How to Prove the Antisymmetry of Riemann Tensor Lower Indices

Understanding the Riemann Tensor's Antisymmetry Property

What We're Solving:

We need to prove that the Riemann tensor satisfies R^ρ_[σμν] = 0, where the square brackets indicate complete antisymmetrization over the three lower indices σ, μ, and ν. This is a fundamental symmetry property that the Riemann tensor must satisfy.

The Approach:

This proof relies on understanding the Bianchi identities and the existing symmetry properties of the Riemann tensor. We'll use the fact that the Riemann tensor already has certain known symmetries, and show that when we try to antisymmetrize over three indices, everything cancels out to zero. Think of it like trying to balance three weights that have specific relationships - they end up perfectly canceling each other!

Step-by-Step Solution:

Step 1: Recall the known symmetries of the Riemann tensor The Riemann tensor has these established properties:

• R^ρ_σμν = -R^ρ_σνμ (antisymmetric in the last two indices)
• R_ρσμν = -R_σρμν (antisymmetric in the first two indices when all lowered)
• The first Bianchi identity: R^ρ_σμν + R^ρ_μνσ + R^ρ_νσμ = 0

Step 2: Write out what antisymmetrization means R^ρ_[σμν] = (1/6)[R^ρ_σμν + R^ρ_μνσ + R^ρ_νσμ - R^ρ_σνμ - R^ρ_μσν - R^ρ_νμσ]

This represents all possible permutations of the three indices, with even permutations getting a + sign and odd permutations getting a - sign.

Step 3: Apply the first Bianchi identity From the first Bianchi identity, we know: R^ρ_σμν + R^ρ_μνσ + R^ρ_νσμ = 0

This immediately tells us that the first three terms in our antisymmetrization sum to zero!

Step 4: Use the antisymmetry in the last two indices Now let's look at the remaining terms. Using R^ρ_σμν = -R^ρ_σνμ:

• R^ρ_σνμ = -R^ρ_σμν
• R^ρ_μσν = -R^ρ_μνσ
• R^ρ_νμσ = -R^ρ_νσμ

Step 5: Substitute and simplify Our antisymmetrization becomes: R^ρ_[σμν] = (1/6)[(R^ρ_σμν + R^ρ_μνσ + R^ρ_νσμ) - (-R^ρ_σμν - R^ρ_μνσ - R^ρ_νσμ)] = (1/6)[0 - (-0)] = (1/6)[0 + 0] = 0

The Answer:

R^ρ_[σμν] = 0

The antisymmetric part of the Riemann tensor over any three indices is identically zero. This happens because the Bianchi identity creates a perfect balance between the terms, and the existing antisymmetry properties ensure that all terms cancel out completely.

Memory Tip:

Think of this as the "three's company rule" - the Riemann tensor can handle being antisymmetric over two indices just fine, but when you try to make it antisymmetric over three indices, the Bianchi identity steps in and says "not possible!" The tensor's existing symmetries create a perfect cancellation that always results in zero.

This property is actually quite profound - it reflects deep geometric constraints on how curvature can behave in spacetime. Pretty amazing how the mathematics of curved space naturally builds in these consistency conditions!