Background independence
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Background independence is a condition in theoretical physics, especially in quantum gravity, that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. The different configurations (or backgrounds) should be obtained as different solutions of the underlying equations.
Although physics of string theory can be showed to be background-independent - there exists one string theory only, and this theory has many solutions - the current formulations of this theory don't make this independence manifest because they usually require the physicists to start with a particular solution; i.e. a concrete background. A very different approach to quantum gravity called loop quantum gravity is, at least formally, manifestly background-independent. However physics of loop quantum gravity is not quite background-independent. For example, it requires us to choose a topology of the space that can't be changed. Also, loop quantum gravity seems to violate Lorentz invariance.
The classical background-independent approach to string theory is [[string field theory]]. Although string field theory has been useful to understand tachyon condensation, most string theorists believe that it will never be useful to understand non-perturbative physics of string theory.
The below is an incomplete draft version - work in progress - doesn't include
all the references.
Contents |
The origin of background independence (1912-1916 Einstein's hole argument)
In 1912, while developing general relativity, Einstein realised something he found rather alarming. It was only when this problem was finally resolved in 1915/16 that GR was born and its resolution is what Einstein referred to when he made his remark "beyond my wildest expectations". 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. It starts with an utterly straightforward mathematical observation. Here is written the SHO differential equation twice
Eq(1) <math> {d^2 f(x) \over dx^2} + f(x) = 0 </math>
Eq(2) <math> {d^2 g(y) \over dy^2} + g(y) = 0 </math>
except in Eq(1) the independent variable is x and in Eq(2) the independent variable is <math>y</math>. Once we find out that a solution to Eq(1) is <math>f(x) = \cos x</math>, we immediately know that <math>g(y) = \cos y</math> solves Eq(2). This observation combined with general covariance has profound implications for GR.
Assume pure gravity first. Say we have two coordinate systems, <math>x</math>-coordinates and <math>y</math>-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 <math>x</math> and in the other the independent variable is <math>y</math>. Once we find a metric function <math>g_{ab}(x)</math> that solves the EQM in the <math>x</math>-coordinates we immediately know (by exactly the same reasoning as above!) that the same function written as a function of <math>y</math> solves the EOM in the <math>y</math>-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 <math>t=0</math> 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 <math>\tilde{g}_{ab}</math> whose value at <math>P</math> is given by the value of <math>g_{ab}</math> at <math>P_0</math>, i.e.
Eq(3) <math>\tilde{g}_{ab}(P) = g_{ab}(P_0)</math>.
Now consider a coordinate system which assigns to <math>P</math> the same coordinate values that <math>P_0</math> has in the x-coordinates. We then have
Eq(4) <math> \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), </math>
where <math> u_0,u_1,u_2,u_3 </math>.
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" (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 Rovelli's book) 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
<math> \Rightarrow</math> 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 to be given, we can say:
DYNAMICAL SPACETIME <math> \Rightarrow</math> BACKGROUND INDEPENDENCE
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).
Equivalence of all observers
It is helpful to disscus some analogies regarding Maxwell's equations and special relativity.
Recall the principle of special relativity that all inertial observers are equivalent. This states that we cannot perform an experiment in our own intertial frame (internally) which could tell it apart from any other inertial frame. This is equivalent to saying that all inertial frames must share the same space of solutions.
To better understand this we consider the situation with regards to Maxwell's equations before the advent special relativity, when it was thought that Maxwell's equations only held in the frame at rest with respect to the eather and if you wanted to know the electrodynamical field in any other inertial frame you would do so through a Galilean transformation.
Say we have the, x-inertial, frame which is at rest with respect to the (then believed in) eather and a second, y-interial, frame in uniform motion with respect to the eather. In the x-frame a charge sitting at the origin has the solution
<math>E = q^2 / \vec{x}^2</math> and <math>B =0</math>
to Maxwell's equations. Now, say instead we have a charged particle sitting at the origin of the y-inertial frame, what is the electric and magnetic field it produces? We solve Maxwell's equations in the x-frame and find a non-spherically symmetric electric field with a non-zero magnetic field. Now what if you are in the y-frame moving along with the elctron? According to the belief of the day, you would see the same electric and magnetic field as the observer in the x-frame does (except they wound not be in motion with respect your frame).
So say you are in a box and wanted to find out if you are in motion with respect to the eather, what you could do is bring with you charged particle and measure its electromagnetic field!
In general, if you have any solution to Maxwell's equations which is not a solution in a frame moving with respect to the eather you would be able to device experiments to tell if your frame was in motion. Michelson Morley devised ... but they got a negative result!
Einstein said that Maxwell's equations hold in all inertial frames! If
Eq(5) <math>E = q^2 / \vec{x}^2</math>
is a solution in one x-frame then
Eq(6) <math>E = q^2 / \vec{y}^2</math>
in <math>y</math>-frame. Take any solution in the <math>x</math>-frame and write it as a function of y instead, then this must also be a (distinct) solution to Maxwell's equation! If this wasn't you would be able to device experiments to tell your interial frame apart from others!
In particular, if an electrodynamical wave travelling at speed <math>c</math> is a solution to Maxwell's equations in one inertial frame then it must be a solution in all inertial frames. Of course, this was used by Einstein to derive the Lorentz transformation equations.
Einsten generalised the principle of special relativity (partly motivated by the principle that gravity can be removed by going to a frame in free-fall) to hold for all observers: The principle of general relativity states that all observers are equivalent. This is equivalent to saying that all reference frames (coordinate systems) must share the same space of solutions. If not all reference frames (coordinate systems) shared the same space of solutions then one could perform experiments to tell apart one reference frame (coordinate system) from another.
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, the situation is different for GR because the metric is not fixed and non-dymanic.
There should not be a preferred reference frame which has a physical role to play:
Often we hear the argument that general covariance has no content because any theory can be written in a coordinate independent way, and as such this principle cannot take you to the general relativistic equations of motion. But a similar argument apllies to special relativity: it is easy to show that Newton's equations of motion can be written in a Lorentz covariant form. As a consequence the requirement of Lorentz covariance cannot take you to the special relativistic genralization of Newton's equations. What really makes the difference between Newtons' theory and special relativity is that in Newton's theory there is a preffered inertial frame with a physical role to play (i.e. the frame at rest with repect to an eather) whereas in special relativity there isn't!
In special relativity, if we were to introduce a preffered inertial frame with a physical role to play this would reintroduce the eather to some interial observers. In general relativity, if there were a preffered coordinate system with a physical role to play, this would reintroduce gravity as a force to some observers and we would loose the equivalence principle! (see page ?? of Penrose's new book)
Observables in background independent theories
Any quantity whose defintion is dependent on a coordinate system cannot be an observable in GR. For example consider a surface which is defined by a set of points in some coordinate system, the area of the surface is given by
<math> A = \int_\Sigma dS.</math>
This quantity may be is invariant under coordinate transformations, however, it is not invariant under an active diffeomorphisms because under such a transformation the surface stays where it is while the metric gets dragged across the manifold, the new metric imposes a different spacetime geometry and as so asigns a different area to the surface.
The area of a surface defined by a physical object, such as a table, is an observable. Under an active transformation the surface's world sheet gets dragged across together with the metric. To see this, note that the world sheet of the table is found by calculating the geodesics of the particles making up the table, under an active diffeomorphism we obtain a new metric and the geodesics of the particles must once again be solved for this new metric.
The coordinate time and spatial coordinates can in principle be discarded from the formulation of the theory without loss of physical content, because results of real gravitational experiments are always expressed in coordinate-free form.
What we learn from GR is that coordinates do not have any meaning independent of observations, invloving clocks and light signals. Clocks and other reference objects are concrete physical objects also in generally covariant theories. What is predicted by GR are relationships between mearsurable quantities.
GPS coordinates - they represent directly measurable quantities.
Clocks are complicated functions of the gravitational field. Up to recently the
only know observerables were known for aymptotically flat spacetimes, where
active diffeomorphisms reduce to Lorentz transformations. B. Dittrich has,
extending Rovelli's partial and complete observables, developed explicite
expressions for Hamiltonian Constrained Systems [6], including general
relativity [7].
Observerables from the Master constraint method, [8]
Rovelli et al are trying to recover low energy physics from the full non-perturbative quantum theory by setting up the formulism for calculating and interpreting background indepenent scattering amplitudes.
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. However, besdie the theories achievements, a majour (and potenially fatal) open problem in LQG is to demostrate that it has the correct semi-classical limit.
Relating to the theories inability, so far, of recovering the low energy physics, the theroy lacks the formulism to calculate scattering amplitudes. However, recently (end of 2005) Rovelli et al have made significant progress in to putting together the formulism to calculate background independent scattering amplitudes (this is no easy task!) and Rovelli has obtained Newton's law from the fully non-perturbative quantum theory. However, it is still early days and the work is explorative with many not so weak assumptions and approximations, so this result is not yet convincing established.
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.
More on background independent theories can be found in Smolin's paper [11].
Background independent formulations of string theory
I think at some point Witten thought the best way to arrive at a background independent theory of string theory was with twister theory; in twister theory spacetime itself is secondary entity, constructed from twistor space. (cant remember reference though)
Common Misunderstandings
Often the general relatvist will use terms which have a different meaning to many people in the rest of 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. Very often 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.
Even though the full content of GR is that it discards the very notion of space-time, a general relatvist may continue to use the terms "space" and "time" but it should be understood that they do so only to indicate certain aspects of the gravitational field.
Notes
understand the idea, compare Eq(1) and Eq(2) to Einstein's vacuum field equations:
Eq(6) <math> R_{ab} ( {\partial^2 \over \partial x_e \partial x_f} g_{cd}(x) ,{\partial g_{cd}(x) \over \partial x_e}, g_{cd} (x) ) = 0 </math>
Eq(7) <math> R_{ab} ( {\partial \over \partial y_e \partial y_f} \tilde{g}_{cd} (y) ,{\partial \tilde{g}_{cd} (y) \over \partial y_e} , \tilde{g}_{cd} (y) ) = 0 </math>
(It should be noted that for the purposes of the argument above, the precise form of the EQM is unimportant. The argument only rests on the EQM being the same for all observers).
"The stage dissapears and becomes one of the actors".
Spacetime (the stage) dissapears and its place is the gravitational field (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.
References
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[3] Einstein,A. (1916). Die Grundlage der allgemeinen Relativit?atstheorie, Annalen der Physik 49, 769 822; (1952) translation by W. Perrett and G. B. Je rey, The Foundation of the General Theory of Relativity, in The Principle of Relativity, (pp. 117 118). New York: Dover.
[4] L.Lusanna, M. Pauri (Parma Univ.) Explaining Leibniz-equivalence away: dis-solution of the Hole Argument and physical individuation of point-events
[5]
[6] B. Dittrich, Partial and Complete Observables for Hamiltonian Constrained Systems, [gr-qc/0411013.
[7] B. Dittrich, Partial and Complete Observables for Canonical General Relativity}, [gr-qc/0507106].
[8]
[9] Leonardo Modesto, Carlo Rovelli, {\em Particle scattering in loop quantum gravity}, Phys.Rev.Lett. 95 (2005) 191301, available at [gr-qc/0502036].
[10] C. Rovelli, Graviton propagator from background-independent quantum
gravity}, [gr-qc/0508124].
[11] L. Smolin, The case for background independence}, [hep-th/0507235]
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