Friday, March 25, 2016

The polarization of gravitational waves

In general relativity, a plane gravitational wave, such as that apparently detected by the LIGO apparatus in September 2015, is a type of transverse shear wave in the geometry of space.

To understand this, first consider the concept of a transverse wave in general relativity.

Recall that observers in general relativity are represented by timelike curves, and instantaneous observers correspond to particular points along timelike curves.

For an instantaneous observer, represented by the tangent vector $Z$ to a timelike curve at a point $z$, there is a local version of Euclidean space, dubbed the local rest-space $R = Z^\bot$, and defined as the set of (spacelike) vectors orthogonal to $Z$.

A plane gravitational wave travels in a spatial direction specified by a propagation vector $k \in R = Z^\bot$, and distorts the geometry of space in the two-dimensional plane $T$ orthogonal to $k$ in the observer's local rest-space $R$. It is in this sense that a gravitational wave is a transverse wave.

In particular, a plane gravitational wave is also a shear wave, and understanding this requires an explanation of the polarization of gravitational waves.

In the simplest case, a linearly-polarized gravitational wave alternately stretches space in one direction $e_x \in T$, and compresses it in a direction $e_y \in T$ at right-angles to $e_x$, in a manner which distorts circles into ellipses, but preserves spatial areas.

However, linearly polarized plane gravitational waves are nothing more than very special cases, and the purpose of this post is largely to put linear polarization into context.

But before digging a little deeper, it's worthwhile first to recall the characteristics of an electromagnetic plane wave, and its possible polarizations.

Just like a gravitational wave, an electromagnetic plane wave has a direction of propagation $k$. The electric $E$ and magnetic fields $B$ are then defined by perpendicular vectors of oscillating magnitude in a plane which is orthogonal to the propagation vector $k$. However, it is the direction in which the electric field vector points which defines the plane of polarization.

In the case of linear polarization, the plane of the electric field vector is constant. The electric field merely oscillates back-and-forth within this plane.

However, the most general case of an electromagnetic plane wave is one which is elliptically polarized. This is a superposition of two perpendicular plane waves, which may differ in either phase or amplitude. The polarization direction of one is separated by 90 degrees from the polarization direction of the other. The net effect is that the tip of the resultant electric field vector will sweep out an ellipse in the plane orthogonal to the direction of propagation.

If the relative phases of the component waves differ by 90 degrees, and the amplitudes of the two components are the same, then this reduces to the special case of circular polarization. In this event, the tip of the resultant electric field vector will sweep out a circle in the plane orthogonal to the direction of propagation.

One important distinction between gravitational waves and electromagnetic waves is that, whilst the most general case of an electromagnetic wave is defined as a linear combination of two components oriented at 90 degrees to each other, the most general case of a plane gravitational wave is defined as a linear combination of two components oriented at 45 degrees to each other.

To understand this, first note that the wave-fronts of a plane gravitational wave are represented by a foliation of space-time into a 1-parameter family of null hypersurfaces, each of which $\mathscr{W}$ is defined by a particular value of the function $\phi = t - z$.

This assumes that the z-coordinate is aligned with the direction of propagation of the wave. In general, one might be interested in surfaces with a constant value of $\omega (t - k \cdot x)$, with $\omega$ being the wave frequency and $k$ being the propagation vector.

Tangent to these null hypersurfaces $\mathscr{W}$ is a null vector field $Y$ which defines the space-time propagation vector of the gravitational wave (Sach and Wu, General relativity for mathematicians, 1977, p244). The projection of the null vector field $Y$ into an observer's local rest-space at a point provides the spatial propagation vector $k$.

If one imagines space-time as a 2-dimensional plane, with the time axis $t$ as the vertical axis, and the spatial direction $z$ as the horizontal axis, then the null hypersurfaces of constant $\phi$ correspond to diagonal lines running from the bottom left to the top-right. These represent a gravitational wave passing from the left to the right of the diagram. An observer corresponds to a timelike curve, tracing a path from the bottom to the top of the diagram.

In Christian Reisswig's diagram below, (taken from a different application), the null hypersurfaces are those labelled as $u$=constant, and the worldline of an observer corresponds to that labelled as $R_\Gamma$.

As the proper time of the observer elapses, the observer's worldline intersects a sequence of the null hypersurfaces. This corresponds to the different phases of the wave passing through the observer's point-in-space. Hence $\phi$ can be thought of as defining the phase of a plane gravitational wave.

In terms of the metric tensor, a gravitational wave is typically represented as a perturbation $h_{\mu\nu}$ on a background space-time geometry $\bar{g}_{\mu\nu}$: $$ g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu} $$ The perturbation is represented as follows: $$ h_{\mu\nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & h_+(\phi) & h_\times(\phi) & 0 \\ 0 & h_\times(\phi) & -h_+(\phi) & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \; . $$ The two components, or polarizations, of the wave are denoted as $h_+(\phi)$ and $h_\times(\phi)$. They form a net polarization tensor $h(\phi)$, which can be extracted from the metric tensor above, and written as follows: $$ h(\phi) = h_+(\phi)(e_x \otimes e_x - e_y \otimes e_y) + h_\times(\phi)(e_x \otimes e_y + e_y \otimes e_x) $$ Now, suppose that the source of a gravitational wave is a gravitationally bound system consisting of two compact objects (i.e., black holes or neutron stars). The plane of that orbital system will be inclined at an angle $\iota$ between 0 and 90 degrees to the line-of-sight of the observer. The case $\iota$ = 0 corresponds to a system which is face-on to the observer, and the case $\iota = \pi/2$ corresponds to a system which is edge-on to the observer.

The time-variation of a plane gravitational wave emitted by such a compact binary system, passing through a distant observer's point-of-view, is effectively specified by the phase-dependence of the two components of the wave: $$ h_+(\phi) = A(1+ \cos^2\iota) \cos (\phi) \\ h_\times(\phi) = -2A \cos \iota \sin \phi $$ $A$ determines the amplitude of the wave.

This is the general case, corresponding to elliptical polarization. The orbital paths of the stars or black holes in the binary system will appear as ellipses. In terms of the basis vectors in which the metric tensor perturbation is expressed, $e_x$ is determined by the long axis of the ellipse, and $e_y$ is perpendicular to $e_x$ in the plane orthogonal to the line-of-sight.

There are two special cases: when the system is face-on, the gravitational wave exhibits circular polarization; and when the system is edge-on, the wave exhibits linear polarization.

To make this explicit, consider first the case where the source of the wave is edge-on to the observer. $\iota = \pi/2$, hence $\cos^2 \iota = \cos \iota = 0$, and it follows that: $$ h_+(\phi) = A(1+ \cos^2\iota) \cos (\phi) = A \cos \phi \\ h_\times(\phi) = -2A \cos \iota \sin \phi = 0 $$ One of the polarization components has vanished altogether, hence from the perspective of the distant observer, space alternately stretches and contracts along a fixed pair of perpendicular axes. One of these axes, $e_x$, is determined by the orientation of the orbital plane of the source system, seen edge-on, and the other, $e_y$, is the axis perpendicular to $e_x$ in the plane orthogonal to the line-of-sight. The polarization tensor reduces to: $$\eqalign{ h(\phi) &= h_+(\phi)(e_x \otimes e_x - e_y \otimes e_y) \cr &= A \cos \phi(e_x \otimes e_x - e_y \otimes e_y)} $$ The negative sign associated with $e_y \otimes e_y$ entails that as space is stretching in direction $e_x$, it is contracting in direction $e_y$. This linear polarization is the simplest special case of a plane gravitational wave, as beautifully demonstrated in the animation below from Markus Possel:

In the other special case, the case of a face-on system, $\iota$ = 0. It follows that $\cos^2 \iota = \cos \iota = 1$, hence: $$ h_+(\phi) = A(1+ \cos^2\iota) \cos (\phi) = A \cos \phi + A \cos \phi = 2A \cos \phi \\ h_\times(\phi) = -2A \cos \iota \sin \phi = -2A \sin \phi $$ In this case, then, the two components have equal amplitude, $2A$, and differ by virtue of the fact that the $h_\times$ component lags 90 degrees behind the $h_+$ component. This is the case of circular polarization. As seen in the Markus Possel animation below, the net effect is to produce a rotation of the shear axes.