phypy

A python library to automate certain calculations in physics

Features

Note that this is only a list of the available features and not a documentation. Some examples can be found below. For additional information visit the Wiki.

Note furthermore, that calculations can be done symbolically with the help of sympy.

General Relativity

The chosen sign convention is (+---). Given the covariant metric tensor the following objects can be calculated automatically.

• contravariant metric tensor
• determinant of the covariant metric tensor
• Christoffel symbols
• Ricci tensor
• Ricci scalar
• co- and contravariant partial derivative (partial_mu / partial^mu)
• Visualization of the polarization modes of the gravitational wave in first order of perturbation

Predefined objects:

• Minkowski metric
• Schwarzschild metric
• Friedmann-Robertson-Walker metric
• gravitational wave, infinitesimal spatial distance between neighbouring geodesics

Quantum Field Theory

If no metric is specified the Minkowski metric is assumed.

• Minimal coupled Klein-Gordon equation in the background of a given GR metric
• Wick contractions

Requirements

In order to make phypy work, one has to install some external packages.

• numpy for obvious reasons

python -m pip install numpy

• sympy for symbolic calculations

python -m pip install sympy

• matplotlib for plotting

python -m pip install matplotlib

Usage

Feel free to use any python file, e.g. calculations.py, and make sure that you have imported everything:

from imports import *

Example 1: Christoffel symbols of the Schwarzschild metric

metric = SchwarzschildMetric()
metric.christoffel_symbols(retC=False, simplify=True)
print(metric.c0)

By setting retC=False the function christoffel_symbols doent return the Christoffel symbols, but assign them to the class. By simplify=True the sympy.simplify method will be used before assigning the Christoffel symbols to the class. metric.c0 is the Christoffel symbol Gamma^0_{ij} in a matrix-format.

Example 2: Define your own metric

t, x, y, z = sy.symbols("t x y z", real=True)
matrix = np.array([
    [t, 0, 0, 0],
    [0, -x, 0, 0],
    [0, 0, -y, 0],
    [0, 0, 0, -z**2]
])
metric = Metric(matrix=matrix, t=t, x=x, y=y, z=z)
metric.ricci_scalar(retR=False, simplify=True)
print(metric.ricciscalar)

Example 3: Ricci scalar of FRW metric

metric = FRWMetric()
ricciscalar = metric.ricci_scalar(retR=True, simplify=True)
print(ricciscalar)

Example 4: Klein-Gordon equation in FRW metric

name = "phi"
phi = RealScalarField4D(name=name, metric=FRWMetric())
kg = phi.klein_gordon(retK=True, simplify=True, latex=False)
print(kg)

Example 5: Wick contractions

Find all Wick contractions of <0|T phi_1 phi_2 phi_3 phi_3|0>.

names = ["phi_1", "phi_2", "phi_3", "phi_3"]
fields = [RealScalarField4D(name=name) for name in names]
WickContraction(fields=fields)

The output looks like this:

<0|T[1, 2, 3, 3]|0> =

1 x [[1, 2], [3, 3]] +
2 x [[1, 3], [2, 3]]

This should be read as: <0|T phi_1 phi_2 phi_3 phi_3|0> = 1x<0|T phi_1 phi_2|0><0|T phi_3 phi_3|0> + 2x<0|T phi_1 phi_3> <0|T phi_2 phi_3|0>.