Conformal Field Theory and Virasoro Algebra. Note Quote.

Realization of the Virasoro algebra

There are a few reasons why Conformal Field Theories (CFTs) are very interesting to study: The first is that at fixed points of Renormalization Group flows, or at second order phase transitions, a quantum field theory is scale invariant. Scale invariance is a weaker form of conformal invariance, and it turns out in all cases that we know of scale invariance of a quantum field theory actually ends up implying the larger symmetry of conformal invariance. The second reason is that the requirement that a theory is conformally invariant is so restrictive that many things can be solved for that would otherwise be intractable. As an example, conformal invariance fixes 2- and 3-point functions entirely. In an ordinary quantum field theory, especially one at strong coupling, these would be hard or impossible to calculate at all. A third reason is string theory. In string theory, the worldsheet theory describing the string’s excitations is a CFT, so if string theory is correct, then in some sense conformal invariance is really one of the most fundamental features of the elemental constituents of reality. And through string theory we have the most precise and best-understood gauge/gravity dualities (the AdS/CFT dualities) that also involve CFT’s.

A Conformal Field Theory (CFT) is a Quantum Field Theory (QFT) in which conformal rescaling of the metric acts by conjugation. For the family of morphisms Dg

D[ehg] = ec·α[h] L−1[h|B1] Dg L[h|B2] —– (1)

The analogous statement (conjugating the state on each boundary) is true for any Σ.

Here L is a linear operator depending only on the restriction of h to one of the boundaries of the annulus. All the dependence on the conformal rescaling away from the boundary is determined by a universal (independent of the particular Conformal Field Theory) functional α[h] ∈ R, which appears in an overall multiplicative factor ec·α[h]. The quantity c, called “Virasoro central charge”.

The corresponding operators L[h] form a semigroup, with a self-adjoint generator H. Then, since according to the axioms of QFT the spectrum of H is bounded below, we can promote this to a group action. This can be used to map any of the Hilbert spaces Hd to a single Hl for a fixed value of l, say l = 1. We will now do this and use the simpler notation H ≅ H1,

How do we determine the L[h]? First, we uniformize Σ – in other words, we find a complex diffeomorphism φ from our surface with boundary Σ to a constant curvature surface. We then consider the restriction of φ to each of the boundary components Bi, to get an element φi of Diff S1 × R+, where the R+ factor acts by an overall rescaling. We then express each φi as the exponential of an element li in the Lie algebra Diff S1, to find an appropriate projective representation of this Lie algebra on H.

Certain subtleties are in order here: The Lie algebra Diff S1 which appears is actually a subalgebra of a direct sum of two commuting algebras, which act independently on “left moving” and “right moving” factors in H. Thus, we can write H as a direct sum of irreps of this direct sum algebra,

H = ⊕iHL,i ⊗ HR,i —– (2)

Each of these two commuting algebras is a central extension of the Lie algebra Diff S1, usually called the Virasoro algebra or Vir.

Now, consider the natural action of Diff S1 on functions on an S1 parameterized by θ ∈ [0, 2π). After complexification, we can take the following set of generators,

ln = −ieinθ ∂/∂θ n ∈ Z —– (3)

which satisfy the relations

[lm, ln] = (m − n)lm+n —– (4)

The Virasoro algebra is the universal central extension of this, with generators Ln with n ∈ Z, c ∈ R, and the relations

[Lm, Ln] = (m − n)Lm+n + c/12 n(n2 − 1)δm+n,0 —– (5)

The parameter c is again the Virasoro central charge. It is to be noted that the central extension is required in any non-trivial unitary CFT. Unitarity and other QFT axioms require the Virasoro representation to act on a Hilbert space, so that L−n = Ln. In particular, L0 is self-adjoint and can be diagonalized. Take a “highest weight representation,” in which the spectrum of L0 is bounded below. The L0 eigenvector with the minimum eigenvalue, h, is by definition the “highest weight state”, or a state |h⟩, so that

L0|h⟩ = h|h⟩ —– (6)

and normalize it so that ⟨h|h⟩ = 1. Since this is a norm in a Hilbert space, we conclude that h ≥ 0, with equality only if L−1|h⟩ = 0. In fact, L−1|0⟩ = 0 can be related to the translation invariance of the vacuum. Rephrasing this in terms of local operators, instead of in terms of states, take Σ to be the infinite cylinder R × S1, or equivalently the punctured complex plane C with the complex coordinate z. In a CFT the component Tzz of the stress tensor can be expressed in terms of the Virasoro generators:

Tzz ≡ T(z) = ∑n∈Z Lnz−n−2 —– (7)

The component Tz̄z̄ is antiholomorphic and can be similarly expressed in terms of the generators L̄n of the second copy of the Virasoro algebra:

Tz̄z̄ ≡ T(z̄) = ∑n∈Zn−n−2 —– (8)

The mixed component Tzz̄ = Tz̄z is a c-number which vanishes for a flat metric. The state corresponding to T(z) is L−2|0⟩.

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

F1.large

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:

qμαβ = Γμαβ – Γ’μαβ

wμαβ = uμαβ – u’μαβ —– (1)

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

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds

can be covariantly recast as:

LH = dα(Pβμuαβμ)ds + 1/2κ[gβμρβσΓσρμ – ΓαασΓσβμ) – 2∧]√g ds

= dα(Pβμwαβμ)ds + 1/2κ[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ)  – 2∧]√g ds —– (2)

The first expression for LH 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[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ)  – 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 Land 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:

L1B = 1/2κ[√g(R – 2∧) – dα(√g gμνwαμν) – √g'(R’ – 2∧)]ds

= 1/2κ[(R’ – 2∧)(√g – √g’) + √g gβμ(qρβσqσρμ – qαασqσβμ)]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 L1B 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 L1B 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 L1B is thence a Lagrangian

L1B : J2Lor(M) xM J1Lor(M) → Am(M)

Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise

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The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re εt Γε|M = Vol|M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ Tx X, i.e.,

φ|V ≤ Vol|V —– (3)

then φ will be a calibration in the sense of implying the global statement

φ ≤ ∫Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ωp can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re eΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dimR M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ωp calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

  1. M is a p-dimensional complex submanifold of X.
  2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
  3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
  4. F satisfies Im e(ω|M + ils2F)p = 0 for some real constant φ, where ls is the correction.

The second kind of BPS D-brane, based on the Re eΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e Ω|M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of TxM ⊂ TxX is contained in TxM. Equivalently, one requires ker ωM to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ωM.

(3)  Let F M be the vector bundle T M/ ker ωM. It follows from the first two conditions that ωM and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ−1f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dimR M. Let Ω be the holomorphic trivialization of KX. One requires that Im eΩ|M ∧ Fr/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.

How Black Holes Emitting Hawking Radiation At Best Give Non-Trivial Information About Planckian Physics: Towards Entanglement Entropy.

The analogy between quantised sound waves in fluids and quantum fields in curved space-times facilitates an interdisciplinary knowhow transfer in both directions. On the one hand, one may use the microscopic structure of the fluid as a toy model for unknown high-energy (Planckian) effects in quantum gravity, for example, and investigate the influence of the corresponding cut-off. Examining the derivation of the Hawking effect for various dispersion relations, one reproduces Hawking radiation for a rather large class of scenarios, but there are also counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawkings result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics.

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On the other hand, the emergence of an effective geometry/metric allows us to apply the vast amount of universal tools and concepts developed for general relativity (such as horizons), which provide a unified description and better understanding of (classical and quantum) non-equilibrium phenomena (e.g., freezing and amplification of quantum fluctuations) in condensed matter systems. As an example for such a universal mechanism, the Kibble-Zurek effect describes the generation of topological effects due to the amplification of classical/thermal fluctuations in non-equilibrium thermal phase transitions. The loss of causal connection underlying the Kibble-Zurek mechanism can be understood in terms of an effective horizon – which clearly indicates the departure from equilibrium. The associated breakdown of adiabaticity leads to an amplification of thermal fluctuations (as in the Kibble-Zurek mechanism) as well as quantum fluctuations (at zero temperature). The zero-temperature version of this amplification mechanism is completely analogous to the early universe and becomes particularly important for the new and rapidly developing field of quantum phase transitions.

Furthermore, these analogue models might provide the exciting opportunity of measuring the analogues of these exotic effects – such as Hawking radiation or the generation of the seeds for structure formation during inflation – in actual laboratory experiments, i.e., experimental quantum simulations of black hole physics or the early universe. Even though the detection of these exotic quantum effects is partially very hard and requires ultra-low temperatures etc., there is no (known) principal objection against it. The analogue models range from black and/or white hole event horizons in flowing fluids and other laboratory systems over apparent horizons in expanding Bose–Einstein condensates, for example, to particle horizons in quantum phase transitions etc.

However, one should stress that the analogy reproduces the kinematics (quantum fields in curved space-times with horizons etc.) but not the dynamics, i.e., the effective geometry/metric is not described by the Einstein equations in general. An important and strongly related problem is the correct description of the back-reaction of the quantum fluctuations (e.g., phonons) onto the background (e.g., fluid flow). In gravity, the impact of the (classical or quantum) matter is usually incorporated by the (expectation value of) energy-momentum tensor. Since this quantity can be introduced at a purely kinematic level, one may use the same construction for phonons in flowing fluids, for example, the pseudo energy-momentum tensor. The relevant component of this tensor describing the energy density (which is conserved for stationary flows) may become negative as soon as the flow velocity exceeds the sound speed. These negative contributions explain the energy balance of the Hawking radiation in black hole analogues as well as super-radiant scattering. However, the (expectation value of the) pseudo energy-momentum tensor does not determine the quantum back-reaction correctly.

One should not neglect to mention the occurrence of a horizon in the laboratory – the Unruh effect. A uniformly accelerated observer cannot see half of the (1+1- dimensional) space-time, the two Rindler wedges are completely causally disconnected by the horizon(s). In each wedge, one may introduce a set of observables corresponding to the measurements made by the observers confined to this wedge – thereby obtaining two equivalent copies of observables in one wedge. In terms of these two copies, the Minkowski vacuum is an entangled state which yields the usual phenomena (thermo-field formalism) including the Unruh effect – i.e., the uniformly accelerated observer experiences the Minkowski vacuum as a thermal bath: For rather general quantum fields (Bisognano-Wichmann theorem), the quantum state ρ obtained by restricting the Minkowski vacuum to one of the Rindler wedges behaves as a mixed state ρ = exp{−2πHˆτ/κ}/Z, where Hˆτ corresponds to the Hamiltonian generating the proper (co-moving wristwatch) time τ measured by the accelerated observer and κ is the analogue to the surface gravity and determines the acceleration.

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Space-time diagram with a trajectory of a uniformly accelerated observer and the resulting particle horizons. The observer is confined to the right Rindler wedge (region x > |ct| between the two horizons) and cannot influence or be influenced by all events in the left Rindler wedge (x < |ct|), which is completely causally disconnected.

The thermal character of this restricted state ρ arises from the quantum correlations of the Minkowski vacuum in the two Rindler wedges, i.e., the Minkowski vacuum is a multi-mode squeezed state with respect the two equivalent copies of observables in each wedge. This is a quite general phenomenon associated with doubling the degrees of freedom and describes the underlying idea of the thermo-field formalism, for example. The entropy of the thermal radiation in the Unruh and the Hawking effect can be understood as an entanglement entropy: For the Unruh effect, it is caused by averaging over the quantum correlations between the two Rindler wedges. In the black hole case, each particle of the outgoing Hawking radiation has its infalling partner particle (with a negative energy with respect to spatial infinity) and the entanglement between the two generates the entropy flux of the Hawking radiation. Instead of accelerating a detector and measuring its excitations, one could replace the accelerated observer by an accelerated scatterer. This device would scatter (virtual) particles from the thermal bath and thereby create real particles – which can be interpreted as a signature of Unruh effect.

Time and World-Lines

Let γ: [s1, s2] → M be a smooth, future-directed timelike curve in M with tangent field ξa. We associate with it an elapsed proper time (relative to gab) given by

∥γ∥= ∫s1s2 (gabξaξb)1/2 ds

This elapsed proper time is invariant under reparametrization of γ and is just what we would otherwise describe as the length of (the image of) γ . The following is another basic principle of relativity theory:

Clocks record the passage of elapsed proper time along their world-lines.

Again, a number of qualifications and comments are called for. We have taken for granted that we know what “clocks” are. We have assumed that they have worldlines (rather than worldtubes). And we have overlooked the fact that ordinary clocks (e.g., the alarm clock on the nightstand) do not do well at all when subjected to extreme acceleration, tidal forces, and so forth. (Try smashing the alarm clock against the wall.) Again, these concerns are important and raise interesting questions about the role of idealization in the formulation of physical theory. (One might construe an “ideal clock” as a point-size test object that perfectly records the passage of proper time along its worldline, and then take the above principle to assert that real clocks are, under appropriate conditions and to varying degrees of accuracy, approximately ideal.) But they do not have much to do with relativity theory as such. Similar concerns arise when one attempts to formulate corresponding principles about clock behavior within the framework of Newtonian theory.

Now suppose that one has determined the conformal structure of spacetime, say, by using light rays. Then one can use clocks, rather than free particles, to determine the conformal factor.

Let g′ab be a second smooth metric on M, with g′ab = Ω2gab. Further suppose that the two metrics assign the same lengths to timelike curves – i.e., ∥γ∥g′ab = ∥γ∥gab ∀ smooth, timelike curves γ: I → M. Then Ω = 1 everywhere. (Here ∥γ∥gab is the length of γ relative to gab.)

Let ξoa be an arbitrary timelike vector at an arbitrary point p in M. We can certainly find a smooth, timelike curve γ: [s1, s2] → M through p whose tangent at p is ξoa. By our hypothesis, ∥γ∥g′ab = ∥γ∥gab. So, if ξa is the tangent field to γ,

s1s2 (g’ab ξaξb)1/2 ds = ∫s1s2 (gabξaξb)1/2 ds

∀ s in [s1, s2]. It follows that g′abξaξb = gabξaξb at every point on the image of γ. In particular, it follows that (g′ab − gab) ξoa ξob = 0 at p. But ξoa was an arbitrary timelike vector at p. So, g′ab = gab at our arbitrary point p. The principle gives the whole story of relativistic clock behavior. In particular, it implies the path dependence of clock readings. If two clocks start at an event p and travel along different trajectories to an event q, then, in general, they will record different elapsed times for the trip. This is true no matter how similar the clocks are. (We may stipulate that they came off the same assembly line.) This is the case because, as the principle asserts, the elapsed time recorded by each of the clocks is just the length of the timelike curve it traverses from p to q and, in general, those lengths will be different.

Suppose we consider all future-directed timelike curves from p to q. It is natural to ask if there are any that minimize or maximize the recorded elapsed time between the events. The answer to the first question is “no.” Indeed, one then has the following proposition:

Let p and q be events in M such that p ≪ q. Then, for all ε > 0, there exists a smooth, future directed timelike curve γ from p to q with ∥γ ∥ < ε. (But there is no such curve with length 0, since all timelike curves have non-zero length.)

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If there is a smooth, timelike curve connecting p and q, there is also a jointed, zig-zag null curve connecting them. It has length 0. But we can approximate the jointed null curve arbitrarily closely with smooth timelike curves that swing back and forth. So (by the continuity of the length function), we should expect that, for all ε > 0, there is an approximating timelike curve that has length less than ε.

The answer to the second question (“Can one maximize recorded elapsed time between p and q?”) is “yes” if one restricts attention to local regions of spacetime. In the case of positive definite metrics, i.e., ones with signature of form (n, 0) – we know geodesics are locally shortest curves. The corresponding result for Lorentzian metrics is that timelike geodesics are locally longest curves.

Let γ: I → M be a smooth, future-directed, timelike curve. Then γ can be reparametrized so as to be a geodesic iff ∀ s ∈ I there exists an open set O containing γ(s) such that , ∀ s1, s2 ∈ I with s1 ≤ s ≤ s2, if the image of γ′ = γ|[s1, s2] is contained in O, then γ′ (and its reparametrizations) are longer than all other timelike curves in O from γ(s1) to γ(s2). (Here γ|[s1, s2] is the restriction of γ to the interval [s1, s2].)

Of all clocks passing locally from p to q, the one that will record the greatest elapsed time is the one that “falls freely” from p to q. To get a clock to read a smaller elapsed time than the maximal value, one will have to accelerate the clock. Now, acceleration requires fuel, and fuel is not free. So the above proposition has the consequence that (locally) “saving time costs money.” And proposition before that may be taken to imply that “with enough money one can save as much time as one wants.” The restriction here to local regions of spacetime is essential. The connection described between clock behavior and acceleration does not, in general, hold on a global scale. In some relativistic spacetimes, one can find future-directed timelike geodesics connecting two events that have different lengths, and so clocks following the curves will record different elapsed times between the events even though both are in a state of free fall. Furthermore – this follows from the preceding claim by continuity considerations alone – it can be the case that of two clocks passing between the events, the one that undergoes acceleration during the trip records a greater elapsed time than the one that remains in a state of free fall. (A rolled-up version of two-dimensional Minkowski spacetime provides a simple example)

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Two-dimensional Minkowski spacetime rolledup into a cylindrical spacetime. Three timelike curves are displayed: γ1 and γ3 are geodesics; γ2 is not; γ1 is longer than γ2; and γ2 is longer than γ3.

The connection we have been considering between clock behavior and acceleration was once thought to be paradoxical. Recall the so-called “clock paradox.” Suppose two clocks, A and B, pass from one event to another in a suitably small region of spacetime. Further suppose A does so in a state of free fall but B undergoes acceleration at some point along the way. Then, we know, A will record a greater elapsed time for the trip than B. This was thought paradoxical because it was believed that relativity theory denies the possibility of distinguishing “absolutely” between free-fall motion and accelerated motion. (If we are equally well entitled to think that it is clock B that is in a state of free fall and A that undergoes acceleration, then, by parity of reasoning, it should be B that records the greater elapsed time.) The resolution of the paradox, if one can call it that, is that relativity theory makes no such denial. The situations of A and B here are not symmetric. The distinction between accelerated motion and free fall makes every bit as much sense in relativity theory as it does in Newtonian physics.

A “timelike curve” should be understood to be a smooth, future-directed, timelike curve parametrized by elapsed proper time – i.e., by arc length. In that case, the tangent field ξa of the curve has unit length (ξaξa = 1). And if a particle happens to have the image of the curve as its worldline, then, at any point, ξa is called the particle’s four-velocity there.

Causal Isomorphism as a Diffeomorphism. Some further Rumination on Philosophy of Science. Thought of the Day 82.0

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes that are both future- and past-distinguishing, and let φ : M → M′ be a ≪-causal isomorphism. Then φ is a diffeomorphism and preserves gab up to a conformal factor; i.e. φ⋆(g′ab) is conformally equivalent to gab.

Under the stated assumptions, φ must be a homeomorphism. If a spacetime (M, gab) is not just past and future distinguishing, but strongly causal, then one can explicitly characterize its (manifold) topology in terms of the relation ≪. In this case, a subset O ⊆ M is open iff, ∀ points p in O, ∃ points q and r in O such that q ≪ p ≪ r and I+(q) ∩ I(r) ⊆ O (Hawking and Ellis). So a ≪-causal isomorphism between two strongly causal spacetimes must certainly be a homeomorphism. Then one invokes a result of Hawking, King, and McCarthy that asserts, in effect, that any continuous ≪-causal isomorphism must be smooth and must preserve the metric up to a conformal factor.

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The following example shows that the proposition fails if the initial restriction on causal structure is weakened to past distinguishability or to future distinguishability alone. We give the example in a two-dimensional version to simplify matters. Start with the manifold R2 together with the Lorentzian metric

gab = (d(at)(db)x) − (sinh2t)(dax)(dbx)

where t, x are global projection coordinates on R2. Next, form a vertical cylinder by identifying the point with coordinates (t, x) with the one having coordinates (t, x + 2). Finally, excise two closed half lines – the sets with respective coordinates {(t, x): x = 0 and t ≥ 0} and {(t, x): x = 1 and t ≥ 0}. Figure shows, roughly, what the null cones look like at every point. (The future direction at each point is taken to be the “upward one.”) The exact form of the metric is not important here. All that is important is the indicated qualitative behavior of the null cones. Along the (punctured) circle C where t = 0, the vector fields (∂/∂t)a and (∂/∂x)a both qualify as null. But as one moves upward or downward from there, the cones close. There are no closed timelike (or null) curves in this spacetime. Indeed, it is future distinguishing because of the excisions. But it fails to be past distinguishing because I(p) = I(q) for all points p and q on C. For all points p there, I(p) is the entire region below C. Now let φ be the bijection of the spacetime onto itself that leaves the “lower open half” fixed but reverses the position of the two upper slabs. Though φ is discontinuous along C, it is a ≪-causal isomorphism. This is the case because every point below C has all points in both upper slabs in its ≪-future.

Ruminations on Philosophy of Science: A Case of Volume Measure Respecting Orientation

Let M be an n–dimensional manifold (n ≥ 1). An s-form on M (s ≥ 1) is a covariant field αb1…bs that is anti-symmetric (i.e., anti-symmetric in each pair of indices). The case where s = n is of special interest.

Let αb1…bn be an n-form on M. Further, let ξi(i = 1,…,n) be a basis for the tangent space at a point in M with dual basis ηi(i=1,…,n). Then αb1…bn can be expressed there in the form

αb1…bn = k n! η1[b1…ηnbn] —– (1)

where

k = αb1…bnξ1b1…ξnbn

(To see this, observe that the two sides of equation (1) have the same action on any collection of n vectors from the set {ξ1b, . . . , ξnb}.) It follows that if αb1…bn and βb1…bn are any two smooth, non-vanishing n-forms on M, then

βb1…bn = f αb1…bn

for some smooth non-vanishing scalar field f. Smooth, non-vanishing n-forms always exist locally on M. (Suppose (U, φ) is a chart with coordinate vector fields (γ⃗1)a, . . . , (γ⃗n)a, and suppose ηib(i = 1, . . . , n) are dual fields. Then η1[b1…ηnbn] qualifies as a smooth, non-vanishing n-form on U.) But they do not necessarily exist globally. Suppose, for example, that M is the two-dimensional Möbius strip, and αab is any smooth two-form on M. We see that αab must vanish somewhere as follows.

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A 2-form αab on the Möbius strip determines a “positive direction of rotation” at every point where it is non-zero. So there cannot be a smooth, non-vanishing 2-form on the Möbius strip.

Let p be any point on M at which αab ≠ 0, and let ξa be any non-zero vector at p. Consider the number αab ξa ρb as ρb rotates though the vectors in Mp. If ρb = ±ξb, the number is zero. If ρb ≠ ±ξb, the number is non-zero. Therefore, as ρb rotates between ξa and −ξa, it is always positive or always negative. Thus αab determines a “positive direction of rotation” away from ξa on Mp. αab must vanish somewhere because one cannot continuously choose positive rotation directions over the entire Möbius strip.

M is said to be orientable if it admits a (globally defined) smooth, non- vanishing n-form. So far we have made no mention of metric structure. Suppose now that our manifold M is endowed with a metric gab of signature (n+, n). We take a volume element on M (with respect to gab) to be a smooth n-form εb1…bn that satisfies the normalization condition

εb1…bn εb1…bn = (−1)nn! —– (2)

Suppose εb1…bn is a volume element on M, and ξi b (i = 1,…,n) is an orthonormal basis for the tangent space at a point in M. Then at that point we have, by equation (1),

εb1…bn = k n! ξ1[b1 …ξbn] —– (3)

where

k = εb1…bn ξ1b1…ξnbn

Hence, by the normalization condition (2),

(−1)nn! = (k n! ξ1[b1 …ξbn]) (k n! ξ1[b1 …ξbn])

= k2 n!2 1/n! (ξ1b1 ξ1b1) … (ξnbn ξnbn) = k2 (−1)n

So k2 = 1 and, therefore, equation (3) yields

εb1…bn ξ1b1…ξnbn = ±1 —– (4)

Clearly, if εb1…bn is a volume element on M, then so is −εb1…bn. It follows from the normalization condition (4) that there cannot be any others. Thus, there are only two possibilities. Either (M, gab) admits no volume elements (at all) or it admits exactly two, and these agree up to sign.

Condition (4) also suggests where the term “volume element” comes from. Given arbitrary vectors γ1a , . . . , γna at a point, we can think of εb1…bn γ1b1 … γnbn as the volume of the (possibly degenerate) parallelepiped determined by the vectors. Notice that, up to sign, εb1…bn is characterized by three properties.

(VE1) It is linear in each index.

(VE2) It is anti-symmetric.

(VE3) It assigns a volume V with |V | = 1 to each orthonormal parallelepiped.

These are conditions we would demand of any would-be volume measure (with respect to gab). If the length of one edge of a parallelepiped is multiplied by a factor k, then its volume should increase by that factor. And if a parallelepiped is sliced into two parts, with the slice parallel to one face, then its volume should be equal to the sum of the volumes of the parts. This leads to (VE1). Furthermore, if any two edges of the parallelepiped are coalligned (i.e., if it is a degenerate parallelepiped), then its volume should be zero. This leads to (VE2). (If for all vectors ξa, εb1…bn ξb1 ξb2 = 0, then it must be the case that εb1 …bn is anti-symmetric in indices (b1, b2). And similarly for all other pairs of indices.) Finally, if the edges of a parallelepiped are orthogonal, then its volume should be equal to the product of the lengths of the edges. This leads to (VE3). The only unusual thing about εb1…bn as a volume measure is that it respects orientation. If it assigns V to the ordered sequence γ1a , . . . , γna, then it assigns (−V) to γ2a, γ1a, γ3a,…,γna, and so forth.

Conformal Factor. Metric Part 3.

Part 1 and Part 2.

Suppose gab is a metric on a manifold M, ∇ is the derivative operator on M compatible with gab, and Rabcd is associated with ∇. Then Rabcd (= gam Rmbcd) satisfies the following conditions.

(1) Rab(cd) = 0.

(2) Ra[bcd] = 0.

(3) R(ab)cd = 0.

(4) Rabcd = Rcdab.

Conditions (1) and (2) follow directly from clauses (2) and (3) of proposition, which goes like

Suppose ∇ is a derivative operator on the manifold M. Then the curvature tensor field Rabcd associated with ∇ satisfies the following conditions:

(1) For all smooth tensor fields αa1…arb1 …bs on M,

2∇[cd] αa1…arb1 …bs = αa1…arnb2…bs Rnb1cd +…+ αa1…arb1…bs-1n Rnbscd – αna2…arb1…bs Ra1ncd -…- αa1…ar-1nb1…bs Rarncd.

(2) Rab(cd) = 0.

(3) Ra[bcd] = 0.

(4) ∇[mRa|b|cd (Bianchi’s identity).

And by clause (1) of that proposition, we have, since ∇agbc = 0,

0 = 2∇[cd]gab = gnbRnacd + ganRnbcd = Rbacd + Rabcd.

That gives us (3). So it will suffice for us to show that clauses (1) – (3) jointly imply (4). Note first that

0 = Rabcd + Radbc + Racdb

= Rabcd − Rdabc − Racbd.

(The first equality follows from (2), and the second from (1) and (3).) So anti-symmetrization over (a, b, c) yields

0 = R[abc]d −Rd[abc] −R[acb]d.

The second term is 0 by clause (2) again, and R[abc]d = −R[acb]d. So we have an intermediate result:

R[abc]d = 0 —– (1)

Now consider the octahedron in the figure below.

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Using conditions (1) – (3) and equation (1), one can see that the sum of the terms corresponding to each triangular face vanishes. For example, the shaded face determines the sum

Rabcd + Rbdca + Radbc = −Rabdc − Rbdac − Rdabc = −3R[abd]c = 0

So if we add the sums corresponding to the four upper faces, and subtract the sums corresponding to the four lower faces, we get (since “equatorial” terms cancel),

4Rabcd −4Rcdab = 0

This gives us (4).

We say that two metrics gab and g′ab on a manifold M are projectively equivalent if their respective associated derivative operators are projectively equivalent – i.e., if their associated derivative operators admit the same geodesics up to reparametrization. We say that they are conformally equivalent if there is a map : M → R such that

g′ab = Ω2gab

is called a conformal factor. (If such a map exists, it must be smooth and non-vanishing since both gab and g′ab are.) Notice that if gab and g′ab are conformally equivalent, then, given any point p and any vectors ξa and ηa at p, they agree on the ratio of their assignments to the two; i.e.,

(g′ab ξa ξa)/(gab ηaηb) =  (gab ξa ξb)/(g′ab ηaηb)

(if the denominators are non-zero).

If two metrics are conformally equivalent with conformal factor, then the connecting tensor field Cabc that links their associated derivative operators can be expressed as a function of Ω.

Metric. Part 1.

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A (semi-Riemannian) metric on a manifold M is a smooth field gab on M that is symmetric and invertible; i.e., there exists an (inverse) field gbc on M such that gabgbc = δac.

The inverse field gbc of a metric gab is symmetric and unique. It is symmetric since

gcb = gnb δnc = gnb(gnm gmc) = (gmn gnb)gmc = δmb gmc = gbc

(Here we use the symmetry of gnm for the third equality.) It is unique because if g′bc is also an inverse field, then

g′bc = g′nc δnb = g′nc(gnm gmb) = (gmn g′nc) gmb = δmc gmb = gcb = gbc

(Here again we use the symmetry of gnm for the third equality; and we use the symmetry of gcb for the final equality.) The inverse field gbc of a metric gab is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A−1 depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field gab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field gbc on M such that gabgbc = δac.

(2) ∀ p in M, and all vectors ξa at p, if gabξa = 0, then ξa =0.

(When the conditions obtain, we say that gab is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξa at any point p, if gab ξa = 0, it follows that ξc = δac ξa = gbc gab ξa = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (Mp)a to (Mp)b defined by ξa → gab ξa is an injective linear map. Since (Mp)a and (Mp)b have the same dimension, it must be surjective as well. So the map must have an inverse gbc defined by gbc(gab ξa) = ξc or gbc gab = δac.

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In the presence of a metric gab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξa at some point, we write gab ξa as ξb; and given a covariant vector ηb, we write gbc ηb as ηc. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor αcab at a point, for example, how should we write gmc αcab? As αmab? Or as αamb? Or as αabm? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write αabc or αacb or αcab. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write gmc αabc as αabm; gmc αacb as αamb; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors αabc and αacb (at some point) need not be equal. Here is an example. Suppose ξ1a, ξ2a, … , ξna is a basis for the tangent space at a point p. Further suppose αabc = ξia ξjb ξkc at the point. Then αacb = ξia ξjc ξkb. Hence, lowering indices, we have αabc =ξia ξjb ξkc but αacb =ξia ξjc ξib at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δba and the Riemann curvature field Rabcd (associated with some derivative operator). Our convention will be to write these as δab and Rabcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δab = δba. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field αb:

δbaαb = (gbngamδnmb = gbnganαb = gbngnaαb = δabαb

Now suppose gab is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1a, ξ2a,…, ξna for the tangent space at p such that

gabξia ξib = +1 if 1≤i≤m

gabξiaξib = −1 if m<i≤n

gabξiaξjb = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of gab at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of gab

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose gab has signature (m, n − m), and ξ1a, ξ2a, . . . , ξna is an orthonormal basis at a point. Further, suppose μa and νa are vectors there. If

μa = ∑ni=1 μi ξia and νa = ∑ni=1 νi ξia, then it follows from the linearity of gab that

gabμa νb = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

gabμaνb = μ1ν1 +…+ μnνn

And where it is Lorentzian,

gab μaνb = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose gab and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξa and λa is a smooth field on γ. Both ∇ and gab determine a criterion of “constancy” for λa. λa is constant with respect to ∇ if ξnnλa = 0 and is constant with respect to gab if gab λa λb is constant along γ – i.e., if ξnn (gab λa λb = 0. It seems natural to consider pairs gab and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with gab if, for all γ and λa as above, λa is constant w.r.t. gab whenever it is constant with respect to ∇.

Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(Mm, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where Mm is a complete manifold of dimension m. We assume that supM |Rm|g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (Mm, g) be a complete Ricci-flat manifold, x0 is a point on M such that d(x0, x) < A. Suppose the ball B(x0, r0) is A−1−non-collapsed, i.e., r−m0|B(x0, r0)| ≥ A−1, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r0 and d(x, x0) < Ar0? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3−mA−m−1 such that B(x, r) is κ-non-collapsed, i.e., r−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure

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We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time.

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However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r2) = (4π)-m/2 r-m ∫M e-d2(y, x)/4r2 dvy —– (3)

which is almost the volume ratio of Bg(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r2, if one replace d2(y, x)/4r2 in the above formula by the reduced distance l((x, t), (y, t − r2)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.