5. Applications in Physics A. Thermodynamics B. Hamiltonian mechanics C. Electromagnetism D. Dynamics of a perfect fluid E. Cosmology

A. Thermodynamics 5.1Simple systems 5.2Maxwell and other mathematical identities 5.3Composite thermodynamic systems: Caratheodory's theorem

5.1.Simple Systems 1-component fluid: 1-forms on 2-D manifold with coordinates (V,U).  Frobenius' theorem (§4.26) → (1st law) (δQ is not exact ) δQ is a 1-form in 2-space → its ideal is closed (see Ex 4.31(b)) 2nd law:holds for all systems in thermodynamic equilibrium.

5.2. Maxwell and Other Mathematical Identities → Switching to (S,V) gives → (Maxwell identity) Switching to (T,V) gives → → →

→

5.3. Composite Thermodynamic Systems: Caratheodory's Theorem 1-forms on 2N-D manifold with coordinates (V i,U i ). Frobenius’ theorem:  (See Ex 4.30 ) δQ integrable → equilibrium submanifolds defined by S = constant. → Equilibrium states on different submanifolds cannot be bridged adiabatically. Question: is the converse true? i.e., Not every state reachable adiabatically → S exisits ? c.f. 2nd law: Heat can’t be transfer from cold to hot regions in a closed system without doing work. Caratheodory: 2nd law → S exisits.

Proof: If δQ is not integrable, then  V, W, s.t. in the neighborhod of P, but i.e., the subspace K P of vector fields that annul δQ do not form a hypersurface. e.g.,  δQ is not integrable → Any states near P can be reached adiabatically. Not all states near P can be reached adiabatically → δQ is integrable

B. Hamiltonian Mechanics 4.Hamiltonian Vector Fields 5.Canonical Transformation 6.Map between Vectors & 1-forms Provided by ω 7.Poisson Bracket 8.Many Particle System: Symplectic Forms 9.Linear Dynamical Systems: Symplectic Inner Product & Conserved Quantities 10.Fibre Bundle Structure of the Hamiltonian Equations See Frankel

5.4. Hamiltonian Vector Fields Lagrangian: Momenta: Hamiltonian: Euler-Lagrange eq: Hamilton’s eqs: Let M be the manifold with coordinates ( q i ). Then the set ( q i, q i,t ) is the tangent bundle T(M). The set ( q i, p i ) can be taken as the cotangent bundle T*(M), or a symplectic manifold with the Poincare (symplectic) 2-form

Definition: Symplectic Forms: A 2-form on M 2n is symplectic ( M is then a symplectic manifold ) if ω is non-degenerate, i.e., i X ω is non-singular. Definition: Interior Product: by s.t.if α is a 0-form if α is a 1-form if α is a p-form Properties: Components:

Eq(4.67): → U is called a Hamiltonian vector field. If U is tangent to system trajectory (Conservative system)

5.5. Canonical Transformation is canonical ifA coordinate transformation This can be achieved through a generating function F. E.g., given F = F(q,Q ) s.t. we have&

5.6. Map between Vectors & 1-forms Provided by ω Define or However, ω is not a metric since See Ex 5.3 Ex 5.5: A hamiltonian vector field corresponds to an exact 1-form, i.e. Note difference in order of indices with eq(5.27)

5.7. Poisson Bracket Let then → → Note sign difference with eq(5.31), which can be traced to eq(5.27)

5.8. Many Particle System: Symplectic Forms Symplectic Form Phase space = Symplectic manifold Ex 5.6  8 Symplectic = German for plaiting together

5.9. Linear Dynamical Systems: Symplectic Inner Product & Conserved Quantities Linear system: Linearity: if&are solutions, so is  Solution sub-manifold is also a vector space. A manifold that is also a vector space is a flat manifold (M is isomorphic to R n )

Treating elements of phase space as vectors, we set The (anti-symmetric) symplectic inner product is defined as The symplectic inner product is time-independent if Y (i) are solution curves. Proof: Reminder:

For time independent T i j and V i j, we have → → i.e., ifis a solution, so is Define the canonical energy by If Y is a solution, then E C is conserved on solution curves.

Let U be a vector field on configuration space s.t.c.f. Ex 5.8 The canonical U-momentum is defined by P U is conserved on solution curves. Ifis a solution, so is For the Klein-Gordon eq. 4-current density = j μ is conserved: Setting we have ( DoF =  )

5.10. Fibre Bundle Structure of the Hamiltonian Equations q i defines a configuration space manifold M. Evolution of system is a curve q i (t) on M. Lagrangian L( q i, q i, t ) is a function on the tangent bundle T(M). is a 1-form field on M,i.e., phase space is a cotangent bundle T*(M) Proof: Consider a new set of coordinates and new momenta Since q i, t and Q j, t are elements of the (tangent) fibre of T(M), they transform like contravariant vectors, i.e., →( p i & P j are 1-forms )

Phase space { q i, p i } is a cotangent bundle T*(M). H is a function on T*(M). The symplectic formis coordinate free. Proof: → →  where →  since QED Reminder: System with constraints leads to non-trivial bundles.

C. Electromagnetism 11.Rewriting Maxwell’s Equations Using Differential Forms 12.Charge & Topology 13.The Vector Potential 14.Plane Waves: A Simple Example

5.11. Rewriting Maxwell’s Equations Using Differential Forms Maxwell’s equations in vacuum with sources, Gaussian units with c = 1: Faraday 2-form:  → 

corresponds to the homogeneous eqs.  i,j,k cyclic.

 Inhomogeneous eqs: Metric volume form = →

i,j,k cyclic.

 Inhomogeneous eqs are given by Magnetic monopole:

Alternative Approach See §7.2, Frankel §4.6, Flanders

→ → where

→ →  Inhomogeneous eqs are given by Ex 5.14

12. Charge & Topology Charge = Topology 1. Wheeler: Wormhole (handle) → Pair of  charges. Objections: a.Origin of wormhole unknown. b.Linkage of distant pair of charge unacceptable. 2. Sorkin: Wormhole creating pair of nearby charges of same sign.

5.13. The Vector Potential ← F is invariant under a gauge transformation: A cannot be defined in region with magnetic monopole. Ex 5.16

5.14. Plane Waves: A Simple Example Let → → Static fields ignored

D. Dynamics of a Perfect Fluid 15.Role of Lie Derivatives 16.The Comoving Time-Derivative 17.Equation of Motion 18.Conservation of Vorticity

5.15. Role of Lie Derivatives Perfect fluid: No viscosity. No heat conduction (adiabatic). Quantities conserved in any fluid element ( local conservation laws ) : Mass. Entropy. Vorticity. Conservation laws are more transparent within the framework of Lie derivatives.

5.16. The Comoving Time-Derivative Equation of continuity: where τ is the volume 3-form: = time-derivative operator in a frame travelling with the fluid element. Let ( x, y, z, t ) be the coordinates of a fluid particle in the Galilean space-time. The tangent U to the “world-line” of the fluid particle is ( parameter of world-line = t ) The time-derivative operator in a frame travelling with the fluid element is L U. Proof :

i = x, y, z.  W W  if W is purely spatial, i.e., W t = 0 if W is purely spatial This holds if W is replaced by any purely spatial ( n 0 ) tensor. Reminder: The Galilean space-time is a fibre bundle with t as base. Ex 5.19

5.17. Equation of Motion Adiabatic flow: specific entropy S conserved → Euler’s equation of motion ( see Landau & Lifshitz, “Fluid Mechanics”, §2 ) : ( p = pressure, Φ = gravitational potential ) In Cartesian coordinates: Equation valid only in Cartesian coordinates because : Index mismatch (allowable only in orthonormal bases).  j V i is a tensor only for transformations with coordinate independent Λ i’ j.

Usual remedy is to introduce a covariant derivative (see Chap 6). An alternative approach via Lie derivative is as follows. Index mismatch can be resolved using V i = V i for Cartesian coordinates : is not a tensor equation in general coordinates. Non-tensorial transformation behavior is resolved using → ( d involves only spatial derivatives )

5.18. Conservation of Vorticity ( d involves only spatial derivatives ) d both sides → Since Case I : p = p(ρ ) → ( Helmholtz circulation theorem )

Case II : p = p(ρ, S ) Since  → Since any two 3-forms are proportional in our 3-D space, we can write α = some scalar function, τ = volume 3-form ( Ertel’s Theorem ) d S  both sides of gives →

→ Ex 5.21 : Ex.5.22

E. Cosmology 19.The Cosmological Principle 20.Lie Algebra of Maximal Symmetry 21.The Metric of a Spherically Symmetric 3-Space 22.Construction of the Killing Vectors 23.Open, Closed, & Flat Universes

5.19. The Cosmological Principle General relativity → Cosmology Assuming universe to be homogeneous & isotropic in the large scale, D.G. → only 3 cosmology models (different initial metrics) are possible: Flat, Open, Closed. This result can be derived without using general relativity or Riemannian geometry. Mass distribution of the universe: Small scale [ 10  15 m (nuclear) ~ m (interstellar) ] : lumpy. Star cluster = Galaxy : lumpy Cluster of galaxies ( 10 1 – 10 3 galaxies ) : lumpy Cluster of galaxy clusters = Supercluster : lumpy Beyond superclusters : homogeneous & isotropic

Definition of homogeneity Let G be the isometry Lie group of manifold S with metric tensor field g. The Lie algebra G of G is that of the Killing vector fields of g. Elements of G are diffeomorphisms of S onto itself. The action of G on S is transitive if  P, Q  S,  g  G s.t. g(P) = Q. A manifold S is homogeneous if its isometry group acts transitively on it, i.e., the geometry is the same everywhere on S. Elements of G which leaves a point P on S fixed form a subgroup H P of G. H P is called the isotropy group of P. Since the universe is evolving, the “observed” homogeneity is an interpolation to the “present time”. Spacetime is thus treated as a foliation with leaves of constant time hypersurfaces. A hypersurface is space-like is g is positive-definite on all vectors tangent to it.

The isotropy group H P of P maps any curve through P to another curve through it.  H P : T P → T P (c.f. adjoint representation of a Lie group) A cosmology model M is a homogeneous cosmology if it has a foliation of homogeneous space-like hypersurfaces. Similarly for isotropic cosmology. The universe is observed to be homogeneous on the large scale about us. Cosomological principle: likewise for all observers in the universe. Ex 5.23 A manifold S is isotropic about P if its H P = SO(m). If S is isotropic about all P, it is isotropic.

5.20. Lie Algebra of Maximal Symmetry Let S be a 3-D manifold & ξ a Killing vector field on it, i.e., → where= Christoffel symbol

 ξ is over-determined for n > 1. → A general g may have no Killing vector fields. Task: Find criteria for g to have the maximal set of Killing vectors. is symmetric in i & j → ½ n(n+1) eqs for n variables ξ j.  k eq. gives i → j → k →i : (1)+(3)  (2) : (1) (2) (3) where

 For a given g, if ξ i and ξ i, j at point P are known, then all higher derivatives of ξ at P are known. → ξ is known in any neighborhood of P where ξ is analytic. Hence, a Killing vector field on S is determined given some appropriate values Given ξ i, the symmetric part of ξ i, j is given by at a single point P  S. Number of independent choices of η i is n. That of A i j is ½ n(n  1). → Maximal number of Killing vector fields is ½ n(n+1). In which case, M is maximally symmetric.

A maximally symmetric connected manifold is homogeneous. Proof : S is maximally symmetric →  a Killing field whose tangent at P = any desired value. The 1-par Lie group associated with the Killing field maps P to any point Q in some coord patch U of P. By extending the map across different coord patches, P can be mapped to any Q in S. Thus, the isometry group G maps P to any Q in S. → G acts transitively on S (S is homogeneous ).

Let G be the isotropy group of P. → P is fixed under any action of G. → The Killing fields associated with G vanish at P. If V and W are any 2 Killing fields of G, then [V, W] = 0 at P. Hence, the Lie algebra of G is a subalgebra of that of the isometry group. Ex.5.24: The isotropy group of a space-like S is SO(m). I.e., a maximally symmetric space-like manifold is isotropic.

5.21. The Metric of a Spherically Symmetric 3-Space Let S be a space-like 3-manifold. If the isotropy group of S is SO(3), then S is spherically symmetric everywhere. The Killing vectors of SO(3) define spheres S 2 by their integral curves. → they foliate S. Spherical coordinates: r labels different leaves; (θ, φ) = coord on each leaf. Metric of S induces metric on each S 2 → volume 2-form & its integral (total area). Intrinsic definition of r :→ Caution: r defined this way need not be monotonically increasing everywhere. E.g., 2-manifold S 2 (leaves are circles) : r 1 st increases, then decreases when moving away from P towards P.

At every point Q on a leaf S 2,  a unit normal vector n s.t. ( n is orthogonal to S 2 ) ( n is normalized ) The unit normal vector field is C  everywhere except at the poles where θ= 0 or π. The poles (θ= 0 ) on different spheres can be related by demanding that they lie on the integral curve of n through the pole of an arbitrarily chosen sphere. Example: 2-manifold S 2 n is orthogonal to the leaves S 1. Poles (φ = 0 ) on different leaves lie on integral curve of n.

θ & φ are constant on any integral curve of the unit normal vector field. → Integral curves of n are coordinate lines of r. Since  θ &  φ are tangent to S 2 : → i.e., where f(r) is to be determined by the rest of the isometries of S. Ex 5.25

5.22. Construction of the Killing Vectors Any vector field on S can be written as sum over repeated indices implied where (see §4.29) With we have

 If V is to be a Killing vector, it components must satisfy the Killing eq i.e., where g is the 3-D metric tensor

→ ( no summation on i & j ) g diagonal →

→

where

Non-trivial solutions requires

→ G:G: → F:F:  If m  0, then Hence, the only solutions are l = 0, or 1. or i.e., If m = 0, then, which can only be satisfied by &

For l = 0 :→ For l = 1:→ For l  2 : →& η 1m & ζ 1m are determined by the rest of the Killing eqs, i.e., K rr = 0 K rθ = 0 K r φ = 0

→ The divergence (see §4.16) of a vector is given by For a vector on S 2,so that Treating K = (K rθ, K rφ ) as a 1-form on S 2, we have

→ →

→  l > 0 → → →

→ Summary: For a non-trivial solution, l = 1 and → → →  →→ →

→ V m = const 

Robertson-Walker model : (homogeneous, isotropic 3-space) See I.D.Lawrie, “A Unified Grand Tour of Theoretical Physics”, 2nd ed., Chap 14. i.e., t = proper time. Obervers with fixed r, θ & φ (comoving) are in free fall. If C  0, then gives where Case C = 0 is the same as k = 0 with [ Spatial section is Euclidean (flat) ] Open, Closed, & Flat Universes

Surface of fixed r coordinate is a sphere with physical radius Circumference of the equator ( θ = π/2 ) of the sphere is 0  r < 1 Closed Flat Open Hubble’s law: