# Curved spacetime and the scope of general relativity > General relativity relates spacetime geometry to stress-energy. The metric determines spacetime intervals and causal structure. Course: [Advanced physics methods and spacetime](https://studybrix.com/en/courses/advanced-physics-methods/index.md) Level: advanced Prerequisites: Special relativity, tensors and differential geometry as prerequisites. ## Notes ### Geometry General relativity relates spacetime geometry to stress-energy. The metric determines spacetime intervals and causal structure. ### Free fall A freely falling ideal test body follows a geodesic when non-gravitational forces and self-force effects are neglected. ### Local frames In a sufficiently small freely falling frame, gravitational effects can be locally removed to first order; tidal curvature effects generally remain across finite regions. ### Scope This is a conceptual bridge, not a tensor-calculus course. Coordinate effects must be distinguished from invariant curvature and physical observables. ## Practice questions ### 1. What geometric object determines spacetime intervals? A. The metric B. A colour map C. Only temperature D. Only a coordinate label Answer: The metric Explanation: The metric contracts coordinate displacements to determine the interval. ### 2. In the ideal test-particle approximation, free fall follows what? A. A path independent of the metric B. A spacetime geodesic C. A Euclidean straight line in every chart D. Any path with constant coordinates Answer: A spacetime geodesic Explanation: Geodesics generalize inertial motion in curved spacetime. ### 3. Why does a coordinate singularity not automatically imply divergent physical curvature? Sample answer / marking guidance: Coordinates can fail while invariant curvature quantities remain finite. One must inspect invariant or operational quantities rather than a single coordinate component. Explanation: Credit chart dependence and invariants. ### 4. Which effect cannot generally be removed throughout a finite region by a single freely falling frame? A. The choice of time units B. All coordinate names C. A constant chart offset D. Tidal effects associated with curvature Answer: Tidal effects associated with curvature Explanation: Curvature describes relative gravitational effects across finite separation. ### 5. Explain why locally inertial does not mean spacetime is globally flat. Sample answer / marking guidance: A local frame can remove certain connection effects at a point, while nonzero curvature prevents doing so throughout an extended region. Explanation: Credit pointwise versus extended geometry. ### 6. In a model metric ds²=4dt²−dx² with other coordinates fixed, a path has dx=0 and dt=3. Find positive ds in model units. Answer: 6 Explanation: ds²=4×9=36, so ds=6. ### 7. What assumption permits neglecting a test particle's effect on the background geometry? A. Its mass must be negative B. It has no trajectory C. Its speed is zero in every frame D. Its self-gravitational influence is negligible Answer: Its self-gravitational influence is negligible Explanation: The test-body approximation treats the background as fixed. ### 8. Name two prerequisites for a quantitative study of Einstein field equations. Sample answer / marking guidance: Tensor calculus and differential geometry, together with special relativity and differential equations, are appropriate prerequisites. Explanation: Credit two substantive mathematical or physical prerequisites; this lesson itself does not replace them. ## Further reading - https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/ - https://openstax.org/details/books/university-physics-volume-3 Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.