Hole argument
Encyclopedia
In general relativity
, the hole argument is a "paradox" which much troubled Albert Einstein
while developing his famous field equation.
It is incorrectly interpreted by some philosophers as an argument
against manifold
substantialism, a doctrine that the manifold
of events in spacetime
are a "substance" which exists independently of the matter within it. Physicists disagree with this interpretation, and view the argument as a confusion about gauge invariance and gauge fixing
instead.
determine 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 determined uniquely 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.
, 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 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:
He considered this the deepest insight of general relativity. When asked by reporters to summarize his theory, he said:
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. However, it has been suggested that
loop quantum gravity violates background dependence by introducing a preferred frame of reference (`spin foams').
Perturbative string theory
(in addition to a number of non-perturbative
formulations) is not `obviously' background independent, because it depends on
boundary conditions at infinity, similarly to how perturbative general relativity is not `obviously' background dependent. However some sectors of string theory admit formulations in which background independence is manifest, including most notably the AdS/CFT. It is believed that string theory is background independent in general, even if many useful formulations do not make it manifest.
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.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the hole argument is a "paradox" which much troubled Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
while developing his famous field equation.
It is incorrectly interpreted by some philosophers as an argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
against manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
substantialism, a doctrine that the manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
of events in spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
are a "substance" which exists independently of the matter within it. Physicists disagree with this interpretation, and view the argument as a confusion about gauge invariance and gauge fixing
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
instead.
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 determined uniquely 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 RelativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, 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 background independence for some theories of quantum gravity
Loop quantum gravityLoop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
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. However, it has been suggested that
loop quantum gravity violates background dependence by introducing a preferred frame of reference (`spin foams').
Perturbative string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
(in addition to a number of non-perturbative
formulations) is not `obviously' background independent, because it depends on
boundary conditions at infinity, similarly to how perturbative general relativity is not `obviously' background dependent. However some sectors of string theory admit formulations in which background independence is manifest, including most notably the AdS/CFT. It is believed that string theory is background independent in general, even if many useful formulations do not make it manifest.
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.