Out of My Later Years: The Scientist, Philosopher, and Man Portrayed Through His Own Words by Albert Einstein Read Free Book Online
Authors: Albert Einstein
special theory of relativity. This is empty space without electromagnetic field and without matter. It is completely determined by its “metric” property: Let dx 0 , dy 0 , dz 0 , dt 0 be the coordinate differences of two infinitesimally near points (events); then
(1) ds 2 = dx 0 2 + dy 0 2 + dz 0 2 - c 2 dt 0 2
is a measurable quantity which is independent of the special choice of the inertial system. If one introduces in this space the new coordinates x 1 , x 2 , x 3 , x 4 through a general transformation of coordinates, then the quantity ds 2 for the same pair of points has an expression of the form
(2) ds 2 = ∑g ik dx i dx k (summed for i and k from 1 to 4)
where g ik = g ki . The g ik which form a “symmetric tensor” and are continuous functions of x 1 . . . x 4 then describe according to the “principle of equivalence” a gravitational field of a special kind (namely one which can be re transformed to the form (1)). From Riemann’s investigations on metric spaces the mathematical properties of this g ik field can be given exactly (“Riemann-condition”). However, what we are looking for are the equations satisfied by “general” gravitational fields. It is natural to assume that they too can be described as tensor-fields of the type g ik , which in general do not admit a transformation to the form (1), i.e., which do not satisfy the “Riemann condition,” but weaker conditions, which, just as the Riemann condition, are independent of the choice of coordinates (ie., are generally invariant). A simple formal consideration leads to weaker conditions which are closely connected with the Riemann condition. These conditions are the very equations of the pure gravitational field (on the outside of matter and at the absence of an electromagnetic field).
These equations yield Newton’s equations of gravitational mechanics as an approximate law and in addition certain small effects which have been confirmed by observation (deflection of light by the gravitational field of a star, influence of the gravitational potential on the frequency of emitted light, slow rotation of the elliptic circuits of planets—perihelion motion of the planet Mercury). They further yield an explanation for the expanding motion of galactic systems, which is manifested by the red-shift of the light omitted from these systems.
The general theory of relativity is as yet incomplete insofar as it has been able to apply the general principle of relativity satisfactorily only to gravitational fields, but not to the total field. We do not yet know with certainty, by what mathematical mechanism the total field in space is to be described and what the general invariant laws are to which this total field is subject. One thing, however, seems certain: namely, that the general principle of relativity will prove a necessary and effective tool for the solution of the problem of the total field.
11
E = MC 2
IN ORDER TO UNDERSTAND the law of the equivalence of mass and energy, we must go back to two conservation or “balance” principles which, independent of each other, held a high place in pre-relativity physics. These were the principle of the conservation of energy and the principle of the conservation of mass. The first of these, advanced by Leibnitz as long ago as the seventeenth century, was developed in the nineteenth century essentially as a corollary of a principle of mechanics.
Drawing from Dr. Einstein’s manuscript.
Consider, for example, a pendulum whose mass swings back and forth between the points A and B. At these points the mass m is higher by the amount h than it is at C, the lowest point of the path (see drawing). At C, on the other hand, the lifting height has disappeared and instead of it the mass has a velocity v. It is as though the lifting height could be converted entirely into velocity, and vice versa. The exact relation would be expressed as mgh = m/2 v 2 , with g representing the
(1) ds 2 = dx 0 2 + dy 0 2 + dz 0 2 - c 2 dt 0 2
is a measurable quantity which is independent of the special choice of the inertial system. If one introduces in this space the new coordinates x 1 , x 2 , x 3 , x 4 through a general transformation of coordinates, then the quantity ds 2 for the same pair of points has an expression of the form
(2) ds 2 = ∑g ik dx i dx k (summed for i and k from 1 to 4)
where g ik = g ki . The g ik which form a “symmetric tensor” and are continuous functions of x 1 . . . x 4 then describe according to the “principle of equivalence” a gravitational field of a special kind (namely one which can be re transformed to the form (1)). From Riemann’s investigations on metric spaces the mathematical properties of this g ik field can be given exactly (“Riemann-condition”). However, what we are looking for are the equations satisfied by “general” gravitational fields. It is natural to assume that they too can be described as tensor-fields of the type g ik , which in general do not admit a transformation to the form (1), i.e., which do not satisfy the “Riemann condition,” but weaker conditions, which, just as the Riemann condition, are independent of the choice of coordinates (ie., are generally invariant). A simple formal consideration leads to weaker conditions which are closely connected with the Riemann condition. These conditions are the very equations of the pure gravitational field (on the outside of matter and at the absence of an electromagnetic field).
These equations yield Newton’s equations of gravitational mechanics as an approximate law and in addition certain small effects which have been confirmed by observation (deflection of light by the gravitational field of a star, influence of the gravitational potential on the frequency of emitted light, slow rotation of the elliptic circuits of planets—perihelion motion of the planet Mercury). They further yield an explanation for the expanding motion of galactic systems, which is manifested by the red-shift of the light omitted from these systems.
The general theory of relativity is as yet incomplete insofar as it has been able to apply the general principle of relativity satisfactorily only to gravitational fields, but not to the total field. We do not yet know with certainty, by what mathematical mechanism the total field in space is to be described and what the general invariant laws are to which this total field is subject. One thing, however, seems certain: namely, that the general principle of relativity will prove a necessary and effective tool for the solution of the problem of the total field.
11
E = MC 2
IN ORDER TO UNDERSTAND the law of the equivalence of mass and energy, we must go back to two conservation or “balance” principles which, independent of each other, held a high place in pre-relativity physics. These were the principle of the conservation of energy and the principle of the conservation of mass. The first of these, advanced by Leibnitz as long ago as the seventeenth century, was developed in the nineteenth century essentially as a corollary of a principle of mechanics.
Drawing from Dr. Einstein’s manuscript.
Consider, for example, a pendulum whose mass swings back and forth between the points A and B. At these points the mass m is higher by the amount h than it is at C, the lowest point of the path (see drawing). At C, on the other hand, the lifting height has disappeared and instead of it the mass has a velocity v. It is as though the lifting height could be converted entirely into velocity, and vice versa. The exact relation would be expressed as mgh = m/2 v 2 , with g representing the
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