The Laplace-Beltrami derivative operator

Along with the idea that the metric of spacetime is dynamical, the other big idea of general relativity is that physics should be independent of any particular coordinate system, in the sense that any physically relevant mathematical object should be defined in such a way that in a different coordinate system it is an object of the same type. This is called the "Principle of General Covariance." (Covariance = co-variance = varying along with - the idea is that as the coordinate system is allowed to change, the physically relevant quantities will change along with it in a well-defined way). The allowed kind of transformation law for physically relevant quantities is called the tensor transformation law, and it tells you how vectors, differential forms, and other types of tensors are allowed to transform under coordinate transformations.

First, let's say our manifold $M$ has coordinates $x^{\mu}$ (where $\mu$ runs from 0 to 3). Suppose we make a change of coordinates to a new set, $y^{\rho}$ ($\rho = 0, \dots , 3$), $y^{\rho} = y^{\rho} \left( x^{\mu} \right)$. Then by the chain rule, we have

\begin{align} \frac{\partial}{\partial x^{\mu}} = \frac{\partial y^{\rho}}{\partial x^{\mu}} \frac{\partial}{\partial y^{\rho}} \end{align}

(we're using the Einstein summation convention here). So the vector $\mathbf{v} = v^{\mu} \partial_{\mu}$ transforms under the coordinate transformation:

\begin{align} v^{\mu} \frac{\partial}{\partial x^{\mu}} = v^{\mu} \frac{\partial y^{\rho}}{\partial x^{\mu}} \frac{\partial}{\partial y^{\rho}}, \end{align}

so the components of $\mathbf{v}$ in the new coordinate system (i.e. with respect to the new basis tangent vectors $\frac{\partial}{\partial y^{\rho}}$) are

\begin{align} \tilde{v}^{\rho} = v^{\mu} \frac{\partial y^{\rho}}{\partial x^{\mu}} \end{align}

(again using Einstein summation convention).

You can similarly work out the transformation for a differential 1-form $\mathbf{\omega} = \omega_{\mu} dx^{\mu}$ to be

\begin{align} \tilde{\omega}_{\rho} = \omega_{\mu} \frac{\partial x^{\mu}}{\partial y^{\rho}} \end{align}

and for the metric,

\begin{align} \tilde{g}_{\rho \sigma} = g_{\mu \nu} \frac{\partial x^{\mu}}{\partial y^{\rho}} \frac{\partial x^{\nu}}{\partial y^{\sigma}} \end{align}

The important thing is that these are just curvilinear generalizations of the way you transform vectors and matrices under rotations and flips in the plane or $\mathbb{R}^3$. Although the components of the vectors or 1-forms or metrics alter during the transformation, they do so in a clean way that doesn't privilege any coordinate system over any other. All this stuff is very nicely done in Chapter 1 of Baez and Muniain, Gauge Fields, Knots, and Gravity.

Now since derivative operators are necessary to describe time evolution and spatial constraints in physics, we quickly notice that we need derivative operators more cleverly designed than just plain coordinate derivatives $\frac{\partial}{\partial x^{\mu}}$ to construct physical laws. If we just used the coordinate derivatives, we would be implicitly privileging one coordinate system over another, because $\frac{\partial}{\partial x^{\mu}}$ is only a simple coordinate derivative with respect to a particular coordinate system. If we transform to another coordinate system, it will be some nontrivial linear combination of coordinate derivatives. The coordinate direction $x^{\mu}$ has no more intrinsic physical meaning than the coordinate direction $y^{\rho}$.

In order to address this issue, we create derivative operators which treat all coordinate systems democratically, and transform in an allowed manner with coordinate transformations. In particular, the curved-space version of the Laplacian is called the Laplace-Beltrami operator, and is defined like this, acting on a function $f: M \to \mathbb{R}$:

\begin{align} \Delta_{LB} f = \frac{1}{\sqrt{\lvert g \rvert}} \partial_{\mu} \left( \sqrt{ \lvert g \rvert } g^{\mu \nu} \partial_{\nu} f \right) \end{align}

where $\sqrt{ \lvert g \rvert }$ is the square root of the absolute value of the determinant of the metric $g_{\mu \nu}$.

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