GraviPy - tutorial¶

Coordinates and MetricTensor¶

To start working with the gravipy package you must load the package and to initialize a pretty-printing mode in IPython environment

The next step is to choose coordinates and to define a metric tensor of a particular space. Let's take, for example, the Schwarzschild metric - vacuum solution to the Einstein's field equations which describes the gravitational field of a spherical mass distribution.

Each component of any tensor object, can be computed by calling the appropriate instance of the GeneralTensor subclass with indices as arguments. The covariant indices take positive integer values (1, 2, ..., dim). The contravariant indices take negative values (-dim, ..., -2, -1).

Matrix representation of a tensor can be obtained in the following way

Predefined Tensor Classes¶

The GraviPy package contains a number of the Tensor subclasses that can be used to calculate a tensor components. The Tensor subclasses available in the current version of GraviPy package are

The Christoffel symbols¶

The first one is the Christoffel class that represents Christoffel symbols of the first and second kind. (Note that the Christoffel symbols are not tensors) Components of the Christoffel objects are computed from the below formula

\[ \Gamma_{\rho \mu \nu} = g_{\rho \sigma}\Gamma^{\sigma}_{\ \mu \nu} = \frac{1}{2}(g_{\rho \mu, \nu} + g_{\rho \nu, \mu} - g_{\mu \nu, \rho})\]

Let's create an instance of the Christoffel class for the Schwarzschild metric g and compute some components of the object

Each component of the Tensor object is computed only once due to memoization procedure implemented in the Tensor class. Computed value of a tensor component is stored in components dictionary (attribute of a Tensor instance) and returned by the next call to the instance.

The above dictionary consists of two elements because the symmetry of the Christoffel symbols is implemented in the Christoffel class.

If necessary, you can clear the components dictionary

The Matrix representation of the Christoffel symbols is the following

You can get help on any of classes mentioned before by running the command

Try also "Christoffel?" and "Christoffel??"

The Ricci tensor¶

\[ R_{\mu \nu} = \frac{\partial \Gamma^{\sigma}_{\ \mu \nu}}{\partial x^{\sigma}} - \frac{\partial \Gamma^{\sigma}_{\ \mu \sigma}}{\partial x^{\nu}} + \Gamma^{\sigma}_{\ \mu \nu}\Gamma^{\rho}_{\ \sigma \rho} - \Gamma^{\rho}_{\ \mu \sigma}\Gamma^{\sigma}_{\ \nu \rho} \]

Contraction of the Ricci tensor \(R = R_{\mu}^{\ \mu} = g^{\mu \nu}R_{\mu \nu}\)

The Riemann tensor¶

\[ R_{\mu \nu \rho \sigma} = \frac{\partial \Gamma_{\mu \nu \sigma}}{\partial x^{\rho}} - \frac{\partial \Gamma_{\mu \nu \rho}}{\partial x^{\sigma}} + \Gamma^{\alpha}_{\ \nu \sigma}\Gamma_{\mu \rho \alpha} - \Gamma^{\alpha}_{\ \nu \rho}\Gamma_{\mu \sigma \alpha} - \frac{\partial g_{\mu \alpha}}{\partial x^{\rho}}\Gamma^{\alpha}_{\ \nu \sigma} + \frac{\partial g_{\mu \alpha}}{\partial x^{\sigma}}\Gamma^{\alpha}_{\ \nu \rho} \]

Some nonzero components of the Riemann tensor are

You can also display the matrix representation of the tensor

Contraction of the Riemann tensor \(R_{\mu \nu} = R^{\rho}_{\ \mu \rho \nu}\)

The Einstein tensor¶

\[ G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R \]

Geodesics¶

\[ w_{\mu} = \frac{Du_{\mu}}{d\tau} = \frac{d^2x_{\mu}}{d\tau^2} - \frac{1}{2}g_{\rho \sigma, \mu} \frac{dx^{\rho}}{d\tau}\frac{dx^{\sigma}}{d\tau} \]

Please note that instantiation of a Geodesic class for the metric \(g\) automatically turns on a Parametrization mode for the metric \(g\).
Then all coordinates are functions of a world line parameter \(\tau\)

Parametrization mode can be deactivated by typing

Derivatives¶

Partial derivative¶

All instances of a GeneralTensor subclasses inherits partialD method which works exactly the same way as SymPy diff method.

The only difference is that computed value of partialD is saved in "partial_derivative_components" dictionary an then returned by the next call to the partialD method.

Covariant derivative¶

Covariant derivative components of the tensor T can be computed by the covariantD method from the formula

\[ \nabla_{\sigma} T_{\mu}^{\ \nu} = T_{\mu \ ;\sigma}^{\ \nu} = \frac{\partial T_{\mu}^{\ \nu}}{\partial x^{\sigma}} - \Gamma^{\rho}_{\ \mu \sigma}T_{\rho}^{\ \nu} + \Gamma^{\nu}_{\ \rho \sigma}T_{\mu}^{\ \rho}\]

Let's compute some covariant derivatives of a scalar field C

All covariantD components of every Tensor object are also memoized

Proof that the covariant derivative of the metric tensor \(g\) is zero

Bianchi identity in the Schwarzschild spacetime

\[ R_{\mu \nu \sigma \rho ;\gamma} + R_{\mu \nu \gamma \sigma ;\rho} + R_{\mu \nu \rho \gamma ;\sigma} = 0\]

User-defined tensors¶

To define a new scalar/vector/tensor field in some space you should extend the Tensor class or create an instance of the Tensor class.

Tensor class instantiation¶

Let's create a third-rank tensor field living in the Schwarzshild spacetime as an instance of the Tensor class

Until you define (override) the _compute_covariant_component method of the S object, all of \(4^3\) components are arbitrary functions of coordinates

Let's assume that tensor S is the commutator of the covariant derivatives of some arbitrary vector field V and create a new _compute_covariant_component method for the object S

One can check that the well known formula is correct

\[ V_{\mu ;\nu \rho} - V_{\mu ;\rho \nu} = R^{\sigma}_{\ \mu \nu \rho}V_{\sigma} \]

Another way of tensor creation is to make an instance of the Tensor class with components option. Tensor components stored in Matrix object are writen to the components dictionary of the instance by this method.

Tensor class extension¶

As an example of the Tensor class extension you can get the source code of any of the predefined Tensor subclasses