Asking what a force is is a very good question with no good answer. One eighteenth century philosopher (if only I could remember who) said that we do not know what forces are but “time has domesticated them”.

At around the time of Newton, what we now call physics changed from being something based in an ontology—a theory of what the world was made of to explain why it worked a certain way—into something mathematised where the equations accurately predict the behaviour of the world but there is no ontology other than that the universe is a universe where those equations hold. Modern physics still fundamentally works this way (see Maxwell’s equations, quantum physics, etc). Compare to, for example, Descartes with his theory of corpuscles and (totally wrong) billiard ball mechanics, or just about any other ontology from before him.

> One eighteenth century philosopher (if only I could remember who) said that we do not know what forces are but “time has domesticated them”.

Interestingly, I could only find one other reference to that phrase[1]:

One 18th century natural philosopher wrote that forces were incomprehensible but "time has domesticated them."

[1] Why do we study geometry https://www.dpmms.cam.ac.uk/~piers/F-I-G_opening_ppr.pdf

Newton's first law provides a guide: "Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it."

So, the definition of a force as something that causes an object to change its movement from a "straight line" comes to us from Newton's laws.

> Why does the existence of a transformation that makes movement under a supposed force actually follow a straight line mean it's not really a force?

It's not just the fact that the transformation exists that means we don't really consider gravity to be a force. It's that the transformation exists, and provides useful predictions about the universe that turn out to be backed by experiment.

Under the theory of general relativity gravity isn't a force, because the fundamental premises of general relativity assume that gravity is actually a distortion of spacetime caused by mass.

We say that "gravity" isn't a force simply because the predictions made by general relativity have been validated by a number of experimental observations—at least on the macro scale.

> For the other forces (e.g. electromagnetism) can we say that there's _no way_ to exhibit a transformation that causes charged particles travel on "straight" lines?

I'm certainly not an expert, so I can't really comment on this. My best guess is that there doesn't exist any such transformation that causes charged particles to travel on "straight" lines that makes experimental predictions as well as our current scientific theories.

Actually it's quite easy to cast electromagnetism into a purely geometrical form, you just add a few extra curled up dimensions to a flattish spacetime in general relativity and you basically get Maxwell's equations for free. There are other reasons that's not a great theory (IIRC sources and self-action get nasty and you have to make arbitrary choices about scale, etc), but it's straightforward to get force laws out of pure geometry.

So what you are saying is that if we assumed that all particles, including light are magnetic, and everything that has a mass, emits a corresponding magnetic field with a strength relative to its mass, we could not form a similar theory of "general magnetic relativity" in which the frame of reference under magnetic fields would behave in a similar way it does for gravity?

That seems kinda odd. What exactly would provide this difference? How is magnetism different from gravity? Is it that gravity doesn't have a mass, but magnetic fields do?

I think (and, again, I'm absolutely not an expert) that a fundamental difference between the two is that from our perspective gravity "effects" particles with 0 mass, while the electromagnetic field does not effect particles with 0 charge.

So a photon, which has 0 mass is still bent by gravity. Everything that we've observed that moves through spacetime is bent by gravity. That's why we say that gravity is a warping of spacetime itself, where electromagnetism isn't.

If you tried to build the "general electromagnetic theory of relativity", then 0 charged particles wouldn't follow a straight line on a geodesic of spacetime. With gravity, everything follows a straight-line on the geodesic of a curved straight line, regardless of its mass.

As to why such a difference exists between gravity and electromagnetism, that's well above my pay grade.

There's no emitting magnetic fields, no diverging fields allowed. The right analogy is between mass and electric charge, only mass is limited to positive values.

The effect of a spacetime transformation isn't just to redefine straight lines along which particles move. It means measurements (e.g. lengths, areas, time intervals) are different depending on where you are in the spacetime. The are forces don't come with these "extra" effects - an EM field doesn't stretch and contract space.

However, there are a lot of parallels between electromagnetism and relativity! Quite often relativity effects are introduced with an EM analogy, e.g. gravitational waves (which have polarisation) and electromagnetic waves ie photons (which also have polarisation). Note though it really is an analogy - they are fundamentally different things in both reality and mathematical form.

The main difference though is that gravitation (probably) doesn't have a mass of its own, while EM fields do. Plus, matter does not react to EM fields in the way it does towards gravitation. I.e. light is not "pulled" by EM fields. However, these are technicalities. If all matter reacted to EM fields the same way it would to gravitation, would that make EM fields no force either? Or put another way, gravitation act universally on all particles, while EM fields do not. That necessarily has consequences when it comes to relativity. However it seems odd to argue that general relativity would exclude gravitation from being a force. If it acted only on a subset of particles, it would likely be in the same position as EM fields, and suddenly become a force again?

> The main difference though is that gravitation (probably) doesn't have a mass of its own, while EM fields do.

Both gravity and EM fields have energy which is what couples to the gravitational field. Neither of the fields has mass, though.

> If all matter reacted to EM fields the same way it would to gravitation, would that make EM fields no force either? Or put another way, gravitation act universally on all particles, while EM fields do not. That necessarily has consequences when it comes to relativity. However it seems odd to argue that general relativity would exclude gravitation from being a force. If it acted only on a subset of particles, it would likely be in the same position as EM fields, and suddenly become a force again?

This is very well thought. Indeed, the equivalence principle, the fact that gravity couples to everything in exactly the same way (and that includes gravity itself as per the previous clarification) lurks behind our ability to reinterpret gravity in a geometric fashion. After all, if something didn't interact with gravity in the same way as everything else we could establish an experiment to differentiate if a spaceship is accelerating or stationary under a gravitational field (see Einstein's mental experiment) by measuring how that thing behaves. And that same fact would stop us from interpreting gravity as curvature of spacetime itself.

To your last point, speaking of forces is probably antiquated anyway, although still in use partly for historical reasons partly abuse of terminology. Preferably we should use the term "interactions", after all some of the "forces" do not result in push or pull as we usually understand a force in Newtonian mechanics but in things like color change. Others, like the gravitational "force" can be expressed entirely as spacetime geometry. But discussing semantics is quite pointless so as long as everyone understands in what way the term "force" is an abuse of terminology it's OK to keep using it.

Also a layperson here. Can you give an example of how we can tell that EM fields don't "stretch" space, but gravity does? Is it just about how light behaves in those fields or is there something more to it?

As mentioned below, one way to think about it is that EM only affects charged particles (and depends on their charge), whereas because gravity is acting on the underlying space-time it has a universal effect on everything (including light).

We can pretty much boil EM down to: like charges repel, unlike attract, strength is charge1*charge2/distance^2. What about magnetic field, photons, QFT etc?? None of this exhibits effects which could be described as stretching space-time either.

But we cannot do the same with gravity. An explanation like the above but for gravity (which is traditional Newtonian) leaves out many, now observed effects such as:

-> time dilation (GPS relies on this calculation) (measures time stretching and contracting)

-> gravitational waves (LIGO) (measures space stretching and contracting)

How do we _know_ any of this? People propose theories, those theories are then tested against experiment. AFAIK to date there is no experimental evidence suggesting EM stretches space, and no theory proposed that includes such an effect and correctly matches experimental data. That's the most holistic answer (but unfortunately one you just have to believe unless you have a lot of spare time!)

Kaluza-Klein theory is an attempt to exhibit a transformation that charged particles travel on "straight" lines in a 5-dimensional space. The fifth dimension is supposed to be a very small circle. There are some experimental implications, but the circle is so small that we can't detect them (it's similar with string theories).

String theory tries to do the same thing with other forces; so it's not clear whether there's such a transformation. Forces are technically simpler than embedding extra dimensions that there are no evidence for.

> For the other forces (e.g. electromagnetism) can we say that there's _no way_ to exhibit a transformation that causes charged particles travel on "straight" lines?

Gravity is peculiar in that there is only one type of charge and it's exactly equal to the inertia quantity. If you try to do something similar to the other forces, you'll get really complicated models, with hidden dimensions and things that don't interact the same way with them.

An intuitive way to tell if something is a "real" force (as proposed to a pseudo- or fictitious force) - can someone subject to that force feel it, or equivalently, can an accelerometer measure the acceleration it produces?

When you accelerate or decelerate in a vehicle, you feel it. But when you jump out of a plane and accelerate towards the ground, you don't feel a force.

Noticing this fact was a big part of what led Einstein to his equivalence principle and General Relativity.

If all the particles in your body were made magnetic and you jump out of a plane on a weightless magnetic planet, how would it feel different than jumping out of a plane on Earth?

In my high school we were taught the other way - that the gravity on you in free fall is a real force and centrifugal force going round a corner is not.

It depends a bit how you define words like real.

What about when you splat on the ground? Is that not a force? Where did it come from?

That involves a force which comes from encountering an object that's not in free fall with you.

Specifically the electrons in the atoms of the ground and the electrons of the atoms of yourself repel each other when they come together.

“So pushing just two atoms close to each other takes energy, as all their electrons need to go into unoccupied high-energy states. Trying to push all the table-atoms and finger-atoms together demands an awful lot of energy – more than your muscles can supply. You feel that, as resistance to your finger, which is why and how the table feels solid to your touch.“ https://theconversation.com/if-atoms-are-mostly-empty-space-...

Doesn't directly answer what is a force, but perhaps you'll find this perspective interesting: https://en.wikipedia.org/wiki/Static_forces_and_virtual-part...

I think a big difference is that other forces (electroweak, strong) can be thought as mediated by exchange of some virtual bosons. Quantum gravity would use gravitons, but this isn’t how GR works.