they merely exist. The s-diagram is drawn in a region which is not ordinary space, but is a sort of blend of space and imaginary time. Space and time are combined in it into a single absolute quantity s equal to yd2 c2T2. The separation of s into two parts d and cT is a purely relative operation of no real significance. 

To this it may be replied that the s-diagram is purely imaginary. Actual space is of three dimensions and so far as we know a space of four dimensions is impossible. Also the s-diagram involves imaginary time so that it could not be drawn to scale even if there were a space of four dimensions to draw it in. The s-diagram is interesting and it shows up very clearly the relative character of space and time, but it is difficult to see how such a purely imaginary mathematical conception can be supposed to represent reality. 

The imaginary part of e of one RF-sample with different amounts of adsorbed water is measured to verify the theory on the influence of water on the impedance spectra of the RF aerogels. Immediately after exposing the sample to an atmosphere with 100% relative humidity the impedance spectrum is measured 3 times in dry atmosphere with decreasing sample weight (i.e. water content) as parameter. 

This means of course, that at such frequencies the effect of the permanent moment has vanished, and that only the part of the dielectric constant due to deformation of the atoms and molecules remains. According to this theory non-polar substances, for which fi equals zero, should not show anomalous dispersion, and this is borne out by experiment. The effect will become evident at frequencies at which the product, o>t, has an appreciable value, From equation (28) for the time of relaxation, t, anomalous dispersion should appear at relatively lower frequencies (a) if the molecules have large radii, r, (b) if the viscosity, 77, is high, and (c) if the temperature is low, all of which have been observed experimentally. Since equation (27) is complex it follows that the dielectric constant, D, has a real and an imaginary part. Debye lias shown that the real part, Df, which corresponds to the measured dielectric constant, follows the equation... 

Prom [72] we recall the existence of a center manifold Ai associated to a rigidly rotating reference spiral, alias a relative eqnilibrium SE 2)u see also (3.1)-(3.7) above. In fact, the center manifold M. accounts for all solutions which remain in a neighborhood of the relative equilibrium SE 2)ut, for all positive and negative times. In particular, all bifurcations due to point spectrum on the imaginary axis of the linearization L from... 

As already described, a magic-tee is a four port device in which two of the ports are isolated, i.e., have infinite impedance between them if the impedances at the other two ports are matched. Thus, the device can be used to match a resonant circuit to 50 Q or, for that matter, to measure the impedance at resonance of the tank circuit. The tank circuit under question will have impedance R at resonance when the output of the magic-tee, easily monitored on an oscilloscope, is an absolute minimum. (See the figure under magic-tee on p. 393.) Unfortunately, this is easier said than done because there are relative minima when the capacitors are adjusted to give an impedance which is, for example, equal to R in magnitude but with an imaginary component. When you play around with a scheme like this for the first time, it will be useful to let R be variable so that it can be adjusted for a... 

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