Metric And Geometry

Question 2

What is the metric tensor?

Answer 2

A (0,2) symmetric tensor g_{μν} that defines distances and angles in spacetime.

Question 3

What is the line element in curved spacetime?

Answer 3

ds² = g_{μν} dx^μ dx^ν.

Question 4

What is meant by a ‘signature’ of a metric?

Answer 4

The signs of the metric’s eigenvalues; usually (-,+,+,+) in GR.

Question 5

What is the physical meaning of the metric?

Answer 5

It determines spacetime intervals, causal structure, and curvature.

Question 6

How does the metric raise or lower indices?

Answer 6

Use g_{μν} or its inverse g^{μν} to move indices up or down.

Question 7

What is the inverse metric?

Answer 7

g^{μν}, satisfying g^{μλ}g_{λν} = δ^μ_ν.

Question 8

Is the metric always symmetric?

Answer 8

Yes, g_{μν} = g_{νμ} by definition.

Question 9

What is a coordinate basis?

Answer 9

A basis formed by ∂/∂x^μ and dx^μ at each point in spacetime.

Question 10

How is proper time calculated?

Answer 10

dτ² = -ds² = -g_{μν} dx^μ dx^ν for timelike paths.

Question 11

What does it mean for two vectors to be orthogonal?

Answer 11

Their inner product using the metric is zero: g_{μν} A^μ B^ν = 0.

Question 12

What is a null vector?

Answer 12

A vector for which g_{μν}V^μV^ν = 0.

Question 13

What is the determinant of the metric used for?

Answer 13

In volume integrals and defining the Levi-Civita tensor.

Question 14

What is the proper length in spacetime?

Answer 14

Spatial length between two events at equal time, ds² > 0.

Question 15

What defines spacetime curvature?

Answer 15

How the metric varies from point to point.

Question 16

What is meant by a ‘flat’ metric?

Answer 16

A metric whose curvature tensors vanish; e.g., Minkowski space.

Question 17

How is a change of coordinates reflected in the metric?

Answer 17

Metric components transform with the coordinate basis.

Question 18

What is the Minkowski metric?

Answer 18

η_{μν} = diag(-1, 1, 1, 1), the flat spacetime metric.

Question 19

What is the metric of spherical coordinates in flat space?

Answer 19

ds² = -dt² + dr² + r²dθ² + r²sin²θ dφ².

Question 20

What is a coordinate transformation?

Answer 20

A mapping x^μ → x’^μ that changes the metric components.

Question 21

What is meant by curvature being intrinsic?

Answer 21

It can be detected by measurements within the space, without embedding.

Question 22

What is a geodesic in terms of the metric?

Answer 22

A path that extremizes proper time or length.

Question 23

What is a metric-compatible connection?

Answer 23

A connection ∇ such that ∇λ g{μν} = 0.

Question 24

What is a manifold?

Answer 24

A topological space that is locally like ℝ⁴ and supports coordinate charts.

Question 25

What is a coordinate chart?

Answer 25

A smooth assignment of coordinates to a neighborhood on a manifold.

Question 26

What is a tangent space?

Answer 26

The space of all tangent vectors at a point in the manifold.

Question 27

What is a metric space?

Answer 27

A space equipped with a metric that defines distances.

Question 28

What is the covariant form of the metric?

Answer 28

g_{μν}, with lower indices, used in ds² = g_{μν} dx^μ dx^ν.

Question 29

What is the contravariant form of the metric?

Answer 29

g^{μν}, used for raising indices and inner products.

Question 30

What is the length of a spacelike curve?

Answer 30

ℓ = ∫√(g_{μν} dx^μ dx^ν) for spacelike paths.

Question 31

What is the arc length in GR?

Answer 31

The proper distance or time measured along a worldline.

Question 32

What are light cones?

Answer 32

Surfaces defined by ds² = 0; determine causal structure.

Question 33

What is a conformally flat metric?

Answer 33

One that can be written as ds² = Ω²(x) η_{μν} dx^μ dx^ν.

Question 34

What is the volume element in GR?

Answer 34

√|g| d⁴x where g is the determinant of the metric.

Question 35

What is metric compatibility?

Answer 35

A condition ∇_λ g_{μν} = 0 that preserves inner products under parallel transport.

Question 36

What is parallel transport?

Answer 36

Moving a vector along a curve while preserving its direction and length.

Question 37

What happens if g_{μν} has off-diagonal terms?

Answer 37

Coordinates are not orthogonal; can lead to cross terms in ds².

Question 38

How is curvature encoded in the metric?

Answer 38

Through second derivatives of the metric and the Riemann tensor.

Question 39

Can you always find coordinates that make g_{μν} = η_{μν}?

Answer 39

Yes, locally at a point using Riemann normal coordinates.

Question 40

What is the significance of a diagonal metric?

Answer 40

Simplifies calculations; indicates orthogonal coordinates.