General Relativity,the Hole Argument,

Einstein, Hilbert,

Causality, …

John D. NortonDepartment of History

and Philosophy of ScienceUniversity of Pittsburgh

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Einsteinand theHole Argument

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Chronology

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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.

The Hole Argument

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General covariance

means all transformations are admissible.

Transformation generate new solutions from old.

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???

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“§12. 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…”

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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…”

Einstein in the 1950s on Causality

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"...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.

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The Point-Coincidence Argument

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

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“…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.

David Hilbert

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

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“…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.

Causality as restriction of spacetime coordinate systems to “proper” coordinate systems.(Modern: 4th coordinate adapted to timelike curves.)

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“…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.”

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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 potentialsgoverned by10 independent equations.

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“…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).

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“…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 g, qs in the present, all asssertion about them for the future follow

necessarily and uniquely, in so far as they have physical meaning.”

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“…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 g, qs have certain determined values or satisfy certain determined relations in the future…”

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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…”

What’s Next?

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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