 The Meaning of Relativity, Lecture 1. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org, recording by Amelia Chesley. The Meaning of Relativity, Lecture 1. Space and Time in Pre-Relativity Physics. The theory of relativity is intimately connected with the theory of space and time. I shall therefore begin with a brief investigation of the origin of our ideas of space and time. Although in doing so, I know that I introduce a controversial subject. The object of all science, whether natural science or psychology, is to coordinate our experiences and to bring them into a logical system. How are our customary ideas of space and time related to the character of our experiences? The experiences of an individual appear to us arranged in a series of events. In this series, the single events which we remember appear to be ordered according to the criterion of earlier and later, which cannot be analyzed further. This exists therefore for the individual, an eye time, or subjective time. This in itself is not measurable. I can indeed associate numbers with the events in such a way that a greater number is associated with the later event than with an earlier one. But the nature of this association may be quite arbitrary. This association I can define by means of a clock by comparing the order of events furnished by the clock with the order of the given series of events. We understand by a clock something which provides a series of events which can be counted and which has other properties of which we shall speak later. By the aid of speech different individuals can to a certain extent compare their experiences. In this way it is shown that certain sense perceptions of different individuals correspond to each other, while for other sense perceptions no such correspondence can be established. We are accustomed to regard as real those sense perceptions which are common to different individuals and which therefore are in a measure impersonal. The natural sciences and in particular the most fundamental of them, physics, deal with such sense perceptions. The conception of physical bodies, in particular of rigid bodies, is a relatively constant complex of such sense perceptions. A clock is also a body or a system in the same sense with the additional property that the series of events which accounts is formed of elements, all of which can be regarded as equal. The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences. Beyond this they have no legitimacy. I'm convinced that philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means, but is in a sense a creation of the human mind, without which no science is possible. Nevertheless, this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition. We now come to our concepts and judgments of space. It is essential here also to pay strict attention to the relation of experience to our concepts. It seems to me that Poincaré clearly recognized the truth in the account he gave in his book, La Sance et le Hepothés. Among all the changes which we can perceive in a rigid body, those are marked by their simplicity which can be made reversibly by an arbitrary motion of the body. Poincaré calls these changes in position. By means of simple changes in position, we can bring two bodies into contact. The theorems of congruence, fundamental in geometry, have to do with the laws that govern such changes in position. For the concept of space, the following seems essential. We can form new bodies by bringing bodies B, C, up to body A. We see that we continue body A. We can continue body A in such a way that it comes into contact with any other body X. The ensemble of all continuations of body A, we can designate as the space of the body A. Then it is true that all bodies are in the space of the arbitrarily chosen body A. In this sense, we cannot speak of space in the abstract, but only of the space belonging to a body A. The Earth's crust plays such a dominant role in our daily life in judging the relative positions of bodies that it has led to an abstract conception of space, which certainly cannot be defended. In order to free ourselves from this fatal error, we shall speak only of bodies of reference or space of reference. It was only through the theory of general relativity that refinement of these concepts became necessary as we shall see later. I shall not go into detail concerning those properties of the space of reference which led to our conceiving points as elements of space and space as a continuum. Nor shall I attempt to analyze further the properties of space which justify the conception of continuous series of points or lines. If these concepts are assumed together with their relation to the solid bodies of experience, then it is easy to say what we mean by the three dimensionality of space. To each point, three numbers, x1, x2, x3, coordinates, may be associated in such a way that this association is uniquely reciprocal, and that x1, x2, and x3 vary continuously when the point describes a continuous series of points, a line. It is assumed in pre-relativity physics that the laws of the orientation of ideal rigid bodies are consistent with Euclidean geometry. What this means may be expressed as follows. Two points marked on a rigid body form an interval. Such an interval can be oriented at rest relatively to our space of reference in a multiplicity of ways. If now the points of this space can be referred to coordinates x1, x2, x3 in such a way that the differences of the coordinates delta x1, delta x2, delta x3 of the two ends of the interval furnish the same sum of squares. S squared equals delta x1 squared plus delta x2 squared plus delta x3 squared. For every orientation of the interval, then the space of reference is called Euclidean and the coordinates Cartesian. It is sufficient indeed to make this assumption in limit for an infinitely small interval. Involved in this assumption, there are some which are rather less special to which we must call attention on account of their fundamental significance. In the first place, it is assumed that one can move an ideal rigid body in an arbitrary manner. In the second place, it is assumed that the behavior of ideal rigid bodies towards orientation is independent of the material of the bodies and their changes of position in the sense that if two intervals can once be brought into coincidence, they can always and everywhere be brought into coincidence. Both of these assumptions which are of fundamental importance for geometry and especially for physical measurements naturally arise from experience. In the theory of general relativity, their validity needs to be assumed only for bodies and spaces of reference which are infinitely small compared to astronomical dimensions. The quantity as we call the length of the interval in order that this may be uniquely determined it is necessary to fix arbitrarily the length of a definite interval. For example, we can put it equal to one unit of length. Then the lengths of all other intervals may be determined. If we make the x v linearly dependent upon parameter lambda x v equals a v plus lambda b v, we obtain a line which has all the properties of the straight lines of Euclidean geometry. In particular, it easily follows that by laying off n times the interval s upon a straight line, an interval length of n times s is obtained. A length therefore means the result of a measurement carried out along a straight line by means of a unit measuring rod. It has significance which is as independent of the system of coordinates as that of a straight line as will appear in the sequel. We come now to a train of thought which plays an analogous role in the theories of special and general relativity. We ask the question, besides the Cartesian coordinates which we have used, are there other equivalent coordinates? An interval has a physical meaning which is independent of the choice of coordinates and so has the spherical surface which we obtain as the locus of the endpoints of all equal intervals that we lay off from an arbitrary point of our space of reference. If x v as well as x prime v v from one to three are Cartesian coordinates of our space of reference, then the spherical surface will be expressed in our two systems of coordinates by the equations. Sum delta x v squared equals constant. Sum delta x prime v squared equals constant. How must the x prime v be expressed in terms of the x v in order that equations two and two a may be equivalent to each other? Regarding the x prime v expressed as functions of the x v, we can write by Taylor's theorem for small values of the delta x v. Delta x prime v equals the summation of dou x prime v over dou x alpha times delta x alpha plus one half summation of dou squared x prime v over dou x alpha dou x beta times delta x alpha delta x beta and so forth. If we substitute to a in this equation and compare with equation one, we see that the x prime v must be linear functions of the x v if we therefore put x prime v equals alpha v plus summation of b v alpha times x alpha. Or delta x prime v equals the summation b v alpha delta x alpha. And then the equivalence of equations two and two a is expressed in the form sum delta x prime v squared equals lambda sum delta x v squared lambda being independent of delta x v. And therefore follows that lambda must be a constant. If we put lambda equal to one to be and three a furnish the conditions. Sum b v alpha b v beta equals dou alpha beta in which dou alpha beta equals one or dou alpha beta equals zero according as alpha equals beta or alpha does not equal beta. The conditions are called the conditions of orthogonality and the transformations three and four linear orthogonal transformations. If we stipulate that s squared equals the sum of delta x v squared shall be equal to the square of the length in every system of coordinates. And if we always measure with the same unit scale, then lambda must be equal to one. And therefore the linear orthogonal transformations are the only ones by means of which we can pass from one Cartesian system of coordinates in our space of reference to another. We see that in applying such transformations, the equations of a straight line become equations of a straight line. Reversing equations three a by multiplying both sides by b v beta and summing for all these we obtain some b v beta delta x prime v equals some b v alpha b v beta delta x alpha equals some dou alpha beta delta x alpha equals delta x beta. The same coefficients b also determine the inverse substitution of delta x v. Geometrically b v alpha is the cosine of the angle between the x prime v axis and the x alpha axis. To sum up, we can say that in the Euclidean geometry, there are, in a given space of reference, preferred systems of coordinates, the Cartesian systems, which transform into each other by linear orthogonal transformations. The distance s between two points of our space of reference, measured by a measuring rod, is expressed in such coordinates in a particularly simple manner. The whole of geometry may be founded upon this conception of distance. In the present treatment, geometry is related to actual things, rigid bodies, and its theorems are statements concerning the behavior of these things, which may prove to be true or false. One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. This is satisfactory to the pure mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The question as to whether Euclidean geometry is true or not, does not concern him, but for our purpose, it is necessary to associate the fundamental concepts of geometry with natural objects. Without such an association, geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not. That Euclidean geometry from this point of view affirms something more than the mere deductions derived logically from definitions may be seen from the following simple consideration. Between n points space, there are n times n minus 1, all divided by two distances, s mu v. Between these and the three n coordinates, we have relations s mu v squared equals x1 mu minus x1 v all squared plus x2 mu minus x2 v all squared plus and so on. From these n, n minus 1 all over two equations, the three n coordinates may be eliminated. And from this elimination, at least n times n minus 1 all over two minus three n equations in the s mu v will result. Footnote. In reality, there are n times n minus 1 all over two minus three n plus six equations. End of footnote. Since the s mu v are measurable quantities and by definition are independent of each other, these relations between s mu v are not necessary a priori. From the foregoing, it is evident that the equations of transformation three and four have a fundamental significance in Euclidean geometry in that they govern the transformation from one Cartesian system of coordinates to another. The Cartesian systems of coordinates are characterized by the property that in them the measurable distance between two points s is expressed by the equation s squared equals the sum of delta x v squared. If k xv and k prime xv are two Cartesian systems of coordinates, then sum of delta xv squared equals sum of delta x prime v squared. The right hand side is identically equal to the left hand side on account of the equations of the linear orthogonal transformation. The right hand side differs from the left hand side only in that the xv are replaced by the x prime v. This is expressed by the statement that sum delta xv squared is an invariant with respect to linear orthogonal transformations. It is evident that in the Euclidean geometry only such and all such quantities have an objective significance independent of the particular choice of the Cartesian coordinates as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariance which has to do with the laws that govern the form of invariance is so important for analytical geometry. As the second example of geometrical invariant, consider a volume. This is expressed by v equals integral dx1 dx2 dx3. By means of Jacobi's theorem, we may write integral dx prime 1 dx prime 2 dx prime 3 equals integral of dou x prime 1 x prime 2 x prime 3 all over dou x1 x2 x3 times dx1 dx2 dx3. Where the integrand in the last integral is the functional determinant of the x prime v with respect to the xv. And this by equation 3 is equal to the determinant b mu v of the coefficients of substitution b v alpha. If we form the determinant of the dou mu alpha from equation 4, we obtain by means of the theorem of multiplication of determinants. 1 equals the absolute value of dou alpha beta equals the absolute value of the sum b v alpha b v beta equals the absolute value of b mu v squared. And b mu v equals plus or minus one. If we limit ourselves to those transformations which have the determinant plus one, and only these arise from the continuous variations of the systems of coordinates, then v is an invariant. Footnote. There are thus two kinds of Cartesian systems which are designated as right handed and left handed systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them. Invariants, however, are not the only forms by means of which we can give expression to the independence of the particular choice of the Cartesian coordinates. Vectors and tensors are other forms of expression. Let us express the fact that the point with the current coordinates xv lies upon a straight line. We have xv minus a v equals lambda b v v from one to three. Without limiting the generality, we can put sum of b v squared equals one. If we multiply the equations by b beta v, compare equation 3a and equation 5, and sum for all the v's, we get x prime beta minus a prime beta equals lambda b prime beta. Where we have written b prime beta equals the sum of b beta v b v and a prime beta equals the sum of b beta v a v. These are the equations of straight lines with respect to a second Cartesian systems of coordinates k prime. They have the same form as the equations with respect to the original system of coordinates. It is therefore evident that straight lines have a significance which is independent of the system of coordinates. Formally, this depends upon the fact that the quantities xv minus a v all minus lambda b v are transformed as the components of an interval delta xv. The ensemble of three quantities defined for every system of Cartesian coordinates and which transform as the components of an interval is called a vector. If the three components of a vector vanish for one system of Cartesian coordinates, they vanish for all systems because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. This behavior of the equations of a straight line can be expressed by saying that the equation of a straight line is covariant with respect to linear orthogonal transformations. We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let p0 be the center of a surface of the second degree p any point on the surface and xi v the projections of the interval p0 p upon the coordinate axes. Then the equation of the surface is sum a mu v chi mu chi v equals one. In this and in analogous cases we shall omit the sign of summation and understand that the summation is to be carried out for those indices that appear twice. We thus write the equation of the surface a mu v chi mu chi v equals one. The quantities a mu v determine the surface completely for a given position of the center with respect to the chosen system of Cartesian coordinates. From the known law of transformation for the chi v for linear orthogonal transformations, we easily find the law of transformation for the a mu v. A prime o rho b o mu b rho v a mu v. This transformation is homogenous and of the first degree in the a mu v. On account of this transformation the a mu v are called components of a tensor of the second rank, the latter on account of the double index. If all the components a mu v of a tensor with respect to any system of Cartesian coordinates vanish, they vanish with respect to every other Cartesian system. The form and the position of the surface of the second degree is described by this tensor a. Analytic tensors of higher rank, number of indices, may be defined. It is possible and advantageous to regard vectors as tensors of rank one and invariance scalars as tensors of rank zero. In this respect, the problem of the theory of invariance may be so formulated according to what laws may new tensors be formed from given tensors. We shall consider these laws now in order to be able to apply them later. We shall deal first only with the properties of tensors with respect to the transformation from one Cartesian system to another in the same space of reference by means of linear orthogonal transformations. As the laws are wholly independent of the number of dimensions, we shall leave this number and indefinite at first definition. If a figure is defined with respect to every system of Cartesian coordinates in a space of reference of n dimensions by the n to the alpha numbers a mu v p alpha equaling the number of indices, then these numbers are the components of a tensor of rank alpha if the transformation law is a prime mu prime v prime fo prime and so forth equals b mu prime mu b v prime v b fo prime fo and so forth a mu v fo and so forth. Remark, from this definition it follows that a mu v fo and so forth equals b mu c v d fo and so forth is an invariant provided that b c d and so forth are vectors. Conversely, the tensor character of a may be inferred if it is known that the expression 8 leads to an invariant for an arbitrary choice of the vectors b c etc. Addition and subtraction. By addition and subtraction of the corresponding components of tensors of the same rank, a tensor of equal rank results a mu v fo etc plus or minus b mu v fo etc equals c mu v fo etc. The proof follows from the definition of a tensor given above. Multiplication. From a tensor of rank alpha and a tensor of rank beta, we may obtain a tensor of rank alpha plus beta by multiplying all the components of the first tensor by all the components of the second tensor. T mu v fo etc alpha beta etc equals a mu v fo etc b alpha beta lambda etc. Contraction. A tensor of rank alpha minus 2 may be obtained from the rank of alpha by putting two definite indices equal to each other and then summing for this single index. T fo and so forth equals a mu mu fo and so forth equals the sum of a mu mu fo and so forth. The proof is a alpha mu mu fo and so forth equals b mu alpha b mu beta b fo lambda and so forth. A alpha beta lambda equals dou alpha beta b fo lambda and so forth. A alpha beta lambda and so forth equals b fo lambda and so forth. A alpha alpha lambda and so forth. In addition to these elementary rules of operation, there is also the formation of tensors by differentiation. T mu v fo etc alpha equals dou a mu v fo and so forth all over dou x alpha. New tensors in respect to linear orthogonal transformations may be formed from tensors according to these rules of operation. Symmetrical properties of tensors. Tensors are called symmetrical or skew symmetrical in respect to two of their indices mu and v. If both the components which resolve from interchanging the indices mu and v are equal to each other or equal with opposite signs. Condition for symmetry. A mu v fo equals a v mu fo. Condition for skew symmetry. A mu v fo equals negative a v mu fo. Theorem. The character of symmetry or skew symmetry exists independently of the choice of coordinates and in this lies its importance. The proof follows from the equation defining tensors. Special tensors. One. The quantities dou fo sigma are tensor components, fundamental tensor. Proof. If in the right hand side of the equation of transformation a prime mu v equals b mu alpha b v beta a alpha beta. We substitute for a alpha beta the quantities dou alpha beta which are equal to one or zero according as alpha equals beta or alpha does not equal beta. We get a prime mu v equals b mu alpha b v alpha equals dou mu v. The justification for the last sign of equality becomes evident if one applies equation four to the inverse substitution equation five. Two. There is a tensor dou mu v fo and so forth skew symmetrical with respect to all pairs of indices whose rank is equal to the number of dimensions and. Whose components are equal to positive one or negative one according as mu v fo etc is an even or odd pre mutation of 123 etc. The proof follows with the aid of the theorem proved above the absolute value of b fo sigma equals one. These few simple theorems form the apparatus from the theory of invariance for building the equations of pre relatively physics and the theory of special relativity. We have seen that in pre relativity physics in order to specify relations in space, the body of reference or a space of reference is required. And in addition to Cartesian system of coordinates, we can fuse both these concepts into a single one by thinking of a Cartesian system of coordinates as a cubicle framework formed of rods each of unit length. The coordinates of the lattice points of this frame are integral numbers. It follows from the fundamental relation s squared equals delta x one squared plus delta x two squared plus delta x three squared. That the members of such a space lattice are all of unit length to specify relations in time we require in addition a standard clock placed at the origin of our Cartesian system of coordinates for frame of reference. If an event takes place anywhere we can assign it to three coordinates x me and a time t as soon as we have specified the time of the clock at the origin which is simultaneous with the event. We therefore give an objective significance to the statement of the simultaneity of distant events. While previously we have been concerned only with the simultaneity of two experiences of an individual. The time so specified is that all events independent of the position of the system of coordinates in our space of reference, and is therefore an invariant with respect to the transformation. It is postulated that the system of equations expressing the laws of pre relativity physics is covariant with the respect to the transformation three, as are the relations of Euclidean geometry. The isotropy and homogeneity of space is expressed in this way. Footnote. The laws of physics could be expressed even in case there were any unique direction in space in such a way as to be covariant with respect to the transformation three, but such an expression would in this case be unsuitable. If there were a unique direction in space, it would simplify the description of natural phenomena to orient the system of coordinates in a definite way in this direction. But if, on the other hand, there is no unique direction in space, it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of coordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity. And footnote. We shall now consider some of the more important equations of physics from this point of view. The equations of motion of a material particle are m d squared xv over dt squared equal xv. dxv is a vector, dt and therefore also 1 over dt and invariant. Thus dxv over dt is a vector in the same way it may be shown that d squared xv over dt squared is a vector. In general, the operation of differentiation with respect to time does not alter the tensor character. Since m is an invariant tensor of rank 0, m d squared xv over dt squared is a vector or tensor of rank 1 by the theorem of the multiplication of tensors. If the force xv has a vector character, the same holds for the difference m times d squared xv over dt squared minus xv. These equations of motion are therefore valid in every other system of Cartesian coordinates in the space of reference. In the case where the forces are conservative, we can easily recognize the vector character of xv. For a potential energy, phi exists, which depends only upon the mutual distances of the particles and is therefore an invariant. The vector character of the force xv equals negative dou phi all over dou xv is then a consequence of our general theorem about the derivative of a tensor of rank 0. Multiplying by the velocity a tensor of rank 1, we obtain the tensor equation m times d squared xv over dt squared minus xv all multiplied by dxv over dt equals 0. By contraction and multiplication by the scalar dt, we obtain the equation of kinetic energy d times mq squared over 2 equals xv dxv. If chi v denotes the difference of the coordinates of the material particle and a point fixed in space, then the chi v have the character of vectors. We evidently have d squared xv over dt squared equals d squared chi v over dt squared so that the equations of motion of the particle may be written m times d squared chi v over dt squared minus xv equals 0. Multiplying this equation by chi mu, we obtain a tensor equation m d squared chi v over dt squared minus xv all multiplied by chi mu equals 0. Contracting the tensor on the left and taking the time average, we obtain the varial theorem, which we shall not consider further. By interchanging the indices and subsequent subtraction, we obtain after a simple transformation the theorem of moments d over dt times m times chi mu times d chi v over dt minus chi v times d chi mu over dt equals chi mu xv minus chi v x mu. It is evident in this way that the moment of a vector is not a vector, but a tensor. On account of their skew symmetrical character, there are not nine, but only three independent equations of this system. The possibility of replacing skew symmetrical tensors of the second rank in space of three dimensions by vectors depends upon the formation of the vector. A mu equals one half a sigma rho dou sigma rho mu. If we multiply the skew symmetrical tensor of rank two by the special skew symmetrical tensor dou introduced above and contract twice, a vector results whose components are numerically equal to those of the tensor. These are the so-called axial vectors, which transform differently from a right-handed system to a left-handed system from the delta xv. There is a gain in picturesqueness in regarding a skew symmetrical tensor of rank two as a vector in space of three dimensions, but it does not represent the exact nature of the corresponding quantity so well as considering it a tensor. We consider next the equations of motion as a continuous medium. Let foe be the density, uv the velocity components consider as functions of the coordinates, and the time xv the volume forces per unit of mass, and pv sigma the stresses upon a surface perpendicular to the sigma axis in the direction of increasing xv. Then the equations of motion are by Newton's law foe duv over dt equals negative dou pv sigma over dou x sigma plus foe xv, in which duv over dt is the acceleration of the particle which at time t has the coordinates xv. If we express this acceleration by partial differential coefficients, we obtain after dividing by foe dou uv over dt plus dou uv over dx sigma times u sigma equals negative one over foe times dou pv sigma over dou x sigma plus xv. We must show that this equation holds independently of the special choice of the Cartesian system of coordinates. uv is a vector and therefore dou uv over dou t is also a vector. dou uv over dou x sigma is a tensor of rank two. dou uv over dou x sigma times u rho is a tensor of rank three. The second term on the left results from a contraction of the indices sigma and rho. The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector, it is necessary for pv sigma to be a tensor. Then by differentiation and contraction, dou pv sigma over dou x sigma results and is therefore a vector as it also is after multiplication by the reciprocal scalar one over foe. That pv sigma is a tensor and therefore transforms according to the equation p prime mu v equals b mu alpha bv beta p alpha beta is proved in mechanics by integrating this equation over an infinitely small tetrahedron. It is also proved there by application of the theorem of moments to an infinitely small parallel pdn that pv sigma equals p sigma v and hence that the tensor of the stress is a symmetrical tensor. From what has been said it follows that with the aid of the rules given above, the equation is covariant with respect to orthogonal transformations in space, rotational transformations. And the rules according to which the quantities in the equation must be transformed in order that the equation may be covariant also become evident. The covariance of the equation of continuity dou foe over dou t plus dou fo uv over dou xv equals zero requires from the foregoing no particular discussion. We shall also test for covariance the equations which express the dependence of the stress components upon the properties of the matter and set up these equations for the case of a compressible viscous fluid with the aid of the conditions of covariance. If we neglect the viscosity, the pressure p will be a scalar and will depend only upon the density and the temperature of the fluid. The contribution to the stress tensor is then evidently p dou mu v in which dou mu v is the special symmetrical tensor. This term will also be present in the case of a viscous fluid, but in this case there will also be pressure terms which depend upon the space derivatives of the uv. We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be alpha multiplied by dou u mu over dou xv plus dou uv over dou xmu plus beta dou mu v times dou u alpha over dou x alpha. For dou u alpha over dou x alpha is a scalar. For physical reasons, no slipping. It is assumed that for symmetrical dilations in all directions, for example, when dou u1 over dou x1 equals dou u2 over dou x2 equals dou u3 over dou x3, dou u1 over dou x2, etc. equals zero. There are no frictional forces present from which it follows that beta equals negative two-thirds alpha. If only dou u1 over dou x3 is different from zero, let p31 equal negative alpha multiplied by dou u1 over dou x3 by which a is determined. We then obtain for the complete stress tensor p mu v equals p dou mu v minus alpha multiplied by dou u mu over dou xv plus dou uv over dou xmu minus two-thirds multiplied by dou u1 over dou x1 plus dou u2 over dou x2 plus dou u3 over dou x3 multiplied by dou mu v. The heuristic value of the theory of invariance, 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. Dou h3 over dou x2 minus dou h2 over dou x3 equals 1 over c dou e1 over dou t plus 1 over ci1. Dou h1 over dou x3 minus dou h3 over dou x1 equals 1 over c times dou e2 over dou t plus 1 over ci2. Dou h2 over dou x1 minus dou h1 over dou x2 equals 1 over c times dou e3 over dou t plus 1 over ci3. Dou e1 over dou x1 plus dou e2 over dou x2 plus dou e3 over dou x3 equals 4. Dou e3 over dou x2 minus dou e2 over dou x3 equals negative 1 over c times dou h1 over dou t. Dou e1 over dou x3 minus dou e3 over dou x1 equals negative 1 over c dou h2 over dou t. Dou e2 over dou x1 minus dou e1 over dou x2 equals negative 1 over c times dou h3 over dou t. Dou h1 over dou x1 plus dou h2 over dou x2 plus dou h3 over dou x3 equals 0. 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. Footnote, these considerations will make the reader familiar with tensor operations. Without the special difficulties of the four-dimensional treatment, corresponding considerations in the theory of special relativity, Manowski's interpretation of the field will then offer fewer difficulties. End footnote. The equations may, however, easily be 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, and h3 respectively. Paying attention to the skew symmetry of h mu v, the first three equations of 19 and 20 may be written in the form of dou h mu v over dou xv equals 1 over c times dou e mu over dou t plus 1 over c i mu dou e mu over dou xv minus dou e v over dou x mu equals 1 over c times dou h mu v over dou t. 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 dou e v over dou xv equals foe dou h mu v over dou x foe plus dou h v foe over dou x mu plus dou h foe mu over dou xv equals 0. The last equation is a skew symmetrical tensor equation of the third rank. The skew symmetry of the left-hand side with the respect to every pair of indices may easily be proved if attention is paid to the skew symmetry of h mu v. 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. End of the Meaning of Relativity, Lecture 1 Excerpt from Chapter 4, Names and Sentences of the First Fleeters from Convict Life in New South Wales and Van Demon's Land. This is a Libervox recording. All Libervox recordings are in the public domain. For more information or to volunteer, please visit Libervox.org. Excerpt from Chapter 4, Names and Sentences of the First Fleeters by Charles White A correct list, The Lifers, 14 and 7 Years Convicts A prophetic song, Story of the Last Survivor The following is a correct list of the convicts who were sent out in the First Fleet, showing the periods for which they were transported, transported for life. George Barbsby Robert Bales Thomas Barrett William Blatherhorn James Cox William Davis Joseph Donage Thomas Eccles Thomas Gearing John Harris Joseph Hall James Heading John Hill William Hilt John Kellan Alias Keeling David Kilpac George List Thomas Limpis Mary Long Mary Marshall Hannah Mullins Richard Partridge Sarah Perry Charles Pete John Ponty Ann Reed Thomas Riesdale Alias Crowder John Rugblass John Ruffler James Shears Joseph Tussaud John Welsh Edward Whitten Charles Whitten John Whitten John Whitten John Whitten Charles Wilson Samuel Woodham and John Wolcott Transported for 14 years Samuel Day Francis Davis William Hogg Margaret Jones John Jones Thomas Jones Jeremiah Leahy Joseph Long Ann Lynch Joseph Marshall Betty Mason Lydia Munro Joseph Owen Isaac Rogers Daniel Spencer John Stogdell James Underwood Mary Wade Alias Cackle John Stogdell James Underwood Alias Cacklane and Mary Wickham Transported for 7 years Robert Abel Henry Abrams Esther Abrahams Mary Abel Alias Tilly Thomas Akers John Adams Mary Abel John Adams Mary Adams Richard Agley John Allen William Allen Charles Allen Susanna Allen Mary Allen Jasmine Allen Alias Boddington Mary Allen Alias Conner John Anderson John Anderson John Anderson Eliza Anderson John Anderson Fanny Anderson John Archer John Arscott George Alkinson Sarah Ault John Iners Alias Agnew John Iris James Bartlett Hennessey Henry Barnett Alias Barnard Alias Burton Stefan Barnes George Bannister John Bursard George Barland James Balding Alias William Elizabeth Basin James Bailey John Basely James Bailie James Bailie John Basely Thomas Baker Katyn Batley Samuel Barbsby John Ball John Berry Daniel Barrett Elizabeth Barber Roof Baldwin Alias Bowyer Martha Baker William Bell Samuel Barber Baker William Bell Samuel Vanier Jacob Bellett Ann Beardsley John Bast Elizabeth Beckford Thomas Bellamy James Byrd Samuel Byrd Joseph Bishop John Bingham Alias Bangham Eliza Bingham Alias Mooring Eliza Byrd Alias Winifred William Blackall William Blunt Francis Blake James Blodeworth Susanna Blanchett Peter Bond John Boyd James Blodeworth Susanna Blanchett Peter Bond John Boyle William Bogus William Bond Mary Bond Rebecca Bolton Jane Bonner Mary Bolton James Brown William Brown John Bringley John Brindley Richard Brown William Brindley John Brindley Brown William Brogue James Bradley James Bradley Thomas Brown William Bradbury Thomas Bryant William Bryant Thomas Brown John Bradford James Branigan Robert Bruce William Brown John Bradford John Brown John Bryant William Brewer William Bryce Curtis Brand Michael Bryant Lucy Brand Alias Wood Mary Branham Elizabeth Bruce James Berlay Peter Byrne Patrick Byrne Simon Byrne John Busley Margaret Byrne Mary Burkitt Sarah Burdow Joseph Carver James Castle James Campbell Alias George James Campbell John Carney Francis Carti Ann Kerry Richard Carter Alias Michael Cartwright Henry Cable Mary Coral John Sesser William Shields Thomas Shaddick William Church William Chaff Samuel Chinnery Edward Chanin Richard Carter Chanin Richard Clo Thomas Clements John Clark Alias Hozer William Clark John Clark Mary Cleaver George Clear Elizabeth Clark William Connolly Edward McCormick James Corden Joseph Colling Joseph Colling William Cole John Matthew Cox Richard Collier William Connolly Cornelius Connolly Ishmael Coleman John Coffin Elizabeth Cole James Cope Ann Combs Elizabeth Cole Elizabeth Coley Charlotte Cook Mary Cooper Ann Colpitz John Cross John Cropper William Cross John Creamer Jane Creek Edward Cunningham James Bryan Cullen John Cullyhorn John Cullyhorn John Cullyhorn John Cullyhorn Jacob Cudlip Alias Norris John Cuss Alias Hannaboy William Cucko Aaron Davis Richard Day Edward Davies Samuel Davis William Davis James Davis Daniel Daniels James Daly John Davidson Richard Davis Ann Daly Alias Ann Warbutton Margaret Darnell Ann Davis Elizabeth Dalton Rebecca Davidson Margaret Davidson Sarah Davies Mary Davies Michael Denison Barnaby Denison Patrick Delaney Thomas Dixon Alias Ralf Kahl Timothy Diskell Mary Dixon Mary Dickinson William Douglas Ferdinand Dowland James Dodding Alias Doring William Dring Elizabeth Dudgins Jane Dundis Ann Dutton Leonard Dare Mary Dykes William Earl William Eagleton Alias Bones Mary Dixon William Duglass Ferdinand Dowland James Dodding Alias Bones Mary Eaton Alias Shepard Rachel Early Martha Eaton William Edmonds William Edwards George Eggleston Peter Alam William Elliott Joseph Elliott Deborah Alam Nicholas English John Everett Matthew Irvingham William Evans Elizabeth Evans Philip Farrell William Farley Ann Farmer Benjamin Fentum John Ferguson Thomas Fillesi Jane Fitzgerald Alias Phillips William Field John Finlow Alias Ervingham John Finlow Alias Herve Jane Field Elizabeth Fitzgerald Edward Flynn Thebe Flarty Francis Focus Robert Forrester William Foyle Ann Fowles Margaret Fownes Ann Forbes James Freeman Robert Freeman William Francis George Francisco George Fry Catherine Fryer Alias Pryor William Fraser Ellen Frazier John Fuller Francis Gardner Edward Garth Francis Garland Susanna Garth Mary Gable Olive Gaston Olive Gascoigne George Gess Annie George Thomas Glenton Daniel Gordon Edward Goodwin Andrew Goodwin John Gould Charles Gray Samuel Griffles Alias Briscoe Alias Butcher Nicholas Greenwell John Greenwell John Green Thomas Griffiths Charles Granger James Grace Hannah Green Mary Gloves Mary Green Ann Green Mary Greenwood William Gunter John Handford John Hatcher William Houghfield Richard Houghfield Richard Hawks William Harris John Hatch John Hartley John Hart Joseph Haynes Henry Halfaway Dennis Hayes Samuel Hall Joseph Harbine Joshua Harper George Hayton Alias Clayton Joseph Harrison John Hart John Hayes Joseph Hatham Joseph Harrison William Hamlin John Hall John Haddon William Harris Cooper Handy William Haynes Elizabeth Hervey William Hatham William Hatham Elizabeth Hervey Margaret Hall Frances Hart Mary Harrison Thomas Heddington John Herbert Catherine Hart John Herbert Dorothy Hanland Alias Gray Sarah Hall Maria Hamilton Mary Harrison Esther Harwood Alias Howard Elizabeth Hayward Elizabeth Hall Jane Herbert Alias Rose Alias Jenny Russell Catherine Henry William Hindley Alias Platt Ottowell Henry William Hindley Alias Platt Ottowell Henry Ottowell Hendel John Hill Thomas Hill Thomas Hill Elizabeth Hipsley Mary Hill Job Hollister Thomas Howell William Holmes James Holloway Thomas Howard John Howard James Howell James Howell Howard James Hortop William Holland Susanna Holmes Elizabeth Hologen Hugh Hughes Edward Humphrey William Husband John Hughes Jeremiah Hurley William Hubbard Henry Humphreys Thomas Hughes James Hortop Hugh's James Hussie John Hudson Francis Ann Hughes Susanna Huffwell Mary Humphreys Thomas Highlids James Ignam John Irving Ann Annette William Jackson David Jacobs John Jacobs Hannah Jackson James Jameson Jane Jackson Alias Esther Roberts Mary Jackson Robert Jeffries John Jeffries Robert Jenkins Alias Brown John Jep William Jenkins John Jep John Jep John Jep William Jenkins Francis Joseph Thomas Jones Charles Johnson Edward Jones Thomas Joseph William Johnson Stefan Johns Edward Johnson Richard Jones William Jones Catherine Johnson Mary Johnson Thomas Kelly Martha Kennedy Thomas Kidney Wilham Kilby John King Edward Kimberley John Knowler Andrew Knowland David Lenke Richard Lane John Laurel William Lane William Lane William Lane William Lane James Larn John Lambef Henry Laveau Flora Lara Caroline Laycock John Leary Stefan Legrove George Legg Elizabeth Lee Isaac Lemon Elizabeth Leonard Joseph Levy Amelia Levin Levy Amelia Levy Sophia Lewis Samuel Lightfoot John Limberner Elizabeth Locke John Lockely Joseph Longstreet Mary Love Nathaniel Lucas Humphrey Lynch John Lide John Law James McRee James McRee John Massentire John Mansfield Stefan Martin John Martin Abraham Martin Thomas Martin Anne Martin James Martin Will Marney William Mariner John Merritt Jane Marriott Mary Marshall Joseph Marshall Susanna Mason Anne Mather Thomas Matten Richard May Sarah McCormick Mary McCormick Eleanor McCabe Richard McDede Alexander McDonald James McDonald James McDonnell James McDonnell James McDonnell Redmond McGrath Francis McLean Thomas McLean Edward McLean Charles McLaughlin William McNamar Jane Meach William Meach Jacob Messiah John Maynell Samuel Midgley Richard Middleton Charles Milton Matthew Mills Mary Mitchcraft D'Faniel Mitchell Mary Mitchell Samuel Mobs John Mullins Charles Mood John Mooden William Moore William Morgan Robert Morgan Richard Morgan John Morley Joseph Morley John Marispy John Morris Mary Morton John Mortimer John Mowbray John Mowbray John Mowbray John Mowbray John Mowbray John Mowbray Edward Moyle Jesse Mullick Stefan Mullis John Monroe James Murphy William Murphy John Neal James Neal Elizabeth Needham Robert Nettleton John Newland John Nicholas Phoebe Norton Robert Nunn John O'Craft James Ogden William O'Kee Thomas O'Field Isabella O'Field Peter Opley Thomas Orford Elizabeth Osborn Thomas Osborn John Owen John Owls Paul Paid Joseph Paget John Henry Palmer William Payne Elizabeth Parker Mary Parker John Parker William Parr Edward Perry William Parrish Peter Parris Jane Parkinson Ann Parsley Peter Parrish Peter Parrish Ann Parsley Sarah Partridge John Pierce James Poulet Joshua Peck John Penny Edward Perkins Richard Percival Edward Bearcroft Perret John Petrie John Petit John Pethrick William Philomor Mary Phillips Richard Phillips Roger Pfeifield Mary Fine Samuel Piggit Mary Piles Mary Pinder Elizabeth Pipkin William Platt Jane Poole David Pope John Powell Elizabeth Pipkin William Platt Luke John Power William Power Ann Poelle Elizabeth Powley John Price James Price Thomas Price Thomas Prichard John Ramsey William Radford John Randall William Reed Bartholomew Reardon John Randall Reardon Charles Repeat George Raymond John Rice James Richard James Richard David Richard Hardwick Richardson John Richardson James Richardson Samuel Richardson William Richardson John Richardson William Richardson John Richards William Rickson Edward Risbee Henry Roach John Robert William Roberts William Roberts William Robinson George Robinson George Robinson George Robinson Thomas Robinson John Robbins Daniel Roberts Robbins Daniel Rogers Mary Rolt John Romain Anthony Rope Isabella Rossin Walton Roos John Rowe William Rowe James Roos or Roos John Russell Robert Roof Jenny Rose or Russell John Ryan William Saltmash Peter Sampson William Sands Thomas Sanderson Ann Sandlin Robert Scattergood Elizabeth Scott Samuel Selfhire John Seymour George Sharp Joseph Shaw William Shearman Robert Shepherd William Shore John Shore Robert Sideway John Silverthorn Sarah Slater John Small Richard Smart Daniel Smart Ann Smith Ann Smith Ann Smith Catherine Smith Catherine Smith Catherine Smith Catherine Smith Edward Smith Edward Smith Hannah Smith James Smith John Smith John Smith Mary Smith Thomas Smith William Smith William Smith William Smith William Smith William Smith William Snalham Henry Sparks John Spencer Mary Spence Charlotte Sprigmore Mary Springham James Squires William Stanley Thomas Stanton John Morris Steffens Robert Steffens Margaret Stewart John Stoke Martin Stone Charles Stone Henry Stone James Stowe Thomas Stretch James Strong John Summers Joshua Taylor Henry Taylor Sarah Taylor Cornelius Teague Thomas Hilton Tenet James Tenchaw Elizabeth Takri John Thomas John Thomas James Thomas Elizabeth Thomas William Thompson William Thompson James Thompson Anne Thornton James Taudy Thomas Till Thomas Tilley Nicholas Todd John Trace Susanna Trippett Joseph Trotter Moses Tucker Thomas Tummins John Turner John Turner Ralph Turner Mary Turner Thomas Turner Anne Twiffield William Twinham William Terrell John Usher Edward Vandell William Vickrey Henry Vincent Richard Waddecombe Mary Wade Benjamin Wager Ellen Wainwright James Wauburn John Walker William Wall William Walsh John Ward Anne Ward Charlotte Ware William Waterhouse Mary Watkins John Watson Thomas Watson John Welsh Benjamin West John Westwood Edward Westlale Samuel Wheeler George Whitaker William Whitting John Wifehammer Samuel Wilcox John Wilding Charles Williams John Williams John Williams James Williams Peter Williams Robert Williams Daniel Williams Francis Williams Mary Williams Peter Wilson William Wilton George Wood Mark Wood Peter Woodcock Francis Woodcock William Worsdell Anne Wright Benjamin Wright James Wright Joseph Wright Thomas Wright William Wright Lucy Wood Thomas Yardsley Nancy Yates John Young Simon Young Elizabeth Youngson George Youngson This list is given not for the purpose of gratifying any morbid curiosity, but simply to preserve the names of those who were the real pioneers of the colony. To many of these men, with others who were not marked with the felons brand, belonged the credit of making the initial movements in that work of colonization which has spread so widely and with such marvelous rapidity through this vast continent. They toiled and suffered to an extent which the mind fails to compass, and from their toil and suffering, their sprang the first shoots of that industrial and commercial life which to-day is throbbing with a vigor not exceeded by any nation upon the face of the earth. And of excerpt from Chapter 4 Names and Sentences of the First Fleeters by Charles White