# If gravity isn’t a force how does it cause motion? [duplicate]

In general relativity, gravity is the effect of curving space-time caused by mass and is described as an acceleration.

So what exactly is causing this pull towards the earth?

Is a force not required for movement?

Is acceleration and force just interchangeable so the force is just the mass times that acceleration?

What is causing us to “slide” down this curve of space and time?

I feel like this is kind of vague and I am not sure if I got across what I really wanted to ask, but if you could explain how exactly gravity causes motion I would really appreciate it.

• Does this answer your question? How exactly does curved space-time describe the force of gravity? Sep 6 at 2:03
• It’s a good question and no one here or The so-called already answer have answered it. Every answer is just kicking the can down the road and not addressing the cause. Words like warping, travel, follow, curved, physically motivated, walks straight, and movement are all verbs describing real physical actions. Nobody wants to explain how matter or mass are supposed to curve space or space time. Or explain why an object following a straight line should accelerate. Why an object following a straight line should stop and go the opposite way? Everyone’s ignoring the physical aspect of it all. Sep 6 at 18:52

Is a force not required for movement?

A force is not required for movement. A force is required for (proper) acceleration. An object in free fall does not accelerate as can easily be confirmed by an accelerometer attached to the object.

So what exactly is causing this pull towards the earth?

There is no pull downwards towards the earth. It is the surface of the earth that is accelerating upwards, again as can easily be confirmed by an accelerometer on the ground.

If spacetime were flat then the fact that the surface of the earth is accelerating upwards everywhere on Earth would imply that Earth would be expanding. However, the spacetime is curved which allows the surface of the Earth to accelerate upwards without the Earth expanding.

What is causing us to “slide” down this curve of space and time?

That is inertia. We go in a straight line at constant speed, as measured by an accelerometer unless acted on by a real force. This is called inertia.

• I know gravity is the 4-velocity, however, I tried thinking of gravity as the centrifugal force where the object is seeing the side of the spinning container (the gravitating mass) coming closer even though they are not moving towards it by accelerating. This would be of course in 3 + 1 dimensions where you are only moving through time until you experience a gradient in time that makes you see the mass coming towards you in space. Is the centrifugal analogy a correct way to thinking about gravity? Sep 6 at 10:37
• Really nice answer. Sep 7 at 4:28

All objects follow paths in curved spacetime, and these paths are called geodesics. An object falling in the earth's surface follows a geodesic that depends on its velocity and not its mass (light and other massless objects, where $$v=c$$ always, follow geodesics).

Suppose you define a Euclidean coordinate system on earth with you at the origin, and suppose you throw an object up into the air at $$t = 0$$. The position of the object in spacetime is given by functions of an affine parameter, so that the path the object takes makes it appear like it is accelerating to earth, giving rise to the idea that gravity is a force. In reality, the motion of the object in the coordinate system is described by the geodesic equation $$\frac{d^2x^{\alpha}}{dt^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}=0$$ where $$x^\mu \,, \mu\in[0,1,2,3]$$ is the position of the object in the coordinate system and $$t$$ is the proper time or the time measured by a clock following the object.

The first term on the LHS is the acceleration of the object in our coordinate system. The second term describes the effects of gravity where $$\Gamma^\alpha_{\mu\nu}$$ are called the Christoffel, or connection, symbols, and the other two are velocity. These symbols encode the effects of the curvature of spacetime. It shows us that the curvature of spacetime changes the acceleration of the object, based on its velocity through spacetime.

If there is no spacetime curvature, then all of the Christoffel symbols vanish $$\Gamma^\alpha_{\mu\nu}=0\leftrightarrow \frac{d^2x^{\alpha}}{dt^2}=0$$ and there is no acceleration (Christoffel symbols may not completely vanish in non-Euclidean coordinates, but this changes none of the physics). The important point is that the acceleration of the object is completely determined by the curvature of spacetime$$^1$$.

You may have heard this many times in general relativity, and that is matter curves spacetime, and the effect of the curvature of spacetime is to cause gravitational force. But it isn't a force that acts on the object, but instead it’s that the object follows a geodesic in spacetime.

$$^1$$ And the Einstein field equations tell us that the amount of spacetime curvature depends on the matter and energy content in the region of the spacetime, or $$G_{\mu \nu }+\Lambda g_{\mu\nu}= 8\pi G T_{\mu \nu }$$ where the LHS represents the curvature of spacetime and the RHS represents the stress–mass/energy–momentum content of the spacetime.

1. Any body has a constant speed in space-time; when you at rest in space, you moving with a constant speed through time; when you moving in space, your motion in time is slowed down. The total space-time vector is always constant.
2. Gravity warps the space-time, squeezing the time, so the only choice the body has to keep the space-time speed constant is to change the position (since its time movement can’t fit anymore to the “time in a plane space-time”) so that it starts moving which we consider as acceleration.
3. The only direction it could move is ahead to Earth, since the curvature of space-time in direction to Earth increases, other directions are shorter.

You may find more detailed explanation here: Acceleration due to Gravity & Space-Time Continuum Curvature (General Relativity Vs. Newton)

– Community Bot
Sep 6 at 1:53
• @Community, would the link I've added be enough or you need more formal source, if so, please specify which will suffice? Sep 6 at 2:03
• Community is not a person, but a bot. This is an automated comment left by anonymous reviewer. I'm sorry for the inconvenience Sep 6 at 5:29
• @Prallax, I am lost even more ) . If Community is a bot, why are you sorry? Sep 7 at 21:01
• Because since a recent update of SE, that allowed users to post anonymous comments through Community, there has been some confusion, especially for new users. See this meta post physics.meta.stackexchange.com/q/13810 Sep 7 at 21:07

Is a force not required for movement?

We can summarize the evolution of the notion of inertia in 3 steps:

1. In the world of our daily experience, objects stay at rest unless some force moves them. But on the other hand, if we throw something away, even after losing contact (so without the force of my hand) it keeps moving for a while. So, it seems that a force is not always necessary.

2. Objects move in uniform rectilinear motion unless some force (friction for example) changes that motion. In that case, a falling (and accelerated) object must be under a force, called gravity. It is a peculiar force however, because a load cell mesures no force in a free falling object.

3. Objects move following trajectories called geodesics in spacetime, unless some force changes its motion. It is the concept of GR.

You will find that gravity behaves exactly like a low pressure system. That could be a low pressure system in the weather, where air (and blowing leaves) are sucked into the centre of the low. Or low pressure in water, where a cork will shoot from the bottom of a lake to the top. So the motion of something falling is just that object following the pressure gradient from high pressure to low pressure.

But what is the pressure in? I would conjecture that it is a low pressure of time. That is, time runs slower near a planet, so all objects want to move where time runs slowest.

All objects fall at the same speed, from a feather to an anvil, because falling has nothing to do with the object itself. It only has to do with the pressure above and below the object.

So don't think about the standard bowling ball in a trampoline model of warped spacetime. Think more about a weather pressure map where the wind is sucked into the middle. This is what causes objects to fall. In this image a spinning black hole leads to frame dragging. You can see that it looks exactly like a spinning tornado.  And these two images show merging galaxies, compared with merging typhones. The similarities are striking.