Level 4 · Advanced undergraduate teaching kit · Third- and fourth-year university
General relativity
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Stress-energy source and energy-condition ledger
What source components and invariant tests are required by a proposed spacetime geometry?
Setup
Use the stress-energy laboratory. Choose one metric profile, record the inferred density and principal stresses in the declared frame, then vary one geometric scale at a time.
Predict first
- 1. Predict how sharper geometry changes derivative-based source scales.
- 2. Predict the null-energy result when density plus one principal pressure is negative.
| Variable | Role | Unit |
|---|---|---|
| Metric amplitude and length scale | geometry inputs | declared model units |
| Energy density | inferred source | energy/volume |
| Principal pressures | inferred source | energy/volume |
| Null-energy contractions | invariant/frame-declared tests | energy/volume |
Observation columns
Analyze
- 1. Which source component dominates?
- 2. How does the required scale respond to halving the length scale?
- 3. Why must the observer frame or tetrad be declared?
- 4. Does an energy-condition violation demonstrate constructibility?
Conclusion frame
For geometry scale ___, the inferred source required density ___ and minimum null contraction ___ in frame ___; this establishes ___ but not ___.
Instructor guide · 60–80 minutes
Teach the investigation, not the interface
Learning target: Learners translate geometry into a frame-declared source ledger and keep energy-condition diagnostics separate from engineering feasibility.
Prepare
- • Review Gμν=8πGTμν/c⁴ component by component.
- • Define the orthonormal frame used by the model.
- • Prepare one dimensional scaling estimate.
Facilitation moves
- • Require source units and frame labels.
- • Separate local contractions from volume-integrated budgets.
- • Ask what matter model could realize the inferred tensor.
Accessibility and participation
- • Use a component ledger alongside tensor notation.
- • Read index contractions aloud in observer language.
- • Offer a scaling-first route before full equations.
Evidence of learning
- • A frame-declared source ledger
- • A valid NEC calculation
- • A geometry-versus-realizability distinction
Misconception checks
Writing a metric automatically supplies a physical source.
Einstein's equation infers a stress-energy tensor; realizability, stability, preparation, and backreaction remain separate problems.
Negative coordinate density is an invariant fact.
Physical density is measured by a specified observer contraction, not an unlabeled coordinate component.
Extension
Construct a candidate scalar-field source and test whether its stress-energy reproduces the required eigenvalue pattern.