The Law of Relativity

⁵ But it is not enough that the Euclidean (or non-Euclidean) geometry can ever be directly contradicted by experiment.

Nor could it happen that it can only agree with experiment by a violation of the principle of sufficient reason, and of that of the relativity of space.

Consider any material system.

We have to consider 2 things:

1. The “state” of the various bodies of this system such as their temperature, electric potential, etc.
2. Their position in space

This position has 2 parts:

1. The mutual distances of these bodies that define their relative positions
2. The conditions which define the absolute position of the system and its absolute orientation in space.

The law of the phenomena produced in this system will depend on:

• on the state* of these bodies, and
• on their mutual distances

But because of the relativity and the inertia of space, they will not depend on the absolute position and orientation of the system.

In other words, the state of the bodies and their mutual distances at any moment will solely depend on the state of the same bodies and on their mutual distances at the initial moment. But it will in no way depend on the absolute initial position of the system and of its absolute initial orientation.

This is what we shall call the law of relativity.

*Superysics Note: This is consistent with both Descartes’ Rule 1 of Movement and Quantum Mechanics which is also state-based. And so, we can safely replace Einstein’s Relativity with that of Poincare and use it as foudation for anti-gravity (spacetime) technologies that are impossible with Einstein.

So far I have spoken as a Euclidean geometer. But an experiment requires an interpretation on:

• the Euclidean hypothesis, and
• the non-Euclidean hypothesis*.

*Superphysics Note: Euclid’s Elements is about the 5 Elements and not a book about shapes. Therefore the universe is only Euclidean. Riemann geometry is a mix of Euclidean principles.

We have interpreted a series of experiments on the Euclidean hypothesis. These experiments thus interpreted do not violate this “law of relativity.”

We now interpret them on the non-Euclidean hypothesis. This is always possible, only the non-Euclidean distances of our different bodies in this new interpretation will not generally be the same as the Euclidean distances in the primitive interpretation.

Will our experiment interpreted in this new manner be still in agreement with our “law of relativity”?

If not, could we say that the experiment has proved the falsity of non-Euclidean geometry?

In fact, to apply the law of relativity in all its rigour, it must be applied to the entire universe.

If we were to consider only a part of the universe, and if the absolute position of this part were to vary, the distances of the other bodies of the universe would equally vary.

Their influence on the part of the universe considered might therefore increase or diminish. This might modify the laws of the phenomena which take place in it.

Hence, our law of relativity may be enunciated as follows:

The readings that we can make with our instruments at any given moment will depend only on the readings that we were able to make on the same instruments at the initial moment.

Such an enunciation is independent of all interpretation by experiments.

If the law is true in the Euclidean interpretation, it will be also true in the non-Euclidean interpretation.

I have spoken above of the data which define the position of the different bodies of the system.

I might also have spoken of those which define their velocities. I should then have to distinguish:

• the velocity with which the mutual distances of the different bodies are changing, and
• the velocities of translation and rotation of the system; that is to say, the velocities with which its absolute position and orientation are changing.

The law of relativity would have to be enunciated as follows:

Unfortunately, the law thus enunciated does not agree with experiments.

Suppose a man was in a planet with a sky that was constantly covered with thick clouds that he could never see the other stars.

On that planet he would live as if it were isolated in space.

But he would notice that it revolves, either:

• by measuring its ellipticity (which is ordinarily done by means of astronomical observations, but which could be done by purely geodesic means), or
• by repeating the experiment of Foucault’s pendulum.

This shows the absolute rotation of this planet.

From this, Newton concluded the existence of absolute space. I cannot accept this.

However that may be, the difficulty is the same for both Euclid’s geometry and for Lobatschewsky’s.

I need not therefore trouble about it further, and I have only mentioned it incidentally.

To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.