Covariant Differentiation

Abstract

The theory of forms is a theory of antisymmetric tensors. In such a theory we need an antisymmetric version of the covariant derivative such that the derivative of a form is a form. Hence in this chapter we first introduce the covariant derivative and then the antisymmetric exterior derivative. The relativistic Euler–Lagrange equations are introduced and applied to deduce the equation of motion of free particles in curved spacetime—the geodesic equation. Since gravity is not reckoned as a force in the general theory of relativity “free particles” in this theory correspond to particles acted upon by gravity in Newton’s theory. The fundamental equations of electromagnetism in form language are also presented in this chapter.

Exercises

• 5.1 Dual forms

Let $\{ \vec{e}_{{\hat{i}}} \}$ be a Cartesian basis in the three-dimensional Euclidean space. Using a vector $\vec{a} = a^{{\hat{i}}} \vec{e}_{{\hat{i}}}$ there are two ways of constructing a form:

• (i) By constructing a 1-form from its covariant components $a_{j} = g_{ji} a^{i}$:

  $$\underline{A} = a_{{\hat{i}}} \underline{{{\text{d}}x}}^{{\hat{i}}} .$$

• (ii) By constructing a 2-form from its dual components, defined by $a_{{\hat{i}\,\hat{j}}} = \varepsilon_{{\hat{i}\,\hat{j}\,\hat{k}}} a^{{\hat{k}}}$:

  $$\underline{a} = \frac{1}{2}a_{{\hat{i}\,\hat{j}}} \underline{{{\text{d}}x}}^{{\hat{i}}} \wedge \underline{{{\text{d}}x}}^{{\hat{j}}} .$$

  We write this form as $\underline{a} = { \star }\underline{A}$ where ${ \star }$ means to take the dual form.

• (a) Given the vectors $\vec{a} = \vec{e}_{x} + 2\vec{e}_{y} - \vec{e}_{z}$ and $\vec{b} = 2\vec{e}_{x} - 3\vec{e}_{y} + \vec{e}_{z}$.

  Find the corresponding 1-forms $\underline{A}$ and $\underline{B}$, and the dual 2-forms $\underline{a} = { \star }\underline{A}$ and $\underline{b} = { \star }\underline{B}$, and also the dual form $\theta$ to the 1-form $\underline{\sigma } = \underline{{{\text{d}}x}} - 2 \underline{{{\text{d}}y}}$.

• (b) Take the exterior product $\underline{A} \wedge \underline{B}$ and show that

  $$\theta_{{\hat{i}\,\hat{j}}} = \varepsilon_{{\hat{i}\,\hat{j}\,\hat{k}}} C^{{\hat{k}}} ,$$

  where $\underline{\theta } = \underline{A} \wedge \underline{B}$ and $\vec{C} = \vec{a} \times \vec{b}$.

• (c) Show that the exterior product $\underline{A} \wedge { \star }\underline{B}$ is given by the 3-form

  $$\underline{A} \wedge { \star }\underline{B} = (\vec{a} \cdot \vec{b})\underline{{{\text{d}}x}} \wedge \underline{{{\text{d}}y}} \wedge \underline{{{\text{d}}z}} .$$

• (d) Show that the exterior derivative of a 1-form, $\underline{{{\text{d}}A}}$, corresponds to the curl $\nabla \times \vec{A}$ of the corresponding vector.

• (e) Finally show that

  $$\underline{\text{d}} * \underline{{{\text{d}}f}} = \nabla^{2} f\underline{{{\text{d}}x}} \wedge \underline{{{\text{d}}y}} \wedge \underline{{{\text{d}}z}}$$

  for a scalar field f.

• 5.2 Differential operators in spherical coordinates

We consider an Euclidean three-dimensional space with Cartesian coordinates $\left( {x,y,z} \right)$ and spherical coordinates $\left( {r,\theta ,\phi } \right)$. The transformation from the spherical to the Cartesian coordinates is

$$x = r\sin \theta \cos \phi , \quad y = r\sin \theta \sin \phi , \quad z = r\cos \theta$$

• (a) Find the components of the metric tensor and the form of the line-element in spherical coordinates.

• (b) Let f be a scalar field. The gradient of f is given by

  $$\nabla f = \frac{\partial f}{\partial r}\vec{e}_{{\hat{r}}} + \frac{\partial f}{\partial \theta }\vec{e}_{{\hat{\theta }}} + \frac{\partial f}{\partial \phi }\vec{e}_{{\hat{\phi }}} ,$$

  where $\vec{e}_{{\hat{i}}}$ are the orthonormal basis vectors formed from the coordinate basis vectors in the spherical coordinate system.

  Find the expressions for the gradient of f in coordinate basis.

• (c) In a coordinate system with orthogonal coordinate basis vectors, the curl of a vector field is given by

  $$\begin{aligned} \nabla \times \vec{A} & = \frac{1}{{\sqrt {g_{22} g_{33} } }}\left( {\frac{{\partial A^{{\hat{3}}} }}{{\partial x^{2} }} - \frac{{\partial A^{2} }}{{\partial x^{3} }}} \right)\vec{e}_{1} + \frac{1}{{\sqrt {g_{11} g_{33} } }}\left( {\frac{{\partial A^{{\hat{1}}} }}{{\partial x^{3} }} - \frac{{\partial A^{{\hat{3}}} }}{{\partial x^{1} }}} \right)\vec{e}_{2} \\ & \quad + \frac{1}{{\sqrt {g_{11} g_{22} } }}\left( {\frac{{\partial A^{{\hat{2}}} }}{{\partial x^{1} }} - \frac{{\partial A^{{\hat{1}}} }}{{\partial x^{2} }}} \right)\vec{e}_{3} . \\ \end{aligned}$$

  Find an expression for the curl in spherical coordinates. (The division by the factors $\sqrt {g_{ii} g_{jj} }$ is a normalization of the area of a surface element normal to the basis vector $\vec{e}_{k} ,k \ne i,j$.)

• (d) The divergence of a vector field can in general (in an arbitrary basis) be defined by

  $$\left( {\nabla \cdot \vec{A}} \right)\underline{\varepsilon } = \underline{\text{d}} * \underline{A} ,$$

  where

  $$\underline{\varepsilon } = \sqrt {\left| g \right|} \underline{\omega }^{1} \wedge \underline{\omega }^{2} \wedge \underline{\omega }^{3} ,$$

  Is the volume form, and $\left| g \right|$ is the determinant of the matrix formed by the components of the metric tensor. The volume form represents an invariant volume element.

  Find the expression for the divergence of $\vec{A}$ in spherical coordinate.

  Finally find the expression for the Laplacian of f in the spherical coordinate system.

• 5.3 Spatial geodesics in a rotating frame of reference

Our point of departure is the line-element (4.20) for 3-space in a rotating reference frame,

$$\text{d}\ell^{2} = \text{d} r^{2} + \frac{{r^{2} \text{d} \theta^{2} }}{{1 - r^{2} \omega^{2} /c^{2} }} + \text{d} z^{2} .$$

We shall consider geodesics in the two-dimensional surface with $z =$ constant. The task is to calculate the shortest curve between two points with the same distance from the axis using the Lagrangian equations with the Lagrange function $L = \left( {1/2} \right)\dot{\ell }^{2}$, where the dot denotes differentiation with respect to an invariant parameter representing the arc length along the curve.

• (a) Find the form of the 2-vector identity for the tangent vectors of the curve.

• (b) Find an expression for the momentum $p_{\theta }$ conjugate to the cyclic coordinate $\theta$ of L.

• (c) Find the differential equation for the geodesic curves.

• (d) Use the boundary condition that the point on the curve closest to the axis has a distance $r_{0}$ from the axis, to show that

  $$p_{\theta } = \frac{{r_{0} }}{{\sqrt {1 - r_{0}^{2} \omega^{2} /c^{2} } }}.$$

• (e) Show that the differential equation of the curve can be written as

  $$\frac{{ \text{d}r}}{{r\sqrt {r^{2} - r_{0}^{2} } }} - \frac{{\omega^{2} }}{{c^{2} }}\frac{{r \text{d} r}}{{\sqrt {r^{2} - r_{0}^{2} } }} = \frac{{ \text{d} \theta }}{{r_{0} }}.$$

  Integrate this equation and find the equation of the curve. Finally draw the curve.

• 5.4 Christoffel symbols in a uniformly accelerated reference frame

• (a) Use the coordinate transformation (4.80)–(4.82) and the formula (5.27) to calculate the non-vanishing Christoffel symbols in the coordinate system of Chap. 4 co-moving with a uniformly accelerated reference frame.

• (b) Use Eq. (5.65) to calculate the same Christoffel symbols as in (a).

• 5.5 Relativistic vertical projectile motion

A particle is thrown vertically upwards with velocity v from the origin of the coordinate system in the gravitational field of a uniformly accelerated reference frame.

Calculate the maximal height of the particle.

• 5.6 The geodesic equation and constants of motion

• (a) Show that the geodesic equation can be written in the following form: $\frac{{ \text{d}u_{\alpha } }}{{\text{d}s}} - \frac{1}{2}\frac{{\partial g_{\beta \gamma } }}{{\partial x^{\alpha } }}u^{\beta } u^{\gamma } = 0$.

• (b) Assume that the metric is static and the space is cylindrically symmetric with cylindrical coordinates $\left( {r,\theta ,z} \right)$. What constants of motion are there then for a free particle?