General Relativity Symbolic Utilities

The pystein package contains utilities for computing symbolic utilities for computing various quantities that arise in general relativity. Presently, this package is essentially a sympy extension that computes components of tensors directly.

Symbolic Tools

The pystein package makes use of sympy to compute symbolic curvature equations (EFE).

Example Computation: FLRW Cosmology

# Load the predefined FLRW metric
from pystein import metric, gravity
from pystein import utilities

flrw = metric.flrw().subs({'c': 1})
flrw

efe_00 = utilities.full_simplify(gravity.einstein_equation(0, 0, flrw))
efe_00

# Can simplify notation using "dots"
metric.simplify_deriv_notation(efe_00, flrw, use_dots=True)

Symbolic Tools

Coordinate Systems

from sympy.diffgeom import Manifold, Patch
from pystein import coords

# The pystein CoordinateSystem extends the sympy.diffgeom api to make parameters more accessible
M = Manifold('M', dim=2)
P = Patch('origin', M)
cs = coords.CoordSystem('cartesian', P, ['x', 'y'])
cs

# In sympy it is difficult to access underlying parameters, but the new base_symbols function makes it easy:
cs.base_symbols()

Metrics

# Assembling a metric is easy
from sympy import Array, symbols
from pystein import metric
from pystein.utilities import tensor_pow as tpow

# Metrics can be created either from a (Matrix, Coords) combo or from a TwoForm Expression
# Let's create a metric from a twoform expression, using the basis of oneforms from the coordinate system
a, b = symbols('a b')  # some constants to use in the metric
dx, dy = cs.base_oneforms()
form = a ** 2 * tpow(dx, 2) + b ** 2 * tpow(dy, 2)
g1 = metric.Metric(twoform=form)  # Note: don't have to specify coords since implied by basis of one-forms

# Notice that the Metric class will represent itself as a twoform
g1

# Now let's create the same metric from a matrix
# First let's create a Matrix
matrix = Array([[a ** 2, 0], [0, b ** 2]])
matrix

# Creating a Metric from a matrix also requires you to specify the coordinate system (so the axes can be labeled)
g2 = metric.Metric(matrix=matrix, coord_system=cs)

# Note that the Metric class automatically computes the two-form and uses it for representation
g2

# Metrics can be inverted, and produce other metrics
g3 = g2.inverse
g3

Curvature

# Now let's compute curvature terms
from sympy import Function
from pystein import curvature

# Let's create a metric with some curvature..
x, y = cs.base_symbols()  # grab the coordinate parameters
F = Function('F')(x, y)  # Define an arbitrary function that depends on x and y
g4 = metric.Metric(twoform=F ** 2 * tpow(dx, 2) + b ** 2 * tpow(dy, 2))

curvature.ricci_tensor_component(0, 0, g4).doit()

Matter

# Let's compute the matter stress energy tensor of a perfect fluid in 1D
from pystein import matter

# Need to quickly redefine the coordinates to have a temporal coordinate
t, x, y = symbols('t x y')
M = Manifold('M', dim=3)
P = Patch('origin', M)
cs = coords.CoordSystem('OneDim', P, [t, x, y])

dt, dx, dy = cs.base_oneforms()
Q = Function('Q')(t, y)  # Define an arbitrary function that depends on x and y
S = Function('S')(t, x)  # Define an arbitrary function that depends on x and y
g5 = metric.Metric(twoform=- Q ** 2 * tpow(dt, 2) + b ** 2 * tpow(dx, 2) + S ** 2 * tpow(dy, 2), components=(Q, S, b))
g5

# Now use the matter module to create the stress energy tensor for perfect fluid
T = matter.perfect_fluid(g5)
T

utilities.clean_expr(curvature.einstein_tensor_component(0, 0, g5))

# Note that in the limit Q -> 1
g5_lim = g5.subs({Q: 1})
T_lim = matter.perfect_fluid(g5_lim)
T_lim

utilities.clean_expr(curvature.einstein_tensor_component(0, 0, g5_lim))

Gravity

# One can also directly compute the Einstein Equations
from pystein import gravity

utilities.clean_expr(gravity.einstein_equation(0, 0, g5, T))

# Similarly in the limit:
utilities.clean_expr(gravity.einstein_equation(0, 0, g5_lim, T_lim))

Full Example: FLRW Cosmology

# Load the predefined FLRW metric
flrw = metric.flrw(cartesian=True)
flrw

T = matter.perfect_fluid(flrw)
efe_00 = utilities.clean_expr(gravity.einstein_equation(0, 0, flrw, T).doit())
efe_00

# Simplify derivative notation:
metric.simplify_deriv_notation(efe_00, flrw)

# Can also use "dots"
metric.simplify_deriv_notation(efe_00, flrw, use_dots=True)

Numeric Tools

The pystein package contains some limited numerical utilities, including:

• ability to numerically integrate the geodesic equations geodesic.numerical_geodesic
• convenience functions to compute multiple geodesics from a variety of initial conditions (2D)

These utilities are compatible with the symbolic tools thanks to sympy.lambdify, which is used to convert symbolic equations into numeric equations.

*Note that the numeric tools in pystein are still in beta.

Example Geodesic Usage

Construct a metric from a twoform

M = Manifold('M', dim=2)
P = Patch('origin', M)

rho, phi, a = sympy.symbols('rho phi a', nonnegative=True)
cs = coords.CoordSystem('schw', P, [rho, phi])
drho, dphi = cs.base_oneforms()
ds2 = a ** 2 * ((1 / (1 - rho ** 2)) * tpow(drho, 2) + rho ** 2 * tpow(dphi, 2))
g = metric.Metric(twoform=ds2)
g

Compute the symbolic geodesic equations

full_simplify(geodesic.geodesic_equation(0, sympy.symbols('lambda'), g))

Numerically integrate the geodesic equations

init = (numpy.sin(numpy.pi / 4), 0.0, numpy.cos(numpy.pi / 4), numpy.pi / 4)
lambdas = numpy.arange(0, 2.1, 0.001)
df = geodesic.numerical_geodesic(g, init, lambdas)
df.head()