Pseudo-Riemannian Manifolds¶
A pseudo-Riemannian manifold is a pair \((M,g)\) where \(M\) is a real
differentiable manifold \(M\) (see
DifferentiableManifold
)
and \(g\) is a field of non-degenerate symmetric bilinear forms on \(M\), which is
called the metric tensor, or simply the metric (see
PseudoRiemannianMetric
).
Two important subcases are
Riemannian manifold: the metric \(g\) is positive definite, i.e. its signature is \(n = \dim M\);
Lorentzian manifold: the metric \(g\) has signature \(n-2\) (positive convention) or \(2-n\) (negative convention).
On a pseudo-Riemannian manifold, one may use various standard
operators
acting on scalar and tensor
fields, like grad()
or
div()
.
All pseudo-Riemannian manifolds, whatever the metric signature, are implemented
via the class PseudoRiemannianManifold
.
Example 1: the sphere as a Riemannian manifold of dimension 2
We start by declaring \(S^2\) as a 2-dimensional Riemannian manifold:
sage: M = Manifold(2, 'S^2', structure='Riemannian')
sage: M
2-dimensional Riemannian manifold S^2
We then cover \(S^2\) by two stereographic charts, from the North pole and from the South pole respectively:
sage: U = M.open_subset('U')
sage: stereoN.<x,y> = U.chart()
sage: V = M.open_subset('V')
sage: stereoS.<u,v> = V.chart()
sage: M.declare_union(U,V)
sage: stereoN_to_S = stereoN.transition_map(stereoS,
....: [x/(x^2+y^2), y/(x^2+y^2)], intersection_name='W',
....: restrictions1= x^2+y^2!=0, restrictions2= u^2+v^2!=0)
sage: W = U.intersection(V)
sage: stereoN_to_S
Change of coordinates from Chart (W, (x, y)) to Chart (W, (u, v))
sage: stereoN_to_S.display()
u = x/(x^2 + y^2)
v = y/(x^2 + y^2)
sage: stereoN_to_S.inverse().display()
x = u/(u^2 + v^2)
y = v/(u^2 + v^2)
We get the metric defining the Riemannian structure by:
sage: g = M.metric()
sage: g
Riemannian metric g on the 2-dimensional Riemannian manifold S^2
At this stage, the metric \(g\) is defined as a Python object but there remains to
initialize it by setting its components with respect to the vector frames
associated with the stereographic coordinates. Let us begin with the frame
of chart stereoN
:
sage: eU = stereoN.frame()
sage: g[eU, 0, 0] = 4/(1 + x^2 + y^2)^2
sage: g[eU, 1, 1] = 4/(1 + x^2 + y^2)^2
The metric components in the frame of chart stereoS
are obtained by
continuation of the expressions found in \(W = U\cap V\) from the known
change-of-coordinate formulas:
sage: eV = stereoS.frame()
sage: g.add_comp_by_continuation(eV, W)
At this stage, the metric \(g\) is well defined in all \(S^2\):
sage: g.display(eU)
g = 4/(x^2 + y^2 + 1)^2 dx⊗dx + 4/(x^2 + y^2 + 1)^2 dy⊗dy
sage: g.display(eV)
g = 4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du⊗du
+ 4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dv⊗dv
The expression in frame eV
can be given a shape similar to that in frame
eU
, by factorizing the components:
sage: g[eV, 0, 0].factor()
4/(u^2 + v^2 + 1)^2
sage: g[eV, 1, 1].factor()
4/(u^2 + v^2 + 1)^2
sage: g.display(eV)
g = 4/(u^2 + v^2 + 1)^2 du⊗du + 4/(u^2 + v^2 + 1)^2 dv⊗dv
Let us consider a scalar field \(f\) on \(S^2\):
sage: f = M.scalar_field({stereoN: 1/(1+x^2+y^2)}, name='f')
sage: f.add_expr_by_continuation(stereoS, W)
sage: f.display()
f: S^2 → ℝ
on U: (x, y) ↦ 1/(x^2 + y^2 + 1)
on V: (u, v) ↦ (u^2 + v^2)/(u^2 + v^2 + 1)
The gradient of \(f\) (with respect to the metric \(g\)) is:
sage: gradf = f.gradient()
sage: gradf
Vector field grad(f) on the 2-dimensional Riemannian manifold S^2
sage: gradf.display(eU)
grad(f) = -1/2*x ∂/∂x - 1/2*y ∂/∂y
sage: gradf.display(eV)
grad(f) = 1/2*u ∂/∂u + 1/2*v ∂/∂v
It is possible to write grad(f)
instead of f.gradient()
, by importing
the standard differential operators of vector calculus:
sage: from sage.manifolds.operators import *
sage: grad(f) == gradf
True
The Laplacian of \(f\) (with respect to the metric \(g\)) is obtained either
as f.laplacian()
or, thanks to the above import, as laplacian(f)
:
sage: Df = laplacian(f)
sage: Df
Scalar field Delta(f) on the 2-dimensional Riemannian manifold S^2
sage: Df.display()
Delta(f): S^2 → ℝ
on U: (x, y) ↦ (x^2 + y^2 - 1)/(x^2 + y^2 + 1)
on V: (u, v) ↦ -(u^2 + v^2 - 1)/(u^2 + v^2 + 1)
Let us check the standard formula \(\Delta f = \mathrm{div}( \mathrm{grad}\, f )\):
sage: Df == div(gradf)
True
Since each open subset of \(S^2\) inherits the structure of a Riemannian
manifold, we can get the metric on it via the method metric()
:
sage: gU = U.metric()
sage: gU
Riemannian metric g on the Open subset U of the 2-dimensional Riemannian
manifold S^2
sage: gU.display()
g = 4/(x^2 + y^2 + 1)^2 dx⊗dx + 4/(x^2 + y^2 + 1)^2 dy⊗dy
Of course, gU
is nothing but the restriction of \(g\) to \(U\):
sage: gU is g.restrict(U)
True
Example 2: Minkowski spacetime as a Lorentzian manifold of dimension 4
We start by declaring a 4-dimensional Lorentzian manifold \(M\):
sage: M = Manifold(4, 'M', structure='Lorentzian')
sage: M
4-dimensional Lorentzian manifold M
We define Minkowskian coordinates on \(M\):
sage: X.<t,x,y,z> = M.chart()
We construct the metric tensor by:
sage: g = M.metric()
sage: g
Lorentzian metric g on the 4-dimensional Lorentzian manifold M
and initialize it to the Minkowskian value:
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1
sage: g.display()
g = -dt⊗dt + dx⊗dx + dy⊗dy + dz⊗dz
sage: g[:]
[-1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1]
We may check that the metric is flat, i.e. has a vanishing Riemann curvature tensor:
sage: g.riemann().display()
Riem(g) = 0
A vector field on \(M\):
sage: u = M.vector_field(name='u')
sage: u[0] = cosh(t)
sage: u[1] = sinh(t)
sage: u.display()
u = cosh(t) ∂/∂t + sinh(t) ∂/∂x
The scalar square of \(u\) is:
sage: s = u.dot(u); s
Scalar field u.u on the 4-dimensional Lorentzian manifold M
Scalar products are taken with respect to the metric tensor:
sage: u.dot(u) == g(u,u)
True
\(u\) is a unit timelike vector, i.e. its scalar square is identically \(-1\):
sage: s.display()
u.u: M → ℝ
(t, x, y, z) ↦ -1
sage: s.expr()
-1
Let us consider a unit spacelike vector:
sage: v = M.vector_field(name='v')
sage: v[0] = sinh(t)
sage: v[1] = cosh(t)
sage: v.display()
v = sinh(t) ∂/∂t + cosh(t) ∂/∂x
sage: v.dot(v).display()
v.v: M → ℝ
(t, x, y, z) ↦ 1
sage: v.dot(v).expr()
1
\(u\) and \(v\) are orthogonal vectors with respect to Minkowski metric:
sage: u.dot(v).display()
u.v: M → ℝ
(t, x, y, z) ↦ 0
sage: u.dot(v).expr()
0
The divergence of \(u\) is:
sage: s = u.div(); s
Scalar field div(u) on the 4-dimensional Lorentzian manifold M
sage: s.display()
div(u): M → ℝ
(t, x, y, z) ↦ sinh(t)
while its d’Alembertian is:
sage: Du = u.dalembertian(); Du
Vector field Box(u) on the 4-dimensional Lorentzian manifold M
sage: Du.display()
Box(u) = -cosh(t) ∂/∂t - sinh(t) ∂/∂x
AUTHORS:
Eric Gourgoulhon (2018): initial version
REFERENCES:
- class sage.manifolds.differentiable.pseudo_riemannian.PseudoRiemannianManifold(n, name, metric_name='g', signature=None, base_manifold=None, diff_degree=+ Infinity, latex_name=None, metric_latex_name=None, start_index=0, category=None, unique_tag=None)¶
Bases:
sage.manifolds.differentiable.manifold.DifferentiableManifold
PseudoRiemannian manifold.
A pseudo-Riemannian manifold is a pair \((M,g)\) where \(M\) is a real differentiable manifold \(M\) (see
DifferentiableManifold
) and \(g\) is a field of non-degenerate symmetric bilinear forms on \(M\), which is called the metric tensor, or simply the metric (seePseudoRiemannianMetric
).Two important subcases are
Riemannian manifold: the metric \(g\) is positive definite, i.e. its signature is \(n = \dim M\);
Lorentzian manifold: the metric \(g\) has signature \(n-2\) (positive convention) or \(2-n\) (negative convention).
INPUT:
n
– positive integer; dimension of the manifoldname
– string; name (symbol) given to the manifoldmetric_name
– (default:'g'
) string; name (symbol) given to the metricsignature
– (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; ifsignature
is not provided, \(S\) is set to the manifold’s dimension (Riemannian signature)base_manifold
– (default:None
) if notNone
, must be a differentiable manifold; the created object is then an open subset ofbase_manifold
diff_degree
– (default:infinity
) degree \(k\) of differentiabilitylatex_name
– (default:None
) string; LaTeX symbol to denote the manifold; if none is provided, it is set toname
metric_latex_name
– (default:None
) string; LaTeX symbol to denote the metric; if none is provided, it is set tometric_name
start_index
– (default: 0) integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g. coordinates in a chartcategory
– (default:None
) to specify the category; ifNone
,Manifolds(RR).Differentiable()
(orManifolds(RR).Smooth()
ifdiff_degree
=infinity
) is assumed (see the categoryManifolds
)unique_tag
– (default:None
) tag used to force the construction of a new object when all the other arguments have been used previously (withoutunique_tag
, theUniqueRepresentation
behavior inherited fromManifoldSubset
, viaDifferentiableManifold
andTopologicalManifold
, would return the previously constructed object corresponding to these arguments).
EXAMPLES:
Pseudo-Riemannian manifolds are constructed via the generic function
Manifold()
, using the keywordstructure
:sage: M = Manifold(4, 'M', structure='pseudo-Riemannian', signature=0) sage: M 4-dimensional pseudo-Riemannian manifold M sage: M.category() Category of smooth manifolds over Real Field with 53 bits of precision
The metric associated with
M
is:sage: M.metric() Pseudo-Riemannian metric g on the 4-dimensional pseudo-Riemannian manifold M sage: M.metric().signature() 0 sage: M.metric().tensor_type() (0, 2)
Its value has to be initialized either by setting its components in various vector frames (see the above examples regarding the 2-sphere and Minkowski spacetime) or by making it equal to a given field of symmetric bilinear forms (see the method
set()
of the metric class). Both methods are also covered in the documentation of methodmetric()
below.The metric object belongs to the class
PseudoRiemannianMetric
:sage: isinstance(M.metric(), sage.manifolds.differentiable.metric. ....: PseudoRiemannianMetric) True
See the documentation of this class for all operations available on metrics.
The default name of the metric is
g
; it can be customized:sage: M = Manifold(4, 'M', structure='pseudo-Riemannian', ....: metric_name='gam', metric_latex_name=r'\gamma') sage: M.metric() Riemannian metric gam on the 4-dimensional Riemannian manifold M sage: latex(M.metric()) \gamma
A Riemannian manifold is constructed by the proper setting of the keyword
structure
:sage: M = Manifold(4, 'M', structure='Riemannian'); M 4-dimensional Riemannian manifold M sage: M.metric() Riemannian metric g on the 4-dimensional Riemannian manifold M sage: M.metric().signature() 4
Similarly, a Lorentzian manifold is obtained by:
sage: M = Manifold(4, 'M', structure='Lorentzian'); M 4-dimensional Lorentzian manifold M sage: M.metric() Lorentzian metric g on the 4-dimensional Lorentzian manifold M
The default Lorentzian signature is taken to be positive:
sage: M.metric().signature() 2
but one can opt for the negative convention via the keyword
signature
:sage: M = Manifold(4, 'M', structure='Lorentzian', signature='negative') sage: M.metric() Lorentzian metric g on the 4-dimensional Lorentzian manifold M sage: M.metric().signature() -2
- metric(name=None, signature=None, latex_name=None, dest_map=None)¶
Return the metric giving the pseudo-Riemannian structure to the manifold, or define a new metric tensor on the manifold.
INPUT:
name
– (default:None
) name given to the metric; ifname
isNone
or matches the name of the metric defining the pseudo-Riemannian structure ofself
, the latter metric is returnedsignature
– (default:None
; ignored ifname
isNone
) 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; ifsignature
is not provided, \(S\) is set to the manifold’s dimension (Riemannian signature)latex_name
– (default:None
; ignored ifname
isNone
) LaTeX symbol to denote the metric; ifNone
, it is formed fromname
dest_map
– (default:None
; ignored ifname
isNone
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, the identity map is assumed (case of a metric tensor field on \(U\))
OUTPUT:
instance of
PseudoRiemannianMetric
EXAMPLES:
Metric of a 3-dimensional Riemannian manifold:
sage: M = Manifold(3, 'M', structure='Riemannian', start_index=1) sage: X.<x,y,z> = M.chart() sage: g = M.metric(); g Riemannian metric g on the 3-dimensional Riemannian manifold M
The metric remains to be initialized, for instance by setting its components in the coordinate frame associated to the chart
X
:sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1 sage: g.display() g = dx⊗dx + dy⊗dy + dz⊗dz
Alternatively, the metric can be initialized from a given field of nondegenerate symmetric bilinear forms; we may create the former object by:
sage: X.coframe() Coordinate coframe (M, (dx,dy,dz)) sage: dx, dy, dz = X.coframe()[1], X.coframe()[2], X.coframe()[3] sage: b = dx*dx + dy*dy + dz*dz sage: b Field of symmetric bilinear forms dx⊗dx+dy⊗dy+dz⊗dz on the 3-dimensional Riemannian manifold M
We then use the metric method
set()
to makeg
being equal tob
as a symmetric tensor field of type(0,2)
:sage: g.set(b) sage: g.display() g = dx⊗dx + dy⊗dy + dz⊗dz
Another metric can be defined on
M
by specifying a metric name distinct from that chosen at the creation of the manifold (which isg
by default, but can be changed thanks to the keywordmetric_name
inManifold()
):sage: h = M.metric('h'); h Riemannian metric h on the 3-dimensional Riemannian manifold M sage: h[1,1], h[2,2], h[3,3] = 1+y^2, 1+z^2, 1+x^2 sage: h.display() h = (y^2 + 1) dx⊗dx + (z^2 + 1) dy⊗dy + (x^2 + 1) dz⊗dz
The metric tensor
h
is distinct from the metric entering in the definition of the Riemannian manifoldM
:sage: h is M.metric() False
while we have of course:
sage: g is M.metric() True
Providing the same name as the manifold’s default metric returns the latter:
sage: M.metric('g') is M.metric() True
In the present case (
M
is diffeomorphic to \(\RR^3\)), we can even create a Lorentzian metric onM
:sage: h = M.metric('h', signature=1); h Lorentzian metric h on the 3-dimensional Riemannian manifold M
- open_subset(name, latex_name=None, coord_def={}, supersets=None)¶
Create an open subset of
self
.An open subset is a set that is (i) included in the manifold and (ii) open with respect to the manifold’s topology. It is a differentiable manifold by itself. Moreover, equipped with the restriction of the manifold metric to itself, it is a pseudo-Riemannian manifold. Hence the returned object is an instance of
PseudoRiemannianManifold
.INPUT:
name
– name given to the open subsetlatex_name
– (default:None
) LaTeX symbol to denote the subset; if none is provided, it is set toname
coord_def
– (default: {}) definition of the subset in terms of coordinates;coord_def
must a be dictionary with keys charts in the manifold’s atlas and values the symbolic expressions formed by the coordinates to define the subset.supersets
– (default: onlyself
) list of sets that the new open subset is a subset of
OUTPUT:
instance of
PseudoRiemannianManifold
representing the created open subset
EXAMPLES:
Open subset of a 2-dimensional Riemannian manifold:
sage: M = Manifold(2, 'M', structure='Riemannian') sage: X.<x,y> = M.chart() sage: U = M.open_subset('U', coord_def={X: x>0}); U Open subset U of the 2-dimensional Riemannian manifold M sage: type(U) <class 'sage.manifolds.differentiable.pseudo_riemannian.PseudoRiemannianManifold_with_category'>
We initialize the metric of
M
:sage: g = M.metric() sage: g[0,0], g[1,1] = 1, 1
Then the metric on
U
is determined as the restriction ofg
toU
:sage: gU = U.metric(); gU Riemannian metric g on the Open subset U of the 2-dimensional Riemannian manifold M sage: gU.display() g = dx⊗dx + dy⊗dy sage: gU is g.restrict(U) True
- volume_form(contra=0)¶
Volume form (Levi-Civita tensor) \(\epsilon\) associated with
self
.This assumes that
self
is an orientable manifold, with a preferred orientation; seeorientation()
for details.The volume form \(\epsilon\) is a \(n\)-form (\(n\) being the manifold’s dimension) such that, for any vector frame \((e_i)\) that is orthonormal with respect to the metric of the pseudo-Riemannian manifold
self
,\[\epsilon(e_1,\ldots,e_n) = \pm 1\]There are only two such \(n\)-forms, which are opposite of each other. The volume form \(\epsilon\) is selected as the one that returns \(+1\) for any right-handed vector frame with respect to the chosen orientation of
self
.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 ofDiffForm
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 methodup()
); the output is then an instance ofTensorField
, with the appropriate antisymmetries, or of the subclassMultivectorField
if \(k=n\)
EXAMPLES:
Volume form of the Euclidean 3-space:
sage: M = Manifold(3, 'M', structure='Riemannian', start_index=1) sage: X.<x,y,z> = M.chart() sage: g = M.metric() sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1 sage: eps = M.volume_form(); eps 3-form eps_g on the 3-dimensional Riemannian manifold M sage: eps.display() eps_g = dx∧dy∧dz
Raising the first index:
sage: eps1 = M.volume_form(1); eps1 Tensor field of type (1,2) on the 3-dimensional Riemannian manifold M sage: eps1.display() ∂/∂x⊗dy⊗dz - ∂/∂x⊗dz⊗dy - ∂/∂y⊗dx⊗dz + ∂/∂y⊗dz⊗dx + ∂/∂z⊗dx⊗dy - ∂/∂z⊗dy⊗dx sage: eps1.symmetries() no symmetry; antisymmetry: (1, 2)
Raising the first and second indices:
sage: eps2 = M.volume_form(2); eps2 Tensor field of type (2,1) on the 3-dimensional Riemannian manifold M sage: eps2.display() ∂/∂x⊗∂/∂y⊗dz - ∂/∂x⊗∂/∂z⊗dy - ∂/∂y⊗∂/∂x⊗dz + ∂/∂y⊗∂/∂z⊗dx + ∂/∂z⊗∂/∂x⊗dy - ∂/∂z⊗∂/∂y⊗dx sage: eps2.symmetries() no symmetry; antisymmetry: (0, 1)
Fully contravariant version:
sage: eps3 = M.volume_form(3); eps3 3-vector field on the 3-dimensional Riemannian manifold M sage: eps3.display() ∂/∂x∧∂/∂y∧∂/∂z