[sclarisse]

As others have noted, the speed of light (in a vacuum) will always constant. To be nice and pedantic, you can always slow it down by making that light go through a medium, where it will go slower (e.g. any prism that makes a pretty rainbow). It’s a cheat of sorts, but the speed of light isn’t some special property of light: it’s a property of spacetime. There is a maximum speed of anything at all, and unimpeded light goes this speed. For most astrophysics purposes this doesn’t worry us much, as space is essentially empty.

You can also “slow light down” by just making it go further: a few clever mirrors will do this easily. This is even more of a cheat, as the light itself isn’t any slower, it just gets where it was going a little later because it went further.

In a sense that’s what’s happening here: the spacetime is being stretched on one axis and squeezed on another, as gravitational waves pass through it. It isn’t by much, which is why we need a whole galaxy to measure it.

[raattgift]

We don't need a whole galaxy; pulsar timing arrays (PTAs) would be happier if there were stable millisecond pulsars much closer than the ones they're using, as it would control for some uncertainties particularly in "red noise" which arises from the interstellar medium (ISM) as well as the configuration of the neutron star itself. Closer pulsars would have less of an ISM contribution to the red noise (and thus less red noise).

The nearest millisecond pulsar is about 500 light years away. There are closer "classical" pulsars. Most of the millisecond pulsars used by PTA collaborations are a few thousand light years away.

Maybe a better way of thinking about the gravitational wave is that it alters a form of gravitational potential along a fraction of the worldline of an element representing a pulse, causing light to run a little uphill (redshifting) or downhill (blueshifting). cf. the Harvard tower experiment (Pound-Rebka). This doesn't seem very geleneral-relativistic but hey looking for a Poisson equation in the Newtonian approximation goes back to Einstein. The Earth and the array pulsars are moving sufficiently slowly with respect to each other, and low-redshift SMBHBs are OK where (handwave handwave) linearized gravity is OK, so we could work likek Einstein in the Newtonian limit with some care. A more modern approach to calculating a Poisson-equation gravity potential analogue is a technically annoying trip through the spacetime decomposition formalism, but once it's there it's conceptually useful, and for gravitation physicists it can be related to the g-induction and g-field in gravitoelectromagnetism.

There are other ways to describe the elephant <https://en.wikipedia.org/wiki/Blind_men_and_an_elephant> though; an equivalence to acceleration (one can change the direction component of the velocity vector for electromagnetic waves); calculating time-dependent (and here that means depends on the orbital phase of the supermassive black hole binary) null geodesics; and so forth. This is the gift of general covariance: we can choose practically whatever coordinates we want, which highlights different components of the covariant tensors used in relativistic theories (e.g. the electromagnetic field tensor F_{\mu\nu} or the Einstein curvature tensor G_{\mu\nu}).

Finally, the concept of https://en.wikipedia.org/wiki/Retarded_time would probably be useful in this thread and others like it that try to understand how what we see here-and-now arose in the past there-and-then. "Perhaps surprisingly - electromagnetic fields and forces acting on charges depend on their history, not their mutual separation" -- the same is true for gravitational fields and forces acting on masses, and this is what drives some of the above Poisson-equation thinking and the GEM formal analogy to the electromagnetic Heaviside-Maxwell equations.

PS: for experts, re my handwave and GEM, see Ruggiero 2022, https://link.springer.com/article/10.1007/s10714-022-02983-8 (open access).