Vector time-like

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... 

We now consider the representation r of induced by the irreducible representation y of . We pose the question when F is broken up into its irreducible parts, how many times will each irreducible representation rw appear To decide this, we first observe that the independent aj > which form a basis for F can all be generated from a single j > of r by application of the elements of , including those of. For the vectors of r, this follows from the irreducibility of y, for the others from the nature of the induction process. Since the e-operators form a complete set in U, one can equally well say that the basis vectors of r are generated by applying all the e-operators to a single j >. Now suppose that y appears c times in the subduced representation TW ( ) . Choose a basis for such that the first c basis vectors transform like j > under . In this basis we have... 

For Z may be zero, but need not be. It follows that we can generate from j > at most c vectors transforming like the fc th basis vector of fM. When the operations of are applied to these, each may separately generate the representation I M, but it cannot be generated more times than this. Thus, the representation fW appears in T at most the same number of times that y appears in the subduced representation of r( ), and its decomposition into irreducible components is... 

If A] is phase-free, as discussed in Section III, and in Ref. 15, there are no longitudinal electric field components. This also occurs if A,-3"1 is zero [17]. The B(3) field is then a Fourier sum over modes with operators a qaq and is perpendicular to the plane defined by A and /1<2>. The four-dimensional dual to this term is defined on a time-like surface, following Crowell [17], which can be interpreted as E under dyad vector duality in three dimensions. The ( field vanishes because of the nonexistence of the raising and lowering operators l3 , . The BM is nonzero because of the occurrence of raising and lowering... 

The only nonzero components of the PL vectors and are the longitudinal and time-like components. It follows that since is null, its magnitude is zero, and so 1 and Evl are null. This result is, in turn, consistent with the fact that the PL vector is a pseudovector, whereas is a null vector whose dual is null. 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

In the past, comparisons between NMR relaxation and MD simulations have concentrated on internal motions, since these often involve sub-nanosecond time scales that could be examined with limited computer resources. In this approach, overall rotational motion is removed by an rms fitting procedure (for example, on backbone atoms in regular secondary structure), and computing time-correlation functions from the result. Typical results are shown in the upper panel of Figure 8.1 similar plots have been presented many times before [4,12,10,11]. Many backbone vectors are like Thr 49, and decay in less than 0.1 ns to a plateau value which can be identified as the order parameter for that vector. Most regions of regular secondary structure resemble this, although there can be exceptions, and there is potentially important information in the decay rates and plateau values that are obtained. 

Exactly like the proper time all other vectors such as the momentum or force vectors have one time-like and three space-like components. The 4-velocity... 

The matrix elements of r, and a are of rank-one, r is the exact analogue of the isospin-changing El operator and the conserved-vector-current (CVC) theorem can be invoked to relate the matrix element of a to that of r. Further, [r, spin-dipole operator. The helicity operator 75 is the time-like component of the rank-zero axial current - also known as the axial charge - discussed in Section 1. We thus have five independent matrix elements, two each of rank zero (RO) and rank one (Rl) and one of rank two (R2) which we denote as... 

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