Length contraction

“ Each portion of the circumference is moving in the direction of its length, and might be expected to undergo the FitzGerald contraction due to its velocity; each portion of a radius is moving transversely and would therefore have no longitudinal contraction. ”

“ The writing-down of lengths for balance-sheet purposes is the FitzGerald contraction. The shortening of the moving rod is true, but it is not really true. It is not a statement about reality (the absolute) but it is a true statement about appearances in our frame of reference. [1] An object has different lengths in the different space-frames, and any 6-foot man will have a length 3 feet in some frame or other. The statement that the length of the rapid traveller is 3 feet is true, but it does not indicate any special peculiarity about the man ”

“ The volume of a body is to be transformed by the Lorentz transformation because of the same reason as the length parallel to the velocity. Since by that, in every system is equal to the volume in the co-moving system, if follows that the equation, which serves as the transformation formula for the volume of a body with velocity relative to and relative to , is:
Another consequence of contraction is, that the form of a body is differently seen in different systems. ”

“ And yet, if we retain all our rods as equally valid, whether in motion or at rest, we obtain conflicting results which are such as to deprive the geometry of space of all significance. When we analyse this anomalous situation we find that the following facts stand out clearly: Rods remain unmodified whether in motion or at rest, and yet their length varies. The truth is that length is not a definite characteristic of a body; it appears as a shadow of a something else projected into the space of our frame. When the rod is at rest in our frame, its length, or the shadow, presents a maximum ”

“ Obviously, physical length is not what we once thought it to be; it can in no wise be immanent in the body, since a body has no determinate length until the relative motion of the observer has been specified. A length is therefore but the expression of a relationship between the observer and the observed, and the two partners of the relationship must be specified before the length can have any meaning. ”

“ Quite similar relations occur with respect to measurements of the length of a moving rod. If it is at rest in and located parallel to the -axis, then its rest length is determined by the difference of the -coordinates of its endpoints, thus:
those can be determined by applying a measuring rod also at rest in . ”

“ A cord, whether laid on itself or extended in only one direction, and though conceived of the utmost conceivable thinness, can not be conceived with less than three dimensions. Nor can a line or a point. When we think of a mathematical plane or line or point, we do nothing more than fix our attention on length and breadth, regardless of thickness ”

“ The second feature common to all concepts brought out by the detailed discussion of length is that, as we approach the experimentally attainable limit, concepts lose their individuality, fuse together, and become fewer in number, as we have seen that at dimensions of the order of the diameter of an electron the concepts of length and the electric field vectors fuse into an amorphous whole. ”

“ We thus see that a meter-stick, which, when held perpendicular to its line of motion, has the same length as a meter-stick at rest, will be shortened when turned parallel to the line of motion in the ratio , and indeed any moving body must be shortened in the direction of its motion in the same ratio. ”

“ It is a mere matter of geometry to discover what the β particle's lengths and times become when referred to the partitions which we have drawn across the world. But when we calculate the consequential change of mass resulting from the changes of length and time, we find that it should be increased in precisely the proportion indicated by the most refined experiments. ”

“ In order to form an exact idea of their real dimensions, we must always remember that, at this distance, four fifths of a second is equal to 140 kilometres, and consequently these apparent threads, whose seeming width is at most not more than one or two seconds, are in reality immense currents, being, about the middle, of 600 or 700 kilometres in width, while their length is at least equal to the diameter of the terrestrial globe. ”

“ In particular, if we examine the definition of absolute time in the light of experiment, we find nothing in nature with such properties.
The new attitude toward a concept is entirely different. We may illustrate by considering the concept of length: what do we mean by the length of an object? We evidently know what we mean by length if we can tell what the length of any and every object is, and for the physicist nothing more is required. To find the length of an object, we have to perform certain physical operations. ”

“ We see, then, if we elevate the centre of gravity precisely the requisite quantity, by employing a sufficient length of the compensating material, that although the distance from the centre of gravity to the point of suspension is lessened, yet the distance from the point of suspension to the centre of oscillation will suffer no change. ”

“ A very sound definition; but unfortunately it does not agree with the meaning of distance in general use. When the relativist wishes to refer to this length, he calls it the proper-length; in non-relativity physics it does not seem to have been used at all. You see it is not convenient to send your apparatus hurling through the laboratory—after a pair of α particles, for example. And you could scarcely measure the length of a wave of light by this convention. [1] So the physicist refers his lengths to apparatus at rest on the earth ”

“ Secondly, if an observer at rest in measure the radius and circumference of the rotating disc, he will obtain the same value for the radius as when the disc is at rest, but since, when he measures the circumference of the disc, the scale lies along the direction of motion, it suffers contraction, and, consequently, will divide more often into the circumference than if the scale and the disc were at rest. ”

“ It merely regulates the relationships or defines the laws of the mutual dispositions of bodies, and these laws may remain unmodified even though the absolute shape and size of the bodies vary. Restricting ourselves to the problem of absolute size, we shall find that with the finite universe the metrical field yields a universal standard of length given by the radius of the finite universe. As referred to this natural gauge, an object presents a definite size, but obviously there is nothing absolute about the gauge itself. ”

“ For the globe is to its circumscribed cylinder as two to three; and therefore the force which can destroy all the motion of the cylinder while the same cylinder is describing the length of four of its diameters, will destroy all the motion of the globe while the globe is describing two thirds of this length, that is, eight third parts of its own diameter. ”

“ But, if we demand, in particular, that each point of the manifold may be fixed by means of rectangular Cartesian co-ordinates, whereby particular assumptions are made concerning the possible ways of placing the measuring rods, then the line-element, [Pg 26] expressed in these special variables, assumes the form Hitherto this expression has always been introduced for the length of the line-element in all physical laws. ”

“ Consequently the length that we arrive at by measurement is the “proper” length, that is to say, the length as estimated by an observer who shares the motion of the body. We never, in ordinary life, have to tackle the problem of [Pg 65] measuring a body which is in continual motion. And even if we did, the velocities of visible bodies on the earth are so small relatively to the earth that the anomalies dealt with by the theory of relativity would not appear. But in astronomy, or in the investigation of atomic structure, we are faced with problems which cannot be tackled in this way. ”

“ No, you must not define length by means of a rigid scale, and define a rigid scale by means of length.
Phys. I admit I am hazy about strict definitions. There is not time for everything; and there are so many interesting things to find out in physics, which take up my attention. Are you so sure that you are prepared with a logical definition of all the terms you use? ”

“ About the time that we discover the necessity for a complete vacuum, we discover that the gauges themselves are atomic in structure, that they have no definite boundaries, and therefore no definite length, but that the length is a hazy thing, varying rapidly in time between certain limits. ”

“ Thanks to studies thus systematically pursued, we have succeeded in the course of a hundred years in increasing the precision of measures in the proportion of a thousand to one, and we may ask ourselves whether such an increase will continue in the future. No doubt progress will not be stayed; but if we keep to the definition of length by a material standard, it would seem that its precision cannot be considerably increased. ”

“ Now, if a parallelogram be drawn with a constant diameter, it is seen that by increasing the inclination of one of the sides to the diameter (and even decreasing its length) the inclination of the other side to the diameter may also be in creased. The most favourable case for such a change is when the side whose inclination is increased is nearly as long as the diameter. ”

“ We can no more transfer the length of the ten-foot rod, constituted in its first position by the first real modes, to the place of the length constituted in its second position by the second real modes, than we are able to do so for intervals themselves, which we compare by measurement. ”

“ Since the straight diameter is easily measured with great accuracy, when he had the area he could readily have found the circumference by working backward the rule announced by Archimedes, viz.: that the area of a circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude is equal to the radius. ”

“ He further showed that if a man were to measure the length of a rod, for instance, on the earth and then were able to carry the rod and his measuring apparatus to Mars, the sun, or to Arcturus he would obtain the same numerical value for the length in all places and at all times. ”

“ As long nothing is known about these influences, as long nothing can be said about the contraction of the volume. Most directly, this can be seen at slightly compressible bodies, as in gases, but it applies equally well to solid bodies. ”

“ Unfortunately to construct and preserve a metre or yard is also a task which is either impossible, or what comes nearly to the same thing, cannot be shown to be possible. Passing over the practical difficulty of defining the ends of the standard length with complete accuracy, whether by dots or lines on the surface, or by the terminal points of the bar, we have no means of proving that substances remain of invariable dimensions. ”

“ It is found that the kind of material—metal, stone or wood—makes no difference to the experiment. The contraction must be the same for all kinds of matter; the expected delay depends only on the ratio of the speed of the aether current to the speed of light, and the contraction which compensates it must be equally definite. ”

“ For the earth's orbital motion the contraction amounts to one part in 200 million; in fact the earth's diameter in the direction of its motion is always shortened by 21⁄2 inches, the transverse diameter being unaffected. ”

“ These differences are due to the fact that the form of the child is altering with growth, that other linear dimensions are varying somewhat differently from length or stature, and that consequently the growth in bulk or weight is following a more complicated law.
Our curve of growth, whether for weight or length, is a direct picture of velocity, for it represents, as a connected series, the successive epochs of time at which successive weights or lengths are attained. ”

“ By making the cuts more central the length of the segments are reduced, thereby causing less curvature of the extreme ends and more nearly equalizing the extent of curvature both ways from the concentric form. This equalization is what causes the reduction in the middle error and its absence in ordinary balances is what causes the larger error. ”

“ The uniform contraction in one dimension of the moving material system, the possibility of which we have thus been led to entertain, suffices of itself for the discussion of some interesting optical problems. ”

“ Now, this differs only by 2,944 inches, or by 82 yards, from 500,500,000 such inches, and this would be the whole error on a length of 8,000 miles which would arise from the adoption of this precise round number of inches for its length, or from making the inch, so defined, our fundamental unit of length. After pointing out that the calculation required for correlating a dimension so stated with the earth's axis is a shorter one than that required for correlating such dimension with the quadrant of a meridian. ”

“ To see the force of the explanation we must, however, realise the relativity of extension. Extension which is not relative to something in the surroundings has no meaning. Imagine yourself alone in the midst of nothingness, and then try to tell me how large you are. The definiteness of extension of the standard rod can only be a definiteness of its ratio to some other extension. But we are speaking now of the extension of a rod placed in empty space, so that every standard of reference has been removed except extensions belonging to and implied by the metric of the region. ”