def _del_derived(self):
r"""
Delete the derived quantities.
"""
# First delete the derived quantities pertaining to the mother class:
TensorField._del_derived(self)
# then deletes the inverse automorphism:
self._inverse = None
def _del_derived(self):
r"""
Delete the derived quantities.
"""
# First delete the derived quantities pertaining to the mother class:
TensorField._del_derived(self)
# then deletes the inverse automorphism:
self._inverse = None
def _del_derived(self):
r"""
Delete the derived quantities
"""
TensorField._del_derived(self)
self._exterior_derivative = None
class Metric(SymBilinFormField):
r"""
Base class for pseudo-Riemannian metrics on a differentiable manifold.
INPUT:
- ``domain`` -- the manifold domain on which the metric is defined
(must be an instance of class :class:`Domain`)
- ``name`` -- name given to the metric
- ``signature`` -- (default: None) signature `S` of the metric as a single
integer: `S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the number of
positive terms (resp. number of negative terms) in any diagonal writing
of the metric components; if ``signature`` is not provided, `S` is set to
the manifold's dimension (Riemannian signature)
- ``latex_name`` -- (default: None) LaTeX symbol to denote the metric; if
none, it is formed from ``name``
EXAMPLES:
Metric on a 2-dimensional manifold::
sage: m = Manifold(2, 'M', start_index=1)
sage: c_xy.<x,y> = m.chart('x y')
sage: g = Metric(m, 'g') ; g
pseudo-Riemannian metric 'g' on the 2-dimensional manifold 'M'
sage: latex(g)
g
A metric is a special kind of tensor field and therefore inheritates all the
properties from class :class:`TensorField`::
sage: isinstance(g, TensorField)
True
sage: g.tensor_type
(0, 2)
sage: g.symmetries() # g is symmetric:
symmetry: (0, 1); no antisymmetry
sage: isinstance(g, SymBilinFormField) # a metric is actually a field of symmetric bilinear forms:
True
Setting the metric components in the manifold's default frame::
sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-x
sage: g[:]
[ x + 1 x*y]
[ x*y -x + 1]
sage: g.view()
g = (x + 1) dx*dx + x*y dx*dy + x*y dy*dx + (-x + 1) dy*dy
Metric components in a frame different from the manifold's default one::
sage: c_uv.<u,v> = m.chart('u v') # new chart on M
sage: xy_to_uv = CoordChange(c_xy, c_uv, x+y, x-y) ; xy_to_uv
coordinate change from chart (M, (x, y)) to chart (M, (u, v))
sage: uv_to_xy = xy_to_uv.inverse() ; uv_to_xy
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: m.atlas
[chart (M, (x, y)), chart (M, (u, v))]
sage: m.frames
[coordinate frame (M, (d/dx,d/dy)), coordinate frame (M, (d/du,d/dv))]
sage: g.comp(c_uv.frame)[:] # metric components in frame c_uv.frame expressed in M's default chart (x,y)
[ 1/2*x*y + 1/2 1/2*x]
[ 1/2*x -1/2*x*y + 1/2]
sage: g.view(c_uv.frame)
g = (1/2*x*y + 1/2) du*du + 1/2*x du*dv + 1/2*x dv*du + (-1/2*x*y + 1/2) dv*dv
sage: g.comp(c_uv.frame)[:, c_uv] # metric components in frame c_uv.frame expressed in chart (u,v)
[ 1/8*u^2 - 1/8*v^2 + 1/2 1/4*u + 1/4*v]
[ 1/4*u + 1/4*v -1/8*u^2 + 1/8*v^2 + 1/2]
sage: g.view(c_uv.frame, c_uv)
g = (1/8*u^2 - 1/8*v^2 + 1/2) du*du + (1/4*u + 1/4*v) du*dv + (1/4*u + 1/4*v) dv*du + (-1/8*u^2 + 1/8*v^2 + 1/2) dv*dv
The inverse metric is obtained via :meth:`inverse`::
sage: ig = g.inverse() ; ig
tensor field 'inv_g' of type (2,0) on the 2-dimensional manifold 'M'
sage: ig[:]
[ (x - 1)/(x^2*y^2 + x^2 - 1) x*y/(x^2*y^2 + x^2 - 1)]
[ x*y/(x^2*y^2 + x^2 - 1) -(x + 1)/(x^2*y^2 + x^2 - 1)]
sage: ig.view()
inv_g = (x - 1)/(x^2*y^2 + x^2 - 1) d/dx*d/dx + x*y/(x^2*y^2 + x^2 - 1) d/dx*d/dy + x*y/(x^2*y^2 + x^2 - 1) d/dy*d/dx - (x + 1)/(x^2*y^2 + x^2 - 1) d/dy*d/dy
"""
def __init__(self, domain, name, signature=None, latex_name=None):
SymBilinFormField.__init__(self, domain, name, latex_name)
# signature:
ndim = self.manifold.dim
if signature is None:
signature = ndim
else:
if not isinstance(signature, (int, Integer)):
raise TypeError("The metric signature must be an integer.")
if (signature < - ndim) or (signature > ndim):
raise ValueError("Metric signature out of range.")
if (signature+ndim)%2 == 1:
if ndim%2 == 0:
raise ValueError("The metric signature must be even.")
else:
raise ValueError("The metric signature must be odd.")
self._signature = signature
# the pair (n_+, n_-):
self._signature_pm = ((ndim+signature)/2, (ndim-signature)/2)
self._indic_signat = 1 - 2*(self._signature_pm[1]%2) # (-1)^n_-
Metric._init_derived(self)
def _repr_(self):
r"""
Special Sage function for the string representation of the object.
"""
description = "pseudo-Riemannian metric '%s'" % self.name
description += " on the " + str(self.domain)
return description
def _new_instance(self):
r"""
Create a :class:`Metric` instance with the same signature.
"""
return Metric(self.domain, 'unnamed', signature=self._signature)
def _init_derived(self):
r"""
Initialize the derived quantities
"""
# Initialization of quantities pertaining to the mother class:
SymBilinFormField._init_derived(self)
# inverse metric:
inv_name = 'inv_' + self.name
inv_latex_name = self.latex_name + r'^{-1}'
self._inverse = TensorField(self.domain, 2, 0, inv_name,
inv_latex_name, sym=(0,1))
self._connection = None # Levi-Civita connection (not set yet)
self._weyl = None # Weyl tensor (not set yet)
self._determinants = {} # determinants in various frames
self._sqrt_abs_dets = {} # sqrt(abs(det g)) in various frames
self._vol_forms = [] # volume form and associated tensors
def _del_derived(self):
r"""
Delete the derived quantities
"""
# First the derived quantities from the mother class are deleted:
SymBilinFormField._del_derived(self)
# The inverse metric is cleared:
self._inverse.components.clear()
self._inverse._del_derived()
# The connection and Weyl tensor are reset to None:
self._connection = None
self._weyl = None
# The dictionary of determinants over the various frames is cleared:
self._determinants.clear()
self._sqrt_abs_dets.clear()
# The volume form and the associated tensors is deleted:
del self._vol_forms[:]
def signature(self):
r"""
Signature of the metric.
OUTPUT:
- signature `S` of the metric, defined as the integer
`S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the number of
positive terms (resp. number of negative terms) in any diagonal
writing of the metric components
EXAMPLES:
Signatures on a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: g = Metric(M, 'g') # if not specified, the signature is Riemannian
sage: g.signature()
2
sage: h = Metric(M, 'h', signature=0)
sage: h.signature()
0
"""
return self._signature
def set(self, symbiform):
r"""
Defines the metric from a field of symmetric bilinear forms
INPUT:
- ``symbiform`` -- field of symmetric bilinear forms
"""
if not isinstance(symbiform, SymBilinFormField):
raise TypeError("The argument must be a symmetric bilinear form.")
if symbiform.manifold != self.manifold:
raise TypeError("The manifold of the symmetric bilinear form " +
"differs from that of the metric.")
self.domain = symbiform.domain
self._init_derived()
self.components.clear()
for frame in symbiform.components:
self.components[frame] = symbiform.components[frame].copy()
def inverse(self):
r"""
Return the inverse metric.
EXAMPLES:
Inverse metric on a 2-dimensional manifold::
sage: m = Manifold(2, 'M', start_index=1)
sage: c_xy.<x,y> = m.chart('x y')
sage: g = Metric(m, 'g')
sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-x
sage: g[:] # components in the manifold's default frame
[ x + 1 x*y]
[ x*y -x + 1]
sage: ig = g.inverse() ; ig
tensor field 'inv_g' of type (2,0) on the 2-dimensional manifold 'M'
sage: ig[:]
[ (x - 1)/(x^2*y^2 + x^2 - 1) x*y/(x^2*y^2 + x^2 - 1)]
[ x*y/(x^2*y^2 + x^2 - 1) -(x + 1)/(x^2*y^2 + x^2 - 1)]
If the metric is modified, the inverse metric is automatically updated::
sage: g[1,2] = 0 ; g[:]
[ x + 1 0]
[ 0 -x + 1]
sage: g.inverse()[:]
[ 1/(x + 1) 0]
[ 0 -1/(x - 1)]
"""
from sage.matrix.constructor import matrix
from component import CompFullySym
from vectorframe import CoordFrame
from utilities import simplify_chain
# Is the inverse metric up to date ?
for frame in self.components:
if frame not in self._inverse.components:
# the computation is necessary
manif = self.manifold
dom = self.domain
if isinstance(frame, CoordFrame):
chart = frame.chart
else:
chart = dom.def_chart
si = manif.sindex
nsi = manif.dim + si
try:
gmat = matrix(
[[self.comp(frame)[i, j, chart].express
for j in range(si, nsi)] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
gmat_inv = gmat.inverse()
cinv = CompFullySym(frame, 2)
for i in range(si, nsi):
for j in range(i, nsi): # symmetry taken into account
cinv[i, j, chart] = simplify_chain(
gmat_inv[i-si,j-si])
self._inverse.components[frame] = cinv
return self._inverse
def connection(self, name=None, latex_name=None):
r"""
Return the unique torsion-free affine connection compatible with
``self``.
This is the so-called Levi-Civita connection.
INPUT:
- ``name`` -- (default: None) name given to the Levi-Civita connection;
if none, it is formed from the metric name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Levi-Civita connection; if none, it is formed from the symbol
`\nabla` and the metric symbol
OUTPUT:
- the Levi-Civita connection, as an instance of
:class:`LeviCivitaConnection`.
EXAMPLES:
Levi-Civitation connection associated with the Euclidean metric on
`\RR^3`::
sage: m = Manifold(3, 'R^3', start_index=1)
sage: # Let us use spherical coordinates on R^3:
sage: c_spher.<r,th,ph> = m.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2 # the Euclidean metric
sage: g.connection()
Levi-Civita connection 'nabla_g' associated with the pseudo-Riemannian metric 'g' on the 3-dimensional manifold 'R^3'
sage: g.connection()[:] # Christoffel symbols in spherical coordinates
[[[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]]]
Test of compatibility with the metric::
sage: Dg = g.connection()(g) ; Dg
tensor field 'nabla_g g' of type (0,3) on the 3-dimensional manifold 'R^3'
sage: Dg == 0
True
sage: Dig = g.connection()(g.inverse()) ; Dig
tensor field 'nabla_g inv_g' of type (2,1) on the 3-dimensional manifold 'R^3'
sage: Dig == 0
True
"""
if self._connection is None:
if name is None:
name = 'nabla_' + self.name
if latex_name is None:
latex_name = r'\nabla_{' + self.latex_name + '}'
self._connection = LeviCivitaConnection(self, name, latex_name)
return self._connection
def christoffel(self, chart=None):
r"""
Christoffel symbols of ``self`` with respect to a chart.
INPUT:
- ``chart`` -- (default: None) chart with respect to which the
Christoffel symbolds are required; if none is provided, the
manifold's default chart is assumed.
OUTPUT:
- the set of Christoffel symbols in the given chart, as an instance of
:class:`CompWithSym`
EXAMPLES:
Christoffel symbols of the flat metric on `\RR^3` with respect to
spherical coordinates::
sage: m = Manifold(3, 'R3', r'\RR^3', start_index=1)
sage: X.<r,th,ph> = m.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, r^2*sin(th)^2
sage: g.view() # the standard flat metric expressed in spherical coordinates
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: Gam = g.christoffel() ; Gam
3-indices components w.r.t. the coordinate frame (R3, (d/dr,d/dth,d/dph)), with symmetry on the index positions (1, 2)
sage: print type(Gam)
<class 'sage.geometry.manifolds.component.CompWithSym'>
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: Gam[1,2,2]
-r
sage: Gam[2,1,2]
1/r
sage: Gam[3,1,3]
1/r
sage: Gam[3,2,3]
cos(th)/sin(th)
sage: Gam[2,3,3]
-cos(th)*sin(th)
"""
if chart is None:
frame = self.domain.def_chart.frame
else:
frame = chart.frame
return self.connection().coef(frame)
def riemann(self, frame=None, name=None, latex_name=None):
r"""
Return the Riemann curvature tensor associated with the metric.
This method is actually a shortcut for ``self.connection().riemann()``
The Riemann curvature tensor is the tensor field `R` of type (1,3)
defined by
.. MATH::
R(\omega, u, v, w) = \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:
- ``frame`` -- (default: None) vector frame in which the computation
must be performed; if none is provided, the computation is performed
in a frame in which the metric components are known, privileging the
domain's default frame.
- ``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
:class:`TensorField`
EXAMPLES:
Riemann tensor of the standard metric on the 2-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: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.view() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.riemann()
tensor field 'Riem(g)' of type (1,3) on the 2-dimensional manifold 'S^2'
sage: g.riemann()[:]
[[[[0, 0], [0, 0]], [[0, sin(th)^2], [-sin(th)^2, 0]]],
[[[0, (cos(th)^2 - 1)/sin(th)^2], [1, 0]], [[0, 0], [0, 0]]]]
In dimension 2, the Riemann tensor can be expressed entirely in terms of
the Ricci scalar `r`:
.. MATH::
R^i_{\ \, jlk} = \frac{r}{2} \left( \delta^i_{\ \, k} g_{jl}
- \delta^i_{\ \, l} g_{jk} \right)
This formula can be checked here, with the r.h.s. rewritten as
`-r g_{j[k} \delta^i_{\ \, l]}`::
sage: g.riemann() == -g.ricci_scalar()*(g*IdentityMap(m)).antisymmetrize([2,3])
True
"""
return self.connection().riemann(frame, name, latex_name)
def ricci(self, frame=None, name=None, latex_name=None):
r"""
Return the Ricci tensor associated with the metric.
This method is actually a shortcut for ``self.connection().ricci()``
The Ricci tensor is the tensor field `Ric` of type (0,2)
defined from the Riemann curvature tensor `R` by
.. MATH::
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:
- ``frame`` -- (default: None) vector frame in which the computation
must be performed; if none is provided, the computation is performed
in a frame in which the metric components are known, privileging the
domain's default frame.
- ``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 :class:`SymBilinFormField`
EXAMPLES:
Ricci tensor of the standard metric on the 2-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: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.view() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.ricci()
field of symmetric bilinear forms 'Ric(g)' on the 2-dimensional manifold 'S^2'
sage: g.ricci()[:]
[ 1 0]
[ 0 sin(th)^2]
sage: g.ricci() == a^(-2) * g
True
"""
return self.connection().ricci(frame, name, latex_name)
def ricci_scalar(self, frame=None, name=None, latex_name=None):
r"""
Return the Ricci scalar associated with the metric.
The Ricci scalar is the scalar field `r` defined from the Ricci tensor
`Ric` and the metric tensor `g` by
.. MATH::
r = g^{ij} Ric_{ij}
INPUT:
- ``frame`` -- (default: None) vector frame in which the computation
must be performed; if none is provided, the computation is performed
in a frame in which the metric components are known, privileging
the domain's default frame.
- ``name`` -- (default: None) name given to the Ricci scalar;
if none, it is set to "r(g)", where "g" is the metric's name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Ricci scalar; if none, it is set to "\\mathrm{r}(g)", where "g"
is the metric's name
OUTPUT:
- the Ricci scalar `r`, as an instance of :class:`ScalarField`
EXAMPLES:
Ricci scalar of the standard metric on the 2-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: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.view() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.ricci_scalar()
scalar field 'r(g)' on the 2-dimensional manifold 'S^2'
sage: g.ricci_scalar().expr()
2/a^2
"""
return self.connection().ricci_scalar(frame, name, latex_name)
def weyl(self, name=None, latex_name=None):
r"""
Return the Weyl conformal tensor associated with the metric.
The Weyl conformal tensor is the tensor field `C` of type (1,3)
defined as the trace-free part of the Riemann curvature tensor `R`
INPUT:
- ``name`` -- (default: None) name given to the Weyl conformal tensor;
if none, it is set to "C(g)", where "g" is the metric's name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Weyl conformal tensor; if none, it is set to "\\mathrm{C}(g)", where
"g" is the metric's name
OUTPUT:
- the Weyl conformal tensor `C`, as an instance of :class:`TensorField`
EXAMPLES:
Checking that the Weyl tensor identically vanishes on a 3-dimensional
manifold, for instance the hyperbolic space `H^3`::
sage: m = Manifold(3, 'H^3', start_index=1)
sage: X.<rh,th,ph> = m.chart(r'rh:[0,+oo):\rho th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: g = Metric(m, 'g')
sage: b = var('b')
sage: g[1,1], g[2,2], g[3,3] = b^2, (b*sinh(rh))^2, (b*sinh(rh)*sin(th))^2
sage: g.view() # standard metric on H^3:
g = b^2 drh*drh + b^2*sinh(rh)^2 dth*dth + b^2*sin(th)^2*sinh(rh)^2 dph*dph
sage: C = g.weyl() ; C
tensor field 'C(g)' of type (1,3) on the 3-dimensional manifold 'H^3'
sage: C == 0
True
"""
from rank2field import IdentityMap
if self._weyl is None:
n = self.manifold.dim
if n < 3:
raise ValueError("The Weyl tensor is not defined for a " +
"manifold of dimension n <= 2.")
delta = IdentityMap(self.domain)
riem = self.riemann()
ric = self.ricci()
rscal = self.ricci_scalar()
# First index of the Ricci tensor raised with the metric
ricup = ric.up(self, 0)
# The identity map is expressed in a frame in which the Riemann
# tensor is known
delta.comp(riem.pick_a_frame())
aux = self*ricup + ric*delta - rscal/(n-1)* self*delta
self._weyl = riem + 2/(n-2)* aux.antisymmetrize([2,3])
if name is None:
self._weyl.name = "C(" + self.name + ")"
else:
self._weyl.name = name
if latex_name is None:
self._weyl.latex_name = r"\mathrm{C}\left(" + self.latex_name \
+ r"\right)"
else:
self._weyl.latex_name = latex_name
return self._weyl
def determinant(self, frame=None):
r"""
Determinant of the metric components in the specified frame.
INPUT:
- ``frame`` -- (default: None) vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the domain's default frame is used. If a chart is
provided, the associated coordinate frame is used
OUTPUT:
- the determinant `\det (g_{ij})`, as an instance of :class:`ScalarField`
EXAMPLES:
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x y')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.determinant() # determinant in M's default frame
sage: s.expr()
-x^2*y^2 - (x + 1)*y + x + 1
Determinant in a frame different from the default's one::
sage: Y.<u,v> = M.chart('u v')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: g.comp(Y.frame)[:, Y]
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.determinant(Y.frame).expr()
-1/4*x^2*y^2 - 1/4*(x + 1)*y + 1/4*x + 1/4
sage: g.determinant(Y.frame).expr(Y)
-1/64*u^4 - 1/64*v^4 + 1/32*(u^2 + 2)*v^2 - 1/16*u^2 + 1/4*v + 1/4
A chart can be passed instead of a frame::
sage: g.determinant(X) is g.determinant(X.frame)
True
sage: g.determinant(Y) is g.determinant(Y.frame)
True
The metric determinant depends on the frame::
sage: g.determinant(X.frame) == g.determinant(Y.frame)
False
"""
from sage.matrix.constructor import matrix
from scalarfield import ScalarField
from utilities import simple_determinant, simplify_chain
manif = self.manifold
dom = self.domain
if frame is None:
frame = dom.def_frame
if frame in dom.atlas:
# frame is actually a chart and is changed to the associated
# coordinate frame:
frame = frame.frame
if frame not in self._determinants:
# a new computation is necessary
resu = ScalarField(dom)
gg = self.comp(frame)
i1 = manif.sindex
for chart in gg[[i1, i1]].express:
gm = matrix( [[ gg[i, j, chart].express
for j in manif.irange()] for i in manif.irange()] )
detgm = simplify_chain(simple_determinant(gm))
resu.add_expr(detgm, chart=chart)
self._determinants[frame] = resu
return self._determinants[frame]
def sqrt_abs_det(self, frame=None):
r"""
Square root of the absolute value of the determinant of the metric
components in the specified frame.
INPUT:
- ``frame`` -- (default: None) vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the domain's default frame is used. If a chart is
provided, the associated coordinate frame is used
OUTPUT:
- `\sqrt{|\det (g_{ij})|}`, as an instance of :class:`ScalarField`
EXAMPLES:
Standard metric in the Euclidean space `\RR^3` with spherical
coordinates::
sage: m = Manifold(3, 'M', 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 = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.view()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: g.sqrt_abs_det().expr()
r^2*sin(th)
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x y')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.sqrt_abs_det() ; s
scalar field on the 2-dimensional manifold 'M'
sage: s.expr()
sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
Determinant in a frame different from the default's one::
sage: Y.<u,v> = M.chart('u v')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: g.comp(Y.frame)[:, Y]
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.sqrt_abs_det(Y.frame).expr()
1/2*sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
sage: g.sqrt_abs_det(Y.frame).expr(Y)
1/8*sqrt(-u^4 - v^4 + 2*(u^2 + 2)*v^2 - 4*u^2 + 16*v + 16)
A chart can be passed instead of a frame::
sage: g.sqrt_abs_det(Y) is g.sqrt_abs_det(Y.frame)
True
The metric determinant depends on the frame::
sage: g.sqrt_abs_det(X.frame) == g.sqrt_abs_det(Y.frame)
False
"""
from sage.functions.other import sqrt
from scalarfield import ScalarField
from utilities import simplify_chain
manif = self.manifold
dom = self.domain
if frame is None:
frame = dom.def_frame
if frame in dom.atlas:
# frame is actually a chart and is changed to the associated
# coordinate frame:
frame = frame.frame
if frame not in self._sqrt_abs_dets:
# a new computation is necessary
detg = self.determinant(frame)
resu = ScalarField(dom)
for chart in detg.express:
x = self._indic_signat * detg.express[chart].express # |g|
x = simplify_chain(sqrt(x))
resu.add_expr(x, chart=chart)
self._sqrt_abs_dets[frame] = resu
return self._sqrt_abs_dets[frame]
def volume_form(self, contra=0):
r"""
Volume form (Levi-Civita tensor) `\epsilon` associated with the metric.
This assumes that the manifold is orientable.
The volume form `\epsilon` is a `n`-form (`n` being the manifold's
dimension) such that for any vector basis `(e_i)` that is orthonormal
with respect to the metric,
.. MATH::
\epsilon(e_1,\ldots,e_n) = \pm 1
There are only two such `n`-forms, which are opposite of each other.
The volume form `\epsilon` is selected such that the domain's default
frame is right-handed with respect to it.
INPUT:
- ``contra`` -- (default: 0) number of contravariant indices of the
returned tensor
OUTPUT:
- if ``contra = 0`` (default value): the volume `n`-form `\epsilon`, as
an instance of :class:`DiffForm`
- if ``contra = k``, with `1\leq k \leq n`, the tensor field of type
(k,n-k) formed from `\epsilon` by raising the first k indices with the
metric (see method :meth:`TensorField.up`); the output is then an
instance of :class:`TensorField`, with the appropriate antisymmetries
EXAMPLES:
Volume form on `\RR^3` with spherical coordinates::
sage: m = Manifold(3, 'M', 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 = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.view()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: eps = g.volume_form() ; eps
3-form 'eps_g' on the 3-dimensional manifold 'M'
sage: eps.view()
eps_g = r^2*sin(th) dr/\dth/\dph
sage: eps[[1,2,3]] == g.sqrt_abs_det()
True
sage: latex(eps)
\epsilon_{g}
The tensor field of components `\epsilon^i_{\ \, jk}` (``contra=1``)::
sage: eps1 = g.volume_form(1) ; eps1
tensor field of type (1,2) on the 3-dimensional manifold 'M'
sage: eps1.symmetries()
no symmetry; antisymmetry: (1, 2)
sage: eps1[:]
[[[0, 0, 0], [0, 0, r^2*sin(th)], [0, -r^2*sin(th), 0]],
[[0, 0, -sin(th)], [0, 0, 0], [sin(th), 0, 0]],
[[0, 1/sin(th), 0], [-1/sin(th), 0, 0], [0, 0, 0]]]
The tensor field of components `\epsilon^{ij}_{\ \ k}` (``contra=2``)::
sage: eps2 = g.volume_form(2) ; eps2
tensor field of type (2,1) on the 3-dimensional manifold 'M'
sage: eps2.symmetries()
no symmetry; antisymmetry: (0, 1)
sage: eps2[:]
[[[0, 0, 0], [0, 0, sin(th)], [0, -1/sin(th), 0]],
[[0, 0, -sin(th)], [0, 0, 0], [1/(r^2*sin(th)), 0, 0]],
[[0, 1/sin(th), 0], [-1/(r^2*sin(th)), 0, 0], [0, 0, 0]]]
The tensor field of components `\epsilon^{ijk}` (``contra=3``)::
sage: eps3 = g.volume_form(3) ; eps3
tensor field of type (3,0) on the 3-dimensional manifold 'M'
sage: eps3.symmetries()
no symmetry; antisymmetry: (0, 1, 2)
sage: eps3[:]
[[[0, 0, 0], [0, 0, 1/(r^2*sin(th))], [0, -1/(r^2*sin(th)), 0]],
[[0, 0, -1/(r^2*sin(th))], [0, 0, 0], [1/(r^2*sin(th)), 0, 0]],
[[0, 1/(r^2*sin(th)), 0], [-1/(r^2*sin(th)), 0, 0], [0, 0, 0]]]
sage: eps3[1,2,3]
1/(r^2*sin(th))
sage: eps3[[1,2,3]] * g.sqrt_abs_det() == 1
True
"""
if self._vol_forms == []:
# a new computation is necessary
manif = self.manifold
dom = self.domain
ndim = manif.dim
eps = DiffForm(dom, ndim, name='eps_'+self.name,
latex_name=r'\epsilon_{'+self.latex_name+r'}')
ind = tuple(range(manif.sindex, manif.sindex+ndim))
eps[[ind]] = self.sqrt_abs_det(dom.def_frame)
self._vol_forms.append(eps) # Levi-Civita tensor constructed
# Tensors related to the Levi-Civita one by index rising:
for k in range(1, ndim+1):
epskm1 = self._vol_forms[k-1]
epsk = epskm1.up(self, k-1)
if k > 1:
# restoring the antisymmetry after the up operation:
epsk = epsk.antisymmetrize(range(k))
self._vol_forms.append(epsk)
return self._vol_forms[contra]
class Metric(SymBilinFormField):
r"""
Base class for pseudo-Riemannian metrics on a differentiable manifold.
INPUT:
- ``manifold`` -- the manifold on which the metric is defined
- ``name`` -- name given to the metric
- ``signature`` -- (default: None) signature `S` of the metric as a single
integer: `S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the number of
positive terms (resp. number of negative terms) in any diagonal writing
of the metric components; if ``signature`` is not provided, `S` is set to
the manifold's dimension (Riemannian signature)
- ``latex_name`` -- (default: None) LaTeX symbol to denote the metric; if
none, it is formed from ``name``
EXAMPLES:
Metric on a 2-dimensional manifold::
sage: m = Manifold(2, 'M', start_index=1)
sage: c_xy = Chart(m, 'x y', 'xy')
sage: g = Metric(m, 'g') ; g
pseudo-Riemannian metric 'g' on the 2-dimensional manifold 'M'
sage: latex(g)
g
A metric is a special kind of tensor field and therefore inheritates all the
properties from class :class:`TensorField`::
sage: isinstance(g, TensorField)
True
sage: g.tensor_type
(0, 2)
sage: g.symmetries() # g is symmetric:
symmetry: (0, 1); no antisymmetry
sage: isinstance(g, SymBilinFormField) # a metric is actually a field of symmetric bilinear forms:
True
Setting the metric components in the manifold's default frame::
sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-x
sage: g[:]
[ x + 1 x*y]
[ x*y -x + 1]
sage: g.show()
g = (x + 1) dx*dx + x*y dx*dy + x*y dy*dx + (-x + 1) dy*dy
Metric components in a frame different from the manifold's default one::
sage: c_uv = Chart(m, 'u v', 'uv') # new chart on M
sage: xy_to_uv = CoordChange(c_xy, c_uv, x+y, x-y) ; xy_to_uv
coordinate change from chart 'xy' (x, y) to chart 'uv' (u, v)
sage: uv_to_xy = xy_to_uv.inverse() ; uv_to_xy
coordinate change from chart 'uv' (u, v) to chart 'xy' (x, y)
sage: m.get_atlas()
{'xy': chart 'xy' (x, y), 'uv': chart 'uv' (u, v)}
sage: m.frames
{'xy_b': coordinate basis 'xy_b' (d/dx,d/dy), 'uv_b': coordinate basis 'uv_b' (d/du,d/dv)}
sage: g.comp('uv_b')[:] # metric components in frame 'uv_b' expressed in M's default chart (x,y)
[ 1/2*x*y + 1/2 1/2*x]
[ 1/2*x -1/2*x*y + 1/2]
sage: g.show('uv_b')
g = (1/2*x*y + 1/2) du*du + 1/2*x du*dv + 1/2*x dv*du + (-1/2*x*y + 1/2) dv*dv
sage: g.comp('uv_b')[:, 'uv'] # metric components in frame 'uv_b' expressed in chart (u,v)
[ 1/8*u^2 - 1/8*v^2 + 1/2 1/4*u + 1/4*v]
[ 1/4*u + 1/4*v -1/8*u^2 + 1/8*v^2 + 1/2]
sage: g.show('uv_b', 'uv')
g = (1/8*u^2 - 1/8*v^2 + 1/2) du*du + (1/4*u + 1/4*v) du*dv + (1/4*u + 1/4*v) dv*du + (-1/8*u^2 + 1/8*v^2 + 1/2) dv*dv
The inverse metric is obtained via :meth:`inverse`::
sage: ig = g.inverse() ; ig
tensor field 'inv_g' of type (2,0) on the 2-dimensional manifold 'M'
sage: ig[:]
[ (x - 1)/(x^2*y^2 + x^2 - 1) x*y/(x^2*y^2 + x^2 - 1)]
[ x*y/(x^2*y^2 + x^2 - 1) -(x + 1)/(x^2*y^2 + x^2 - 1)]
sage: ig.show()
inv_g = (x - 1)/(x^2*y^2 + x^2 - 1) d/dx*d/dx + x*y/(x^2*y^2 + x^2 - 1) d/dx*d/dy + x*y/(x^2*y^2 + x^2 - 1) d/dy*d/dx - (x + 1)/(x^2*y^2 + x^2 - 1) d/dy*d/dy
"""
def __init__(self, manifold, name, signature=None, latex_name=None):
SymBilinFormField.__init__(self, manifold, name, latex_name)
# signature:
ndim = self.manifold.dim
if signature is None:
signature = ndim
else:
if not isinstance(signature, (int, Integer)):
raise TypeError("The metric signature must be an integer.")
if (signature < -ndim) or (signature > ndim):
raise ValueError("Metric signature out of range.")
if (signature + ndim) % 2 == 1:
if ndim % 2 == 0:
raise ValueError("The metric signature must be even.")
else:
raise ValueError("The metric signature must be odd.")
self._signature = signature
# the pair (n_+, n_-):
self._signature_pm = ((ndim + signature) / 2, (ndim - signature) / 2)
self._indic_signat = 1 - 2 * (self._signature_pm[1] % 2) # (-1)^n_-
# inverse metric:
inv_name = 'inv_' + self.name
inv_latex_name = self.latex_name + r'^{-1}'
self._inverse = TensorField(self.manifold,
2,
0,
inv_name,
inv_latex_name,
sym=(0, 1))
self._connection = None # Levi-Civita connection (not set yet)
self._weyl = None # Weyl tensor (not set yet)
self._determinants = {} # determinants in various frames
self._sqrt_abs_dets = {} # sqrt(abs(det g)) in various frames
self._vol_forms = [] # volume form and associated tensors
def _repr_(self):
r"""
Special Sage function for the string representation of the object.
"""
description = "pseudo-Riemannian metric '%s'" % self.name
description += " on the " + str(self.manifold)
return description
def _new_instance(self):
r"""
Create a :class:`Metric` instance with the same signature.
"""
return Metric(self.manifold, 'unnamed', signature=self._signature)
def _del_derived(self):
r"""
Delete the derived quantities
"""
# First the derived quantities from the mother class are deleted:
SymBilinFormField._del_derived(self)
# The inverse metric is cleared:
self._inverse.components.clear()
self._inverse._del_derived()
# The connection and Weyl tensor are reset to None:
self._connection = None
self._weyl = None
# The dictionary of determinants over the various frames is cleared:
self._determinants.clear()
self._sqrt_abs_dets.clear()
# The volume form and the associated tensors is deleted:
del self._vol_forms[:]
def signature(self):
r"""
Signature of the metric.
OUTPUT:
- signature `S` of the metric, defined as the integer
`S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the number of
positive terms (resp. number of negative terms) in any diagonal
writing of the metric components
EXAMPLES:
Signatures on a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: g = Metric(M, 'g') # if not specified, the signature is Riemannian
sage: g.signature()
2
sage: h = Metric(M, 'h', signature=0)
sage: h.signature()
0
"""
return self._signature
def inverse(self):
r"""
Return the inverse metric.
EXAMPLES:
Inverse metric on a 2-dimensional manifold::
sage: m = Manifold(2, 'M', start_index=1)
sage: c_xy = Chart(m, 'x y', 'xy')
sage: g = Metric(m, 'g')
sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-x
sage: g[:] # components in the manifold's default frame
[ x + 1 x*y]
[ x*y -x + 1]
sage: ig = g.inverse() ; ig
tensor field 'inv_g' of type (2,0) on the 2-dimensional manifold 'M'
sage: ig[:]
[ (x - 1)/(x^2*y^2 + x^2 - 1) x*y/(x^2*y^2 + x^2 - 1)]
[ x*y/(x^2*y^2 + x^2 - 1) -(x + 1)/(x^2*y^2 + x^2 - 1)]
If the metric is modified, the inverse metric is automatically updated::
sage: g[1,2] = 0 ; g[:]
[ x + 1 0]
[ 0 -x + 1]
sage: g.inverse()[:]
[ 1/(x + 1) 0]
[ 0 -1/(x - 1)]
"""
from sage.matrix.constructor import matrix
from component import CompFullySym
from utilities import simplify_chain
# Is the inverse metric up to date ?
for frame_name in self.components:
if frame_name not in self._inverse.components:
# the computation is necessary
manif = self.manifold
if frame_name[-2:] == '_b':
# coordinate basis
chart_name = frame_name[:-2]
else:
chart_name = manif.def_chart.name
si = manif.sindex
nsi = manif.dim + si
try:
gmat = matrix([[
self.comp(frame_name)[i, j, chart_name].express
for j in range(si, nsi)
] for i in range(si, nsi)])
except KeyError:
continue
gmat_inv = gmat.inverse()
cinv = CompFullySym(manif, 2, frame_name)
for i in range(si, nsi):
for j in range(i, nsi): # symmetry taken into account
cinv[i, j,
chart_name] = simplify_chain(gmat_inv[i - si,
j - si])
self._inverse.components[frame_name] = cinv
return self._inverse
def connection(self, name=None, latex_name=None):
r"""
Return the unique torsion-free affine connection compatible with
``self``.
This is the so-called Levi-Civita connection.
INPUT:
- ``name`` -- (default: None) name given to the Levi-Civita connection;
if none, it is formed from the metric name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Levi-Civita connection; if none, it is formed from the symbol
`\nabla` and the metric symbol
OUTPUT:
- the Levi-Civita connection, as an instance of
:class:`LeviCivitaConnection`.
EXAMPLES:
Levi-Civitation connection associated with the Euclidean metric on
`\RR^3`::
sage: m = Manifold(3, 'R^3', start_index=1)
sage: # Let us use spherical coordinates on R^3:
sage: c_spher = Chart(m, r'r:positive, th:positive:\theta, ph:\phi', 'spher')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2 # the Euclidean metric
sage: g.connection()
Levi-Civita connection 'nabla_g' associated with the pseudo-Riemannian metric 'g' on the 3-dimensional manifold 'R^3'
sage: g.connection()[:] # Christoffel symbols in spherical coordinates
[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]],
[[0, 1/r, 0], [1/r, 0, 0], [0, 0, -sin(th)*cos(th)]],
[[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]]
Test of compatibility with the metric::
sage: Dg = g.connection()(g) ; Dg
tensor field 'nabla_g g' of type (0,3) on the 3-dimensional manifold 'R^3'
sage: Dg == 0
True
sage: Dig = g.connection()(g.inverse()) ; Dig
tensor field 'nabla_g inv_g' of type (2,1) on the 3-dimensional manifold 'R^3'
sage: Dig == 0
True
"""
if self._connection is None:
if name is None:
name = 'nabla_' + self.name
if latex_name is None:
latex_name = r'\nabla_{' + self.latex_name + '}'
self._connection = LeviCivitaConnection(self, name, latex_name)
return self._connection
def christoffel(self, chart_name=None):
r"""
Christoffel symbols of ``self`` with respect to a chart.
INPUT:
- ``chart_name`` -- (default: None) name of the chart; if none is
provided, the manifold's default chart is assumed.
OUTPUT:
- the set of Christoffel symbols in the given chart, as an instance of
:class:`CompWithSym`
EXAMPLES:
Christoffel symbols of the flat metric on `\RR^3` with respect to
spherical coordinates::
sage: m = Manifold(3, 'R3', r'\RR^3', start_index=1)
sage: X = Chart(m, r'r:positive th:\theta ph:\phi','spher')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, r^2*sin(th)^2
sage: g.show() # the standard flat metric expressed in spherical coordinates
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: Gam = g.christoffel() ; Gam
3-indices components w.r.t. the coordinate basis 'spher_b' (d/dr,d/dth,d/dph), with symmetry on the index positions (1, 2)
sage: print type(Gam)
<class 'sage.geometry.manifolds.component.CompWithSym'>
sage: Gam[:]
[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]],
[[0, 1/r, 0], [1/r, 0, 0], [0, 0, -sin(th)*cos(th)]],
[[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]]
sage: Gam[1,2,2]
-r
sage: Gam[2,1,2]
1/r
sage: Gam[3,1,3]
1/r
sage: Gam[3,2,3]
cos(th)/sin(th)
sage: Gam[2,3,3]
-sin(th)*cos(th)
"""
if chart_name is None:
frame_name = self.manifold.def_chart.frame.name
else:
frame_name = self.manifold.atlas[chart_name].frame.name
return self.connection().coef(frame_name)
def riemann(self, frame_name=None, name=None, latex_name=None):
r"""
Return the Riemann curvature tensor associated with the metric.
This method is actually a shortcut for ``self.connection().riemann()``
The Riemann curvature tensor is the tensor field `R` of type (1,3)
defined by
.. MATH::
R(\omega, u, v, w) = \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:
- ``frame_name`` -- (default: None) string containing the name of the
vector frame in which the computation must be performed; if none is
provided, the computation is performed in a frame for which the
metric components are known, privileging the manifold's default
frame.
- ``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
:class:`TensorField`
EXAMPLES:
Riemann tensor of the standard metric on the 2-sphere::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher = Chart(m, r'th:\theta, ph:\phi', 'spher')
sage: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.show() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.riemann()
tensor field 'Riem(g)' of type (1,3) on the 2-dimensional manifold 'S^2'
sage: g.riemann()[:]
[[[[0, 0], [0, 0]], [[0, sin(th)^2], [-sin(th)^2, 0]]],
[[[0, (cos(th)^2 - 1)/sin(th)^2], [1, 0]], [[0, 0], [0, 0]]]]
In dimension 2, the Riemann tensor can be expressed entirely in terms of
the Ricci scalar `r`:
.. MATH::
R^i_{\ \, jlk} = \frac{r}{2} \left( \delta^i_{\ \, k} g_{jl}
- \delta^i_{\ \, l} g_{jk} \right)
This formula can be checked here, with the r.h.s. rewritten as
`-r g_{j[k} \delta^i_{\ \, l]}`::
sage: g.riemann() == -g.ricci_scalar()*(g*IdentityMap(m)).antisymmetrize([2,3])
True
"""
return self.connection().riemann(frame_name, name, latex_name)
def ricci(self, frame_name=None, name=None, latex_name=None):
r"""
Return the Ricci tensor associated with the metric.
This method is actually a shortcut for ``self.connection().ricci()``
The Ricci tensor is the tensor field `Ric` of type (0,2)
defined from the Riemann curvature tensor `R` by
.. MATH::
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:
- ``frame_name`` -- (default: None) string containing the name of the
vector frame in which the computation must be performed; if none is
provided, the computation is performed in a frame for which the
metric components are known, privileging the manifold's default
frame.
- ``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 :class:`SymBilinFormField`
EXAMPLES:
Ricci tensor of the standard metric on the 2-sphere::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher = Chart(m, r'th:\theta, ph:\phi', 'spher')
sage: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.show() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.ricci()
field of symmetric bilinear forms 'Ric(g)' on the 2-dimensional manifold 'S^2'
sage: g.ricci()[:]
[ 1 0]
[ 0 sin(th)^2]
sage: g.ricci() == a^(-2) * g
True
"""
return self.connection().ricci(frame_name, name, latex_name)
def ricci_scalar(self, frame_name=None, name=None, latex_name=None):
r"""
Return the Ricci scalar associated with the metric.
The Ricci scalar is the scalar field `r` defined from the Ricci tensor
`Ric` and the metric tensor `g` by
.. MATH::
r = g^{ij} Ric_{ij}
INPUT:
- ``frame_name`` -- (default: None) string containing the name of the
vector frame in which the computation must be performed; if none is
provided, the computation is performed in a frame for which the
metric components are known, privileging the manifold's default
frame.
- ``name`` -- (default: None) name given to the Ricci scalar;
if none, it is set to "r(g)", where "g" is the metric's name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Ricci scalar; if none, it is set to "\\mathrm{r}(g)", where "g"
is the metric's name
OUTPUT:
- the Ricci scalar `r`, as an instance of :class:`ScalarField`
EXAMPLES:
Ricci scalar of the standard metric on the 2-sphere::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher = Chart(m, r'th:\theta, ph:\phi', 'spher')
sage: a = var('a') # the sphere radius
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2
sage: g.show() # standard metric on the 2-sphere of radius a:
g = a^2 dth*dth + a^2*sin(th)^2 dph*dph
sage: g.ricci_scalar()
scalar field 'r(g)' on the 2-dimensional manifold 'S^2'
sage: g.ricci_scalar().expr()
2/a^2
"""
return self.connection().ricci_scalar(frame_name, name, latex_name)
def weyl(self, name=None, latex_name=None):
r"""
Return the Weyl conformal tensor associated with the metric.
The Weyl conformal tensor is the tensor field `C` of type (1,3)
defined as the trace-free part of the Riemann curvature tensor `R`
INPUT:
- ``name`` -- (default: None) name given to the Weyl conformal tensor;
if none, it is set to "C(g)", where "g" is the metric's name
- ``latex_name`` -- (default: None) LaTeX symbol to denote the
Weyl conformal tensor; if none, it is set to "\\mathrm{C}(g)", where
"g" is the metric's name
OUTPUT:
- the Weyl conformal tensor `C`, as an instance of :class:`TensorField`
EXAMPLES:
Checking that the Weyl tensor identically vanishes on a 3-dimensional
manifold, for instance the hyperbolic space `H^3`::
sage: m = Manifold(3, 'H^3', start_index=1)
sage: X = Chart(m, r'rh:positive:\rho th:\theta ph:\phi', 'rtp-coord')
sage: g = Metric(m, 'g')
sage: b = var('b')
sage: g[1,1], g[2,2], g[3,3] = b^2, (b*sinh(rh))^2, (b*sinh(rh)*sin(th))^2
sage: g.show() # standard metric on H^3:
g = b^2 drh*drh + b^2*sinh(rh)^2 dth*dth + b^2*sin(th)^2*sinh(rh)^2 dph*dph
sage: C = g.weyl() ; C
tensor field 'C(g)' of type (1,3) on the 3-dimensional manifold 'H^3'
sage: C == 0
True
"""
from rank2field import IdentityMap
if self._weyl is None:
n = self.manifold.dim
if n < 3:
raise ValueError("The Weyl tensor is not defined for a " +
"manifold of dimension n <= 2.")
delta = IdentityMap(self.manifold)
riem = self.riemann()
ric = self.ricci()
rscal = self.ricci_scalar()
# First index of the Ricci tensor raised with the metric
ricup = ric.up(self, 0)
# The identity map is expressed in a frame in which the Riemann
# tensor is known
delta.comp(riem.pick_a_frame())
aux = self * ricup + ric * delta - rscal / (n - 1) * self * delta
self._weyl = riem + 2 / (n - 2) * aux.antisymmetrize([2, 3])
if name is None:
self._weyl.name = "C(" + self.name + ")"
else:
self._weyl.name = name
if latex_name is None:
self._weyl.latex_name = r"\mathrm{C}\left(" + self.latex_name + r"\right)"
else:
self._weyl.latex_name = latex_name
return self._weyl
def determinant(self, frame_name=None):
r"""
Determinant of the metric components in the specified frame.
INPUT:
- ``frame_name`` -- (default: None) name of the vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the manifold's default frame is used. If a chart name is
provided, the associated coordinate basis is used
OUTPUT:
- the determinant `\det (g_{ij})`, as an instance of :class:`ScalarField`
EXAMPLES:
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X = Chart(M, 'x y', 'xy')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.determinant() # determinant in M's default frame
sage: s.expr()
-x^2*y^2 - (x + 1)*y + x + 1
Determinant in a frame different from the default's one::
sage: Y = Chart(M, 'u v', 'uv')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart 'uv' (u, v) to chart 'xy' (x, y)
sage: g.comp('uv_b')[:, 'uv']
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.determinant('uv_b').expr()
-1/4*x^2*y^2 - 1/4*(x + 1)*y + 1/4*x + 1/4
sage: g.determinant('uv_b').expr('uv')
-1/64*u^4 - 1/16*u^2 - 1/64*v^4 + 1/32*(u^2 + 2)*v^2 + 1/4*v + 1/4
The name of a chart can be passed instead of the name of a frame::
sage: g.determinant('xy') is g.determinant('xy_b')
True
sage: g.determinant('uv') is g.determinant('uv_b')
True
The metric determinant depends on the frame::
sage: g.determinant('xy_b') == g.determinant('uv_b')
False
"""
from sage.matrix.constructor import matrix
from scalarfield import ScalarField
from utilities import simple_determinant, simplify_chain
manif = self.manifold
if frame_name is None:
frame_name = manif.def_frame.name
if frame_name in manif.atlas:
# frame_name is actually the name of a chart and is changed to the
# name of the associated coordinate basis:
frame_name = manif.atlas[frame_name].frame.name
if frame_name not in self._determinants:
# a new computation is necessary
resu = ScalarField(manif)
gg = self.comp(frame_name)
i1 = manif.sindex
for chart_name in gg[[i1, i1]].express:
gm = matrix(
[[gg[i, j, chart_name].express for j in manif.irange()]
for i in manif.irange()])
detgm = simplify_chain(simple_determinant(gm))
resu.set_expr(detgm,
chart_name=chart_name,
delete_others=False)
self._determinants[frame_name] = resu
return self._determinants[frame_name]
def sqrt_abs_det(self, frame_name=None):
r"""
Square root of the absolute value of the determinant of the metric
components in the specified frame.
INPUT:
- ``frame_name`` -- (default: None) name of the vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the manifold's default frame is used. If a chart name is
provided, the associated coordinate basis is used
OUTPUT:
- `\sqrt{|\det (g_{ij})|}`, as an instance of :class:`ScalarField`
EXAMPLES:
Standard metric in the Euclidean space `\RR^3` with spherical
coordinates::
sage: m = Manifold(3, 'M', start_index=1)
sage: c_spher = Chart(m, r'r:positive th:positive:\theta ph:\phi', 'spher')
sage: assume(th>=0); assume(th<=pi)
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.show()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: g.sqrt_abs_det().expr()
r^2*sin(th)
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X = Chart(M, 'x y', 'xy')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.sqrt_abs_det() ; s
scalar field on the 2-dimensional manifold 'M'
sage: s.expr()
sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
Determinant in a frame different from the default's one::
sage: Y = Chart(M, 'u v', 'uv')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart 'uv' (u, v) to chart 'xy' (x, y)
sage: g.comp('uv_b')[:, 'uv']
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.sqrt_abs_det('uv_b').expr()
1/2*sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
sage: g.sqrt_abs_det('uv_b').expr('uv')
1/8*sqrt(-u^4 - 4*u^2 - v^4 + 2*(u^2 + 2)*v^2 + 16*v + 16)
The name of a chart can be passed instead of the name of a frame::
sage: g.sqrt_abs_det('uv') is g.sqrt_abs_det('uv_b')
True
The metric determinant depends on the frame::
sage: g.sqrt_abs_det('xy_b') == g.sqrt_abs_det('uv_b')
False
"""
from sage.functions.other import sqrt
from scalarfield import ScalarField
from utilities import simplify_chain
manif = self.manifold
if frame_name is None:
frame_name = manif.def_frame.name
if frame_name in manif.atlas:
# frame_name is actually the name of a chart and is changed to the
# name of the associated coordinate basis:
frame_name = manif.atlas[frame_name].frame.name
if frame_name not in self._sqrt_abs_dets:
# a new computation is necessary
detg = self.determinant(frame_name)
resu = ScalarField(manif)
for chart_name in detg.express:
x = self._indic_signat * detg.express[chart_name].express # |g|
x = simplify_chain(sqrt(x))
resu.set_expr(x, chart_name=chart_name, delete_others=False)
self._sqrt_abs_dets[frame_name] = resu
return self._sqrt_abs_dets[frame_name]
def volume_form(self, contra=0):
r"""
Volume form (Levi-Civita tensor) `\epsilon` associated with the metric.
This assumes that the manifold is orientable.
The volume form `\epsilon` is a `n`-form (`n` being the manifold's
dimension) such that for any vector basis `(e_i)` that is orthonormal
with respect to the metric,
.. MATH::
\epsilon(e_1,\ldots,e_n) = \pm 1
There are only two such `n`-forms, which are opposite of each other.
The volume form `\epsilon` is selected such that the manifold's default
frame is right-handed with respect to it.
INPUT:
- ``contra`` -- (default: 0) number of contravariant indices of the
returned tensor
OUTPUT:
- if ``contra = 0`` (default value): the volume `n`-form `\epsilon`, as
an instance of :class:`DiffForm`
- if ``contra = k``, with `1\leq k \leq n`, the tensor field of type
(k,n-k) formed from `\epsilon` by raising the first k indices with the
metric (see method :meth:`TensorField.up`); the output is then an
instance of :class:`TensorField`, with the appropriate antisymmetries
EXAMPLES:
Volume form on `\RR^3` with spherical coordinates::
sage: m = Manifold(3, 'M', start_index=1)
sage: c_spher = Chart(m, r'r:positive th:positive:\theta ph:\phi', 'spher')
sage: assume(th>=0); assume(th<=pi)
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.show()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: eps = g.volume_form() ; eps
3-form 'eps_g' on the 3-dimensional manifold 'M'
sage: eps.show()
eps_g = r^2*sin(th) dr/\dth/\dph
sage: eps[[1,2,3]] == g.sqrt_abs_det()
True
sage: latex(eps)
\epsilon_{g}
The tensor field of components `\epsilon^i_{\ \, jk}` (``contra=1``)::
sage: eps1 = g.volume_form(1) ; eps1
tensor field of type (1,2) on the 3-dimensional manifold 'M'
sage: eps1.symmetries()
no symmetry; antisymmetry: (1, 2)
sage: eps1[:]
[[[0, 0, 0], [0, 0, r^2*sin(th)], [0, -r^2*sin(th), 0]],
[[0, 0, -sin(th)], [0, 0, 0], [sin(th), 0, 0]],
[[0, 1/sin(th), 0], [-1/sin(th), 0, 0], [0, 0, 0]]]
The tensor field of components `\epsilon^{ij}_{\ \ k}` (``contra=2``)::
sage: eps2 = g.volume_form(2) ; eps2
tensor field of type (2,1) on the 3-dimensional manifold 'M'
sage: eps2.symmetries()
no symmetry; antisymmetry: (0, 1)
sage: eps2[:]
[[[0, 0, 0], [0, 0, sin(th)], [0, -1/sin(th), 0]],
[[0, 0, -sin(th)], [0, 0, 0], [1/(r^2*sin(th)), 0, 0]],
[[0, 1/sin(th), 0], [-1/(r^2*sin(th)), 0, 0], [0, 0, 0]]]
The tensor field of components `\epsilon^{ijk}` (``contra=3``)::
sage: eps3 = g.volume_form(3) ; eps3
tensor field of type (3,0) on the 3-dimensional manifold 'M'
sage: eps3.symmetries()
no symmetry; antisymmetry: (0, 1, 2)
sage: eps3[:]
[[[0, 0, 0], [0, 0, 1/(r^2*sin(th))], [0, -1/(r^2*sin(th)), 0]],
[[0, 0, -1/(r^2*sin(th))], [0, 0, 0], [1/(r^2*sin(th)), 0, 0]],
[[0, 1/(r^2*sin(th)), 0], [-1/(r^2*sin(th)), 0, 0], [0, 0, 0]]]
sage: eps3[1,2,3]
1/(r^2*sin(th))
sage: eps3[[1,2,3]] * g.sqrt_abs_det() == 1
True
"""
if self._vol_forms == []:
# a new computation is necessary
manif = self.manifold
ndim = manif.dim
eps = DiffForm(manif,
ndim,
name='eps_' + self.name,
latex_name=r'\epsilon_{' + self.latex_name + r'}')
ind = tuple(range(manif.sindex, manif.sindex + ndim))
eps[[ind]] = self.sqrt_abs_det(manif.def_frame.name)
self._vol_forms.append(eps) # Levi-Civita tensor constructed
# Tensors related to the Levi-Civita one by index rising:
for k in range(1, ndim + 1):
epskm1 = self._vol_forms[k - 1]
epsk = epskm1.up(self, k - 1)
if k > 1:
# restoring the antisymmetry after the up operation:
epsk = epsk.antisymmetrize(range(k))
self._vol_forms.append(epsk)
return self._vol_forms[contra]
def _del_derived(self):
r"""
Delete the derived quantities
"""
TensorField._del_derived(self)
self._exterior_derivative = None
def _del_derived(self):
r"""
Delete the derived quantities
"""
TensorField._del_derived(self)
self._del_dependencies()
def _del_derived(self):
r"""
Delete the derived quantities
"""
TensorField._del_derived(self)
self._del_dependencies()