Einstein

1
Einsteins Methods
John D. Norton Department of History and
Philosophy of Science University of Pittsburgh
2
What can we know of how Einstein made his
discoveries?
1905 Special relativity 1905 Light quantum 1905
Atoms/Brownian motion 1906 Specific heats 1909
Wave particle duality 1907-1915 General
relativity 1916 Gravitational waves 1916 A and
B coefficients 1917 Relativistic
cosmology 1924-25 Bose Einstein statistics and
more
Inscrutable flashes of insight or methodical
exploration?
3

Michel Janssen, John Norton, Jürgen Renn, Tilman
Sauer, John Stachel, et al. General Relativity
in the Making Einstein's Zurich Notebook. Vol.
1 The Genesis of General Relativity Documents
and Interpretation. Vol. 2 The Genesis of
General Relativity. Vol. 3
Vol.4. Dordrecht Kluwer.
The view of Einsteins work on general relativity
as driven by the tension of physical and formal
ways of thinking was developed in a collaborative
research group working on Einsteins Zurich
Notebook of 1912-1913 at the Max Planck Institut
für Bildungsforschung and then at the Max Planck
Institut für Wissenschaftgeschichte.
4
Physical versus Formal
approach
approach
Considerable overlap. Often both are the same
inferences in different guises.
5
Physical approach illustrated
Based on Einsteins 1905 magnet-conductor thought
experiment.
Principle of relativity requires that the
electromagnetic field manifests as different
mixtures of magnetic field B and electric field E
according to motion of observer.
6
Formal approach illustrated
Write Maxwells equations using four-vector and
six-vector (now antisymmetric second rank tensor)
quantities and operators of Minkowskis 1908
spacetime, geometrical approach. Satisfaction of
the principle of relativity is automatic.
Frame dependence of decomposition of
electromagnetic field is a consequence of
spacetime geometry.
7
Evolution of Einsteins approaches
1902-1904 statistical physics 1905 Brownian
motion 1905 Light quantum 1905 Special
relativity 1906 Specific heats 1909 Wave particle
duality 1907-1915 General
relativity 1916 A and B coefficients 1917
Relativististic cosmology 1924-25
Bose-Einstein statistics 1935 EPR
Physical
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Einsteins early distain for higher mathematics
in physics
Special relativity, light quantum use only
calculus of many variables. Marked reluctance to
adopt Minkowskis four-dimensional methods. He
does not use them until 1912. Quip I can
hardly understant Laues book 1911 textbook on
special relativity that used Minkowskis
methods. Four-dimensional methods disparaged as
superfluous learnedness.
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Abrahams 1912 theory of gravity
Abrahams theory is the simplest mathematically
delivered by four-dimensional methods.
uict
where cc(F) Einsteins idea!
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General relativity begins to turn the tide
In 1912, Einstein began work on the precursor to
general relativity, the Entwurf theory of 1913
with the mathematical assistance of Marcel
Grossmann, who introduced Einstein to Ricci and
Levi-Civitas absolute differential calculus
(now called tensor calculus).
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Einstein and Grossmanns Entwurf 1913
Complete framework of general theory of
relativity. Gravity as curvature of spacetime
geometry. One thing is missing
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The Einstein Equations are approached
Einstein and Grossman present gravitational field
equations that are not generally covariant and
have no evident geometrical meaning.
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Einsteins Zurich Notebook
A notebook of calculation Einstein kept while he
worked on the Entwurf theory with Grossmann.
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Inside the cover
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Einstein connects gravity and curvature of
spacetime.
p. 39L
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The physical approach to energy-momentum
conservation
p.5R
17
and the formal approach to energy-momentum
conservation.
Is the conservation law
of the form
p.5R
18
The formal approach to the gravitational field
equations
Following pages Einstein shows how to select
coordinate systems so that they do vanish.
p. 14L
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Failure of the formal approach
Einstein finds multiple problems with the
gravitational field equations based on the
Riemann curvature tensor.
p. 21R
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Entwurf gravitational field equations
Derived from a purely physical approach.
Energy-momentum conservation.
pp. 26L-R
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Einsteins short-lived methodological moral of
1914
The physical approach is superior to the formal
approach.
At the moment I do not especially feel like
working, for I had to struggle horribly to
discover what I described above. The general
theory of invariants was only an impediment. The
direct route proved to be the only feasible one.
It is just difficult to understand why I had to
grope around for so long before I found what was
so near at hand.
Einstein to Besso, March 1914
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Einstein snatches triumph from near disaster
Fall 1915.

• David Hilbert in Göttingen applies formal methods
  to general field equations for Einsteins theory
• ... and Einstein knows it.
• Communications to the Göttingen Academy

Einstein realizes his Entwurf field equations
are wrong and returns to seek generally covariant
equations. Communications to the Prussian
Academy
Nov. 4 Almost generally covariant field
equations Nov. 11 Almost generally covariant
field equations Nov. 18 Explanation of Mercurys
perihelion motion Nov. 26 Einstein equations
Nov. 20 Something very close to Einsteins
equations
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Einsteins new methodological moral
Triumph of formal methods over physical
considerations.
I had already taken into consideration the only
possible generally covariant equations, which now
prove to be the right ones, three years ago with
my friend Grossmann. Only with heavy hearts did
we detach ourselves from them, since the physical
discussion had apparently shown their
incompatibility with Newton's law.
Hardly anyone who has truly understood it can
resist the charm of this theory it signifies a
real triumph of the method of the general
differential calculus, founded by Gauss, Riemann,
Christoffel, Ricci and Levi-Civita.
Einstein to Hilbert Nov 18, 1915
This time the most obvious was correct however
Grossmann and I believed that the conservation
laws would not be satisfied and that Newton's law
would not come out in the first approximation.
Communication to Prussian Academy of Nov. 4, 1915
Einstein to Besso, Dec. 10, 1915
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Hesitations
Except for the agreement with reality, it is in
any case a grand intellectual achievement.
Einstein to Hermann Weyl, Apr. 6, 1918, on Weyls
mathematically most natural, geometric
unification of gravity and electromagnetism
It seems to me that you overrate the value of
formal points of view. These may be valuable when
an already found truth needs to be formulated,
but fail always as heuristic aids.
Einstein to Felix Klein, 1917, on the conformal
invariance of Maxwells equations
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Einsteins manifesto of June 10, 1933
Herbert Spenser Lecture, "On the Methods of
Theoretical Physics," University of Oxford
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Einsteins search for unified field theory
I have learned something else from the theory of
gravitation no collection of empirical facts
however comprehensive can ever lead to the
setting up of such complicated equations as
non-linear field equations of the unified field.
A theory can be tested by experience, but there
is no way from experience to the construction of
a theory. Equations of such complexity as are the
equations of the gravitational field can be found
only through the discovery of a logically simple
mathematical condition that determines the
equations completely or almost completely. Once
one has obtained those sufficiently strong formal
conditions, one requires only little knowledge of
facts for the construction of the theory in the
case of the equations of gravitation it is the
four-dimensionality and the symmetric tensor as
expression for the structure of space that,
together with the invariance with respect to the
continuous transformation group, determine the
equations all but completely.
Autobiographical Notes, 1946
27
A concluding puzzle
"If you want to find out anything from the
theoretical physicists about the methods they
use, I advise you to stick closely to one
principle don't listen to their words, fix your
attention on their deeds."
Einsteins manifesto begins
To lessen the shock of the extraordinary view he
is about to present that so fully contradicts the
then present mainstream of philosophical thought?
Or to induce you to look at what
physicists--Einstein and others--actually do so
you come to reject the mainstream in favor of
Einsteins view?