Hamiltonian analysis of BHT massive gravity

0.21 MB·English

by  M. Blagojević

Hamiltonian analysis of BHT massive gravity
M. Blagojevi´c and B. Cvetkovi´c
∗
University of Belgrade, Institute of Physics,
1 P. O. Box 57, 11001 Belgrade, Serbia
1
0
2

n
a Abstract
J

4 We study the Hamiltonian structure of the Bergshoeff-Hohm-Townsend (BHT)
2
massive gravity with a cosmological constant. In the space of coupling constants

(Λ ,m2), our canonical analysis reveals the special role of the condition Λ /m2 = 1.
] 0 0
c In this sector, the dimension of the physical phase space is found to be N∗6 =−4,
q
- which corresponds to two Lagrangian degree of freedom. When applied to the AdS
r
g asymptotic region, the canonical approach yields the conserved charges of the BTZ
[ black hole, and central charges of the asymptotic symmetry algebra.

2
v
6 1 Introduction
9
5
2 Thenewtheoryofmassivegravityinthreedimensions(3D),recentlyproposedbyBergshoeff,
.
0 Hohm and Townsend (BHT) [1, 2], is defined by adding the parity invariant, curvature-
1
squared terms to the Einstein-Hilbert action. With the cosmological constant Λ and the
0 0
1 sign of the Einstein-Hilbert term σ = 1, the action takes the form
: ±
v
Xi I = a d3x√g σR 2Λ + 1 K , K := Rˆ Rˆij 3R2, (1.1)
− 0 m2 ij − 8
r Z (cid:18) (cid:19)
a
where Rˆ is the Ricci tensor and R the scalar curvature. At the linearized level in asymp-
ij
totically Minkowskian spacetime, the BHT gravity is equivalent to the Pauli-Fierz theory
for a free massive spin-2 field. The action (1.1) ensures the absence of ghosts (negative
energy modes), and the unitarity in flat space [3]; moreover, the theory is renormalizable
[4]. In the AdS background and for generic values of the coupling constants, the unitarity
of the massive gravitons is found to be in conflict with the positivity of central charges in
the boundary CFT [5, 2]. One should also note that the BHT theory possesses a number of
exact solutions [6, 7, 8], its AdS sector is studied in [5, 9], central charges are discussed in
[5, 2, 10], and supersymmetric extension in [11] .
It is interesting to observe that the particle content of the BHT gravity depends on the
values of coupling constants. Thus, if we consider a maximally symmetric vacuum state
defined by G = Λ η , where G is the Einstein tensor and Λ the effective cosmological
ij eff ij ij eff
constant, this configuration solves the BHT field equations if Λ solves a simple quadratic
eff
∗Email addresses: [email protected], [email protected]
1
equation. For Λ /m2 = 1, two solutions for Λ coincide, and the two massive modes
0 eff
−
degenerate with each other [2, 5]. In that case, there is an extra gauge symmetry at the
linearized level which allows massive modes to become partially massless [2, 12, 13]. The
modes corresponding to Λ /m2 = 3 are also found to be special, but they remain massive
0
[2].
Motivated by the fact that the nature of physical modes in the BHT gravity has been
studied only in the linear approximation, see also [14], we use here the constrained Hamil-
tonian approach to clarify the dynamical content of the BHT gravity nonperturbatively. In
particular, we will find out a natural role of the condition Λ /m2 = 1 in the canonical
0
6 −
consistency procedure.
The paper is organized as follows. In section 2, we give a brief account of the basic
dynamical features of the BHT gravity in the Lagrangian formalism, and describe the BTZ
blackholesolution. Insection3, weapplyDirac’smethodforconstraineddynamical systems
[15] to make a consistent canonical analysis of the BHT gravity. Then, in section 4, we
classify the constraints and find that the theory exhibits two local Lagrangian degrees of
freedom. To obtain this result, we used a condition which, when applied to maximally
symmetric solutions, takes the form Λ /m2 = 1, corresponding to the case of massive
0
6 −
gravitons. In section 5, we find the form of the gauge generator, showing thereby that the
obtained classification of constraints is correct. In section 6, we briefly describe the AdS
asymptoticstructure byimposing theBrown-Henneaux asymptoticconditions, findtheform
of the improved generators and the corresponding conserved quantities, and calculate the
central charges of the asymptotic symmetry. Finally, section 7 is devoted to concluding
remarks, while appendices contain some technical details.
Our conventions are given by the following rules: the Latin indices refer to the local
Lorentz frame, the Greek indices refer to the coordinate frame; the middle alphabet letters
(i,j,k,...;µ,ν,λ,...) run over 0,1,2, the first letters of the Greek alphabet (α,β,γ,...) run
over 1,2; the metric components in the local Lorentz frame are η = (+, , ); totally
ij
− −
antisymmetric tensorεijk andtherelatedtensor densityεµνρ arebothnormalizedasε012 = 1.
2 Lagrangian dynamics in the first order formalism
The BHT massive gravity with a cosmological constant is formulated as a gravitational
theory in Riemannian spacetime. Instead of using the standard Riemannian formalism,
with an action defined in terms of the metric as in (1.1), we find it more convenient to
use the triad field and the spin connection as fundamental dynamical variables. Such an
approach can be naturally described in the framework of Poincar´e gauge theory [16], where
basic gravitational variables are the triad field bi and the Lorentz connection Aij = Aji
−
(1-forms), and the corresponding field strengths are the torsion Ti and the curvature Rij
(2-forms). After introducing the notation Aij =: εij ωk and Rij =: εij Rk, we have:
k k
− −
1
Ti = dbi +εi ωj bk, Ri = dωi + εi ωj ωk.
jk jk
∧ 2 ∧
The antisymmetry of Aij ensures that the underlying geometric structure corresponds to
Riemann-Cartan geometry, in which bi is an orthonormal coframe, g := η bi bj is the
ij
⊗
metric of spacetime, ωi is the Cartan connection, and Ti,Ri are the torsion and the Cartan
2
curvature, respectively. For T = 0, this geometry reduces to Riemannian. In what follows,
i
we will omit the wedge product sign for simplicity.
∧
The description of the BHT massive gravity can be technically simplified as follows.
(a) Weusethetriadfieldbi andthespinconnectionωi asindependent dynamicalvariables.
(b) The Riemannian nature of the connection is ensured by imposing the vanishing of
torsion with the help of the Lagrange multiplier λi = λi dxµ.
µ
(c) Finally, by introducing an auxiliary field fi = fi dxµ, we transform the term K into
µ
an expression linear in curvature.
These modifications lead to a new formulation of the BHT massive gravity, classically equiv-
alent to (1.1):
1 1
L = a 2σbiR Λ ε bibjbk + L +λiT . (2.1a)
i − 3 0 ijk m2 K i
(cid:18) (cid:19)
Here, the piece L is linear in curvature and depends on the auxiliary field fi:
K
1
L = R fi V , V := f ⋆ fi f bi = ǫˆ, (2.1b)
K i K K i K
− 4 − V
(cid:0) (cid:1)
where f = fk and ǫˆ = b0b1b2 is the volume 3-form. In the component notation, with
k
R = Gk ε , L takes the well-known form [1]:
imn i kmn K
1
L = f Gik ǫˆ, := (f fik f2).
K ik K K ik
−V V 4 −
(cid:0) (cid:1)
The form of V = V (bi,fi) ensures that after using the field equations to eliminate fi, L
K K K
reduces to Kǫˆ(Appendix A).
The field equations
Variation with respect to bi,ωi,fi,λi, yields the BHT field equations:
1
a 2σR Λ ε bjbk Θ + λ = 0, (2.2a)
i − 0 ijk − m2 i ∇ i
(cid:18) (cid:19)
1
a 2σT + f +ε λmbn = 0, (2.2b)
i i imn
m2∇
(cid:18) (cid:19)
2R ⋆ fi f bi = 0, (2.2c)
i
− −
T = 0, (2.2d)
i (cid:0) (cid:1)
where Θ = ∂L /∂bi is the energy-momentum current (2-form) associated to L , and
i K K
− ∇
is the covariant derivative: for a 1-form X , X = dX +ε ωjXk.
i i i ijk
∇
The last equation ensures that spacetime is Riemannian. The third equation implies:
fi fbi = ⋆Ri = 2Gi bk,
k
−
2f = R, fi = 2Li = 2Li bk, (2.3)
k
where G is the Einstein tensor, and L the Schouten tensor:
ij ij
1 1
G := Rˆ η R, L := Rˆ η R.
ij ij ij ij ij ij
− 2 − 4
3
Introducing the Cotton 2-form C = L , the second equation reads
i i
∇
2a
C +ε λmbn = 0.
i imn
m2
Next, we introduce the Cotton tensor C by C = Ck ǫˆ , where ǫˆ = 1ε bmbn, and note
ij i i k k 2 kmn
that the previous equation, combined with Ci = 0, implies:
i
2a
λ = C , C = ε mn L ,
ij ij ij i m nj
m2 ∇
2a
λ = ( C )bmbn.
i m in
∇ m2 ∇
Now, the first field equation takes the form:
1 2
2σR Λ ε bmbn Θ + ( C )bmbn = 0. (2.4a)
i 0 imn i m in
− − m2 m2 ∇
We can express the energy-momentum current Θ in terms of the corresponding energy-
i
momentum tensor n as (Appendix A)
i
T
1
Θ = nˆǫ , n := δn f (fkn fηkn).
i Ti n Ti i VK − 2 ik −
Expanding (2.4a) in the dual basis ǫˆj, with R = 2G ǫˆj, yields:
i ij
1
σG Λ η K = 0, (2.4b)
ij − 0 ij − 2m2 ij
where
K := 2( C )εmn
ij ij m in j
T − ∇
= Kη 2L Gk 2( C )εmn .
ij ik j m in j
− − ∇
These equations coincide with those found in [5, 2] (Appendix A).
We display here a set of algebraic consequences of the field equations:
f = f , (2.5a)
ij ji
λ = λ , λ = 0, (2.5b)
ij ji
1
σf +3Λ + = 0, (2.5c)
0 K
2m2V
where we used n = . Consider now a maximally symmetric solution, for which
n K
T V
R = Λ ε , Rˆ = 2Λ η , R = 6Λ . (2.6)
ijk eff ijk ij eff ij eff
− −
Equation (2.5c) with f = 2L implies that the effective cosmological constant Λ
km km eff
satisfies the quadratic equation
Λ2 +4m2σΛ 4m2Λ = 0,
eff eff − 0
which yields
Λ = 2m2 σ 1+Λ /m2 . (2.7)
eff 0
− ±
(cid:16) p (cid:17)
4
BTZ black hole solution
IntheAdSsectoroftheBHTgravity, withΛ = 1/ℓ2, thereexistsamaximallysymmetric
eff
−
solution, locally isomorphic to the BTZ black hole [1, 17, 18].
In the Schwartzschild-like coordinates xµ = (t,r,ϕ), the BTZ black hole solution is
defined in terms of the lapse and shift functions, respectively:
r2 16G2J2 4GJ
N2 = 8Gm + + 0 , N = 0 ,
− 0 ℓ2 r2 ϕ r2
(cid:18) (cid:19)
where m ,J are the integration parameters and Λ = 1/ℓ2. The triad filed has the
0 0 eff
−
simple diagonal form
b0 = Ndt, b1 = N−1dr, b2 = r(dϕ+N dt) , (2.8a)
ϕ
while the Riemannian connection reads:
r
ω˜0 = Ndϕ, ω˜1 = N−1N dr, ω˜2 = dt rN dϕ. (2.8b)
ϕ ϕ
− −ℓ2 −
Then, using (2.6) and C = 0, the field equations imply that the Lagrange multiplier λi
ij
vanishes, while the auxiliary field fi is proportional to the triad field:
1
λi = 0, fi = bi. (2.8c)
ℓ2
3 Hamiltonian and constraints
In local coordinates xµ, the component form of the Lagrangian density reads:
1 a 1
= aεµνρ σbi R Λ εijkb b b + + εµνρλi T , (3.1a)
L µ iνρ − 3 0 iµ jν κρ m2LK 2 µ iνρ
(cid:18) (cid:19)
where the term is conveniently represented in the first order formalism as
K
L
1
= εµνρfi R b , (3.1b)
K µ iνρ K
L 2 − V
where b = det(bi ).
µ
Primary constraints. Fromthedefinitionofthecanonicalmomenta(π µ,Π µ,p µ,P µ),
i i i i
conjugate to the basic dynamical variables (bi ,ωi ,λi ,fi ), respectively, we obtain the
µ µ µ µ
primary constraints:
φ 0 := π 0 0, φ α := π α ε0αβλ 0,
i i i i iβ
≈ − ≈
1
Φ 0 := Π 0 0, Φ α := Π α 2aε0αβ σb + f 0.
i i i i iβ iβ
≈ − 2m2 ≈
(cid:18) (cid:19)
p µ 0, P µ 0. (3.2)
i i
≈ ≈
The PB algebra of the primary constraints is displayed in Appendix B.
5
After noting that the term b is bilinear in the variables bi and fi , one can conve-
K 0 0
V
niently represent the canonical Hamiltonian as
a
= bi +ωi +fi +λi + b +∂ Dα,
c 0 i 0 i 0 i 0 i K α
H H K R T m2 V
where
= ε0αβ aσR aΛ ε bj bk + λ ,
i iαβ 0 ijk α β α iβ
H − − ∇
a
= ε0αβ (cid:0)aσT + f +ε bj λk (cid:1) ,
i iαβ α iβ ijk α β
K − m2∇
a (cid:16) (cid:17)
= ε0αβR ,
i iαβ
R −2m2
1
= ε0αβT ,
i iαβ
T −2
a
Dα = ε0αβ ωi 2aσb + f +bi λ .
0 iβ m2 iβ 0 iβ
h (cid:16) (cid:17) i
Secondary constraints. Going over to the total Hamiltonian,
= +ui φ µ +vi Φ µ +wi p µ +zi P µ,
T c µ i µ i µ i µ i
H H
where (ui ,vi ,wi ,zi ) are canonical multipliers, we find that the consistency conditions
µ µ µ µ
of the primary constraints π 0, Π 0, p 0 and P 0 yield the secondary constraints:
i i i i
a
ˆ := + b 0 0,
i i i
H H m2 T ≈
0,
i
K ≈ a
ˆ := + b(f 0 fh 0) 0,
i i i i
R R 2m2 − ≈
0. (3.3)
i
T ≈
They correspond to the µ = 0 components of the field equations (2.2). Using the relation
1
= bi 0 +fi (f 0 fh 0),
K 0 i 0 i i
V T 2 −
the canonical Hamiltonian can be rewritten in the form
= bi ˆ +ωi +fi ˆ +λi +∂ Dα. (3.4)
c 0 i 0 i 0 i 0 i α
H H K R T
The consistency of the remaining primary constraints X := (φ α,Φ α,p α,P α) leads to
A i i i i
the determination of the multipliers (ui ,vi ,wi ,zi ) (Appendix B). However, we find it
α α α α
more convenient to continue our analysis in the reduced phase space formalism. Using the
second class constraints X , we can eliminate the momenta (π a,Π α,p α,P α) and construct
A i i i i
the reduced phase space R , in which the basic nontrivial Dirac brackets (DB) take the
1
following form (Appendix B):
m2
bi ,λj ∗ = ηijε δ, ωi ,fj ∗ = ηijε δ
{ α β}1 0αβ { α β}1 a 0αβ
λi ,fj ∗ = 2m2σηijε δ. (3.5)
{ α β}1 − 0αβ
6
The remaining DBs are the same as the corresponding Poisson brackets.
In R , the total Hamiltonian takes the simpler form:
1
= +ui φ 0 +vi Φ 0 +wi p 0 +zi P 0, (3.6)
T c 0 i 0 i 0 i 0 i
H H
is given by (3.4), and the consistency conditions (3.3) remain unchanged.
c
H
Tertiary constraints. The consistency conditions of the secondary constraints can be
written in the form:
a 1
ˆ ,H ∗ (b µ) ε b(fmµ fhmµ)λn ,
{Hi T}1 ≈ m2∇µ Ti − 2 imn − µ
,H ∗ 0,
{Ki T}1 ≈
1
,H ∗ bε fjk,
{Ti T}1 ≈ −2 ijk
a
ˆ ,H ∗ [b(f µ fh µ)] , (3.7)
{Ri T}1 ≈ 2m2∇µ i − i
where, on the right-hand side, we use the symbolic notation φ˙ := φ,H ∗. The result is
{ T}1
obtained with the help of the canonical algebra of constraints, displayed in Appendix C. By
using (bh µ) 0, the divergence of b µ can be represented in the form
µ i i
∇ ≈ T
1 1
(b µ) bh µ (f fmn f2) b(f fjµ ffµ ) .
µ i i µ mn µ ji i
∇ T ≈ 4 ∇ − − 2∇ −
(cid:2) (cid:3)
The third relation in (3.7) yields the following tertiary constraints:
θ := f f 0, (3.8a)
0β 0β β0
− ≈
θ := f f 0. (3.8b)
αβ αβ βα
− ≈
They represent Hamiltonian counterparts of the Lagrangian relations (2.5a).
ˆ ˆ
To find an explicit form of the consistency conditions for and , we have to replace
i i
˙ H R
the time derivatives φ by their canonical expressions φ,H . To do that, we introduce the
T
{ }
following change of variables in :
T
H
π 0′ := π 0 +f kP 0, zi ′ := zi fi uk , (3.9)
i i i k 0 0 k 0
−
whereupon the (π 0,P 0) piece of takes the form
i i T
H
ui π 0 +zi P 0 = ui π 0′ +zi ′P 0.
0 i 0 i 0 i 0 i
Besides, we introduce the generalized multipliers
Ui = ui +εimnω b ,
µ µ m0 nµ
Zi = zi +εimnω f ,
µ µ m0 nµ
which correspond, on-shell, to bi and fi , respectively; moreover, we define
0 µ 0 µ
∇ ∇
Z′i = Zi fi Um .
µ µ m µ
−
7
The consistency condition of ˆ , multiplied first by bi and then by bi , yields:
i 0 β
R
Uν (f 0 f) f µU0 (Z′α fU0 )+b−1bi [b(f α fh α)] = 0, (3.10a)
ν 0 0 µ α 0 0 α i i
− − − − ∇ −
g0µZ′ +f 0Uα (f α fδα)U0 +b−1bi [b(f α fh α)] = 0. (3.10b)
βµ β α − β − β α β∇α i − i
Thefirstrelations, inwhichthearbitrarymultipliers U andZ′ arecancelled, containsonly
k0 k0
the determined multipliers U and Z′ . Using the expressions for U and Z′ calculated
kα kα kα kα
with the help of Appendix B, one finds that this relation reduces to an identity (Appendix
D). The second relation defines the two components Z′ = bk Z′ of Z′ .
β0 β k0 k0
The consistency condition of ˆ in conjunction with (3.10) yields:
i
H
m2
(fj0h α fjαh 0)Z′ +fjα f fjkh α f + ε (fjn fηjn)λk 0.
i − i jα ∇α ji − i ∇α jk a ijk − n ≈
Substituting here the expression for the determined multiplier Z′ , we find:
jα
fε λjk = 0.
ijk
Thus, for f = 0, we obtain three tertiary constraints:
6
ψ := λ λ 0, (3.11a)
0β 0β β0
− ≈
ψ := λ λ 0. (3.11b)
αβ αβ βα
− ≈
– The consistency conditions of the secondary constraints determine Z′ and produce
β0
the tertiary constraints θ and ψ .
µν µν
Quartic constraints. The consistency of ε0αβθ reads
αβ
4m2
ε0αβθ ,H ∗ bλ 0,
{ αβ T}1 ≈ a ≈
and we have a new, quartic constraint, the canonical counterpart of (2.5b):
χ := λ 0. (3.12a)
≈
The consistency condition of θ is identically satisfied (Appendix D):
0β
θ ,H ∗ = z′ z′ 0. (3.12b)
{ 0β T}1 0β − β0 ≈
The consistency of ε0αβψ reads:
αβ
1
ε0αβψ ,H ∗ 4ab σf +3Λ + 0.
{ αβ T}1 ≈ − 0 2m2VK ≈
(cid:18) (cid:19)
Thus, we have a new quartic constraint:
1
ϕ := σf +3Λ + 0, (3.13a)
0 2m2VK ≈
as expected from (2.5c).
8
To interpret the consistency condition for ψ , we introduce the notation
0β
π 0′′ := π′0 +λk p 0, wi ′ := wi uk λ .
i i i k 0 0 − 0 ik
The, the (π 0,P 0,p 0) piece of the Hamiltonian takes the form
i i i
ui π 0 +wi p 0 +zi P 0 = ui π 0′′ +wi ′p 0 +zi ′P 0,
0 i 0 i 0 i 0 i 0 i 0 i
and we have:
ψ ,H ∗ = w′ w¯′ 0. (3.13b)
{ 0β T}1 0β − β0 ≈
Hence, the multipliers w′ are determined.
β0
– The consistency conditions of the tertiary constraints determine w′ and produce the
0β
quartic constraints χ and ϕ.
End of the consistency procedure. The consistency condition of the quartic con-
straint χ determines the multiplier w′ :
00
χ,H ∗ = w′µ 0,
{ T}1 µ ≈
g00w′ +g0βw¯′ +hiαw¯′ = 0, (3.14)
00 β0 iα
where w¯′ = w¯ λ u¯k .
iα iα − ik α
The consistency condition for the quartic constraint ϕ has the form:
ϕ,H ∗ = Ωµνz′ 0,
{ T}1 µν ≈
1
Ωµν := σgµν + (fµν fgµν) . (3.15)
4m2 −
This relation determines the multiplier z′ , provided the coefficient Ω00 does not vanish.
00
– The consistency conditions for the quartic constraints determine w′ and z′ .
00 00
This finally completes the consistency procedure. At the end, we wish to stress that the
completion of this process is achieved by employing the following extra conditions:
f = 0, (3.16a)
6
Ω00 = 0. (3.16b)
6
Dynamical interpretation of these conditions is discussed in the next section.
4 Classification of constraints
Among the primary constraints, those that appear in with arbitrary multipliers are first
T
H
class (FC):
π 0′′,Π 0 = FC, (4.1a)
i i
while the remaining ones, p 0 and P 0, are second class.
i i
Going to the secondary constraints, we use the following simple theorem:
9
If φ is a FC constraint, then φ,H ∗ is also a FC constraint.
{ T}1
The proof relies on using the Jacoby identity. The theorem implies that the secondary
constraints ¯ := π 0′′,H ∗ and ¯ = Π 0,H ∗ are FC. After a straightforward but
Hi −{ i T}1 Ki −{ i T}1
lengthy calculation, we obtain:
¯ = ˆ′′ +h µ( λ )bk pj0 +h µ( f )bk Pj0,
Hi Hi i ∇µ jk 0 i ∇µ jk 0
¯ = ε (λj pk0 bj λk pn0) ε (fj Pk0 bj fk Pn0), (4.1b)
i i ijk 0 0 n ijk 0 0 n
K K − − − −
where ˆ′′ := ˆ +fk ˆ +λk . As before, the time derivative φ˙ is a short for φ,H ∗.
Hi Hi iRk iTk { T}1
ThetotalHamiltoniancanbeexpressedintermsoftheFCconstraints(uptoanignorable
square of constraints) as follows:
ˆ = bi ¯ +ωi ¯ +ui π 0′′ +vi Π 0. (4.2)
T 0 i 0 i 0 i 0 i
H H K
Inwhatfollows, wewillshowthatthecompleteclassificationofconstraintsinthereduced
space R is given as in Table 1, provided the conditions (3.16) are satisfied.
1
Table 1. Classification of contraints in R
1
First class Second class
Primary π 0′′,Π 0 p 0,P 0
i i i i
Secondary ¯ , ¯ , ˆ′
Hi Ki Ti Ri
Tertiary θ ,θ ,ψ ,ψ
0β αβ 0β αβ
Quartic χ,ϕ
Here, ˆ′ is a suitable modification of ˆ , defined so that it does not contain f :
Ri Ri i0
ab
ˆ′ = + (g00h α g0αh 0)f +g0αf h 0fα .
Ri Ri 2m2 i − i 0α iα − i α
(cid:2) (cid:3)
To prove the content of Table 1, we need to verify the second-class nature of the con-
straints in the last column. This can be done by calculating the determinant of their DBs.
In order to simplify the calculation, we divide the procedure into three simpler steps, as de-
scribed in Appendix E: (i) we start with the subset of 6 constraints Y := (θ ,ϕ,Pα0,P 0)
A 0β 0
and show that they are second class since the determinant of Y ,Y ∗ is nonsingular; then,
{ A B}1
(ii) we extend our considerations to Z := (ψ ,χ,pα0,p 0), and show that these 6 con-
A 0β 0
strains are also second class; finally, (iii) we show in the same manner that the remaining 8
constraints W := ( , ˆ′, 1ε0αβψ , 1ε0αβθ ) are second class.
A Ti Ri 2 αβ 2 αβ
Thus, all 20 constraints (Y ,Z ,W ) are second class.
A B C
Note, however, that this result is valid only if the condition (3.16b) is satisfied, as shown in
Appendix E.
When the classifcation of constraints is complete, the number of independent dynamical
degrees of freedom in the phase space R is given by the formula:
1
N∗ = N 2N N ,
1 2
− −
10