Curvature: Riemann, Ricci, and Einstein tensors

Question 2

What does curvature in GR describe?

Answer 2

How spacetime is deformed by the presence of mass-energy.

Question 3

What is the Riemann curvature tensor?

Answer 3

R^ρ_{σμν} measures how much vectors are rotated by parallel transport around loops.

Question 4

How is the Riemann tensor defined?

Answer 4

R^ρ_{σμν} = ∂μ Γ^ρ{νσ} - ∂ν Γ^ρ{μσ} + Γ^ρ_{μλ}Γ^λ_{νσ} - Γ^ρ_{νλ}Γ^λ_{μσ}.

Question 5

How many indices does the Riemann tensor have?

Answer 5

Four: R^ρ_{σμν}.

Question 6

Is the Riemann tensor a tensor?

Answer 6

Yes, it transforms properly under coordinate transformations.

Question 7

What is the Ricci tensor?

Answer 7

A contraction of the Riemann tensor: R_{μν} = R^λ_{μλν}.

Question 8

What is the Ricci scalar?

Answer 8

R = g^{μν}R_{μν} — a full contraction of the Ricci tensor.

Question 9

What is the Einstein tensor?

Answer 9

G_{μν} = R_{μν} - 1/2 g_{μν}R.

Question 10

Why is the Einstein tensor important?

Answer 10

It appears on the left-hand side of the Einstein field equations.

Question 11

What are the symmetries of the Riemann tensor?

Answer 11

Antisymmetric in last two indices; R_{ρσμν} = -R_{ρσνμ} and others.

Question 12

How many independent components does Riemann have in 4D?

Answer 12

20 independent components.

Question 13

How many components does the Ricci tensor have in 4D?

Answer 13

10 independent components (symmetric 4×4 matrix).

Question 14

What is the Bianchi identity?

Answer 14

∇λ R^ρ{σμν} + ∇μ R^ρ{σνλ} + ∇ν R^ρ{σλμ} = 0.

Question 15

What is the contracted Bianchi identity?

Answer 15

∇^μ G_{μν} = 0.

Question 16

Why is ∇^μ G_{μν} = 0 important?

Answer 16

Ensures energy-momentum conservation: ∇^μ T_{μν} = 0.

Question 17

What does R_{μν} = 0 signify?

Answer 17

A vacuum solution to Einstein’s field equations.

Question 18

What does G_{μν} = 0 imply?

Answer 18

Spacetime is empty (no matter-energy).

Question 19

What is the curvature tensor for flat spacetime?

Answer 19

All components are zero.

Question 20

What does a non-zero Riemann tensor indicate?

Answer 20

Intrinsic curvature of spacetime.

Question 21

What kind of curvature does the Riemann tensor measure?

Answer 21

Sectional curvature — curvature in a specific 2D plane.

Question 22

What does the Ricci tensor describe?

Answer 22

Volume distortion — convergence/divergence of geodesics.

Question 23

What is the geometric meaning of the Ricci scalar?

Answer 23

Total curvature averaged over all directions.

Question 24

What is the geometric meaning of Einstein tensor?

Answer 24

Encodes curvature effects due to matter and energy.

Question 25

How is the Riemann tensor related to tidal forces?

Answer 25

It describes how free-fall trajectories deviate.

Question 26

Is the Ricci tensor symmetric?

Answer 26

Yes, R_{μν} = R_{νμ}.

Question 27

What does a constant Riemann tensor indicate?

Answer 27

Maximally symmetric spacetime (e.g., de Sitter, Minkowski).

Question 28

What is meant by sectional curvature?

Answer 28

The curvature in a 2D surface spanned by two vectors.

Question 29

What is the Weyl tensor?

Answer 29

The traceless part of the Riemann tensor — measures tidal distortions in vacuum.

Question 30

Why is curvature tensor antisymmetric in μν?

Answer 30

Because parallel transporting around μ-ν loop leads to antisymmetric behavior.

Question 31

Does Riemann fully determine curvature?

Answer 31

Yes, it contains all local curvature information.

Question 32

Can the Ricci scalar be negative?

Answer 32

Yes — its sign depends on the spacetime geometry.

Question 33

What determines the motion of test particles?

Answer 33

Geodesic deviation equation, which uses the Riemann tensor.

Question 34

What is curvature singularity?

Answer 34

A point where the curvature scalars (e.g., R) diverge.

Question 35

What is the Kretschmann scalar?

Answer 35

K = R_{μνρσ} R^{μνρσ}, used to detect singularities.

Question 36

Is the Einstein tensor divergence-free?

Answer 36

Yes, ∇^μ G_{μν} = 0.

Question 37

What does G_{μν} ∝ T_{μν} mean?

Answer 37

Spacetime curvature is sourced by energy and momentum.

Question 38

Why is curvature nonlocal?

Answer 38

Because it's determined by how vectors change under parallel transport around finite loops.

Question 39

What is the Petrov classification?

Answer 39

A classification of Weyl tensor algebraic types — not covered in this course.

Question 40

What is the role of curvature in GR?

Answer 40

It replaces the gravitational force with geometry.

Question 41

What happens when Riemann = 0 but Γ ≠ 0?

Answer 41

The spacetime is flat but described in curvilinear coordinates.