This was originally a Quora post of mine, but I think it belongs here.

Imagine for a moment that we are back in the 19th century. I am chatting with physicists and introduce to them a radically modern theory. Expressed in words, it simply states: In the vicinity of a massive body, clocks tick slower.

Of course my suggestion would be met with incredulity, but at least some of the more open-minded physicists of the era would be willing to give me the benefit of the doubt: "How much slower, kind sir?" they would ask. Well... let's do the math. The time measured by distant observers—let's call it "coordinate time"—is $t$. The time measured by a clock—let's call it "proper time"—should be denoted by the Greek letter $\tau$. I propose, boldly, that near a compact mass the ratio of an infinitesimal time interval as measured by distant observers, $dt$, vs. the interval between the same two events measured by the local clock, $d\tau$, are related by the expression,
$$\frac{dt}{d\tau} = 1+\frac{GM}{c^2r},$$
where $c$ is a very large constant speed, much larger than any sensible velocity $v$. Alright, says the skeptical 19th-century physicist, let us plug this into the Lagrangian action integral that governs the motion of a test particle. Normally, a free particle's Lagrangian is of course just its kinetic energy, so the action is
$$S=\int \tfrac{1}{2}mv^2~dt.$$
I'd interject at this point: "Wait, it is true that we can add any constant to this integrand get the same equations of motion in the end, right? So just for my benefit, please add $mc^2$. You will see soon why." The skeptical physicist reluctantly agrees:
$$S=\int (\tfrac{1}{2}mv^2+mc^2)~dt.$$
But as per my earlier proposal, we know that $dt=(dt/d\tau)d\tau=(1+GM/c^2r)d\tau$ so we can rewrite this expression in terms of proper time:
$$S=\int (\tfrac{1}{2}mv^2+mc^2)\left(1+\frac{GM}{c^2r}\right)~d\tau,$$
or, after expanding the product,
$$S=\int \left(\tfrac{1}{2}mv^2+\frac{GMm}{r}+mc^2+\frac{GMm}{r}\frac{v^2}{c^2}\right)~d\tau.$$
We then agree that when $v\ll c$ (since most speeds are much, much less than the large, constant value $c$ we chose) the last term in those parentheses can be dropped, and we can of course also ignore the constant term $mc^2$. What we are left with is
$$S=\int \left(\tfrac{1}{2}mv^2+\frac{GMm}{r}\right)~d\tau,$$
which the 19th century physicist instantly recognizes as the Lagrangian action functional of a particle in a gravitational field! He then instantly derives from this the corresponding Euler-Lagrange equation, which is well-known inverse square law:
$$m\frac{dv}{d\tau}+\frac{GMm}{r^2} = 0,$$
where $dv/d\tau$ is of course the acceleration of the mass $m$ measured with respect to its proper time $\tau$.

In other words, we just deduced gravity using nothing more than the assumption that clocks tick slower near masses. The only "sleight of hand," if it can be called that, was including the constant term $mc^2$ in the correct spot, but in the end that term vanished from the equations anyway.

So there we have it. A 19th century version of general relativity (no absolute time) without filling our heads with badly misleading concepts from the popular literature, images of "curved spacetime" or the impression that this "spacetime" is some substance with physical properties that can be stretched, curved, or otherwise manipulated, as opposed to merely the geometric background we use in our mathematical descriptions when we map the relationships of things.

Instead, I just talked about clocks. I simply discarded, following Einstein's insight, the notion of "absolute time". Instead, I stressed that identical clocks tick at different rates depending on where they are with respect to massive objects. From this, the law of gravity follows. Sure, general relativity adds more to this, and yes, "spatial curvature" also plays a (tiny) role, and yes, this simplistic derivation loses its validity in the presence of very strong gravitational fields but nonetheless, I hope it clearly illustrates my main point: "spacetime" is not a thing. Time is, but only insofar as counting the ticks of a clock amounts to time. And "curved" is a metaphor: What matters is what we measure, the counting of the ticks of a clock.