Tensors and Metric

Introduces tensors, metric, indices, and contractions in spacetime geometry.

Conceptual context

In general relativity, coordinates are labels rather than physical rulers. Two observers may assign different numbers to the same events and still agree on their physical separation. The metric provides that agreement. Its components, including $g₀₀$, $g₀₁$, and $g₁₁$, convert the increments $dx⁰$ and $dx¹$ into the scalar $ds²$. The essential distinction is between coordinate-dependent components and geometric quantities that remain invariant.

🟢 Essential level

A tensor is a geometric entity whose components change in a coordinated way when the reference system changes. The metric is a special tensor because it acts as a local measuring rule. The quantities $dx⁰$ and $dx¹$ only describe changes in coordinate labels; by themselves, they are not physical distances. The metric combines them and produces $ds²$, which has coordinate-independent geometric meaning.

The components $g₀₀$ and $g₁₁$ weight the contributions of each direction. The off-diagonal component $g₀₁$ mixes both directions, showing that coordinate axes need not be decoupled. Raising or lowering indices also depends on the metric: $v₀$ and $v₁$ can combine to produce $v⁰$. Index position therefore expresses a geometric operation, not a visual decoration.

A tensor therefore preserves one geometric object while allowing many coordinate descriptions.

The tensor calculus that made this language possible was developed in 1900 by Gregorio Ricci-Curbastro and his former student Tullio Levi-Civita, who published the "absolute differential calculus" in a paper that went nearly unnoticed for fifteen years, until Einstein adopted it as the exact mathematical framework he needed to express general relativity.

🔵 Formal level

Basic formulas

The local interval in two coordinates is obtained from:

Metric interval

This relation combines $g₀₀$, $g₀₁$, and $g₁₁$ with $dx⁰$ and $dx¹$. For a symmetric metric, the cross contribution appears twice and is represented by a factor of two. The result $ds²$ has dimensions of length squared when coordinates are measured in length and metric components are dimensionless. If $dx⁰$ is fixed, the dependence on $dx¹$ is quadratic; concavity depends on $g₁₁$, while the linear term depends on $g₀₁$.

Real student questions

Does the metric always represent curvature? No. A flat spacetime may have variable metric components in non-Cartesian coordinates.

Why can $ds²$ be negative? Because relativistic geometry is not Euclidean. The sign depends on the signature and classifies causal separation.

Does raising an index change the physical vector? It changes the representation, not the underlying geometric entity. $v⁰$ and $v₀$ belong to dual descriptions related by the metric.

Does zero $Tr(T)$ imply a zero tensor? No. $T⁰₀$ and $T¹₁$ may cancel.

Does a cross term represent a force? Not necessarily. $g₀₁$ represents coordinate mixing, not a force.

Final synthesis

The metric converts coordinate separations into physical geometry. Components depend on the basis, while $ds²$ and $Tr(T)$ expose invariant structure. Understanding indices, contractions, and local validity turns formulas and graphs into meaningful physics.