Waves in Space...and Time!!...Help

Matt, a well phrased question, and boy are you in famous company. From the Ancients through to Gallileo, Newton, Leibnitz, Mach, Poincare, Einstein, Lorentz .....

Is space a separately defined entity from it contents? ( See.... I've already begged the question with my 'natural' intuition ). Would the laws of physics hold if the universe was empty? This underlies many of those 'imagine a ship in space far from any material influence' type thought experiments that Relativitians love.

I think that resolves to this:
- Yes, it would be handy to be outside of space and time to best observe and resolve this stuff.
- We're not.
- One can create a working hypothesis on whatever suitable basis pleases us.
- You may have yours and I mine, but if they explain and predict in exact correspondence, they are equivalent and of equal scientific merit. I ignore any issues of ease of use, teachability, calculability etc for the moment. Have a chat to Occam if you like, and stroll through the gallery of 'logical positivism' while you're at it.

The 'modern' view is pretty well as follows. To describe events we have a reference frame with a metric. The specification of that frame includes origins for it's co-ordinates - 3 spatial numbers ie. an 'over there' and 1 time number ie. a 'then'. It also assumes stuff like symmetry, orthogonality or independence of co-ordinates and similiar operational stuff which we'll ignore. So you start your clock 'now', and mark chalk on the ground somewhere 'here' and measure all else with regard to that. But we need a scale, and for that you need a measuring stick - the metric. With more than one event to measure ( be boring with only one! ) we want a 'distance' between events which are separated in space and/or time. Examples: (A) An electron in my town yesterday, and a positron in my town today ( same place but different times ). (B) An electron in my town today and a positron in your town today ( same time but different places ). (C) An electron in my town yesterday, and a positron in your town today ( different places and different times ). (D) An electron in my town today, and a positron in my town today ( same place and same time ). You get the idea, and yes I've ignored comparison and synchronizing issues.

Now, we normally think of measurement like I said above - I have a clock for time and I have a ruler for space. Now cop this - a light beam can serve both purposes - and it doesn't need calibrating. Take a Lorentz type metric ( ^ is exponent, * is multiply ):

L(x, y, z, t) = x^2 + y^2 + z^2 - c^2*t^2

where x, y, x and t are the respective space and time DIFFERENCES between TWO events measured with respect to a GIVEN reference frame as specified above. So you in your frame ( different to mine in detail but constructed conceptually as above ) do you're x's, y's ,x's and t's for the SAME two events. You calculate your value of L = L(MATT), and I'll do mine L = L(MIKE). Guess what? Yup, the're the same ( in a Special Relavistic context anyway ). Note we used the same value of 'c' even though our x's etc differ. That calibrates your metric ( L ) to mine.
Now imagine I'm a ( or THE if you like ) multidimensional God ( MDG ) strolling around my multidimensional backyard looking 'down' at us toolheads at EAH. I could, if so minded, whip out the Lorentz ruler that's hanging off my toolbelt ( and I have others ... ) and measure away to see if we are up to spec ( shades of the Reverend Abbott and Flatland here ). We can't, but we can conceptually 'construct' in effect the same MDG ruler out of our clocks and measuring sticks. MDG uses the same 'c' too.
So where have I taken you, if you're still reading....
(A) I'll be a nasty MDG. I'll choose a cube 4km on a side, with you in it, stop your clock, expand/shrink everything within PROPORTIONALLY including you and your metrics ( and thus the means of constructing mine ) and then restart your clock ( faster/slower if I like ), shoot the breeze with my pals for a while of both your and my time, then stop your clock again, revert the above scaling ( time included ) and then restart your clock. Being an MDG then my clock doesn't stop. Provided you have no way of comparing outside of that cube in space and outside of that time interval you will be unaffected in your measurements, and hence in any construction of my metric compared to the case of if I had left you alone. Read that again.
(B) I'll be a real nice MDG. Leave you alone.
(C) I'll be a 50/50 MDG, not stopping your clock but progressively deforming the cube over some of your time ( and my time ). My Lorentz ruler will measure scale changes, you'll ( by construction ) deduce that too.
If case A has/will happened/happen we never did/will know ( discard ).
Case B I cannot prove, as absence of evidence is not evidence for absence ( ditto ).
For us EAH toolheads, case A and B are the same. For MDG they are different.
Case C is gravitational wave measurement. At any instant during a wave's passage one part of the detector is deforming differently ( in space AND time ) from the other. That's why there are TWO orthogonal arms to COMPARE. I'm using two ( identically constructed space-time metrics ) rulers pointing in DIFFERENT directions. The light-beams ARE the rulers and each is unalterably calibrated to 'c'.
As Ben says, count the wave phases, which is effectively done by allowing the returning light from each arm to interfere ( each counts the other ). Isn't light helpful like that? So we watch and wait and see if they quarrel....
As Mark says time compare or distance compare as you please ( c = wavelength*frequency ).

I've ignored a blizzard of confounding real life stuff, and probably mass murdered vast pages of technical definitions ....