Tensors and the metric on a curved manifold · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Writing physics that survives relabelling

Science · Astronomy & Astrophysics · ages 23-24
Name ______________________   Date ____________

  1. What does a Riemannian metric directly determine?

    • Only the names assigned to coordinates
    • Lengths and angles
    • Only a preferred origin for coordinates

  2. What changes when the same physical event is described in new coordinates?

    • Its coordinate labels
    • The physical event being described
    • The physical laws acting on the event

  3. A vector can have different components in two coordinate systems while remaining the same vector.

    Circle one:   True   False

  4. In the flat polar metric ds²=dr²+r²dθ², follow a circle of radius 8 through an angle of 1/4 radian. What is the arc length?

    • 2 length units
    • 32 length units
    • 1/32 length units

  5. For g=diag(4,9) and contravariant components V=(3,1), what are the lowered components?

    • V₁=3/4, V₂=1/9
    • V₁=3, V₂=1
    • V₁=12, V₂=9

  6. On a two-dimensional surface with r>0, the full metric is ds²=dr²+r²dθ². What does its r dependence prove?

    • The surface has positive intrinsic curvature
    • It does not by itself prove curvature
    • The surface must have negative intrinsic curvature

  7. A nonzero one-dimensional vector component is 1. Under x′=2x, a student keeps it equal to 1. What is missing?

    • The component must become 2
    • The metric must always remain numerically fixed
    • Every coordinate change leaves every component fixed

  8. With g=1 and V=3, squared length is 9. After x′=2x, V′=6. Which metric component preserves the squared length?

    • g′=1, keeping the original component
    • g′=4, squaring the coordinate scale
    • g′=1/4
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Writing physics that survives relabelling W1-mt_9r9SG3NcZR-s1

  1. Lengths and angles · The metric evaluates lengths and inner products, which also determine angles.
  2. Its coordinate labels · A coordinate change relabels the same event. It does not replace the event.
  3. True · The components transform with the coordinate system; the geometric vector stays fixed.
  4. 2 length units · At fixed radius, arc length is r|Δθ|=8×1/4=2.
  5. V₁=12, V₂=9 · Lowering uses the metric, not its inverse: 4×3=12 and 9×1=9.
  6. It does not by itself prove curvature · These are the polar components of a flat plane's metric.
  7. The component must become 2 · A contravariant vector component follows the Jacobian: V′=2V=2.
  8. g′=1/4 · The transformed metric is 1/4, so (1/4)×6²=9.
Worksheet · LightMySky