For the polar metric g=diag(1,r²) in coordinates (r,θ), use Γ^r_{θθ}=−(1/2)∂_r g_{θθ}. What is this symbol at r=5?

• 5, with the derivative sign reversed
• −10, without the factor 1/2
• −5

How can you compare vectors at two different points using a connection?

• Compare their coordinate components without a path
• Transport one along a chosen path
• Treat their coordinate labels as the transport rule

With an affine parameter, a geodesic parallel-transports its own tangent.

Circle one:   True   False

Use dimensionless radius r and affine parameter λ, with θ measured in radians. Dots mean λ derivatives. For r¨−r(θ˙)²=0, r=2 and θ˙=3, what is r¨?

• 6
• 18
• −18

In the flat polar metric g=diag(1,r²), r=4, V^θ=2 and ∂_θ V^r=0 at a point. What is ∇_θ V^r?

• −8
• 0, from the component derivative alone
• 8, from changing the correction's sign

What does the affine geodesic equation set to zero?

• Every coordinate velocity along the entire path
• The metric components along the entire path
• Coordinate acceleration plus its connection term

In a flat plane, a path has fixed polar radius r>0 and constant θ˙≠0. Is it a geodesic?

• Yes, because the radius remains fixed throughout
• No; −r(θ˙)² is nonzero
• Yes, because polar coordinate lines are all geodesics

An ideal freely falling capsule has nonzero coordinate acceleration. What can its onboard accelerometer read?

• Zero proper acceleration
• The coordinate acceleration in every coordinate system
• A force caused solely by changing coordinate labels