def test_hash():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a,b,c,d,e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
p_type = p.args[1]
t1 = p(a)*q(b)
t2 = p(a)*p(b)
assert hash(t1) != hash(t2)
t3 = p(a)*p(b) + g(a,b)
t4 = p(a)*p(b) - g(a,b)
assert hash(t3) != hash(t4)
assert a.func(*a.args) == a
assert Lorentz.func(*Lorentz.args) == Lorentz
assert g.func(*g.args) == g
assert p.func(*p.args) == p
assert p_type.func(*p_type.args) == p_type
assert p(a).func(*(p(a)).args) == p(a)
assert t1.func(*t1.args) == t1
assert t2.func(*t2.args) == t2
assert t3.func(*t3.args) == t3
assert t4.func(*t4.args) == t4
assert hash(a.func(*a.args)) == hash(a)
assert hash(Lorentz.func(*Lorentz.args)) == hash(Lorentz)
assert hash(g.func(*g.args)) == hash(g)
assert hash(p.func(*p.args)) == hash(p)
assert hash(p_type.func(*p_type.args)) == hash(p_type)
assert hash(p(a).func(*(p(a)).args)) == hash(p(a))
assert hash(t1.func(*t1.args)) == hash(t1)
assert hash(t2.func(*t2.args)) == hash(t2)
assert hash(t3.func(*t3.args)) == hash(t3)
assert hash(t4.func(*t4.args)) == hash(t4)
def check_all(obj):
return all([isinstance(_, Basic) for _ in obj.args])
assert check_all(a)
assert check_all(Lorentz)
assert check_all(g)
assert check_all(p)
assert check_all(p_type)
assert check_all(p(a))
assert check_all(t1)
assert check_all(t2)
assert check_all(t3)
assert check_all(t4)
tsymmetry = tensorsymmetry([2], [1], [1, 1, 1])
assert tsymmetry.func(*tsymmetry.args) == tsymmetry
assert hash(tsymmetry.func(*tsymmetry.args)) == hash(tsymmetry)
assert check_all(tsymmetry)
def test_hash():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
p_type = p.args[1]
t1 = p(a) * q(b)
t2 = p(a) * p(b)
assert hash(t1) != hash(t2)
t3 = p(a) * p(b) + g(a, b)
t4 = p(a) * p(b) - g(a, b)
assert hash(t3) != hash(t4)
assert a.func(*a.args) == a
assert Lorentz.func(*Lorentz.args) == Lorentz
assert g.func(*g.args) == g
assert p.func(*p.args) == p
assert p_type.func(*p_type.args) == p_type
assert p(a).func(*(p(a)).args) == p(a)
assert t1.func(*t1.args) == t1
assert t2.func(*t2.args) == t2
assert t3.func(*t3.args) == t3
assert t4.func(*t4.args) == t4
assert hash(a.func(*a.args)) == hash(a)
assert hash(Lorentz.func(*Lorentz.args)) == hash(Lorentz)
assert hash(g.func(*g.args)) == hash(g)
assert hash(p.func(*p.args)) == hash(p)
assert hash(p_type.func(*p_type.args)) == hash(p_type)
assert hash(p(a).func(*(p(a)).args)) == hash(p(a))
assert hash(t1.func(*t1.args)) == hash(t1)
assert hash(t2.func(*t2.args)) == hash(t2)
assert hash(t3.func(*t3.args)) == hash(t3)
assert hash(t4.func(*t4.args)) == hash(t4)
def check_all(obj):
return all([isinstance(_, Basic) for _ in obj.args])
assert check_all(a)
assert check_all(Lorentz)
assert check_all(g)
assert check_all(p)
assert check_all(p_type)
assert check_all(p(a))
assert check_all(t1)
assert check_all(t2)
assert check_all(t3)
assert check_all(t4)
tsymmetry = tensorsymmetry([2], [1], [1, 1, 1])
assert tsymmetry.func(*tsymmetry.args) == tsymmetry
assert hash(tsymmetry.func(*tsymmetry.args)) == hash(tsymmetry)
assert check_all(tsymmetry)
def test_special_eq_ne():
# test special equality cases:
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, d0, d1, i, j, k = tensor_indices('a,b,d0,d1,i,j,k', Lorentz)
# A, B symmetric
A, B = tensorhead('A,B', [Lorentz] * 2, [[1] * 2])
p, q, r = tensorhead('p,q,r', [Lorentz], [[1]])
t = 0 * A(a, b)
assert t == 0
assert t == S.Zero
assert p(i) != A(a, b)
assert A(a, -a) != A(a, b)
assert 0 * (A(a, b) + B(a, b)) == 0
assert 0 * (A(a, b) + B(a, b)) == S.Zero
assert 3 * (A(a, b) - A(a, b)) == S.Zero
assert p(i) + q(i) != A(a, b)
assert p(i) + q(i) != A(a, b) + B(a, b)
assert p(i) - p(i) == 0
assert p(i) - p(i) == S.Zero
assert A(a, b) == A(b, a)
def test_mul():
from sympy.abc import x
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
sym = tensorsymmetry([1] * 2)
t = TensMul.from_data(S.One, [], [], [])
assert str(t) == '1'
A, B = tensorhead('A B', [Lorentz] * 2, [[1] * 2])
t = (1 + x) * A(a, b)
assert str(t) == '(x + 1)*A(a, b)'
assert t.types == [Lorentz]
assert t.rank == 2
assert t.dum == []
assert t.coeff == 1 + x
assert sorted(t.free) == [(a, 0, 0), (b, 1, 0)]
assert t.components == [A]
t = A(-b, a) * B(-a, c) * A(-c, d)
t1 = tensor_mul(*t.split())
assert t == t(-b, d)
assert t == t1
assert tensor_mul(*[]) == TensMul.from_data(S.One, [], [], [])
t = TensMul.from_data(1, [], [], [])
zsym = tensorsymmetry()
typ = TensorType([], zsym)
C = typ('C')
assert str(C()) == 'C'
assert str(t) == '1'
assert t.split()[0] == t
raises(ValueError, lambda: TIDS.free_dum_from_indices(a, a))
raises(ValueError, lambda: TIDS.free_dum_from_indices(-a, -a))
raises(ValueError, lambda: A(a, b) * A(a, c))
t = A(a, b) * A(-a, c)
raises(ValueError, lambda: t(a, b, c))
def test_riemann_invariants():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11 = \
tensor_indices('d0:12', Lorentz)
# R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1]
# T_c = -R^{d0 d1}_{d0 d1}
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
t = R(d0, d1, -d1, -d0)
tc = t.canon_bp()
assert str(tc) == '-R(L_0, L_1, -L_0, -L_1)'
# R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} *
# R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10}
# can = [0,2,4,6, 1,3,8,10, 5,7,12,14, 9,11,16,18, 13,15,20,22,
# 17,19,21<F10,23, 24,25]
# T_c = R^{d0 d1 d2 d3} * R_{d0 d1}^{d4 d5} * R_{d2 d3}^{d6 d7} *
# R_{d4 d5}^{d8 d9} * R_{d6 d7}^{d10 d11} * R_{d8 d9 d10 d11}
t = R(-d11,d1,-d0,d5)*R(d6,d4,d0,-d5)*R(-d7,-d2,-d8,-d9)* \
R(-d10,-d3,-d6,-d4)*R(d2,d7,d11,-d1)*R(d8,d9,d3,d10)
tc = t.canon_bp()
assert str(
tc
) == 'R(L_0, L_1, L_2, L_3)*R(-L_0, -L_1, L_4, L_5)*R(-L_2, -L_3, L_6, L_7)*R(-L_4, -L_5, L_8, L_9)*R(-L_6, -L_7, L_10, L_11)*R(-L_8, -L_9, -L_10, -L_11)'
def test_fun():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
t = q(c) * p(a) * q(b) + g(a, b) * g(c, d) * q(-d)
assert t(a, b, c) == t
assert t - t(b, a, c) == q(c) * p(a) * q(b) - q(c) * p(b) * q(a)
assert t(b, c, d) == q(d) * p(b) * q(c) + g(b, c) * g(d, e) * q(-e)
t1 = t.fun_eval((a, b), (b, a))
assert t1 == q(c) * p(b) * q(a) + g(a, b) * g(c, d) * q(-d)
# check that g_{a b; c} = 0
# example taken from L. Brewin
# "A brief introduction to Cadabra" arxiv:0903.2085
# dg_{a b c} = \partial_{a} g_{b c} is symmetric in b, c
dg = tensorhead('dg', [Lorentz] * 3, [[1], [1] * 2])
# gamma^a_{b c} is the Christoffel symbol
gamma = S.Half * g(a,
d) * (dg(-b, -d, -c) + dg(-c, -b, -d) - dg(-d, -b, -c))
# t = g_{a b; c}
t = dg(-c, -a,
-b) - g(-a, -d) * gamma(d, -b, -c) - g(-b, -d) * gamma(d, -a, -c)
t = t.contract_metric(g, True)
assert t == 0
t = q(c) * p(a) * q(b)
assert t(b, c, d) == q(d) * p(b) * q(c)
def __new__(cls, dim=4, eps_dim=None, dummy_fmt="L"):
if (dim, eps_dim) in _LorentzContainer.lorentz_types:
return _LorentzContainer.lorentz_types[(dim, eps_dim)]
new_L = TensorIndexType("Lorentz", dim=dim, eps_dim=eps_dim, dummy_fmt=dummy_fmt)
_LorentzContainer.lorentz_types[(dim, eps_dim)] = new_L
return new_L
def test_tensor_element():
L = TensorIndexType("L")
i, j, k, l, m, n = tensor_indices("i j k l m n", L)
A = TensorHead("A", [L, L], TensorSymmetry.no_symmetry(2))
a = A(i, j)
assert isinstance(TensorElement(a, {}), Tensor)
assert isinstance(TensorElement(a, {k: 1}), Tensor)
te1 = TensorElement(a, {Symbol("i"): 1})
assert te1.free == [(j, 0)]
assert te1.get_free_indices() == [j]
assert te1.dum == []
te2 = TensorElement(a, {i: 1})
assert te2.free == [(j, 0)]
assert te2.get_free_indices() == [j]
assert te2.dum == []
assert te1 == te2
array = Array([[1, 2], [3, 4]])
assert te1.replace_with_arrays({A(i, j): array}, [j]) == array[1, :]
def test_TensorIndexType():
D = Symbol('D')
G = Metric('g', False)
Lorentz = TensorIndexType('Lorentz', metric=G, dim=D, dummy_fmt='L')
m0, m1, m2, m3, m4 = tensor_indices('m0:5', Lorentz)
sym2 = tensorsymmetry([1] * 2)
sym2n = tensorsymmetry(*get_symmetric_group_sgs(2))
assert sym2 == sym2n
g = Lorentz.metric
assert str(g) == 'g(Lorentz,Lorentz)'
assert Lorentz.eps_dim == Lorentz.dim
TSpace = TensorIndexType('TSpace')
i0, i1 = tensor_indices('i0 i1', TSpace)
g = TSpace.metric
A = tensorhead('A', [TSpace] * 2, [[1] * 2])
assert str(A(i0, -i0).canon_bp()) == 'A(TSpace_0, -TSpace_0)'
def test_bug_correction_tensor_indices():
# to make sure that tensor_indices does not return a list if creating
# only one index:
from sympy.tensor.tensor import tensor_indices, TensorIndexType, TensorIndex
A = TensorIndexType("A")
i = tensor_indices('i', A)
assert not isinstance(i, (tuple, list))
assert isinstance(i, TensorIndex)
def test_TensorManager():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
LorentzH = TensorIndexType('LorentzH', dummy_fmt='LH')
i, j = tensor_indices('i,j', Lorentz)
ih, jh = tensor_indices('ih,jh', LorentzH)
p, q = tensorhead('p q', [Lorentz], [[1]])
ph, qh = tensorhead('ph qh', [LorentzH], [[1]])
Gsymbol = Symbol('Gsymbol')
GHsymbol = Symbol('GHsymbol')
TensorManager.set_comm(Gsymbol, GHsymbol, 0)
G = tensorhead('G', [Lorentz], [[1]], Gsymbol)
assert TensorManager._comm_i2symbol[G.comm] == Gsymbol
GH = tensorhead('GH', [LorentzH], [[1]], GHsymbol)
ps = G(i) * p(-i)
psh = GH(ih) * ph(-ih)
t = ps + psh
t1 = t * t
assert t1 == ps * ps + 2 * ps * psh + psh * psh
qs = G(i) * q(-i)
qsh = GH(ih) * qh(-ih)
assert ps * qsh == qsh * ps
assert ps * qs != qs * ps
n = TensorManager.comm_symbols2i(Gsymbol)
assert TensorManager.comm_i2symbol(n) == Gsymbol
assert GHsymbol in TensorManager._comm_symbols2i
raises(ValueError, lambda: TensorManager.set_comm(GHsymbol, 1, 2))
TensorManager.set_comms((Gsymbol, GHsymbol, 0), (Gsymbol, 1, 1))
assert TensorManager.get_comm(n, 1) == TensorManager.get_comm(1, n) == 1
TensorManager.clear()
assert TensorManager.comm == [{
0: 0,
1: 0,
2: 0
}, {
0: 0,
1: 1,
2: None
}, {
0: 0,
1: None
}]
assert GHsymbol not in TensorManager._comm_symbols2i
nh = TensorManager.comm_symbols2i(GHsymbol)
assert GHsymbol in TensorManager._comm_symbols2i
def test_contract_automatrix_and_data():
numpy = import_module('numpy')
if numpy is None:
return
L = TensorIndexType('L')
S = TensorIndexType('S')
G = tensorhead('G', [L, S, S], [[1] * 3], matrix_behavior=True)
def G_data():
G.data = [[[1]]]
raises(ValueError, G_data)
L.data = [1, -1]
raises(ValueError, G_data)
S.data = [[1, 0], [0, 2]]
G.data = [
[[1, 2],
[3, 4]],
[[5, 6],
[7, 8]]
]
m0, m1, m2 = tensor_indices('m0:3', L)
s0, s1, s2 = tensor_indices('s0:3', S)
assert (G(-m0).data == numpy.array([
[[1, 4],
[3, 8]],
[[-5, -12],
[-7, -16]]
])).all()
(G(m0) * G(-m0)).data
G(m0, s0, -s1).data
c1 = G(m0, s0, -s1)*G(-m0, s1, -s2)
c2 = G(m0) * G(-m0)
assert (c1.data == c2.data).all()
del L.data
del S.data
del G.data
assert L.data is None
assert S.data is None
assert G.data is None
def test_TensorHead():
assert TensAdd() == 0
# simple example of algebraic expression
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
# A, B symmetric
A = tensorhead('A', [Lorentz] * 2, [[1] * 2])
assert A.rank == 2
assert A.symmetry == tensorsymmetry([1] * 2)
def test_TensorType():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = tensorsymmetry([1] * 2)
A = tensorhead('A', [Lorentz] * 2, [[1] * 2])
assert A.typ == TensorType([Lorentz] * 2, sym)
assert A.types == [Lorentz]
typ = TensorType([Lorentz] * 2, sym)
assert str(typ) == "TensorType(['Lorentz', 'Lorentz'])"
raises(ValueError, lambda: typ(2))
def test_contract_metric1():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p = tensorhead('p', [Lorentz], [[1]])
t = g(a, b) * p(-b)
t1 = t.contract_metric(g)
assert t1 == p(a)
A, B = tensorhead('A,B', [Lorentz] * 2, [[1] * 2])
# case with g with all free indices
t1 = A(a, b) * B(-b, c) * g(d, e)
t2 = t1.contract_metric(g)
assert t1 == t2
# case of g(d, -d)
t1 = A(a, b) * B(-b, c) * g(-d, d)
t2 = t1.contract_metric(g)
assert t2 == D * A(a, d) * B(-d, c)
# g with one free index
t1 = A(a, b) * B(-b, -c) * g(c, d)
t2 = t1.contract_metric(g)
assert t2 == A(a, c) * B(-c, d)
# g with both indices contracted with another tensor
t1 = A(a, b) * B(-b, -c) * g(c, -a)
t2 = t1.contract_metric(g)
assert t2 == A(a, b) * B(-b, -a)
t1 = A(a, b) * B(-b, -c) * g(c, d) * g(-a, -d)
t2 = t1.contract_metric(g)
t2 = t2.contract_metric(g)
assert t2 == A(a, b) * B(-b, -a)
t1 = A(a, b) * g(-a, -b)
t2 = t1.contract_metric(g)
assert t2 == A(a, -a)
assert not t2.free
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
g = Lorentz.metric
raises(ValueError, lambda: g(a, -a).contract_metric(g)) # no dim
def test_riemann_cyclic_replace():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m0, m1, m2, m3 = tensor_indices('m:4', Lorentz)
symr = tensorsymmetry([2, 2])
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
t = R(m0, m2, m1, m3)
t1 = riemann_cyclic_replace(t)
t1a = -S.One / 3 * R(m0, m3, m2, m1) + S.One / 3 * R(
m0, m1, m2, m3) + Rational(2, 3) * R(m0, m2, m1, m3)
assert t1 == t1a
def test_tensorsymmetry():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = tensorsymmetry([1] * 2)
sym1 = TensorSymmetry(get_symmetric_group_sgs(2))
assert sym == sym1
sym = tensorsymmetry([2])
sym1 = TensorSymmetry(get_symmetric_group_sgs(2, 1))
assert sym == sym1
sym2 = tensorsymmetry()
assert sym2.base == Tuple() and sym2.generators == Tuple(Permutation(1))
raises(NotImplementedError, lambda: tensorsymmetry([2, 1]))
def test_indices():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
assert a.tensortype == Lorentz
assert a != -a
A, B = tensorhead('A B', [Lorentz] * 2, [[1] * 2])
t = A(a, b) * B(-b, c)
indices = t.get_indices()
L_0 = TensorIndex('L_0', Lorentz)
assert indices == [a, L_0, -L_0, c]
raises(ValueError, lambda: tensor_indices(3, Lorentz))
raises(ValueError, lambda: A(a, b, c))
def test_hash():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
t1 = p(a) * q(b)
t2 = p(a) * p(b)
assert hash(t1) != hash(t2)
t3 = p(a) * p(b) + g(a, b)
t4 = p(a) * p(b) - g(a, b)
assert hash(t3) != hash(t4)
def test_div():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m0, m1, m2, m3 = tensor_indices('m0:4', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
t = R(m0, m1, -m1, m3)
t1 = t / S(4)
assert str(t1) == '1/4*R(m0, L_0, -L_0, m3)'
t = t.canon_bp()
assert not t1._is_canon_bp
t1 = t * 4
assert t1._is_canon_bp
t1 = t1 / 4
assert t1._is_canon_bp
def test_noncommuting_components():
numpy = import_module("numpy")
if numpy is None:
skip("numpy not installed.")
euclid = TensorIndexType('Euclidean')
euclid.data = [1, 1]
i1, i2, i3 = tensor_indices('i1:4', euclid)
a, b, c, d = symbols('a b c d', commutative=False)
V1 = tensorhead('V1', [euclid] * 2, [[1]] * 2)
V1.data = [[a, b], (c, d)]
V2 = tensorhead('V2', [euclid] * 2, [[1]] * 2)
V2.data = [[a, c], [b, d]]
vtp = V1(i1, i2) * V2(-i2, -i1)
assert vtp == a**2 + b * c + c * b + d**2
assert vtp != a**2 + 2 * b * c + d**2
Vc = b * V1(i1, -i1)
assert Vc.expand() == b * a + b * d
def test_noncommuting_components():
numpy = import_module("numpy")
if numpy is None:
skip("numpy not installed.")
euclid = TensorIndexType('Euclidean')
euclid.data = [1, 1]
i1, i2, i3 = tensor_indices('i1:4', euclid)
a, b, c, d = symbols('a b c d', commutative=False)
V1 = tensorhead('V1', [euclid] * 2, [[1]]*2)
V1.data = [[a, b], (c, d)]
V2 = tensorhead('V2', [euclid] * 2, [[1]]*2)
V2.data = [[a, c], [b, d]]
vtp = V1(i1, i2) * V2(-i2, -i1)
assert vtp == a ** 2 + b * c + c * b + d ** 2
assert vtp != a**2 + 2*b*c + d**2
Vc = b * V1(i1, -i1)
assert Vc.expand() == b * a + b * d
def test_add2():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m, n, p, q = tensor_indices('m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
A = tensorhead('A', [Lorentz] * 3, [[3]])
t1 = 2 * R(m, n, p, q) - R(m, q, n, p) + R(m, p, n, q)
t2 = t1 * A(-n, -p, -q)
assert t2 == 0
t1 = S(2) / 3 * R(m, n, p, q) - S(1) / 3 * R(m, q, n, p) + S(1) / 3 * R(
m, p, n, q)
t2 = t1 * A(-n, -p, -q)
assert t2 == 0
t = A(m, -m, n) + A(n, p, -p)
assert t == 0
def test_canonicalize3():
D = Symbol('D')
Spinor = TensorIndexType('Spinor', dim=D, metric=True, dummy_fmt='S')
a0, a1, a2, a3, a4 = tensor_indices('a0:5', Spinor)
C = Spinor.metric
chi, psi = tensorhead('chi,psi', [Spinor], [[1]], 1)
t = chi(a1) * psi(a0)
t1 = t.canon_bp()
assert t1 == t
t = psi(a1) * chi(a0)
t1 = t.canon_bp()
assert t1 == -chi(a0) * psi(a1)
def __new__(cls, symbol, coords, matrix, **kwargs):
"""
Create a new Metric object.
Parameters
----------
symbol : str
Name of the tensor and the symbol to denote it by when printed.
coords : iterable
List of ~sympy.Symbol objects to denote the coordinates by which
derivatives are taken with respect to.
matrix : (list, tuple, ~sympy.Matrix, ~sympy.Array)
Matrix representation of the tensor to be used in substitution.
Can be of any type that is acceptable by ~sympy.Array.
Examples
--------
>>> from sympy import diag, symbols
>>> from einsteinpy.symbolic.tensor import indices, expand_tensor
>>> from einsteinpy.symbolic.metric import Metric
>>> t, x, y, z = symbols('t x y z')
>>> eta = Metric('eta', [t, x, y, z], diag(1, -1, -1, -1))
>>> mu, nu = indices('mu nu', eta)
>>> expr = eta(mu, nu) * eta(-mu, -nu)
>>> expand_tensor(expr)
4
"""
array = Array(matrix)
if array.rank() != 2 or array.shape[0] != array.shape[1]:
raise ValueError(
"matrix must be square, received matrix of shape {}".format(
array.shape))
obj = TensorIndexType.__new__(
cls,
symbol,
metric=cls._MetricId(symbol, False),
dim=array.shape[0],
dummy_fmt=symbol,
**kwargs,
)
obj = AbstractTensor.__new__(cls, obj, array)
obj.metric = Tensor(obj.name, array, obj, covar=(-1, -1))
obj.coords = tuple(coords)
ReplacementManager[obj] = array
return obj
def test_canonicalize_no_dummies():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a, b, c, d', Lorentz)
sym1 = tensorsymmetry([1])
sym2 = tensorsymmetry([1] * 2)
sym2a = tensorsymmetry([2])
# A commuting
# A^c A^b A^a
# T_c = A^a A^b A^c
S1 = TensorType([Lorentz], sym1)
A = S1('A')
t = A(c) * A(b) * A(a)
tc = t.canon_bp()
assert str(tc) == 'A(a)*A(b)*A(c)'
# A anticommuting
# A^c A^b A^a
# T_c = -A^a A^b A^c
A = S1('A', 1)
t = A(c) * A(b) * A(a)
tc = t.canon_bp()
assert str(tc) == '-A(a)*A(b)*A(c)'
# A commuting and symmetric
# A^{b,d}*A^{c,a}
# T_c = A^{a c}*A^{b d}
S2 = TensorType([Lorentz] * 2, sym2)
A = S2('A')
t = A(b, d) * A(c, a)
tc = t.canon_bp()
assert str(tc) == 'A(a, c)*A(b, d)'
# A anticommuting and symmetric
# A^{b,d}*A^{c,a}
# T_c = -A^{a c}*A^{b d}
A = S2('A', 1)
t = A(b, d) * A(c, a)
tc = t.canon_bp()
assert str(tc) == '-A(a, c)*A(b, d)'
# A^{c,a}*A^{b,d}
# T_c = A^{a c}*A^{b d}
t = A(c, a) * A(b, d)
tc = t.canon_bp()
assert str(tc) == 'A(a, c)*A(b, d)'
def test_riemann_products():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
d0, d1, d2, d3, d4, d5, d6 = tensor_indices('d0:7', Lorentz)
a0, a1, a2, a3, a4, a5 = tensor_indices('a0:6', Lorentz)
a, b = tensor_indices('a,b', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
# R^{a b d0}_d0 = 0
t = R(a, b, d0, -d0)
tc = t.canon_bp()
assert tc == 0
# R^{d0 b a}_d0
# T_c = -R^{a d0 b}_d0
t = R(d0, b, a, -d0)
tc = t.canon_bp()
assert str(tc) == '-R(a, L_0, b, -L_0)'
# R^d1_d2^b_d0 * R^{d0 a}_d1^d2; ord=[a,b,d0,-d0,d1,-d1,d2,-d2]
# T_c = -R^{a d0 d1 d2}* R^b_{d0 d1 d2}
t = R(d1, -d2, b, -d0) * R(d0, a, -d1, d2)
tc = t.canon_bp()
assert str(tc) == '-R(a, L_0, L_1, L_2)*R(b, -L_0, -L_1, -L_2)'
# A symmetric commuting
# R^{d6 d5}_d2^d1 * R^{d4 d0 d2 d3} * A_{d6 d0} A_{d3 d1} * A_{d4 d5}
# g = [12,10,5,2, 8,0,4,6, 13,1, 7,3, 9,11,14,15]
# T_c = -R^{d0 d1 d2 d3} * R_d0^{d4 d5 d6} * A_{d1 d4}*A_{d2 d5}*A_{d3 d6}
V = tensorhead('V', [Lorentz] * 2, [[1] * 2])
t = R(d6, d5, -d2, d1) * R(d4, d0, d2, d3) * V(-d6, -d0) * V(-d3, -d1) * V(
-d4, -d5)
tc = t.canon_bp()
assert str(
tc
) == '-R(L_0, L_1, L_2, L_3)*R(-L_0, L_4, L_5, L_6)*V(-L_1, -L_4)*V(-L_2, -L_5)*V(-L_3, -L_6)'
# R^{d2 a0 a2 d0} * R^d1_d2^{a1 a3} * R^{a4 a5}_{d0 d1}
# T_c = R^{a0 d0 a2 d1}*R^{a1 a3}_d0^d2*R^{a4 a5}_{d1 d2}
t = R(d2, a0, a2, d0) * R(d1, -d2, a1, a3) * R(a4, a5, -d0, -d1)
tc = t.canon_bp()
assert str(
tc) == 'R(a0, L_0, a2, L_1)*R(a1, a3, -L_0, L_2)*R(a4, a5, -L_1, -L_2)'
def test_substitute_indices():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i, j, k, l, m, n, p, q = tensor_indices('i,j,k,l,m,n,p,q', Lorentz)
A, B = tensorhead('A,B', [Lorentz] * 2, [[1] * 2])
t = A(i, k) * B(-k, -j)
t1 = t.substitute_indices((i, j), (j, k))
t1a = A(j, l) * B(-l, -k)
assert t1 == t1a
p = tensorhead('p', [Lorentz], [[1]])
t = p(i)
t1 = t.substitute_indices((j, k))
assert t1 == t
t1 = t.substitute_indices((i, j))
assert t1 == p(j)
t1 = t.substitute_indices((i, -j))
assert t1 == p(-j)
t1 = t.substitute_indices((-i, j))
assert t1 == p(-j)
t1 = t.substitute_indices((-i, -j))
assert t1 == p(j)
def test_TensExpr():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
g = Lorentz.metric
A, B = tensorhead('A B', [Lorentz] * 2, [[1] * 2])
raises(ValueError, lambda: g(c, d) / g(a, b))
raises(ValueError, lambda: S.One / g(a, b))
raises(ValueError, lambda: (A(c, d) + g(c, d)) / g(a, b))
raises(ValueError, lambda: S.One / (A(c, d) + g(c, d)))
raises(ValueError, lambda: A(a, b) + A(a, c))
t = A(a, b) + B(a, b)
raises(NotImplementedError, lambda: TensExpr.__mul__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__add__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__radd__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__sub__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__rsub__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__div__(t, 'a'))
raises(NotImplementedError, lambda: TensExpr.__rdiv__(t, 'a'))
raises(NotImplementedError, lambda: A(a, b)**2)
raises(NotImplementedError, lambda: 2**A(a, b))
raises(NotImplementedError, lambda: abs(A(a, b)))
def test_riemann_cyclic():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i, j, k, l, m, n, p, q = tensor_indices('i,j,k,l,m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz] * 4, [[2, 2]])
t = R(i,j,k,l) + R(i,l,j,k) + R(i,k,l,j) - \
R(i,j,l,k) - R(i,l,k,j) - R(i,k,j,l)
t2 = t * R(-i, -j, -k, -l)
t3 = riemann_cyclic(t2)
assert t3 == 0
t = R(i, j, k, l) * (R(-i, -j, -k, -l) - 2 * R(-i, -k, -j, -l))
t1 = riemann_cyclic(t)
assert t1 == 0
t = R(i, j, k, l)
t1 = riemann_cyclic(t)
assert t1 == -S(1) / 3 * R(i, l, j, k) + S(1) / 3 * R(
i, k, j, l) + S(2) / 3 * R(i, j, k, l)
t = R(i, j, k, l) * R(-k, -l, m,
n) * (R(-m, -n, -i, -j) + 2 * R(-m, -j, -n, -i))
t1 = riemann_cyclic(t)
assert t1 == 0
def test_canonicalize2():
D = Symbol('D')
Eucl = TensorIndexType('Eucl', metric=0, dim=D, dummy_fmt='E')
i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14 = \
tensor_indices('i0:15', Eucl)
A = tensorhead('A', [Eucl] * 3, [[3]])
# two examples from Cvitanovic, Group Theory page 59
# of identities for antisymmetric tensors of rank 3
# contracted according to the Kuratowski graph eq.(6.59)
t = A(i0, i1, i2) * A(-i1, i3, i4) * A(-i3, i7, i5) * A(-i2, -i5, i6) * A(
-i4, -i6, i8)
t1 = t.canon_bp()
assert t1 == 0
# eq.(6.60)
#t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i2,i5,i6)*A(-i3,i7,i8)*A(-i6,-i7,i9)*
# A(-i8,i10,i13)*A(-i5,-i10,i11)*A(-i4,-i11,i12)*A(-i3,-i12,i14)
t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i2,i5,i6)*A(-i3,i7,i8)*A(-i6,-i7,i9)*\
A(-i8,i10,i13)*A(-i5,-i10,i11)*A(-i4,-i11,i12)*A(-i9,-i12,i14)
t1 = t.canon_bp()
assert t1 == 0
def test_epsilon():
Lorentz = TensorIndexType('Lorentz', dim=4, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
epsilon = Lorentz.epsilon
p, q, r, s = tensorhead('p,q,r,s', [Lorentz], [[1]])
t = epsilon(b, a, c, d)
t1 = t.canon_bp()
assert t1 == -epsilon(a, b, c, d)
t = epsilon(c, b, d, a)
t1 = t.canon_bp()
assert t1 == epsilon(a, b, c, d)
t = epsilon(c, a, d, b)
t1 = t.canon_bp()
assert t1 == -epsilon(a, b, c, d)
t = epsilon(a, b, c, d) * p(-a) * q(-b)
t1 = t.canon_bp()
assert t1 == epsilon(c, d, a, b) * p(-a) * q(-b)
t = epsilon(c, b, d, a) * p(-a) * q(-b)
t1 = t.canon_bp()
assert t1 == epsilon(c, d, a, b) * p(-a) * q(-b)
t = epsilon(c, a, d, b) * p(-a) * q(-b)
t1 = t.canon_bp()
assert t1 == -epsilon(c, d, a, b) * p(-a) * q(-b)
t = epsilon(c, a, d, b) * p(-a) * p(-b)
t1 = t.canon_bp()
assert t1 == 0
t = epsilon(c, a, d, b) * p(-a) * q(-b) + epsilon(a, b, c,
d) * p(-b) * q(-a)
t1 = t.canon_bp()
assert t1 == -2 * epsilon(c, d, a, b) * p(-a) * q(-b)
def test_no_metric_symmetry():
# no metric symmetry; A no symmetry
# A^d1_d0 * A^d0_d1
# T_c = A^d0_d1 * A^d1_d0
Lorentz = TensorIndexType('Lorentz', metric=None, dummy_fmt='L')
d0, d1, d2, d3 = tensor_indices('d:4', Lorentz)
A = tensorhead('A', [Lorentz] * 2, [[1], [1]])
t = A(d1, -d0) * A(d0, -d1)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)'
# A^d1_d2 * A^d0_d3 * A^d2_d1 * A^d3_d0
# T_c = A^d0_d1 * A^d1_d0 * A^d2_d3 * A^d3_d2
t = A(d1, -d2) * A(d0, -d3) * A(d2, -d1) * A(d3, -d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)*A(L_2, -L_3)*A(L_3, -L_2)'
# A^d0_d2 * A^d1_d3 * A^d3_d0 * A^d2_d1
# T_c = A^d0_d1 * A^d1_d2 * A^d2_d3 * A^d3_d0
t = A(d0, -d1) * A(d1, -d2) * A(d2, -d3) * A(d3, -d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_2)*A(L_2, -L_3)*A(L_3, -L_0)'
def test_hidden_indices_for_matrix_multiplication():
L = TensorIndexType('Lorentz')
S = TensorIndexType('Matind')
m0, m1, m2, m3, m4, m5 = tensor_indices('m0:6', L)
s0, s1, s2 = tensor_indices('s0:3', S)
A = tensorhead('A', [L, S, S], [[1], [1], [1]], matrix_behavior=True)
B = tensorhead('B', [L, S], [[1], [1]], matrix_behavior=True)
D = tensorhead('D', [L, L, S, S], [[1, 1], [1, 1]], matrix_behavior=True)
E = tensorhead('E', [L, L, L, L], [[1], [1], [1], [1]], matrix_behavior=True)
F = tensorhead('F', [L], [[1]], matrix_behavior=True)
assert (A(m0)) == A(m0, S.auto_left, -S.auto_right)
assert (B(-m1)) == B(-m1, S.auto_left)
A0 = A(m0)
B0 = B(-m0)
B1 = B(m1)
assert (B1*A0*B0) == B(m1, s0)*A(m0, -s0, s1)*B(-m0, -s1)
assert (B0*A0) == B(-m0, s0)*A(m0, -s0, -S.auto_right)
assert (A0*B0) == A(m0, S.auto_left, s0)*B(-m0, -s0)
C = tensorhead('C', [L, L], [[1]*2])
assert (C(True, True)) == C(L.auto_left, -L.auto_right)
assert (A(m0)*C(m1, -m0)) == A(m2, S.auto_left, -S.auto_right)*C(m1, -m2)
assert (C(True, True)*C(True, True)) == C(L.auto_left, m0)*C(-m0, -L.auto_right)
assert A(m0) == A(m0)
assert B(-m1) == B(-m1)
assert A(m0) - A(m0) == 0
ts1 = A(m0)*A(m1) + A(m1)*A(m0)
ts2 = A(m1)*A(m0) + A(m0)*A(m1)
assert ts1 == ts2
assert A(m0)*A(m1) + A(m1)*A(m0) == A(m1)*A(m0) + A(m0)*A(m1)
assert A(m0) == (2*A(m0))/2
assert A(m0) == -(-A(m0))
assert 2*A(m0) - 3*A(m0) == -A(m0)
assert 2*D(m0, m1) - 5*D(m1, m0) == -3*D(m0, m1)
D0 = D(True, True, True, True)
Aa = A(True, True, True)
assert D0 * Aa == D(L.auto_left, m0, S.auto_left, s0)*A(-m0, -s0, -S.auto_right)
assert D(m0, m1) == D(m0, m1, S.auto_left, -S.auto_right)
raises(ValueError, lambda: C(True))
raises(ValueError, lambda: C())
raises(ValueError, lambda: E(True, True, True, True))
# test that a delta is automatically added on missing auto-matrix indices in TensAdd
assert F(m2)*F(m3)*F(m4)*A(m1) + E(m1, m2, m3, m4) == \
E(m1, m2, m3, m4)*S.delta(S.auto_left, -S.auto_right) +\
F(m2)*F(m3)*F(m4)*A(m1, S.auto_left, -S.auto_right)
assert E(m1, m2) + F(m1)*F(m2) == E(m1, m2) + F(m1)*F(m2)*L.delta(L.auto_left, -L.auto_right)
assert E(m1, m2)*A(m3) + F(m1)*F(m2)*F(m3) == \
E(m1, m2, L.auto_left, -L.auto_right)*A(m3, S.auto_left, -S.auto_right) +\
F(m1)*F(m2)*F(m3)*L.delta(L.auto_left, -L.auto_right)*S.delta(S.auto_left, -S.auto_right)
assert L.delta() == L.delta(L.auto_left, -L.auto_right)
assert S.delta() == S.delta(S.auto_left, -S.auto_right)
assert L.metric() == L.metric(L.auto_left, -L.auto_right)
assert S.metric() == S.metric(S.auto_left, -S.auto_right)
assert hash(tsymmetry.func(*tsymmetry.args)) == hash(tsymmetry)
assert check_all(tsymmetry)
### TEST VALUED TENSORS ###
numpy = import_module('numpy')
if numpy:
minkowski = Matrix((
(1, 0, 0, 0),
(0, -1, 0, 0),
(0, 0, -1, 0),
(0, 0, 0, -1),
))
Lorentz = TensorIndexType('Lorentz', dim=4)
Lorentz.data = minkowski
i0, i1, i2, i3, i4 = tensor_indices('i0:5', Lorentz)
E, px, py, pz = symbols('E px py pz')
A = tensorhead('A', [Lorentz], [[1]])
A.data = [E, px, py, pz]
B = tensorhead('B', [Lorentz], [[1]], 'Gcomm')
B.data = range(4)
AB = tensorhead("AB", [Lorentz] * 2, [[1]]*2)
AB.data = minkowski
ba_matrix = Matrix((
(1, 2, 3, 4),
(5, 6, 7, 8),
