General relativity
Connect spacetime curvature to stress-energy, observables, and the limits of proposed metrics.
Derive the structure of Einstein's equation, solve and interpret standard spacetimes, calculate weak-field and orbital tests, and use energy conditions to audit exotic geometries without confusing solutions with engineering demonstrations.
Before you begin
- • Tensor calculus and differential geometry
- • Lagrangian mechanics
- • Special relativity
By the end, you can
- • Interpret Einstein's equation component by component.
- • Calculate observables in Schwarzschild and weak-field spacetimes.
- • Use geodesic deviation and curvature to describe tidal gravity.
- • Evaluate stress-energy and energy-condition requirements of proposed metrics.
Interactive model
Explore before calculating
Live laboratory
Stress-energy source explorer
Change a local diagonal source and audit common energy conditions. The matrix is one local matter model; solving Einstein's equation still requires a spacetime metric, derivatives, boundaries, and conservation.
Radial NEC ρ+pr: 4.0 · pass
Tangential NEC ρ+pt: 4.0 · pass
Weak condition: pass
Dominant condition: pass
Strong condition: pass
Passing or failing an energy condition classifies this source model; it does not establish that a desired geometry exists, is stable, or can be engineered. Quantum violations also remain constrained by magnitude, duration, distribution, and backreaction.
Level 4 · Advanced undergraduate teaching kit
Record the investigation. Teach the reasoning.
A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.
Learner record
Stress-energy source and energy-condition ledger
What source components and invariant tests are required by a proposed spacetime geometry?
Download learner recordInstructor guide
Teach for evidence, not button pushing
Learners translate geometry into a frame-declared source ledger and keep energy-condition diagnostics separate from engineering feasibility.
Download instructor guideLesson 1 of 3
Einstein's equation and the Newtonian limit
Why does the field equation equate a divergence-free curvature tensor with stress-energy?
The Einstein tensor combines Ricci curvature so its covariant divergence vanishes identically. Stress-energy conservation therefore fits the geometric side structurally.
The coupling 8πG/c⁴ is fixed by recovering Poisson gravity in the weak, slow, static limit. A cosmological constant is a separate geometric term with observational consequences.
Worked example
Estimate the weak-field time dilation for potential Φ with |Φ|≪c².
- 1. Use g₀₀≈−(1+2Φ/c²).
- 2. For a stationary clock, dτ≈√(−g₀₀)dt.
- 3. Expand the square root to first order.
dτ≈(1+Φ/c²)dt; clocks deeper in a negative potential run more slowly.
Try it
Field-equation term ledger
Materials: Einstein equation and dimensional-analysis worksheet
- 1. Label geometric and matter terms.
- 2. Check units after restoring c.
- 3. Take the covariant divergence.
- 4. State which quantities are locally conserved.
Notice: The equation is a constrained tensor relation, not a slogan that matter simply bends a rubber sheet.
Check your understanding: Does ∇_μT^{μν}=0 mean ordinary coordinate components are constant?
Answer: No.
Covariant conservation includes connection terms and expresses local energy-momentum balance in curved spacetime.
Lesson 2 of 3
Standard solutions and precision tests
Which invariant predictions connect a metric solution to observations?
The Schwarzschild exterior predicts gravitational redshift, light deflection, perihelion advance, and a horizon structure. Coordinate singularities must be separated from curvature singularities.
Rotating sources require the Kerr family, which adds frame dragging and an ergoregion. Real tests infer parameters through light, clocks, or trajectories with instrumental and environmental models.
Worked example
Find the Schwarzschild radius of one solar mass to order of magnitude.
- 1. Use r_s=2GM/c².
- 2. Insert G≈6.67×10⁻¹¹, M≈2×10³⁰, c≈3×10⁸.
- 3. Evaluate numerator and denominator.
r_s≈3 km.
Try it
Observable-to-metric map
Materials: Table of redshift, lensing, orbit, and timing observations
- 1. Name the measured quantity.
- 2. Identify the metric component or geodesic effect involved.
- 3. List nuisance parameters.
- 4. State whether the test probes weak or strong gravity.
Notice: No experiment measures a metric component in isolation; it measures an invariant comparison modeled through a metric.
Check your understanding: Why is r=2GM/c² not a curvature singularity in Schwarzschild spacetime?
Answer: Curvature invariants remain finite there and regular coordinates cross it.
The divergence in Schwarzschild coordinates is a chart artifact.
Lesson 3 of 3
Energy conditions and proposed geometries
What does a desired geometry demand from its source, and which demand is physically unresolved?
Given a metric, Einstein's equation can be read backward to calculate its required stress-energy. Null, weak, dominant, and strong energy conditions summarize inequalities satisfied by many classical matter models but are not identical statements.
Quantum fields can violate some pointwise conditions in restricted settings. That does not automatically provide macroscopic, persistent, controllable negative energy or a complete propulsion mechanism.
Worked example
For a null vector k^μ, what quantity tests the null energy condition?
- 1. Construct T_μνk^μk^ν.
- 2. Evaluate it for all null directions.
- 3. Check whether it is nonnegative.
The NEC holds when T_μνk^μk^ν≥0 for every null k^μ.
Try it
Metric engineering audit
Materials: One proposed warp or wormhole metric
- 1. Compute or inspect the implied T_μν.
- 2. Map energy-condition violations.
- 3. Estimate scale and gradients.
- 4. List stability, creation, and backreaction questions.
Notice: A geometry can be exact mathematics while its source remains unphysical, unknown, or experimentally unsupported.
Check your understanding: Does a measured microscopic negative energy density prove a traversable wormhole can be built?
Answer: No.
Magnitude, duration, spatial distribution, controllability, stability, and backreaction remain separate requirements.
Formula-to-meaning deck
Read the equation in ordinary language.
G_μν+Λg_μν=(8πG/c⁴)T_μν
Spacetime curvature and the cosmological term are related to local stress-energy.
r_s=2GM/c²
A nonrotating mass sets a characteristic horizon radius.
D²ξ^μ/Dτ²=−R^μ_ναβu^νξ^αu^β
Geodesic deviation turns curvature into measurable relative acceleration.
Independent practice
Problem set
Work each problem before opening its hint and solution.
1. Compute r_s for Earth using M≈6×10²⁴ kg.
Reveal hint
Use 2GM/c².
Reveal solution
Approximately 9 mm.
2. In the weak field, estimate Δf/f for height h near Earth.
Reveal hint
Use ΔΦ≈gh and Δf/f≈ΔΦ/c².
Reveal solution
Δf/f≈gh/c².
3. If T_μνk^μk^ν<0 for one null direction, which condition fails?
Reveal hint
Match the defining contraction.
Reveal solution
The null energy condition.
Derivation studio
Build the result, line by line.
Keep the assumptions visible so the mathematics remains auditable.
Starting point
Gravitational redshift in a static metric
ds²=g₀₀c²dt² for stationary clocks
- 1. Set spatial displacements to zero.
- 2. Relate proper time to coordinate time.
- 3. Compare two clocks sharing the same coordinate-time interval.
- 4. Invert periods to obtain frequency ratio.
f₂/f₁=√[g₀₀(x₁)/g₀₀(x₂)] with a consistent sign convention
Clock comparison is invariant even though g₀₀ depends on the chosen static chart.
Starting point
Newtonian limit of the geodesic equation
ẍ^ρ+Γ^ρ_μνẋ^μẋ^ν=0
- 1. Assume slow motion so temporal velocity dominates.
- 2. Assume a weak static metric g₀₀≈−(1+2Φ/c²).
- 3. Compute Γ^i_00 to first order.
- 4. Convert proper-time derivatives to coordinate time.
d²x^i/dt²≈−∂^iΦ
Newtonian gravitational acceleration is the weak-field limit of geodesic motion.
Computational notebook
Turn the model into an experiment.
Relativistic orbit and photon laboratory
Which observables distinguish Newtonian, Schwarzschild, and parameter-perturbed trajectories?
Inputs
- • Central mass and initial conditions
- • Timelike or null trajectory
- • Metric parameters and integration tolerance
Algorithm
- 1. Construct conserved energy and angular momentum.
- 2. Integrate radial and angular motion.
- 3. Measure precession or deflection.
- 4. Run convergence and parameter-sensitivity checks.
Evidence to produce
- • Orbit or ray plots
- • Precession/deflection table
- • Numerical convergence and uncertainty report
Paper-reading studio
Interrogate the source, not its reputation.
Reconstruct the assumptions, reproduce one calculation, and stop at the boundary of the reported evidence.
Audit an exotic metric paper
Which claims follow from the metric, which require a source model, and which require experiment?
- 1. Separate the metric ansatz from its interpretation.
- 2. Reproduce the required stress-energy contraction.
- 3. Check asymptotics, horizons, and curvature invariants.
- 4. List unmodeled creation, stability, and control requirements.
Calculation to reproduce: Recompute one energy-condition diagnostic or characteristic energy scale from the paper's own parameters.
Evidence boundary: Solving Einstein's equation for a proposed geometry demonstrates mathematical consistency, not technological feasibility or experimental realization.
Continue into the evidence