General Relativity, the Hole Argument, Einstein

1. General Relativity, the Hole Argument, Einstein, Hilbert, Causality, … John D. Norton Department of History and Philosophy of

Science
University of Pittsburgh
1

2. Einstein and the Hole Argument

2

3. Chronology

1907. Einstein begins work on general relativity.
Summer 1912. The big breakthrough. Einstein teams up with Marcel
Grossmann and begins working with the Ricci-Levi-Civita calculus (“tensor
calculus”)
Summer 1913. The first sketch (“Entwurf…”) of general relativity is
completed. Its gravitational field equations are not generally covariant!
Later in 1913. Einstein formulates the hole argument to show that a generally
covariant theory would be physically uninteresting.
November 1915. Einstein hurriedly returns to general covariance and
completes the theory at the same time as Hilbert publishes on the theory.
Late 1915, early 1916. Einstein explains the failure of the hole argument
with his “point-coincidence” argument.
3

4. The Hole Argument

General
covariance
means all
transformations
are admissible.
Choose transformation that
only differs from identity in
a small region of spacetime.
(The “hole”.)
Same fields outside the hole.
Different fields inside the hole.
Determinism fails???
Transformation
generate new
solutions from old.
4

5.

5

6.

Ҥ12. Proof of the
Necessity of a Restriction on
the Choice of Coordinates.”
“… i. e. generally covariant
differential equations for the
gravitational field cannot
uniquely determine the
processes in the latter…”
6

7.

Previous p. 1066
“…We will show, however,
that this requirement [of
general covariance] must be
restricted, if we want the law
of causality to be satisfied
completely…”
7

8. Einstein in the 1950s on Causality

"...the laws of the external world were
also taken to be complete, in the
following sense: If the state of the
objects is completely given at a certain
time, then their state at any other time
is completely determined by the laws
of nature. This is just what we mean
when we speak of 'causality.' Such was
approximately the framework of the
physical thinking a hundred years ago.”
Albert Einstein, "Physics, Philosophy, and Scientific
Progress," International Congress of Surgeons,
Cleveland, Ohio, 1950; printed in Physics Today, June
2005, pp.46-48.
8

9.

9

10. The Point-Coincidence Argument

The PointCoincidence
Argument
“All our space-time verifications invariably
amount to a determination of space-time
coincidences.”
10

11.

“…a coincidence is characterized by
the identity of the co-ordinates…”
Equality of co-ordinates in
one co-ordinate system
Equality of co-ordinates in
another co-ordinate system
Not said in print,
but in letters.
Therefore nothing physical is left undetermined
by the construction of the hole argument.
11

12. David Hilbert

“Die Grundlagen der Physik”
Erste Mitteilung November 20, 1915
Zweite Mitteilung December 23 1916
12

13.

13

14.

Causality as restriction of spacetime
coordinate systems to “proper” coordinate
systems.
(Modern: 4th coordinate adapted to timelike curves.)
“…we turn to the problem of causality in
the new physics.”
“…if two world points … can stand in the relation
of cause and effect, it should not be possible to
transform them to simultaneity…” [which means
the same x4 coordinate]
Hence define:
“proper” coordinate systems,
“proper” coordinate transformations.
14

15.

“…two world points on a time[like]
line can never be transformed to the
same value of the time coordinate x4;
that is, to simultaneous.”
“So we see that the concepts of cause and
effect that ground the principle of causality
lead to no inner contradiction at all in the new
physics, as long as … we restrict ourselves
everywhere to the use of proper spacetime
coordinates.”
15

16.

Causality as determinism
= well-posed Cauchy problem
(modern expression!).
“…principle of causality…
…principal content…
…in physical theories..
…knowledge of physical magnitudes and
their time derivatives in the present
determines these magnitudes for the future
uniquely…”
Fails in general relativity.
14 potentials
governed by
10 independent equations.
16

17.

“…in the new physics of general
relativity…it is in no way possible any
more to deduce uniquely from knowledge
of physical magnitudes in the present and
past to their values in the future…”
Hilbert’s version of the hole argument.
Initial system= electron at rest for all
time, past, present and future.
Transformation smoothly sets electron
into motion for future only (and drags metric
and electromagnetic fields).
17

18.

“…the essence of the new principle of
relativity…
each individual assertion in physics
can be ascribed invariant character,
just in case it has physical meaning…
“…an assertion in physics that does not
remain invariant under every, arbitrary
transformation of the coordinate
systems, we must designate as physically
meaningless …”
“…i.e. the principle of causality holds
in this formulation:
From the knowledge of the 14 physical
potentials gmn, qs in the present, all
asssertion about them for the future follow
necessarily and uniquely, in so far as
they have physical meaning.”
18

19.

“…There are very many forms in
which physically meaningful, i.e.
invariant, assertions can be brought
into mathematical expression…”
1.
“…by means of an invariant
coordinate system…”
2.
“… there exists a coordinate
system in which the 14 potentials
gmn, qs have certain determined
values or satisfy certain
determined relations in the
future…”
19

20.

3.
“…an assertion is invariant and
thus always has physical
meaning if it remains valid in
every, arbitrary coordinate
system…”
“…example…Einstein’s momentum
energy equation of divergence
character…”
20

21. What’s Next?

Causation as
determinism
Causal
connection as
non-spacelike
connection.
Dies a slow death with the
indeterminism of quantum theory
Becomes the new notion of
causation in relativity.
• Reichenbach 1920’s: causal theory of time.
•Timelike and lightlike connected becomes
causally connected
21