From God’s Perspective, There Are No Fields…Justified Newtonian, Unjustified Relativistic Claim. Note Quote.

Electromagnetism is a relativistic theory. Indeed, it had been relativistic, or Lorentz invariant, before Einstein and Minkowski understood that this somewhat peculiar symmetry of Maxwell’s equations was not accidental but expressive of a radically new structure of time and space. Minkowski spacetime, in contrast to Newtonian spacetime, doesn’t come with a preferred space-like foliation, its geometric structure is not one of ordered slices representing “objective” hyperplanes of absolute simultaneity. But Minkowski spacetime does have an objective (geometric) structure of light-cones, with one double-light-cone originating in every point. The most natural way to define a particle interaction in Minkowski spacetime is to have the particles interact directly, not along equal-time hyperplanes but along light-cones

In other words, if z_i(τ_􏱁i)  and z_j(τ_j􏱁) denote the trajectories of two charged particles, it wouldn’t make sense to say that the particles interact at “equal times” as it is in Newtonian theory. It would however make perfectly sense to say that the particles interact whenever

(z^μ_i z^μ_j)(z^μ_i z^μ_j) = (z_i – z_j)² = 0 —– (1)

For an observer finding himself in a universe guided by such laws it might then seem like the effects of particle interactions were propagating through space with the speed of light. And this observer may thus insist that there must be something in addition to the particles, something moving or evolving in spacetime and mediating interactions between charged particles. And all this would be a completely legitimate way of speaking, only that it would reflect more about how things appear from a local perspective in a particular frame of reference than about what is truly and objectively going on in the physical world. From “Gods perspective” there are no fields (or photons, or anything of that kind) – only particles in spacetime interacting with each other. This might sound hypothetical, but, it actually is not entirely fictitious. for such a formulation of electrodynamics actually exists and is known as Wheeler-Feynman electrodynamics, or Wheeler-Feynman Absorber Theory. There is a formal property of field equations called “gauge invariance” which makes it possible to look at things in several different, but equivalent, ways. Because of gauge invariance, this theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called “advanced” wave). The effect of all of these “advanced” disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object). So, in this view fields do not have a real existence independent of the sources that emit and absorb them. It is defined by the principle of least action.

Wheeler–Feynman electrodynamics and Maxwell–Lorentz electrodynamics are for all practical purposes empirically equivalent, and it may seem that the choice between the two candidate theories is merely one of convenience and philosophical preference. But this is not really the case since the sad truth is that the field theory, despite its phenomenal success in practical applications and the crucial role it played in the development of modern physics, is inconsistent. The reason is quite simple. The Maxwell–Lorentz theory for a system of N charged particles is defined, as it should be, by a set of mathematical equations. The equation of motion for the particles is given by the Lorentz force law, which is

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:

describing the acceleration of a charged particle in an electromagnetic field. The electromagnetic field, represented by the field-tensor F^μν, is described by Maxwell’s equations. The homogenous Maxwell equations tell us that the antisymmetric tensor F^μν (a 2-form) can be written as the exterior derivative of a potential (a 1-form) A^μ(x), i.e. as

F^μν = ∂^μ A^ν – ∂^ν A^μ —– (2)

The inhomogeneous Maxwell equations couple the field degrees of freedom to matter, that is, they tell us how the charges determine the configuration of the electromagnetic field. Fixing the gauge-freedom contained in (2) by demanding ∂_μA^μ(x) = 0 (Lorentz gauge), the remaining Maxwell equations take the particularly simple form:

□ A^μ = – 4π j^μ —– (3)

where

□ = ∂_μ∂^μ

is the d’Alembert operator and j^μ the 4-current density.

The light-cone structure of relativistic spacetime is reflected in the Lorentz-invariant equation (3). The Liénard–Wiechert field at spacetime point x depends on the trajectories of the particles at the points of intersection with the (past and future) light-cones originating in x. The Liénard–Wiechert field (the solution of (3)) is singular precisely at the points where it is needed, namely on the world-lines of the particles. This is the notorious problem of the electron self-interaction: a charged particle generates a field, the field acts back on the particle, the field-strength becomes infinite at the point of the particle and the interaction terms blow up. Hence, the simple truth is that the field concept for managing interactions between point-particles doesn’t work, unless one relies on formal manipulations like renormalization or modifies Maxwell’s laws on small scales. However, we don’t need the fields and by taking the idea of a relativistic interaction theory seriously, we can “cut the middle man” and let the particles interact directly. The status of the Maxwell equation’s (3) in Wheeler–Feynman theory is now somewhat analogous to the status of Laplace’s equation in Newtonian gravity. We can get to the Gallilean invariant theory by writing the force as the gradient of a potential and having that potential satisfy the simplest nontrivial Galilean invariant equation, which is the Laplace equation:

∆V(x, t) = ∑_iδ(x – x_i(t)) —– (4)

Similarly, we can get the (arguably) simplest Lorentz invariant theory by writing the force as the exterior derivative of a potential and having that potential satisfy the simplest nontrivial Lorentz invariant equation, which is (3). And as concerns the equation of motion for the particles, the trajectories, if, are parametrized by proper time, then the Minkowski norm of the 4-velocity is a constant of motion. In Newtonian gravity, we can make sense of the gravitational potential at any point in space by conceiving its effect on a hypothetical test particle, feeling the gravitational force without gravitating itself. However, nothing in the theory suggests that we should take the potential seriously in that way and conceive of it as a physical field. Indeed, the gravitational potential is really a function on configuration space rather than a function on physical space, and it is really a useful mathematical tool rather than corresponding to physical degrees of freedom. From the point of view of a direct interaction theory, an analogous reasoning would apply in the relativistic context. It may seem (and historically it has certainly been the usual understanding) that (3), in contrast to (4), is a dynamical equation, describing the temporal evolution of something. However, from a relativistic perspective, this conclusion seems unjustified.

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Graviton Fields Under Helicity Rotations. Thought of the Day 156.0

Einstein described gravity as equivalent to curves in space and time, but physicists have long searched for a theory of gravitons, its putative quantum-scale source. Though gravitons are individually too weak to detect, most physicists believe the particles roam the quantum realm in droves, and that their behavior somehow collectively gives rise to the macroscopic force of gravity, just as light is a macroscopic effect of particles called photons. But every proposed theory of how gravity particles might behave faces the same problem: upon close inspection, it doesn’t make mathematical sense. Calculations of graviton interactions might seem to work at first, but when physicists attempt to make them more exact, they yield gibberish – an answer of “infinity.” This is the disease of quantized gravity. With regard to the exchange particles concept in the quantum electrodynamics theory and the existence of graviton, let’s consider a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the gravitational field. But when we define the graviton relative to the photon, it is necessary to explain the properties and behavior of photon in the gravitational field. The fields around a “ray of light” are electromagnetic waves, not static fields. The electromagnetic field generated by a photon is much stronger than the associated gravitational field. When a photon is falling in the gravitational field, it goes from a low layer to a higher layer density of gravitons. We should assume that the graviton is not a solid sphere without any considerable effect. Graviton carries gravity force, so it is absorbable by other gravitons; in general; gravitons absorb each other and combine. This new view on graviton shows, identities of graviton changes, in fact it has mass with changeable spin.

When we derive various supermultiplets of states, at the noninteracting level, these states can easily be described in terms of local fields. But, at the interacting level, there are certain ambiguities that withdraw as a result of different field representations describing the same massless free states. So the proper choice of the field representation may be subtle. The supermultiplets can then be converted into supersymmetric actions, quadratic in the fields. For selfdual tensor fields, the action must be augmented by a duality constraint on the corresponding field strength. For the graviton field,

The linearized Einstein equation for g_μν = η_μν + κh_μν implies that (for D ≥ 3)

R_μν ∝ ∂²h_μν + ∂_μ∂_νh – ∂_μ∂^ρh_νρ – ∂_ν∂^ρh_ρμ = 0 —– (1)

where h ≡ h_μμ and R_μν is the Ricci tensor. To analyze the number of states implied by this equation, one may count the number of plane-wave solutions with given momentum q^μ. It then turns out that there are D arbitrary solutions, corresponding to the linearized gauge invariance h_μν → h_μν + ∂_μξ_ν + ∂_νξ_μ, which can be discarded. Many other components vanish and the only nonvanishing ones require the momentum to be lightlike. Thee reside in the fields h_ij, where the components i, j are in the transverse (D-2) dimensional subspace. In addition, the trace of h_ij must be zero. Hence, the relevant plane-wave solutions are massless and have polarizations (helicities) characterized by a symmetric traceless 2-rank tensor. This tensor comprises 1/2D(D-3), which transform irreducibly under the SO(D-2) helicity group of transverse rotations. For the special case of D = 6 spacetime dimensions, the helicity group is SO(4), which factorizes into two SU(2) groups. The symmetric traceless representation then transforms as a doublet under each of the SU(2) factors and it is thus denoted by (2,2). As for D = 3, there are obviously no dynamic degrees of freedom associated with the gravitational field. When D = 2 there are again no dynamic degrees of freedom, but here (1) should be replaced by R_μν = 1/2g_μνR.

Gauge Fixity Towards Hyperbolicity: The Case For Equivalences. Part 2.

The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

Let then Γ’ be any (torsionless) reference connection. Introducing the following relative quantities, which are both tensors:

q^μ_αβ = Γ^μ_αβ – Γ’^μ_αβ

w^μ_αβ = u^μ_αβ – u’^μ_αβ —– (1)

For any linear torsionless connection Γ’, the Hilbert-Einstein Lagrangian

L_H: J²Lor(m) → ∧^o_m(M)

L_H: L_H(g^αβ, R_αβ)ds = 1/2κ (R – 2∧)√g ds

can be covariantly recast as:

L_H = d_α(P^βμu^α_βμ)ds + 1/2κ[g^βμ(Γ^ρ_βσΓ^σ_ρμ – Γ^α_ασΓ^σ_βμ) – 2∧]√g ds

= d_α(P^βμw^α_βμ)ds + 1/2κ[g^βμ(R’_βμ + q^ρ_βσq^σ_ρμ – q^α_ασq^σ_βμ)  – 2∧]√g ds —– (2)

The first expression for L_H shows that Γ’ (or g’, if Γ’ are assumed a priori to be Christoffel symbols of the reference metric g’) has no dynamics, i.e. field equations for the reference connection are identically satisfied (since any dependence on it is hidden under a divergence). The second expression shows instead that the same Einstein equations for g can be obtained as the Euler-Lagrange equation for the Lagrangian:

L₁ = 1/2κ[g^βμ(R’_βμ + q^ρ_βσq^σ_ρμ – q^α_ασq^σ_βμ)  – 2∧]√g ds —– (3)

which is first order in the dynamical field g and it is covariant since q is a tensor. The two Lagrangians L_H and L₁, are thence said to be equivalent, since they provide the same field equations.

In order to define the natural theory, we will have to declare our attitude towards the reference field Γ’. One possibility is to mimic the procedure used in Yang-Mills theories, i.e. restrict to variations which keep the reference background fixed. Alternatively we can consider Γ’ (or g’) as a dynamical field exactly as g is, even though the reference is not endowed with a physical meaning. In other words, we consider arbitrary variations and arbitrary transformations even if we declare that g is “observable” and genuinely related to the gravitational field, while Γ’ is not observable and it just sets the reference level of conserved quantities. A further important role played by Γ’ is that it allows covariance of the first order Lagrangian L₁, . No first order Lagrangian for Einstein equations exists, in fact, if one does not allow the existence of a reference background field (a connection or something else, e.g. a metric or a tetrad field). To obtain a good and physically sound theory out of the Lagrangian L₁, we still have to improve its dependence on the reference background Γ’. For brevity’s sake, let us assume that Γ’ is the Levi-Civita connection of a metric g’ which thence becomes the reference background. Let us also assume (even if this is not at all necessary) that the reference background g’ is Lorentzian. We shall introduce a dynamics for the reference background g’, (thus transforming its Levi-Civita connection into a truly dynamical connection), by considering a new Lagrangian:

L_1B = 1/2κ[√g(R – 2∧) – d_α(√g g^μνw^α_μν) – √g'(R’ – 2∧)]ds

= 1/2κ[(R’ – 2∧)(√g – √g’) + √g g^βμ(q^ρ_βσq^σ_ρμ – q^α_ασq^σ_βμ)]ds —– (4)

which is obtained from L₁ by subtracting the kinetic term (R’ – 2∧) √g’. The field g’ is no longer undetermined by field equations, but it has to be a solution of the variational equations for L_1B w. r. t. g, which coincide with Einstein field equations. Why should a reference field, which we pretend not to be observable, obey some field equation? Field equations are here functional to the role that g’ plays in our framework. If g’ has to fix the zero value of conserved quantities of g which are relative to the reference configuration g’ it is thence reasonable to require that g’ is a solution of Einstein equations as well. Under this assumption, in fact, both g and g’ represent a physical situation and relative conserved quantities represent, for example, the energy “spent to go” from the configuration g’ to the configuration g. To be strictly precise, further hypotheses should be made to make the whole matter physically meaningful in concrete situations. In a suitable sense we have to ensure that g’ and g belong to the same equivalence class under some (yet undetermined equivalence relation), e.g. that g’ can be homotopically deformed onto g or that they satisfy some common set of boundary (or asymptotic) conditions.

Considering the Lagrangian L_1B as a function of the two dynamical fields g and g’, first order in g and second order in g’. The field g is endowed with a physical meaning ultimately related to the gravitational field, while g’ is not observable and it provides at once covariance and the zero level of conserved quantities. Moreover, deformations will be ordinary (unrestricted) deformations both on g’ and g, and symmetries will drag both g’ and g. Of course, a natural framework has to be absolute to have a sense; any further trick or limitation does eventually destroy the naturality. The Lagrangian L_1B is thence a Lagrangian

L_1B : J²Lor(M) x_M J¹Lor(M) → A_m(M)

Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.

Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

L_H: J²Lor(m) → ∧^o_m(M)

L_H: L_H(g^αβ, R_αβ)ds = 1/2κ (R – 2∧)√g ds —– (1)

where

R = g^αβ R_αβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

R_αβ – 1/2 Rg_αβ = -∧g_αβ —— (2)

Lagrangian momenta is defined by:

p_αβ = ∂L_H/∂g^αβ = 1/2κ (R_αβ – 1/2(R – 2∧)g_αβ)√g

P^αβ = ∂L_H/∂R_αβ = 1/2κ g^αβ√g —– (3)

Thus the covariance identity is the following:

d_α(L_Hξ^α) = p_αβ£_ξg^αβ + P^αβ£_ξR_αβ —– (4)

or equivalently,

∇_α(L_Hξ^α) = p_αβ£_ξg^αβ + P^αβ∇_ε(£_ξΓ^ε_αβ – δ^ε_β£_ξΓ^λ_αλ) —– (5)

where ∇_ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(L_H, ξ) = W(L_H, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(L_H, ξ) = [P^αβ£_ξΓ^ε_αβ – P^αε£_ξΓ^λ_αλ – L_Hξ^ε]ds_ε = √g/2κ(g^αβg^εσ – g^σβg^εα) ∇_α£_ξg^βσds_ε – √g/2κξ^εRds_ε = √g/2κ[(3/2R^α_λ – (R – 2∧)δ^α_λ)ξ^λ + (g^βγδ^α_λ – g^α(γδ^β)_λ ∇_βγξ^λ]ds_α —– (7)

W(L_H, ξ) = √g/κ(R_αβ – 1/2(R – 2∧)g_αβ)∇(^αξ^β)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(L_H, ξ) = ε'(L_H, ξ) + Div U(L_H, ξ) —– (9)

where we set

ε'(L_H, ξ) = √g/κ(R^α_β – 1/2(R – 2∧)δ^α_β)ξ^β)ds_α

U(L_H, ξ) = 1/2κ ∇[^βξ^α] √gds_αβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫_∞ U(L_H, ∂_t, g)

J = ∫_∞ U(L_H, ∂_φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂_t and ∂_φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(L_H, ∂_φ) is the superpotential for J while 2U(L_H, ∂_t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’^α_βμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

Grand Unification Theory/(Anti-GUT): Emerging Symmetry, Topology in a Momentum Space. Thought of the Day 129.0

Quantum phase transition between two ground states with the same symmetry but of different universality class – gapless at q < q_c and fully gapped at q > q_c – as isolated point (a) as the termination point of first order transition (b)

There are two schemes for the classification of states in condensed matter physics and relativistic quantum fields: classification by symmetry (GUT scheme) and by momentum space topology (anti-GUT scheme).

For the first classification method, a given state of the system is characterized by a symmetry group H which is a subgroup of the symmetry group G of the relevant physical laws. The thermodynamic phase transition between equilibrium states is usually marked by a change of the symmetry group H. This classification reflects the phenomenon of spontaneously broken symmetry. In relativistic quantum fields the chain of successive phase transitions, in which the large symmetry group existing at high energy is reduced at low energy, is in the basis of the Grand Unification models (GUT). In condensed matter the spontaneous symmetry breaking is a typical phenomenon, and the thermodynamic states are also classified in terms of the subgroup H of the relevant group G. The groups G and H are also responsible for topological defects, which are determined by the nontrivial elements of the homotopy groups π_n(G/H).

The second classification method reflects the opposite tendency – the anti Grand Unification (anti-GUT) – when instead of the symmetry breaking the symmetry gradually emerges at low energy. This method deals with the ground states of the system at zero temperature (T = 0), i.e., it is the classification of quantum vacua. The universality classes of quantum vacua are determined by momentum-space topology, which is also responsible for the type of the effective theory, emergent physical laws and symmetries at low energy. Contrary to the GUT scheme, where the symmetry of the vacuum state is primary giving rise to topology, in the anti-GUT scheme the topology in the momentum space is primary while the vacuum symmetry is the emergent phenomenon in the low energy corner.

At the moment, we live in the ultra-cold Universe. All the characteristic temperatures in our Universe are extremely small compared to the Planck energy scale E_P. That is why all the massive fermions, whose natural mass must be of order E_P, are frozen out due to extremely small factor exp(−E_P/T). There is no matter in our Universe unless there are massless fermions, whose masslessness is protected with extremely high accuracy. It is the topology in the momentum space, which provides such protection.

For systems living in 3D space, there are four basic universality classes of fermionic vacua provided by topology in momentum space:

(i)  Vacua with fully-gapped fermionic excitations, such as semiconductors and conventional superconductors.

(ii)  Vacua with fermionic excitations characterized by Fermi points – points in 3D momentum space at which the energy of fermionic quasiparticle vanishes. Examples are provided by the quantum vacuum of Standard Model above the electroweak transition, where all elementary particles are Weyl fermions with Fermi points in the spectrum. This universality class manifests the phenomenon of emergent relativistic quantum fields at low energy: close to the Fermi points the fermionic quasiparticles behave as massless Weyl fermions, while the collective modes of the vacuum interact with these fermions as gauge and gravitational fields.

(iii)  Vacua with fermionic excitations characterized by lines in 3D momentum space or points in 2D momentum space. We call them Fermi lines, though in general it is better to characterize zeroes by co-dimension, which is the dimension of p-space minus the dimension of the manifold of zeros. Lines in 3D momentum space and points in 2D momentum space have co-dimension 2: since 3−1 = 2−0 = 2. The Fermi lines are topologically stable only if some special symmetry is obeyed.

(iv) Vacua with fermionic excitations characterized by Fermi surfaces. This universality class also manifests the phenomenon of emergent physics, though non-relativistic: at low temperature all the metals behave in a similar way, and this behavior is determined by the Landau theory of Fermi liquid – the effective theory based on the existence of Fermi surface. Fermi surface has co-dimension 1: in 3D system it is the surface (co-dimension = 3 − 2 = 1), in 2D system it is the line (co- dimension = 2 − 1 = 1), and in 1D system it is the point (co-dimension = 1 − 0 = 1; in one dimensional system the Landau Fermi-liquid theory does not work, but the Fermi surface survives).

The possibility of the Fermi band class (v), where the energy vanishes in the finite region of the 3D momentum space and thus zeroes have co-dimension 0, and such topologically stable flat band may exist in the spectrum of fermion zero modes, i.e. for fermions localized in the core of the topological objects. The phase transitions which follow from this classification scheme are quantum phase transitions which occur at T = 0. It may happen that by changing some parameter q of the system we transfer the vacuum state from one universality class to another, or to the vacuum of the same universality class but different topological quantum number, without changing its symmetry group H. The point q_c, where this zero-temperature transition occurs, marks the quantum phase transition. For T ≠ 0, the second order phase transition is absent, as the two states belong to the same symmetry class H, but the first order phase transition is not excluded. Hence, there is an isolated singular point (q_c, 0) in the (q, T) plane, or the end point of the first order transition. The quantum phase transitions which occur in classes (iv) and (i) or be- tween these classes are well known. In the class (iv) the corresponding quantum phase transition is known as Lifshitz transition, at which the Fermi surface changes its topology or emerges from the fully gapped state of class (i). The transition between the fully gapped states characterized by different topological charges occurs in 2D systems exhibiting the quantum Hall and spin-Hall effect: this is the plateau-plateau transition between the states with different values of the Hall (or spin-Hall) conductance. The less known transitions involve nodes of co-dimension 3 and nodes of co-dimension 2.

Philosophizing Loops – Why Spin Foam Constraints to 3D Dynamics Evolution?

The philosophy of loops is canonical, i.e., an analysis of the evolution of variables defined classically through a foliation of spacetime by a family of space-like three- surfaces ∑_t. The standard choice is the three-dimensional metric g_ij, and its canonical conjugate, related to the extrinsic curvature. If the system is reparametrization invariant, the total hamiltonian vanishes, and this hamiltonian constraint is usually called the Wheeler-DeWitt equation. Choosing the canonical variables is fundamental, to say the least.

Abhay Ashtekar‘s insights stems from the definition of an original set of variables stemming from Einstein-Hilbert Lagrangian written in the form,

S = ∫e^a ∧ e^b ∧ R^cdε_abcd —– (1)

where, e^a are the one-forms associated to the tetrad,

e^a ≡ e^a_μdx^μ —– (2)

The associated SO(1, 3) connection one-form ϖ^a_b is called the spin connection. Its field strength is the curvature expressed as a two form:

R^a_b ≡ dϖ^a_b + ϖ^a_c ∧ ϖ^c_b —– (3)

Ashtekar’s variables are actually based on the SU(2) self-dual connection

A = ϖ − i ∗ ϖ —– (4)

Its field strength is

F ≡ dA + A ∧ A —– (5)

The dynamical variables are then (A_i, Eⁱ ≡ F⁰ⁱ). The main virtue of these variables is that constraints are then linearized. One of them is exactly analogous to Gauss’ law:

D_iEⁱ = 0 —– (6)

There is another one related to three-dimensional diffeomorphisms invariance,

trF_ijEⁱ = 0 —– (7)

and, finally, there is the Hamiltonian constraint,

trF_ijEⁱE^j = 0 —– (8)

On a purely mathematical basis, there is no doubt that Astekhar’s variables are of a great ingenuity. As a physical tool to describe the metric of space, they are not real in general. This forces a reality condition to be imposed, which is akward. For this reason it is usually prefered to use the Barbero-Immirzi formalism in which the connection depends on a free parameter, γ

Aⁱ_a + ϖⁱ_a + γKⁱ_a —– (9)

ϖ being the spin connection, and K the extrinsic curvature. When γ = i, Ashtekar’s formalism is recovered, for other values of γ, the explicit form of the constraints is more complicated. Even if there is a Hamiltonian constraint that seems promising, was isn’t particularly clear is if the quantum constraint algebra is isomorphic to the classical algebra.

Some states which satisfy the Astekhar constraints are given by the loop representation, which can be introduced from the construct (depending both on the gauge field A and on a parametrized loop γ)

W (γ, A) ≡ trPe^φ_γA —– (10)

and a functional transform mapping functionals of the gauge field ψ(A) into functionals of loops, ψ(γ):

ψ(γ) ≡ ∫DAW(γ, A) ψ(A) —– (11)

When one divides by diffeomorphisms, it is found that functions of knot classes (diffeomorphisms classes of smooth, non self-intersecting loops) satisfy all the constraints. Some particular states sought to reproduce smooth spaces at coarse graining are the Weaves. It is not clear to what extent they also approach the conjugate variables (that is, the extrinsic curvature) as well.

In the presence of a cosmological constant the hamiltonian constraint reads:

ε_ijkE^aiE^bj(F^k_ab + λ/3ε_abcE^ck) = 0 —– (12)

A particular class of solutions expounded by Lee Smolin of the constraint are self-dual solutions of the form

Fⁱ_ab = -λ/3ε_abcE^ci —– (13)

Loop states in general (suitable symmetrized) can be represented as spin network states: colored lines (carrying some SU(2) representation) meeting at nodes where intertwining SU(2) operators act. There is also a path integral representation, known as spin foam, a topological theory of colored surfaces representing the evolution of a spin network. Spin foams can also be considered as an independent approach to the quantization of the gravitational field. In addition to its specific problems, the hamiltonian constraint does not say in what sense (with respect to what) the three-dimensional dynamics evolve.

Dynamics of Point Particles: Orthogonality and Proportionality

Let γ be a smooth, future-directed, timelike curve with unit tangent field ξ^a in our background spacetime (M, g_ab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λ^a be a vector at p. Then there is a natural decomposition of λ^a into components proportional to, and orthogonal to, ξ^a:

λ^a = (λ^bξ_b)ξ^a + (λ^a −(λ^bξ_b)ξ^a) —– (1)

Here, the first part of the sum is proportional to ξ^a, whereas the second one is orthogonal to ξ^a.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λ^a relative to ξ^a (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξ^a is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

k_ab = ξ_a ξ_b —– (2)

h_ab = g_ab − ξ_a ξ_b —– (3)

then the decomposition can be expressed in the form

λ^a = k^a_b λ^b + h^a_b λ^b —– (4)

We can think of k_ab and h_ab as the relative temporal and spatial metrics determined by ξ^a. They are symmetric and satisfy

k^a_bk^b_c = k^a_c —– (5)

h^a_bh^b_c = h^a_c —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′^a. (Since ξ^a and ξ′^a are both future-directed, they are co-oriented; i.e., ξ^a ξ′^a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′^a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥h^a_b ξ′^b∥ / ∥k^a_b ξ′^b∥ —– (7)

For any vector μ^a, ∥μ^a∥ is (μ^aμ_a)^1/2 if μ is causal, and it is (−μ^aμ_a)^1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥k^a_b ξ′^b∥ = (k^a_b ξ′^b k_ac ξ′^c)^1/2 = (k_bc ξ′^bξ′^c)^1/2 = (ξ′^bξ_b)

and

∥h^a_b ξ′^b∥ = (−h^a_b ξ′^b h_ac ξ′^c)^1/2 = (−h_bc ξ′^bξ′^c)^1/2 = ((ξ′^bξ_b)² − 1)^1/2

so

v = ((ξ’^bξ_b)² − 1)^1/2 / (ξ′^bξ_b) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′^bξ_b) = 1/√(1 – v²) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector P^a that is tangent to its worldline there. The length ∥P^a∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if P^a is timelike, we can write it in the form P^a =mξ^a, where m = ∥P^a∥ > 0 and ξ^a is the four-velocity of the particle. No such decomposition is possible when P^a is null and m = ∥P^a∥ = 0.

Suppose a particle O with positive mass has four-velocity ξ^a at a point, and another particle O′ has four-momentum P^a there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose P^a in terms of ξ^a. By equations (4) and (2), we have

P^a = (P^bξ_b) ξ^a + h^a_bP^b —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = P^bξ_b. If O′ has positive mass and P^a = mξ′^a, this yields, by equation (9),

E = m (ξ′^bξ_b) = m/√(1 − v²) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc² and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term h^a_bP^b in the decomposition of P^a, i.e., the component of P^a orthogonal to ξ^a. It follows from equations (8) and (9) that it has magnitude

p = ∥h^a_b mξ′^b∥ = m((ξ′^bξ_b)² − 1)^1/2 = mv/√(1 − v²) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξ^a. Just as we understand ξ^a to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξⁿ∇_nξ^a – the directional derivative of ξ^a in the direction ξ^a – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξ^a. (This is, since ξ^a(ξⁿ∇_nξ_a) = 1/2 ξⁿ∇_n(ξ^aξ_a) = 1/2 ξⁿ∇_n (1) = 0). The magnitude ∥ξⁿ∇_nξ^a∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field F^a on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξⁿ∇_nξ^a satisfies

F^a = mξⁿ∇_nξ^a —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields F_ab. If a particle with mass m > 0, charge q, and four-velocity field ξ^a is present, the force exerted by the field on the particle at a point is given by qF^a_bξ^b. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qF^a_bξ^b = mξ^b∇_bξ^a —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξ^a. But so is the force field on the left, since ξ_a(F^a_bξ^b) = ξ^aξ^bF_ab = ξ^aξ^bF_(ab), and F_(ab) = 0 by the anti-symmetry of F_ab.

Simultaneity

Let us introduce the concept of space using the notion of reflexive action (or reflex action) between two things. Intuitively, a thing x acts on another thing y if the presence of x disturbs the history of y. Events in the real world seem to happen in such a way that it takes some time for the action of x to propagate up to y. This fact can be used to construct a relational theory of space à la Leibniz, that is, by taking space as a set of equitemporal things. It is necessary then to define the relation of simultaneity between states of things.

Let x and y be two things with histories h(x_τ) and h(y_τ), respectively, and let us suppose that the action of x on y starts at τ_x⁰. The history of y will be modified starting from τ_y⁰. The proper times are still not related but we can introduce the reflex action to define the notion of simultaneity. The action of y on x, started at τ_y⁰, will modify x from τ_x¹ on. The relation “the action of x on y is reflected to x” is the reflex action. Historically, Galileo introduced the reflection of a light pulse on a mirror to measure the speed of light. With this relation we will define the concept of simultaneity of events that happen on different basic things.

Besides we have a second important fact: observation and experiment suggest that gravitation, whose source is energy, is a universal interaction, carried by the gravitational field.

Let us now state the above hypothesis axiomatically.

Axiom 1 (Universal interaction): Any pair of basic things interact. This extremely strong axiom states not only that there exist no completely isolated things but that all things are interconnected.

This universal interconnection of things should not be confused with “universal interconnection” claimed by several mystical schools. The present interconnection is possible only through physical agents, with no mystical content. It is possible to model two noninteracting things in Minkowski space assuming they are accelerated during an infinite proper time. It is easy to see that an infinite energy is necessary to keep a constant acceleration, so the model does not represent real things, with limited energy supply.

Now consider the time interval (τ_x¹ − τ_x⁰). Special Relativity suggests that it is nonzero, since any action propagates with a finite speed. We then state

Axiom 2 (Finite speed axiom): Given two different and separated basic things x and y, such as in the above figure, there exists a minimum positive bound for the interval (τ_x¹ − τ_x⁰) defined by the reflex action.

Now we can define Simultaneity as τ_y⁰ is simultaneous with τ_x^1/2 =_Df (1/2)(τ_x¹ + τ_x⁰)

The local times on x and y can be synchronized by the simultaneity relation. However, as we know from General Relativity, the simultaneity relation is transitive only in special reference frames called synchronous, thus prompting us to include the following axiom:

Axiom 3 (Synchronizability): Given a set of separated basic things {x_i} there is an assignment of proper times τ_i such that the relation of simultaneity is transitive.

With this axiom, the simultaneity relation is an equivalence relation. Now we can define a first approximation to physical space, which is the ontic space as the equivalence class of states defined by the relation of simultaneity on the set of things is the ontic space E_O.

The notion of simultaneity allows the analysis of the notion of clock. A thing y ∈ Θ is a clock for the thing x if there exists an injective function ψ : S_L(y) → S_L(x), such that τ < τ′ ⇒ ψ(τ) < ψ(τ′). i.e.: the proper time of the clock grows in the same way as the time of things. The name Universal time applies to the proper time of a reference thing that is also a clock. From this we see that “universal time” is frame dependent in agreement with the results of Special Relativity.

Cosmology: Friedmann-Lemaître Universes

Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying u_a = −t_,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

C_ijkl := R_ijkl + 1/2(R_ikg_jl + R_jlg_ik − R_il g_jk − R_jkg_il) − 1/6R(g_ikg_jl − g_ilg_jk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds² = −dt² + S²(t)dσ², u^a = δ^a₀ (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector u^a = dx^a/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S²(t) where k is the sign of K. The normalized metric dσ² characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

dσ² = dr² + f²(r) dθ² + sin²θdφ² —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor T_ab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector u^a:

T_ab = (μ + p/c²)u_au_b + (p/c²)g_ab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c² are the timelike and spacelike eigenvalues of T_ab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

T^ab_;b = 0 ⇔ μ ̇ + (μ + p/c²)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc²/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c² (w may or may not be constant). Baryons have {p_bar = 0 ⇔ w = 0} and radiation has {p_radc² = μ_rad/3 ⇔ w = 1/3,μ_rad = aT⁴_rad}, which by (5) imply

μ_bar ∝ S⁻³, μ_rad ∝ S⁻⁴, T_rad ∝ S⁻¹ —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c²) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc² ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μ_grav := μ + 3p/c². For ordinary matter this will be positive:

μ + 3p/c² > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇²/S² = κμ/3 + Λ/3 – k/S² —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μ_bar + μ_rad + μ_cdm + μ_ν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t₀ (eg. today) consists of,

• The Hubble constant H₀ := (S ̇/S)₀ = 100h km/sec/Mpc;

• A dimensionless density parameter Ω_i0 := κμ_i0/3H₀² for each type of matter present (labelled by i);

• If Λ ≠ 0, either Ω_Λ0 := Λ/3H²₀, or the dimensionless deceleration parameter q := −(S ̈/S) H⁻²₀.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ω_i0 for each type of matter present, for example

Ω_m0 = Ω_bar0 + Ω_rad0 + Ω_cdm0 + Ω_ν0, —– (10)

and the total density parameter Ω₀ is the sum of that for matter and for the cosmological constant:

Ω₀ = Ω_m0 + Ω_Λ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c² can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q₀ = 1/2 Ω_m0 − Ω_Λ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q₀); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ω_m0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t₀, the spatial curvature is

K₀:= k/S₀² = H₀² (Ω₀ − 1) —– (13)

The value Ω₀ = 1 corresponds to spatially flat universes (K₀ = 0), separating models with positive spatial curvature (Ω₀ > 1 ⇔ K₀ > 0) from those with negative spatial curvature (Ω₀ < 1 ⇔ K₀ < 0).

The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).

Theories of Fields: Gravitational Field as “the More Equal Among Equals”

Descartes, in Le Monde, gave a fully relational definition of localization (space) and motion. According to Descartes, there is no “empty space”. There are only objects, and it makes sense to say that an object A is contiguous to an object B. The “location” of an object A is the set of the objects to which A is contiguous. “Motion” is change in location. That is, when we say that A moves we mean that A goes from the contiguity of an object B to the contiguity of an object C³. A consequence of this relationalism is that there is no meaning in saying “A moves”, except if we specify with respect to which other objects (B, C,. . . ) it is moving. Thus, there is no “absolute” motion. This is the same definition of space, location, and motion, that we find in Aristotle. Aristotle insists on this point, using the example of the river that moves with respect to the ground, in which there is a boat that moves with respect to the water, on which there is a man that walks with respect to the boat . . . . Aristotle’s relationalism is tempered by the fact that there is, after all, a preferred set of objects that we can use as universal reference: the Earth at the center of the universe, the celestial spheres, the fixed stars. Thus, we can say, if we desire so, that something is moving “in absolute terms”, if it moves with respect to the Earth. Of course, there are two preferred frames in ancient cosmology: the one of the Earth and the one of the fixed stars; the two rotates with respect to each other. It is interesting to notice that the thinkers of the middle ages did not miss this point, and discussed whether we can say that the stars rotate around the Earth, rather than being the Earth that rotates under the fixed stars. Buridan concluded that, on ground of reason, in no way one view is more defensible than the other. For Descartes, who writes, of course, after the great Copernican divide, the Earth is not anymore the center of the Universe and cannot offer a naturally preferred definition of stillness. According to malignants, Descartes, fearing the Church and scared by what happened to Galileo’s stubborn defense of the idea that “the Earth moves”, resorted to relationalism, in Le Monde, precisely to be able to hold Copernicanism without having to commit himself to the absolute motion of the Earth!

Relationalism, namely the idea that motion can be defined only in relation to other objects, should not be confused with Galilean relativity. Galilean relativity is the statement that “rectilinear uniform motion” is a priori indistinguishable from stasis. Namely that velocity (but just velocity!), is relative to other bodies. Relationalism holds that any motion (however zigzagging) is a priori indistinguishable from stasis. The very formulation of Galilean relativity requires a nonrelational definition of motion (“rectilinear and uniform” with respect to what?).

Newton took a fully different course. He devotes much energy to criticise Descartes’ relationalism, and to introduce a different view. According to him, space exists. It exists even if there are no bodies in it. Location of an object is the part of space that the object occupies. Motion is change of location. Thus, we can say whether an object moves or not, irrespectively from surrounding objects. Newton argues that the notion of absolute motion is necessary for constructing mechanics. His famous discussion of the experiment of the rotating bucket in the Principia is one of the arguments to prove that motion is absolute.

This point has often raised confusion because one of the corollaries of Newtonian mechanics is that there is no detectable preferred referential frame. Therefore the notion of absolute velocity is, actually, meaningless, in Newtonian mechanics. The important point, however, is that in Newtonian mechanics velocity is relative, but any other feature of motion is not relative: it is absolute. In particular, acceleration is absolute. It is acceleration that Newton needs to construct his mechanics; it is acceleration that the bucket experiment is supposed to prove to be absolute, against Descartes. In a sense, Newton overdid a bit, introducing the notion of absolute position and velocity (perhaps even just for explanatory purposes?). Many people have later criticised Newton for his unnecessary use of absolute position. But this is irrelevant for the present discussion. The important point here is that Newtonian mechanics requires absolute acceleration, against Aristotle and against Descartes. Precisely the same does special relativistic mechanics.

Similarly, Newton introduced absolute time. Newtonian space and time or, in modern terms, spacetime, are like a stage over which the action of physics takes place, the various dynamical entities being the actors. The key feature of this stage, Newtonian spacetime, is its metrical structure. Curves have length, surfaces have area, regions of spacetime have volume. Spacetime points are at fixed distance the one from the other. Revealing, or measuring, this distance, is very simple. It is sufficient to take a rod and put it between two points. Any two points which are one rod apart are at the same distance. Using modern terminology, physical space is a linear three-dimensional (3d) space, with a preferred metric. On this space there exist preferred coordinates xⁱ, i = 1,2,3, in terms of which the metric is just δ_ij. Time is described by a single variable t. The metric δ_ij determines lengths, areas and volumes and defines what we mean by straight lines in space. If a particle deviates with respect to this straight line, it is, according to Newton, accelerating. It is not accelerating with respect to this or that dynamical object: it is accelerating in absolute terms.

Special relativity changes this picture only marginally, loosing up the strict distinction between the “space” and the “time” components of spacetime. In Newtonian spacetime, space is given by fixed 3d planes. In special relativistic spacetime, which 3d plane you call space depends on your state of motion. Spacetime is now a 4d manifold M with a flat Lorentzian metric η_μν. Again, there are preferred coordinates x^μ, μ = 0, 1, 2, 3, in terms of which η_μν = diag[1, −1, −1, −1]. This tensor, η_μν , enters all physical equations, representing the determinant influence of the stage and of its metrical properties on the motion of anything. Absolute acceleration is deviation of the world line of a particle from the straight lines defined by η_μν. The only essential novelty with special relativity is that the “dynamical objects”, or “bodies” moving over spacetime now include the fields as well. Example: a violent burst of electromagnetic waves coming from a distant supernova has traveled across space and has reached our instruments. For the rest, the Newtonian construct of a fixed background stage over which physics happen is not altered by special relativity.

The profound change comes with general relativity (GTR). The central discovery of GR, can be enunciated in three points. One of these is conceptually simple, the other two are tremendous. First, the gravitational force is mediated by a field, very much like the electromagnetic field: the gravitational field. Second, Newton’s spacetime, the background stage that Newton introduced introduced, against most of the earlier European tradition, and the gravitational field, are the same thing. Third, the dynamics of the gravitational field, of the other fields such as the electromagnetic field, and any other dynamical object, is fully relational, in the Aristotelian-Cartesian sense. Let me illustrate these three points.

First, the gravitational field is represented by a field on spacetime, g_μν(x), just like the electromagnetic field A_μ(x). They are both very concrete entities: a strong electromagnetic wave can hit you and knock you down; and so can a strong gravitational wave. The gravitational field has independent degrees of freedom, and is governed by dynamical equations, the Einstein equations.

Second, the spacetime metric η_μν disappears from all equations of physics (recall it was ubiquitous). At its place – we are instructed by GTR – we must insert the gravitational field g_μν(x). This is a spectacular step: Newton’s background spacetime was nothing but the gravitational field! The stage is promoted to be one of the actors. Thus, in all physical equations one now sees the direct influence of the gravitational field. How can the gravitational field determine the metrical properties of things, which are revealed, say, by rods and clocks? Simply, the inter-atomic separation of the rods’ atoms, and the frequency of the clock’s pendulum are determined by explicit couplings of the rod’s and clock’s variables with the gravitational field g_μν(x), which enters the equations of motion of these variables. Thus, any measurement of length, area or volume is, in reality, a measurement of features of the gravitational field.

But what is really formidable in GTR, the truly momentous novelty, is the third point: the Einstein equations, as well as all other equations of physics appropriately modified according to GTR instructions, are fully relational in the Aristotelian-Cartesian sense. This point is independent from the previous one. Let me give first a conceptual, then a technical account of it.

The point is that the only physically meaningful definition of location that makes physical sense within GTR is relational. GTR describes the world as a set of interacting fields and, possibly, other objects. One of these interacting fields is g_μν(x). Motion can be defined only as positioning and displacements of these dynamical objects relative to each other.

To describe the motion of a dynamical object, Newton had to assume that acceleration is absolute, namely it is not relative to this or that other dynamical object. Rather, it is relative to a background space. Faraday, Maxwell and Einstein extended the notion of “dynamical object”: the stuff of the world is fields, not just bodies. Finally, GTR tells us that the background space is itself one of these fields. Thus, the circle is closed, and we are back to relationalism: Newton’s motion with respect to space is indeed motion with respect to a dynamical object: the gravitational field.

All this is coded in the active diffeomorphism invariance (diff invariance) of GR. Active diff invariance should not be confused with passive diff invariance, or invariance under change of coordinates. GTR can be formulated in a coordinate free manner, where there are no coordinates, and no changes of coordinates. In this formulation, there field equations are still invariant under active diffs. Passive diff invariance is a property of a formulation of a dynamical theory, while active diff invariance is a property of the dynamical theory itself. A field theory is formulated in manner invariant under passive diffs (or change of coordinates), if we can change the coordinates of the manifold, re-express all the geometric quantities (dynamical and non-dynamical) in the new coordinates, and the form of the equations of motion does not change. A theory is invariant under active diffs, when a smooth displacement of the dynamical fields (the dynamical fields alone) over the manifold, sends solutions of the equations of motion into solutions of the equations of motion. Distinguishing a truly dynamical field, namely a field with independent degrees of freedom, from a nondynamical filed disguised as dynamical (such as a metric field g with the equations of motion Riemann[g]=0) might require a detailed analysis (for instance, Hamiltonian) of the theory. Because active diff invariance is a gauge, the physical content of GTR is expressed only by those quantities, derived from the basic dynamical variables, which are fully independent from the points of the manifold.

In introducing the background stage, Newton introduced two structures: a spacetime manifold, and its non-dynamical metric structure. GTR gets rid of the non-dynamical metric, by replacing it with the gravitational field. More importantly, it gets rid of the manifold, by means of active diff invariance. In GTR, the objects of which the world is made do not live over a stage and do not live on spacetime: they live, so to say, over each other’s shoulders.

Of course, nothing prevents us, if we wish to do so, from singling out the gravitational field as “the more equal among equals”, and declaring that location is absolute in GTR, because it can be defined with respect to it. But this can be done within any relationalism: we can always single out a set of objects, and declare them as not-moving by definition. The problem with this attitude is that it fully misses the great Einsteinian insight: that Newtonian spacetime is just one field among the others. More seriously, this attitude sends us into a nightmare when we have to deal with the motion of the gravitational field itself (which certainly “moves”: we are spending millions for constructing gravity wave detectors to detect its tiny vibrations). There is no absolute referent of motion in GTR: the dynamical fields “move” with respect to each other.

Notice that the third step was not easy for Einstein, and came later than the previous two. Having well understood the first two, but still missing the third, Einstein actively searched for non-generally covariant equations of motion for the gravitational field between 1912 and 1915. With his famous “hole argument” he had convinced himself that generally covariant equations of motion (and therefore, in this context, active diffeomorphism invariance) would imply a truly dramatic revolution with respect to the Newtonian notions of space and time. In 1912 he was not able to take this profoundly revolutionary step, but in 1915 he took this step, and found what Landau calls “the most beautiful of the physical theories”.