Discovered by the mathematician Bernhard Riemann c.1854, the metric tensor is way to describe the curvature of a surface given its points. Specifically, a collection of 10 numbers at every point on a 4-dimensional surface can fully describe that surface, no matter how many folds (dimensions) the surface contains.

If one were to view 2-dimensional space, you would need a collection of 3 numbers at every point. To view N-dimensional space, the metric tensor would look like a collection of N x N numbers, as on a chessboard. Thusly, unfolding the surface and flattening it out reduces it to 2-dimensions and you get Pythagoras famous formula.

The metric tensor, a mathematical breakthrough, later gave Albert Einstein one of the keys to unlocking his theory of general relativity.
The metric tensor is more precisely a symmetric bilinear form which gives rise to a Riemannian metric. To clarify, you can write is as a symmetric matrix Aij, and then write the metric in the form
ds2 = Aijdxidxj
where x is the coordinate on your manifold. Note that because of the symmetry of A, it will have 3 independent components in 2-d, and 10 independent components in 3-d. The above reduces to the Euclidean metric when A is the identity matrix, and then
ds2 = dx2 + dy2 + dz2
which is a differential statement of Pythagoras's Theorem.

In G.F.B. Riemann's scheme of geometry, the metric tensor must be positive definite, that is to say that A has strictly positive eigenvalues, in order that all distances are measured as being positive. However, in the Special and General Theories of Relativity, 3 of the eigenvalues are positive and 1 is negative, and so it these cases, the metric is said to be pseudo-Riemannian (it had the effect that distances measured inside a light cone all turn out to be zero). If you think about it, it is precisely this condition which makes space and time different (read more about it here).

A bilinear (0,2) tensor field g: V×VR on a manifold M that is symmetric and non-degenerate. One can think of the metric as an object g(_,_) that takes two vectors u and v, and returns a real number, g(u,v). We may associate this number with the scalar product or dot product of u and v, in which case the metric gives shape to an otherwise shapeless topological space by defining the lengths of tangent vectors and the angles between them. The metric is an important concept in differential geometry.

The metric is bilinear in that it is linear in both its arguments:

g(a u + b v, w) = a g(u,w) + b g(v,w)

g(u, a v + b w) = a g(u,v) + b g(u,w)

It is symmetric in that the result is independent on the order in which it operates on two vectors: g(u,v) = g(v,u). We impose this requirement so that u.v = v.u and so that v.v = ||v||2. is well defined.

It is also non-degenerate in that g(u,v) = 0, ∀uVv = 0, i.e. if g(u,v) = 0 for all u then v must be zero, and so with the converse. unperson kindly points out that g(u,v) = 0 implies that either u or v or both are zero, or that they are perpendicular to one another.

As an example, in Euclidean R3 we may write

g(  ,  ) = dx1(  )dx1(  ) + dx2(  )dx2(  ) + dx3(  )dx3(  )

where dxα(  ) are the basis oneforms. Here it is understood that the first oneform in each term acts on the first argument of g, while the second oneform acts on the second argument. Strictly, dx1(  )dx1(  ) is a tensor product dx1(  )⊗dx1(  ). For instance, let u = ∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3 and v = 2 ∂/∂x1 - ∂/∂x2 + ∂/∂x3. Then

g(u,v) =

dx1(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx1(2 ∂/∂x1 - ∂/∂x2 + ∂/∂x3) +
dx2(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx2(2 ∂/∂x1 - ∂/∂x2 + ∂/∂x3) +
dx3(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx3(2 ∂/∂x1 - ∂/∂x2 + ∂/∂x3)

Since dx1(∂/∂x1) = 1 and dx2(∂/∂x1) = 0, etc. So g(u,v) = (1)(2) + (2)(-1) + (3)(1) = 3, or u.v = 3.

We can also employ the short-hand notation g = dx1dx1 + dx2dx2dx3dx3 where it is again understood that the first oneform in each term acts on the first argument of g, and so on. You may sometimes see this written as ds2 = dx2 + dy2 + dz2 which is suggestive of the pythagorean theorem applied to the differentials dx, dy and dz, where ds is some differential length, however this isn't very rigorous although it may serve as a memory aid.

Another way to use the metric is to act on only one vector, leaving the other argument unoccupied, e.g.  g(u,  ). The result is then an object that takes another vector v and returns a real number, with precisely the same properties as a oneform. Thus the metric provides a natural relationship between vectors and oneforms and a way to associate a unique oneform to every vector, e.g. μ = g(u,  ) where μ is the oneform associated with the vector u. For instance, consider again u = ∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3. Then

g(u,  ) =

dx1(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx1 +
dx2(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx2 +
dx3(∂/∂x1 + 2 ∂/∂x2 + 3 ∂/∂x3)dx3

So μ = g(u,  ) = dx1 + 2 dx2 + 3 dx3. Since we are dealing with Euclidean R3 the relationship between a vector and its associated oneform is trivial, as the components are unaltered, however the example serves its illustrative purpose.

It would then be natural to have a way to associate a vector with any oneform such that the mapping is one-to-one and onto. That is, so that every vector u maps to a unique oneform μ, and for every oneform μ there is one and only one vector that maps to it, namely u. To do this we introduce the inverse metric g-1 such that g-1(g(u,  ),  ) = u. We can then say associate with every oneform μ a unique vector u = g-1(μ,  ). Again, in Euclidean R3 we may write

g-1(  ,  ) = ∂/∂x1(  )∂/∂x1(  ) + ∂/∂x2(  )∂/∂x2(  ) + ∂/∂x3(  )∂/∂x3(  )

where g-1 takes two oneforms as arguments and returns a real number, or a single oneform to return a vector.

The metric becomes more useful in other coordinate systems and spaces. For instance, consider the cylindrical parameterization of R3 in terms of r, θ,  and z:

x1 = r cosθ
x2 = r sinθ
x3 = z

Then dx1 = ∂x1/∂r dr + ∂x1/∂θ dθ + ∂x1/∂z dz = cosθdr - r sinθ dθ, dx2 = sinθdr + r cosθ dθ, and dx3 = dz. Then

g = dx1dx1 + dx2dx2 + dx3dx3
= (cosθdr - r sinθ dθ)(cosθ dr - r sinθ dθ) + (sinθdr + r cosθ dθ)(sinθdr + r cosθ dθ) + dz dz
= cos2θ dr dr - r sinθ cosθ dθ dr - r sinθ cosθ dr dθ + r2 sin2θ dθ dθ + sin2θ dr dr + r sinθ cosθ dθ dr + r sinθ cosθ dr dθ + r2 cos2θ dθ dθ + dz dz

Thus g = dr dr + r2 dθ dθ + dz dz

In some coordinate systems the non-degeneracy condition breaks down in certain places, for example, in spherical coordinates g = dr dr + r2 dθ dθ + r2 sin2θ dφ dφ, the metric becomes degenerate where sin2θ = 0, or at θ = 0 or π. In such cases additional parameterizations may be necessary to completely cover the entire manifold.

It is useful to express the metric tensor in abstract index notation, so in terms of basis oneforms eαa, where α denotes the αth basis oneform, we may write:

gab = gαβ eαa eβb

where gαβ are the components of the metric tensor, and greek indices that appear both as superscript and subscript are summed over. In this notation, a denotes part of the tensor that acts on the corresponding vector labeled a, while b denotes the part that acts on the corresponding vector labeled b. We may then write the action of g on two vectors u and v as  gab ua vb. It is understood that u corresponds to the first argument of g, while v corresponds to the second argument of g, regardless of the order that they may appear in the expression, e.g. gab ua vb = gab vb ua = gdc ud vc . Since the metric is symmetric the distinction is not an important one, but is useful when g acts on objects other than vectors. In terms of components and the basis vectors, ua = uα eαa, and vb = vβ eβb, then

gab ua vb = gαβ eαa eβb uγ eγa vδ eδb

Rearranging yields

gab ua vb = gαβ  uγ vδ eαa eγa eδb eβb

Since eαa eγa = δαγ, etc. where δαγ is the Kronecker delta we have

gab ua vb = gαβ  uγ vδ δαγ δδβ

gab ua vb = gαβ  uα vα

We may also write the inverse metric g-1 in terms of the basis vectors eαa as

gab = gαβ eαa eβb

With both the metric and its inverse, we can find the oneform that corresponds to any vector, and vice versa. Thus the metric can be thought of an object that raises or lowers indices of oneforms and vectors. We can then also use the metric and its inverse to raise and lower the indices of higher order tensors. Consider the (1,1) tensor Tab = Tαβ eαa eβb. Suppose we wish to calculate Tab, that is lower the first index while raising the second. We can use gab to lower the first index, and then use gab to raise the second index, as follows:

Tab = gac gdb  Tcd

(Components have been suppressed for simplicity.)

Although the above discussion has mostly dealt with Euclidean R3 in Cartesian coordinates, it is general enough to be extended to other spaces easily. Starting in Rn with the coordinate basis oneforms dxα  in Cartesian coordinates we can express the metric as

gab = δαβ dxαa dxβb

We can then embed a manifold of dimension m ≤ n in Rn using the m coordinates ξα. On the manifold where xα are smooth, invertible functions of ξα, i.e. the Jacobian Determinant is nonzero, we can write dxαa = ∂xα/∂xδδa. Thus

gab = δαβ ∂xα/∂xδδa ∂xβ/∂xεεb, or

gab = δαβ ∂xα/∂xδ ∂xβ/∂xεδa εb,

gab = gδεδa εb,

Thus in the new coordinate system, gδε = δαβ ∂xα/∂xδ ∂xβ/∂xε, and we may write, after relabling indices,

gαβ = Σδ (∂xδ/∂xα )(∂xδ/∂xβ)

Although this formula assumes an embedding into a higher-dimensional space, the concept of the metric is an intrinsic property of the manifold and is an important construct that embodies the curvature of the manifold. The metric tensor can be used to specify the Christoffel Symbols (see Covariant Derivative) as well as the Riemann Curvature tensor. A connection derived from the metric tensor is known as a metric compatible connection. We can also derive a metric compatible volume form (see volume form).

An example of a metic important in physics is the Minkowski metric in Special relativity which describes flat spacetime:

g = - c dt dt + dx1dx1 + dx2dx2 + dx3dx3

where t is the temporal (time) coordinate and xα re spatial coordinates. In this spacetime non-zero intervals can have positive (timelike), negative (spacelike) and zero (null or lightlike) magnitudes. There is also the Schwarzschild  metric in General Relativity which describes spacetimes outside of static spherically symmetric distributions of matter, for instance a non-rotating neutral black hole. In spherical coordinates (with coordinates in which c and G = 0):

g = - (1 - 2M/r) dt dt  + (1-2M/r)-1 dr dr + r2 dθ dθ + r2 sin2θ dφ dφ

Here the coordinate system breaks down at r = 2M which is known as the Schwarzschild Radius for the object. In other words, if the entire mass of the object where compressed to within this radius, becoming a Schwarzschild black hole then the metric would break down at this radius. Although relativistic effects become severe as one approaches this radius, spacetime itself does not actually break down here, which was originally thought to be the case. Instead, once inside the event horizon, an observer's future light cone lies entirely within this radius and the observer becomes trapped. We may reparameterize the space, for example in terms of Kruskal-Szekeres coordinates, in which the metric still remains valid at this distance. However, the singularity at r = 0 remains and is a real singularity.

One may also speak of the signature of a metric1. When the metric is diagonalized, the signature is the number of positive elements minus the number of negative elements. For instance, the Minkowski metric has signature 3-2 = 2. A metric in which all elements are positive is known as a positive definite metric or Riemannian metric. General Relativity deals with metrics with one negative element and are known as Lorentzian metrics. Metrics such as these are known as indefinite where some elements are positive and some are negative. One can also denote the signature as (p,q) where p is the number of positive elements and q is the number of negative elements. More explicitly, one may write the metric in the form, for example, -+++ for a Lorentzian metric.

1. krimson Pointed out that I should mention this distinction.

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