Plane-fronted gravitational wave

Forthe only potentially nonzero Ricci component isThe vacuum equation is therefore the transverse Laplace equation.

For the plane-fronted gravitational wave $d{s}^2=2dudv+d{\mathbf{x}}^2+{\mathbf{x}}^{T}M(u)\mathbf{x}d{u}^2$, synchronized neighboring geodesics parametrized by $u$ have transverse separation $\mathbf{\xi }$ satisfying the displayed equation. Along the central timelike geodesic $(u,v,\mathbf{x})=(\tau ,-\tau /2,0)$ the coordinate basis is parallel transport, so this is also the transverse Jacobi field equation in a transported frame. Vacuum requires $\mathrm{tr}M=0$, producing equal and opposite instantaneous transverse tidal acceleration eigenvalues. The ordinary coordinate equation ${\ddot{\xi }}^v=0$ holds on that central reference, but generally fails on off-axis reference curves.

A homogeneous quadratic polynomial in the transverse coordinates $x,y$ is a harmonic function exactly when the sum of its $x^2$ and $y^2$ coefficients is zero. Thus it has the displayed form. For the Minkowski metric $dudv+d{x}^2+d{y}^2$, a metric perturbation $h=\psi d{u}^2$ has zero trace, zero divergence and $\square h=0$, since ${\partial }_v\psi =0$ and ${\partial }_x^2\psi +{\partial }_y^2\psi =0$. The two coefficient functions describe the two gravitational wave polarizations.

For $d{s}^2=-2dudv+d{x}^2+d{y}^2-2H(u,x,y)d{u}^2$, the nonzero connections are ${\Gamma }_{uu}^v=H_u$, ${\Gamma }_{ux}^v=H_x$, ${\Gamma }_{uy}^v=H_y$, ${\Gamma }_{uu}^x=H_x$, and ${\Gamma }_{uu}^y=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hd{u}^2$ convention.

For a Brinkmann plane wave whose transverse profile is quadratic, Killing fields obtained from solutions of transverse oscillator equations act transitively on every constant-retarded-time wavefront. Each wavefront is therefore a homogeneous null hypersurface.

Rosen coordinates put a plane wave into the transverse-metric formFor diagonal $g_{AB}=\mathrm{diag}(f^2,g^2)$, the only potentially nonzero Ricci tensor component is $R_{uu}=-f^{\prime \prime }/f-g^{\prime \prime }/g$.

For $g=-d{t}^2+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2+d{z}^2$, $u=t-z$, with nowhere-zero smooth $A,B$, take $e^0=dt$, $e^1=Adx$, $e^2=Bdy$, $e^3=dz$, $\ell =du$. In signature $(-,+,+,+)$, Cartan's first structure equation and metric compatibility give the lower-index connection 1-forms ${\omega }_{01}={\omega }_{13}=-(A^{\prime }/A)e^1$ and ${\omega }_{02}={\omega }_{23}=-(B^{\prime }/B)e^2$. Cartan's second structure equation gives ${\Theta }_{01}={\Theta }_{13}=-(A^{\prime \prime }/A)\ell \wedge e^1$ and ${\Theta }_{02}={\Theta }_{23}=-(B^{\prime \prime }/B)\ell \wedge e^2$. All other independent curvature 2-forms vanish: in particular the $12$ component cancels between the time and longitudinal terms. Contracting $R^a{}_{bad}$ yields $\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$. Thus the Vacuum Einstein equations are $A^{\prime \prime }/A+B^{\prime \prime }/B=0$, while nonzero $A^{\prime \prime }/A=-B^{\prime \prime }/B$ gives opposite transverse tidal curvatures and plus polarization relative to these axes.

A gravitational-wave pulse has curvature supported in a bounded interval of retarded time. Spacetime can be flat both before and after the pulse even though freely falling bodies retain a displacement or relative velocity produced while the pulse passed.

The gravitational-wave memory effect is a lasting change in the relative configuration of freely falling test bodies after a transient gravitational wave has passed.

Displacement memory is a persistent change in the separation of freely falling bodies after a gravitational-wave pulse.

Velocity memory is a persistent change in the relative velocity of freely falling bodies after a gravitational-wave pulse.