Minkowski expression

In the flat case of the special theory of relativity the proper time s is given in terms of the space-time displacements by the classical Minkowski expression... 

The relativistic invariance of the electromagnetic field is conveniently expressed in tensor notation. Factorized in Minkowski space the Maxwell equa-... 

If we replace the space part here by expression (5), we still deal the Minkowski spacetime. In fact... 

Notice that the expression equation (26) has the Painleve form a Minkowski ds2 minus the product of a space-time function by the square of a sum of differentials. Furthermore these three sums of differentials given in equations (19), (25) and (26) have the following common point their ds2 can be written... 

The last expression gives the potential matrix in the standard representation. The transformation law (89) gives precisely the Poincare transformation of the electromagnetic field strengths E and B, which can be combined into a tensor field on Minkowski space. 

Electromagnetic theory can be very compactly expressed in Minkowski four-vector notation. Historically, it was this symmetry of Maxwell s equations... 

Maxwell s equations in Lorenz gauge can be expressed by a single Minkowski-space equation... 

Minkowski space-time Universe is a reality, just like the quadrivec-tors in Minkowski s terms, expressed at a conference in Cologne on September 21 (1908). ""The views on space and time that I want to develop for you were horn on the experimental-physical ground. In this lays their strength. Their tendency is to give radical demonstrations. From now on, the space for itself should completely disappear in shadow, considering only the existence of an association of the two. ""... 

Note that we have artificially extracted the electric field in the last equality of Eq. (3.134), which represents an exact relation. Inserting this expression for the electric field into the relation for the Minkowski force as given by Eq. (3.131) directly yields the four-dimensional Lorentz force / in IS,... 

The dissimilarity D(PQ) between fragments P and Q is assessed using the Minkowski metric (equation 2 above), where V(P ) and V(Q ) are the Jth torsion angles for fragments P and Q, and summation is from J = 1 — Nt. Because torsion angles are circular functions, there is a phase restriction of -180< y<180, and the torsional dissimilarity measure must be expressed as ... 

In Chapter 1, we look at the general principles which govern the establishment of the equations of electromagnetism in the case of a simple medium. These equations are expressed in the Minkowski space and then transferred into the usual three-dimensional space. The quantities used in the four-dimensional space are the tensors of the electromagnetic field and the current 4-vector, the momentum-energy tensor. These quantities will also be presented in Chapter 4, where we shall establish the... 

To begin with, here, we shall present the basic principles and the expressions of the classic quantities, such as the proper time and the universal velocity in the Minkowski timespace. The law of d3mamics of the material point is then stated. 

The electromagnetic balance equations are expressed in the Minkowski space by ... 

From a relativistic point of view, the balance is expressed by means of the flux in the Minkowski space. A quantity F which has the flux 4J and the production rate aW will be such that ... 

In the Minkowski space, the balance of a quantity F at the interface expresses the fact that the flux of F at the borders of a four-dimensional spatial domain constituted by the surface (2") delimited by a closed two-dimensional manifold, is equal to the production of that quantity F within that domain. 

Lff denoting the matrix describing the Lorentz transformation. Each of the indices runs from 0 to 3, and the definition in the equation above assumes the Einstein summation convention, which states that two indices denoted by the same symbol should be summed over when they appear in superscript and subscript positions in an expression featuring relativistic quantities of tensor character. Since the underlying space (called Minkowski space) has a somewhat unusual scalar product, characterized by the prescription... 

The adequate distance is obtained with the v -th root of the previous expression and the symbol Z = Ze corresponds to the inward absolute value of a matrix or second rank tensor. A Minkowski distance maybe of easy definition as corresponds to the first order expression of Eq. (25) ... 

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