Firstly I think shades of this question have appeared elsewhere (like here, or here).  Hopefully mine is a slightly different take on it.  If I'm just being thick please correct me.

We always hear about the force of gravity being the odd-one-out of the four forces.  And this argument, whenever it's presented in popular science at least, always hinges on the relative strength of the forces.  Or for a more in depth picture this excellent thread.  But, having had a single, brief semester studying general relativity, I'm struggling to see how it is viewed as a force at all.

A force, as I understand it, involves the interaction of matter particles with each other via a field.  An energy quantisation of the field is the force carrying particle of the field.

In the case of gravity though, particles don't interact with one another in this way.  General relativity describes how space-time is distorted by energy.  So what looked to everyone before Einstein like two orbiting celestial bodies, bound by some long distance force was actually two lumps of energy distorting space-time enough to make their paths through 3D space elliptical. 

Yet theorists are still very concerned with "uniting the 4 forces".  Even though that pesky 4th force has been well described by distortions in space time.  Is there a reason for this that is understandable to a recent physics graduate like myself?   

My main points of confusion:


Why is gravity still viewed as a force?
Is the interaction of particles with space time the force-like interaction?
Is space-time the force field?
If particles not experiencing EM/weak/strong forces merely follow straight lines in higher-dimensional space (what I understand geodesics to be) then how can there be a 4th force acting on them?  


Thanks to anyone who can help shed some light on this for me!

Gravity is viewed as a force because it is a force.

A force $F$ is something that makes objects of mass $m$ accelerate according to $F=ma$. The Moon or the ISS orbiting the Earth or a falling apple are accelerated by a particular force that is linked to the existence of the Earth and we have reserved the technical term "gravity" for it for 3+ centuries.

Einstein explained this gravitational force, $F=GMm/r^2$, as a consequence of the curved spacetime around the massive objects. But it's still true that:

Gravity is an interaction mediated by a field and the field also has an associated particle, exactly like the electromagnetic field.

The field that communicates gravity is the metric tensor field $g_{\mu\nu}(x,y,z,t)$. It also defines/disturbs the relationships for distances and geometry in the spacetime but this additional "pretty" interpretation doesn't matter. It's a field in the very same sense as the electric vector $\vec E(x,y,z,t)$ is a field. The metric tensor has a higher number of components but that's just a technical difference.

Much like the electromagnetic fields may support wave-like solutions, the electromagnetic waves, the metric tensor allows wave-like solutions, the gravitational waves. According to quantum theory, the energy carried by frequency $f$ waves isn't continuous. The energy of electromagnetic waves is carried in units, photons, of energy $E=hf$. The energy of gravitational waves is carried in the units, gravitons, that have energy $E=hf$. This relationship $E=hf$ is completely universal.

In fact, not only "beams" of waves may be interpreted in terms of these particles. Even static situations with a force in between may be explained by the action of these particles – photons and gravitons – but they must be virtual, not real, photons and gravitons. Again, the situations of electromagnetism and gravity are totally analogous.

You ask whether the spacetime is the force field. To some extent Yes, but it is more accurate to say that the spacetime geometry, the metric tensor, is the field.

Concerning your last question, indeed, one may describe the free motion of a probe in the gravitational field by saying that the probe follows the straightest possible trajectories. But where these straightest trajectories lead – and, for example, whether they are periodic in space (orbits) – depends on what the gravitational field (spacetime geometry) actually is. So instead of thinking about the trajectories as "straight lines" (which is not good as a universal attitude because the spacetime itself isn't "flat" i.e. made of mutually orthogonal straight uniform grids), it's more appropriate to think about the trajectories in a coordinate space and they're not straight in general. They're curved and the degree of curvature of these trajectories depends on the metric tensor – the spacetime geometry – the gravitational force field.

To summarize, gravity is a fundamental interaction just like the other three. The only differences between gravity and the other three forces are an additional "pretty" interpretation of the gravitational force field and some technicalities such as the higher spin of the messenger particle and non-renormalizability of the effective theory describing this particle.
