In flat Minkowski spacetime the interval is ds² = -c²dt² + dx² + dy² + dz². In a curved spacetime, which of the following correctly describes how this changes?
AThe signs of the temporal and spatial terms swap, so ds² = c²dt² - dx² - dy² - dz²
BThe constant coefficients (-c², +1, +1, +1) are replaced by functions g_μν(x) that vary from point to point and may include off-diagonal cross terms
CAdditional spatial dimensions are added, extending the interval to five or more terms
DThe speed of light c is replaced by a position-dependent function c(x) while the metric remains diagonal
The metric tensor g_μν generalizes the Minkowski metric by replacing constant coefficients with position-dependent functions and allowing off-diagonal terms (cross terms like g_{tr} dr dt). This is what encodes curvature. The signature (-,+,+,+) is preserved, the dimensionality remains four, and c remains a universal constant — its apparent variation in some coordinate systems (like Schwarzschild coordinates) is a coordinate artifact, not a physical change.
Question 2 True / False
The metric tensor in general relativity plays the same role as the gravitational potential in Newtonian gravity.
TTrue
FFalse
Answer: True
In the Newtonian limit, the g_{00} component of the metric reduces to approximately -(1 + 2Φ/c²), where Φ is the Newtonian gravitational potential. More broadly, the metric tensor encodes all gravitational information: it determines how clocks tick, how rulers measure, how objects fall, and how light propagates. Just as Φ is the single function that specifies Newtonian gravity, g_μν (with its ten independent components in four dimensions) specifies the full gravitational field in GR.
Question 3 Short Answer
Why does the metric tensor have ten independent components in four-dimensional spacetime rather than sixteen?
Think about your answer, then reveal below.
Model answer: The metric tensor is symmetric: g_μν = g_νμ. In four dimensions, a general 4×4 matrix has 16 components, but symmetry means g_μν = g_νμ for all μ,ν, reducing the independent components to 4×5/2 = 10. This symmetry follows from the definition of the metric as the inner product on the tangent space, which is inherently symmetric: ds² = g_μν dx^μ dx^ν = g_νμ dx^ν dx^μ.
The ten independent components of g_μν are the dynamical degrees of freedom of the gravitational field, though not all ten are physical — four can be removed by coordinate (gauge) freedom, and four more are constrained by the Bianchi identities, leaving two true propagating degrees of freedom (the two polarizations of gravitational waves).
Question 4 Short Answer
Explain the physical distinction between a timelike interval (ds² < 0), a spacelike interval (ds² > 0), and a null interval (ds² = 0) in the (-,+,+,+) signature convention.
Think about your answer, then reveal below.
Model answer: A timelike interval (ds² < 0) connects two events that can be visited by a massive particle traveling slower than light; the square root of -ds²/c² gives the proper time elapsed on a clock following that path. A spacelike interval (ds² > 0) separates events that cannot be causally connected — no signal can travel between them. A null interval (ds² = 0) is the path followed by light (or any massless particle) — the proper time along a null path is zero. The metric's signature ensures that at every event in spacetime, the light cone separating these three regions is well-defined.
The causal classification of intervals is one of the metric's most fundamental roles. It defines the light cone structure, which in turn determines causality: causes must precede effects along timelike or null paths. Curvature changes the shapes of light cones from point to point but never changes the local causal structure.