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Physics · lesson 4 of 4

Curved spacetime and the scope of general relativity

General relativity relates spacetime geometry to stress-energy. The metric determines spacetime intervals and causal structure.

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Learn the ideas

Prerequisites: Special relativity, tensors and differential geometry as prerequisites.

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.

Worked examples

Example 1

What geometric object determines spacetime intervals?

The metric contracts coordinate displacements to determine the interval.

The metric

Example 2

In the ideal test-particle approximation, free fall follows what?

Geodesics generalize inertial motion in curved spacetime.

A spacetime geodesic

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 What geometric object determines spacetime intervals?

Show answer and explanation

The metric

The metric contracts coordinate displacements to determine the interval.

2 In the ideal test-particle approximation, free fall follows what?

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A spacetime geodesic

Geodesics generalize inertial motion in curved spacetime.

3 Why does a coordinate singularity not automatically imply divergent physical curvature?

Sample answer and marking guidance

Coordinates can fail while invariant curvature quantities remain finite. One must inspect invariant or operational quantities rather than a single coordinate component.

Credit chart dependence and invariants.

4 Which effect cannot generally be removed throughout a finite region by a single freely falling frame?

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Tidal effects associated with curvature

Curvature describes relative gravitational effects across finite separation.

5 Explain why locally inertial does not mean spacetime is globally flat.

Sample answer and marking guidance

A local frame can remove certain connection effects at a point, while nonzero curvature prevents doing so throughout an extended region.

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.

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6

ds²=4×9=36, so ds=6.

7 What assumption permits neglecting a test particle's effect on the background geometry?

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Its self-gravitational influence is negligible

The test-body approximation treats the background as fixed.

8 Name two prerequisites for a quantitative study of Einstein field equations.

Sample answer and marking guidance

Tensor calculus and differential geometry, together with special relativity and differential equations, are appropriate prerequisites.

Credit two substantive mathematical or physical prerequisites; this lesson itself does not replace them.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.