Question

(a) In three-dimensional rectangular coordinates $$d{s}^2=d{x}^2+d{y}^2+d{z}^2$$ Show that $d{\mathrm{s}}^2$ is invariant under a rotation of axes, that is, show that the change of variables $r^{\prime }=Ar$ with $A$ the rotation matrix gives $$d{s}^2=d{x}^{\prime 2}+d{y}^{\prime 2}+d{z}^{\prime 2}$$ Hint: Differentiate (11.6) and find $d{\mathrm{s}}^2$. (b) Repeat part (a) in the notation of $(11,10)$ to show that $d{s}^2=\sum d{x}_i^2=\sum d{x}_i^2$, Use matrix notation; if $$dr=\left(\begin{array}{l}d{x}_1\\ d{x}_2\\ d{x}_3\end{array}\right)\text{ and }d{r}^{\prime }=Adr,dr=A^{\mathrm{\top }}d{r}^{\prime },\mathrm{etc}$$ then $d{s}^2=d{r}^{T}dr$

Short answer

The metric $d{s}^2=d{x}^2+d{y}^2+d{z}^2$ is invariant under rotation, as shown by transformations in both coordinate and matrix notations.

Step-by-step solution

Understand the Metric

In 3D rectangular coordinates, the metric is given by $$d{s}^2=d{x}^2+d{y}^2+d{z}^2.$$ This represents the squared infinitesimal distance between two points in 3D space.

Apply the Rotation Matrix

Under a rotation, the coordinates transform as $$r^{\prime }=Ar,$$ where A is the rotation matrix. This implies that $$\left(\begin{array}{c}{x}^{\prime }{y}^{\prime }{z}^{\prime }\end{array}\right)=A\left(\begin{array}{c}xyz\end{array}\right).$$

Differentiate to Find Infinitesimal Changes

Differentiate the transformation equation to find the differentials: $$\left(\begin{array}{c}d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }\end{array}\right)=A\left(\begin{array}{c}dxdydz\end{array}\right).$$ This expresses the infinitesimal changes in the new coordinates.

Express the Metric in Transformed Coordinates

Express the metric $d{s}^2$ in terms of the transformed coordinates: $$d{s}^2=d{x}^2+d{y}^2+d{z}^2=(d{x}^{\prime }{)}^2+(d{y}^{\prime }{)}^2+(d{z}^{\prime }{)}^2.$$

Prove Invariance by Matrix Multiplication

In matrix notation,$$dr=\left(\begin{array}{c}dxdydz\end{array}\right)\text{and}d{r}^{\prime }=Adr.$$ The metric in matrix notation is expressed as: $$d{s}^2=(dr{)}^T(dr).$$ Since $dr=A^{T}d{r}^{\prime }$, then the metric becomes: $$d{s}^2=(dr{)}^T(dr)=(A^{T}d{r}^{\prime }{)}^T(A^{T}d{r}^{\prime })=d{r}^{\prime T}A{A}^{T}d{r}^{\prime }.$$ Because rotation matrices are orthogonal, $A{A}^T=I$, yielding: $$d{s}^2=d{r}^{\prime T}d{r}^{\prime }.$$

Summarize Part (a) and (b)

Since $d{s}^2=d{x}^{\prime 2}+d{y}^{\prime 2}+d{z}^{\prime 2}$ and in matrix notation $d{s}^2=d{r}^{\prime T}d{r}^{\prime }$, the metric $d{s}^2$ is invariant under the rotation. Both parts (a) and (b) show this invariance using different notations.

Key concepts

3D Rectangular Coordinates

In three-dimensional space, we describe points using the coordinates $x,y,z$. This system allows us to uniquely pinpoint locations in the 3D space. The formula $$d{s}^2=d{x}^2+d{y}^2+d{z}^2$$ represents the squared infinitesimal distance between two nearby points. This is known as the metric in 3D rectangular coordinates. The idea is to sum up the squared infinitesimal changes in all three directions (x, y, and z) to get the total change in distance.

Rotation Matrix

A rotation matrix is a matrix used to perform a rotation in Euclidean space. In 3D, a rotation matrix $A$ is a 3×3 matrix that, when multiplied with a coordinate vector, rotates that vector around the origin.
For example, under a rotation, the new coordinates $(x^{\prime },y^{\prime },z^{\prime })$ are given by $$\left(\begin{array}{c}{x}^{\prime }{y}^{\prime }{z}^{\prime }\end{array}\right)=A\left(\begin{array}{c}xyz\end{array}\right).$$ Here, $A$ is the rotation matrix. It is important to note that rotation matrices are orthogonal, meaning their inverse is equal to their transpose: $A^T=A^{-1}$.

Metric Tensor

The metric tensor is a fundamental concept in differential geometry and general relativity. In the context of 3D space, it helps us measure distances and angles. The metric tensor tells us how to compute the dot product of vectors. In 3D rectangular coordinates, the metric tensor is quite simple, represented by the identity matrix.
Given in our exercise, the metric is $d{s}^2=d{x}^2+d{y}^2+d{z}^2$. It means we sum the squares of the differential elements in each coordinate direction. This metric remains unchanged (or invariant) under rotations, demonstrating how rotations preserve distances in space.

Orthogonal Transformation

Orthogonal transformations are transformations that preserve the lengths of vectors. Examples include rotations and reflections. In our context, when we rotate our coordinate system, we use an orthogonal transformation to map old coordinates $(x,y,z)$ to new coordinates $(x^{\prime },y^{\prime },z^{\prime })$.
An orthogonal transformation in matrix form has the property $A{A}^T=I$. This property ensures that the metric or distances stay the same before and after the transformation.
So, when we perform such a transformation, it's evident from the matrix multiplication in our solution that our metric $d{s}^2$ remains unchanged: $$d{s}^2=(d{x}^{\prime }{)}^2+(d{y}^{\prime }{)}^2+(d{z}^{\prime }{)}^2,$$ confirming invariance under orthogonal transformations like rotations.