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Read Ebook: The Principle of Relativity by Einstein Albert Minkowski H Hermann Mahalanobis P C Prasanta Chandra Author Of Introduction Etc Bose Satyendranath Translator Saha Meghnad Translator
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Ebook has 749 lines and 77353 words, and 15 pages
See Note 4.
Footnote 4:
See Notes 9 and 12.
Footnote 5:
Note A.
INTRODUCTION.
It is well known that if we attempt to apply Maxwell's electrodynamics, as conceived at the present time, to moving bodies, we are led to asymmetry which does not agree with observed phenomena. Let us think of the mutual action between a magnet and a conductor. The observed phenomena in this case depend only on the relative motion of the conductor and the magnet, while according to the usual conception, a distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force which corresponds to no energy in itself is produced in the conductor; this causes an electric current of the same magnitude and the same career as the electric force, it being of course assumed that the relative motion in both of these cases is the same.
? 1. Definition of Synchronism.
Let us have a co-ordinate system, in which the Newtonian equations hold. For distinguishing this system from another which will be introduced hereafter, we shall always call it "the stationary system."
If a material point be at rest in this system, then its position in this system can be found out by a measuring rod, and can be expressed by the methods of Euclidean Geometry, or in Cartesian co-ordinates.
It may appear that all difficulties connected with the definition of time can be removed when in place of time, we substitute the position of the little hand of my watch. Such a definition is in fact sufficient, when it is required to define time exclusively for the place at which the clock is stationed. But the definition is not sufficient when it is required to connect by time events taking place at different stations,--or what amounts to the same thing,--to estimate by means of time the occurrence of events, which take place at stations distant from the clock.
Now with regard to this attempt;--the time-estimation of events, we can satisfy ourselves in the following manner. Suppose an observer--who is stationed at the origin of coordinates with the clock--associates a ray of light which comes to him through space, and gives testimony to the event of which the time is to be estimated,--with the corresponding position of the hands of the clock. But such an association has this defect,--it depends on the position of the observer provided with the clock, as we know by experience. We can attain to a more practicable result by the following treatment.
We assume that this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold:--
Thus with the help of certain physical experiences, we have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another; and thereby we have arrived at a definition of synchronism and time.
In accordance with experience we shall assume that the magnitude
We have defined time essentially with a clock at rest in a stationary system. On account of its adaptability to the stationary system, we call the time defined in this way as "time of the stationary system."
? 2. On the Relativity of Length and Time.
The following reflections are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following way:--
where, by 'interval of time' we mean time as defined in ?1.
The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod:--just as if the observer, the measuring rod, and the rod to be measured were at rest.
Relativity of Time.
? 3. Theory of Co-ordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity.
To every value of which fully determines the position and time of an event in the stationary system, there correspond a system of values ; now the problem is to find out the system of equations connecting these magnitudes.
Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear.
If we now introduce the condition that ? is a function of co-ordinates, and apply the principle of constancy of the velocity of light in the stationary system, we have
$$ frac + tau )) $$
It is to be noticed that instead of the origin of co-ordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of .
where
Therefore
where
and
with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,
In the transformations we have got an undetermined function ?, and we now proceed to find it out.
? 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks.
A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition--when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes
$$ R sqrt }, R, R. $$
$$ 1 : sqrt }; $$
It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system.
Therefore the clock loses by an amount 1/2 per second of motion, to the second order of approximation.
We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.
? 5. Addition-Theorem of Velocities.
It is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in ? 3 in the equation of motion of the point, we obtain
The law of parallelogram of velocities hold up to the first order of approximation. We can put
and
$$ frac } $$
We see that such a parallel transformation forms a group.
We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electrodynamics.
? 6. Transformation of Maxwell's equations for Pure Vacuum.
On the nature of the Electromotive Force caused by motion in a magnetic field.
The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that
and
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