# How does general relativity theory explain gravitational pull? [duplicate]

I watched some videos on YouTube that explain why gravity is not a force, according to general relativity theory.

I can wrap my head around the idea that spacetime can be curved due to a massive mass, so that objects that enter those curved spacetime from outside, while appearing to be moving in a new curved trajectory, are actually still moving in a straight line as before. And this curved line is called geodesic.

However, I can't explain other things.

For example, if I jump up (away from the direction that the earth is moving in), what force pulls me down to the surface if not gravity? I should keep moving unless there is another force that act on me right?

Or if a small object stands perfectly still in the middle of the empty universe, then suddenly a massive mass appears next to it, will the small object start moving closer to the massive mass and eventually collide with that mass? If so then what pull or push it?

Maybe another way to ask my question is how does object move through spacetime? Does it really like the bent sheet analogy, where object has to follow a certain rules? And rule #1 is that if you're in a curved surface, always run toward the deepest place?

Or if a small object stands perfectly still in the middle of the empty universe, then suddenly a massive mass appears next to it, will the small object start moving closer to the massive mass and eventually collide with that mass? If so then what pull or push it?

Consider a small object that stands perfectly still. It stands still in space, i.e., it has constant coordinates $$(x_0,y_0,z_0)$$, but it "moves" forward in time - it has a "world line" of $$(t, x_0, y_0, z_0)$$ for all $$t$$s. As you said, it appears to be sitting completely still in space.

Now, add the mass. It curves spacetime, not just space. The world line $$(t, x_0, y_0, z_0)$$, which used to be a straight line with constant space coordinates, now becomes a geodesic on that curved 4-dimensional space. With the field equations from general relativity you can calculate what this geodesic is, but the important observation is that it will no longer be a straight line with 3 coordinates constants - those space coordinates will now be different for each $$t$$, i.e., the object begins to move.

• I see an old answer to a duplicate question, physics.stackexchange.com/a/222433/219989 that is similar in spirit to my answer, but with actual formulas (using the specific spacetime deformation that general relativity predicts) that show not just that the small object will move somehow, but specifically how it will move - if the masses are small it will accelerate in the same way that Newton's law suggests. Commented Jul 11, 2023 at 8:36

TLDR

It's all inertia. If you stand on Earth's surface, your body wants to follow a geodesic toward the center of the Earth. Your body wants to fall. But, the surface blocks your path. It gives you a proper acceleration in the direction opposite to "falling." $$f=ma$$ where $$f$$ is the force you feel pressing upward against your feet, $$a$$ is the proper acceleration of the Earth's surface, and $$m$$ is your own inertial mass.

P.S., This is one reason why General Relativity (GR) was considered to be a "win." Prior to GR, there was no explanation for the apparent equivalence between "inertial mass" and "gravitational mass." In GR, they're not equivalent, they actually are the same thing.

• Why are you distinguishing between "equivalent" and "the same thing"? Mathematically, "equivalent" means "the same thing" Commented Jul 11, 2023 at 3:31
• @justhalf 14 == 2 mod 12. They are equivalent, but not identical. I think this is the sense that Solomon means, roughly speaking. Commented Jul 11, 2023 at 6:06
• in mod 12, they are the same thing, that's the point :) If we are not talking in mod 12, then they're not equivalent, but they are equivalent in mod 12, and in mod 12 they're also the same thing. Commented Jul 12, 2023 at 3:10
• @justhalf, Maybe I said it wrong. In Newton's model of the universe, we had to wonder why the $m$ in $F=GMm/r^2$ had the same numerical value as the $m$ in $F=mA$ for the same given object. Nobody ever was able to answer that. In GR, there is no $F=GMm/r^2$. There is only $F=mA$. The same formula (and therefore, the same $m$) explains both the force of gravity and the forces that cause other accelerations. Commented Jul 12, 2023 at 11:54
• @SolomonSlow ah, I see now. That's interesting insight, thanks for sharing it! Commented Jul 12, 2023 at 14:00

What we call geodesic equation is in reality a system of differential equations. We can say that objects follow geodesic paths even without GR. If you jump, only gravity force is acting on you while in the air. So $$F_g = ma = m\frac{d^2r}{dt^2}$$. So, according to Newton's law:$$m\frac{d^2\mathbf r}{dt^2} = -\frac{GMm\mathbf {\hat r}}{r^2} \implies \frac{d^2\mathbf r}{dt^2} + \frac{GMm\mathbf {\hat r}}{r^2} = 0$$ If we define a potential function $$\Phi = -\frac{GM}{r}$$ the Newton vectorial differential equation can be seen as a geodesic equation. There are 3 equations, one for each vector component: $$\frac{d^2\mathbf r}{dt^2}+ \nabla \phi = 0$$

Nobel prize winner Kip Thorne referred to what he calls Einstein's Law of Time Warps. Thorne said "Things like to live where they age most slowly. Gravity pulls them there. The Earth's mass warps time according Einstein. It slows time near the surface of the Earth. And this time warp is what produces gravity."

So I'll give you a different way to think about it. A planet warps space. This warping of space causes a dilation of time. Put together we have the warping of spacetime as most would call it. So, according to Thorne, gravity is caused by the dilation of time, rather than the warping of space (which of course is the initiating cause of the dilation of time.)

So why does time dilation cause things to fall? The answer would come down to the fact that everything wants to give up energy (or increase entropy under the second law of thermodynamics). By moving into a region of space with slower time, an object is giving up energy and increases entropy.

DO NOT think of gravity like the bowling ball in a trampoline model. This model is seriously flawed. Rather, think more about a low pressure system in the weather. The wind, and your bed sheets on the clothes line, gets sucked into the center of a low pressure system. So if you consider the centre of a planet as the centre of the low pressure system of time, you have a reason why things fall. This model works much better.

• Works much better… for what? It doesn't seem to make any experimental predictions. Commented Jul 10, 2023 at 22:51
• Why is this downvoted? It's basically correct, if a bit elliptical. A note: the warping of space does not cause a warping of time. Mass simply warps time and space. Commented Jul 10, 2023 at 23:36
• @CharlesHudgins you say that mass simply warps time and space. Now you have to ponder how that occurs. That's the missing part of this whole discussion Commented Jul 11, 2023 at 3:52
• No, that's a question for the GUT people to figure out. GR simply (and in fact only) tells us that the stress energy tensor is proportional to the Einstein tensor (a measure a curvature of space and time). A corollary of this is that to first order (in a precise sense) things fall toward slower time. Now why exactly that should be the way things happen to first order, I don't have a snappy explanation for. Commented Jul 11, 2023 at 4:03
• @CharlesHudgins and hence my thoughts on the relationship between warping of space and warping of time. Someone had to come up with a reasonable solution. Commented Jul 11, 2023 at 15:08

# Cyliverse

The "cyliverse" is a 1-dimensional universe comprised of a circle that evolves over time as a cylinder. Objects within this universe are points that can move back and forth along the circle. An object at rest has a worldline that just runs straight up the cylinder. Two objects at rest near each other form two parallel worldlines running up the cylinder. Hopefully this is very clear in your mind.

# Curvature

Now, imagine that the cyliverse has a "dome" at some point. This dome is a half-sphere with the same radius as the cyliverse. Now, this dome does not describe all objects in the cyliverse. Rather, it only describes inertial objects. That is, objects that are not accelerating. What you notice is that if objects are just coasting, when they reach the dome, they all start coming together, as if the universe is heading for a Big Crunch. This is also equivalent to all objects falling into a black hole with no discernable center or location.

Any two straight lines you draw on this dome will meet at the pole. That corresponds to worldlines in which any two objects at rest will end up moving towards each other for no apparent reason. Note that we have not invoked any explicit forces to cause this motion. We did not conjure a rocket or asteroid or laser. All we did was curve spacetime. And all the objects have to do is not try to move. And yet, by sitting still, they end up moving anyway. That's because spacetime itself is curved in such a way that forces this motion.

This is a very crude illustration that is a little bit misleading, but it should still give you a sense of what it means for curved spacetime to cause gravitational motion.

# A Star is Born

A more realistic picture would be to imagine us manifesting a large mass somewhere on the cylinder. If we look at the ring of the cyliverse as a clock, we can say that there is now a large mass at 3 o'clock. What does that do to the worldlines near it? Well, instead of remaining straight, it tilts them all towards 3 o'clock. And the closer the line is to 3 o'clock, the more it tilts. Lines far away (like at 9 o'clock) are curved so little you can't even tell the difference. We could say that the curvature mostly disappears around 1 and 5 o'clock. What does it mean for the worldlines to be curved? It means that an object at rest sitting at 2 o'clock will move towards 3 o'clock. It's like someone took all the worldlines around 3 o'clock and pinched them together in the future. But they didn't just pinch them at one spot. They pinched them at every spot, but for worldlines that started a particular distance in the past.

So if you are at rest at 2 o'clock, it might take 1000 s to fall into the star at 3 o'clock. That means that your worldline starting at 2 o'clock at $$t_0$$ curves towards 3 o'clock, intersecting with it at $$t_0+1000$$. But there is another worldline at 2 o'clock, $$t_0+8500$$, and it curves into 3 o'clock at $$t_0+9500$$, etc. So the cyliverse is warped in this way, where all the worldlines around 3 o'clock curve into it no matter what time they start.

In order to not fall into the star, you have to get off the geodesic by applying a force away from it. You can "hover" a fixed distance from the star by applying thrusters which exactly counteract the curvature of your worldline. But you must expend a force to do so, because the inertial path will bring you towards the star.

# Conclusion

The curvature of space causes motion for inertial objects by decreasing their relative distance in the future. Gravity curves space by causing inertial worldlines to bend towards the gravitational source. And thus, inertial objects on these worldlines will naturally come closer together, because that is what their geodesics do. A train follows the track around a curve because that is where the track leads it. Objects in a gravitational field also "follow the track", and are "pushed along" by time.

For example, if I jump up (away from the direction that the earth is moving in), what force pulls me down to the surface if not gravity? I should keep moving unless there is another force that act on me right?

The "natural" state is to be in 'free-fall' in a reference frame where you feel no forces acting on you at all. The force you feel just standing on the Earth is the force of all that compacted matter preventing you from following your natural path through space-time. After your feet leave the ground when you jump, you are nearly in that state (not accounting for air resistance). You are moving away from the Earth and slow down in relation to the Earth and eventually start heading towards the Earth, all without forces acting on you. When you hit the ground, you then the force of that compact matter again begins to interfere with your natural trajectory in your reference frame as you feel it accelerating you away from your natural path. See how all that change in direction is all relative to something else, the Earth.

I think the problem comes from our brains evolving on a planet with gravity, and spending all our lives (most of us anyway) in that system where the ground (or water, or aircraft) is pushing against you, applying a force to keep you from following your natural path through space-time. The Earth is always nearby and nearly always applying that constant force to us.