Metric Tensor: Definition and Index Manipulation

1. The Metric Tensor: An Inner Product on Tangent Space

The metric tensor provides a way to compute an inner product (a generalization of the dot product) between any two vectors in the same tangent space $T_{p}M$. Since this structure is defined smoothly across the entire manifold, it gives us a way to measure lengths and angles everywhere.

Definition: The Metric Tensor

A metric tensor $g$ on a manifold $M$ is a smooth tensor field of type (0,2) that is:

1. Symmetric: For any two vector fields $X,Y$, $g(X,Y)=g(Y,X)$.
2. Non-degenerate: If $g_p(X_p,Y_p)=0$ for all $Y_p\in T_{p}M$ at a point $p$, then it must be that $X_p=0$.

The non-degeneracy condition is crucial; it ensures that the metric is a faithful "measuring device" and that no non-zero vector has zero "length" with respect to every other vector (with an important exception for null vectors in a Lorentzian metric, which we'll see shortly).

The following resource gives a crisp, formal definition of the metric and its properties.

In a coordinate basis $\{{\partial }_{\mu }\}$ for $T_{p}M$ and $\{d{x}^{\nu }\}$ for $T_p^{\ast }M$, the metric tensor can be written as:
$g=g_{\mu \nu }d{x}^{\mu }\otimes d{x}^{\nu }$
where the components $g_{\mu \nu }$ are given by evaluating the metric on the basis vectors:
$g_{\mu \nu }=g({\partial }_{\mu },{\partial }_{\nu })$
The symmetry of $g$ implies that the matrix of components is symmetric: $g_{\mu \nu }=g_{\nu \mu }$.

The most fundamental physical interpretation of the metric is its role in defining the infinitesimal distance or line element, $ds$. For an infinitesimal displacement vector $dX=d{x}^{\mu }{\partial }_{\mu }$, the squared length is:
$d{s}^2=g(dX,dX)=g(d{x}^{\mu }{\partial }_{\mu },d{x}^{\nu }{\partial }_{\nu })=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$
This equation is the heart of metric geometry. It tells us how to convert abstract coordinate displacements $d{x}^{\mu }$ into a physical, invariant length.

2. Metric Signature and Spacetime Structure

Because the metric components $g_{\mu \nu }(p)$ form a real, symmetric matrix at each point $p$, it can be diagonalized. The non-degeneracy condition ensures that none of the eigenvalues are zero. The signature of the metric is the count of positive and negative eigenvalues. This signature is constant across a connected manifold.

The signature defines the fundamental character of the geometry:

• Riemannian Metric: The signature is $(+,+,\dots ,+)$. All eigenvalues are positive. This metric is positive-definite: $g(X,X)>0$ for any non-zero vector $X$. This is the geometry of ordinary curved spaces.
• Lorentzian Metric: The signature has one negative sign and the rest positive. For the 4-dimensional spacetime of General Relativity, we will use the convention $(-,+,+,+)$. This is the geometry of spacetime.

The Lorentzian signature is what distinguishes spacetime from space. It partitions the tangent space at every point into three distinct types of vectors based on their "norm," $g(X,X)$:

• Timelike vectors: $g(X,X)<0$. These represent the worldlines of massive particles, which travel slower than light. The set of all timelike vectors forms the interior of the light cone.
• Null (or Lightlike) vectors: $g(X,X)=0$. These represent the paths of massless particles, like photons, traveling at the speed of light. They form the boundary of the light cone. Note that for a null vector $X$, $X\ne 0$, but its norm is zero. This is possible because the metric is not positive-definite.
• Spacelike vectors: $g(X,X)>0$. These represent spatial separation. Events separated by a spacelike interval cannot be causally connected. They lie outside the light cone.

The following reading elaborates on this crucial concept, formally distinguishing Riemannian and Lorentzian manifolds.

3. Raising and Lowering Indices: The Musical Isomorphisms

So far, we have viewed the tangent space $T_{p}M$ (home of vectors) and the cotangent space $T_p^{\ast }M$ (home of covectors) as separate entities. The metric tensor provides a canonical, basis-independent isomorphism between them. This allows us to treat a vector as a covector and vice-versa. This mechanism is called raising and lowering indices.

Why is this important? It gives us a way to uniquely pair vectors and covectors. In physics, this is essential for contracting tensors to form scalars, which are the basis of any invariant physical law (e.g., Lagrangians).

The following video provides an excellent motivation and step-by-step explanation of this process.

Let's formalize the operations described in the video.

Lowering Indices (The "Flat" Map)

Given a vector $X\in T_{p}M$, we can define a unique corresponding covector, denoted $X_{♭}\in T_p^{\ast }M$. Its action on an arbitrary vector $Y\in T_{p}M$ is defined by the metric:
$X_{♭}(Y):=g(X,Y)$
This defines a linear map $♭:T_{p}M\to T_p^{\ast }M$. In components, if $X=X^{\mu }{\partial }_{\mu }$, the components of the covector $X_{♭}=X_{\nu }d{x}^{\nu }$ are found by:
$X_{\nu }=g_{\nu \mu }{X}^{\mu }$
(Note the summation over $\mu$). The metric tensor components $g_{\nu \mu }$ have "lowered" the contravariant index $\mu$ to a covariant index $\nu$.

Raising Indices (The "Sharp" Map)

To perform the inverse operation, we need the inverse metric tensor. Its components, denoted $g^{\mu \nu }$, are defined by the matrix inverse of $g_{\mu \nu }$:
$g^{\mu \rho }{g}_{\rho \nu }={\delta }_{\nu }^{\mu }$
where ${\delta }_{\nu }^{\mu }$ is the Kronecker delta. The inverse metric is a type (2,0) tensor, $g^{-1}=g^{\mu \nu }{\partial }_{\mu }\otimes {\partial }_{\nu }$.

Given a covector $\omega \in T_p^{\ast }M$, we can define its corresponding vector, denoted ${\omega }^{♯}\in T_{p}M$, via the inverse map $♯:T_p^{\ast }M\to T_{p}M$. In components, if $\omega ={\omega }_{\nu }d{x}^{\nu }$, the components of the vector ${\omega }^{♯}={\omega }^{\mu }{\partial }_{\mu }$ are found by:
${\omega }^{\mu }=g^{\mu \nu }{\omega }_{\nu }$
The inverse metric components $g^{\mu \nu }$ have "raised" the covariant index $\nu$ to a contravariant index $\mu$.

These $♭$ and $♯$ maps are sometimes called musical isomorphisms, a term you will appreciate given your interest in the formal structures. The notation comes from music, where a flat symbol ($♭$) lowers a note's pitch and a sharp symbol ($♯$) raises it.

Conclusion

This lesson introduced the central object of metric geometry. We have moved from an abstract manifold to a semi-Riemannian manifold—a space where we can measure distances, angles, and define causality.

Key Takeaways:

• The metric tensor $g$ is a symmetric, non-degenerate (0,2) tensor field that functions as a generalized inner product on each tangent space.
• The line element $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ relates coordinate intervals to physical, invariant distances.
• The signature of the metric determines the type of geometry. For General Relativity, the Lorentzian signature $(-,+,+,+)$ is key, partitioning vectors into timelike, null, and spacelike categories.
• The metric and its inverse provide a canonical way to convert vectors to covectors and back again. This is implemented via lowering indices with $g_{\mu \nu }$ and raising indices with $g^{\mu \nu }$.

We now have a static geometric structure. But physics is about dynamics and change. We need tools to differentiate tensor fields. In the next lesson, we will introduce our first notion of a derivative on a manifold: the Lie derivative. It describes how a tensor field changes as it is dragged along the flow of a vector field. This concept is independent of the metric and provides a different perspective on change compared to the metric-compatible derivative we will encounter later.