Space-like

Consider the polyad = 6 of the Hamiltonian ( Al.2.7). This polyad contains the set of levels conventionally assigned as [6, 0, ], [5, 1],. . ., [0, 6], If a Hamiltonian such as ( Al.2.7) described the spectrum, the polyad would have a pattern of levels with monotonically varymg spacing, like that shown in figure Al.2.8. 

Entropy is a measure of disorder in materials. Relaxed polymer chains with a random conformation (shape in space), like cooked spaghetti or a box of fishing worms, have a high degree of entropy, which is favored by Mother Nature. If the chains are stretched out (stressed), the number of conformations the chains can have in space is limited, and the entropy is reduced (see Figure B). The ratio of final length to initial length is denoted a. 

We can differentiate between two cases, namely (a)p is space-like and (b) p is time-like. In the case that p is space-like, due to the invariance of R we can choose without loss of generality a coordinate system in which p = (0,p), in which case Im R(p) — 0, since... 

Upon multiplying this equation by f(x)yu and integrating over the space-like surface x0 = t, we obtain... 

If / (x) is a wave packet solution that vanishes for large spatial distances, then the surface integral over 8 extends over the two space-like surfaces alf o2 bounding Q (see Fig. 10-8). 

Never draw the head of an arrow going off into space, like this ... 

According to the basic ideas concerning ionic atmospheres, the ions contained in them are in random thermal motion, uncoordinated with the displacements of the central ion. But at short distances between the central ion m and an oppositely charged ion j of the ionic atmosphere, electrostatic attraction forces will develop which are so strong that these two ions are no longer independent but start to move together in space like one particle (i.e., the ion pair). The total charge of the ion pair... 

Four-vectors for which the square of the magnitude is greater than or equal to zero are called space-like when the squares of the magnitudes are negative they are known as time-like vectors. Since these characteristics arise from the dot products of the vectors with reference to themselves, which are world scalars, the designations are invariant under Lorentz transformation[17], A space-like 4-vector can always be transformed so that its fourth component vanishes. On the other hand, a time-like four-vector must always have a fourth component, but it can be transformed so that the first three vanish. The difference between two world points can be either space-like or time-like. Let be the difference vector... 

The condition for a time-like difference vector is equivalent to stating that it is possible to bridge the distance between the two events by a light signal, while if the points are separated by a space-like difference vector, they cannot be connected by any wave travelling with the speed c. If the spatial difference vector r i — r2 is along the z axis, such that In — r2 = z — z2, under a Lorentz transformation with velocity v parallel to the z axis, the fourth component of transforms as... 

If X is space-like and the events are designated such that t2 > 11, then c(ti — f2) < z — z2, and it is therefore possible to find a velocity v < c such that ic(t[ — t 2) = X vanishes. Physically the vanishing of X means that if the distance between two events is space-like, then one can always find a Lorentz system in which the two events have the same time coordinate in the selected frame. On the other hand, for time-like separations between events one cannot find a Lorentz transformation that will make them simultaneous, or change the order of the time sequence of the two events. The concepts "future" and "past" are invariant and causality is preserved. That the sequence of events with space-like separations can be reversed does not violate causality. As an example it is noted that no influence eminating from earth can affect an object one light-year away within the next year. 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

Any stationary point is said to trace out a world line parallel to the time axis. A point in motion follows a worldline at an angle 9 with respect to t, with a maximum of 0 = r/4 rad for a velocity of v = c. The space-like region outside the so-called light cone is usually ignored as physically meaningless since v > c. There is no explanation of why a stationary point has a world line and why it s always time-like and never space-like. 

The vibrational levels corresponding to n = 0,1,2... are evenly spaced. Like the particle confined to a line segment, the harmonic oscillator also has zero-point energy Eq = hu. 

Many of the processes responsible for isotope fractionations in the Earth s atmosphere may also occur in the atmospheres of other planetary systems, such as the atmospheric escape of atoms and molecules to outer space. Likely unique to Earth are isotope fractionations related to biological processes or to interactions with the ocean. One aspect of atmospheric research which has great potential for the application of stable isotope investigations is the study of anthropogenic pollution. 

References in submitted manuscripts are double-spaced (like the rest of the text), even though the references will be single-spaced in the final publication. 

The integration can be performed in reciprocal space, like the reciprocal-space evaluation of the electrostatic potential discussed in chapter 8. According to Parseval s rule (discussed in chapter 5),... 

Note that the term wormhole is used in two different senses in the physics literature. The first kind of wormhole is made of quantum foam. Because of the foam-like structure of space, countless wormholes may connect different parts of space, like little tubes. In fact, the theory of superspace suggests that... 

Kant s first published paper, Thoughts on the True Estimation of Living Forces (1747), suggests that he was curious about the fourth dimension. In his paper, Kant asks, Why is our space three-dimensional He uses physics to remind us that forces like gravity seem to move through space like expanding spheres that is, their strength various inversely with the square of the distance. Kant reasoned that if God chose to make a world where forces varied inversely with the cube of the distance, God would have required a space of four dimensions. 

Kuchar, K. (1976) Geometry of hyperspace, 1. Journal of Mathematical Physics. 17(5) 777-91. (The author defines hyperspace as an infinitedimensional manifold of all space-like hypersurfaces drawn in a given Riemannian spacetime.)... 

Not only may hyperspace play a role on the galactic and universal size scale, but it may help characterize the ultrasmall. Physicist John A. Wheeler has suggested that empty space may be filled with countless tiny wormholes connecting different parts of space, like little tubes that run outside of space and back in again at some distant point. Wheeler describes these wormholes as running through superspace, which seems similar to what science fiction has called hyperspace for over a half-century. 

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