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Background independence
Background independence
Background independence
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Background independence is a condition in theoretical physics 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. In particular this means that it must be possible not to refer to a specific coordinate system—the theory must be coordinate-free. In addition, the different spacetime configurations (or backgrounds) should be obtained as different solutions of the underlying equations.

Description

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Background independence is a loosely defined property of a theory of physics. Roughly speaking, it limits the number of mathematical structures used to describe space and time that are put in place "by hand". Instead, these structures are the result of dynamical equations, such as Einstein field equations, so that one can determine from first principles what form they should take. Since the form of the metric determines the result of calculations, a theory with background independence is more predictive than a theory without it, since the theory requires fewer inputs to make its predictions. This is analogous to desiring fewer free parameters in a fundamental theory.

So background independence can be seen as extending the mathematical objects that should be predicted from theory to include not just the parameters, but also geometrical structures. Summarizing this, Rickles writes: "Background structures are contrasted with dynamical ones, and a background independent theory only possesses the latter type—obviously, background dependent theories are those possessing the former type in addition to the latter type."[1]

In general relativity, background independence is identified with the property that the metric of spacetime is the solution of a dynamical equation.[2] In classical mechanics, this is not the case, the metric is fixed by the physicist to match experimental observations. This is undesirable, since the form of the metric impacts the physical predictions, but is not itself predicted by the theory.

Manifest background independence

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Manifest background independence is primarily an aesthetic rather than a physical requirement. It is analogous and closely related 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.

Although making a property manifest is only aesthetic, it is a useful tool for making sure the theory actually has that property. For example, if a theory is written in a manifestly Lorentz-invariant way, one can check at every step to be sure that Lorentz invariance is preserved. Making a property manifest also makes it clear whether or not the theory actually has that property. The inability to make classical mechanics manifestly Lorentz-invariant does not reflect a lack of imagination on the part of the theorist, but rather a physical feature of the theory. The same goes for making classical mechanics or electromagnetism background-independent.

Theories of quantum gravity

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Because of the speculative nature of quantum-gravity research, there is much debate as to the correct implementation of background independence. Ultimately, the answer is to be decided by experiment, but until experiments can probe quantum-gravity phenomena, physicists have to settle for debate. Below is a brief summary of the two largest quantum-gravity approaches.

Physicists have studied models of 3D quantum gravity, which is a much simpler problem than 4D quantum gravity (this is because in 3D, quantum gravity has no local degrees of freedom). In these models, there are non-zero transition amplitudes between two different topologies,[3] or in other words, the topology changes. This and other similar results lead physicists to believe that any consistent quantum theory of gravity should include topology change as a dynamical process.

String theory

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String theory is usually formulated with perturbation theory around a fixed background. While it is possible that the theory defined this way is locally background-invariant, if so, it is not manifest, and it is not clear what the exact meaning is. One attempt to formulate string theory in a manifestly background-independent fashion is string field theory, but little progress has been made in understanding it.

Another approach is the conjectured, but yet unproven AdS/CFT duality, which is believed to provide a full, non-perturbative definition of string theory in spacetimes with anti-de Sitter asymptotics. If so, this could describe a kind of superselection sector of the putative background-independent theory. But it would still be restricted to anti-de Sitter space asymptotics, which disagrees with the current observations of our Universe. A full non-perturbative definition of the theory in arbitrary spacetime backgrounds is still lacking.

Topology change is an established process in string theory.

Loop quantum gravity

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A very different approach to quantum gravity called loop quantum gravity is fully non-perturbative and manifestly background-independent: geometric quantities, such as area, are predicted without reference to a background metric or asymptotics (e.g. no need for a background metric or anti-de Sitter asymptotics), only a given topology.

See also

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References

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Further reading

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Background independence

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from Grokipedia
Background independence is a in asserting that a physical theory's equations and their solutions do not rely on a pre-existing, fixed background structure—such as a static metric—for their formulation or interpretation; instead, all geometric and spatiotemporal elements emerge dynamically from the theory's own . This contrasts with background-dependent theories, like those in or standard on flat , where a non-dynamical arena is presupposed to define fields and their evolution. exemplifies background independence, as its treat curvature as a dynamical entity sourced by and , without invoking an absolute or fixed geometry. The concept traces its roots to historical debates on the nature of space and time, notably the relational versus absolute views championed by and , respectively, and later revived in Ernst Mach's principles influencing Albert Einstein's development of in 1915. In background-independent frameworks, physical distinctions arise solely from relational properties among entities, adhering to principles like the , which prohibits differentiating configurations that differ only by a non-physical background shift. This relational strategy eliminates arbitrary fixed structures, ensuring that laws evolve with the system they describe, as seen in diffeomorphism-invariant formulations where coordinate choices lack intrinsic physical meaning. In the pursuit of , background independence is deemed essential for a complete theory, as background-dependent approaches—such as perturbative —fail to fully reconcile with general relativity's dynamical , leading to issues like the . Approaches like and causal embody this principle by quantizing relational networks of , predicting emergent at low energies without presupposing a continuum background. Definitions formalize it variably: one views it as theories where no physical significance attaches to configurations' positions along continuous symmetries, rendering auxiliary fields unphysical via Machian variations; another requires that distinct physical states correspond to inequivalent . These criteria guide evaluations of candidates, emphasizing full background independence where exhausts all degrees of freedom.

Conceptual Foundations

Definition and Principles

Background independence refers to the foundational principle in whereby the laws and predictions of a are formulated without relying on a pre-existing, fixed structure or ; instead, emerges dynamically as a consequence of the interactions among the theory's fundamental entities. This approach ensures that the theory's content alone determines the structure of , avoiding any absolute or non-dynamical elements that could privilege a particular background. A core principle of background independence is diffeomorphism invariance, the mathematical requirement that the theory's equations remain form-invariant under arbitrary smooth, invertible transformations of coordinates (diffeomorphisms). This invariance embodies the idea that no coordinate choice is physically preferred, allowing the theory to describe phenomena relationally rather than absolutely. In contrast, theories on fixed backgrounds—such as Newtonian gravity, which assumes an absolute and time, or , which presupposes a flat Minkowski metric—treat as a rigid, unchanging arena in which physical fields evolve, thereby introducing a non-dynamical structure that the theory cannot alter. In a background-independent framework, the metric tensor, which encodes the of , functions as a dynamical field governed by the same equations as and other fields, rather than serving as a static backdrop. This treatment places all physical content—geometry, , and fields—on equal dynamical footing, eliminating any privileged background and ensuring that the theory's predictions arise solely from the relational dynamics among its constituents. Thus, background independence fosters a holistic view where is not presupposed but generated by the theory itself, distinguishing it from foreground-dependent descriptions that subordinate to a pre-defined stage.

Historical Origins

The concept of background independence traces its philosophical roots to the late 19th century through Ernst Mach's critique of Newtonian mechanics, where he posited that a body's originates not from an absolute space but from its interaction with the total distribution of matter in the , thereby challenging the notion of a fixed, privileged background. This idea profoundly influenced , who credited as a key motivator in developing . Einstein formalized background independence in his 1915-1916 development of , where the geometry of is dynamically determined by the distribution of matter and energy, eliminating any fixed or absolute background structure. In his seminal 1916 review article, Einstein emphasized the , stating that "Of all imaginable spaces R₁, R₂, etc., in any kind of motion relatively to one another, there is none which we may look upon as privileged a priori without reviving the above-mentioned epistemological objection," underscoring the absence of a pre-existing metric independent of physical content. In the post-Einstein era, advanced this perspective during the 1950s and 1960s through his program of , which sought to describe all physical phenomena purely in terms of geometry without a fixed background, famously summarizing as "space-time tells matter how to move; matter tells space-time how to curve." This approach highlighted the relational nature of , where geometry emerges solely from matter's influence. Concurrently, in the early 1960s, Richard Arnowitt, Stanley Deser, and Charles W. Misner developed the ADM formalism—a canonical Hamiltonian formulation of —that explicitly incorporated coordinate independence, reinforcing background independence by treating as a dynamic entity free from a priori metric structures. By the and , as efforts to quantize intensified, background independence emerged as a central criterion for evaluating viable theories, distinguishing approaches that preserve the relational dynamics of from those reliant on fixed backgrounds, amid debates over unifying with .

Background Independence in Classical Gravity

Role in General Relativity

In , background independence is realized through the , which couple the of to the distribution of and without presupposing a fixed background metric. The equations take the form Gμν=8πTμνG_{\mu\nu} = 8\pi T_{\mu\nu}, where GμνG_{\mu\nu} is the encoding the of and TμνT_{\mu\nu} is the stress-energy tensor representing and content. This interdependence means that emerges dynamically from the interaction between and , rather than serving as a pre-existing arena. Coordinate independence in general relativity ensures that physical predictions remain unchanged under arbitrary choices of coordinates, reflecting the absence of a preferred frame. This is achieved through diffeomorphism invariance, where solutions to the field equations are invariant under smooth mappings of the manifold. Passive diffeomorphisms correspond to mere relabeling of coordinates without altering the physical configuration, while active diffeomorphisms physically relocate points on the manifold but preserve the equivalence class of solutions. Together, these symmetries enforce that no absolute coordinate system is imposed, allowing the theory to describe gravity relationally. General relativity exhibits a gauge structure analogous to that of Yang-Mills theories, but with the diffeomorphism group as the gauge group, which underpins its background freedom. In the ADM (Arnowitt-Deser-Misner) formalism, this manifests through first-class constraints: the Hamiltonian constraint generates time reparametrizations, and the momentum constraints generate spatial diffeomorphisms, both enforcing the theory's invariance under coordinate transformations without reference to an external background. These constraints eliminate redundant , ensuring that the physical content is diffeomorphism-invariant and independent of any fixed structure. A canonical example is the , which describes the geometry around a spherically symmetric, non-rotating and emerges directly as a solution to the vacuum (Tμν=0T_{\mu\nu} = 0, so Gμν=0G_{\mu\nu} = 0) without assuming any prior background metric. The metric ds2=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2dΩ2ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2 arises solely from the and the field equations, illustrating how vacuum curvature is self-consistently determined. Unlike Newtonian , which posits an absolute as an unchanging background independent of , generalizes this to a relational where relations are dynamically shaped by the distribution of and , eliminating absolute elements like fixed and time. This shift, inspired in part by that inertial frames arise from the global distribution of , underscores the theory's departure from preconceived geometric fixtures.

Manifest Background Independence

Manifest background independence refers to formulations of gravity theories where the diffeomorphism invariance is explicitly incorporated into the structure of the action or Hamiltonian from the outset, eliminating the need for additional constraints to enforce it after the fact. In such approaches, the dynamical variables themselves generate the without reliance on a fixed background metric or . In the canonical formulation of , manifest background independence is achieved through the ADM decomposition or Ashtekar variables. In the ADM formalism, the is parameterized by the spatial metric qabq_{ab} and its conjugate momentum, with the total Hamiltonian vanishing due to the first-class constraints (H=0H = 0), which generate time reparameterizations and spatial diffeomorphisms on the of metrics, ensuring that physical configurations are diffeomorphism-invariant. These constraints arise naturally from the . Key techniques to make background independence explicit include covariant phase space methods, which construct the directly from solutions to the while preserving full covariance without spacetime splitting. The use of tetrads and spin connections reformulates in terms of gauge fields, rendering diffeomorphism invariance manifest through the connection's transformation properties. For instance, in the 3+1 decomposition, the lapse function NN and shift vector NaN^a parameterize arbitrary slicings, ensuring no preferred time coordinate and enforcing relational dynamics. These manifest formulations offer advantages over non-manifest ones, such as facilitating path integral quantization by providing a natural measure on the space of geometries without extraneous structures. They also avoid introducing spurious that could arise from incomplete in background-dependent setups. A specific example is the Holst action, a modification of the Palatini given by S=116πG∫(eI∧eJ∧FIJ+12βeI∧eJ∧∗FIJ)S = \frac{1}{16\pi G} \int (e^I \wedge e^J \wedge F_{IJ} + \frac{1}{2\beta} e^I \wedge e^J \wedge {}^*F_{IJ}), where eIe^I are tetrads, FIJF_{IJ} is the curvature of the , ∗{}^* denotes the dual, and β\beta is the Barbero-Immirzi parameter. This action explicitly incorporates the Lorentzian structure through the additional term, maintaining equivalence to on-shell while making local Lorentz invariance and covariance manifest at the level of the first-order formalism.

Background Independence in Quantum Theories

String Theory

String theory, in its perturbative formulation, initially appears to rely on a fixed background , as exemplified by the , which governs the dynamics of the string's embedded in a target equipped with a prescribed metric and other background fields. This setup defines the path integral over configurations, where the background metric enters explicitly as an input parameter. However, the achieves a deeper level of background independence through the conformal invariance of the , which requires the vanishing of the beta functions for the background fields, effectively making the background dynamical and self-consistent. Modular invariance of the partition function further ensures that the is free from anomalies across different topologies, reinforcing this independence without presupposing a specific structure. Background independence is further realized via dualities that equate theories defined on seemingly distinct spacetimes. T-duality, for instance, maps the bosonic string theory on a circle of radius RR to an equivalent theory on a circle of radius 1/R1/R, demonstrating that physical observables are invariant under exchanges of momentum and winding modes, thus transcending the initial background choice. Similarly, S-duality relates strong and weak coupling regimes across different string theories, such as Type IIB self-duality, allowing the theory to probe non-perturbative regimes where background dependence is resolved by equivalence classes of geometries. These symmetries highlight how string theory unifies diverse backgrounds, with the critical dimension—26 for the bosonic string and 10 for superstrings—emerging dynamically from the requirement of anomaly cancellation in the worldsheet quantum theory, rather than being imposed externally. A pivotal advancement occurred in the with the discovery of a web of dualities, notably articulated by in 1995, which interconnected the five consistent superstring theories and elevated them to limits of a single underlying framework. In the strong-coupling limit of Type IIA , this unification manifests as in 11 dimensions, where the different string backgrounds are resolved into a cohesive structure incorporating membranes and resolving perturbative dependencies on specific metrics. Despite these achievements, challenges persist: perturbative formulations remain tied to chosen backgrounds order by order, though non-perturbative extensions via dualities suggest full within the vast landscape of string vacua, comprising 1050010^{500} or more metastable solutions stabilized by fluxes and branes. This landscape conjecture posits that all consistent backgrounds are interconnected, providing a pathway to complete background , albeit without a fully manifest definition to date. The background independence challenge in superstring theory is particularly acute in cosmology, where the theory often assumes specific backgrounds, such as flat spacetime, rather than achieving true background independence like in general relativity. This limitation makes it insufficient for providing a natural explanation of Big Bang initial conditions or inflation.

Loop Quantum Gravity

Loop quantum gravity (LQG) provides a non-perturbative, canonical quantization of general relativity that inherently incorporates background independence by treating spacetime geometry as dynamical at the quantum level. The formulation begins with the Ashtekar variables, which recast the Einstein equations as a gauge theory of SU(2) connections, where the basic phase space variables are holonomies along edges and fluxes through surfaces. These variables facilitate a background-free description, as the connection replaces the metric, and the theory proceeds without presupposing a fixed spacetime manifold. Quantum states of geometry are represented by spin networks, cylindrical functions over graphs labeled by SU(2) representations, which encode discrete quanta of spatial geometry without reliance on a continuous background metric. Background independence in LQG is enforced through the quantum implementation of the constraint, ensuring that physical wavefunctions are invariant under spatial . The constraint operator acts on spin network states by deforming the underlying graphs, projecting onto -invariant subspaces where states depend only on relational properties of the . This leads to operators for area and that yield purely discrete spectra, such as the area eigenvalue A=8πγℓP2j(j+1)A = 8\pi \gamma \ell_P^2 \sqrt{j(j+1)}
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