# Why shouldn't gravity be a force?

Why shouldn't gravity be a force?

I am interested to know the reasons why we shouldn't treat gravity as a force in, for example, General Relativity. Won't we be able to model it accurately by treating it as only a force?

Curved spacetime is currently considered to be the model which reflects best gravity including the equivalence principle, and it is complying excellently with the needs of astronomy.

Einstein did not "prove" that spacetime is curved, but he used it as a model for his description of gravity. And it is working so fine that nearly everybody uses curved spacetime for explaining gravitation and general relativity.

However, we have to remember that curved spacetime is only a model, and you can also imagine gravity as a field (in compliance with general relativity) see e.g. this article.

Yes, we would not be able to really calculate it, or model it well, or probably at all. General Relativity (GR) takes the Equivalence Principle seriously. It uses the mass (call it gravitational mass, really it includes anything that in some way contributes to the so called energy-stress tensor, so radiation also etc) as the source of gravity, and uses that to calculate the geometry of spacetime (with some appropriate boundary/initial/final conditions), and then all particles travel in geodesics of that spacetime.

The equations for the spacetime as function of the stress energy tensor are Einstein's equations. See https://en.wikipedia.org/wiki/Einstein_field_equations

So the mass just enters in in creating the spacetime, upon which everything travels then in geodesics (if no other field or force is present). The only difference on particle masses is if their mass is 0, ie, if they are radiation, or massless particles like the photon. Then, they still travel in geodesics, but they are so-called light-like or null geodesics (locally their speed will be c, the line element $ds^2$ will be zero, i.e. null). The other thing is the Einstein equations are nonlinear, gravity in essence interacts with itself, and that is hard or impossible to do in a force equation, though possible and done other non-linear theories in Quantum Field Theory. It is one (actually that it interacts with all forms of energy) of the reasons that trying to quantize GR leads to a non-renormalizable quantum theory.

In Newtonian physics you calculate a force from the sources creating the force, maybe through a field, and then use the force divided by m (except if m = 0, where it has nothing to say) to get the acceleration, and then the trajectory. GR gets the trajectory directly. GR says it is not a force, but a property of spacetime, and how particles move in it.

That is why force is not a useful entity in GR. Some people still use the term, conceptually, to mean the effect of gravity through spacetime, but it's easy to get confused.

We should discuss two viewpoints: Newtonian and Einsteinian.

In Newton's view gravity is a force that causes massive bodies to be accelerated. However, in Einstein's view gravity is a manifestation of the curvature of spacetime. Despite the fact that these views are conceptually very different, they yield the same predictions in most of cosmological contexts. In the limit of deep potential minima (or strong spatial curvature), only general relativity yields the correct results.

Principle of equivalence:

The gravitational force acting between two objects is $$F=-\frac{GM_g m_g}{r^2}$$ where $m_g$ is gravitational mass. The negative sign in above expression implies that gravity is always an attractive force. According to Newton's second law of motion force and acceleration are related by $$F = m_i a$$ where $m_i$ is the inertial mass that determines the resistance acceleration by any force.

As it turns out, gravitational mass and inertial mass are equal. $$m_g = m_i$$ However, there is no reason for them to be equal, it is completely coincidence (In Newtonian point of view). This is what motivated Einstein to devise his theory. In GR, curvature is a property of spacetime itself and curved spacetime tells mass-energy how to move. Therefore, gravitational acceleration of an object should be independent of mass and composition, it is just following geodesic that is dictated by the geometry of spacetime.

• Could you tell me in what case does $m_g \neq m_i$? – non-sensical Sep 3 '16 at 12:51

If you jump down what do you feel? You feel that you are weightless. At the same time it is obvious that you get accelerated.

What do you feel when you push the accelerator pedal of your car? You feel that you are heavier than usual. At the same time it is obvious that you get accelerated.

As you know force is defined as the product of mass and acceleration. From our everyday experience we define a positive acceleration as something that makes us heavier. So the first example contradicts this experience.

So if you sit in a carousel and swing around you are accelerated (since any change in direction is an acceleration) and get heavier. Flying in the ISS you swing around (the earth) but feel not any acceleration and you stay weightless. To give the phenomenon of acceleration in a gravitational field a name Einstein named the gravitation not a force but a bending the space and locally slowing down or speeding up time.

What I told you is what is our experience today. The genius of Einstein is that he derived this from thought experiments and the guess of equations in the General Relativity.