Intuition for why mass and energy curve the space-time fabric and for why this relationship is linear? The force of gravity does not exist
I understand(-ish) that, following general relativity, an apple falls onto earth, not because there is a force pulling earth and the apple toward each other but because, both the earth and the apple are travelling through time in the same direction and that their mass curve the space-time fabric causing the geodesic of the movement through time to "fall" down the same location in space.
Why does mass and energy curve space-time?
I don't understand why mass (and energy) curve space-time. I am "familiar" (I never saw any explanation about it) with Einstein equation putting $T_{\mu\nu}$ (Mass and energy) and $G_{\mu\nu}$ (space-time geometry) in a linear relationship but I fail to get an intuitive explanation for why such relationship between $T_{\mu\nu}$ and $G_{\mu\nu}$ exist and why it has to be linear. Note that I don't really understand the true meaning of $T_{\mu\nu}$ and $G_{\mu\nu}$ but only the vague description that I made of them.
Question
In short, my question is
Can you give me an intuition for why mass and energy curve the space-time fabric and for why this relationship is linear?
 A: Imagine you don't know of any kind of gravity at all, just all the other interactions matter can have. When you write down an action in Lagrangian field theory, you can use it to derive the zero-gravity equations of motion. Now pretend instead that gravity due to spacetime curvature is the only thing you do know about: again, what happens follows from a stationary action principle. So how do we make a fuller theory that uses both pieces of information? Well, we just add two actions together, with a proportionality constant that governs the strength of gravity. (In theory there could be an extra interaction term in the action, but in general relativity there isn't.) In short, the reason $T_{\mu\nu}-\kappa^{-1}G_{\mu\nu}=0$ comes from varying the full action is because the full action is of the form $S_{\text{Gives}T}-\kappa^{-1}S_{\text{Gives}G}$.
A: There are several reasons:

*

*What we know is that matter interacts. Gravity, whatever it is, is a certain interaction which is different from all the other interaction we know. So there must exist an equation to describe the dynamics given a matter distribution and ignoring all other interaction (we can limit ourselves to EM interaction since the other two are "quantum-effects").

*The fact that the space-time is "curved" (not in the sense of Riemannian curvature, rather from a demagogical perspective) is a natural consequence of searching for a theory which DOES NOT privilege a certain system of coordinates. Think about what happens if you change coordinates in the plane to the surface of a sphere, (by for instance a stereographic projection), you "curved" your plane, but just because you changed point of view. But, since Einstein pointed out that gravity is an expression of a change of coordinates, then what is caused by matter is something pretty similar to what is done by changing coordinates. Ergo, the space-time must be curved by matter!

*The linearity in the equation is because the Ricci tensor is 2nd order in the metric and, as in every fundamental equation of physics, we want to stay up to the second order in the solution; putting higher orders into our equation just makes weaker our system, since we need more and more fixed initial conditions, which is a pretty hard problem even though we stopped just at the 2nd order.

*Finally, we set $T_{\mu\nu}$ and $R_{\mu\nu}$ because we need the "right symmetries" in our field equation and also because we know that where "there is no matter" there is gravity instead (e.g. Schwarzschild solution), so for $T_{\mu\nu}=0$ we still want gravity.

A: The intuition you are looking probably does not exist.
You have reached the bottom of physics.
The empirical level.
Physics is a science. This means that the ultimate authority is observations of the real world. You cannot possibly infer that matter warps spacetime just by sitting down and thinking really hard about what feels right or intuitive (that is mysticism, not science).
We have seen apples falling to Earth, planets orbiting the Sun and even detected clocks running at slightly different speeds when hurtling around the Earth on GPS satellites. General Relativity provides a description of these phenomena in terms of a thing called "spacetime" being warped. There is nothing logically inconsistent about a universe where matter does not warp spacetime (a gravity-free universe), its just not what we see.
Now it is possible that one day someone will find some kind of deeper theory from which GR can be derived, and maybe in that theory the curvature will emerge as a result of something else. But that deeper theory will itself offer some postulates that are specifically cooked up to match observation, just like GR does with those tensors. I don't think there is much chance that the deeper theories postulates are going to be logically self-evident or intuitive.
A: If you consider the dependence of the curvature on the mass-energy tensor, then the lowest-order nonvanishing term has to be first order in order to recover Newtonian gravity in the weak field limit.
It is possible to have higher-order terms as well, which would become nonnegligible for strong gravitational fields, such as in the very early universe, soon after the big bang. Such theories do exist and are viable. They include f(R) gravity and Brans-Dicke gravity. (Brans-Dicke gravity is equivalent to a certain type of f(R) gravity.) It's true, as pointed out in the answer by Bellem, that such theories will in general contain higher-order derivatives, which is problematic. But theorists are tricky, and they have found ways to make certain kinds of f(R) gravity work for certain purposes.
