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:

• [KN1963]

• [Lee1997]

• [ONe1983]

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 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, (∂/∂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 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, (∂/∂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 of DifferentiableManifold)

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 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

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)

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