The Meaning of Relativity Four lectures delivered at Princeton University, May, 1921 By: Albert Einstein (1879-1955)
First Page: uid. The contribution to the stress tensor is then evidently p in which is the special symmetrical tensor. This term will also be present in the case of a viscous uid. But in this case there will also be pressure terms, which depend upon the space derivatives of the u. We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be ff@u @x @u @x fi @uff @xff (for @uff @xff is a scalar). For physical reasons (no slipping) it is assumed that for symmetrical dilatations in all directions, i.e. when @u1 @x1 = @u2 @x2 = @u3 @x3 ; @u1 @x2 ; etc., = 0; there are no frictional forces present, from which it follows that fi = 􀀀 2 3 ff. If only @u1 @x3 is different from zero, let p31 = 􀀀ff @u1 @x3 , by which ff is determined. We then obtain for the complete stress tensor, p = p􀀀ff@u @x @u @x􀀀 2 3@u1 @x1 @u2 @x2 @u3 @x3: (18) PRE RELATIVITY PHYSICS 23 The heuristic value of the theory of invariants, which arises from the isotropy of space (equivalence of all directions), becomes evident from this example. We consider, finally, Maxwell's equations in the form which are the foundation of the electron theory of Lorentz. @h3 @x2 􀀀 @h2 @x3 = 1 c @e1 @t 1 c i1; @h1 @x3 􀀀 @h3 @x1 = 1 c @e2 @t 1 c i2; @h2 @x1 􀀀 @h1 @x2 = 1 c @e3 @t 1 c i3; @e1 @x1 @e2 @x2 @e3 @x3 = ; 9> >>>>>=>>>>>>; (19) @e3 @x2 􀀀 @e2 @x3 = 􀀀 1 c @h1 @t ; @e1 @x3 􀀀 @e3 @x1 = 􀀀 1 c @h2 @t ; @e2 @x1 􀀀 @e1 @x2 = 􀀀 1 c @h3 @t ; @h1 @x1 @h2 @x2 @h3 @x3 = 0: 9>>>>>>=> >>>>>; (20) i is a vector, because the current density is defined as the density of electricity multiplied by the vector velocity of the electricity. According to the first three equations it is evident that e is also to be regarded as a vector. Then h cannot be regarded as a vector. The equations may, however, easily be These considerations will make the reader familiar with tensor opera THE MEANING OF RELATIVITY 24 interpreted if h is regarded as a skew symmetrical tensor of the second rank. In this sense, we write h23, h31, h12, in place of h1, h2, h3 respectively. Paying attention to the skew symmetry of h, the first three equations of (19) and (20) may be written in the form @h @x = 1 c @e @t 1 c i; (19a) @e @x 􀀀 @e @x = 1 c @h @t : (20a) In contrast to e, h appears as a quantity which has the same type of symmetry as an angular velocity. The divergence equations then take the form @e @x = ; (19b) @h @x @h @x @h @x = 0: (20b) The last equation is a skew symmetrical tensor equation of the third rank (the skew symmetry of the left hand side with respect to every pair of indices may easily be proved, if attention is paid to the skew symmetry of h). This notation is more natural than the usual one, because, in contrast to the latter, it is applicable to Cartesian left handed systems as well as to right handed systems without change of sign. tions without the special diculties of the four dimensional treatment; cor responding considerations in the theory of special relativity (Minkowski's interpretation of the field) will then offer fewer diculties. LECTURE II THE THEORY OF SPECIAL RELATIVITY The previous considerations concerning the configuration of rigid bodies have been founded, irrespective of the assumption as to the validity of the Euclidean geometry, upon the hypothesis that all directions in space, or all configurations of Cartesian systems of co ordinates, are physically equivalent. We may express this as the \principle of relativity with respect to direction," and it has been shown how equations (laws of nature) may be found, in accord with this principle, by the aid of the calculus of tensors... Continue reading book >>

The Meaning of Relativity Four lectures delivered at Princeton University, May, 1921 By: Albert Einstein (1879-1955)

First Page:

uid. The contribution to the stress tensor is then evidently p in which is the special symmetrical tensor. This term will also be present in the case of a viscous uid. But in this case there will also be pressure terms, which depend upon the space derivatives of the u. We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be ff@u @x

@u @x fi @uff @xff (for @uff @xff is a scalar). For physical reasons (no slipping) it is assumed that for symmetrical dilatations in all directions, i.e. when @u1 @x1 = @u2 @x2 = @u3 @x3 ; @u1 @x2 ; etc., = 0; there are no frictional forces present, from which it follows that fi = 􀀀 2 3 ff. If only @u1 @x3 is different from zero, let p31 = 􀀀ff @u1 @x3 , by which ff is determined. We then obtain for the complete stress tensor, p = p􀀀ff@u @x

@u @x􀀀 2 3@u1 @x1

@u2 @x2

@u3 @x3: (18) PRE RELATIVITY PHYSICS 23 The heuristic value of the theory of invariants, which arises from the isotropy of space (equivalence of all directions), becomes evident from this example. We consider, finally, Maxwell's equations in the form which are the foundation of the electron theory of Lorentz. @h3 @x2 􀀀 @h2 @x3 = 1 c @e1 @t

1 c i1; @h1 @x3 􀀀 @h3 @x1 = 1 c @e2 @t

1 c i2; @h2 @x1 􀀀 @h1 @x2 = 1 c @e3 @t

1 c i3; @e1 @x1

@e2 @x2

@e3 @x3 = ; 9> >>>>>=>>>>>>; (19) @e3 @x2 􀀀 @e2 @x3 = 􀀀 1 c @h1 @t ; @e1 @x3 􀀀 @e3 @x1 = 􀀀 1 c @h2 @t ; @e2 @x1 􀀀 @e1 @x2 = 􀀀 1 c @h3 @t ; @h1 @x1

@h2 @x2

@h3 @x3 = 0: 9>>>>>>=> >>>>>; (20) i is a vector, because the current density is defined as the density of electricity multiplied by the vector velocity of the electricity. According to the first three equations it is evident that e is also to be regarded as a vector. Then h cannot be regarded as a vector. The equations may, however, easily be These considerations will make the reader familiar with tensor opera THE MEANING OF RELATIVITY 24 interpreted if h is regarded as a skew symmetrical tensor of the second rank. In this sense, we write h23, h31, h12, in place of h1, h2, h3 respectively. Paying attention to the skew symmetry of h, the first three equations of (19) and (20) may be written in the form @h @x = 1 c @e @t

1 c i; (19a) @e @x 􀀀 @e @x = 1 c @h @t : (20a) In contrast to e, h appears as a quantity which has the same type of symmetry as an angular velocity. The divergence equations then take the form @e @x = ; (19b) @h @x

@h @x

@h @x = 0: (20b) The last equation is a skew symmetrical tensor equation of the third rank (the skew symmetry of the left hand side with respect to every pair of indices may easily be proved, if attention is paid to the skew symmetry of h). This notation is more natural than the usual one, because, in contrast to the latter, it is applicable to Cartesian left handed systems as well as to right handed systems without change of sign. tions without the special diculties of the four dimensional treatment; cor responding considerations in the theory of special relativity (Minkowski's interpretation of the field) will then offer fewer diculties. LECTURE II THE THEORY OF SPECIAL RELATIVITY The previous considerations concerning the configuration of rigid bodies have been founded, irrespective of the assumption as to the validity of the Euclidean geometry, upon the hypothesis that all directions in space, or all configurations of Cartesian systems of co ordinates, are physically equivalent. We may express this as the \principle of relativity with respect to direction," and it has been shown how equations (laws of nature) may be found, in accord with this principle, by the aid of the calculus of tensors... Continue reading book >>

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