Question: Why do we need to understand gravitational waves to understand mass?
About the article "Gravitational wave", section "Radiation from other sources".

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 of neutrinos and gravitational waves.
From the near-simultaneous observations of neutrinos and photons from SN 1987A, we know that the Shapiro delay for high-energy neutrinos is the same as that for photons to within 10% (consistent with recent estimates of the neutrino mass which imply that those neutrinos were moving at very close to the speed of light).
Since gravitational waves have not been directly detected, we don't have any data on the Shapiro delay for gravitational waves.
In general relativity and other metric theories of gravity, the Shapiro delay for gravitational waves is expected to be the same as that for light (and neutrinos).
However in theories such as Teves and other modified GR theories which reproduce Milgrom's law and avoid the need for dark matter, the Shapiro delay for gravitational waves is much smaller than that for neutrinos or photons. 
Since formula_4 in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy.
But the gravitational field can do work, so we must expect the gravitational field itself to possess energy, and it does.
However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest".
The fact that the gravitational field itself possesses energy yields a way to understand the nonlinearity of the Einstein field equation: this gravitational field energy itself produces more gravity.
This means that the gravitational field outside the Sun is a bit "stronger" according to general relativity than it is according to Newton's theory. 
 1953: P. C. Vaidya Newtonian time in general relativity, Nature, 171, p260.
1956: John Lighton Synge publishes the first relativity text emphasizing spacetime diagrams and geometrical methods, 1957: Felix A. E. Pirani uses Petrov classification to understand gravitational radiation, 1957: Richard Feynman introduces sticky bead argument,  1957 – John Wheeler discusses the breakdown of classical general relativity near singularities and the need for quantum gravity 1959: Pound–Rebka experiment, first precision test of gravitational redshift, 1959: Lluís Bel introduces Bel–Robinson tensor and the Bel decomposition of the Riemann tensor, 1959: Arthur Komar introduces the Komar mass, 1959: Richard Arnowitt, Stanley Deser and Charles W. Misner developed ADM formalism.  
  ""Our belief is that we have to hold up the standard of abstinence, and we do that in all of our teaching about sexuality by saying that sexual activity outside of marriage is wrong.
Now that's a very high bar to set and I understand that.
And I don't regret that – I subscribe to it and I teach it.
I understand why people find it difficult and disagree with it.
I understand why they do.
I don't agree with them... What I would say is that people who disagree with us can disagree without being disagreeable.""
A gravitational-wave detector is any device designed to measure gravitational waves, tiny distortions of spacetime that are predicted by Einstein's theory of general relativity.
The existence of gravitational radiation is a prediction of general relativity.
Gravitational waves are perturbations in the curvature of spacetime caused by accelerated masses.
Since the 1960s gravitational-wave detectors have been built and constantly improved.
The present-day generation of resonant mass antennas and laser interferometers has reached the necessary sensitivity to detect gravitational waves from sources in the Milky Way.
As of early 2014, no direct detection of gravitational waves has yet been accomplished, although a number of experiments have now added to the evidence that gravitational waves are more than mathematical anomalies of relativity calculations (ref.
the 1993 Nobel Prize in Physics).
On 17 March 2014, astronomers at the Harvard-Smithsonian Center for Astrophysics announced the apparent detection of gravitational waves, which, if confirmed, may provide strong evidence for inflation and the Big Bang.