## 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

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:

[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:

[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:

[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:

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

## Muthusubramanian NV 1:33 pm

onJune 4, 2009 Permalink | Log in to ReplyMachee, use

\begin{bmatrix}

1 & 0 \\

0 & 1 \\

\end{bmatrix}

This is an identity matrix of order 2. very simple 🙂

## Naveen 9:54 am

onJune 11, 2009 Permalink | Log in to Replytypo: (t + dt*, x + dx, y + dy, z + dz)… 🙂

## Akarsh Simha 10:21 am

onJune 11, 2009 Permalink | Log in to ReplyThanks. Fixed.

M-x weblogger-start-entry

[Put password]

M-x weblogger-prev-entry

C-s t + dy

C-? t

C-x C-s

C-X C-c

😉

## SG 9:14 am

onJune 14, 2009 Permalink | Log in to ReplyThese as some remarks with respect to the sphere that you mention.

1. First, it is either a 2D sphere or a sphere in 3D.

2. The line-element in the non-LaTeX form is not correct. The row to the left should be [ dr d \theta d\phi] and a similar correction needs to be done in the column entry as well. Is there any particular reason to choose this non-LaTeX method of writing matrices?

## Akarsh 2:41 pm

onJune 16, 2009 Permalink | Log in to Reply@SG: Ouch! Yes.

Thanks for the corrections.

I chose the non-LaTeX method because I didn’t know how to format matrices side-by-side in LaTeX and decided to put-off learning for a later stage.

## SG 4:18 pm

onJune 18, 2009 Permalink | Log in to ReplyHere is how you do it in LaTeX:

$$

\begin{pmatrix} dr & d\theta & d\phi \end{pmatrix}

\begin{pmatrix}

1 & 0 & 0 \\

0 & r^2 & 0 \\

0 & 0 & r^2 \sin^2\theta

\end{pmatrix}

\begin{pmatrix} dr \\ d\theta \\ d\phi \end{pmatrix}

$$

The above example illustrates the use of pmatrix environment (there are other ones like matrix etc.).