Translation:Lorentz Contraction

by Max von Laue

Lorentz contraction

By M. v. Laue.

In the many attempts (that have to be welcomed) of the professional philosophers, to take position on the theory of relativity in the way presented by Einstein, we almost always encounter a strange misunderstanding as to the role of Lorentz contraction. It can be explained from the way by which the theory arrives at the conclusion of this contraction, and it finds its special support in the formulation of a famous lecture by Minkowski; for according to this formulation, Lorentz contraction in relativity theory is "purely a gift from above". Consequently, we would have expected a far more severe assessment of the theory especially from the philosophic side, indeed its rejection due to lack of scientific method. A "gift from above" - what is this but a new expression of "a wonder"? Fortunately, only the given formulation, not the theory itself, must be criticized. And this we want to explain more closely, to clarify the matter also at a place accessible to philosophic readers.

At first, we only have to deal with the restricted relativity theory, according to which a group of coordinate systems are equally valid for physics, which can be derived from each other by means of the known Lorentz transformation of space and time coordinates. If $K$ and $K'$ are such systems, and if a body rests in $K'$ with respect to which any dimension (parallel to the direction of motion of $K'$ relative to $K$ and given by two marks at the body) has the length $l^{0}$ with respect to $K'$, then the same dimension has a length smaller than $l^{0}$ with respect to $K$. This is given by a simple mathematical conclusion from the transformation equations. Because the conditions of the body with respect to $K$ and $K'$ are only distinguished by the velocity, which it has with respect to $K$, but not with respect to $K'$, then this means: A body brought by us into motion from a state of rest, contracts in the direction of motion. So far as the relativity principle (as it is pronounced by the Lorentz transformation) holds in reality, all bodies must show this behavior.

What is unsatisfactory for some in this way of conclusion, is the fact that it only gives a reason for its recognition, but not a cause; it doesn't follow the causal relation of reality. Before we attempt to fill this gap, we still want to emphasize that such a deviation of the line of reasoning from the causal interconnection is nothing unusual in physics. On the contrary, its value for science (especially as regards the most general natural laws) rests on the fact, that it relieves us from the effort, to follow the often less vivid causal-connections of nature in all details. For example, if we conclude from the energy principle, that the unification

$2H_{2}+O_{2}=2H_{2}O\,$

gives the same sum from heat and work, whether it happens under an explosion of gases in a calorimetric bomb or in a galvanic element connected with a supply of electric current, then we also say very little about the play of the atoms, which lead in both cases to the same energetic end result. And such examples can also be easily given in great number with respect to the second thermodynamic theorem, the theorem of the conservation of momentum etc.. Although the causal connection is not shown, the demand that it is still principally possible (although maybe only during the further progress of science) to specify it in all details, still remains with justification of course. And we have to completely uphold this demand also in the case of Lorentz contraction.

We can also fulfill this demand easily, when we become aware, what a reference systems is representing for the older physics including the restricted theory of relativity. For that, it is surely the best to deal with the question, as to how a valid system of this theory can be found. (When we have one, the Lorentz transformation gives us the other ones). Surely this can most vividly be made by a though experiment described by L. Lange. We investigate the motion of three bodies, which are not subjected to forces emanating from one point, in any reference system. If their trajectories are straight lines in this system, then this system is a valid one, an "inertial system". This system is therefore a physical object that can be established by observation, and when one is bothered by the impracticability of that thought experiment, then we simply allude to the procedures of the astronomers which sought – for the theory of planetary motion – a suitable (i.e. for the application of mechanical and gravitational laws) coordinate system from the planetary motions themselves, and which they found with great accuracy according to general conviction. And equally well as any other object that can be found by observation, it is thus physically real and can exert physical actions as their cause.

Already Newton's dynamics had to admit this view. Indeed, everything that it had to say about the centrifugal forces (or similar things) in a rotating system, tantamounts to this admission. However, here we rather want to discuss an example from electrodynamics, which (although older than the restricted relativity theory) was absorbed by it without changes.

The example be so simple as possible. Two electric point charges are resting in a valid system at first, and at this occasion they exert the known Coulomb forces upon each other. We bring them to a common velocity (invariable with respect to direction and magnitude) against the same system, at an invariable distance from one another. From the electrodynamic equations it can be easily concluded, that the force between them has changed.

What is the cause of this change? Nothing changed as regards their location and motion against each other, but only their relation to the reference system. And only this reference system can be considered as the cause for the change in force. This is not surprising after the things said above. That this conclusion wasn't drawn before the relativity theory, was only due to the fact that this reference system was previously materialized as "aether". But relativity theory has banished the idea of a corporeal aether.

Now we come back to the body, which at first rests in system $K$, but afterwards it obtains a motion against $K$. Its shape is the result of the equilibrium existing between the forces acting from atom to atom (this word is used in the widest sense, thus with the inclusion of electrons or similar things). If we set the body in motion with invariable shape, i.e. in the previous state of the atoms against each other, then these forces can be changed in the same way, as the forces between the charges in the previous example. They needed not to be of electromagnetic nature therefore. The consequence will be, that the previous shape of the body doesn't correspond to an equilibrium state, on the contrary, we must apply external forces to maintain its shape. If the force is missing, as it is assumed in the discussion about the Lorentz contraction, then the shape must vary. In which way? – this can only be given by the current reasoning when the changes of force are known. If they are of electromagnetic nature, then a theorem proven by H. A. Lorentz says, that Lorentz contraction exactly follows. If one doesn't want to introduce this assumption, one consequently knows too little about them to carry out such a conclusion independently of relativity theory. Here (at the current state of our knowledge), where the possibility is missing to follow the causal connections independently from the relativity principle, this principle intervenes in a helpful way; it teaches us that always the same contraction must occur.

The dynamics developed from the relativity principle can certainly also completely explain the causal connection now. But by that, nothing new is achieved. One only takes from it what was implicitly inserted into it previously. The difference against electrodynamics, which achieves the same independently from the relativity principle, is not a principal one though, but solely lies in the current state of physical research. We simply have much more complete and precise knowledge in the theory of electricity, than in mechanics.

Until now, we only have spoken about the restricted relativity theory. The general theory denies the existence of coordinate systems, which have by themselves the preferred state of an inertial system. The attempt by Lange should be without result, if all bodies, except the test bodies, are taken away. Just because those large masses of the fixed stars are on the whole at rest in the astronomical coordinate system, it has something in advantage before the others. Nevertheless, this only inessentially changes the view on Lorentz contraction previously presented. Because there now exists at every place of space another object, which is principally measurable and thus physically real, namely the tensor of the measuring determination with its ten components. Although they are in general also variable from place to place and moment to moment, they can still be transformed (in areas fully sufficient for a physical experiment in terms of space and time) by suitable choice of coordinates into these particularly simple values (±1 or 0), which shall dominate at every place according to the restricted relativity theory. Thus also at this place, its valid systems obtain reality, though this reality is not original but derived. But by that, the view of Lorentz contraction discussed above isn't essentially changed but only becomes more deeper.