In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.
*The preceding memoir by Lorentz was not at this time known to the author.
†i.e. to the first approximation.
*We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.
*“Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated at the place under discussion.”
*The equations of the Lorentz transformation may be more simply deduced directly from the condition that in virtue of those equations the relation x2 + Y2 + z2 = c2t2 shall have as its consequence the second relation ξ2 + η2 ζ2 = c2τ2.
*That is, a body possessing spherical form when examined at rest.
*Not a pendulum-clock, which is physically a system to which the Earth belongs. This case had to be excluded.
*If, for example, X = Y = Z = L = M = O, and N ≠ O, then from reasons of symmetry it is clear that when v ohanges sign without changing its numerical value, Y′ must also change sign without changing its numerical value.
*The definition of force here given is not advantageous, as was first shown by M. Planck. It is more to the point to define force in such a way that the laws of momentum and energy assume the simplest form.
DOES THE INERTIA OF A
BODY DEPEND UPON ITS
ENERGY-CONTENT?
BY
A. EINSTEIN
Translated from “Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?” Annalen der Physik, 17, 1905.
The results of the previous investigation lead to a very interesting conclusion, which is here to be deduced.
I based that investigation on the Maxwell-Hertz equations for empty space, together with the Maxwellian expression for the electromagnetic energy of space, and in addition the principle that:—
The laws by which the states of physical systems alter are independent of the alternative, to which of two systems of coordinates, in uniform motion of parallel translation relatively to each other, these alterations of state are referred (principle of relativity).
With these principles* as my basis I deduced inter alia the following result (§ 8):—
Let a system of plane waves of light, referred to the system of coordinates (x, y, z), possess the energy l; let the direction of the ray (the wave-normal) make an angle ϕ with the axis of x of the system. If we introduce a new system of co-ordinates (ξ, η, ζ) moving in uniform parallel translation with respect to the system (x, y, z), and having its origin of co-ordinates in motion along the axis of x with the velocity v, then this quantity of light—measured in the system (ξ, η, ζ)—possesses the energy
where c denotes the velocity of light. We shall make use of this result in what follows.
Let there be a stationary body in the system (x, y, z), and let its energy—referred to the system (x, y, z)—be E0. Let the energy of the body relative to the system (ξ, η, ζ), moving as above with the velocity v, be H0.
Let this body send out, in a direction making an angle ϕ with the axis of x, plane waves of light, of energy L measured relatively to (x, y, z), and simultaneously an equal quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z). The principle of energy must apply to this process, and in fact (by the principle of relativity) with respect to both systems of co-ordinates. If we call the energy of the body after the emission of light E1 or H1 respectively, measured relatively to the system (x, y, z) or (ξ, η, ζ) respectively, then by employing the relation given above we obtain
By subtraction we obtain from these equations
The two differences of the form H – E occurring in this expression have simple physical significations. H and E are energy values of the same body referred to two systems of co-ordinates which are in motion relatively to each other, the body being at rest in one of the two systems (system (x, y, z)). Thus it is clear that the difference H – E can differ from the kinetic energy K of the body, with respect to the other system (ξ, η, ζ), only by an additive constant C, which depends on the choice of the arbitrary additive constants of the energies H and E. Thus we may place
since C does not change during the emission of light. So we have
The kinetic energy of the body with respect to (ξ, η, ζ) diminishes as a result of the emission of light, and the amount of diminution is independent of the properties of the body. Moreover, the difference K0 – K1, like the kinetic energy of the electron (§ 10), depends on the velocity.
Neglecting magnitudes of fourth and higher orders we may place
From this equation it directly follows that:—
If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that
The mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by L/9 × 1020, the energy being measured in ergs, and the mass in grammes.
It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.
If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.
*The principle of the constancy of the velocity of light is of course contained in Maxwell’s equations.
ON THE INFLUENCE OF
GRAVITATION ON THE
PROPAGATION OF LIGHT
BY
A. EINSTEIN
Translated from “Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes,” Annalen der Physik, 35, 1911.
In a memoir published four years ago* I tried to answer the question whether the propagation of light is influenced by gravitation. I return to this theme, because my previous presentation of the subject does not satisfy me, and for a stronger reason, because I now see that one of the most important consequences of my former treatment is capable of being tested experimentally. For it follows from the theory here to be brought forward, that rays of light, passing close to the sun, are deflected by its gravitational field, so that the angular distance between the sun and a fixed star appearing near to it is apparently increased by nearly a second of arc.
In the course of these reflexions further results are yielded which relate to gravitation. But as the exposition of the entire group of considerations would be rather difficult to follow, only a few quite elementary reflexions will be given in the following pages, from which the reader will readily be able to inform himself as to the suppositions of the theory and its line of thought. The relations here deduced, even if the theoretical foundation is sound, are valid only to a first approximation.
§ 1. A HYPOTHESIS AS TO THE PHYSICAL NATURE OF THE GRAVITATIONAL FIELD
In a homogeneous gravitational field (acceleration of gravity γ) let there be a stationary system of co-ordinates K, orientated so that the lines of force of the gravitational field run in the negative direction of the axis of z. In a space free of gravitational fields let there be a second system of coordinates K′, moving with uniform acceleration (γ) in the positive direction of its axis of z. To avoid unnecessary complications, let us for the present disregard the theory of relativity, and regard both systems from the customary point of view of kinematics, and the movements occurring in them from that of ordinary mechanics.
Relatively to K, as well as relatively to K′, material points which are not subjected to the action of other material points, move in keeping with the equations
For the accelerated system K′ this follows directly from Galileo’s principle, but for the system K, at rest in a
homogeneous gravitational field, from the experience that all bodies in such a field are equally and uniformly accelerated. This experience, of the equal falling of all bodies in the gravitational field, is one of the most universal which the observation of nature has yielded; but in spite of that the law has not found any place in the foundations of our edifice of the physical universe.
But we arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K′ are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system;* and it makes the equal falling of all bodies in a gravitational field seem a matter of course.
As long as we restrict ourselves to purely mechanical processes in the realm where Newton’s mechanics holds sway, we are certain of the equivalence of the systems K and K′.
But this view of ours will not have any deeper significance unless the systems K and K′ are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K′. By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field. We shall now show, first of all, from the standpoint of the ordinary theory of relativity, what degree of probability is inherent in our hypothesis.
§ 2. ON THE GRAVITATION OF ENERGY
One result yielded by the theory of relativity is that the inertia mass of a body increases with the energy it contains; if the increase of energy amounts to E, the increase in inertia mass is equal to E/c2, when c denotes the velocity of light. Now is there an increase of gravitating mass corresponding to this increase of inertia mass? If not, then a body would fall in the same gravitational field with varying acceleration according to the energy it contained. That highly satisfactory result of the theory of relativity by which the law of the conservation of mass is merged in the law of conservation of energy could not be maintained, because it would compel us to abandon the law of the conservation of mass in its old form for inertia mass, and maintain it for gravitating mass.
But this must be regarded as very improbable. On the other hand, the usual theory of relativity does not provide us with any argument from which to infer that the weight of a body depends on the energy contained in it. But we shall show that our hypothesis of the equivalence of the systems K and K′ gives us gravitation of energy as a necessary consequence.
Let the two material systems S1 and S2, provided with instruments, of measurement, be situated on the z-axis of K at the distance h from each other,* so that the gravitation potential in S2 is greater than that in S1 by γh. Let a definite quantity of energy E be emitted from S2 towards S1. Let the quantities of energy in S1 and S2 be measured by contrivances which—brought to one place in the system z and there compared—shall be perfectly alike. As to the process of this conveyance of energy by radiation we can make no a priori assertion, because we do not know the influence of the gravitational field on the radiation and the measuring instruments in S1 and S2.
But by our postulate of the equivalence of K and K′ we are able, in place of the system K in a homogeneous gravitational field, to set the gravitation-free system K′, which moves with uniform acceleration in the direction of positive z, and with the z-axis of which the material systems S1 and S2 are rigidly connected.
We judge of the process of the transference of energy by radiation from S2 to S1 from a system K0, which is to be free from acceleration. At the moment when the radiation energy E2 is emitted from S2 toward S1, let the velocity of K′ relatively to K0 be zero. The radiation will arrive at S1 when the time h/c has elapsed (to a first approximation). But at this moment the velocity of S1 relatively to K0 is γh/c = v. Therefore by the ordinary theory of relativity the radiation arriving at S1 does not possess the energy E2, but a greater energy E1, which is related to E2 to a first approximation by the equation*
FIG. 1.
By our assumption exactly the same relation holds if the same process takes place in the system K, which is not accelerated, but is provided with a gravitational field. In this case we may replace γh by the potential Φ of the gravitation vector in S2, if the arbitrary constant of Φ in S1 is equated to zero. We then have the equation
This equation expresses the law of energy for the process under observation. The energy E1 arriving at S1 is greater than the energy E2, measured by the same means, which was emitted in S2, the excess being the potential energy of the mass E2/c2 in the gravitational field. It thus proves that for the fulfilment of the principle of energy we have to ascribe to the energy E, before its emission in S2, a potential energy due to gravity, which corresponds to the gravitational mass E/c2. Our assumption of the equivalence of K and K′ thus removes the difficulty mentioned at the beginning of this paragraph which is left unsolved by the ordinary theory of relativity.
The meaning of this result is shown particularly clearly if we consider the following cycle of operations:—
1. The energy E, as measured in S2, is emitted in the form of radiation in S2 towards S1, where, by the result just obtained, the energy E(1 + γh/c2), as measured in S1, is absorbed.
2. A body W of mass M is lowered from S2 to S1, work Mγh being done in the process.
3. The energy E is transferred from S1 to the body W while W is in S1. Let the gravitational mass M be thereby changed so that it acquires the value M′.
4. Let W be again raised to S2, work M′γh being done in the process.
5. Let E be transferred from W back to S2.
The effect of this cycle is simply that S1 has undergone the increase of energy Eγh/c2, and that the quantity of energy M′γh – Mγh has been conveyed to the system in the form of mechanical work. By the principle of energy, we must therefore have
or
The increase in gravitational mass is thus equal to E/c2, and therefore equal to the increase in inertia mass as given by the theory of relativity.
The result emerges still more directly from the equivalence of the systems K and K′, according to which the gravitational mass in respect of K is exactly equal to the inertia mass in respect of K′; energy must therefore possess a gravitational mass which is equal to its inertia mass. If a mass M0 be suspended on a spring balance in the system K′, the balance will indicate the apparent weight M0γ on account of the inertia of M0. If the quantity of energy E be transferred to M0, the spring balance, by the law of the inertia of energy, will indicate (M0 + E/c2)γ. By reason of our fundamental assumption exactly the same thing must occur when the experiment is repeated in the system K, that is, in the gravitational field.
§ 3. TIME AND THE VELOCITY OF LIGHT IN THE GRAVITATIONAL FIELD
If the radiation emitted in the uniformly accelerated system K′ in S2 toward S1 had the frequency v2 relatively to the clock in S2, then, relatively to S1, at its arrival in S1 it no longer has the frequency v2 relatively to an identical clock in S1, but a greater frequency v1, such that to a first approximation
For if we again introduce the unaccelerated system of reference K0, relatively to which, at the time of the emission of light, K′ has no velocity, then S1, at the time of arrival of the radiation at S1, has, relatively to K0, the velocity γh/c, from which, by Doppler’s principle, the relation as given results immediately.
In agreement with our assumption of the equivalence of the systems K′ and K, this equation also holds for the stationary system of co-ordinates K, provided with a uniform gravitational field, if in it the transference by radiation takes
place as described. It follows, then, that a ray of light emitted in S2 with a definite gravitational potential, and possessing at its emission the frequency v2—compared with a clock in S2—will, at its arrival in S1, possess a different frequency v1—measured by an identical clock in S1. For γh we substitute the gravitational potential Φ of S2—that of S1 being taken as zero—and assume that the relation which we have deduced for the homogeneous gravitational field also holds for other forms of field. Then
This result (which by our deduction is valid to a first approximation) permits, in the first place, of the following application. Let v0 be the vibration-number of an elementary light-generator, measured by a delicate clock at the same place. Let us imagine them both at a place on the surface of the Sun (where our S2 is located). Of the light there emitted, a portion reaches the Earth (S1), where we measure the frequency of the arriving light with a clock U in all respects resembling the one just mentioned. Then by (2a),
where Φ is the (negative) difference of gravitational potential between the surface of the Sun and the Earth. Thus according to our view the spectral lines of sunlight, as compared with the corresponding spectral lines of terrestrial sources of light, must be somewhat displaced toward the red, in fact by the relative amount
If the conditions under which the solar bands arise were exactly known, this shifting would be susceptible of measurement. But as other influences (pressure, temperature) affect the position of the centres of the spectral lines, it is difficult to discover whether the inferred influence of the gravitational potential really exists.*
A Stubbornly Persistent Illusion Page 4
