While time as a dimenauon can be seen as a mathematical formality (it manifests as a different physical quantity in our observations), it does behave much (but not exactly) like the spatial dimensions.

The entirety of GR, especially the space curvature part is formulated without embedding space and time into a higher dimensional space of any kind. This avoids any speculation about extra dimensions "around" our universe. It also causes the enormous mathematical complexity of GR.

Back to the topic of gravitational waves: these arise out of Einsteins field equation which describes space-time curvature as a result of the distribution of mass and energy in space. These equations show the property that certain small deviations of curvature from their resting state propagate through space as waves. These curvature changes can be measured as changed in physical length of solid objects. This is what LIGO achieved. There is no room in current theories for any different kind of propagation medium for gravitational waves.

I seems to me that you have some difficulties understanding SR and GR. This is understandable because both describe phenomena that are not directly observable in our everyday lives.

Chapter 2

The Formulation of Quantum Mechanics

2.1

Basic Theoretical Concepts

Every physical theory involves some basic physical concepts, a mathematical formalism, and set of correspondence rules which map the physical concepts onto the mathematical objects that represent them. The correspondence rules are first used to express a physical problem in mathematical terms. Once the mathematical version of the problem is formulated, it may be solved by purely mathematical techniques that need not have any physical interpretation. The formal solution is then translated back into the physical world by means of the correspondence rules.

I cannot measure space-time with ruler, not touch it, nor walk 10 seconds ago, nor curve it. It's just mathematical abstraction. When we translate space-time back into physics, we will have coordinates of points in 3D space and time points, which we can measure with ruler and clock.

Same for gravity waves. If we want to translate this back into physics, we will need to reinterpret results of Michelson experiment. He has two predictions: 1) ether is stationary, 2) ether is attached to planets and objects like atmosphere (see Master of Light book). He failed to find ether wind, thus theory of static ether was abandoned. But he also struggled to find any deviations of light travel time, so that led Einstein to think that c is constant, so he developed his GR theory. Michelson-Morley experiment was redone with precision of up to 1E-17 (distance to Alpha-Centaur is just 4E16m), which was good for GR. But then LIGO found that speed of light in vacuum is NOT constant. Thus, GR is not valid anymore. So we should return to theory of Ether, or you need carefully explain to me why you still think that c==constant and GR is valid.

I understand that you may be invested heavily into learning of mainstream theories, but every theory has limited life. If you think that I will change my mind if I will spend some more years on learning of SR and GR, then I should inform you that I think that formulas in these theories are mostly correct, but they lack interpretation, which lead to lack of intuition. I tested my theory on students and it's working well. I can explain what Ether, double slit experiment, atom, and many other things in short time. I hope, with intuition and formulas they will do much more than with just formulas.