Metric Tensor

After a discussion with Henry de Valence (hdevalence) on IRC, who wanted to know something about the metric tensor, I thought I should put down some stuff that I explained to him. This is like a “Why a metric tensor” article.

A quadratic form, is a homogeneous polynomial of degree 2, like

x^2 + y^2 + 2xy

We can always represent such a polynomial of degree 2 by a symmetric matrix:

[x y][1 1][x]
     [1 1][y]

[I’m sorry, I still don’t know how to type matrices in LaTeX, and I want to put-off learning that for a later time]

So, let’s try to represent the distance between two points, which is a quadratic form, as a matrix. Let’s consider very small distances – between two closely separated points: (x, y, z) and (x + dx, y + dy, z + dz) where dx, dy and dz are “very small” distances.

Ordinary 3-D “Euclidian” Space, Pythagoras theorem reads:

ds^2 = dx^2 + dy^2 + dz^2

[dx  dy  dz][1 0 0][ dx ]
            [0 1 0][ dy ]
            [0 0 1][ dz ]

Special relativity tells us that time is just like a “coordinate” and we shouldn’t treat it separately. Space and time are now put together and called “space-time” and there are 4 coordinates. The “distance” (Space-Time interval) between two points (t, x, y, z) and (t + dt, x + dx, y + dy, z + dz) is defined like this:

ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2

[dt dx dy dz][ -c^2           ][ dt ]
             [       1        ][ dx ]
             [           1    ][ dy ]
             [              1 ][ dz ]

Just like distances don’t change under rotations, space-time intervals don’t change under “Lorentz Transformations” in Special Relativity, which is why it is useful to have the notion of such a weird distance. Ordinary 3-D distances do change under “Lorentz Transformations” – as is demonstrated by the effect of “Length Contraction”.

The matrix that we’ve written in the above is (a representation of what is) called a “metric tensor”.

We can write the metric tensor for the surface of a “two-sphere”. That refers to the surface of the ordinary sphere in three dimensions, that we are accustomed to. It’s called a two-sphere because we are talking about the surface of a sphere in 3-D – which is 2-dimensional. Note that this metric varies from point to point (it’s a function of r and \theta) and hence, is a tensor field:

ds^2 = dr^2 + r^2 d\theta^2 + r^2 sin^2(\theta) d\phi^2

[dr d \theta d \phi][ 1                          ][ dr       ]
                    [    r^2                     ][ d \theta ]
                    [          r^2 sin^2(\theta) ][ d \phi   ]

Finally, any “Riemannian Manifold” (curved spacetime) can be described by the metric tensor and its derivatives. So the metric tensor plays a central role in General Relativity. Formally, space-time intervals are written in the following manner:

ds^2 = dx^\mu dx^\nu g_{\mu \nu}

The above follows a notation called “Einstein Summation Convention”, which my 17-year-old friend already knew, which can be read from other sources.