1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
|
#!/usr/bin/env python
"""
This example calculates the Ricci tensor from the metric and does this
on the example of Schwarzschild solution.
If you want to derive this by hand, follow the wiki page here:
https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution
Also read the above wiki and follow the references from there if
something is not clear, like what the Ricci tensor is, etc.
"""
from sympy import (exp, Symbol, sin, dsolve, Function,
Matrix, Eq, pprint, solve)
def grad(f, X):
a = []
for x in X:
a.append(f.diff(x))
return a
def d(m, x):
return grad(m[0, 0], x)
class MT:
def __init__(self, m):
self.gdd = m
self.guu = m.inv()
def __str__(self):
return "g_dd =\n" + str(self.gdd)
def dd(self, i, j):
return self.gdd[i, j]
def uu(self, i, j):
return self.guu[i, j]
class G:
def __init__(self, g, x):
self.g = g
self.x = x
def udd(self, i, k, l):
g = self.g
x = self.x
r = 0
for m in [0, 1, 2, 3]:
r += g.uu(i, m)/2 * (g.dd(m, k).diff(x[l]) + g.dd(m, l).diff(x[k])
- g.dd(k, l).diff(x[m]))
return r
class Riemann:
def __init__(self, G, x):
self.G = G
self.x = x
def uddd(self, rho, sigma, mu, nu):
G = self.G
x = self.x
r = G.udd(rho, nu, sigma).diff(x[mu]) - G.udd(rho, mu, sigma).diff(x[nu])
for lam in [0, 1, 2, 3]:
r += G.udd(rho, mu, lam)*G.udd(lam, nu, sigma) \
- G.udd(rho, nu, lam)*G.udd(lam, mu, sigma)
return r
class Ricci:
def __init__(self, R, x):
self.R = R
self.x = x
self.g = R.G.g
def dd(self, mu, nu):
R = self.R
x = self.x
r = 0
for lam in [0, 1, 2, 3]:
r += R.uddd(lam, mu, lam, nu)
return r
def ud(self, mu, nu):
r = 0
for lam in [0, 1, 2, 3]:
r += self.g.uu(mu, lam)*self.dd(lam, nu)
return r.expand()
def curvature(Rmn):
return Rmn.ud(0, 0) + Rmn.ud(1, 1) + Rmn.ud(2, 2) + Rmn.ud(3, 3)
nu = Function("nu")
lam = Function("lambda")
t = Symbol("t")
r = Symbol("r")
theta = Symbol(r"theta")
phi = Symbol(r"phi")
# general, spherically symmetric metric
gdd = Matrix((
(-exp(nu(r)), 0, 0, 0),
(0, exp(lam(r)), 0, 0),
(0, 0, r**2, 0),
(0, 0, 0, r**2*sin(theta)**2)
))
g = MT(gdd)
X = (t, r, theta, phi)
Gamma = G(g, X)
Rmn = Ricci(Riemann(Gamma, X), X)
def pprint_Gamma_udd(i, k, l):
pprint(Eq(Symbol('Gamma^%i_%i%i' % (i, k, l)), Gamma.udd(i, k, l)))
def pprint_Rmn_dd(i, j):
pprint(Eq(Symbol('R_%i%i' % (i, j)), Rmn.dd(i, j)))
# from Differential Equations example
def eq1():
r = Symbol("r")
e = Rmn.dd(0, 0)
e = e.subs(nu(r), -lam(r))
pprint(dsolve(e, lam(r)))
def eq2():
r = Symbol("r")
e = Rmn.dd(1, 1)
C = Symbol("CC")
e = e.subs(nu(r), -lam(r))
pprint(dsolve(e, lam(r)))
def eq3():
r = Symbol("r")
e = Rmn.dd(2, 2)
e = e.subs(nu(r), -lam(r))
pprint(dsolve(e, lam(r)))
def eq4():
r = Symbol("r")
e = Rmn.dd(3, 3)
e = e.subs(nu(r), -lam(r))
pprint(dsolve(e, lam(r)))
pprint(dsolve(e, lam(r), 'best'))
def main():
print("Initial metric:")
pprint(gdd)
print("-"*40)
print("Christoffel symbols:")
pprint_Gamma_udd(0, 1, 0)
pprint_Gamma_udd(0, 0, 1)
print()
pprint_Gamma_udd(1, 0, 0)
pprint_Gamma_udd(1, 1, 1)
pprint_Gamma_udd(1, 2, 2)
pprint_Gamma_udd(1, 3, 3)
print()
pprint_Gamma_udd(2, 2, 1)
pprint_Gamma_udd(2, 1, 2)
pprint_Gamma_udd(2, 3, 3)
print()
pprint_Gamma_udd(3, 2, 3)
pprint_Gamma_udd(3, 3, 2)
pprint_Gamma_udd(3, 1, 3)
pprint_Gamma_udd(3, 3, 1)
print("-"*40)
print("Ricci tensor:")
pprint_Rmn_dd(0, 0)
e = Rmn.dd(1, 1)
pprint_Rmn_dd(1, 1)
pprint_Rmn_dd(2, 2)
pprint_Rmn_dd(3, 3)
print("-"*40)
print("Solve Einstein's equations:")
e = e.subs(nu(r), -lam(r)).doit()
l = dsolve(e, lam(r))
pprint(l)
lamsol = solve(l, lam(r))[0]
metric = gdd.subs(lam(r), lamsol).subs(nu(r), -lamsol) # .combine()
print("metric:")
pprint(metric)
if __name__ == "__main__":
main()
|
