The Gravity of Light

Abstract

The gravitational field of an idealized plane-wave solution of the Maxwell equations can be described in closed form. After discussing this particular solution of the Einstein-Maxwell equations, the motion of neutral test particles, which are sensitive only to the gravitational background field, is analyzed. This is followed by a corresponding analysis of the dynamics of neutral fields in the particular Einstein-Maxwell background, considering scalars, Majorana spinors and abelian vector fields, respectively.

Appendix: Spinors and the Dirac Algebra

Spinor fields in curved space-time are most easily described in the tangent Minkowski space, using the vierbein formulation to translate the results to the curved space-time manifold. We use the flat-space representation of the Dirac algebra in which $\gamma _5$ is diagonal:

(15.56)

such that for $a = (0,1,2,3)$

(15.57)

The generators of the Lorentz transformations on spinors are defined by

$$\displaystyle \begin{aligned} \sigma _{ab} = \frac{1}{4} \left[ \gamma _a, \gamma _b \right], {} \end{aligned} $$

(15.58)

with commutation relations

$$\displaystyle \begin{aligned} \left[ \sigma _{ab}, \sigma _{cd} \right] = \eta_{ad} \sigma _{bc} - \eta_{ac} \sigma _{bd} - \eta_{bd} \sigma _{ac} + \eta_{bc} \sigma _{ad}. {} \end{aligned} $$

(15.59)

Hermitean conjugation is achieved by

$$\displaystyle \begin{aligned} \gamma _a^{\dagger} = \gamma _0 \gamma _a \gamma _0. {} \end{aligned} $$

(15.60)

The charge conjugation operator C is defined by

$$\displaystyle \begin{aligned} C = C^{\dagger} = C^{-1} = - C^T = \gamma _2 \gamma _0 = \left( \begin{array}{cc} \sigma _2 & 0 \\ 0 & - \sigma _2 \end{array} \right), {} \end{aligned} $$

(15.61)

such that

$$\displaystyle \begin{aligned} C^{-1} \gamma _a C = - \gamma _a^T. {} \end{aligned} $$

(15.62)

If the spinor $\Psi$ is a solution of the Dirac equation in Minkowski space

$$\displaystyle \begin{aligned} \left( \gamma \cdot \partial + m \right) \Psi = 0, {} \end{aligned} $$

(15.63)

then this is also true for the charge-conjugate:

(15.64)

The Majorana constraint $\Psi ^c = \Psi$ reduces the number of independent spinor components from 4 to 2 complex ones. This makes it easy to work in terms of 2-component spinors $(\chi , \eta )$ which are eigenspinors of $\gamma _5$, by the decomposition

(15.65)