Independent Components of a 4D Riemann Tensor et al.

- 6 mins read

Introduction

I was at the tutorial session for the General Relativity course. We were discussing 2-sphere manifold, calculating the Riemann tensor, Ricci tensor, Ricci scalar and Einstein Tensor. The Instructor then asked us to compute the independent components of the Riemann Tensor in 4-Dimensions as a fun painful exercise to get a hang of it. And here we are!

Riemann Tensor in 4-Dimensions

We have some (anti)symmetries we can exploit to reduce the number of independent components of the said tensor. Let me list them below:

note

$\forall \ i,j,k,l = \{1,2,3,4\}$, the following holds

  1. $R_{ijkl} = - R_{jikl} \rightarrow R_{[ij]kl}$

  2. $R_{ijkl} = - R_{ijlk} \rightarrow R_{ij[kl]}$

  3. $R_{ijkl} = R_{klij} \rightarrow R_{([ij][kl])}$

  4. Bianchi Identity: $R_{ijkl} + R_{iklj} + R_{iljk} = 0$

Let us use just the first two constraints and see how many variables we can reduce. I was lazy to count it by hand, so here is a Python code that will do just that for us.

import itertools

def generate_riemann_components():
    # The Riemann tensor has 4 indices, each ranging from 0 to 3 (for 4 dimensions)
    indices = range(4)

    # We need to consider the antisymmetry of the first two indices and the last two indices
    # So we generate combinations for the first two indices and the last two indices
    first_two = list(itertools.combinations(indices, 2))
    last_two = list(itertools.combinations(indices, 2))

    # Generate all possible combinations of the first two and last two indices
    components = []
    for (i, j) in first_two:
        for (k, l) in last_two:
            # The Riemann tensor is antisymmetric in the first two indices and the last two indices
            # So we only consider the case where i < j and k < l
            components.append(((i, j), (k, l)))

    return components

# Generate and print the components
riemann_components = generate_riemann_components()
for component in riemann_components:
    print(f"R_{component[0][0]}{component[0][1]}{component[1][0]}{component[1][1]}")

Running this script provides us with the following list:

note

$R_{0101}$, $R_{0102}$, $R_{0103}$, $R_{0112}$, $R_{0113}$, $R_{0123}$, $R_{0201}$, $R_{0202}$, $R_{0203}$

$R_{0212}$, $R_{0213}$, $R_{0223}$, $R_{0301}$, $R_{0302}$, $R_{0303}$, $R_{0312}$, $R_{0313}$, $R_{0323}$

$R_{1201}$, $R_{1202}$, $R_{1203}$, $R_{1212}$, $R_{1213}$, $R_{1223}$, $R_{1301}$, $R_{1302}$, $R_{1303}$

$R_{1312}$, $R_{1313}$, $R_{1323}$, $R_{2301}$, $R_{2302}$, $R_{2303}$, $R_{2312}$, $R_{2313}$, $R_{2323}$

Initially, we had 256 components for the Riemann Tensor, now using the first two anti-symmetries, we have reduced it to just 36 components. Let us use the third symmetry to reduce the number of components further. As before, I wrote another Python code to achieve exactly the same:

import itertools

def generate_riemann_components():
    # The Riemann tensor has 4 indices, each ranging from 0 to 3 (for 4 dimensions)
    indices = range(4)

    # We need to consider the antisymmetry of the first two indices and the last two indices
    # So we generate combinations for the first two indices and the last two indices
    first_two = list(itertools.combinations(indices, 2))
    last_two = list(itertools.combinations(indices, 2))

    # Generate all possible combinations of the first two and last two indices
    components = set()
    for (i, j) in first_two:
        for (k, l) in last_two:
            # The Riemann tensor is antisymmetric in the first two indices and the last two indices
            # So we only consider the case where i < j and k < l
            component = ((i, j), (k, l))

            # Check if swapping the first two and last two indices gives a different component
            swapped_component = ((k, l), (i, j))

            # Only add the component if its swapped version is not already in the set
            if swapped_component not in components:
                components.add(component)

    return components

# Generate and print the components
riemann_components = generate_riemann_components()
for component in riemann_components:
    print(f"R_{component[0][0]}{component[0][1]}{component[1][0]}{component[1][1]}")

Running the above script, we have the following:

note

$R_{1213}$, $R_{0312}$, $R_{2323}$, $R_{0323}$, $R_{0213}$, $R_{0113}$, $R_{0202}$

$R_{1323}$, $R_{0102}$, $R_{0203}$, $R_{0103}$, $R_{0313}$, $R_{1212}$, $R_{1223}$

$R_{0303}$, $R_{1313}$, $R_{0212}$, $R_{0223}$, $R_{0112}$, $R_{0123}$, $R_{0101}$

We have started with 36, and reduced it to 21 components. Now, using the Bianchi Identity, we can write only one equation:

$$R_{0123} +R_{0102​}+R_{0103​} = 0$$

We rewrite the above as follows:

$$R_{0123}​=−R_{0102}​−R_{0103​}$$

Turning the term $R_{0123}$ to depend on the terms $R_{0102}​$ and $R_{0103​}$. This leaves us with 20 independent components for Riemann tensor in 4 dimensions. They are:

note

$R_{1213}$, $R_{0312}$, $R_{2323}$, $R_{0323}$, $R_{0213}$, $R_{0113}$, $R_{0202}$, $R_{1323}$, $R_{0102}$, $R_{0203}$

$R_{0103}$, $R_{0313}$, $R_{1212}$, $R_{1223}$, $R_{0303}$, $R_{1313}$, $R_{0212}$, $R_{0223}$, $R_{0112}$, $R_{0101}$

Ricci Tensor

Most often, we use Ricci tensor rather than the Riemann tensor. We define this as:

$$R_{ij} = R_{ ilj}^l$$

This means the Ricci tensor is symmetric $R_{(ij)}$, leaving us with 10 independent components, from the 16 total components it has in 4 dimensions. Those 10 components are, and the related 20 components of the Riemann tensor to the Ricci tensor are:

note

Independent Ricci Tensor components:

$R_{00}, R_{11}, R_{22}, R_{33}, R_{01}, R_{02}, R_{03}, R_{12}, R_{13}, R_{23}$

Computing them from the Riemann Tensor components from the above follows as:

  1. $R_{00} = R_{0101} + R_{0202} + R_{0303}$
  2. $R_{11} = R_{0112} + R_{1212} + R_{1313}$
  3. $R_{22} = R_{0212} + R_{1212} + R_{2323}$
  4. $R_{33} = R_{0313} + R_{1313} + R_{2323}$
  5. $R_{01} = R_{0102} + R_{0113}$
  6. $R_{02} = R_{0112} + R_{0223}$
  7. $R_{03} = R_{0113} + R_{0213}$
  8. $R_{12} = R_{0123} + R_{1213}$
  9. $R_{13} = R_{0123} + R_{1223}$
  10. $R_{23} = R_{0213} + R_{1223}$

What’s Next?

Exploring the Riemann Tensor in 4-Dimensions has been a fascinating journey, blending the elegance of mathematical theory with the practicality of computational tools. Starting from 256 components and reducing them to just 20 independent ones using symmetries and identities showcases the power of mathematical reasoning and computational verification.

The Riemann Tensor is not just a theoretical construct; it has profound implications in understanding the fabric of spacetime(for a lack of a better phrase, you will know what I mean in an upcoming article) and the dynamics of gravitational fields. As we delve deeper into the intricacies of general relativity, we uncover the underlying principles that govern our universe.

In general relativity, the Ricci tensor, with its 10 independent components, succinctly captures the essence of spacetime curvature, directly influencing the gravitational dynamics dictated by matter and energy distributions. These components are pivotal in formulating the Einstein tensor, which, despite its intricate derivation, also boasts 10 degrees of freedom. This alignment underscores the Einstein tensor’s role in the Einstein field equations, effectively linking the geometric properties of spacetime to the physical presence of mass and energy, thereby shaping our understanding of the universe’s fundamental structure.

After three paragraphs of flowery language, I conclude this article with the computations as a reference to look back, and no more. We are better off using a package like xAct for Mathematica to do these calculations.