Translation:The Principle of Relativity and the Fundamental Equations of Mechanics

by Max Planck

The Principle of Relativity and the Fundamental Equations of Mechanics;

by Max Planck.

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The "principle of relativity" recently introduced by H.A. Lorentz^[1] and in even more general form by A. Einstein^[2], states that of two reference frames (x, y, z, t) and (x ', y', z ', t') connected by the relations

${\begin{cases}x'={\frac {c}{\sqrt {c^{2}-v^{2}}}}(x-vt),\ y'=y,\ z'=z,\\\\t'={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left(t-{\frac {v}{c^{2}}}x\right)\end{cases}}$

1)

(c is the speed of light in vacuum), none can be used with greater justification than the other one as regards the fundamental equations of mechanics and electrodynamics, i.e., none of them can be called as "at rest" with greater justification than the other one. This implies (if this should be verified in general) such a great simplification of all the problems of electrodynamics of moving bodies, that the question of its admissibility deserves to be put to the forefront of any theoretical research in this field. Of course, this question already seems to be settled by the recent important measurements by W. Kaufmann^[3] in a negative sense, so any further investigation would be superfluous. However, in view of the complicated theory of these experiments I would not completely exclude the possibility, that the principle of relativity on closer elaboration might just prove compatible with the observations. Also as regards the concerns that according to the relativity principle a moving electron would be subject to a specific deformation work, I would attach no decisive importance, because in general we can add this work to the kinetic energy of the electron. However, the question of an electrodynamic explanation of inertia remains open; but instead there arises, on the other hand, the advantage that it's not necessary to ascribe to the electron neither a spherical form nor even any other form in order to arrive at a certain dependence of inertia on speed.

However that may be: a physical idea of that simplicity and universality, as contained in the relativity principle, deserves more than to be examined in a single way, and if it is incorrect it deserves to be led ad absurdum; and that can done in no better way than by exploring the consequences to which it leads. So perhaps at least from this point of view, the following investigation can provide some benefit. Therein the task is treated to determine the preferred form of the fundamental equations of mechanics, which take the place of the usual Newtonian equations of motion of a free mass point:

$m{\ddot {x}}=X,\ m{\ddot {y}}=Y,\ m{\ddot {z}}=Z,$

2)

when the relativity principle should have general validity.

According to this principle, those simple equations are only valid for a stationary point $({\dot {x}}=0,\ {\dot {y}}=0,\ {\dot {z}}=0)$. For a finite velocity of the point:

$q={\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}$

3)

they need an extension. However, for arbitrary values of q one could, simply by definition, set the quantities X, Y, Z equal to the product of mass and acceleration, and denote them as the components of the moving force, as this indeed directly happens in many descriptions of mechanics. But then the moving force, as defined, would have no independent physical meaning, in particular, its simple relation to the potential energy would be lost. For since according to the principle of relativity, also in the "primed" frame of reference the equations:

$m{\ddot {x'}}=X',\ m{\ddot {y}}'=Y',\ m{\ddot {z}}'=Z'$

should be valid in general, thus the relation between X, Y, Z and X', Y', Z' would result in very complicated equations, which one has to derive by means of valid relations between ${\ddot {x}}$ and ${\ddot {x}}'$ etc. according to 1), and that excludes a simple physical meaning of these quantities.

To learn about the general relation between acceleration and moving force, it is advisable to start with a special case in which one knows the connection between the components of the moving force in both reference frames; one such case is the effect of an electromagnetic field in vacuum on a charged mass point m with the quantity of electricity e. Thus for the electric and magnetic field strengths in both reference frames 1) we have the relations^[4]

${\begin{cases}{\mathfrak {E}}'_{x'}={\mathfrak {E}}_{x}&{\mathfrak {H}}'_{x'}={\mathfrak {H}}_{x}\\\\{\mathfrak {E}}'_{y'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {E}}_{y}-{\frac {v}{c}}{\mathfrak {H}}_{z}\right)&{\mathfrak {H}}'_{y'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {H}}_{y}+{\frac {v}{c}}{\mathfrak {E}}_{z}\right)\\\\{\mathfrak {E}}'_{z'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {E}}_{z}+{\frac {v}{c}}{\mathfrak {H}}_{y}\right)&{\mathfrak {H}}'_{z'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {H}}_{z}-{\frac {v}{c}}{\mathfrak {E}}_{y}\right)\end{cases}}$

4)

We imagine that the particle is located at the origin of the coordinates of the "unprimed" system (x, y, z, t) and has the velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$ with respect to this system, and ask for the equations of motion. This question can be answered unequivocally by the fact that we first think of the mass points as the origin of a new reference frame, which moves against the original frame with the constant velocity components ${\dot {x}},{\dot {y}},{\dot {z}}$. The x-axis of this system may coincide with the direction of the velocity q of the mass point, whose size is expressed by 3). Then the mass point is at rest in the new reference frame and in this frame the equations of motion in the simple form 2) apply to it, and for the moving force the product of the electric charge e and the electric field strength has to be used. Now we transform the equations of motion to a second reference frame whose x-axis again coincides with the direction of velocity q, but which is now at rest in frame (x, y, z, t). For this we use on the one hand as regards the acceleration components, the relations 1), on the other hand as regards to the force components, the relations 4) by setting everywhere q in place of v. Finally, by a simple rotation of the coordinate axes we can enter the system (x, y, z, t), and by performing all these elementary calculations, we obtain the equations of motion in the form:

${\begin{cases}{\frac {m{\ddot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}=e{\mathfrak {E}}_{x}-{\frac {e{\dot {x}}}{c^{2}}}({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z})\\\\+{\frac {e}{c}}\left({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y}\right)\ {\mathsf {etc}}.\end{cases}}$

5)

From the general admissibility of these three equations one can convince himself afterwards directly by the consideration that the equations have to remain true as regards the principle of relativity, if the primed quantities instead of the unprimed quantities are written throughout, and the constants c, e and m remain the same. In fact, this is confirmed in a very general way in consequence of the result of relations 1) and 4), for any value of v.

Now we want to bring the equations of motion to its simplest form. If we multiply them respectively by ${\dot {x}},{\dot {y}},{\dot {z}}$ and sum up, then it follows:

$e({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z})={\frac {m({\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}})}{\left(1-{\frac {q^{2}}{c^{2}}}\right)^{\frac {3}{2}}}},$

and this results substituted in 5) gives, if one also puts:

$e{\mathfrak {E}}_{x}+{\frac {e}{c}}({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y})=X,\ {\mathsf {etc}}.$ ${\frac {d}{dt}}\left\{{\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}\right\}=X,\ {\mathsf {etc}}.$

6)

These equations contain the solution of the problem, they form that generalization of Newton's equations of motion 2), which is required by the principle of relativity.

Comparing them with the Lagrangian equations of motion:

${\frac {d}{dt}}\left({\frac {\partial H}{\partial {\dot {x}}}}\right)=X,\ {\mathsf {etc}}.$

7)

where H represents the kinetic potential, we obtain:

$H=-mc^{2}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}+const$

8)

The expression for the kinetic force is obtained, if the equations 7) are respectively multiplied and summed up by ${\dot {x}}\ dt,{\dot {y}}\ dt,{\dot {z}}\ dt$. Then it follows:

$d\left({\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H\right)=X\ dx+Y\ dy+Z\ dz,$

and from this relation the expression of the kinetic force L of the mass point is developed:

$L={\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H={\frac {mc^{2}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}+const.$

The equations of motion 7) can also be represented in the form of the Hamiltonian principle:

$\int _{t_{0}}^{t_{1}}(\delta H+A)dt=0,$

whereby the time t, as well as the start and end position, remain unchanged and the virtual work is designated by A:

$A=X\delta x+Y\delta y+Z\delta z.$

At last we set up the Hamiltonian canonical equations of motion. For this task, the "momentum coordinates" ξ, η, ζ are introduced, where:

$\xi ={\frac {\partial H}{\partial {\dot {x}}}}={\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$, etc.

If we now consider the kinetic force L as a function of ξ, η, ζ, and if we set the abbreviation: $\xi ^{2}+\eta ^{2}+\zeta ^{2}=\varrho ^{2}$, it follows:

$L=mc^{2}{\sqrt {1+{\frac {\varrho ^{2}}{m^{2}c^{2}}}}}+const$

and the Hamiltonian equations of motion become:

${\begin{matrix}{\frac {d\xi }{dt}}&=&X,\quad &{\frac {d\eta }{dt}}&=&Y,\quad &{\frac {d\zeta }{dt}}&=&Z,\\\\{\frac {dx}{dt}}&=&{\frac {\partial L}{\partial \xi }},\quad &{\frac {dy}{dt}}&=&{\frac {\partial L}{\partial \eta }},\quad &{\frac {dz}{dt}}&=&{\frac {\partial L}{\partial \zeta }}.\end{matrix}},$

All these relations are valid for the reference system (x, y, z, t) used here, as well as for any other reference system (x', y', z', t'), which is connected with it by equations 1).

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1. H. A. Lorentz, Versl. Kon. Akad. v. Wet. Amsterdam 1904, S. 809.
2. A. Einstein, Ann. d. Phys. (4) 17, 891, 1905.
3. W. Kaufmann, Sitzungsber. d. preuß. Akad. d. Wiss. 1905, S. 949; Ann. d. Phys. (4) 19, 487, 1906.
4. A. Einstein, l. c., S. 909.