Plane Gravitational Waves

A plane gravitational wave is the local Fourier building block of a freely propagating weak field. Its null wave vector fixes a two-dimensional transverse screen; the trace-free symmetric tensors on that screen form a two-dimensional polarization space. Plane waves are not realistic global fields—they have infinite transverse extent and energy—but they make propagation, polarization, phase, and convention choices exact enough to serve as a reference calculation throughout the volume.

Required background. Weak-Field Gravity is used for the linearized vacuum Einstein equation and gauge content. Waves, Green Functions, and Geometric Optics is used for Fourier modes, null characteristics, and superposition.

Helpful background. Einstein Equations supplies the vacuum field law being linearized. Equivalence Principles and Gravitational Measurement is used when the coordinate perturbation is translated into local tidal measurements.

Null propagation from the field equation

In a vacuum region, Lorenz gauge reduces the trace-reversed perturbation to

$$\Box\bar h_{\mu\nu}=0, \qquad \partial^\mu\bar h_{\mu\nu}=0.$$

For a complex Fourier representative

$$\bar h_{\mu\nu}=A_{\mu\nu}e^{ik_\rho x^\rho},$$

the physical field is its real part. Substitution gives

$$k^\rho k_\rho=0, \qquad k^\mu A_{\mu\nu}=0.$$

Thus the wave covector is null and the Lorenz-gauge amplitude is transverse in spacetime. Residual gauge freedom removes the remaining time, longitudinal, and trace pieces, leaving a TT amplitude on the transverse screen.

Choose propagation toward increasing $z$. With the site time dependence $e^{-i\omega t}$,

$$k^\mu=(\omega,0,0,\omega), \qquad k_\mu=(-\omega,0,0,\omega),$$

so

$$e^{ik_\rho x^\rho}=e^{-i\omega(t-z)}=e^{-i\omega u}, \qquad u=t-z.$$

The null condition follows immediately: $k^2=-\omega^2+\omega^2=0$. Reversing the direction changes $u$ to $t+z$; changing only the sign in the spatial phase without changing the declared propagation direction is inconsistent.

The plane-wave construction and its residual-gauge reduction are given in Flanagan and Hughes 2005, §2.1, eqs. (2.17)–(2.28), pp. 6–8, PDF.

Plus and cross tensors

Let $(\boldsymbol e_x,\boldsymbol e_y)$ be an oriented orthonormal basis of the screen perpendicular to $\boldsymbol n=\boldsymbol e_z$. Define

$$e^+_{ij}=e_{x i}e_{x j}-e_{y i}e_{y j}, \qquad e^\times_{ij}=e_{x i}e_{y j}+e_{y i}e_{x j}.$$

Both tensors are transverse and traceless, and their contractions obey

$$e^A_{ij}e^{B,ij}=2\delta^{AB}, \qquad A,B\in\{+,\times\}.$$

The most general plane TT wave traveling in the $+z$ direction is therefore

$$h^{\mathrm{TT}}_{ij}(u) =h_+(u)e^+_{ij}+h_\times(u)e^\times_{ij} =\begin{pmatrix} h_+&h_\times&0\\ h_\times&-h_+&0\\ 0&0&0 \end{pmatrix}.$$

Any twice differentiable functions of $u=t-z$ solve the homogeneous wave equation because $\Box f(u)=0$. A monochromatic wave is the special case

$$h_A(u)=\Re\!\left[H_A e^{-i\omega u}\right], \qquad A\in\{+,\times\}.$$

The amplitudes $H_A$ can be complex because their relative phase contains physical polarization information; only the final metric perturbation is real.

Spin two and polarization states

Introduce the complex strain

$$h=h_+-i h_\times.$$

Define $m^i=(e_x^i+i e_y^i)/\sqrt2$, so that $h=h^{\mathrm{TT}}_{ij}\bar m^i\bar m^j$. Under the phase rotation $m^i\mapsto e^{i\psi}m^i$, the components transform as

$$h\mapsto e^{-2i\psi}h.$$

The factor of two means that rotating the screen convention by $\pi$ returns the same plus/cross components, while a quarter-turn changes the sign of both. A rotation by $\pi/4$ interchanges the plus and cross axes up to sign. This spin-weight statement is about components under a basis change; it does not add or remove a physical degree of freedom.

Several familiar polarization states are choices of the two complex amplitudes:

• linear polarization: $H_+$ and $H_\times$ have the same phase, so a screen rotation can place the wave entirely in one real tensor;
• circular polarization: the amplitudes have equal magnitude and a relative phase of $\pm\pi/2$;
• elliptical polarization: the generic unequal-magnitude, nonzero-phase case.

The labels “right” and “left” for circular waves depend on whether one looks along or opposite the propagation direction and on the orientation of the screen. A calculation should state that viewing convention rather than attach a handedness label to the sign alone.

Curvature and physical equivalence

For this TT representative,

$$R_{0i0j}^{(1)}=-\frac12\ddot h^{\mathrm{TT}}_{ij}.$$

Two metric perturbations related by a linearized gauge transformation have the same $R^{(1)}_{\alpha\beta\gamma\delta}$. Two plus/cross pairs related only by a rotation of the screen also describe the same tensor in different components. By contrast, changing the relative phase or amplitude while holding the screen fixed changes the tidal history and therefore the physical polarization state.

The tidal-response page solves the motion of test bodies under this curvature. Later, Plus and Cross Polarizations generalizes the screen construction and separates the two GR tensor modes from the kinematically possible modes of other metric theories.

This source-free plane wave is a propagation and polarization baseline. It does not determine the nonlinear compact-binary motion that generated a signal, and it contains no detector noise or inference model. Those inputs belong respectively to Relativistic Two-Body Dynamics and Gravitational-Wave Astronomy.

Common pitfalls

Confusing $k^\mu$ with $k_\mu$. With signature $(-,+,+,+)$, lowering the time index changes its sign. The phase $k_\mu x^\mu=-\omega t+kz$ then identifies propagation toward $+z$.

Treating complex strain as a complex metric. Complex notation packages two real components and their phases. The physical perturbation is real.

Assigning absolute meaning to plus and cross. They depend on the oriented screen basis. The two-dimensional tensor subspace and its curvature are invariant; its named components are not.

Exercises

1. Arbitrary waveform. Show directly that $f(t-z)$ solves the flat-space homogeneous wave equation for any twice differentiable function $f$.

Solution

Let $u=t-z$. Then $\partial_t^2 f=f''(u)$ and $\partial_z^2 f=f''(u)$, while the transverse derivatives vanish. Hence

$$\Box f=-\partial_t^2f+\nabla^2f=-f''+f''=0.$$

The result depends only on the null character of surfaces $u=\text{constant}$, not on monochromaticity. The same cancellation holds component by component for $h_+(u)$ and $h_\times(u)$, so an arbitrary pulse profile can be built by Fourier superposition without changing its propagation direction.

2. Polarization norm. Using the matrices above, verify $e^A_{ij}e^{B,ij}=2\delta^{AB}$ and recover

$$h^{\mathrm{TT}}_{ij}h_{\mathrm{TT}}^{ij} =2(h_+^2+h_\times^2).$$

Solution

The plus tensor has entries $1$ and $-1$, so its squared contraction is $2$. The cross tensor has two off-diagonal unit entries, also giving $2$. Their contraction vanishes because they have support in different matrix entries. Expanding $h_{ij}h^{ij}$ therefore gives $2h_+^2+2h_\times^2$ with no cross term. The result is unchanged by a rotation of the screen basis.

References

• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.