# Is gravity a force given that it derives from curved massless space-time?

One of the answers to a similar question regarding gravity concluded that gravity is an "observed effect" of the curvature of space-time. I read this (and other answers) to imply that gravity results from the curvature of space-time and not directly from the masses producing that curvature. Assuming that space-time is massless, where is the $$m$$ in $$F = m a ,$$ and without it how can gravity be considered to be a force?

The following answer was provided in response to a related question. It was closest to providing an answer to the question I tried to pose. My apologies for not citing the author. I'm a layman, this was my first question on the forum and I was not able to retrace the thread that led me to that answer.

However, general relativity gives a much deeper picture of gravity as a description of the curvature of space-time, so, in a way, gravity is an observed effect of the curvature of space-time, or, if you like, an observed effect of the distribution of mass and energy.

• As is famously said, matter (and energy and momentum) tells space(time) how to curve, and space(time) tells matter (and energy and momentum) how to move. The original quote is due to [Wheeler] of which I took the liberty to modify it just a little bit. : en.wikiquote.org/wiki/John_Archibald_Wheeler – Dvij D.C. Sep 26 '18 at 2:05
• If you observe an object moving with acceleration $a$ in a gravitational field, then $m$ in the equation $F=ma$ is the mass of this object, but not the mass of the source of the field. – safesphere Sep 26 '18 at 6:57
• Nowhere in GR is space-time assumed to have it's own mass. Masses exist within space-time, create curvature and Gravity is really force free motion along the geodesics of the curved spacetime. There are a lot of convoluted concepts in this question making a truly meaningful answer difficult. – ggcg Apr 19 '20 at 23:10

No, in general relativity gravity is not a force. Instead general relativity describes how objects move which are in free fall und thus don't feel a force. Such objects follow curves which are called geodesics. In curved space-time neighboring geodesics are accelerating relativ to each other (away or towards each other), check up geodesic deviation. Whereas in flat space-time geodesics move parallel or linearly to each other.

The source of gravity and hence the source of space-time curvature is the stress-energy-tensor, more precisely its components like energy density, pressure and others. E.g. energy density can be due to a mass.

If "space-time is massless" then there are two possibilities in FRW cosmology, depending on whether or not we have to respect vacuum energy density due to the cosmological constant $$\Lambda$$: Without that the stress-energy-tensor is identical zero and the space-time is flat. With non-zero $$\Lambda$$ the space-time is curved and the universe expands exponentially.

If you have some physics background you must know that there are "effective forces" which do not correspond to a fundamental one , as we think of gravity. The centrifugal force is one such. It is an effective force, not a fundamental one, due to basic conservation laws for matter in motion, conservation of energy, momentum and angular momentum.

In a sense, newtonian gravity is an effective force from general relativity, and this can be demonstrated, as for example in this paper.

In this paper, we shall briefly explore general relativity, the branch of physics concerned with spacetime and gravity. The theory of general relativity states that gravity is the manifestation of the curvature of spacetime, so before we study the physics of relativity, we will need to discuss some necessary geometry. We will conclude by seeing how relativistic gravity reduces to New- tonian gravity when considering slow-moving particles in weak, unchanging gravitational fields.

You say:

Assuming that space-time is massless, where is the m in F=ma, and without it how can gravity be considered to be a force?

This is a wrong assumption. If there are no masses everything is flat, no curvature .

Well gravity has all the characteristics of a force. Actually, it is the most fundamental of all the forces. Curving of space is how GR describes it.

A body in free fall does not feel the force because the force is acting on each and every part of the body and there is nothing to feel. Feeling is a result of relative pressure/stretch of different parts of the body under stress. We feel the force while standing on the ground because our feet/skeleton makes us feel it. Even under free fall - think about spaghettification!

To Anna v's point in her answer on centrifugal force - I consider centrifugal force, as fundamental as gravity. Because the centrifugal force is due to inertia which is as basic as gravity. In my opinion, gravity and inertia are two sides of same coin. Think of the equivalence principle.