Tensors and the Metric on a Curved Manifold · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Writing physics that survives relabelling

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

  1. What does the metric hold?

    • Only the coordinate names
    • The geometry
    • Nothing but the origin

  2. What are coordinates?

    • Labels, not physics
    • Physical forces themselves
    • Fixed universal truths

  3. Objects are distinguished by how their components transform.

    Circle one:   True   False

  4. How do you find a distance in non-Cartesian coordinates?

    • Read it off the metric
    • Use the flat formula regardless
    • Guess from the labels

  5. How do you move between the two index positions of one object?

    • Rename the coordinates
    • Delete one index
    • Raise and lower indices

  6. Two descriptions differ only by relabelling. What do you conclude?

    • One of them is wrong
    • Same physics
    • Labels always change physics

  7. A classmate calls any list of numbers a tensor. What is missing?

    • The tensor transform law
    • Lists can never hold numbers
    • Tensors forbid all numbers

  8. An equation looks broken after a coordinate change. What do you check first?

    • Whether the ink is dark enough
    • Whether coordinates feel nice
    • How each object transformed
LightMySky · lightmysky.comW1-mt_9r9SG3NcZR-s1

Answer key

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

Writing physics that survives relabelling W1-mt_9r9SG3NcZR-s1

  1. The geometry · The metric is the geometry's container.
  2. Labels, not physics · Coordinates label; they never are the physics.
  3. True · The transform law is the sorting test.
  4. Read it off the metric · The metric fits the coordinates you have.
  5. Raise and lower indices · The metric converts the two forms.
  6. Same physics · Relabelling changes nothing real.
  7. The tensor transform law · The transform law, not the list, makes the tensor.
  8. How each object transformed · Blame the labels before the physics.
Worksheet · LightMySky