Riemann Tensor Symmetries and Bianchi Identities

Hello! Let's continue our exploration of the mathematical structure of curved spacetime.

In our last lesson, we introduced the Riemann curvature tensor, $R^{ρ}_{σ μν}$ , as the fundamental object that quantifies curvature. We defined it through the commutator of covariant derivatives and saw that it's constructed from the Christoffel symbols and their derivatives. However, in four dimensions, this tensor has $4^{4} = 256$ components, a daunting number to work with.

Today, we will tame this beast. Our goal is to derive the algebraic symmetries of the Riemann tensor and the Bianchi identities. These identities are not just mathematical conveniences; they reveal the deep internal structure of curvature and are absolutely essential for formulating the laws of gravity. They drastically reduce the number of independent components from 256 to a much more manageable 20 in 4D.

1. The Fully Covariant Riemann Tensor

The symmetries of the Riemann tensor are most apparent when all its indices are in the covariant (lower) position. We obtain this (0,4) tensor by lowering the first index of the standard (1,3) Riemann tensor using the metric:

R_{ρ σ μν} = g_{ρ λ} R^{λ}_{σ μν}

This fully covariant form will be our primary object of study for its symmetries.

2. The Algebraic Symmetries

The fully covariant Riemann tensor, assuming the metric-compatible and torsion-free Levi-Civita connection, possesses three fundamental algebraic symmetries and a cyclic identity.

Antisymmetry in the first two indices: $R_{ρ σ μν} = - R_{σ ρ μν}$
Antisymmetry in the last two indices: $R_{ρ σ μν} = - R_{ρ σ ν μ}$
Pair interchange symmetry: $R_{ρ σ μν} = R_{μν ρ σ}$
First Bianchi Identity (Cyclic Identity): The sum over cyclic permutations of the last three indices vanishes: $R_{ρ σ μν} + R_{ρ μν σ} + R_{ρ ν σ μ} = 0$

Let's explore the derivation of these properties. As you appreciate deep mathematical structure, we will see how these symmetries are not arbitrary but are direct consequences of the properties of our connection (torsion-free, metric-compatible) and the Jacobi identity.

The following video provides an excellent, step-by-step derivation of these symmetries from the coordinate-free definitions. It connects each symmetry to a specific underlying assumption about our geometry, which will resonate with your desire for rigorous foundations.

Tensor Calculus 23: Riemann Curvature Tensor Components and Symmetries

The video 'Riemann Curvature Tensor Components and Symmetries' by eigenchris walks through the proofs for the main algebraic symmetries. It's a clear, methodical presentation that builds from the ground up.

Watch from 06:53 to 16:48. Pay close attention to how each symmetry is derived: Antisymmetry in last two indices (06:53): This follows directly from the definition of the Riemann tensor via the commutator. First Bianchi Identity (08:14): Notice how this relies on the connection being torsion-free. Antisymmetry in first two indices (10:12): This derivation cleverly uses metric compatibility. Pair interchange symmetry (14:18): This final symmetry is shown to be a consequence of the previous three.

A Deeper, Formal Proof

For a more abstract and powerful derivation, which I believe you will appreciate, we can work entirely with vector fields $X, Y, Z, W$ . The document below provides proofs that are the epitome of mathematical elegance in differential geometry.

Bianchi identities

These lecture notes on the Bianchi identities from Nagoya University offer a highly formal and rigorous treatment. They prove the symmetries using the abstract definitions of curvature and the Jacobi identity, completely independent of any coordinate system.

I recommend you study Lemma 3 and Lemma 4 (pages 3-5). Lemma 3 proves the first Bianchi identity in the form R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0, leveraging the torsion-free condition and the Jacobi identity for Lie brackets. Lemma 4 then defines the fully covariant tensor R(X,Y,Z,W) = g(R(X,Y)Z, W) and meticulously proves the three algebraic symmetries we listed above.

This formal approach beautifully demonstrates how the properties of the Riemann tensor are inextricably linked to the foundational axioms of Riemannian geometry.

The Physicist's Shortcut: Riemann Normal Coordinates

There is another powerful technique for deriving these symmetries, which relies on a clever choice of coordinates. At any point $p$ , we can always find a Riemann Normal Coordinate (RNC) system where $g_{μν} (p) = η_{μν}$ and, crucially, $Γ_{μν}^{λ} (p) = 0$ . In this local inertial frame, the formula for the Riemann tensor simplifies enormously to:

R_{ρ σ μν} (p) = \frac{1}{2} (\partial_{σ} \partial_{μ} g_{ρ ν} + \partial_{ρ} \partial_{ν} g_{σ μ} - \partial_{σ} \partial_{ν} g_{ρ μ} - \partial_{ρ} \partial_{μ} g_{σ ν})

From this simplified expression, all four algebraic properties can be verified by simple inspection. Since they are tensor identities, if they hold in one coordinate system (at point $p$ ), they must hold in all coordinate systems everywhere. This method is detailed in Section 3.4 of the Sean Carroll notes (LINK) you have from the previous lesson.

Test your understanding!

Using the algebraic symmetries, prove that for the Levi-Civita connection, the only non-zero contraction of the Riemann tensor $R^{ρ}_{σ μν}$ is the Ricci tensor, $R_{σ ν} = R^{μ}_{σ μν}$ . Specifically, show that contracting on the first and last indices yields the negative of the Ricci tensor, i.e., $R^{μ}_{σ ν μ} = - R_{σ ν}$ .

Show answer

We want to show $R^{μ}_{σ ν μ} = - R^{μ}_{σ μν}$ .
Let's start with the component form $R^{μ}_{σ ν μ}$ . We can't directly apply the symmetries until the indices are all covariant.

Raise the first index and lower the third: $R_{λσ ν μ} = g_{λ α} R^{α}_{σ ν μ}$ .
Now contract $λ$ with $μ$ : $R^{μ}_{σ ν μ} = g^{μ λ} R_{λσ ν μ}$ .

Now let's use the symmetries on $R_{λσ ν μ}$ :

Use pair-interchange symmetry: $R_{λσ ν μ} = R_{ν μ λσ}$ .
Use antisymmetry of the first pair: $R_{ν μ λσ} = - R_{μν λσ}$ .

Substituting back:

R^{μ}_{σ ν μ} = g^{μ λ} R_{λσ ν μ} = g^{μ λ} (- R_{μν λσ}) = - g^{μ λ} R_{μν λσ}

Now, $g^{μ λ} R_{μν λσ}$ is the expression for raising the first index of $R_{μν λσ}$ and contracting it with the first index. Wait, let's be more careful.
Let's just use the index positions:

R^{μ}_{σ ν μ} = g^{μα} R_{α σ ν μ}

Apply symmetries to the term inside:

$R_{α σ ν μ} = - R_{α σ μν}$ (antisymmetry in last pair)
$- R_{α σ μν} = - R_{μν α σ}$ (pair interchange)
$- R_{μν α σ} = R_{ν μα σ}$ (antisymmetry in first pair)

Let's try a more direct route:

Start with $R_{λσ ν μ}$ .
Antisymmetry in last pair: $R_{λσ ν μ} = - R_{λσ μν}$ .
First Bianchi Identity on $R_{λσ μν}$ : $R_{λσ μν} + R_{λ μν σ} + R_{λ ν σ μ} = 0$ .
Rearranging gives: $- R_{λσ μν} = R_{λ μν σ} + R_{λ ν σ μ}$ .
So, $R_{λσ ν μ} = R_{λ μν σ} + R_{λ ν σ μ}$ .

Now contract with $g^{λ ν}$ (this is a different contraction, but let's see). Let's stick to the original contraction.
We want to relate $R^{μ}_{σ ν μ}$ to $R^{μ}_{σ μν}$ .
Consider $R_{σ ν} = R^{μ}_{σ μν}$ .
And $R^{μ}_{σ ν μ}$ .
Let's use the first Bianchi identity in its other form: $R^{ρ}_{[σ μν]} = 0$ . For a (1,3) tensor, this is $R^{ρ}_{σ μν} + R^{ρ}_{μν σ} + R^{ρ}_{ν σ μ} = 0$ .
Let's contract this with $δ_{ρ}^{σ}$ .
$R^{σ}_{σ μν} + R^{σ}_{μν σ} + R^{σ}_{ν σ μ} = 0$
The first term is $R_{μν}$ , the Ricci tensor.
The second term is $R^{σ}_{μν σ}$ .
The third term is $R^{σ}_{ν σ μ} = R_{ν μ}$ . Since the Ricci tensor is symmetric ( $R_{ν μ} = R_{μν}$ , which is itself a consequence of the Riemann symmetries), this is also $R_{μν}$ .

So we have: $R_{μν} + R^{σ}_{μν σ} + R_{μν} = 0$ .
This gives $R^{σ}_{μν σ} = - 2 R_{μν}$ . This wasn't what was asked, but is another useful identity.

Let's retry the original proof.
We want to show $R^{μ}_{σ ν μ} = - R_{σ ν}$ .
Start with $R^{μ}_{σ ν μ}$ .
From the definition, $R^{ρ}_{σ μν}$ is antisymmetric in $μ, ν$ . So $R^{μ}_{σ ν μ} = - R^{μ}_{σ μν}$ .
This is the simplest way. Antisymmetry in the last two lower indices of $R_{ρ σ μν}$ translates to antisymmetry in the last two indices of $R^{ρ}_{σ μν}$ .
Therefore, $R^{μ}_{σ ν μ} = - R^{μ}_{σ μν} = - R_{σ ν}$ . The proof is that simple, relying only on the antisymmetry in the last two indices.

3. Consequence: The Number of Independent Components

These symmetries impose powerful constraints. By systematically applying them, we can determine the true number of degrees of freedom in the Riemann tensor.

The two antisymmetry properties mean we are dealing with pairs of indices.
The pair-swapping symmetry means we have a symmetric matrix of these pairs.
The first Bianchi identity adds one more constraint for every set of four distinct indices.

The final number of independent components in $n$ dimensions is:

N = \frac{n ^{2} ( n ^{2} - 1 )}{12}

For our 4-dimensional spacetime:

N = \frac{4 ^{2} ( 4 ^{2} - 1 )}{12} = \frac{16 ( 15 )}{12} = \frac{4 \cdot 15}{3} = 20

So, the seemingly 256 components are reduced to just 20 independent functions. This is a profound simplification that makes computations in General Relativity feasible.

4. The Differential Bianchi Identity

Beyond the algebraic symmetries that constrain the tensor at a single point, there is a crucial differential identity that relates the curvature at neighboring points. This is often called the Second Bianchi Identity.

It states that the cyclic sum of the covariant derivative of the Riemann tensor is zero:

\nabla_{λ} R^{ρ}_{σ μν} + \nabla_{μ} R^{ρ}_{σ ν λ} + \nabla_{ν} R^{ρ}_{σ λ μ} = 0

This can be written more compactly using antisymmetrization notation as $\nabla_{[λ} R^{ρ}_{σ μν]} = 0$ .

Derivation of the Differential Bianchi Identity — A depiction of the rigorous derivation of the second Bianchi identity on a board. This identity is a cornerstone for the derivation of Einstein's Field Equations.

Just as with the algebraic symmetries, there are multiple ways to prove this.

Symmetry Properties of the Riemann Curvature Tensor - corrected

Let's return to Robert Davie's channel. He provides a concise derivation of the second Bianchi identity using the Riemann Normal Coordinate shortcut.

Watch from 10:24 to 12:44. The logic is the same as before: the identity is derived in the simplified RNC frame where \Gamma=0, and because it's a tensor equation, it must hold true in all frames.

And for the formal, coordinate-free proof that you value:

Bianchi identities

The Nagoya University notes provide another beautiful proof for the second Bianchi identity, again starting from first principles and the Jacobi identity.

Study the proof of Proposition 5 on pages 6-7. It's a masterful demonstration of formal manipulation, starting from the fact that the torsion is zero and repeatedly applying the definitions of curvature and the Leibniz rule for covariant derivatives.

This differential identity is arguably the single most important identity in General Relativity. Its twice-contracted form is the mathematical key to guaranteeing the conservation of energy and momentum, which will lead us directly to the Einstein Field Equations.

Conclusion

This lesson was a deep dive into the mathematical heart of the Riemann tensor. We have uncovered its fundamental structure.

Key Takeaways:

The fully covariant Riemann tensor $R_{ρ σ μν}$ has a rich symmetric structure: it is antisymmetric in its first two and last two indices, symmetric under the interchange of these pairs, and obeys a cyclic identity (the first Bianchi identity).
These algebraic symmetries reduce the number of independent components from 256 to 20 in four dimensions.
The Riemann tensor also satisfies a crucial differential identity, the Second Bianchi Identity: $\nabla_{[λ} R^{ρ}_{σ μν]} = 0$ . This identity relates the curvature in a region of spacetime.

We now have the tools not only to define curvature but also to understand its intricate internal rules. The next step is to see how this complex tensor can be "decomposed" into simpler, more physically intuitive pieces.

In our next lesson, we will perform the Ricci decomposition of the Riemann tensor into the Ricci tensor, the Ricci scalar, and the Weyl tensor. This will allow us to separate the parts of curvature that govern volume changes (related to matter) from the part that governs shape distortion (related to tidal forces and gravitational waves). The twice-contracted Bianchi identity will be our starting point for this crucial next step.

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