## A1      Newtonian Foundations of Classical Physics

Newton published his great work “Philosophiae naturalis principia mathematica”  in 1687 ([01], English edition [02] or [03]). In this work he used Euclidean geometry to embed the previous work of the 'giants' Kepler, Galileo, Descartes and Huygens into a uniform and a comprehensive theory, which today we call Newtonian mechanics. Kepler’s laws of planetary motion, Galileo’s law of relativity of all uniform motion, Descartes’ law of conservation of momentum, Huygen’s analysis of circular motion and the motion of the heavenly and terrestrial bodies were all reduced to 3 axioms and 1 force law. In his book Newton successfully applies his theory to compute the flattening of the earth and of Jupiter, to justify the tides and much more.

The success of his mathematical ideas affected the course of human thought far beyond the natural sciences, and the following two centuries only increased this success. It was therefore even more difficult to abandon the fundamental conceptions on which Newton’s theory builds – the concepts of time, of space and of the inertial mass of a body.

Newtonian Time

The master formulated it beautifully in his book [03-408]:

“Although time, space, place and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception. And this is the source of certain preconceptions; to eliminate them it is useful to distinguish these quantities into absolute and relative, true and apparent, mathematical and common.

Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. Relative, apparent, and common time is any sensible and external measure (precise or imprecise) of duration by means of motion; such a measure – for example, an hour, a day, a month, a year – is commonly used instead of true time.”

And later [03-410]:

“In astronomy, absolute time is distinguished from relative time by the equation of common time. For natural days, which are commonly considered equal for the purpose of measuring time, are actually unequal. Astronomers correct this inequality in order to measure celestial motions on the basis of a truer time. It is possible that there is no uniform motion by which time may have an exact measure. All motions can be accelerated and retarded, but the flow of absolute time cannot be changed.”

Newton's concept of a true, absolute, mathematical time applies for all observers at all places equivalently. Time flows continuously and regularly. Its course cannot be affected by heat, acceleration or gravity. Two different observers always measure the same time interval for the same procedure (up to inaccuracies due to the incompleteness of their clocks). Atomic clocks tick in the laboratory exactly as they would in orbit. And a clock is perfect, if the first derivative of the indicated time to true time is constant, that is when the second derivative is zero.

This true and absolute time implies that the simultaneity of two events is an absolute fact. It is independent of the location or of the state of movement of the observers. Time flows the same for all, just as the sun shines the same on all, just and unjust alike.

Newtonian Space

Concerning space and location things are somewhat more complicated, although Newton likewise postulates a true and absolute space [03-408f]:

“Absolute space, of its own nature without reference to anything external, always remains homogeneous and immovable. Relative space is any movable measure or dimension of this absolute space; such a measure or dimension is determined by our senses from the situation of` the space with respect to bodies and is popularly used for immovable space, as in the case of space under the earth or in the air or in the heavens, where the dimension is determined from the situation of the spare with respect to the earth. Absolute and relative space are the same in species and in magnitude, but they do not always remain the same numerically. For example, if the earth moves, the space of our air, which in a relative sense and with respect to the earth always remains the same, will now he one part of the absolute space into which the air passes, now another part of it, and thus will be changing continually in an absolute sense.”

And somewhat later [03-410]:

“Just as the order of the parts of time is unchangeable, so, too, is the order of the parts of space. Let the parts of space move from their places, and they will move (so to speak) from themselves. For times and spaces are, as it were, the places of themselves and of all things. All things are placed in time with reference to order of succession and in space with reference to order of position. It is of the essence of spaces to be places, and for primary places to move is absurd. They are therefore absolute places, and it is only changes of position from these places that are absolute motions.”

Contrary to absolute time, to which we are helplessly subjugated, we can freely move in absolute space. Newton clearly sees that one must consider three cases:

1. Accelerated movement along a straight line: This is easily identified by the force of inertia it gives rise to.
2. Rotation relative to absolute space: This is recognizable by centrifugal force, to which it gives rise. It is this absoluteness of the rotation (rotation relative to what exactly??), which convinced Newton of the existence of absolute space. In his famous description of the bucket experiment [03-412f] he stresses that he himself performed this experiment.
3. Uniform movement along a straight line: This is characterized in that no additional forces arise. Therefore, in principle, it cannot be determined whether one is at rest in absolute space or whether one is moving with constant speed.

This leads us to the important idea of the inertial frame.

Inertial Frame of Reference

Spatial coordinate systems, which are not accelerated and which do not rotate, are called inertial frames of reference (or simply inertial frames). These are the coordinate systems, which rest in Newton's absolute space or which move uniformly therein. Such coordinate systems are suitable for describing mechanical processes without the need to introduce 'fictitious forces' (also called ‘pseudo forces’ or ‘inertial forces’).

An often found definition of an inertial frame is the following: Inertial frames are coordinate systems, which do not move relative to the fixed stars. Why is this definition, if taken as stated, useless?

Lengths, distances and angles can be measured however in arbitrary (i.e., non-inertial) coordinate systems. All observers measure the same value, just as they would measure the same length of time for a dynamic event (e.g., a roof tile falling to the road). The length of an object depends in no way on how fast it is moving.

Newtonian Mass

Each material body possesses a quantity of mass, with which the body resits to being accelerated:  F = m·a. The force needed for a given acceleration a is a direct measure of this inertial mass m of the body. Newton differentiates carefully between inertial mass and gravitational mass, and he performed his own experiments, to verify that the inertial mass and the weight of a body are always proportional to each other:

“.… I mean this quantity whenever I use the term ‘body’ or ‘mass’ in the following pages. It can always be known from a body’s weight. For - by making very accurate experiments with pendulums - I have found it to be proportional to the weight, as will be shown below. ”    [03-404]

This material quantity is, of course, independent of the movement of the body, just like it is independent of the air pressure or  temperature. The mass of a body is a constant, which is assigned to it, as long as it is not divided in any way.

The fact that inertial and gravitational mass of a body are proportional to each other (whereby the constant of proportionality depends only on the selected units and therefore could also be 1) was a fact Newton could not explain and which he even distrustfully questioned with his pendulum experiments [03-700ff]. In section G2 we will see that for Einstein the equality of the inertial and gravitational mass is no longer a fact to be explained but, rather, a fundamental axiom of a new theory.

Space, time and mass are the fundamental ideas, on which Newton developed his mechanics. All other physical quantities can be derived from these three (e.g. try it yourself for the pressure!). The fact that space, time and mass are fundamental is reflected also in the fact that the associated units (second, meter and kilogram) were until 1983 base quantities. This also explains why the pendulum, yardstick and weight stone are the insignia of the American Institute of Physics:

In the context of relativity theory all three of these basic quantities will actually turn out to be ‘relative'; little remains of Newton's absolute time and absolute space. Also the independence of the inertial mass from the reference system must be abandoned. A somewhat hard formulation would be: the basic assumptions of Newton turned out to be prejudices. To recognize this after the enormous success of Newtonian mechanics required a certain boldness!