General Relativity Tensor Operations

1. Define Metric and Basis

The function to define metric and basis is essential in setting up the framework for tensor calculations in General Relativity.

    g_{a}_{b} := [[g00, ....], [g10, ....], ... , [..., gNN]], [var1, ...., varN]
            

This syntax defines the metric components and the basis coordinates for the tensor operations.

Writing Tensor Expressions

Once the metric and basis are defined, tensor expressions can be written using this structure:

    G_{mu}_{nu}
            

This outputs the entered metric components:

    [[g00, ....], [g10, ....], ... , [..., gNN]]
            

2. Get and Assign Specific Components from Tensors

This functionality allows for retrieving or assigning specific components of a tensor.

    Input: G_{mu:1}_{nu:1}
    Output: The (1,1) component of the metric components
            

It works similarly for any tensor and its individual components:

    R_{nu:1}_{b:2}_{mu}_{d}
    R_{nu:1}_{b}_{mu}_{d}
    R_{nu:1}_{b:2}_{mu:1}_{d:2}
            

To assign a specific component to a variable:

    Comp = G_{mu:1}_{nu:1}
            

3. Tensor Expressions Functionality

Tensor expressions can be performed using the defined metric and basis:

    G^{mu}^{nu}*R_{nu}_{b}_{c}_{d}
    R^{a}_{b}_{a}_{d}
    G^{mu}^{nu}*R_{nu}_{b}_{mu}_{d}
            

This functionality is key for solving complex equations in the field of General Relativity.

Advanced Tensor Operations

This section delves into more advanced aspects of tensor operations, demonstrating the capabilities of the calculator in handling complex tensor equations, specifically relating to Riemann tensors.

Functionality Overview

The advanced tensor operation functionality involves several key steps:

1. Identification of variables and tensor types for user-defined tensors.
2. Computation of the right-hand side (RHS) of the tensor equation.
3. Verification that the indices match, focusing on the covariance of symbols.
4. Rearrangement of the computed tensor to match the user-defined index structure on the left-hand side (LHS) of the equation.
5. Storage of the computed tensor in cache, with components and indices as defined by the user.

Example: Riemann Tensor Computation

An example of such an operation is the computation of Riemann tensor components:

        Coordinates := [t, r, theta, phi]
        g_{mu}_{nu} := [[-(1 - (2 * G * M) / (r)), 0, 0, 0],[0, 1 / (1 - (2 * G * M) / (r)), 0, 0],[0, 0, r**2, 0],[0, 0, 0, r**2 * sin(theta) ** 2]]
        T^{a}_{m}_{b}_{n} = d_{b}*C^{a}_{n}_{m} + C^{a}_{b}_{l}*C^{l}_{n}_{m} - d_{n}*C^{a}_{b}_{m} - C^{a}_{n}_{l}*C^{l}_{b}_{m}
        T^{a}_{m}_{b}_{n}
            

In this example, the Riemann tensor components are computed and assigned to a tensor T, following the specific user-defined structure for indices.