Physicists: Stop using the word “metric” to mean so many different things. A metric tensor is NOT the same object as a metric, it is NOT the same object as its matrix representation, and it is NOT the same object as its associated line element. You should not use those words interchangeably, they are not equivalent structures.
A metric tensor is a tensor field.
If local coordinates are known:
The matrix representation of a metric tensor is a matrix.
The line element is a function of a metric.
In mathematics, the word metric refers to a fairly general function which defines ‘distance’ between elements in a set (it takes in elements of a set, and produces a real number). Riemannian and pseudo-Riemannian metrics (there are many more kinds of classification of metric too) have different conditions on those functions, but that’s more detail than is required here.
A metric tensor is a function defined on a manifold (a vector space) that takes in two tangent vectors and produces a scalar quantity. Metric tensors are used to define the angle between and length of tangent vectors (somewhat analogous to the dot product of vectors in Euclidean space)
Defining a metric versus a metric tensor:
Consider a smooth manifold of dimension n. For every point x in our manifold, there is a vector space called a tangent space (a tangent space contains all of the tangent vectors to our manifold at the specific point x).
Now, a metric at our point x is a function gx(Xx,Yx), which takes in the two tangent vectors Xx and Yx (at x), and outputs a real number. The metric function must also be bilinear, symmetric, and nondegenerate, but we don’t need to go into further details.
Now we can define a metric tensor, g, on our manifold: The metric tensor assigns a metric, gx, to every point x in the manifold (such that it varies smoothly with x in the manifold). The metric tensor is then:
g(X,Y)(x) = gx(Xx,Yx)
For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space). A metric tensor is not the same as a metric (it’s more analogous to an ‘infinitesimal’ metric function), but it is usually understood in differential geometry and related areas in physics that when one says “metric”, they really mean “metric tensor”. Mathematically, they are not equivalent objects, but integration of a metric tensor does induce a metric function.
Most of the time when actually doing physics, we don’t want such a general object. If local coordinates are known, the metric tensor can be expressed in a variety of more useful forms.
If we are in a region of the manifold where we have defined a local coordinate system, ie. xμ (where μ runs from 0 to 3), we can re-write our metric tensor [field] as:
g = gμνdxμ⊗dxν
where, gμν are real-valued functions, and dxμ are one-forms.
If we have local coordinates defined, we can then represent the metric tensor in matrix form, where, for our four-dimensional spacetime, we will have a 4x4 matrix with elements gμν.
In our local coordinates, if we take dxμ to be an infinitesimal coordinate displacement, we can write out a line element: ds2 = gμνdxμdxν. The line element, we know, is incredibly useful, as it provides us with an invariant quantity and also imparts information about causal structure.
EDIT: A note from The Unapologetic Mathematician that I should add: "the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector. However, the polarization identities will allow you to recover the bilinear function from the quadratic one."
Why does this matter? Well, for starters, general relativity is really all about your frame of reference and choice of coordinates. Some structures are unchanged regardless of your choice of coordinates (ie. the metric function & metric tensor), and some structures change with change in coordinates (ie. the matrix representation of a metric and the associated line element).
Just a couple of (well cited) offenders:
C. Brans and R. H. Dicke, Mach's Principle and a Relativistic Theory of Gravitation . Phys. Rev. 124, 925 (1961), Cited 1,139 times.
“As in general relativity the metric tensor is written as
gij= ηij + hij …”
EDIT: If I included more of the quote, it would have been obvious that local coordinates had already been chosen and they weren't writing out a general metric tensor, but a coordinate specific object. The reference is cited for context. Abstract index notation for tensors uses indices to indicate the type of tensor, rather than to indicate components in a particular basis
As I said above, gij is not the metric tensor, or a tensor at all, but a set of real-valued function specified for a local coordinate system (gij are also the matrix elements in the matrix representation - in those coordinates - of the metric tensor). The same goes for ηij and hij as well.
Tullio Regge and John A. Wheeler, Stability of a Schwarzschild Singularity . Phys. Rev. 108, 1063 (1957), Cited 476 times (two authors I respect immensely)
“Schwarzchild found long ago the solution of Einstein equations for the metric around a fixed spherically symmetrical center-of-mass:
ds2 = -(1-3m*/r)dT2 + (1 – 2m*/r)-1 dr2 + r2(dθ+sin2θdφ2) …”
This is the line element, not the metric.
Brandon Carter, Global Structure of the Kerr Family of Gravitational Fields . Phys. Rev. 174, 1559 (1968), Cited 383 times
“The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by
ds2 = ρ2dθ2 – 2a sin2θdrdφ + 2drdu + …”
Again, this is a line element, not a metric tensor.
Marshall N. Rosenbluth, William M. MacDonald, and David L. Judd, Fokker-Planck Equation for an Inverse-Square Force. Phys. Rev. 107, 1 (1957), Cited 263 times.
“Let the expression for distance between two points whose coordinates differ by dq1, dq2, and dq3 be
Where aμν is a metric tensor…”
Again, aμν is not a metric tensor, but a coefficient, when working in local coordinates from this (local coordinate specific) representation of the metric tensor: aμνdxμ⊗dxν…
It isn’t that hard to say “line element”, or “matrix representation in local coordinates…”, or “matrix element in local coordinates…” instead of “metric tensor” or "metric" so why don't we?