Gravitational vs Inertial forces In D'inverno's relativity book, it says that if we describe flat space using a general coordinate system, the equation of motion of a free particle would have terms involving the connection. Those terms would encode inertial forces. It then says that, to generalize this point of view to include gravity, gravity as well as inertial forces are to be described by the connection, but the metric can no longer be flat, the reason being that otherwise there would be no distinction between gravity and inertial forces. But my understanding was that this was the point of the equivalence principle: to treat gravity as an inertial force, since it doesn't depend on any properties of the body or its composition. What is the difference between gravity and inertial forces, at least from the book's point of view? Is my understanding of the equivalence principle wrong?
 A: Rather than going straight to words such as "inertia" it may be helpful to focus your thought first of all on the equivalence principle. This comes in two forms, commonly called "weak equivalence principle" and "strong equivalence principle". The weak equivalence principle is best stated (IMO) as a statement about universality of acceleration under gravity. It is the statement that in any given gravitational field all bodies have the same acceleration under gravity alone. (In more technical language, they have the same worldline). The strong equivalence principle extends this. It is the observation that in the limit of a small region of spacetime all physical effects tend to the ones you would get in flat spacetime (i.e. the physics of special relativity).
Notice, therefore, that the equivalence principle can be stated without explicitly mentioning the concept of inertia. You don't even need to mention any gravitational force!
Having said all that, when we observe an acceleration under gravity, it is convenient to multiply that acceleration by the inertial mass of the body that is accelerating, and thus get a "gravitational force". But you should ask yourself: accelerating relative to what? If our system of coordinates is itself in freefall (and the distances and times involved are small) then there won't be any acceleration of any freely falling object relative to that reference frame. So the only way to get an "acceleration due to gravity" (and hence a "force of gravity") is to employ a reference frame which is not in freefall. Thus the "force of gravity" does indeed turn out to have exactly the same character as an inertial force.
So the above description is basically agreeing with you. What D'Inverno is then pointing out is that in the case of flat spacetime you might as well say there is no gravity. So the physics is such that anyone can, if they want, insist that we only employ the term "gravity" for cases where spacetime is not flat.
The non-flat character of spacetime is revealed by things such as
geodesic deviation. That is a higher-order effect connected to spacetime curvature. The presence or absence of curvature at any event in spacetime is an objective property which can be measured. So if you like you can make a choice of terminology in which the term "inertial force" is reserved for the case where the local curvature is zero (i.e. we have flat spacetime) and the term "gravity" is reserved for the case where the local curvature is not zero. However that way of using the terminology has not caught on, so in the standard use of terms one may well say that  "gravitational force" is an inertial force, for the reason I gave above.
And of course the physical behaviours are what they are, independent of our use of terms.
A: Before I get to the principle of equivalence I want to get the following out of the way: the expression 'inertial force' is awkward.
The idea of 'force' is that you have two objects exerting a force upon each other. (You can have assemblies of multiple objects, of course, it's just that two objects is the minimum.)
Among the simplest examples of force would be the Coulomb force between, say, two protons.
Third law force pair
One way to demarcate the extent of the concept of force is to use Newton's third law as definition. It is a force when it can be described as a third law force pair.
Inertia cannot be categorized as a force, because inertia is about a single object. For a single object: a force is required to cause change of velocity. For inertia the idea of a third law force pair is not applicable.
If inertia would be a counterforce in the sense of Newton's third law then motion would be impossible; every force would be countered by an equal and opposite force. Inertia opposes change of velocity, but does not prevent it.
Inertia is like inductance. (Self-)inductance manifests itself as opposition to change of current strength. At the same time: (self-)inductance cannot prevent change of current strength. The inductance mechanism of opposition to change of current strength is that an electromagnetic response is elicited by change of current strength. So it is inherently impossible for (self-)inductance to prevent change. Instead change is opposed; the stronger the (self-)inductance, the slower the change of current strength. In the case of Inertia: Inertia offers opposition to change of velocity, but does not prevent it.
Summerizing:
Inertia is in a category of it's own.
Any attempt to think of inertia in terms of force leads to self-contradiction.



The Principle of Equivalence
To set the stage of for the principle of equivalence I will discuss some aspects of Maxwell's concept of the electromagnetic Aether. Of course, Maxwell's Aether is a thing of the past, but in its time it served Maxwell purposes well. The reasons why it served Maxwell's purposes are interesting, and relevant.
The problem that Maxwell faced was: how to concieve of an entity that combines the following capabilities: it must have the capacity to act as the mediator of the electromagnetic interaction, and it must support propagation of electromagnetic waves.
What Maxwell conveived was an entity that is omnipresent in space. This omnipresent entity has a null state, in that null state Maxwell's Aether is in a uniform state. In that uniform state Maxwell's Aether does not affect motion of an electric charge moving through it at uniform velocity.
Next: Presence of a source of Coulomb force induces a stressed state in Maxwell's Aether. While the uniform state does not affect motion of electric charge, the stressed state does. This state of stress is then acting as the mediator of electromagnetic interaction. The distribution of the stressed state is then referred to as 'the electromagnetic field'.
Maxwell concieved of this Aether to have an extremely strong tendency to return to the uniform state. Also, rate of change of the state would tend to persist.
Oscillation has the following two requirements:
-Tendency to return to a particular state (generally designated as 'state of lowest potential energy)
-A property that a rate of change of the state tends to persist. (This tendency is generally designated as 'kinetic energy')
That is how Maxwell's Aether supports propagation of electromagnetic waves.
That is a very rough outlone of how Maxwell solved the problem of concieving of a single entity that acts as mediator of electromagnetic interaction, and supports propagation of electromagnetic waves.


Inertia and gravity
In the case of inertia and gravity we have the following problem:
Can we conceive of an entity that does both of the following things:
-Act as the mediator of gravitational interaction
-Accommodate the equivalence of gravitational and inertial mass

Einstein's expressed his solution to the above problem in the form of the principle of equivalence.
Inertia is omnipresent in the universe. Inertia is assumed to be perfectly uniform. Inertia does not affect the motion of objects in uniform motion.
Assume that a source of gravitational interaction induces a stressed state in the omnipresent inertia. This stressed state is then thought of as introducing a bias in the normally uniform inertia. This stressed state is then acting as the mediator of gravitational interaction.
Prior to the introduction of GR the phenomena of inertia and gravity were thought of as separate entities.
GR introduced a fundamental unification.
John Stachel, director of the centre for Einstein studies, proposed in the 1990's the following name for the field that is described with the Einstein Field Equations: The inertio-gravitational field
The name 'theory of the inertio-gravitational field' makes everything fall into place. In that sense it's too bad it hasn't yet found wide adoption.
Example of a recent article in which the name 'inertio-gravitational field is used:
Dennis Lehmkuhl:
'Why Einstein did not believe that General Relativity geometrizes gravity' (PDF-document, 231 KB) Download from sciencedirect.com (no paywall) (also available for download from PhilSciArchive)
Inertia and geometry
Prior to the introduction of relativistic physics it was assumed that the phenomenon of Inertia has the same symmetries and uniformity as Euclidean space.
The introduction of Special Relativity modified that view as follows:
In terms of Special Relativity Inertia has the same symmetries and uniformity as Minkowski spacetime.
The point is: there is no experiment that can distinguish between the properties of Inertia and the properties of spacetime.
In physics there is the following rule of thumb:
If no experiment exists that can distinguish between A and B then it makes no sense to formulate a theory in which A and B are treated as distinct. Instead: treat A and B as one and the same.
In terms of special relativity no distinction is made between the properties of inertia and the properties of Minkowski spacetime; no experiment exists that can distinguish between the two.


Finally: about gravitational waves (the waves detected by the LIGO experiment)
It took the physics community many decades to work out whether GR implies that the inertio-gravitational field also supports propagation of gravitational waves.
The detection of gravitaional waves corroborates the view that gravitational interaction is mediated by a stressed state of the inertio-gravitational field.
In the case of gravitational waves there is an oscillation of stressed state, carrying energy.
A: Although there are similarities between gravity and inertia, I have always thought of these as simply happenstance.  The main difference between the two is, what I feel, the single most important aspect of gravity; its direct connection to time dilation.  You can imagine the incredible inertial force of a bullet in the barrel of a rifle; something like 4x10^5 m/s^2 which is getting darned close to the gravity near a black hole.  But there is no associated time dilation of the bullet. So while the equivalence principle is a good model, it is far from complete.
