# How exactly does curved space-time describe the force of gravity?

I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or trampoline bending under an object, which causes other objects to come together, but I always thought this was a hopelessly circular explanation because the blanket only bends because of "real" gravity pulling the object down and then pulling the other objects down the sloped blanket. In other words, to me, it seems that curved space wouldn't have any actual effect on objects unless there's already another force present.

So how is curved space-time itself actually capable of exerting a force (without some source of a fourth-dimensional force)?

I apologize for my ignorance in advance, and a purely mathematical explanation will probably go over my head, but if it's required I'll do my best to understand.

• In many "video" explanations of general relativity curvature of Time is omited, time is certainly not easy to graph with the blanket example, but sometimes it's not even mentioned, perhaps lack of self-questioning of the explainer, then it's a good question +1
– HDE
May 16, 2011 at 17:28
• I would modify this question as follows: If we could put a particle in orbit around a star with no other planets or satellites and then use a fictional device to cancel all the inertia of the particle, it is obvious that the curve of space-time is towards the star but what is not obvious is what would make the particle begin to move towards the star after all its momentum/inertia were canceled. Gravity is not a force so how would the particle 'know' that it needs to start accelerating towards the star? Sep 29, 2015 at 20:39
• The blanket/trampoline isn't meant to explain anything in the sense of suggesting an underlying mechanism. It's a way of thinking about an esoteric subject far removed from ordinary experience in terms of something more familiar. "Vectors are like arrows" doesn't mean vectors are made of obsidian or fired from bows. In any case, the blanket/trampoline is entirely wrong as a model of curved space in general relativity, though it is a surprisingly accurate model of Newtonian gravity: see this answer. Feb 4, 2019 at 8:37
• Nothing is as instructive as reading. Especially this: archive.org/details/TheClassicalTheoryOfFields Apr 29, 2019 at 18:27

Luboš's answer is of course perfectly correct. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve).

Image a 2-sphere (a surface of a ball). If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. south poles).

Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. E.g. for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. to the precession of the perihelion of the Mercury).

So much for the explanation of how curved space-time (discussion above was just about space; if you introduce special relativity into the picture, you'll get also new effects of mixing of space and time as usual). But how does the space-time know it should be curved in the first place? Well, it's because it obeys Einstein's equations (why does it obey these equations is a separate question though). These equations describe precisely how matter affects space-time. They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. for a Sun you'll obtain that trampoline curvature and the planets (which will also produce little dents, catching moons, for example; but forget about those for a moment because they are not that important for the movement of the planet around the Sun) will follow straight lines, moving in ellipses (again, almost ellipses).

• Thanks heaps guys, it's starting to make some sense. So that makes sense to me with moving objects, but I still don't quite understand how it causes objects to accelerate. For example, with your analogy, what if the ants were stationary on the ball? When we lift something off the ground and let go, it accelerates toward the earth. Is this just because that's the "straighest" line through the curved spacetime around the earth? Why must it always be "moving" through a straight line, and what does it mean in terms of curved spacetime for something to be stationary?
– Zac
Jan 16, 2011 at 12:08
• Also @Zac, for something to be stationary in spaceTIME means that it only exists for a single instant in time! Even something that stays stationary in space for all time moves on a curve in spacetime. (think about what an x vs. t plot looks like for a stationary object)
– wsc
Jan 16, 2011 at 18:33
• @wsc: it's really called perihelium in my language (Slovak) so I never imagined it might be something different in English. Anyway, thank you :) Jan 16, 2011 at 19:34
• @Marek: nor did I realize it was different in Slovak; always nice to learn! Anyway, I was using that meaning of stationary since that was what @Zac was using: his question seemed to me to be: 'Sure you have geodesics on curved manifolds, but why do the ants have to move?' Which is a very good question, you just have to remember that time is a coordinate too.
– wsc
Jan 16, 2011 at 20:07
• @AdamHughes: So the crucial difference is that spacetime includes curved time as well. No object can remain truly stationary in spacetime because that would require being stationary in time. An object that magically appears above Earth may begin stationary in the space dimension of spacetime, but it nevertheless continues to "move" through the time dimension. It must follow the geodesic (straightest/shortest possible path) from that point, and the geodesic from the point above Earth points through time but also towards Earth because of its effect on the curvature of spacetime.
– Zac
Aug 4, 2017 at 16:06

There are actually two different parts of general relativity. They're often stated as

1. Spacetime tells matter how to move
2. Matter tells spacetime how to curve

Point #1 is actually straightforward to explain: objects simply travel on the straightest possible paths through spacetime, called geodesics. The paths only seem curved because of the warping of spacetime. If you're a physicist, then you would want to know that that fact can be derived from the principle of extremal action (with all the requisite mathematical details), but if you don't want to wade through the math, hopefully it at least makes sense that objects move on "straight" lines. There is no actual force involved when a massive (or even a massless) object's trajectory curves in response to gravity, because it doesn't take any force to keep something moving on a straight line. (I can definitely expand on this point if you want)

Now, I mentioned that spacetime needs to be warped in order for objects' trajectories to appear curved to us despite them actually being "straight." So the essence of point #2 is, why is spacetime warped in the first place? Physics doesn't have a good answer to that. Technically, we don't have an answer to point #1 either, but the "straight line" argument at least makes it seem plausible; unfortunately, there's no equivalent plausibility argument for why spacetime warps itself around matter. (Perhaps someday we will come up with one) All we can do right now is produce equations that describe how spacetime behaves around matter, namely the Einstein equations which can be written $$G_{\mu\nu} = 8\pi T_{\mu\nu}$$ among other ways.

• I never got why Wheeler wasn't as famous as Feynman. He had that same magical way of reducing things down to really clear, simple statements that made complicated things seem obvious. Apr 17, 2015 at 2:48

The trampoline analogy needs an extra source of gravity - because this is what the laymen, the recipients of the explanation, intuitively understand - but the real general relativity doesn't need any extra "external" gravity.

Instead, general relativity says that the space is getting curved by Einstein's equations, $$G=T$$ where the left-hand side are numbers describing the curvature at a given point and the right hand side is the density of matter and momentum. I omit indices and constants haha. So general relativity says how the spacetime is curved under the influence of matter.

The second part of the story is that general relativity also says how matter moves in external geometry. It moves along "geodesics", lines that are as straight as you can get. $$\delta S_{action\,ie\,proper\,length} = 0$$ This actually means that the objects move along the predicted, seemingly curved trajectories. These trajectories are actually as straight in the curved spacetime as you can get.

Imagine that there is a hemisphere replacing a disk in the trampoline. So there exists a (nearly) straight line on the hemisphere - namely the equator near the junction with the rest of the trampoline. Note that the equator on the Earth is a maximum circle - so it is one of the straightest lines you can draw on the surface of Earth. The same is true for all actual trajectories that objects choose in spacetime of general relativity.

So in the hemisphere-above-trampoline example, particles can orbit around the equator of the attached hemisphere, just like planets, because it is the straightest and most natural line they can choose. I don't use any external gravity to explain the real gravity; instead, I use the principle that particles choose the most natural - the straightest - line they can find in the curved spacetime.

• Roadways are always banked to a certain degree to facilitate turning, but with a limited budget this is never enough to eliminate the need to turn the steering wheel. You might not need to turn the wheel on some race tracks, but you would still need gravity. Correct me if I am wrong, but isn't the only way you are ever going to turn without ANY outside force at any speed is with a bank angle of 180*, and that only ever occurs around a spinless black hole. If the orbital path of the Earth was a (relatively) straight line, then light would orbit around the sun too. Feb 7, 2021 at 9:57

The other answers are more or less correct, but perhaps I can say something more to the point of the question, *How is curved spacetime itself actually capable of exerting a force?

No force whatsoever is needed.

Gravity is not a force. What is a force, anyway? Newton clarified for almost the first time in Science what a force is: First I will say it, then explain it: A force is something which makes the motion of a body deviate from uniform straightline motion.

Newton pointed out that bodies have a tendency, inertia, to continue in whatever direction they are already going, with whatever velocity they have at the moment. That means uniform, rectilineal motion: steady velocity, same direction. Newton actually knew this was what would be later called a geodesic, since « a straight line is the shortest distance between two points ».

Newton then went on to say that to overcome inertia, to overcome this tendency, requires a force: force is what makes a body depart from the geodesic it is (even momentarily) headed on (its direction and speed).

It was then Einstein (and partly Mach before him) who said this does not get to the essence of the question. For Einstein, any coordinate system had to be equally allowable, and in fact, space-time is curved (as already explained by other posters). A body or particle under the influence of gravity actually does travel in a geodesic....i.e., it does what a free particle does. I.e., it does what a particle not under the influence of any force does. So gravity is not a force.

Newton did not realise that space-time could be curved and that then the geodesics would not appear to our sight to be straight lines when projected into space alone. That ellipse you see in pictures of planetary orbits? It is not really there of course since the planet only reaches different points of the ellipse at different times...that ellipse is not what the planet really traverses in space-time, it is the projection of the path of the planet onto a slice of space, it is really only the shadow of the true path of the planet, and seems much more curved than the true path really is.

( ¡ The curvature of space-time in the neighbourhood of the earth is really very small ! The path of the earth in space-time would even appear to be nearly straight to an imaginary Euclidean observer who, in a flat five-dimensional space larger than ours, was looking down on us in our slightly curved four dimensional space-time embedded in their world. It's $$ct$$, remember, so the curving around the ellipse gets distributed over an entire light-year, and appears to be nearly straight...and is straight when one takes into account the slight curvature of space-time.)

Since every particle under the influence of gravity alone moves in a geodesic, it does not experience any force that would make it depart from its inertia and make it depart from this geodesic. So gravity is not a force, but electric forces still do exist. They could overcome the inertia of a charged body and make it deviate from the geodesic it is headed on: change its speed and direction (when speed and direction are measured in curved space-time).

Einstein (and me too) did not want to change the definition of force in this new situation, since after all electric forces are known to exist and are still forces in GR. So the old notion of force still retains its usefulness for things other than gravity. To repeat: if a body is not moving in a geodesic in space-time, you go looking for a force that is overcoming its inertia....but since gravity and curvature of space-time do not make a body depart from a geodesic, neither of them is a force.

See also http://www.einstein-online.info/elementary/generalRT/GeomGravity.html which avoids the trampoline fallacy and has a great image of the great circle.

• Gravity is not a force in GR. Gravity was a force in classical mechanics. Gravity is ________ in quantum theories (Sorry, I don't know enough to fill in the blank.) My point is that all of these realms are models that predict the motion of terrestrial and astronomical objects. Some models (e.g., GR) make better predictions than others (e.g., classical), but do any of them tell us what gravity really is? Sep 21, 2015 at 20:14
• @james large, the answer is no. There isn't a complete theory of gravity. No one knows what causes. Jun 13, 2017 at 16:26

As others mentioned, the main problem with the common visualization is, that it omits the time dimension. In the animation linked below the time-dimension is included to explain how General Relativity differs form Newton's model.

• Another great visual explanation - no trampoline, no fourth dimension, no ants. Apr 1, 2021 at 21:30

It is straightforward to see how the geometry of spacetime describes the force of gravity -- you just need to understand the geodesic equation, which in general relativity describes the paths of things subject to gravity and nothing else. This is the "spacetime affects matter" side of the theory.

To understand why curvature in particular, as a property of the geometry, is important, you need to understand the "matter affects spacetime" side of general relativity. The postulate is that the Gravitational Lagrangian of the theory is equal to the scalar curvature -- this is called the "Einstein-Hilbert Action" --

$$S=\int{\left( {\lambda R + {{\mathcal{L}}_M}} \right)\sqrt { - g}\, d{x^4}} {\text{ }}$$

You set the variation in the action to zero, as with any classical theory, and solve for the equations of motion. The conventional way to do this goes something like this --

$$\int{\left( {\frac{{\delta \left( {\left( {{{\mathcal{L}}_M} + \lambda R} \right)\sqrt { - g} } \right)}}{{\delta {g_{\mu \nu }}}}} \right)\delta {g_{\mu \nu }}\,d{x^4}} = 0$$ $$\sqrt { - g} \frac{{\delta {{\mathcal{L}}_M}}}{{\delta {g_{\mu \nu }}}} + \lambda \sqrt { - g} \frac{{\delta R}}{{\delta {g_{\mu \nu }}}} + \left( {{{\mathcal{L}}_M} + \lambda R} \right)\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} = 0$$ $$\frac{{\delta R}}{{\delta {g_{\mu \nu }}}} + \frac{R}{{\sqrt { - g} }}\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} = - \frac{1}{\lambda }\left( {\frac{1}{{\sqrt { - g} }}{{\mathcal{L}}_M}\frac{{\delta \sqrt { - g} }}{{\delta {g_{\mu \nu }}}} + \frac{{\delta {{\mathcal{L}}_M}}}{{\delta {g_{\mu \nu }}}}} \right)$$

$${R_{\mu \nu }} - \frac{1}{2}R{g_{\mu \nu }} = \frac{1}{{2\lambda }}{T_{\mu \nu }}$$

To fix the value of $\kappa=1/{2\lambda}$, we impose Newtonian gravity at low energies, for which we only consider the time-time component, which Newtonian gravity describes (I'll use $C$ for the gravitational constant, reserving $G$ for the trace of the Einstein tensor) --

$$\begin{gathered} {G_{00}} = \kappa c^4\rho \\ {R_{00}} = {G_{00}} - \frac{1}{2}Gg_{00} \\ \Rightarrow {R_{00}} \approx \kappa \left( {c^4\rho - \frac{1}{2}\frac{1}{{c^2}}c^4\rho c^2} \right) \approx \frac{1}{2}\kappa c^4\rho \\ \end{gathered}$$

Imposing Poisson's law from Newtonian gravity with $\partial^2\Phi$ approximating $\Gamma _{00,\alpha }^\alpha$,

$$4\pi C\rho \approx {\nabla ^2}\Phi \approx \Gamma _{00,\alpha }^\alpha \approx {R_{00}} \approx \frac{\kappa }{2}c^4\rho \\ \Rightarrow \kappa = \frac{{8\pi G}}{{c^4}} \\$$

(The fact that this is possible is fantastic -- it means that simply postulating that spacetime is curved in a certain sense produces a force that agrees with our observations regarding gravity at low energies.) Giving us the Einstein-Field Equation,

$${G_{\mu \nu }} = \frac{{8\pi G}}{{c^4}}{T_{\mu \nu }}$$

• This is not an explanation in "layman's terms"... Jun 22, 2013 at 1:20
• I just think the average person interested in OP's question would not have knowledge of Lagrangians, Tensors etc. Jun 22, 2013 at 13:13
• @Comp_Warrior, the About says the site is for Academics, students, and researchers of physics and astronomy, so the average audience should not consist of laypeople and it is perfectly ok to give technical and advanced answers for the people who can stomach it. Even though it looks like this since quite some time, physics se is not meant to be a popular physics forum such as quora for example ... Jun 22, 2013 at 14:46
• Btw the op says he does not mind technical answers, so why are you insisting on answers exclusively in layman terms? Jun 22, 2013 at 15:00
• @Comp_Warrier what dimension10 says, and the SE system is exactly desined that the op can accept the answer he likes best, maybe a popular one, whereas there can legitimately be other more technical answers too, that are liked by other members of the community. Answers to a question are not only meant to serve the op, but the whole community. So there is absolutely nothing wrong with a question getting answers of different level. It would be nice if you stop discouraging good technical posts which are perfectly legitimate. Jun 22, 2013 at 22:20

A complete replacement of the brief answer I wrote some time ago:

More than one person has brought up the idea of a pair of ants walking on the surface of a sphere. Each ant is moving in what, for it, is a straight line, but the get closer together at an increasing rate until they collide. (Provided they're lined up right.)

This is an excellent metaphor, but it can be confusing because each ant is propelling itself, so it could stop if it wanted, and also they do have to be lined up right when they start or they won't collide. If you hold a rock still and then let go, it starts to move, which seems different from the ant picture.

All of these problems disappear if you realize that they don't call it spacetime for nothing. The surface of the balloon is two-dimensional in the ants-on-a-balloon analogy (and really the ants ought to be two-dimensional themselves, living embedded in the surface of the balloon just as we are embedded in spacetime). But it's wrong to think we are only throwing away one dimension to be able to visualize curved space. The right way to think of the balloon is that it has one dimension of space and one of time, so we're really throwing away two out of the four dimensions.

Each ant is racing headlong into its own future, and it can't stop or even slow down. And the ants can't miss each other, because the paths they follow are really the histories of their lives. The paths are called world lines. Each point on a world line is a time and a place that the ant passed through. If two world lines cross, that means two ants were at the same place at the same time.

This is still confusing, because the balloon is round. Which direction is time, and which direction is space? What happens when the ant goes all the way around the sphere? To make sense of these questions, you have to put a coordinate system on the sphere. For this toy universe, it actually makes sense to use latitude and longitude as coordinates. The south pole is kind of a big bang (take this with a lot of salt) and the north pole is the big crunch in the future (that definitely isn't going to happen in real life). The lines of latitude are the time coordinate, which means time progresses along the lines of longitude.

I think the problem for the layman is understanding why there is motion in spacetime and I think a sort of answer is that we already accept motion through time when we think of time and space as separate. Well we are in motion through spacetime where time and space are not separable and when we move through a region of spacetime that contains matter the shortest spacetime path between two events is the one that includes motion through the space bit as well as the time bit (ie not orthogonal to the space axes). That is experienced as falling under gravity.

I am a physicist and I always used to hate the trampoline/ rubber blanket model with all my heart - as it explains gravity with gravity, as it fails to account for the more of space in the center (the rectangles of the rubber blanket become bigger - but from my point of view they had to become smaller as the cubes at the earth in the pic above) and so on. However, some time ago I read a nice laymen - article about how to vindicate the blanket / trampoline model (I add the reference in the comments as soon as I find it). And here is how:

First, you have to replace the small ball (which is under the influence of the big mass in the center) with a car. Second, you have to attach double-faced adhesive tape around the wheels of the car. And now, you try to move this attached-to-space car along a straight way past the center of gravity. But you will experience: It will move in the direction of the center of gravity. Why? Because there is more space for the wheels nearer to the center of gravity to pass over! And what I like the most: If you curve the space the opposite round, i.e. bend the trampoline/rubber blanket in direction to the sky, the space-attached small car will move in a curve in direction to the center of gravity as well! Now, this trampoline model is independent of the underlying gravity! It just explains the curved path of the car by the curvature of the space it is attached to.

Next step: to include the time coordinate. As the rubber blanket model is only 2D (not 4D as the real space time) we have to sacrifice one dimension of space for the inclusion of the time. Never mind, though, as most of the gravitational forces and fields are spherically symmetric - meaning that there is anyways only one space coordinate that matters: the radius r, the distance from the center of gravity. Now, we are bound to move along the time coordinate, we all are and so the space-attached little car is as well. The model with one space and one time coordinate now looks as a river bed where we are all bound to move parallel to the river, even if we do not move in space (=perpendicular to the river). There is a curvature of time that means time runs slower in the vicinity of the river. As a result, the wheels near the river turn more slowly. This is followed by the movement of the car in the direction to the river as time goes by. Gravity.

Welcome back, trampoline model.

This question is already filled with excellent answers. I have only thought of complimenting what the others have said with a bit of a historic and "human" perspective and train of thought. So to answer the question of exactly how spacetime curvature generates the effect of gravity, I will first take a small detour to the state of classical mechanics at the end of the XIX century.
Newton's laws of motion and the birth of Calculus have been pushing forward physics for over a century. Powerful elegant formulations have been invented, and the century of lights has even yielded the blossom of classical electromagnetism and classical thermodynamics. Even so, at the foundations of classical mechanics there are still some incongruencies, which have namely to do with the equivalence of masses problem, and the relativity problem, which are intimately related, as General Relativity would eventually show.

1. Mass equivalence

Have you learned Newton's laws of motion like this? $$F=m\,a \hspace{1cm} F=-\frac{GMm}{r^2}$$ And have you perhaps at a physics lesson used it, by equating the two, to figure out the acceleration of a free-falling body at the surface of the earth? Well, this wasn't always that trivial. Newton's laws were actually "written"(he actually didn't write formulas, it wasn't a habit at the time) like this: $$F=m_i a \hspace{1cm} F=-\frac{GMm_g}{r^2}$$ Where $$m_i,m_g$$ denote the inertial and gravitational mass. This might seem like a superfluous difference, but it absolutely is not. If you actually read the Principia (Newton's famous book on his mechanics) you will encounter a bunch of crazy definitions of up to three kinds of force and two kinds of mass. This was because gravity and other forces were conceived as being different kinds of phenomena. So inertial mass is the ability of a body to resist for example a push, and gravitational mass is the ability to being gravitating towards a massive body. Now here's the crazy thing that no one could explain. They were actually numerically the same. We knew that from Galileo's experiments on inclined planes and other experiments on pendulums; that these two very different kinds of masses, associated with two different phenomena, were numerically the same, for some reason. However, no one took this as something significant enough to address it seriously. But it is actually a huge deal, and one of the reasons is that it points out to gravity being a different kind of force. Really think about this. The thing that regulates the strength at which gravity operates is the thing that measures the "strength" of motion. No other force is like this: gravity was the only force of which charge is the same as the "strength" of motion.

2.Galilean Relativity Another feature of Newton's mechanics is that it obeys Galilean Relativity. Which yielded a fatal problem for Newton's second law. I don't know how familiar you are with calculus, but I will try to translate in the end in any case. Galilean relativity says that the velocity of two frames of reference add together. That is, if you are moving with respect to some reference frame at velocity v, your reference frame will be $$x'=x-vt$$, where $$x$$ is the original reference frame. That means that if you change frames in these conditions (constant velocity), in Newton's second law, you get: $$F=m\ddot{x}=m(\ddot{x'}+\ddot{vt})=F'$$ In other words, the equations of motion in a boat with constant speed are the same at seashore, while standing still; no experiment you make can distinguish between rest and constant velocity motion. For that reason, we call these kinds of systems (where $$F=F'$$) inertial, because they are equivalent to being still, or inert. I want you to notice that there is a characteristic of inertial movement that is very geometric: inertial movement traces out straight lines. It is very important that you immediately associate straight lines with inertial movement. It is the first link between motion and geometry.
Now notice what happens if you are instead moving with constant acceleration $$g$$, or in other words $$x'=x-v_0t-\frac{1}{2}g\,t^2$$ $$F=m\ddot{x}=m\ddot{x'}+mg$$ wait what? I thought Newton said F is acceleration times mass. Now there's a second term?
It turns out the second law is not valid for this kinds of systems. You have to introduce the effect of a "fictitious" force to account for the extra $$mg$$ term. This is highly distressing. We call these systems $$non-inertial$$, because they are not equivalent to being still. You can actually feel when you are moving. You feel that extra force right? Think of a car accelerating or a bus turning around a corner. You feel pulled against your car seat/ into the walls of the bus. Notice as well that non-inertial movement also has a geometric characteristic: curved lines. It is also very important that you associate non-inertial movement with curved paths. Anyway. This is the way they solved it back then.Add fictitious forces when the systems are not inertial, and call it a day: it will give the right results anyway. But like the mass equivalence problem, this one also points out to gravity being a different kind of force. When you are falling, you don't feel an extra force. Right? When you are falling, you feel weightless. This is because, in your frame, $$m\ddot{x'}=-mg$$ and so $$F'=0$$. So what gives? I thought accelerating frames were non-inertial? Why does a free-falling body have the properties of an inertial observer? Even more shocking! Gravitationally induced motion is curved, the trademark characteristic of non-inertial movement. How on earth can that motion be inertial? Enter: Riemann, Gauss, and the advent of Differential Geometry
Around the middle of the XIX century, Riemann, a great german mathematician, published a paper that overthrew millenia of mathematics, Euclidean geometry. Riemann proposed a different geometry, a geometry of the differential, of calculus, of locality: a mathematical framework where you could represent and study the properties of curved surfaces of many dimensions. Gauss was also onto many of the things that Riemann discovered, beforehand, but didn't have as much courage to go against the current knowledge. He had been, however, thinking about something very relevant to both the problems we discussed earlier. He imagined he was a little bug, living in a flat piece of paper. He went about his day moving from point A to point B in a straight line. But what if the paper was curved up or down somewhere in the middle? Well, Bug Gauss would still move "in a straight line", as he locally wouldn't be able to tell he was in a curved space, as he was very small, but... he would be deflected around/towards the middle, depending on the sign of the curvature. Just like... gravity. In fact, it was overwhelmingly similar to the way gravity seemed to work. For example, how would you classify the motion of the bug? Inertial, or non-inertial? On one hand, the bug felt no acceleration during the proccess. No additional force (just like a free-falling observer). So maybe inertial? Buuut, on the other hand, his motion was curved... like a non inertial observer. Sounds familiar right? It is the same riddle gravity posed us! Inertial observers following curved paths!
Einstein and the happiest thought of his life
Glossing over all the work done on Special Relativity, the role of which, in General Relativity, is central, we arrive at the beginning of the XX century. Einstein is hoping to turn his theory of relativity into a theory of gravity, but he just can't seem to find a connection. Special Relativity is a success: however, it is only valid for inertial observers. One day, though, he does find it, in what he later described as the happiest thought of his life, which we have enunciated previously. A free-falling observer is equivalent to an inertial observer. A purely gravitational system experiences no forces whatsoever. When you fall, it's exactly the same as being still with gravity turned off. Only when you hit the ground do you feel a force. In this framework, Einstein finally was in a position to solve both gravity's problems. For the mass equivalence, well, of course they were equivallent! If a body in which only gravity applies is the same as an inertial body, either in straight line motion or still, then the inertial and gravitational masses have to be equivalent!! Give a push to the inertial body; the resistance is $$m_i$$ But whatever resistance is given by the inertial body has to be the same as the gravitational one, by this principle of equivalence that Einstein postulated. For the relativity problem, well, if the gravitational system is in some sense inertial, then it follows that some modified notion of inertial preserves a modified version of Newton's laws. It has to have the following features: the new modified notion of inertial has to acommodate curved paths; the new notion of force has to be null in these kinds of systems, from every reference frame. This might also sound familiar: this notion of inertial is precisely the same sense in which Gauss's bug in a curved plane was inertial: curved motion but no force. This is where the link from gravity to geometry accomplishes its final step. Construct a geometric structure, akin to the paper in which the bug lived in, such that the paths induced by gravity will be inertial. It had already been shown by the german mathematician Hermann Minkowski that a special structure composed of points of space and time (spacetime) worked very naturally with Special Relativity ( in a sense that I probably shouldn't explain here, as this answer is already too long) so this geometric structure was the perfect candidate. Make it curved, impose the constancy of the speed of light for all observers, make it coincide with Newton's laws of motion for small speeds and weak gravitational fields (the so called non relativistic weak field limit) and ta-da. You have General Relativity.
I have always found that learning things from a historical point of view improves greatly upon my understanding of a subject. We can see why we deal with some things instead of others. When you ask the question: how does spacetime generate gravity. There's a ton of questions already in there. Why do we even speak of spacetime in the first place? When people speak of it, it may seem like it's this metaphysical substance-entity that is invisible all around us. It is not. Spacetime is a concept invented to acommodate the equivalence between motion and geometry. Curved spacetime is a concept invented to acommodate the equivalence between gravity, and curved inertial motion. When we say that spacetime curvature generates gravity, this is really what we mean; that we as mankind have arrived at a model that describes gravity as inertial motion in a curved geometry, rather than non inertial motion in flat geometry, because it is more consistent with our previous models of the laws of physics, which had a few incongruencies. Had Newton defined force in a different way, or inertial in a different way, and we might not have invented spacetime as we know it; perhaps some other version, or none at all.
I know I have not explicitly answered the details of how gravity is a result of curved spacetime. All the answers did a pretty good job at that anyway; energy and momentum curve space time, matter follows geodesics (the generalization I was talking about of inertial path, that can be curved), all that stuff. But hopefully from this exposition you gained a little more insight into what "spacetime curvature generates gravity" means, which is:
Curved spacetime is the geometry in which gravity becomes, in a suitable sense, inertial.

What Einstein's equation tell us, at a basic level, is that the curvature of space-time and stress-energy are the same thing.

In order for this law to be respected it is clear that the stress-energy of a test particle cannot be constant in a space-time with changing curvature.

So, if you can choose a coordinate set in which the stress energy tensor is represented by the mass-energy of the particle, then the practical effect you can observe is changing energy and momenta of the test particle.

When you therefore observe the test particle, you will see it as having changing energy and momenta, and therefore derive a force driving these changes. This is what we call gravity.

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.

• Part of this answer has been quoted in a new question.
– Nat
Sep 26, 2018 at 10:01

Curvature affects motion by making the lines that are as straight as possible end up converging, just line how if you and your friends fly at constant altitude from the north pole, then no matter what directions you go (even if you and your friend head out in very different directions) then you start to converge on the south pole. This is a very good way to describe an effect that is determined by the path and not by the mass of the object taking the path. This is sometimes described as "spacetime tells matter how to move" but really this is just that the straightest possible lines converge when spacetime is curved the right way.

But something not mentioned enough is that while mass, energy, momentum, stress, and pressure are sources of curvature, they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure.

The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself).

So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.

Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there.

So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources.

Having curved spacetime is something we observe. Having gravitational sources that can change the normal or usual way curvature evolves is something else entirely. We can make theories about how the sources evolve, and then the curvature is forced to co-evolve with it, and that's what gravity is about, about gravitational interactions (source and curvature together) changing how the curvature evolves changing the evolution that the curvature otherwise would have evolved a different way.

So there is nothing circular, curvature is observed, and on its own it interacts and affects itself in a particular way (that is also observed), but gravitational sources get to change that and by interacting with the gravitational sources (which we can do) we can ourselves make the curvature change in different ways than it otherwise would!

A question was marked as a duplicate of a duplicate of this question, so I am posting my answer here.

Gravity is due to the curvature of spacetime

I believe it is true. That is what general relativity says, and general relativity has been confirmed in predictions ranging from the existence of black holes to the orbit of Mercury to the bending of light.

Relation between spacetime, curvature, mass and gravity

You say you are confused about how the curvature of spacetime and gravity are related. I am going to explain mainly that in my answer, starting with simpler examples, and moving to more complicated ones.

Okay, let's say you have a sheet of rubber. This is the classic example of spacetime. Let's say you take a bowling ball, and set it on the taut sheet of rubber. It has a large mass (compared to what else we'll be putting on the sheet), therefore the sheet curves a lot for the bowling ball. We now have an image in our head like the one below:

So mass leads to curvature. Then, take a baseball, say, and set it near the bowling ball. It rolls toward the bowling ball, right? This occurs because of the curvature of the sheet. So, then, curvature leads to gravity. So, if an object has large mass, it will curve spacetime dramatically, leading to strong gravity.

This is, of course, an overly simplistic example. It is 2-d, and it doesn't take into account other factors. Let us move to 3-d (keeping in mind the universe is accepted to be at 4-d, ignoring the holographic principle). The mass of a bowling ball now sucks in space around it, sort of like in the picture below:

And now, in this case, we can see (or understand) that more mass still leads to more curvature. The greater the mass, the more spacetime will "contract" around the object. So we still think that mass leads to curvature. Now, if we set an object near this massive object (like the moon next to Earth) it is "sucked in" sort of, by the curvature of spacetime, though of course the moon contracts spacetime around it as well. At this point, we can reasonably still conclude that in 3-d, mass leads to curvature which leads to gravity.

But, as I said earlier, the universe is generally thought of as 4-d. What does our picture look like when we add time? Well, the time dimension is contracted around a massive object. So let us picture our previous example but that the fabric of spacetime has a few clocks embedded in it occasionally. As the space stretches and contracts, so will the clocks (the "time") and so the time on those clocks will be "wrong" - it'll differ from the other clocks. And in this case, as the Earth contracts space and time around it, it changes the time and space (it curves spacetime) and so when another object enters our region of spacetime, it is "sucked in" still, but so is it's time. This is, of course, a very extreme example, but I hope this shows that we can conclude that mass leads to curvature which leads to gravity. And a black hole, is simply so much mass that it leads to so much curvature that the gravity is so strong that light cannot escape.

I hope this helps!

• I like that you added the 3D image - the 2D one confuses many people : ) Feb 27, 2020 at 20:56
• I see this explanation a lot, but I think it poses more questions than answers. The question that arises is why does a smaller ball fall into the pit? In a sheet analogy it is an xy-component of a sheet reaction on a z-force, but where does this z-force come from, as we're trying to explain it firsthand with this exact analogy? Jul 13, 2020 at 3:40
• ... We can instead imagine that a ball on this sheet stands still, like it would do in zero-gravity setup. I think the issue with this explanation is that it doesn't take time into account, thus is self-referential. It is time-part of spacetime that is also bent, and through which our slow satellite ball "flies" at near the speed of light and that gradient makes it fall with time, even with no additional down-force analogies around. Jul 13, 2020 at 3:40

Here's a simple way to think about it:

Newton's first law of motion says that in the absence of any force on a particle, the particle will move in a straight line.

Hence, if we see a particle move in a curved path - that is of it deviates from a curved path - we can say that there is a force on it.

Now, in GR, particles without any forces acting on it move on geodesics. This is the replacement for the notion of straight lines in a curved spacetime. Nevertheless we can detect the deviation from the usual notion of a straight line in flat space.

This deviation will be correlated with the force of gravity as experienced by this object in its local frame.

The rubber sheet analogy is just that, an analogy. However, when understood, it more or less describes gravity.

First take a curved space. This is not in anything. However, by one of Whitney's theorems it can be embedded in flat space. This gives the rubber-sheet geometry. Now, we need some force to drive particles along geodesics. This is simply Newtons first law for curved spaces. We do this by switching on an external gravity source. This gives the rubber sheet analogy of gravity. Its not a circular argument as we're simply using gravity to illustrate gravity.

In reality, no external force of gravity is used to make particles move along geodesics. In fact, as Aristotle would put it, this is their natural motion. Consider Newtons first law: particles move along straight lines. Notice that there is no force here to ensure that they do. Likewise in Einsteins first law, particles also move along geodesics and there is also no force to maintain them in this motion. However, in the rubber-sheet geometry analogy we need an external gravity to ensure that they do.

It's worth adding that curvature is also used to explain all the four forces in the standard model - not just gravity. In a sense, we have a unified theory of classical forces, but not for that of quantum forces - especially quantum gravity.