Hole argument

In general relativity, the hole argument is a "paradox" which much troubled Albert Einstein on the road to his famous field equation. It has recently been reinterpreted by philosophers as an argument against "manifold substantialism", a doctrine that views the manifold of events in spacetime as a "substance" which exists independently of the matter within it.

Einstein's hole argument

In a usual field equation, knowing the source of the field determines the field everywhere. For example, if we are given the current and charge density and appropriate boundary conditions, Maxwell's equations determine the electric and magnetic fields. They do notdetermine the vector potential though, because the vector potential depends on an arbitrary choice of gauge.

Einstein noticed that if the equations of gravity are generally covariant, then the metric cannot be uniquely determined by its sources as a function of the coordinates of spacetime. The argument is obvious: consider a gravitational source, such as the sun. Then there is some gravitational field described by a metric g(r). Now perform a coordinate transformation r-> r' where r' is the same as r for points which are inside the sun but r' is different from r outside the sun. The coordinate description of the interior of the sun is unaffected by the transformation, but the functional form of the metric for coordinate values outside the sun is changed.

This means that one source, the sun, can be the source of many seemingly different metrics. The resolution is immediate: any two fields which only differ by a coordinate transformation are physically equivalent, just as two different vector potentials which differ by a gauge transformation are equivalent. Then all these different fields are not different at all.

There are many variations on this apparent paradox. In one version, you consider an initial value surface with some data and find the metric as a function of time. Then you perform a coordinate transformation which moves points around in the future of the initial value surface, but which doesn't affect the initial surface or any points at infinity. Then you can conclude that the generally covariant field equations don't determine the future uniquely, since this new coordinate transformed metric is an equally valid solution. So the initial value problem is unsolvable in general relativity. This is also true in electrodynamics--- since you can do a gauge transformation which will only affect the vector potential tomorrow. The resolution in both cases is to use extra conditions to fix a gauge.

Meaning of Coordinate Invariance

For the philosophically inclined, there is still some subtlety. If the metric components are considered the dynamical variables of General Relativity, the condition that the equations are coordinate invariant doesn't have any content by itself. All physical theories are invariant under coordinate transformations if formulated properly. It is possible to write down Maxwell's equations in any coordinate system, and predict the future in the same way.

But in order to formulate electromagnetism in an arbitrary coordinate system, one must introduce a description of the space-time geometry which is not tied down to a special coordinate system. This description is a metric tensor at every point, or a connection which defines which nearby vectors are parallel. The mathematical object introduced, the Minkowski metric, changes form from one coordinate system to another, but it isn't part of the dynamics, it doesn't obey equations of motion. No matter what happens to the electromagnetic field, it is always the same. It acts without being acted upon.

In General Relativity, every separate local quantity which is used to describe the geometry is itself a local dynamical field, with its own equation of motion. This produces severe restrictions, because the equation of motion has to be a sensible one. It must determine the future from initial conditions, it must not have runaway instabilities for small perturbations, it must define a positive definite energy for small deviations. If one takes the point of view that coordinate invariance is trivially true, the principle of coordinate invariance simply states that the metric itself is dynamical and its equation of motion does not involve a fixed background geometry.

Einstein's resolution

In 1915, Einstein realized that the hole argument makes an assumption about the nature of spacetime, it presumes that the gravitational field as a function of the coordinate labels is physically meaningful by itself. By dropping this assumption general covariance became compatible with determinism, but now the gravitational field is only physically meaningful to the extent that it alters the trajectories of material particles. While two fields that differ by a coordinate transformation look different mathematically, after the trajectories of all the particles are relabeled in the new coordinates, their interactions are manifestly unchanged. This was the first clear statement of the principle of gauge invariance in physical law.

Einstein believed that the hole argument implies that the only meaningful definition of location and time is through matter. A point in spacetime is meaningless in itself, because the label which one gives to such a point is undetermined. Spacetime points only acquire their physical significance because matter is moving through them. In his words:

"All our spacetime verifications invariably amount to a determination of spacetime coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points." (Einstein, 1916, p.117).

He considered this the deepest insight of general relativity. When asked by reporters to summarize his theory, he said:

"People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

Implications of BI for some theories of quantum gravity

Loop quantum gravity is an approach to quantum gravity which attempts to marry the fundamental principles of classical GR with the minimal essential features of quantum mechanics and without demanding any new hypotheses. Loop quantum gravity people regard background independence as a central tenet in their approach to quantizing gravity - a classical symmetry that ought to be preserved by the quantum theory if we are to be truly quantizing geometry(=gravity). One immediate consequence is that LQG is UV-finite because small and large distances are gauge equivalent.

Perturbative string theory (as well as a number of non-perturbative developments) is not obviously background independent, because it depends onboundary conditions at infinity.

Expanded explanation of consequences of the hole argument to classical and quantum general relativity can be found at [http://members.lycos.co.uk/ianbay/] "General Relativity and Loop Quantum Gravity" by Ian Baynham.

References

* Albert Einstein, H. A. Lorentz, H. Weyl, and H. Minkowski, "The Principle of Relativity" (1916).
*Carlo Rovelli, Quantum Gravity, Published by Cambridge University Press Year=2004 ID=ISBN 0-521-83733-2
*Norton, John, [http://plato.stanford.edu/archives/spr2004/entries/spacetime-holearg The Hole Argument] , The Stanford Encyclopedia of Philosophy (Spring 2004 Edition), Edward N. Zalta (ed.)
* [http://en.wikipedia.org/wiki/User:Mihaela_Iftime#Mihaela_Iftime Iftime, Mihaela] and Stachel, John, "The Hole Argument for Covariant Theories", in GRG Springer (2006), Vol.38, No 8, 1241-1252; e-print available as [http://arxiv.org/abs/gr-qc/0512021 gr-qc/0512021]
*cite book | author=d'Inverno, Ray | title=Introducing Einstein's Relativity | location=Oxford | publisher=Oxford University Press | year=1992 | id=ISBN 0-19-859686-3 See "section 13.6".
* ``Physics Meets Philosophy at the Planck Scale" (Cambridge University Press).
* [http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&e=Long%20Term%20Visitors&cat_id=453&cat_table=3&e=Joy%20Christian%20&f=3&name=Joy%20Christian%20&resident_id=4625 Joy Christian] , "Why the Quantum Must Yield to Gravity", e-print available as [http://arxiv.org/abs/gr-qc/9810078 gr-qc/9810078] . Appears in ``Physics Meets Philosophy at the Planck Scale" (Cambridge University Press).
* Carlo Rovelli and Marcus Gaul, "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance", e-print available as [http://arxiv.org/abs/gr-qc/9910079 gr-qc/9910079] .
* Alan Macdonald, " [http://link.aip.org/link/?AJPIAS/69/223/1 Einstein's hole argument] " American Journal of Physics (Feb 2001) Vol 69, Issue 2, pp. 223-225.

See also

* Philosophy of Space and Time