For a continuous wave of laser light, a falling edge of the sin wave (of the electric field amplitude) from the laser is split to two falling edges (at the splitter mirror) and then go down the two arms. The falling edge of the sin wave behaves like the leading edge of your pulse for the time delay argument.

The second part of your question in which a constant c and oppositely strained $\lambda_1$, $\lambda_2$ yield $\omega_1 \neq \omega_2$ has also puzzled me, but for a different reason. Whereas you conclude it eliminates any GW phase shift between the two paths, I can not understand how the passive splitter mirror in the presence of a constant GW strain turns one input frequency of laser light into two different output frequencies! I asked this in a Physics Stack question: [How Does LIGO’s Splitter Mirror Cause Two Different Frequencies When a GW is Present?][1], but have received no answers.

Because of the splitter mirror, I conclude that in the presence of the GW, $\omega_1 = \omega_2$. Then either:

1. c is the same along both arms and therefore $\lambda_1=\lambda_2$. This contradicts the standard argument on the LIGO website and in LIGO talks which say $\lambda_1 \ne\lambda_2$ because $\lambda$ strains like the arm lengths (I guess by common sense since I don't know a GR argument for it). However, even if $\lambda_1=\lambda_2$, the laser light leading edge will take different times along the two arms, and the interference pattern will change … therefore LIGO can detect GWs.

OR

2. $\lambda_1 \ne\lambda_2$ and therefore $c_1 \ne c_2$. It is strange having two speeds of light since we are used to saying "the speed of light is the same in all reference frames … that are related by a Lorentz transformation that leaves the Minkowski metric unchanged". The strains caused by GWs and Schwarzschild masses change the Minkowski metric (and therefore c) as evidenced by the Shapiro delay, where a far away observer sees light slow down as it passes near the Sun. If $c_1 \ne c_2$, then by an argument slightly similar to yours, one can show that light goes faster along the lengthend arm and slower along the shortend arm such that the laser light leading edge will take the same time along both arms, and the interference pattern will not change … therefore LIGO can not detect GWs.

Since LIGO appears to have detected GWs, argument (1) seems to be correct, and $\lambda$ does not get strained by a GW. The GW leaves c, $\omega$, and $\lambda$ the same along both arms, and just the length of the arms change. [1]: How Does LIGO’s Splitter Mirror Cause Two Different Frequencies When a GW is Present?

For a continuous wave of laser light, a falling edge of the sin wave (of the electric field amplitude) from the laser is split to two falling edges (at the splitter mirror) and then go down the two arms. The falling edge of the sin wave behaves like the leading edge of your pulse for the time delay argument.

The second part of your question in which a constant c and oppositely strained $\lambda_1$, $\lambda_2$ yield $\omega_1 \neq \omega_2$ has also puzzled me, but for a different reason. Whereas you conclude it eliminates any GW phase shift between the two paths, I can not understand how the passive splitter mirror in the presence of a constant GW strain turns one input frequency of laser light into two different output frequencies! I asked this in a Physics Stack question: How Does LIGO’s Splitter Mirror Cause Two Different Frequencies When a GW is Present?, but have received no answers.

Because of the splitter mirror, I conclude that in the presence of the GW, $\omega_1 = \omega_2$. Then either:

1. c is the same along both arms and therefore $\lambda_1=\lambda_2$. This contradicts the standard argument on the LIGO website and in LIGO talks which say $\lambda_1 \ne\lambda_2$ because $\lambda$ strains like the arm lengths (I guess by common sense since I don't know a GR argument for it). However, even if $\lambda_1=\lambda_2$, the laser light leading edge will take different times along the two arms, and the interference pattern will change … therefore LIGO can detect GWs.

OR

2. $\lambda_1 \ne\lambda_2$ and therefore $c_1 \ne c_2$. It is strange having two speeds of light since we are used to saying "the speed of light is the same in all reference frames … that are related by a Lorentz transformation that leaves the Minkowski metric unchanged". The strains caused by GWs and Schwarzschild masses change the Minkowski metric (and therefore c) as evidenced by the Shapiro delay, where a far away observer sees light slow down as it passes near the Sun. If $c_1 \ne c_2$, then by an argument slightly similar to yours, one can show that light goes faster along the lengthend arm and slower along the shortend arm such that the laser light leading edge will take the same time along both arms, and the interference pattern will not change … therefore LIGO can not detect GWs.

Since LIGO appears to have detected GWs, argument (1) seems to be correct, and $\lambda$ does not get strained by a GW. The GW leaves c, $\omega$, and $\lambda$ the same along both arms, and just the length of the arms change. Because the length of the arms have changed, I have been thinking in the Local Lorentz gauge.