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

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 with a pseudo-Riemannian metric \(g\). Let \(C^\infty(M)\) be the algebra of smooth functions \(M\rightarrow \RR\) (cf. DiffScalarFieldAlgebra) and let \(\mathcal{X}(M)\) be the \(C^\infty(M)\)-module of vector fields on \(M\) (cf. VectorFieldModule). The Levi-Civita connection associated with \(g\) is the unique operator

\[\begin{split}\begin{array}{cccc} \nabla: & \mathcal{X}(M)\times \mathcal{X}(M) & \longrightarrow & \mathcal{X}(M) \\ & (u,v) & \longmapsto & \nabla_u v \end{array}\end{split}\]

that

  • is \(\RR\)-bilinear, i.e. is bilinear when considering \(\mathcal{X}(M)\) 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 \(g\): \(\forall (u,v,w)\in \mathcal{X}(M)^3,\ u(g(v,w)) = g(\nabla_u v, w) + g(v, \nabla_u w)\)

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

\[\forall u \in\mathcal{X}(M), \ \nabla_u v = \nabla v(., u)\]

More generally for any tensor field \(t\in T^{(k,l)}(M)\), we have

\[\forall u \in\mathcal{X}(M), \ \nabla_u t = \nabla t(\ldots, u)\]

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 class PseudoRiemannianMetric
  • name – name given to the connection
  • latex_name – (default: None) LaTeX symbol to denote the connection
  • init_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, (d/dr,d/dth,d/dph))
sage: nab.coef()
3-indices components w.r.t. Coordinate frame (R^3, (d/dr,d/dth,d/dph)),
 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 class Components with 3 indices ordered as \((k,i,j)\); for Christoffel symbols, an instance of the subclass CompWithSym 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, (d/dr,d/dth,d/dph)),
 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]]]
sage: # 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]]]
sage: # 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 of DifferentiableManifold)

OUTPUT:

EXAMPLE:

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 name
  • latex_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 set some name and the latex_name to the output.

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 name
  • latex_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

EXAMPLE:

Riemann tensor of the Levi-Civita connection associated with the metric of the hyperbolic plane (Poincare disk model):

sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x:(-1,1) y:(-1,1)')  # Cartesian coord. on the Poincare disk
sage: X.add_restrictions(x^2+y^2<1)
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)
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

EXAMPLE:

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