https://www.kceta.kit.edu/grk1694/img/01_Theory.pdf
(Kostas D. Kokkotas, 2016)
The metric of time g₀₀ undulates in the solution, and there are also other nonzero values on the row 0 and the column 0 of the metric tensor g.
Then people do a change of coordinates:
x -> x + ξ(x).
A suitable function ξ makes the metric tensor simpler, and time becomes orthogonal to spatial coordinates:
-1 0 0 0
0 1 + h+ h× 0
0 h× 1 - h+ 0
0 0 0 1
Let us call the above nice metric g+×. It is the transverse traceless (TT) gauge.
The (small) value h+ is the amplitude of the + polarized gravitational wave, and h× is the amplitude of the × polarized wave.
The coordinate change ξ is called a gauge transformation. It does not affect any physical phenomena.
However, the change of coordinates assumes that we have the entire Minkowski space filled with a plane wave, and that the space does not contain any matter.
Suppose that the gravitational wave encounters an infinitely rigid wall, inside which we use the standard Minkowski coordinates and the standard Minkowski metric. If we try to extend the coordinate transformation inside the wall, the metric no longer is orthogonal inside the wall. That is not what we want.
When the wave hits the wall, we can measure the "true" metric of the wave from the pressures it induces inside the wall. The "true" metric is not affected by a coordinate transformation.
We can make gravitational waves look like the TT metric g+× above, but the "true" metric is revealed when the wave hits obstacles. We cannot glue the metric g+× in a nice way into the "natural" metric which we would like to use inside the obstacle.
Michael J. Koop and Lee Samuel Finn (2013) stress that we are interested in the results of actual physical measurements of gravitational waves. Measurements are gauge and coordinate independent, while the metric depends on the choice of coordinates. They prefer an approach where the geometry of spacetime is specified through the Riemann curvature.
The electromagnetic analogue
Suppose that we have an electromagnetic wave and a small test charge. We can calculate the motion of the test charge from the electric field of the wave.
However, if the test charge is very big, then the electromagnetic wave cannot move it much because that would require more energy and momentum than is available in the wave.
One might use comoving coordinates of small test charges and claim that these are "good" coordinates to use. But the "true" coordinates of space are revealed by putting into space huge charges which the wave does not have enough muscle to move.
A coordinate transformation transforms the gravitational wave, but what does it do to the "interface" to matter?
● ● ) ) ) ) ) ) ) • • •
binary wave matter
black hole
Suppose that we have calculated the wave in some convenient standard coordinates which we also use with a matter obstacle.
Suppose then that we change the coordinates in the wave zone and make the metric of the wave to look nice. We may obtain the TT metric g+×, for example.
But if the "interface" becomes complicated to the coordinates and the metric which we use inside the matter obstacle, then it is not a good idea to change coordinates.
The interface to the binary black hole may become complicated, too.
Are gravitational waves produced by a binary black hole purely spatial and transverse?
The above question appears on the Internet. Most people will answer yes.
However, the answer depends on the coordinates which we use. For an infinite plane wave produced by a binary black hole, if the wave propagates in otherwise empty space, then there exists a TT choice of coordinates which makes the metric of time -1, and time orthogonal to spatial dimensions.
We do not know if such a choice of coordinates exists for a realistic, burst wave produced by a binary black hole merger, even if that burst wave propagates in empty space.
In general, the question if a wave is in the metric of time or in the metric of space depends on the coordinates. If we measure the time signal from a remote clock to undulate in the frequency, we may alternatively interpret that as a result of a change of distance, or a change in the rate of the clock.
Conclusions
When we change coordinates to normalize the metric of time to -1 inside a gravitational wave, and make time orthogonal to spatial dimensions, it is sleight of hand which assumes an infinite plane wave.
What matters is if we can conveniently use the new coordinates to calculate the response of matter to gravitational waves. We need to check if that really is the case with the g+× metric.
Also the change of coordinates may be hard to extend in a nice way to empty space outside the gravitational wave. Outside observers would not see the waves having the nice structure indicated by the g+× metric.
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