The relation between Gauge theories and physical constants can be summed up as follows [1, verbatim]:
In gauge theories, the physical content is gauge invarient: the physical properties of a configuration do not change under a gauge transformation...
Gravitational waves are physical entities which can be measured both indirectly (e.g. the Hulse and Taylor binary pulsar ) and recently directly (i.e. LIGO).
Let the presence of (linearized) physical gravitational waves depends on the Gauge chosen - you need to use the harmonic gauge else you simply get waves that travel at the speed of thought. Eddington had similar arguments in is 1923 book .
I am confused, therefore about the application of gauges here. Our choice of gauge changes the physics. You can choose one gauge and get gravitational waves traveling at the speed of light, but choose another and you don't  i.e. our gauge conditions don't seem like gauge conditions, but cause actual physical difference between solutions. Please can someone explain the resolution and how it is applicable to think of this situation.
 Hemker, P.W. and Wesseling, P. eds., 2012. Multigrid Methods IV: Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6–9, 1993 (Vol. 116). Birkhäuser. (p63)
 Weisberg, J.M. and Taylor, J.H., 2005, July. The relativistic binary pulsar b1913+ 16: Thirty years of observations and analysis. In Binary Radio Pulsars (Vol. 328, p. 25).
 Eddington, A.S., 1930. Mathematical theory of relativity. General Books LLC.