Making Sense of the TT Gauge

As introduced so far, the $TT$ gauge might appear rather abstract and not particularly interesting. Quite the opposite, the $TT$ gauge introduces a number of important advantages and simplifications in the study of gravitational waves. The most important of these is that, in this gauge, the only nonzero components of the Riemann tensor are

$$R_{j0k0}=R_{0j0k} = -R_{j00k} = -R_{0jk0} .$$ (90)

Since, however,

$$R_{j0k0}=-\frac{1}{2} h^{^{\rm TT}}_{jk,00} ,$$ (91)

the use of the $TT$ gauge indicates that a travelling gravitational wave with periodic time behaviour $h^{^{\rm TT}}_{jk} \propto \exp (i \omega t)$ can be associated to a local oscillation of the spacetime curvature, i.e.

$$h^{^{\rm TT}}_{jk,00} \sim -\omega^2 \exp (i \omega t) \sim R_{j0k0} ,$$ (92)

and thus

$$R_{j0k0} = \frac{1}{2} \omega^{2} h^{^{\rm TT}}_{jk} .$$ (93)

To better appreciate the effects of the propagation of a gravitational wave, it is useful to consider the separation between two neighbouring particles $A$ and $B$ on a geodesic motion and how this separation changes in the presence of an incident gravitational wave (see Fig. 6). Note that considering a single particle would not be sufficient to establish the effect of an incident gravitational wave. Its coordinate position does not change at the passage of the wave and it is, indeed, the relative displacement between two adjacent particles that allows for the detection of the measurement.

For this purpose, let us introduce a coordinate system $x^{\hat \alpha }$ in the neighbourhood of particle $A$ so that along the worldline of the particle $A$ the line element will have the form

$$ds^2 = -d\tau^2 + \delta_{\hat i \hat j} dx^{\hat i} dx^{\hat j} + {\cal O}(\vert x^{\hat j}\vert^2) dx^{\hat \alpha } dx^{\hat \beta } ,$$ (94)

where, of course, we are interested exactly in quantifying the ${\cal O}(\vert x^{\hat j}\vert^2)$ terms.

The arrival of a gravitational wave will perturb the geodesic motion of the two particles and produce a nonzero contribution to the geodesic deviation equation. I remind that the changes in the separation four-vector $\boldsymbol {\xi }$ between two geodesic trajectories with tangent four-vector V are expressed through the geodesic deviation equation [cf. equation (32)]

$$V^{\gamma} V^{\beta } \xi^{\alpha }_{ ;\beta \gamma} = -R^{\alpha }_{ \beta \gamma \delta} V^{\beta } \xi^{\gamma} V^{\delta} ,$$ (95)

or, equivalently, as

$$V^{\gamma} V^{\beta } \left(\frac{D^2 \xi^{\alpha }}{D \tau^2} \r... ...t) = -R^{\alpha }_{ \beta \gamma \delta} V^{\beta } \xi^{\gamma} V^{\delta} .$$ (96)

Indicating now with $n^{\hat j}_{_{\rm B}} \equiv x^{\hat j}_{_{\rm B}} - x^{\hat j}_{_{\rm A}} = x^{\hat j}_{_{\rm B}}$ the components of the separation three-vector in the positions of the two particles, the geodesic deviation equation (96) can be written as

Figure: Schematic diagram for the changes in the separation vector between two particles $A$ and $B$ moving along geodesic trajectories produced by the interaction with a gravitational wave propagating along the direction ${\vec k}$.

$$\frac{D^2 x^{\hat j}_{_{\rm B}}}{D \tau^2} = -R^{\hat j}_{ 0 \hat k 0} x^{\hat k}_{_{\rm B}} .$$ (97)

A first simplification to these equations comes from the fact that around the particle $A$ the affine connections vanish (i.e. $\Gamma^{\hat j}_{{\hat \alpha } {\hat \beta }}=0$) and the covariant derivative in (97) can be replaced by an ordinary total derivative. Furthermore, because in the $TT$ gauge the coordinate system $x^{\hat \alpha }$ moves together with the particle $A$, the proper and the coordinate time coincide at first order in the metric perturbation [i.e. $\tau=t$ at ${\cal O}(h^{^{\rm TT}}_{\mu \nu })$]. As a result, equation (97) effectively becomes

$$\frac{d^2 x^{\hat j}_{_{\rm B}}}{d t^2} = \frac{1}{2}\left(\frac{... ...{\rm TT}}_{{\hat j} {\hat k}}} {\partial t^2} \right) x^{\hat k}_{_{\rm B}} ,$$ (98)

and has solution

$$x^{\hat j}_{_{\rm B}}(t) = x^{\hat k}_{_{\rm B}}(0) \left[ \delta_{{\hat j} {\hat k}} + \frac{1}{2} h^{^{\rm TT}}_{{\hat j} {\hat k}}(t)\right] .$$ (99)

Equation (99) has a straightforward interpretation and indicates that, in the reference frame comoving with $A$, the particle $B$ is seen oscillating with an amplitude proportional to $h^{^{\rm TT}}_{{\hat j} {\hat k}}$.

Note that because these are transverse waves, they will produce a local deformation of the spacetime only in the plane orthogonal to their direction of propagation. As a result, if the two particles lay along the direction of propagation (i.e. if ${\vec n} \parallel {\vec k}$), then

$$h^{^{\rm TT}}_{{\hat j} {\hat k}} x^{\hat j}_{_{\rm B}}(0) \propto h^{^{\rm TT}}_{{\hat j} {\hat k}} \kappa^{\hat j}_{_{\rm B}}(0) = 0 ,$$ (100)

and no oscillation will be recorded by $A$ [cf. equation (87)]

Let us now consider a concrete example and in particular a planar gravitational wave propagating in the positive $z$-direction. In this case the only nonzero metric functions will be

$$h^{^{\rm TT}}_{xx} = - h^{^{\rm TT}}_{yy} = {\Re \left\{ A_{+} \exp[-i\omega(t-z)]\right\} } ,$$ (101)

$$h^{^{\rm TT}}_{xy} = h^{^{\rm TT}}_{yx} = {\Re \left\{ A_{\times} \exp[-i\omega(t-z)]\right\}} ,$$ (102)

where $A_{+}$ and $A_{\times}$ represent the two independent modes of polarization.

As in classical electromagnetism, in fact, it is possible to decompose a gravitational wave in two linearly polarized plane waves or in two circularly polarized ones. In the first case, and for a gravitational wave propagating in the $z$-direction, the polarization tensors $+$ (``plus'') and $\times$ (``cross'') are defined as

$${\mathbf e}_{+} \equiv {\vec e}_{x} \otimes {\vec e}_{x} - {\vec e}_{y} \otimes {\vec e}_{y} ,$$ (103)

$${\mathbf e}_{\times} \equiv {\vec e}_{x} \otimes {\vec e}_{x} + {\vec e}_{y} \otimes {\vec e}_{y} .$$ (104)

The deformations that are associated with these two modes of linear polarization are shown in Fig. 7 where the positions of a ring of freely-falling particles are schematically represented at different fractions of an oscillation period. Note that the two linear polarization modes are simply rotated of $\pi/4$.

Figure: Schematic deformations produced on a ring of freely-falling particles by gravitational waves that are linear polarized in the ``$+$'' (``plus'') and ``$\times$'' (``cross'') modes. The continuous lines and the dark filled dots show the positions of the particles at different times, while the dashed lines and the open dots show the unperturbed positions.

In a similar way, it is possible to define two tensors describing the two states of circular polarization and indicate with ${\bf e}_{_{\rm R}}$ the circular polarization that rotates clockwise (see Fig. 8)

Figure: Schematic deformations produced on a ring of freely-falling particles by gravitational waves that are circularly polarized in the $R$ (clockwise) and $L$ (counter-clockwise) modes. The continuous lines and the dark filled dots show the positions of the particles at different times, while the dashed lines and the open dots show the unperturbed positions.

$${\bf e}_{_{\rm R}} \equiv \frac{{\bf e}_{+} + i {\bf e}_{\times}}{\sqrt 2} ,$$ (105)

and with ${\bf e}_{_{\rm L}}$ the circular polarization that rotates counter-clockwise (see Fig. 8)

$${\bf e}_{_{\rm L}} \equiv \frac{{\bf e}_{+} - i {\bf e}_{\times}}{\sqrt 2} .$$ (106)

The deformations that are associated to these two modes of circular polarization are shown in Fig. 8