Speed of gravitational waves and light We all know that speed of gravitational waves (GW) and that of light in space are exactly the same (= $c$).
We also know that space is medium for GW. 
Does that mean space is also the medium for light?
Because both of them are waves, and speed of a wave is property of the medium. 
Is there any other way to justify the identical speed of the two?
 A: As described in this article, whenever we are talking about gravitational waves in cosmology, we actually assume the underlying spacetime to be flat (so just Minkowski metric $\eta^{\mu\nu}$) and a small perturbation $h^{\mu\nu}$ causing ripples in the metric is the graviton field:
$$g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}+O(h^2)$$
Truncating this expansion after the linear term as written above simplifies Einsteins Equations, so that after a few variable transformations and field redefinitions Einsteins Equations in vacuum can be written as
$$\Box \bar h^{\mu\nu}=0$$
Note that here $\bar h^{\mu\nu}$ is related to $h^{\mu\nu}$, and even though it originated as part of the metric - it used to describe spacetime curvature itself - it is now treated as a field propagating in flat spacetime. Therefore, we reduced the picture to special relativity with an extra field.
Interestingly, the photon field $A^\mu$ satisfies the same equation of motion in Lorentz gauge
$$\Box A^\mu=0$$
The fact that both fields satisfy the same Wave equations of motion on the Minkowski background tells us that both excitations travel with the same speed through flat spacetime. It is not appropriate to think of spacetime as a medium, it is just the background space. There is a difference. Look up Aether for more details on that.
EDIT:
If you have doubt about the approximations used, take a look at this publication, which constraints graviton mass to $<10^{-32}eV$. This is 32 orders of magnitude smaller than i.e. electron neutrino mass, and those particles are already moving at effectively the speed of light.
A: You are using the wrong scientific terminology but you are right in a sense. The speed of light is indeed determined by the environment it travels in. We don't call it medium in relativity or physics, we call it spacetime. Spacetime has a geometry, a 4 dimensional (ignoring quantum gravity) manifold that has 1 time dimension and 3 spatial ones. The simplest spacetime is Minkowski, which is flat. The metric of it, that which describes its geometry (though not its global topology) is:
ds^2 = - c^2 * dt^2 + dx^2 + dy^2 + dz^2
That spacetime means that anyone in an inertial reference frame on it will see any massless particle travel at speed c.
For General Relativity (GR)it is curved spacetime, and the metric is similar to the above but each term can have functions of space and time in front, and mixed terms (also with functions in front) like dxdt dxdy and so on. The functions in front are determined by Einsteins gravitational equations which relate the metric (i.e., the spacetime) to the matter content in the spacetime. The interesting thing about GR for your question is that the speed of massless particles, in a LOCAL reference frame is also c in ALL those cases. It is said that they locally obey special relativity.
So you see, it is the spacetime which determines the speed of light, gravitons, and any other massless particle. And it is always c. So in a sense you are right. But it is even more, spacetime also determines the motion of any test particle (i.e., small enough so you can ignore its own gravitational field), and in fact all test particles move exactly the same way in the gravitational field. They move as free falling particles, in spacetime curves determined by the spacetime metric, and INDEPENDENT of their mass. This is the so called principle of equivalence, the spacetime determines the motions of freely falling particles. It's the same reason in Newtonian gravity astronauts float exactly the same inside their space station orbiting the earth, regardless of how heavy or small they are. GR says the same. (But there's plenty other differences). 
Your idea of the medium determining the speed of the waves sort of means in physics that a material medium is involved. It really is not, it is just spacetime. A wireless or electromagnetic medium is a standard term used in wireless and optical communications, an electronic engineering view of electromagnetic propagation. In that case it means the medium, whether air or clouds or glass or fiber, or for cellular wireless the air plus the terrain and objects around from which the waves diffract, reflect, scatter and get absorbed. Those are different concerns and physical media. In physics we are talking mostly, when we talk about c, about empty space, and with gravity possibly other matter around creating that gravity (for cosmological distances plenty of mostly empty space, which however can be curved due to the other matter in the universe. Similarly for the spacetime outside stars, or outside black holes). In all of those the local speed of massless particles is c.
So you are partially right. It is all about the environment in which they travel. Because of the equivalence principle nothing else can make a difference.
A: The general equation of an electromagnetic wave can be written as 
$$ \eta^{\mu\nu}\partial_{\mu}\partial_{\nu} A_{\sigma}=0$$
This can be derived from the vacuum solution of Maxwell equations. The metric for empty space-time can be written as
$$g_{\mu\nu} = \eta_{\mu\nu}+ h_{\mu\nu}$$
The Ricci tensor in first order can be reduced to 
$$R_{\mu\nu} = \partial^{\alpha}\partial_{\alpha}h_{\mu\nu} $$
The vacuum field equations would become 
$$\partial_{\alpha}\partial^{\alpha}h_{\mu\nu}= 0$$
This is the same almost the same as the previous equation. The solutions to both the equations is 
$$h_{\mu\nu}= \epsilon_{\mu\nu}e^{i\bf{kx}}$$
$$A_{\mu} = \epsilon_{\mu}e^{i\bf{kx}}$$
Observing both the equations one can say that both will travel at the same speed $c$.
