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Physics · Advanced university

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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Prerequisites

Special relativity, tensors and differential geometry as prerequisites.

Study 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.

Worked examples

What geometric object determines spacetime intervals?

The metric contracts coordinate displacements to determine the interval.

The metric

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

Geodesics generalize inertial motion in curved spacetime.

A spacetime geodesic

Check your understanding

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

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

    1. The choice of time units
    2. All coordinate names
    3. A constant chart offset
    4. Tidal effects associated with curvature

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

  4. 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.

Answers and marking guidance

Answers and marking notes
  1. 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.

  2. Tidal effects associated with curvature

    Curvature describes relative gravitational effects across finite separation.

  3. 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.

  4. 6

    ds²=4×9=36, so ds=6.

Further reading and review references

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