Levi-Civita Connections¶
The class LeviCivitaConnection
implements the Levi-Civita
connection associated with some pseudo-Riemannian metric on a smooth
manifold.
AUTHORS:
Eric Gourgoulhon, Michal Bejger (2013-2015) : initial version
Marco Mancini (2015) : parallelization of some computations
Marius Gerbershagen (2022) : use the first Bianchi identity in the computation of the Riemann tensor
REFERENCES:
- class sage.manifolds.differentiable.levi_civita_connection.LeviCivitaConnection(metric, name, latex_name=None, init_coef=True)¶
Bases:
sage.manifolds.differentiable.affine_connection.AffineConnection
Levi-Civita connection on a pseudo-Riemannian manifold.
Let \(M\) be a differentiable manifold of class \(C^\infty\) (smooth manifold) over \(\RR\) endowed with a pseudo-Riemannian metric \(g\). Let \(C^\infty(M)\) be the algebra of smooth functions \(M\rightarrow \RR\) (cf.
DiffScalarFieldAlgebra
) and let \(\mathfrak{X}(M)\) be the \(C^\infty(M)\)-module of vector fields on \(M\) (cf.VectorFieldModule
). The Levi-Civita connection associated with \(g\) is the unique operator\[\begin{split}\begin{array}{cccc} \nabla: & \mathfrak{X}(M)\times \mathfrak{X}(M) & \longrightarrow & \mathfrak{X}(M) \\ & (u,v) & \longmapsto & \nabla_u v \end{array}\end{split}\]that
is \(\RR\)-bilinear, i.e. is bilinear when considering \(\mathfrak{X}(M)\) as a vector space over \(\RR\)
is \(C^\infty(M)\)-linear w.r.t. the first argument: \(\forall f\in C^\infty(M),\ \nabla_{fu} v = f\nabla_u v\)
obeys Leibniz rule w.r.t. the second argument: \(\forall f\in C^\infty(M),\ \nabla_u (f v) = \mathrm{d}f(u)\, v + f \nabla_u v\)
is torsion-free
is compatible with \(g\): \(\forall (u,v,w)\in \mathfrak{X}(M)^3,\ u(g(v,w)) = g(\nabla_u v, w) + g(v, \nabla_u w)\)
The Levi-Civita connection \(\nabla\) gives birth to the covariant derivative operator acting on tensor fields, denoted by the same symbol:
\[\begin{split}\begin{array}{cccc} \nabla: & T^{(k,l)}(M) & \longrightarrow & T^{(k,l+1)}(M)\\ & t & \longmapsto & \nabla t \end{array}\end{split}\]where \(T^{(k,l)}(M)\) stands for the \(C^\infty(M)\)-module of tensor fields of type \((k,l)\) on \(M\) (cf.
TensorFieldModule
), with the convention \(T^{(0,0)}(M):=C^\infty(M)\). For a vector field \(v\), the covariant derivative \(\nabla v\) is a type-(1,1) tensor field such that\[\forall u \in\mathfrak{X}(M), \ \nabla_u v = \nabla v(., u)\]More generally for any tensor field \(t\in T^{(k,l)}(M)\), we have
\[\forall u \in\mathfrak{X}(M), \ \nabla_u t = \nabla t(\ldots, u)\]Note
The above convention means that, in terms of index notation, the “derivation index” in \(\nabla t\) is the last one:
\[\nabla_c t^{a_1\ldots a_k}_{\quad\quad b_1\ldots b_l} = (\nabla t)^{a_1\ldots a_k}_{\quad\quad b_1\ldots b_l c}\]INPUT:
metric
– the metric \(g\) defining the Levi-Civita connection, as an instance of classPseudoRiemannianMetric
name
– name given to the connectionlatex_name
– (default:None
) LaTeX symbol to denote the connectioninit_coef
– (default:True
) determines whether the Christoffel symbols are initialized (in the top charts on the domain, i.e. disregarding the subcharts)
EXAMPLES:
Levi-Civita connection associated with the Euclidean metric on \(\RR^3\) expressed in spherical coordinates:
sage: forget() # for doctests only sage: M = Manifold(3, 'R^3', start_index=1) sage: c_spher.<r,th,ph> = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') sage: g = M.metric('g') sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2 sage: g.display() g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph sage: nab = g.connection(name='nabla', latex_name=r'\nabla') ; nab Levi-Civita connection nabla associated with the Riemannian metric g on the 3-dimensional differentiable manifold R^3
Let us check that the connection is compatible with the metric:
sage: Dg = nab(g) ; Dg Tensor field nabla(g) of type (0,3) on the 3-dimensional differentiable manifold R^3 sage: Dg == 0 True
and that it is torsionless:
sage: nab.torsion() == 0 True
As a check, let us enforce the computation of the torsion:
sage: sage.manifolds.differentiable.affine_connection.AffineConnection.torsion(nab) == 0 True
The connection coefficients in the manifold’s default frame are Christoffel symbols, since the default frame is a coordinate frame:
sage: M.default_frame() Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph)) sage: nab.coef() 3-indices components w.r.t. Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph)), with symmetry on the index positions (1, 2)
We note that the Christoffel symbols are symmetric with respect to their last two indices (positions (1,2)); their expression is:
sage: nab.coef()[:] # display as a array [[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]], [[0, 1/r, 0], [1/r, 0, 0], [0, 0, -cos(th)*sin(th)]], [[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]] sage: nab.display() # display only the non-vanishing symbols Gam^r_th,th = -r Gam^r_ph,ph = -r*sin(th)^2 Gam^th_r,th = 1/r Gam^th_th,r = 1/r Gam^th_ph,ph = -cos(th)*sin(th) Gam^ph_r,ph = 1/r Gam^ph_th,ph = cos(th)/sin(th) Gam^ph_ph,r = 1/r Gam^ph_ph,th = cos(th)/sin(th) sage: nab.display(only_nonredundant=True) # skip redundancy due to symmetry Gam^r_th,th = -r Gam^r_ph,ph = -r*sin(th)^2 Gam^th_r,th = 1/r Gam^th_ph,ph = -cos(th)*sin(th) Gam^ph_r,ph = 1/r Gam^ph_th,ph = cos(th)/sin(th)
The same display can be obtained via the function
christoffel_symbols_display()
acting on the metric:sage: g.christoffel_symbols_display(chart=c_spher) Gam^r_th,th = -r Gam^r_ph,ph = -r*sin(th)^2 Gam^th_r,th = 1/r Gam^th_ph,ph = -cos(th)*sin(th) Gam^ph_r,ph = 1/r Gam^ph_th,ph = cos(th)/sin(th)
- coef(frame=None)¶
Return the connection coefficients relative to the given frame.
\(n\) being the manifold’s dimension, the connection coefficients relative to the vector frame \((e_i)\) are the \(n^3\) scalar fields \(\Gamma^k_{\ \, ij}\) defined by
\[\nabla_{e_j} e_i = \Gamma^k_{\ \, ij} e_k\]If the connection coefficients are not known already, they are computed
as Christoffel symbols if the frame \((e_i)\) is a coordinate frame
from the above formula otherwise
INPUT:
frame
– (default:None
) vector frame relative to which the connection coefficients are required; if none is provided, the domain’s default frame is assumed
OUTPUT:
connection coefficients relative to the frame
frame
, as an instance of the classComponents
with 3 indices ordered as \((k,i,j)\); for Christoffel symbols, an instance of the subclassCompWithSym
is returned.
EXAMPLES:
Christoffel symbols of the Levi-Civita connection associated to the Euclidean metric on \(\RR^3\) expressed in spherical coordinates:
sage: M = Manifold(3, 'R^3', start_index=1) sage: c_spher.<r,th,ph> = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') sage: g = M.metric('g') sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2 sage: g.display() g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph sage: nab = g.connection() sage: gam = nab.coef() ; gam 3-indices components w.r.t. Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph)), with symmetry on the index positions (1, 2) sage: gam[:] [[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]], [[0, 1/r, 0], [1/r, 0, 0], [0, 0, -cos(th)*sin(th)]], [[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]]
The only non-zero Christoffel symbols:
sage: gam[1,2,2], gam[1,3,3] (-r, -r*sin(th)^2) sage: gam[2,1,2], gam[2,3,3] (1/r, -cos(th)*sin(th)) sage: gam[3,1,3], gam[3,2,3] (1/r, cos(th)/sin(th))
Connection coefficients of the same connection with respect to the orthonormal frame associated to spherical coordinates:
sage: ch_basis = M.automorphism_field() sage: ch_basis[1,1], ch_basis[2,2], ch_basis[3,3] = 1, 1/r, 1/(r*sin(th)) sage: e = c_spher.frame().new_frame(ch_basis, 'e') sage: gam_e = nab.coef(e) ; gam_e 3-indices components w.r.t. Vector frame (R^3, (e_1,e_2,e_3)) sage: gam_e[:] [[[0, 0, 0], [0, -1/r, 0], [0, 0, -1/r]], [[0, 1/r, 0], [0, 0, 0], [0, 0, -cos(th)/(r*sin(th))]], [[0, 0, 1/r], [0, 0, cos(th)/(r*sin(th))], [0, 0, 0]]]
The only non-zero connection coefficients:
sage: gam_e[1,2,2], gam_e[2,1,2] (-1/r, 1/r) sage: gam_e[1,3,3], gam_e[3,1,3] (-1/r, 1/r) sage: gam_e[2,3,3], gam_e[3,2,3] (-cos(th)/(r*sin(th)), cos(th)/(r*sin(th)))
- restrict(subdomain)¶
Return the restriction of the connection to some subdomain.
If such restriction has not been defined yet, it is constructed here.
INPUT:
subdomain
– open subset \(U\) of the connection’s domain (must be an instance ofDifferentiableManifold
)
OUTPUT:
instance of
LeviCivitaConnection
representing the restriction.
EXAMPLES:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: g = M.metric('g') sage: g[0,0], g[1,1] = 1+y^2, 1+x^2 sage: nab = g.connection() sage: nab[:] [[[0, y/(y^2 + 1)], [y/(y^2 + 1), -x/(y^2 + 1)]], [[-y/(x^2 + 1), x/(x^2 + 1)], [x/(x^2 + 1), 0]]] sage: U = M.open_subset('U', coord_def={X: x>0}) sage: nabU = nab.restrict(U); nabU Levi-Civita connection nabla_g associated with the Riemannian metric g on the Open subset U of the 2-dimensional differentiable manifold M sage: nabU[:] [[[0, y/(y^2 + 1)], [y/(y^2 + 1), -x/(y^2 + 1)]], [[-y/(x^2 + 1), x/(x^2 + 1)], [x/(x^2 + 1), 0]]]
Let us check that the restriction is the connection compatible with the restriction of the metric:
sage: nabU(g.restrict(U)).display() nabla_g(g) = 0
- ricci(name=None, latex_name=None)¶
Return the connection’s Ricci tensor.
This method redefines
sage.manifolds.differentiable.affine_connection.AffineConnection.ricci()
to take into account the symmetry of the Ricci tensor for a Levi-Civita connection.The Ricci tensor is the tensor field \(Ric\) of type (0,2) defined from the Riemann curvature tensor \(R\) by
\[Ric(u, v) = R(e^i, u, e_i, v)\]for any vector fields \(u\) and \(v\), \((e_i)\) being any vector frame and \((e^i)\) the dual coframe.
INPUT:
name
– (default:None
) name given to the Ricci tensor; if none, it is set to “Ric(g)”, where “g” is the metric’s namelatex_name
– (default:None
) LaTeX symbol to denote the Ricci tensor; if none, it is set to “\mathrm{Ric}(g)”, where “g” is the metric’s name
OUTPUT:
the Ricci tensor \(Ric\), as an instance of
TensorField
of tensor type (0,2) and symmetric
EXAMPLES:
Ricci tensor of the standard connection on the 2-dimensional sphere:
sage: M = Manifold(2, 'S^2', start_index=1) sage: c_spher.<th,ph> = M.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') sage: g = M.metric('g') sage: g[1,1], g[2,2] = 1, sin(th)^2 sage: g.display() # standard metric on S^2: g = dth⊗dth + sin(th)^2 dph⊗dph sage: nab = g.connection() ; nab Levi-Civita connection nabla_g associated with the Riemannian metric g on the 2-dimensional differentiable manifold S^2 sage: ric = nab.ricci() ; ric Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold S^2 sage: ric.display() Ric(g) = dth⊗dth + sin(th)^2 dph⊗dph
Checking that the Ricci tensor of the Levi-Civita connection associated to Schwarzschild metric is identically zero (as a solution of the Einstein equation):
sage: M = Manifold(4, 'M') sage: c_BL.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') # Schwarzschild-Droste coordinates sage: g = M.lorentzian_metric('g') sage: m = var('m') # mass in Schwarzschild metric sage: g[0,0], g[1,1] = -(1-2*m/r), 1/(1-2*m/r) sage: g[2,2], g[3,3] = r^2, (r*sin(th))^2 sage: g.display() g = (2*m/r - 1) dt⊗dt - 1/(2*m/r - 1) dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph sage: nab = g.connection() ; nab Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M sage: ric = nab.ricci() ; ric Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M sage: ric == 0 True
- riemann(name=None, latex_name=None)¶
Return the Riemann curvature tensor of the connection.
This method redefines
sage.manifolds.differentiable.affine_connection.AffineConnection.riemann()
to take into account the symmetry of the Riemann tensor for a Levi-Civita connection.The Riemann curvature tensor is the tensor field \(R\) of type (1,3) defined by
\[R(\omega, w, u, v) = \left\langle \omega, \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{[u, v]} w \right\rangle\]for any 1-form \(\omega\) and any vector fields \(u\), \(v\) and \(w\).
INPUT:
name
– (default:None
) name given to the Riemann tensor; if none, it is set to “Riem(g)”, where “g” is the metric’s namelatex_name
– (default:None
) LaTeX symbol to denote the Riemann tensor; if none, it is set to “\mathrm{Riem}(g)”, where “g” is the metric’s name
OUTPUT:
the Riemann curvature tensor \(R\), as an instance of
TensorField
EXAMPLES:
Riemann tensor of the Levi-Civita connection associated with the metric of the hyperbolic plane (Poincaré disk model):
sage: M = Manifold(2, 'M', start_index=1) sage: X.<x,y> = M.chart('x:(-1,1) y:(-1,1)', coord_restrictions=lambda x,y: x^2+y^2<1) ....: # Cartesian coord. on the Poincaré disk sage: g = M.metric('g') sage: g[1,1], g[2,2] = 4/(1-x^2-y^2)^2, 4/(1-x^2-y^2)^2 sage: nab = g.connection() sage: riem = nab.riemann(); riem Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold M sage: riem.display_comp() Riem(g)^x_yxy = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1) Riem(g)^x_yyx = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1) Riem(g)^y_xxy = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1) Riem(g)^y_xyx = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
The same computation parallelized on 2 cores:
sage: Parallelism().set(nproc=2) sage: riem_backup = riem sage: g = M.metric('g') sage: g[1,1], g[2,2] = 4/(1-x^2-y^2)^2, 4/(1-x^2-y^2)^2 sage: nab = g.connection() sage: riem = nab.riemann(); riem Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold M sage: riem == riem_backup True sage: Parallelism().set(nproc=1) # switch off parallelization
- torsion()¶
Return the connection’s torsion tensor (identically zero for a Levi-Civita connection).
See
sage.manifolds.differentiable.affine_connection.AffineConnection.torsion()
for the general definition of the torsion tensor.OUTPUT:
the torsion tensor \(T\), as a vanishing instance of
TensorField
EXAMPLES:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: g = M.metric('g') sage: g[0,0], g[1,1] = 1+y^2, 1+x^2 sage: nab = g.connection() sage: t = nab.torsion(); t Tensor field of type (1,2) on the 2-dimensional differentiable manifold M
The torsion of a Levi-Civita connection is always zero:
sage: t.display() 0