Dear Minister, respected audience,
I have entitled my talk "from Copernicus to Einstein".
It refers to a period of time from the year 1543, in which Copernicus' great work "De Revolutionibus" appeared, up to our century.
I put Copernicus' name in the title since it makes it immediately clear to everyone what topic is being discussed here.
And the name of Einstein, whose great achievement, the Theory of Relativity, seems to me a kind of summing up.
The aim of this talk is to place the general theory of relativity,
which often appears in the contemporary consciousness as a fringe area of physics,
in its central position in relation to one of the absolutely fundamental problems of scientific research.
Actually I could have placed the beginning of the time period mentioned further back, as far as ancient Greece.
Because the first person we know of
who chose a heliocentric point of view for the interpretation of events in the starry sky
is not Copernicus but Aristarch of Samos, a Greek philosopher from the 3rd century B.C.
But Aristarch had no enduring influence on science.
In antiquity the geocentric point of view,
which was developed into a comprehensive system of planetary motion by Claudius Ptolemeus in the 2nd century A.D.,
was utterly victorious.
All the complicated movements which were observed for the planets on the astral sphere,
led Ptolemy with a lot of acumen to their following of particular paths in space which did not seem implausible to him.
The whole of the Middle Ages took the same view, the more so as the Bible seemed to support the idea of the stationary Earth.
Aristarch had been completely forgotten, so Copernicus had to reinvent the heliocentric point of view
which put the sun in the centre of all planetary orbits and ascribes to them an approximate circularity.
In addition, each planet, which now includes the earth, rotates about its own axis.
I have no wish at all to recite once more the history of the struggle between the Ptolemaic and the Copernican systems,
but rather to emphasize the epistemological foundation that was at its centre,
although this was not at all clear to the opposing parties at the time.
The epistemologists among the philosophers have always thought
about the nature of our three-dimensional view of space, as well as our view of time.
It seems to me, from the flood of literature on the subject, the Kantian view stands out,
according to which both the views are forms imprinted on the human mind which human cognition can never transcend.
No science can change anything here.
Physics is forced to accommodate its objects in space and time, whereas psychology is satisfied with the time view.
In this I have to agree with Kant completely.
But then there immediately arises, since physics has to deal with quantitative aspects of the outside world,
the question of the measurement of space and time.
Kant assumed they were both equally given in advance and here we cannot follow him today.
Since both views are continuous.
And in a continuum there is no calibrated measurement system.
This can be most simply seen in a one-dimensional example.
A chain has its own scale, one can simply number the links.
But a completely uniform, continuous thread contains no such scale.
In order to measure it, one has to lay a tape measure next to it like an old-fashioned tailor
and read its divisions against the thread.
For this reason, therefore, there is no a priori unit of time.
With space, despite its three dimensions, things are similar.
Looking at space reveals no division lines or suchlike to provide a basis for units of measure, since it is in fact continuous.
In both cases natural science, in this case physics, must provide the way of measuring,
and this is the problem that our talk will address.
The essence of spatial measurements is embodied by geometry.
The oldest geometry, a wonderful logically closed system constructed by Euclid in ancient times,
used almost exclusively by humanity up to the present day, is based on a small number of axioms
which were regarded across the millennia as self-evident and requiring no further support.
For example, there is the axiom that for a straight line there is one and only one parallel line through any point.
A particularly important proposition there is Pythagoras' theorem,
that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.
He also characterised the Euclidean geometry for himself alone.
Initially this geometry was pursued graphically or in more formal language, synthetically.
We could also say with ruler and compass.
It was a great step forward when Descartes, the great French philosopher of the 17th century, created analytical geometry.
Here the content corresponded exactly to the teachings of Euclid.
But the method is different, using calculations.
For this Descartes constructed a crossed axis system consisting of three straight lines at right angles,
and defined the three coordinates of an arbitrary point as the distances you need
to travel to reach the point in question from the zero point where the axes cross.
These distances are the coordinates of the point.
To calculate the distance from the zero point, one must find the sum of the squares of these coordinates,
which gives the square of the distance.
That is Pythagoras' theorem for three dimensions.
If one wants the distance between two other points, one must proceed correspondingly with the differences of their coordinates.
With this the whole of Euclidean geometry is determined.
While still maintaining Euclidean geometry, physics faces the question of how to select the crossed axes,
which we could also call the coordinate system.
This is a mathematical description of the conflict between Ptolemy and Copernicus.
One can see immediately that this conflict has little obvious relation to physics.
Yes, that it appears to be a conflict about words,
since one can translate any motion data from one coordinate system to another purely mathematically.
It is just a matter of taste whether one prefers this or that coordinate system.
Certainly, among the Copernicans the feeling was very lively
that their circular planetary orbits were preferable
for reasons of simplicity to the Ptolemaic system's paths combined from two circular movements.
But such reasons cannot exercise a logical constraint.
And one should not criticise Osiander, the Nuremberg scholar responsible for the publication of Copernicus' book,
for describing his teaching as a hypothesis in his foreword.
Quite early on people began to introduce more physical considerations into this conflict.
But it was particularly the opponents of Copernicus who did this.
They argued, among other things: We observe in every vehicle that travels fast, in every rotating body,
that everything which is not firmly fixed is blown away, is ejected.
How then should people and animals and other things on the earth remain where they are
if it has both rotational as well as translational movement?
It is easy for us to rebut this with today's physical understanding.
For Galileo it was more difficult, since he had to acquire the necessary insight into physics himself.
And the physical decision was not finally resolved until 1687,
when Isaac Newton derived the mathematical theory of planetary movements
from his laws of motion and the law of attraction between two bodies.
Because included was the important assumption of the Copernican frame of reference
with the Sun at its centre and the axes aligned with the fixed stars.
This fixing of the relations of this thesis was entirely adequate for Newton and his successors, but it is not exact.
The fixed stars are not invariant.
We are well aware of slow alterations in the relative positions of the fixed stars.
But the physicists are burdened by a weightier question:
What right does one have to regard such a reference system as the right one?
The corresponding question also occurred to me: Why is the unit of time which Newton used with success,
determined by the length of a day, the right one?
Newton himself felt uneasy about these questions.
He escaped this predicament by assuming an absolute spatial frame of reference and an absolute time.
But since he could not provide any evidence for them, these concepts had an indefinite, a ghostly air.
But still, in two hundred years no physicist, none of the great philosophers
who concerned themselves with this question, went beyond Newton.
Again and again they tried to hang this reference system as a geometrical construction on material entities,
whether they might be stars or even more remote, undetectable heavenly bodies.
It was not until 1885 that illumination came, namely from a young psychologist in Leipzig named Ludwig Lange.
It almost reminds us of Columbus' egg when he declares that
the reference system necessary for Newtonian mechanics and the associated time unit are completely defined
by the purpose that they are intended to fulfil, namely to provide the basis for Newton's laws of motion.
The experimenter should seek a reference system and a time unit which satisfy this.
Then he has a correct reference system, a correct time unit.
If Newtonian mechanics is valid at all, then such a system must be there to be found.
Since the Galilean principle of inertia,
which holds that in the absence of force the coordinates of a body are a linear function of time,
forms a part of Newtonian mechanics, Lang called such appropriate frames of reference inertial systems,
and an appropriate time unit inertial time.
Both expressions are now established in science.
Ludwig Lange, in my opinion, completely decided the problem of space and time measurement for Newtonian mechanics.
Inertial systems and inertial time remove the ghostly air that adhered to Newton's absolute space and absolute time.
Because they are not mathematical abstractions which must somehow be attached to bodies,
but physical realities in and of themselves, which also have a physical feel to them.
That is to say they lead the free point mass with constant speed in a straight path.
The inertia of a body which mechanics talks about becomes here the effect of a guiding field.
In addition there is, as Newton and even Galileo already knew, more than one mechanically calculated reference system.
From the equations of motion it follows purely mathematically that all reference systems are equivalent
which move with constant velocity and without rotation in relation to an inertial system.
When a physicist speaks of speed, he must therefore always state which inertial system underlies it.
His statement is only meaningful in relation to one such.
This constitutes the Galilean relativity principle as it is known in our time.
A transformation of the coordinates maps one of these equivalent reference systems to another,
a Galileo transformation, where time also plays a role.
But the time itself remains untransformed.
The measurement of time is in this sense absolute for Newtonian mechanics.
After Galileo transformations,
velocities in different reference systems differ in magnitude and direction or at least in magnitude or direction.
In the course of the 19th century, however, Newton established with ever-increasing certainty the astonishing result
that there is a speed which has the same value in all reference systems.
The speed of light, usually known as c, in empty space.
That was not compatible with the Galilean relativity principle,
however, experience also provided a relativity principle for optical and electromagnetic phenomena.
In the course of its annual motion around the Sun, the Earth adopts quite different directions of motion.
But the most exact optical and electromagnetic measurements with light sources on Earth show no sign of this.
Those experiments were a search for a preferred reference system in which the speed of light, unlike in all the others,
would have the same value in all directions.
The luminiferous ether, which was talked about so much in those days,
was nothing other than the hypothetical material basis of such a reference system.
But such a privilege reference system failed escaped detection.
This dilemma, for about two decades, constituted a focus for research in physics, as we would put it today.
The Dutch physicist and Nobel Prize winner Hendrik Antoon Lorentz was one of the leaders in this conflict.
In 1894 he showed in a famous paper that the extension of Maxwell's electrodynamics worked out by him and Joseph Larmor
could explain a large part of these experiments without difficulty, but certainly not all.
In order to accommodate all of them, in 1904 he extended this theory by recasting Newtonian mechanics
in a way that already came very close to the theory of relativity.
The difference was that he made do without any principle of relativity;
one had to add new hypotheses from case to case in order to align a new area of physics with his theory.
But with Lorentz one already finds that replacement for the Galileo transformations,
which carries the name Lorentz transformation and would be fundamental to the theory of relativity.
But with him it is nothing more than a rule for calculation.
Expedient for the solution of particular problems of physics whose extent was initially not clearly defined.
On the same basis, the French mathematician Henri Poincaré carried out some valuable groundwork.
Unlike Lorentz and Poincaré, Einstein postulated a principle of relativity from the very beginning.
That means the equivalence of all reference systems moving linearly in relation to an inertial system.
Then he began with a simultaneous analysis of the expressions.
In the whole of physics up to then this appeared as something absolute,
as it had also reflected an absolute time in Newton's concept.
And Newtonian mechanics was totally consistent with this.
It assumed the existence of rigid bodies, where light signals can be transmitted instantaneously,
without any time lag from one place to another.
In fact there are no completely rigid bodies.
For this reason it was already established practice, long before Einstein,
to transfer time signals optically or via the electric telegraph,
where the propagation time of such signals are so short in earthly conditions that they can be completely ignored in many cases.
For fundamental research however they must be taken into account,
especially if they are not limited to short propagation distances.
Einstein now defined, in conformity with this practice, two clocks at rest in the same inertial system,
separated by distance L and running synchronously,
when a light beam emitted from the first at indicated
time zero hits the second at indicated time L divided by c, the speed of light.
He further asked what the indication is that two events happen simultaneously.
And answered: When the clocks in their location have the same indicated time when they happen.
But now the principle of relativity demands that this indication applies to all inertial systems.
Now when one such moves with the assumed uniform translational motion in relation to the other,
then it is easy to show that two simultaneous events in one system would take place at different times,
perhaps at different settings of any clocks there.
The expression "simultaneous" therefore has only relative validity.
One must only state which frame of reference one is using as the basis.
So time measurement also becomes something relative.
And that also applies to spatial measurement.
For the reason that the length of a moving rod is determined by the simultaneous difference of location of its ends.
The transformation of space and time measurements from one frame of reference to another
then takes place via the aforementioned Lorentz Transformation,
which, unlike the analogous Galileo transformation, does not only relate the space coordinates, but also the time values.
But here it is no longer just an expedient rule for calculation, but a binding natural law of nature.
Time measurements and spatial measurements are thus linked with one another here.
The nicest fruit of this fundamental progress was the law published in the same year of 1905 of the inertia of energy,
expressed in the only too well known formula: energy is mass times the square of the speed of light (E=mc^2).
The name "theory of relativity", which Einstein took over at once from other authors,
has often led to the misunderstanding that this theory relativises the objective physical situation.
It makes it dependent on the point of view that an observer can choose freely.
Kant in particular occasionally expressed strong opposition to this interpretation.
The theory may attribute different durations to different reference systems with the same event.
The measurement of the same body may give different values for different reference systems,
as with its energy, its impulse, its temperature etc.,
but all these values can be unambiguously transformed from one reference frame to the other, so they are essentially the same.
Hermann Minkowski, the mathematician from Göttingen who died at an early age in 1909, introduced,
instead of the three space and one time coordinate, four coordinates of a four-dimensional continuum which he called the world.
Every event was associated with a world point.
In order to know where and when it happened in relation to a particular frame of reference,
one must cut a three-dimensional section through this continuum, this four-dimensional continuum, in reference to it.
Its coordinates indicate the location, the fourth coordinate the time point sought.
The description of an event in the framework of this world therefore contains a description for every possible frame of reference.
The mathematical advantage which the representation of natural processes the space offers
also rests on an extensive equality of all four world coordinates.
But in one point this fails.
If one asks about the calculation of the separation of two world points from the difference of their four coordinates,
one must add the squares of the three spatial coordinates to obtain the square of the separation,
but the square of the time separation should be subtracted.
So this is a modification of Pythagoras' theorem.
Certainly the difference between Euclidean geometry and the four dimensional is not very deep,
the parallel axiom for instance is valid here too.
One also speaks here of a pseudo-Euclidean geometry.
But this mathematical advantage is far from the most important in the expression of Minkowski space,
it acquires a direct physical reality to the extent that the function of the guiding field is transferred to it.
This is what holds the free point of mass on a straight world line and is just another expression for the fact
that a path in three-dimensional space is a straight line which is travelled with constant speed.
But if any forces are active, then a struggle takes place between them and the guiding field.
The body deviates from the straight world line, i.e.
its path in three dimensions becomes curved, its velocity variable.
The theory of special relativity has evoked thousands of publications.
Agreeing, enthusiastic, but also brusquely rejecting.
When the struggle was at its peak, Einstein started on its enhancement.
Two deficiencies led him to this.
First the obvious one, that this theory said nothing about gravitation,
this general attraction of masses in accordance with the Newtonian law.
Secondly, however, there was something missing,
the effect of the body on the guiding field as a reaction to the effect of the guiding field on the body.
Although everywhere else in physics we see a reaction to every action.
Driven by unease with this, Einstein created in seven years of struggle, from 1908 to 1915, the general theory of relativity.
As a starting point, he chose the law of the equality of inertial and gravitational mass.
The term mass appears twice in Newtonian physics.
Firstly, in the law of motion, mass appears as a measure of the resistance with which a body opposes acceleration by a force.
This is the inertial mass.
But secondly, it appears in Newton's law of attraction in
that this makes the gravitational force between two bodies proportional to their masses.
That is the gravitational mass.
Galileo, later Newton and many others have confirmed with ever increasing precision that the two masses are equal to each other,
but this result was not actually anchored in mechanics.
The law of the identity of the two masses was an appendix for them without real inner commitment.
They could have existed even without this law.
Einstein conversely made it into a main pillar of his theory of general relativity.
The sought-for reaction of the body on the guiding field must have been connected with this.
But this was the only lighthouse which experience offered to illuminate the path to be explored.
Apart from that, Einstein had to rely on the compass of mathematics
which could help a bit to follow the initial direction, but was far from sufficient to define the path clearly.
Finally Einstein found this path.
Not without occasional detours and deviations.
That he got through at all, I consider his greatest achievement.
But he had to make a great sacrifice right at the start, namely the abandonment of Euclidean geometry,
of three-dimensional space and thus the pseudo-Euclidean geometry of Minkowski space.
This was too rigid, not flexible enough to satisfy the requirements stated above.
In the mathematical literature Einstein found attempts at a three-dimensional non-Euclidean geometry.
In 1854 Bernhard Riemann had laid the foundations, others had extended them.
But still Einstein had to add some of the mathematics himself.
Quite apart from the generalisation to the four dimensions of the world.
The essence of this can be described thus: that Pythagoras' theorem now applies only on an infinitesimally small scale,
but no longer in general.
Similarly, there are now no parallel straight lines,
the idea of a straight line is replaced with that of a geometrical line
which, as the straightest possible connection between two points, comes as close as possible to the Euclidean straight line.
Preferred coordinate systems, corresponding to the inertial systems of the special theory of relativity,
are completely absent here in principle.
The coordinates, which one cannot do without for the treatment of mathematical problems, have no physical meaning any more.
They degenerate into names for the points in the space.
Only with the determination of the distance between two space points do those mathematical values appear
which characterise geometry.
They are variable from place to place which underlies the deviation from a Euclidean or pseudo-Euclidean geometry,
but also the greatest adaptability to a non-Euclidean or Riemann geometry.
The central point of the general theory of relativity is now formed by Einstein's field equations,
which allow these deviations to be calculated from the masses naturally present and their motion.
The sought-after reaction of the body on the guiding field is expressed in them.
But from the same deviations this theory also extracts the determinants of the gravitational field.
This double use confirms the equality of inertial and gravitational mass.
And now we can add that these field equations contain the whole of mechanics.
Against all earlier conceptions, gravitation is no longer listed among the forces
which can conflict with the effect of the guiding field, but is only an expression of this guiding itself.
A body moves without the influence of forces when it is only under the influence of gravitation.
Its world line is then a geodetic line.
Two large, and previously only externally linked areas of physics, gravitation and mechanics,
have fused here into a single unit.
That means an enormous simplification of the fundamentals of physics.
At the same time, the question about literal space and time measurement
by specification of the geometry of this space finds its most comprehensive answer.
This is where Einstein himself saw the real value of his field equations.
But here we can still ask whether a certain result of the old physics has been sacrificed for this or not.
Is, for example, the special relativity theory not invalidated
because there is no longer a preferred reference system?
A theorem of non-Euclidean geometry states
that one can introduce so-called geodetic coordinates for every point of the continuum observed,
which approximate closely, in a limited area around themselves, to play the same role as in Euclidean geometry.
The non-Euclidean character of that geometry reveals itself only in the observation of larger areas
for which the geodetic coordinates no longer fit properly.
Applied to the theory of relativity, this says that one can, in limited areas of space-time,
continue to apply the inertial system of special relativity.
And experience shows that the limits here are set so high that we will hardly ever come into conflict with them in physics.
Admittedly we have avoided those deviations from the geometry of Euclidean space
which can be observed near strong gravitational centres, e.g. the Sun.
But the inertial systems have a new feature which the general theory of relativity with the named adjustment allows to persist;
it is necessarily free of gravitation.
Mathematically speaking, gravitation has been transformed away by the transition to geodetic coordinates.
As a realisation of such an inertial system we have for example that lift, often mentioned by Einstein,
which has separated from its support and is now in free fall.
That the effect of gravity is no longer perceptible in it, is not at all new.
Inside it, the paths of a free body or a ray of light are exactly straight.
The older physics allowed marked effects for inertial systems.
So here it experiences a correction from the general theory of relativity.
The Copernican reference system, which astronomy uses for the theory of motion,
is not, despite the slightness of the changes which the theory of general relativity make to its metrics,
an inertial system in the sense of the theory of general relativity.
This is a significant difference between Ludwig Lange's view and that of all earlier physicists.
Alternatively, one could imagine an inertial system as a small meteorite flying about independently
without perceptible effects of its own weight.
The special theory of relativity is only applicable, with its theory of space and time measurement,
if there is no effect which propagates locally with a speed differing from that of light.
Because otherwise one could transmit the signals mentioned above with this effect and so define a different kind of simultaneity.
Consequently, gravitation, which Newton treats as immediate action at a distance, must also be propagated at the speed of light.
Astronomers suspected this long before the theory of relativity, but could never produce any evidence to support this view.
The general theory of relativity is in complete harmony with the special theory,
since it can be mathematically derived from Einstein's field equations
that the effects of gravity in temporally changing fields propagate at the speed of light.
For temporally unchanging gravitational fields,
however, mathematical investigations produce a good approximation to Newton's law of attraction.
Therefore Kepler's laws for planetary motion also apply as a good approximation.
Only for the close environment of the Sun and any body of a similar mass do the deviations of the geometry of this space
from the pseudo-Euclidean, derived by Schwarzschild from the field equations, become somewhat relevant.
That is the site of those three Einstein effects which,
however small their scale, deliver strong support for the general theory of relativity.
I will express myself concisely here, I mean only what is known as the perihelion motion of Mercury,
the deflection of light at the Sun and that red-shift of the spectral lines emitted by the Sun or strong gravitational centres,
which was finally observed with the so-called white dwarves, those fixed stars of normal mass but particularly small volume.
In all these examples one sees how tiny the deviations of the geometry of this space are from the pseudo-Euclidean,
even in proximity to the large bodies.
But a gigantic change in our cosmological ideas is under discussion at the moment.
As long as three-dimensional space was assumed to be Euclidean, it had to be assumed to be of infinite extent.
The border region contradicts our a priori view of space.
There is something unsatisfying about that,
because it robs us of all hope of grasping the workings of Nature as a whole for once.
Then there is no such whole.
No matter how far we manage to extend our understanding, behind the border there still lies an infinity of unknowns.
But if Minkowski space and with it our three-dimensional space has a non-Euclidean geometry,
then there is in any case the possibility of thinking of this as a closed space, perhaps a spherical space,
which is unbounded and yet of finite size.
I would bring to mind as an analogy the surface of a sphere, which has no borderlines but still only a limited area.
However you may move in the sphere space, you return to the starting point if you only go far enough.
In fact, Einstein's field equations allow this idea,
under the assumption that matter is uniformly distributed with equal density everywhere.
Such an idea is not exactly correct, but when we take a sufficiently large scale,
such as when the separation between neighbouring galactic systems appears small,
then this idea corresponds well with today's astronomical knowledge.
In 1917 Einstein developed the idea of closed space of constant spatial and temporal curvature,
on the basis of his field equations.
Still, however, the reservation has arisen against this, that such a distribution of matter would be unstable.
But in 1922 Friedmann proposed, also on the basis of Einstein's field equations,
such a space with temporally increasing radius of curvature.
And what then seemed to be pure mathematical speculation acquired the highest physical interest
when the Californian astronomer Hubble in 1928 concluded from spectral observations of the remotest observable clouds,
that these are receding from our Milky Way with velocities which increase with their remoteness from us.
One knows of speeds up to one-fifth the speed of light.
That corresponded exactly with Friedmann's theory.
Experience could also be fitted quantitatively: one arrives at,
e.g., from astronomical observations an age for today's cosmos of 4.5 billion years,
while research into radioactivity, which rests on quite another basis, results in 4.3 billion.
The agreement, given the immense extrapolations which cannot be avoided in both estimates, is fantastic. Let us summarize.
The perception of space and time is indelibly stamped on human perception.
Properties of our cognitive faculties which cannot be altered at all by any experience.
Space and time measurement, on the other hand, are products of experimental science, i.e.
from physics; the long history of this process has recently reached a conclusion in the general theory of relativity,
without any possibility of going beyond this at the moment,
and the credit for this immense progress is undoubtedly due to Albert Einstein alone.