Theory of relativity

Gravitational waves

Gravitational waves are propagating spacetime disturbances, emitted strongly by changing, nonspherical accelerating mass distributions.

h_{ij}^{\mathrm{TT}}(t-z/c)

Definition: Key quantity

The metric and field tensors have coordinate components, but physical predictions must be invariant. The central relation is $h_{ij}^{\mathrm{TT}}(t-z/c)$ .

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Reading the equation

Gravitational waves are propagating spacetime disturbances, emitted strongly by changing, nonspherical accelerating mass distributions.

Example: Limit and interpretation

Consider a weak field, low speeds, and a region far from the source. General relativity should approach the appropriate Newtonian description; near a horizon or in a strong field this approximation fails.

Solution

This is the correspondence principle: the newer theory recovers tested results of the older one in its domain of validity.

Example: Estimating a gravitational wavelength

A nearly circular binary has orbital period $T=0.20\,s$ . In the leading quadrupole approximation, the dominant gravitational-wave frequency is twice the orbital frequency. Estimate the wave frequency and its vacuum wavelength using $c=3.0\times10^8\,m/s$ .

Solution

$f_{orb}=1/T=5\,Hz$ , so $f_{GW}\approx2f_{orb}=10\,Hz$ . The wavelength is $\lambda=c/f_{GW}\approx3.0\times10^7\,m$ . As radiation carries energy away and the orbit shrinks, $T$ decreases and the signal sweeps upward in frequency, producing a chirp.

In the wave zone, the transverse-traceless gauge describes two polarizations, $h_+$ and $h_\times$ . They drive relative distortions of freely falling test masses in orthogonal patterns; an interferometer measures the tiny differential change in arm lengths. A strong source needs a changing mass quadrupole, since conservation laws forbid gravitational monopole and dipole radiation in general relativity. The observed waveform encodes chirp mass, eccentricity, and spin, but inferring these parameters requires source modeling and detector calibration.

In the wave zone, strain falls approximately as the inverse distance from its source, making signals tiny at Earth. Seismic, thermal, and quantum noise turn extraction into a statistical problem. Distant detectors help distinguish astrophysical events from local disturbances and locate the source.

Quick check

What does the central expression in this topic describe?

Which is the most appropriate interpretation?

References

Sean Carroll (2019). Spacetime and Geometry
Misner, Thorne, Wheeler (1973). Gravitation