User:Ian Beynon/Hole argument

Einstein on Background independence - "Beyond my wildest expectations"
It is often said that background independence is unsubstantial or meta-physical. Niether 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".

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 $$f(x) = \cos x$$, we immediately know that $$g(y) = \cos y$$ 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 $$g_{ab}(x)$$ 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 $$g_{ab}$$ at $$P_0$$, 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 $$P_0$$ 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 $$ u_0,u_1,u_2,u_3 $$.

Figure 1



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 diffeomorphisms). 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

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 dissapears and becomes one of the actors..."

Spacetime (the stage) dissapears 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.

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 referes 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 chosing 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.

Implications of BI for some theories of quantum gravity
Loop quantum gravity is an approach to quantum gravity which attempts to marry the fundemental 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 absense 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 challange of combining qunatum mechanics with general relativity is learning how to do physics in the absence of space-time, LQG could be described as the atempt to develop background independent quantum field theories, we can still define physical theories overspace-time, but which are invariant under active diffeomorphisms.