Gravitational Wave - What is waving? Two kinds of wave transmission are:


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*Light waves, where a substance (photon) travels as a wave.

*An attached rope, like at the gym, that is "waved" up and down. Here, no substance travels to a new spot, but adjacent parts transmit the energy to others.
QUESTION: Which method do gravitational waves propagate by? 
 A: A gravitational wave is a wave in a tensor field called the “metric of spacetime”. This metric determines the geometry of spacetime by specifying how far apart spacetime events are. The wave just consists of changes at each point in the values of the components of this field. 
It is very similar to a classical electromagnetic wave, which is just changes in the values of components of a different field, one that has nothing to do with spacetime geometry.
A: First, I am not purporting this to be a complete answer to your question, but rather to only offer some simple insight into the nature of classical electromagnetic waves, and by extension, "classical" gravitational waves, a term I am coining.  I am interpreting the intent of your question about gravity waves to be about the concept described by Einstein, which I know is more complicated than my simple explanation here.  However, it appears that you have a common misconception of electromagnetic waves or light as consisting of a material substance.  To refute the misconception, note that light waves propagate through the vacuum of space, which by definition means no matter.
The nature of the classical electromagnetic wave and by extension, a "classical" gravitational wave rests on the notion that the strength of an electric field or gravitational field at a point depends on the distance from that point to the origin of the field, charge q or mass m.  Imagine fixing an electric charge q at one end of a meter stick, say 0 m, and a field meter at the other end, 1 m.  If you start to move the field meter from 1 m to 0 m, you would find that the strength of the measured field increases.  Conversely, if you fix the field meter and move the charge, you would also measure a change in field strength.
Now imagine placing field meters at every centimeter marking from 0 to 1 m, so that you have 100 field meters.  This time imagine vibrating over one cycle (jostling) the point charge at 0 m just a little bit, say 2 mm, and then back to 0 m.  The other field meters will feel an increase and then decrease in field strength as the charge moves closer to them (moving from 0 to 2 mm) and then moves farther away (moving from 2 mm to 0 mm).
However, do they all feel the increase and decrease of field strength through the one vibration cycle at the same time?  Or do the field meters closer to the 0 m end feel the change first?  It turns out that the field meter closest to the point charge feels a change first, then the next field meter, then the next, until the change is felt at the end.  The updating of the numbers on the meters reports on the propagation of the change in field, in other words an electromagnetic wave.  The movement of the charge "refreshes" the electric field strength, and this refresh moves as a wave at the speed of light.  So there is no movement of matter, just a refresh of the electromagnetic field throughout space.  The wave continues to travel beyond the meter stick of course.  After one year, it has traveled a distance of a light year.
You can imagine a similar dynamic of gravity, which is how I imagine it, with the caveat that the explanation involves additional physics.  It is my understanding however that the gravity wave travels at the same speed of the electromagnetic wave.
A: Let us clear at first that photons are elementary quantum mechanical particles. That classical electromagnetic waves emerge from a confluence of photons can be shown, but classical electromagnetic theory works with the self propagation of the light energy, carried by electric and magnetic fields.

Two kinds of wave transmission are: 1. Light waves, where a substance (photon) travels as a wave.

Lets ignore the quantum frame, as gravity is not yet definitively quantized.



*An attached rope, like at the gym, that is "waved" up and down. Here, no substance travels to a new spot, but adjacent parts transmit the energy to others.

QUESTION: Which method do gravitational waves propagate by?

Like 1., except what is waving is spacetime itself. This was beautifully seen in the experiment at LIGO .
Now it is expected when gravity is quantized that the gravitational waves will be emergent from a confluence of gravitons ( hypothetical at present) the way electromagnetic waves are a confluence, superposition of wavefunctions, of  photons.
