# Why do we still need to think of gravity as a force?

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!

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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.

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@ejrb: it somewhat depends on your framework how special gravity is - eg in string theory gravity is less special than in loop quantum gravity, and the teleparallel reformulation of general relativity makes it into a proper force instead of an inertial one – Christoph Apr 22 '13 at 17:49
Gravity is NOT a 'real' force. Its an inertial force similar to what you experience in braking car. Just because something is accelerating, it doesn't mean there's a real force behind it. – Evil Angel May 5 '13 at 19:04
@SachinShekhar Are you saying its an apparent force, much like coriolis arising due to being in a rotating frame of reference? – Isopycnal Oscillation May 7 '13 at 17:28
@SachinShekhar, and what is a force? Give a definition, please? – Peter Kravchuk May 7 '13 at 17:34
@SachinShekhar the source of gravity is mass and its effect on space-time – Physiks lover Jul 8 '13 at 15:37

At the risk of being chided by physicist for vast oversimplification, can I provide an intuitive answer?

In a rotating frame of reference, centrifugal force exists (as does Coriolis force). Observers in rotating frames see freely moving objects travelling curved paths. They conclude that a force exists, and can even generate a formula for it. An observer outside the rotating frame sees the object moving at constant velocity, and concludes that no force is acting on it. The force is fictitious, but valid inside the rotating frame.

The fictitious force has two interesting qualities, though. First, it has no obvious explanation. Why does something APPEAR to be forced away from the centre of rotation? There are no strings attached, no magnets, no wind blowing out from the centre. No explanation for centrifugal force, no matter how real it feels.

Second, the fictitious force has the astounding feature that it produces the same (apparent) acceleration on all objects, no matter what their mass. (Compare: F=ma, so for a given F, if m is higher, a must be lower.)

Now look at gravity. The Earth pulls objects down without touching them. No strings attached! Also, all objects have the same acceleration due to gravity regardless of mass. So gravity has the hallmarks of a fictitious force.

Now, consider Einstein's GR. In a nutshell (and here's where the physicists might lambaste me, but I'm going for intuition not mathematical validity): Einstein says space-time is curved. We don't perceive the curve. So when we think we are travelling at constant velocity through space-time, we are actually accelerating. Therefore, we should detect a fictitious force. Gravity is that force.

(Remember, a force might be fictitious, but it's very real inside the accelerated frame of reference.)

-Rob

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Yes that is the beauty of GR, the curvature of space-time is not apparent within a local frame of reference and so what appears to be a force proportional to mass can be viewed as masses distorting and moving through the actual curved spacetime. Incidentally centrifugal "force" is also proportional to mass. My issue is that expert theoretical physicists discuss gravity in the same way as other forces -namely with an exchange particle (graviton, a massless spin-2 boson) so are clearly privy to something that wasn't covered in my 12 weeks of GR (as well as plenty of other things too!) :) – ejrb May 5 '13 at 21:30
The language is non-technical, but the explanation is quite wells stated. – dmckee Jul 6 '15 at 20:12

Gravity is not special at all. It seemed to be special at dawn of the 20th century but now the picture is different.

Fields are more than just forces. Fields can have their intrinsic dynamics, solitons, topological features, nontrivial vacuum.

As of force aspect, electromagnetic field makes a 4-force $qF^{\mu\nu}u_{\nu}$, and gravitational field makes a 4-force $-m\Gamma^{\mu}_{\nu\lambda}u^{\nu}u^{\lambda}$. This looks essentialy similar.

Every known field has a Langangian density. Gravity does have one too.

On the other hand, values of gravitational field, that is, $g_{\mu\nu}$, $\Gamma^{\mu}_{\nu\lambda}$ and $R^{\mu}{}_{\nu\lambda\rho}$, can be interpreted as geometrical quantities describing the curved space-time. This seems to be a difference... at first. But the modern theory of fields uses the same notion for other fields too! It says that electromagnetic potential and field strength are geometrical quantities describing the curved space of a special kind - a fiber bundle, whose base is our usual space-time. All gauge fields can be interpreted this way - and all 4 "fundamental forces" are in fact gauge fields.

And this geometrical interpretation does not in any way obstruct quantisation of the field. (It can be understood in the sense of Feynman path integral for a field.) Just the same way as quantisation of electromagnetic field constructs photon - a particle that carries lectromagnetic interaction, quantisation of gravitational field constructs graviton, playing the same role. The problems with quantisation, mentioned everywhere, arise later - at calculating perturbations and gathering them together as a renormalized theory.

You can think of the force picture and of the spacetime curvature picture as of two points of view on the same subject. They do not contradict each other, nor hinder, but instead complement, and help to imagine and analyze different phenomena.

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I like the comparisons between GR and QED tensors, I hadn't seen the similarity before. And the point about being able to view the other forces in terms of geometric quantities. So is the term "fundamental force" just a pseudonym for "gauge field" when modern physicists use it and there are only 4 known gauge fields, hence 4 forces? – ejrb May 8 '13 at 12:13
There might be some non-fundamental gauge fields, for example in condensed matter physics, but for the fundamental physics - yes, they are pretty much synonyms. The term "fundamental force" was coined earlier, and since it is not well-defined, there might be subtle differences. For example, the Higgs field is not a gauge field, but is it a "force"? At least it is coupled to fermion fields similar to the Yukawa meson force field (early model for inter-nucleon interaction). For the basic quantum view of gravity, you can read Feynman Lectures on Gravitation. – firtree May 8 '13 at 13:09

My knowledge of physics does not really extend to these realms, and I apologize if I am wrong or off topic, or talking to far above my head. It is often funny how the mathematization of things can force some level of belief or unification, even though I am just trusting the mathematician.

I understand that the Kaluza-Klein theory from the twenties provides a form of unification of gravity and electromagnetism, By extending general relativity to 5-dimensionnal space, they obtained equations that could be separated into two sets corresponding respectively to Einstein field equations and to Maxwell equations for the electromagnetic field, and some extra ...wikipedia

In other words, the same 5D space-time distortions would then create both the gravity field and the electromagnetic field. Then there is no reason to see one as "a force" and find it unreasonnable for the other. Or to state it differently, the electromagnetic field may also be the result of space distortion.

Now, I suppose there are lots of problems with the Kaluza-Klein theory, which I will not even try to understand. But it seems to me enough to dismiss the idea that gravitation should be of a different nature, or at least it makes it very plausible that it has the same nature.

This has probably been said above in more technical terms. But something like the KK theory, however inadequate, speaks better to a layman like me.

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You're answer is fine (+1). But just note that curvature of 5D spacetime leads to gravity in 5D spacetime but either by considering that little extra dimension as a mathematical tool only, or by considering it as physical, but compactified on a circle, this 5D gravity is equivalent to 4D gravity and 4D EM. I.e. In 5D itslef, it is pretty much, just gravity. P.S. Including the Weak and Strong forces in Kaluza-Klein-theory had problems with fermions, so in came supersymmetry! Which made this new Kaluza-Klein theory 11D supergravity, which is the low-energy classical limit of M-theory.... – centralcharge Jul 6 '13 at 4:49
(contd.). M-theory has no problems But supergravity alone does... So you're right about Kaluza-Klein theory by itselnf having a lot of problems. – centralcharge Jul 6 '13 at 4:50

In classical mechanics, accelerated frames create fictitious forces such as the Coriolis Force. From the equivalence principle, gravity inside an unaccelerated frame is equivalent to an accelerated frame without gravity; it then follows that gravity is equivalent to a fictitious force by the equivalence principle.

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In quantum physics, a force can be represented as a wave or a particle (or vice versa). And, in terms of space-time, gravity is a "negative" wave, or a wave that represents attractive energy which interacts with energy & matter. Since waves can be thought of in quanta, the graviton, a gravitational wave's particle representation, is a unit of attractive force. Therefore, gravity's status as a force is preserved by quantum theory.(This is illustrated when you are accelerating, by the force you feel opposite to the direction of acceleration, relative to the frame of reference that is changing velocity relative to an outer field (note that this was said by einstein,(not by me; I don't see how this illustrates how spacetime works) though not word for word))

Answering your second question, the interaction with irregularities in space-time is the particle-particle interaction. Plain space-time has no energy, and therefore no quanta to affect matter.

Next, you are correct in assuming that space-time is the "force field" that matter interacts with to create gravity. (WARNING: PERSONAL OPINION AHEAD!!!!) I think this is due to matter being contained within space-time as a moving part of it. If it is part of space-time, it's "positive" energy would necessitate a negative counterpart; hence, gravity exists today.

The last point is because space-time includes all of those higher dimensions. This means that any higher dimensions than length, width, height, and time are also affected by gravity.