Hole argument

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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.

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[edit] 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 not 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 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.

[edit] 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.

[edit] Einstein's resolution

The hole argument was resolved in 1915. Einstein had realized that there was a mistaken assumption about the nature of spacetime and in dropping this assumption there would no longer be any incompatibility between general covariance and determinism. To understand this let us see how the hole argument was resolved.

The idea was to define locations using physical objects, for example particles. (The lesson to be learned from the hole argument doesn't depend on whether or not the physical objects affect the gravitation field or not. The important point is that physical objects move along geodesics). For simplicity we consider only test particles. Consider the arrangement in figure 3. We have four particles labeled by A, B, C and D. The particles A and B intersect in i and similarly the particles C and D intersect in j. These particles start at the initial surface and their geodesics are found by solving the equations of motion. After we have performed the active diffeomorphism we need to solve to find the geodesics for the new metric \tilde{g}_{ab}. The distances between such defined locations is deterministic. This is because the trajectories are dragged across together with the metric by the active diffeomorphism. This is because we solve to find the geodesics for the transformed metric. A deterministic quantity is the distance between the two particles in figure 13. So physical geometry is invariably define with respect to matter degrees of freedom (or in principle using degrees of freedom of the gravitational field itself).

What Einstein construed from the solution of the hole argument is that it is meaningful to refer to a location as a place where two freely falling particles intersect; however it is not meanigful to refer to a location as a point in spacetime (a spacetime event) because the distance from one such point to another is in undetermined in GR. That is spacetime points have, in themselves, no physical significance. In Einstein's own 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).

[edit] Einstein on Background independence - "Beyond my wildest expectations"

It is often said that background independence is unsubstantial or meta-physical. Neither of these statements are true. Background independence was introduced by Einstein himself in 1915/16 as the resolution to his "hole argument". It was only when the hole argument was finally resolved that GR was born and in fact this resolution, background independence, is what Einstein was referring to when he made his remark "beyond my wildest expectations".

[edit] The hole argument

Below is given an easy argument which uses only the very basics of GR making it accessible to anyone, and also rather difficult to dismiss. In 1912, while developing general relativity, Einstein realised something he found rather alarming. Here is one version of the argument. It begins with an utterly straightforward mathematical observation. Here is written the SHO differential equation twice

Eq(1) {d^2 f(x) \over dx^2} + f(x) = 0

Eq(2) {d^2 g(y) \over dy^2} + g(y) = 0

except in Eq(1) the independent variable is x and in Eq(2) the independent variable is y. Once we find out that a solution to Eq(1) is f(x) = cosx, we immediately know that g(y) = cosy solves Eq(2). This observation combined with general covariance has profound implications for GR.

Assume pure gravity first. Say we have two coordinate systems, x-coordinates and y-coordinates. [[General covariance]] demands the equations of motion have the same form in both coordinate systems, that is, we have exactly the same differential equation to solve in both coordinate systems except in one the independent variable is x and in the other the independent variable is y. Once we find a metric function gab(x) that solves the EQM in the x-coordinates we immediately know (by exactly the same reasoning as above!) that the same function written as a function of y solves the EOM in the y-coordinates. As both metric functions have the same functional form but belong to different coordinate systems, they impose different spacetime geometries. Thus we have generated a second distinct solution! Now comes the problem. Say the two coordinate systems coincide at first, but at some point after t = 0 we allow them to differ. We then have two solutions, they both have the same initial conditions yet they impose different spacetime geometries. The conclusion is that GR does not determine the proper-time between spacetime points! The argument I have given (or rather a refinement of it) is what's known as Einstein's hole argument. It is straightforward to include matter - we have a larger set of differential equations but they still have the same form in all coordinates systems, the same argument applies and again we obtain two solutions with the same initial conditions which impose different spacetime geometries.

At first sight this doesn't look like good news, Einstein himself was fairly alarmed. In 1912 he published a paper entitled "Towards a theory of gravitation" in which he claims we should abandon general covariance! Before we can go on to the resolution we need to better understand these extra solutions.

It is very important to note that we could not have generated these extra distinct solutions if spacetime were fixed and non-dynamical, and so the resolution (background independence) only comes about when we allow spacetime to be dynamical. We can interpret these extra distinct solutions as follows. For simplicity we first assume there is no matter. Define a metric function \tilde{g}_{ab} whose value at P is given by the value of gab at P0, i.e.

Eq(3) \tilde{g}_{ab}(P) = g_{ab}(P_0).

Now consider a coordinate system which assigns to P the same coordinate values that P0 has in the x-coordinates. We then have

Eq(4) \tilde{g}_{ab} (y_0=u_0,y_1=u_1, y_2=u_2, y_3=u_3) = g_{ab} (x_0=u_0,x_1=u_1, x_2=u_2 , x_3=u_3),

where u0,u1,u2,u3.

Figure 1

Image:Activediffwiki.jpg

When we allow the coordinate values to range over all permissible values, Eq(3) is precisely the condition that the two metric functions have the same functional form! We see that the new solution is generated by dragging the original metric function over the spacetime manifold while keeping the coordinate lines "attached" , see Fig 1. It is important to realise that we are not performing a coordinate transformation here, this is what's known as an active diffeomorphsm (coordinate transformations are called [[passive diffeomorphism]]s). It should be easy to see that when we have matter present, simultaneously performing an active diffeomorphism on the gravitational and matter fields generates the new distinct solution.

It was only in 1915 when Einstein finally resolved the hole argument that GR was born. The resolution (mainly taken from [1]) is as follows. As GR does not determine the distance between spacetime points, how the gravitational and matter fields are located over spacetime, and so the values they take at spacetime points, can have no physical meaning. What GR does determine are the mutual relations that exist between the gravitational field and the matter fields (i.e. the value the gravitational field takes where the matter field takes such and such value). From these mutual relations we can form a notion of matter being located with respect to the gravitational field and vice-versa, (see [1] for exposition). What Einstein discovered was that physical entities are located with respect to one another only and not with respect to the spacetime manifold. This is what background independence is! And what Einstein was referring to when he made his remark "beyond my wildest expectations".

DYNAMICAL SPACETIME + DETERMINISM + GENERAL COORDINATE INVARIANCE \Rightarrow BACKGROUND INDEPENDENCE

General coordinate invariance says that a system does not care which coordinate system you use to describe it and determinism is taken for granted, with these assumed:

DYNAMICAL SPACETIME \Rightarrow BACKGROUND INDEPENDENCE

[edit] A farewell to spacetime

Since the Hole Argument is a direct consequence of the general covariance of GR, this led Einstein to state:

"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion..." (Einstein, 1916, p.117).


It is very tempting to expect that any particular observer's reference frame would carry with it its own notion of length just as an inertial observer does in SR. However, it turns out not to be the case for GR as a result of the metric being dymanical and so not fixed. We learnt from SR that position and motion only have meaning relative to an inertial frame; GR teaches us that there are no background geometric reference systems at all, position and motion have become completely relative!


Recall the saying

"...The stage disappears and becomes one of the actors..."

Spacetime (the stage) disappears and what remains is the gravitational field, which is one field out of a collection of fields, that is, the stage becomes one of the actors! This saying isn't a metaphor for dynamical spacetime in itself but rather a metaphor for the feature that a dynamical theory of spacetime is background independence.

[edit] Common Misunderstandings

Often the general relativist will use terms which have a different meaning to many other people in the physics community, leading to much confusion.

When a general relatvist refers to diffeomorphisms they are most likely referring to active diffeommorphisms and not passive diffeomrophisms (if they are using the coordinate-free geometry formulism then the only diffeomorphisms are active diffeomorphisms!)

When it is said that GR is invariant under diffeomorphisms, it is meant that the theory is invaraiant under active diffeomorphisms. These are the gauge transformations of GR and they should not be confused with the freedom of choosing coordinates on the space-time M. Invariance under coordinate transformations is not a special feature of GR, all physical theories are invaraint under coordinate transformations!

It is sometimes stated that an active diffeomorphism is just a coordinate transformation viewed differently. This is misleading, consider a non-uniform translation in Minkowski spacetime. Under a passive transformation the resulting spacetime is, of course, still Minkowski but under the active transformation the resulting spacetime is no longer Minkowski. (Under a uniform translation the active transformation results in Minkowski spacetime but this is only because of the homogeneity of Minkowski spacetime).

People should be aware of the differing use of the term general covariance. The principle is defined as the condition that the equations of motion should take the same form in all coordinate systems. However, when a general relatvist says that GR is a generally covariant theory they are not emphasing that it is invaraint under general coordinate transformations but rather that the theory is background independent as a direct consequence of coordinate invariance.

[edit] 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 essntial 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. A less immediate consequence is that the theory can be formulated at a level of rigour of mathematical physics, which is invaluable in the absence of experimental guidance.

Other Background independent theories of quantum gravity are [[dynamical triangulations]] and non-commutative geometry.

Perturbative string theory (as well as a number of non-perturbative developments) is not background independent, the scattering matrix they calculate is not invariant under active diffeomorphisms.

Quantisation is the problem of deriving the mathematical framework of a quantum mechanical system from the mathematical framework of the corresponding classical mechanical system. Quantum system exists in the absence of perturtbation theory. Perturbation theory is just one approximation scheme. So that perturbation theory breaks down does not necessarily imply any incompatabilty between quantum mechanics and general relativity! Loop quantum gravity people, for example, would claim that the challenge of combining quantum mechanics with general relativity is learning how to do physics in the absence of space-time, LQG could be described as the attempt to develop background independent quantum field theories, we can still define physical theories overspace-time, but which are invariant under active diffeomorphisms.

Expanded explanation of consequences of the hole argument to classical and quantum general relativity can be found at [1] "General Relativity and Loop Quantum Gravity" by Ian Baynham.

[edit] 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, The Hole Argument, The Stanford Encyclopedia of Philosophy (Spring 2004 Edition), Edward N. Zalta (ed.)
  • 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 gr-qc/0512021
  • d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0-19-859686-3.  See section 13.6.
  • ``Physics Meets Philosophy at the Planck Scale (Cambridge University Press).
  • Joy Christian, Why the Quantum Must Yield to Gravity, e-print available as 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 gr-qc/9910079.
  • Alan Macdonald, Einstein's hole argument American Journal of Physics (Feb 2001) Vol 69, Issue 2, pp. 223-225.

[edit] See also

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