Reading Help Relativity: The Special and General Theory
THE GENERAL THEORY OF RELATIVITY `
` `
` `
` SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY `
` `
` `
` The basal principle, which was the pivot of all our previous `
` considerations, was the special principle of relativity, i.e. the `
` principle of the physical relativity of all uniform motion. Let as `
` once more analyse its meaning carefully. `
` `
` It was at all times clear that, from the point of view of the idea it `
` conveys to us, every motion must be considered only as a relative `
` motion. Returning to the illustration we have frequently used of the `
` embankment and the railway carriage, we can express the fact of the `
` motion here taking place in the following two forms, both of which are `
` equally justifiable : `
` `
` (a) The carriage is in motion relative to the embankment, `
` (b) The embankment is in motion relative to the carriage. `
` `
` In (a) the embankment, in (b) the carriage, serves as the body of `
` reference in our statement of the motion taking place. If it is simply `
` a question of detecting or of describing the motion involved, it is in `
` principle immaterial to what reference-body we refer the motion. As `
` already mentioned, this is self-evident, but it must not be confused `
` with the much more comprehensive statement called "the principle of `
` relativity," which we have taken as the basis of our investigations. `
` `
` The principle we have made use of not only maintains that we may `
` equally well choose the carriage or the embankment as our `
` reference-body for the description of any event (for this, too, is `
` self-evident). Our principle rather asserts what follows : If we `
` formulate the general laws of nature as they are obtained from `
` experience, by making use of `
` `
` (a) the embankment as reference-body, `
` (b) the railway carriage as reference-body, `
` `
` then these general laws of nature (e.g. the laws of mechanics or the `
` law of the propagation of light in vacuo) have exactly the same form `
` in both cases. This can also be expressed as follows : For the `
` physical description of natural processes, neither of the reference `
` bodies K, K1 is unique (lit. " specially marked out ") as compared `
` with the other. Unlike the first, this latter statement need not of `
` necessity hold a priori; it is not contained in the conceptions of " `
` motion" and " reference-body " and derivable from them; only `
` experience can decide as to its correctness or incorrectness. `
` `
` Up to the present, however, we have by no means maintained the `
` equivalence of all bodies of reference K in connection with the `
` formulation of natural laws. Our course was more on the following `
` Iines. In the first place, we started out from the assumption that `
` there exists a reference-body K, whose condition of motion is such `
` that the Galileian law holds with respect to it : A particle left to `
` itself and sufficiently far removed from all other particles moves `
` uniformly in a straight line. With reference to K (Galileian `
` reference-body) the laws of nature were to be as simple as possible. `
` But in addition to K, all bodies of reference K1 should be given `
` preference in this sense, and they should be exactly equivalent to K `
` for the formulation of natural laws, provided that they are in a state `
` of uniform rectilinear and non-rotary motion with respect to K ; all `
` these bodies of reference are to be regarded as Galileian `
` reference-bodies. The validity of the principle of relativity was `
` assumed only for these reference-bodies, but not for others (e.g. `
` those possessing motion of a different kind). In this sense we speak `
` of the special principle of relativity, or special theory of `
` relativity. `
` `
` In contrast to this we wish to understand by the "general principle of `
` relativity" the following statement : All bodies of reference K, K1, `
` etc., are equivalent for the description of natural phenomena `
` (formulation of the general laws of nature), whatever may be their `
` state of motion. But before proceeding farther, it ought to be pointed `
` out that this formulation must be replaced later by a more abstract `
` one, for reasons which will become evident at a later stage. `
` `
` Since the introduction of the special principle of relativity has been `
` justified, every intellect which strives after generalisation must `
` feel the temptation to venture the step towards the general principle `
` of relativity. But a simple and apparently quite reliable `
` consideration seems to suggest that, for the present at any rate, `
` there is little hope of success in such an attempt; Let us imagine `
` ourselves transferred to our old friend the railway carriage, which is `
` travelling at a uniform rate. As long as it is moving unifromly, the `
` occupant of the carriage is not sensible of its motion, and it is for `
` this reason that he can without reluctance interpret the facts of the `
` case as indicating that the carriage is at rest, but the embankment in `
` motion. Moreover, according to the special principle of relativity, `
` this interpretation is quite justified also from a physical point of `
` view. `
` `
` If the motion of the carriage is now changed into a non-uniform `
` motion, as for instance by a powerful application of the brakes, then `
` the occupant of the carriage experiences a correspondingly powerful `
` jerk forwards. The retarded motion is manifested in the mechanical `
` behaviour of bodies relative to the person in the railway carriage. `
` The mechanical behaviour is different from that of the case previously `
` considered, and for this reason it would appear to be impossible that `
` the same mechanical laws hold relatively to the non-uniformly moving `
` carriage, as hold with reference to the carriage when at rest or in `
` uniform motion. At all events it is clear that the Galileian law does `
` not hold with respect to the non-uniformly moving carriage. Because of `
` this, we feel compelled at the present juncture to grant a kind of `
` absolute physical reality to non-uniform motion, in opposition to the `
` general principle of relatvity. But in what follows we shall soon see `
` that this conclusion cannot be maintained. `
` `
` `
` `
` THE GRAVITATIONAL FIELD `
` `
` `
` "If we pick up a stone and then let it go, why does it fall to the `
` ground ?" The usual answer to this question is: "Because it is `
` attracted by the earth." Modern physics formulates the answer rather `
` differently for the following reason. As a result of the more careful `
` study of electromagnetic phenomena, we have come to regard action at a `
` distance as a process impossible without the intervention of some `
` intermediary medium. If, for instance, a magnet attracts a piece of `
` iron, we cannot be content to regard this as meaning that the magnet `
` acts directly on the iron through the intermediate empty space, but we `
` are constrained to imagine -- after the manner of Faraday -- that the `
` magnet always calls into being something physically real in the space `
` around it, that something being what we call a "magnetic field." In `
` its turn this magnetic field operates on the piece of iron, so that `
` the latter strives to move towards the magnet. We shall not discuss `
` here the justification for this incidental conception, which is indeed `
` a somewhat arbitrary one. We shall only mention that with its aid `
` electromagnetic phenomena can be theoretically represented much more `
` satisfactorily than without it, and this applies particularly to the `
` transmission of electromagnetic waves. The effects of gravitation also `
` are regarded in an analogous manner. `
` `
` The action of the earth on the stone takes place indirectly. The earth `
` produces in its surrounding a gravitational field, which acts on the `
` stone and produces its motion of fall. As we know from experience, the `
` intensity of the action on a body dimishes according to a quite `
` definite law, as we proceed farther and farther away from the earth. `
` From our point of view this means : The law governing the properties `
` of the gravitational field in space must be a perfectly definite one, `
` in order correctly to represent the diminution of gravitational action `
` with the distance from operative bodies. It is something like this: `
` The body (e.g. the earth) produces a field in its immediate `
` neighbourhood directly; the intensity and direction of the field at `
` points farther removed from the body are thence determined by the law `
` which governs the properties in space of the gravitational fields `
` themselves. `
` `
` In contrast to electric and magnetic fields, the gravitational field `
` exhibits a most remarkable property, which is of fundamental `
` importance for what follows. Bodies which are moving under the sole `
` influence of a gravitational field receive an acceleration, which does `
` not in the least depend either on the material or on the physical `
` state of the body. For instance, a piece of lead and a piece of wood `
` fall in exactly the same manner in a gravitational field (in vacuo), `
` when they start off from rest or with the same initial velocity. This `
` law, which holds most accurately, can be expressed in a different form `
` in the light of the following consideration. `
` `
` According to Newton's law of motion, we have `
` `
` (Force) = (inertial mass) x (acceleration), `
` `
` where the "inertial mass" is a characteristic constant of the `
` accelerated body. If now gravitation is the cause of the acceleration, `
` we then have `
` `
` (Force) = (gravitational mass) x (intensity of the gravitational `
` field), `
` `
` where the "gravitational mass" is likewise a characteristic constant `
` for the body. From these two relations follows: `
` `
` eq. 26: file eq26.gif `
` `
` `
` If now, as we find from experience, the acceleration is to be `
` independent of the nature and the condition of the body and always the `
` same for a given gravitational field, then the ratio of the `
` gravitational to the inertial mass must likewise be the same for all `
` bodies. By a suitable choice of units we can thus make this ratio `
` equal to unity. We then have the following law: The gravitational mass `
` of a body is equal to its inertial law. `
` `
` It is true that this important law had hitherto been recorded in `
` mechanics, but it had not been interpreted. A satisfactory `
` interpretation can be obtained only if we recognise the following fact `
` : The same quality of a body manifests itself according to `
` circumstances as " inertia " or as " weight " (lit. " heaviness '). In `
` the following section we shall show to what extent this is actually `
` the case, and how this question is connected with the general `
` postulate of relativity. `
` `
` `
` `
` `
` THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS `
` AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY `
` `
` `
`
` `
` `
` SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY `
` `
` `
` The basal principle, which was the pivot of all our previous `
` considerations, was the special principle of relativity, i.e. the `
` principle of the physical relativity of all uniform motion. Let as `
` once more analyse its meaning carefully. `
` `
` It was at all times clear that, from the point of view of the idea it `
` conveys to us, every motion must be considered only as a relative `
` motion. Returning to the illustration we have frequently used of the `
` embankment and the railway carriage, we can express the fact of the `
` motion here taking place in the following two forms, both of which are `
` equally justifiable : `
` `
` (a) The carriage is in motion relative to the embankment, `
` (b) The embankment is in motion relative to the carriage. `
` `
` In (a) the embankment, in (b) the carriage, serves as the body of `
` reference in our statement of the motion taking place. If it is simply `
` a question of detecting or of describing the motion involved, it is in `
` principle immaterial to what reference-body we refer the motion. As `
` already mentioned, this is self-evident, but it must not be confused `
` with the much more comprehensive statement called "the principle of `
` relativity," which we have taken as the basis of our investigations. `
` `
` The principle we have made use of not only maintains that we may `
` equally well choose the carriage or the embankment as our `
` reference-body for the description of any event (for this, too, is `
` self-evident). Our principle rather asserts what follows : If we `
` formulate the general laws of nature as they are obtained from `
` experience, by making use of `
` `
` (a) the embankment as reference-body, `
` (b) the railway carriage as reference-body, `
` `
` then these general laws of nature (e.g. the laws of mechanics or the `
` law of the propagation of light in vacuo) have exactly the same form `
` in both cases. This can also be expressed as follows : For the `
` physical description of natural processes, neither of the reference `
` bodies K, K1 is unique (lit. " specially marked out ") as compared `
` with the other. Unlike the first, this latter statement need not of `
` necessity hold a priori; it is not contained in the conceptions of " `
` motion" and " reference-body " and derivable from them; only `
` experience can decide as to its correctness or incorrectness. `
` `
` Up to the present, however, we have by no means maintained the `
` equivalence of all bodies of reference K in connection with the `
` formulation of natural laws. Our course was more on the following `
` Iines. In the first place, we started out from the assumption that `
` there exists a reference-body K, whose condition of motion is such `
` that the Galileian law holds with respect to it : A particle left to `
` itself and sufficiently far removed from all other particles moves `
` uniformly in a straight line. With reference to K (Galileian `
` reference-body) the laws of nature were to be as simple as possible. `
` But in addition to K, all bodies of reference K1 should be given `
` preference in this sense, and they should be exactly equivalent to K `
` for the formulation of natural laws, provided that they are in a state `
` of uniform rectilinear and non-rotary motion with respect to K ; all `
` these bodies of reference are to be regarded as Galileian `
` reference-bodies. The validity of the principle of relativity was `
` assumed only for these reference-bodies, but not for others (e.g. `
` those possessing motion of a different kind). In this sense we speak `
` of the special principle of relativity, or special theory of `
` relativity. `
` `
` In contrast to this we wish to understand by the "general principle of `
` relativity" the following statement : All bodies of reference K, K1, `
` etc., are equivalent for the description of natural phenomena `
` (formulation of the general laws of nature), whatever may be their `
` state of motion. But before proceeding farther, it ought to be pointed `
` out that this formulation must be replaced later by a more abstract `
` one, for reasons which will become evident at a later stage. `
` `
` Since the introduction of the special principle of relativity has been `
` justified, every intellect which strives after generalisation must `
` feel the temptation to venture the step towards the general principle `
` of relativity. But a simple and apparently quite reliable `
` consideration seems to suggest that, for the present at any rate, `
` there is little hope of success in such an attempt; Let us imagine `
` ourselves transferred to our old friend the railway carriage, which is `
` travelling at a uniform rate. As long as it is moving unifromly, the `
` occupant of the carriage is not sensible of its motion, and it is for `
` this reason that he can without reluctance interpret the facts of the `
` case as indicating that the carriage is at rest, but the embankment in `
` motion. Moreover, according to the special principle of relativity, `
` this interpretation is quite justified also from a physical point of `
` view. `
` `
` If the motion of the carriage is now changed into a non-uniform `
` motion, as for instance by a powerful application of the brakes, then `
` the occupant of the carriage experiences a correspondingly powerful `
` jerk forwards. The retarded motion is manifested in the mechanical `
` behaviour of bodies relative to the person in the railway carriage. `
` The mechanical behaviour is different from that of the case previously `
` considered, and for this reason it would appear to be impossible that `
` the same mechanical laws hold relatively to the non-uniformly moving `
` carriage, as hold with reference to the carriage when at rest or in `
` uniform motion. At all events it is clear that the Galileian law does `
` not hold with respect to the non-uniformly moving carriage. Because of `
` this, we feel compelled at the present juncture to grant a kind of `
` absolute physical reality to non-uniform motion, in opposition to the `
` general principle of relatvity. But in what follows we shall soon see `
` that this conclusion cannot be maintained. `
` `
` `
` `
` THE GRAVITATIONAL FIELD `
` `
` `
` "If we pick up a stone and then let it go, why does it fall to the `
` ground ?" The usual answer to this question is: "Because it is `
` attracted by the earth." Modern physics formulates the answer rather `
` differently for the following reason. As a result of the more careful `
` study of electromagnetic phenomena, we have come to regard action at a `
` distance as a process impossible without the intervention of some `
` intermediary medium. If, for instance, a magnet attracts a piece of `
` iron, we cannot be content to regard this as meaning that the magnet `
` acts directly on the iron through the intermediate empty space, but we `
` are constrained to imagine -- after the manner of Faraday -- that the `
` magnet always calls into being something physically real in the space `
` around it, that something being what we call a "magnetic field." In `
` its turn this magnetic field operates on the piece of iron, so that `
` the latter strives to move towards the magnet. We shall not discuss `
` here the justification for this incidental conception, which is indeed `
` a somewhat arbitrary one. We shall only mention that with its aid `
` electromagnetic phenomena can be theoretically represented much more `
` satisfactorily than without it, and this applies particularly to the `
` transmission of electromagnetic waves. The effects of gravitation also `
` are regarded in an analogous manner. `
` `
` The action of the earth on the stone takes place indirectly. The earth `
` produces in its surrounding a gravitational field, which acts on the `
` stone and produces its motion of fall. As we know from experience, the `
` intensity of the action on a body dimishes according to a quite `
` definite law, as we proceed farther and farther away from the earth. `
` From our point of view this means : The law governing the properties `
` of the gravitational field in space must be a perfectly definite one, `
` in order correctly to represent the diminution of gravitational action `
` with the distance from operative bodies. It is something like this: `
` The body (e.g. the earth) produces a field in its immediate `
` neighbourhood directly; the intensity and direction of the field at `
` points farther removed from the body are thence determined by the law `
` which governs the properties in space of the gravitational fields `
` themselves. `
` `
` In contrast to electric and magnetic fields, the gravitational field `
` exhibits a most remarkable property, which is of fundamental `
` importance for what follows. Bodies which are moving under the sole `
` influence of a gravitational field receive an acceleration, which does `
` not in the least depend either on the material or on the physical `
` state of the body. For instance, a piece of lead and a piece of wood `
` fall in exactly the same manner in a gravitational field (in vacuo), `
` when they start off from rest or with the same initial velocity. This `
` law, which holds most accurately, can be expressed in a different form `
` in the light of the following consideration. `
` `
` According to Newton's law of motion, we have `
` `
` (Force) = (inertial mass) x (acceleration), `
` `
` where the "inertial mass" is a characteristic constant of the `
` accelerated body. If now gravitation is the cause of the acceleration, `
` we then have `
` `
` (Force) = (gravitational mass) x (intensity of the gravitational `
` field), `
` `
` where the "gravitational mass" is likewise a characteristic constant `
` for the body. From these two relations follows: `
` `
` eq. 26: file eq26.gif `
` `
` `
` If now, as we find from experience, the acceleration is to be `
` independent of the nature and the condition of the body and always the `
` same for a given gravitational field, then the ratio of the `
` gravitational to the inertial mass must likewise be the same for all `
` bodies. By a suitable choice of units we can thus make this ratio `
` equal to unity. We then have the following law: The gravitational mass `
` of a body is equal to its inertial law. `
` `
` It is true that this important law had hitherto been recorded in `
` mechanics, but it had not been interpreted. A satisfactory `
` interpretation can be obtained only if we recognise the following fact `
` : The same quality of a body manifests itself according to `
` circumstances as " inertia " or as " weight " (lit. " heaviness '). In `
` the following section we shall show to what extent this is actually `
` the case, and how this question is connected with the general `
` postulate of relativity. `
` `
` `
` `
` `
` THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS `
` AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY `
` `
` `
`