# Background independence

Background independence

Background independence is a condition in theoretical physics, especially in quantum gravity (QG), 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, and in particular to not refer to a specific coordinate system or metric. The different configurations (or backgrounds) should be obtained as different solutions of the underlying equations.

Diffeomorphism invariance and background independence

The argument involves only the very basics of GR, as we will see below. More details and discussions can be found in Rovelli's book or the papers by Rovelli and Gaulcite journal| url=http://arxiv.org/abs/gr-qc/9910079| title=Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance| first= Marcus| last=Gaul| coauthors= Rovelli, Carlo| journal= Lect.Notes Phys.| volume= 541| year=2000| pages=277–324| format=subscription required] and by Smolin.cite journal| url=http://arxiv.org/abs/hep-th/0507235| title= The case for background independence| first=Lee| last=Smolin| id=hep-th/0507235| format= subscription required]

It begins with a utterly trivial mathematical observation. Here is written the differential equation for the simple harmonic oscillator twice

:Eq(1) $frac\left\{d^2 f\left(x\right)\right\}\left\{dx^2\right\} + f\left(x\right) = 0$:Eq(2) $frac\left\{d^2 g\left(y\right)\right\}\left\{dy^2\right\} + g\left(y\right) = 0$

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 $f\left(x\right) = cos x$, we immediately know that $g\left(y\right) = cos y$ 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 $g_\left\{ab\right\}\left(x\right)$ 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, so the same argument applies and again we obtain two solutions with the same initial conditions which impose different spacetime geometries. 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 to the hole argument, background independence, only comes about when we allow spacetime to be dynamical. Before we can go on to understand this resolution we need to better understand these extra solutions. We can interpret these solutions as follows. For simplicity we first assume there is no matter. Define a metric function $ilde\left\{g\right\}_\left\{ab\right\}$ whose value at $P$ is given by the value of $g_\left\{ab\right\}$ at $P_0$, i.e.:Eq(3) $ilde\left\{g\right\}_\left\{ab\right\}\left(P\right) = g_\left\{ab\right\}\left(P_0\right)$.

(see figure 1(a)). Now consider a coordinate system which assigns to $P$ the same coordinate values that $P_0$ has in the x-coordinates (see figure 1(b)). We then have:Eq(4) $ilde\left\{g\right\}_\left\{ab\right\} \left(y_0=u_0,y_1=u_1, y_2=u_2, y_3=u_3\right) = g_\left\{ab\right\}\left(x_0=u_0,x_1=u_1, x_2=u_2 , x_3=u_3\right),$where $u_0,u_1,u_2,u_3$ are the coordinate values of $P_0$ in the x-coordinate system.When we allow the coordinate values to range over all permissible values, Eq(4) 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 diffeomorphism (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.

The resolution to the hole argument (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, however, 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 Rovelli's 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 that is the context for Einstein's remark "beyond my wildest expectations".

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, ..."citequote

The term "active diffeomorphism" has been used, instead of just "diffeomorphism", to emphasize that this is not a case of simple coordinate transformations. It is active diffeomorphisms which 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 as all physical theories are invariant under coordinate transformations. (Indeed, the mathematical definition of a diffeomorphism is a transformation which relates manifolds with equivalent topological and differentiable structure, but not necessarily equivalent metrics. For example, a diffeomorphism can turn a doughnut into a tea cup.)

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent. The equations of LQG are not embedded in, or presuppose, space and time, except for its invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length. At present, it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit, described by general relativity with possible quantum corrections.

Manifest background-independence

This is primarily an aesthetic rather than a physical requirement. It is analogous to requiring in differential geometry that equations be written in a form that is independent of the choice of charts and coordinate embeddings. If a background-independent formalism is present, it can lead to simpler and more elegant equations. However there is no physical content in requiring that a theory be manifestly background-independent - for example, the equations of general relativity can be rewritten in local coordinates without affecting the physical implications.

tring theory

Although physics of string theory can in principle be background-independent , the current formulations of this theory do not make this independence manifest because they require starting with a particular solution and performing a perturbative expansion about this background.

Background independent QG theories

tring theory

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.

Loop quantum gravity

A very different approach to quantum gravity called loop quantum gravity has been claimed to be background-independent. However, this theory has difficulty reproducing Einstein's theory of general relativity. Furthermore, the physics of loop quantum gravity is only background-independent in a weak sense. For example, it requires a fixed choice of the topology and dimensionality of the spacetime, while any consistent quantum theory of gravity should include topology change as a dynamical process. Topology change is an established process in string theory.

History of background independent theories

This dichotomy between background dependent and independent theories is sometimes traced back as far as the antagonism between Newton and Leibniz about absolute vs. relational space. Most physicists would claim that the choice of approach is merely philosophical so far as no different falsifiable claims follow, not unlike the question of interpretations of quantum mechanics. But philosophers of science Imre Lakatos and Elie Zahar have argued that research programs can be driven by metaphysical questions and so adopting the view of background independence may lead to different results.

ee also

*Causal dynamical triangulation
*Loop quantum gravity
*Quantum field theory

* [http://arxiv.org/abs/hep-th/0507235 The case for background independence]

References

* L. Smolin, "The case for background independence", [http://arxiv.org/abs/hep-th/0507235 hep-th/0507235]
* C. Rovelli et al, "Background independence in a nutshell", Class.Quant.Grav. 22 (2005) 2971-2990, [http://arxiv.org/abs/gr-qc/0408079 gr-qc/0408079]
* Edward Witten, "Quantum Background Independence In String Theory", [http://arxiv.org/abs/hep-th/9306122 hep-th/9306122] .
* J. Stachel, "The Meaning of General Covariance: The Hole Story", in J. Earman, A. Janis, G. Massey and N. Rescher (eds.), "Philosophical Problems of the Internal and External Worlds: Essays on the Philosophy of Adolf Grünbaum", University of Pittsburgh Press 1993, ISBN 0-8229-3738-7, pp. 129-160.
* J. Stachel, Changes in the Concepts of Space and Time Brought About by Relativity , in C. C. Gould and R. S. Cohen (eds.), "Artifacts, Representations and Social Practice"; Kluwer Academic 1994, ISBN 0-7923-2481-1, pp. 141-162.
* E. Zahar, "Einstein's Revolution: A Study in Heuristic", ISBN 0-8126-9066-4

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