product of surfaces with a LCW

Computations for the paper: Sufficient conditions for the existence of limiting Carleman weights (section 5)¶

Pablo Angulo-Ardoy, Daniel Faraco, Luis Guijarro http://arxiv.org/abs/1603.04201

This notebook only covers the computations in the proof of lemma 5.2

In [1]:

M = Manifold(4, 'M', r'\mathcal{M}', start_index=1)
print M
X.<t,x,y,z> = M.chart()
print X
g = M.riemannian_metric('g')
print g

4-dimensional differentiable manifold M
Chart (M, (t, x, y, z))
Riemannian metric g on the 4-dimensional differentiable manifold M

In [2]:

F = function('F', x,y,z)
g[1,1], g[2,2], g[3,3], g[4,4] = 1, 1, F(x,y,z), F(x,y,z)
g.display()

/projects/sage/sage-6.10/local/lib/python2.7/site-packages/IPython/core/interactiveshell.py:3066: DeprecationWarning: Calling function('f',x) is deprecated. Use function('f')(x) instead.
See http://trac.sagemath.org/17447 for details.
  exec(code_obj, self.user_global_ns, self.user_ns)
/projects/sage/sage-6.10/local/lib/python2.7/site-packages/IPython/core/interactiveshell.py:3066: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)
See http://trac.sagemath.org/5930 for details.
  exec(code_obj, self.user_global_ns, self.user_ns)

Out[2]:

g = dt*dt + dx*dx + F(x, y, z) dy*dy + F(x, y, z) dz*dz

In [3]:

show(g[:])

(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & F (x, y, z) & 0 \\ 0 & 0 & 0 & F (x, y, z) \end{array})

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & F\left(x, y, z\right) & 0 \\ 0 & 0 & 0 & F\left(x, y, z\right) \end{array}\right)$

The Weyl tensor has type B¶

In [4]:

%time W = g.weyl()

CPU times: user 1min 25s, sys: 1.59 s, total: 1min 27s
Wall time: 1min 30s

In [5]:

W4 = W.down(g,0)

In [6]:

Frame = X.frame()
def g_norm(v):
    return sqrt(g(v,v))
vs  = [Frame[i] for i in [1..4]]
ovs = [v/g_norm(v) for v in vs]
v1,v2,v3,v4 = ovs

N = 4
INDEX_PAIRS = [tuple(c) for c in Combinations([1..N], 2)]

#Bivectors
BASIS = [(vi*vj).antisymmetrize(0,1) for vi,vj in Combinations(ovs, 2)]
for elem in BASIS:
    print elem.display()
d = 6
v12 = BASIS[0]
v13 = BASIS[1]

def WO(bv1, bv2):
    r = 0
    for i,j in INDEX_PAIRS:
        c1 = bv1[i,j]
        for k,l in INDEX_PAIRS:
            c2 = bv2[k,l]
            r += c1*c2*W4[i,j,k,l]
    return r

1/2 d/dt*d/dx - 1/2 d/dx*d/dt
1/2/sqrt(F(x, y, z)) d/dt*d/dy - 1/2/sqrt(F(x, y, z)) d/dy*d/dt
1/2/sqrt(F(x, y, z)) d/dt*d/dz - 1/2/sqrt(F(x, y, z)) d/dz*d/dt
1/2/sqrt(F(x, y, z)) d/dx*d/dy - 1/2/sqrt(F(x, y, z)) d/dy*d/dx
1/2/sqrt(F(x, y, z)) d/dx*d/dz - 1/2/sqrt(F(x, y, z)) d/dz*d/dx
1/2/F(x, y, z) d/dy*d/dz - 1/2/F(x, y, z) d/dz*d/dy

In [7]:

MM = matrix(SR, d)
for i, bvi in enumerate(BASIS):
    for j,bvj in enumerate(BASIS):
        MM[i,j] = WO(bvi, bvj).expr()

In [9]:

show(MM[:3,3:])

(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array})

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$

In [10]:

show(MM[:3,:3])

(\begin{array}{rrr} - \frac{F (x, y, z) D [0] (F) {(x, y, z)}^{2} - F {(x, y, z)}^{2} D [0, 0] (F) (x, y, z) - D [1] (F) {(x, y, z)}^{2} + F (x, y, z) D [1, 1] (F) (x, y, z) - D [2] (F) {(x, y, z)}^{2} + F (x, y, z) D [2, 2] (F) (x, y, z)}{24 F {(x, y, z)}^{3}} & \frac{F (x, y, z) D [0, 1] (F) (x, y, z) - D [0] (F) (x, y, z) D [1] (F) (x, y, z)}{16 F {(x, y, z)}^{\frac{5}{2}}} & \frac{F (x, y, z) D [0, 2] (F) (x, y, z) - D [0] (F) (x, y, z) D [2] (F) (x, y, z)}{16 F {(x, y, z)}^{\frac{5}{2}}} \\ \frac{F (x, y, z) D [0, 1] (F) (x, y, z) - D [0] (F) (x, y, z) D [1] (F) (x, y, z)}{16 F {(x, y, z)}^{\frac{5}{2}}} & \frac{F (x, y, z) D [0] (F) {(x, y, z)}^{2} - F {(x, y, z)}^{2} D [0, 0] (F) (x, y, z) - D [1] (F) {(x, y, z)}^{2} + F (x, y, z) D [1, 1] (F) (x, y, z) - D [2] (F) {(x, y, z)}^{2} + F (x, y, z) D [2, 2] (F) (x, y, z)}{48 F {(x, y, z)}^{3}} & 0 \\ \frac{F (x, y, z) D [0, 2] (F) (x, y, z) - D [0] (F) (x, y, z) D [2] (F) (x, y, z)}{16 F {(x, y, z)}^{\frac{5}{2}}} & 0 & \frac{F (x, y, z) D [0] (F) {(x, y, z)}^{2} - F {(x, y, z)}^{2} D [0, 0] (F) (x, y, z) - D [1] (F) {(x, y, z)}^{2} + F (x, y, z) D [1, 1] (F) (x, y, z) - D [2] (F) {(x, y, z)}^{2} + F (x, y, z) D [2, 2] (F) (x, y, z)}{48 F {(x, y, z)}^{3}} \end{array})

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} -\frac{F\left(x, y, z\right) D[0]\left(F\right)\left(x, y, z\right)^{2} - F\left(x, y, z\right)^{2} D[0, 0]\left(F\right)\left(x, y, z\right) - D[1]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[1, 1]\left(F\right)\left(x, y, z\right) - D[2]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[2, 2]\left(F\right)\left(x, y, z\right)}{24 \, F\left(x, y, z\right)^{3}} & \frac{F\left(x, y, z\right) D[0, 1]\left(F\right)\left(x, y, z\right) - D[0]\left(F\right)\left(x, y, z\right) D[1]\left(F\right)\left(x, y, z\right)}{16 \, F\left(x, y, z\right)^{\frac{5}{2}}} & \frac{F\left(x, y, z\right) D[0, 2]\left(F\right)\left(x, y, z\right) - D[0]\left(F\right)\left(x, y, z\right) D[2]\left(F\right)\left(x, y, z\right)}{16 \, F\left(x, y, z\right)^{\frac{5}{2}}} \\ \frac{F\left(x, y, z\right) D[0, 1]\left(F\right)\left(x, y, z\right) - D[0]\left(F\right)\left(x, y, z\right) D[1]\left(F\right)\left(x, y, z\right)}{16 \, F\left(x, y, z\right)^{\frac{5}{2}}} & \frac{F\left(x, y, z\right) D[0]\left(F\right)\left(x, y, z\right)^{2} - F\left(x, y, z\right)^{2} D[0, 0]\left(F\right)\left(x, y, z\right) - D[1]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[1, 1]\left(F\right)\left(x, y, z\right) - D[2]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[2, 2]\left(F\right)\left(x, y, z\right)}{48 \, F\left(x, y, z\right)^{3}} & 0 \\ \frac{F\left(x, y, z\right) D[0, 2]\left(F\right)\left(x, y, z\right) - D[0]\left(F\right)\left(x, y, z\right) D[2]\left(F\right)\left(x, y, z\right)}{16 \, F\left(x, y, z\right)^{\frac{5}{2}}} & 0 & \frac{F\left(x, y, z\right) D[0]\left(F\right)\left(x, y, z\right)^{2} - F\left(x, y, z\right)^{2} D[0, 0]\left(F\right)\left(x, y, z\right) - D[1]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[1, 1]\left(F\right)\left(x, y, z\right) - D[2]\left(F\right)\left(x, y, z\right)^{2} + F\left(x, y, z\right) D[2, 2]\left(F\right)\left(x, y, z\right)}{48 \, F\left(x, y, z\right)^{3}} \end{array}\right)$

In [ ]: