def init_S2(self):
"""Initialize the metric on the 2-sphere, S2.
"""
theta = sp.symbols(self.S2[0])
phi = sp.symbols(self.S2[1])
g = Metric(index_dict=self.S2)
g.convert_to_shorthand()
g.elements['thetatheta'] = 1
g.elements['phiphi'] = (sp.sin(theta))**2
return g
def init_kerr(self):
"""Initialize the Kerr metric on 4D spacetime.
"""
t = sp.symbols(self.KERR[0])
r = sp.symbols(self.KERR[1])
theta = sp.symbols(self.KERR[2])
phi = sp.symbols(self.KERR[3])
G = sp.symbols('G')
M = sp.symbols('M')
rho = sp.symbols('rho')
a = sp.symbols('a')
Delta = sp.symbols('Delta')
"""
Notes:
------
a = J/M where J is the Komar angular momentum (see 6.48
in Carroll).
"""
Delta_expr = r**2 - 2 * G * M * r + a**2
rho2_expr = r**2 + a**2 * (sp.cos(theta))**2
g = Metric(index_dict=self.KERR)
g.convert_to_shorthand()
g.elements['tt'] = -(1 - 2 * G * M * r / rho**2)
g.elements['tt'] = g.elements['tt'].subs(rho**2, rho2_expr)
g.elements['tphi'] = -2 * G * M * a * r * (sp.sin(theta))**2 / rho**2
g.elements['tphi'] = g.elements['tphi'].subs(rho**2, rho2_expr)
g.elements['phit'] = g.elements['tphi']
g.elements['rr'] = rho**2 / Delta
g.elements['rr'] = g.elements['rr'].subs(rho**2, rho2_expr)
g.elements['rr'] = g.elements['rr'].subs(Delta, Delta_expr)
g.elements['thetatheta'] = rho**2
g.elements['thetatheta'] = g.elements['thetatheta'].subs(
rho**2, rho2_expr)
g.elements['phiphi'] = ((sp.sin(theta))**2 / rho**2) * (
(r**2 + a**2)**2 - a**2 * Delta * (sp.sin(theta))**2)
g.elements['phiphi'] = g.elements['phiphi'].subs(rho**2, rho2_expr)
g.elements['phiphi'] = g.elements['phiphi'].subs(Delta, Delta_expr)
return g
def init_deSitter(self):
"""Initialize the metric on 4D de Sitter spacetime.
"""
t = sp.symbols(self.DESITTER[0])
x = sp.symbols(self.DESITTER[1])
y = sp.symbols(self.DESITTER[2])
z = sp.symbols(self.DESITTER[3])
H = sp.symbols('H')
g = Metric(index_dict=self.DESITTER)
g.convert_to_shorthand()
g.elements['tt'] = -1
g.elements['xx'] = sp.exp(2 * H * t)
g.elements['yy'] = g.elements['xx']
g.elements['zz'] = g.elements['xx']
return g
def init_schwarzchild(self):
"""Initialize Schwarzchild metric.
"""
t = sp.symbols(self.SCHWARZCHILD[0])
r = sp.symbols(self.SCHWARZCHILD[1])
theta = sp.symbols(self.SCHWARZCHILD[2])
phi = sp.symbols(self.SCHWARZCHILD[3])
R = sp.symbols('R')
g = Metric(index_dict=self.SCHWARZCHILD)
g.convert_to_shorthand()
g.elements['tt'] = 1 - R / r
g.elements['rr'] = (1 - R / r)**(-1)
g.elements['thetatheta'] = r**2
g.elements['phiphi'] = r**2 * (sp.sin(theta))**2
return g
def init_schwarzchild_func(self):
"""Initialize Schwarzchild metric
but the variables are now functions
of a parameter.
"""
t = sp.Function(self.SCHWARZCHILD[0])
r = sp.Function(self.SCHWARZCHILD[1])
theta = sp.Function(self.SCHWARZCHILD[2])
phi = sp.Function(self.SCHWARZCHILD[3])
R = sp.symbols('R')
y = sp.symbols('y')
g = Metric(index_dict=self.SCHWARZCHILD)
g.convert_to_shorthand()
g.elements['tt'] = 1 - R * r(y)**(-1)
g.elements['rr'] = (1 - R * r(y)**(-1))**(-1)
g.elements['thetatheta'] = r(y)**2
g.elements['phiphi'] = r(y)**2 * (sp.sin(theta(y)))**2
return g
def init_FRW(self):
"""Initialize the FRW metric on 4D spacetime.
"""
t = sp.symbols('t')
r = sp.symbols('r')
kappa = sp.symbols('k')
theta = sp.symbols('theta')
phi = sp.symbols('phi')
a = sp.Function('a')
g = Metric(index_dict=self.FRW)
g.convert_to_shorthand()
g.elements['tt'] = -1
g.elements['rr'] = a(t)**2 / (1 - kappa * r**2)
g.elements['thetatheta'] = a(t)**2 * r**2
g.elements['phiphi'] = a(t)**2 * r**2 * sp.sin(theta)**2
return g
def read_in_symbol_from_file(self, file_with_path, complex_symbol,
symbol_type):
from sympy.parsing.sympy_parser import parse_expr
"""Read a text file that contains a string version
of a complex symbol and return a complex symbol
with the entries that were stored in the text file.
Assumes that the text file has been created with
``self.write_complex_symbol()``; i.e. this function
is not fit for reading arbitrary text files.
"""
index_list = []
if symbol_type == 'metric':
constructed_symbol = Metric(index_dict=complex_symbol.index_dict)
elif symbol_type == 'christoffel':
constructed_symbol = Christoffel(
index_dict=complex_symbol.index_dict)
elif symbol_type == 'riemann':
constructed_symbol = Riemann(index_dict=complex_symbol.index_dict)
elif symbol_type == 'ricci':
constructed_symbol = Ricci(index_dict=complex_symbol.index_dict)
elif symbol_type == 'einstein':
constructed_symbol = Einstein(index_dict=complex_symbol.index_dict)
else:
raise TypeError(
'Got a bad symbol type when trying to read in a symbol from a text file.'
)
constructed_symbol.convert_to_shorthand()
with open(file_with_path, 'r') as text_file:
text = text_file.read()
for key in complex_symbol.elements:
index_list.append(key)
l = len(index_list)
#print('Here is the index list: {list}.'.format(list=index_list))
for i in range(l):
s = '[{ss}]'.format(ss=index_list[i])
#print('Searching for the end of {s}...'.format(s=s))
start = text.find(s) + len(s) # start of the symbolic string
#print('Found the end of {s} at {loc}.'.format(s=s, loc=start))
if i + 1 < l:
t = '[{tt}]'.format(tt=index_list[i + 1])
#print('Searching for {t}...'.format(t=t))
end = text.find(t)
#print('Found {t} at {loc}.'.format(t=t, loc=end))
symbol_string = text[start:end]
#print('Here is the symbol we found:\n')
#print('\t' + symbol_string)
output_symbol = parse_expr(symbol_string, evaluate=False)
else:
# handle the last symbol, which won't be followed
# by another ['xx'] flag...
symbol_string = text[start:] # go to end of the string!
#print('Here is the symbol we found:\n')
#print('\t' + symbol_string)
output_symbol = parse_expr(symbol_string, evaluate=False)
#print('Adding the following symbol to [{key}]:'.format(key=index_list[i]))
constructed_symbol.elements[index_list[i]] = output_symbol
#print(constructed_symbol.elements[index_list[i]])
return constructed_symbol
