It starts with a simple idea: there is no difference between a force due to acceleration and a force due to gravity. Albert Einstein showed that this is the case if spacetime (he had already shown that space and time were really a common entity in his theory of special relativity) is curved, and massive objects curve spacetime in their vicinity.

Pretty soon, you have black holes, wormholes, and all manner of weirdness.

Slendro is almost right about the acceleration/gravity thing, but he's a step or two too far foward in the theory. One of the implications of general relativity is that gravity is not a force, and acceleration doesn't exert a force anyway.

The theory of general relativity is based on the theory of special relativity along with Minkowski's geometrical formulation of it. It can be summed up in the equation E=mc2, where E is energy, m is mass, and c represents the speed of light.

As I explained in the theory of relativity (go there to check the background info on any assertions I make here), the theory (the special one) is based on two principles:

  1. "The laws of physics should work across all inertial frames of reference" and
  2. One law of physics is that the speed of light, c, is a constant
Now, from inside any given frame of reference, it is assumed to be impossible to tell what the state of motion of that frame is. In a frame accelerating at a constant rate, anyone in that frame will be 'pushed' in the opposite direction to that of the acceleration by a 'force'. Try this in an elevator - when it starts off, note what you feel. When it accelerates up, you feel heavier, when it accelerates down you feel lighter. Incidentally, if you find a good elevator, jumping in the air while it accelerates downwards is a lot of fun, I've got up to 4 seconds air time this way, then got destroyed as it decelerated when I landed.
This 'force' is indistinguishable from gravity. I call it a 'force', but it isn't really. It's just a manifestation of inertiality. An explanation follows.

Inertia is the principle, observed throughout human history but not systematized until Newton, that objects do not change their state of motion unless an outside force acts on them. Objects travelling in inertial motion (motion with no forces acting on them) should continue to move in straight lines. F=ma. Because of this, and because objects were observed to be accelerated towards massive objects (the centre of the Earth, for one. Or the Sun, or whatever), Newton thought of gravity as a force.

However, this is because Newton thought of the universe as 3-dimensional. One of the consequences of Minkowski's geometry (see here) is that we must not think of the laws of physics as applying to a 3-dimensional 'space', but rather of a 4-dimensional spacetime structure. Newton only thought of gravity as a force because he perceived acceleration of objects in three dimensions as being non-inertial motion. However, in 4 dimensions, accelerated motion may be regarded as being inertial. That is, in 3-d an object can change its spatial position over time in inertial motion, add another dimension and an object can change its rate of change of position over time and still be in inertial motion. Phew, that was a conceptual challenge...

Ok, so objects in spacetime can accelerate and still be in inertial motion. This means that instead of having to view gravity as a force, it can instead be viewed as a property of spacetime... this is getting too abstract, no? Ok, let me put it this way. I'll stop trying to explain and just say that in order for spacetime to have this property, it must be curved. That's curved in 4 dimensions, not 3. Look, I don't know, take some psychedelics and get back to me, ok? In Newtonian Physics, objects in inertial motion travel in straight lines at constant velocities. In spacetime, objects can't travel in straight lines because the 'stuff' that the universe is made of is curved, so they have to follow curved paths. Just as a line is the shortest path between two points in a flat medium, the shortest path between two points on a curved surface is called a geodesic, and these are the paths that objects in inertial motion follow under general relativity. What we see as acceleration is actually an object following the shortest path in a set of dimensions we can only see 3/4 of.

How does this relate to gravity? Well, I've said that spacetime is curved, right? It's curved by the presence of mass, kinda like a rubber sheet is curved when you put something heavy on it (only in that case, gravity explains the curvature. In this case, the curvature explains gravity). The geodesics in the vicinity of Earth, for example, all curve in towards the centre of the planet, just like the sheet curves in towards that steel ball I told you to put on it.

The force you feel on your feet (well, probably more like on your ass, cos you're sitting down reading this) is not holding you 'stationary' by countering the 'force' of gravity, rather it is actually accelerating you off the inertial course you'd otherwise follow to the centre of the earth. It's pushing you around in spacetime.

Anyway, onto black holes. Spacetime can be curved by the presence of matter, yes? Loosely speaking, the more mass in an area the greater the curvature of spacetime in that area, ok? If you put a whole lot of mass all in one place, it curves spacetime and so it all falls in towards the other mass, and spacetime gets more radically curved and so it falls faster and closer and spacetime curves more until... infinite curvature. The graph you could draw of the curvature has a spike here, a spike that goes up forever. There is no spacetime as such in this region, the laws of physics no longer work in the singularity that has been created (general relativity is cool in that it predicts an exact circumstance in which it no longer works. In the singularity, the universe as we know it does not exist). For more on black holes visit the appropriate node.

Actually, the fundamental equation of General Relativity is NOT E = mc^2 as some people seem to believe, but rather Einstein's equation, G-alpha-beta = 8*Pi*T-alpha-beta, where G-alpha-beta is the Einstein tensor and T-alpha-beta is the Energy-Momentum tensor. The spacetime metric follows from G-alpha-beta, thereby defining the geometry of spacetime.

I see a big discussion breaking out here, and although I don't like jumping on the bandwagon for these things, I'll try to say a few things to clear up the confusion. First, a mildly technical explanation of the theory:

So you know that spacetime is a Riemannian manifold (ie. a space idetitified by a set of coordinates, usually space and time), and that the metric tensor allows you to measure distances within it. Now, where the real Physics comes in is when you have derivatives of the metric:

  • First order effects come from first derivatives of the metric. Acceleration and Newtonian gravity belong in this category: in Special Relativity, acceleration is simply d2xa/d2τ - the natural extension of what acceleration is in the Newtonian sense. In General Relativity, some extra terms quadratic in the velocity come in, and acceleration is then given by
      d2xa/d2τ + Γabcdxb/dτdxc/dτ
    The equation come from specifying that freely falling (ie. unaccelerated) particles pursue geodesics. Don't worry too much about what the Γ term means, except to bear in mind that it is a thing expressed in terms of the first derivative of the metric. Now, because of all the factors of c knocking around, at low speeds we can forget about most of these terms that have just been added, and what we're left with as an expression for the acceleration the usual d2xa/d2τ minus the acceleration due to the (Newtonian) gravitational field - and thus Newtonial gravity is recovered at small speeds in weak fields.
    It is which recovers the equivalence principle: that gravitational fields behave like accelerations.
  • Second order effects come from the second derivatives of the metric, which naturally characterise the curvature of spacetime (mathematically speaking, these effects come from the presence of the Riemann tensor). Thus, the Einstein field equations - which involve the presence of matter - are expressed in terms of second derivatives of the metric (which quantifies the statement 'matter bends spacetime').
It is a fundamental principle of GR that we can always find local coordinates such that the first order effects vanish - that is, a frame can be accelerated such that the effects of gravity can be anulled.

Now to resolve aozilla's problem: he states that because of the inverse square law, gravity can be distingished from uniform acceleration. The key to the problem is the crucial requirement above that (in general) you can only makes the gravity balance the acceleration locally - as soon as you start thinking about inverse square laws, you are examining the particle's behaviour in a non-local context.
Indeed, if we could make accelerations equivalent to gravitational effect everywhere, then the first derivatives of the metric would vanish everywhere, and then so to would the second order effects. If this were true, then the Einstein field equations would fall apart and there would be no gravity anywhere.

    aozilla replies:Does this imply that the gravity is in essence coming from an infinitely far away source? The gravity is not within the system, but we still can measure exactly how far away the center of mass is...
What I was trying to say above is that the source is not important - the presence of matter produces seconds order effects (curvature) which are not relevent in as far as the Equivalence principle is concerned.
The fact that the gravitational field varies with position (whereas an uniform acceleration does not) is something that you must be observing on a 'larger scale' than that for which the Equivalence principle applies, so how is varies not important inasmuch as your question is concerned.

Okay, so this is how it goes. To understand general relativity, here’s a brief run-through of what Einstein accomplished with special relativity and how he intended to counter its deficits with his ideas of a general theory.

Special relativity is rooted, essentially, in one of the principles of Galilean relativity: to whit, the idea that it is impossible to say whether or not you are moving. For instance, if you were running with respect to a stationary bus, you could just as easily say that you’re standing at rest and that the bus was moving away from you. Similarly, if you were standing at rest and a train whooshes by, you could just as easily say that you’re in motion and it’s actually the train that’s stationary with respect to you: there is no way to determine which one of you is really in motion. That is what the principle of Galilean relativity states: there is no test with which you can determine whether or not you are in motion. There is only relative motion; determining absolute motion is impossible.

Einstein decided that this principle was a fundamental physical law, and tried to hypothesize what would happen if you always measured the speed of light to be constant, keeping the laws of physics the same in every frame of reference. After all, it’s a clear violation of absolute motion: light is always moving; its speed is fixed, and it can never be at rest. Furthermore, if you didn’t measure it to be moving at the same speed always, you could use it as a test to determine if you’re moving: an observer at rest while you’re in motion calls out a different speed than the one you measure, and suddenly you know you’re in motion, simply by working out the math. Fair enough. So Einstein rolled up his sleeves, and came up with thought experiment after thought experiment to see what would happen. The results? Time dilation and Lorentz contraction (the relevant thought experiments that showed this I won’t go into, as you’re already familiar with them). In your own frame of reference, moving at a particular, invariant speed, you would observe fundamental quantities to be quite different from another observer moving at another speed. If you tried to see whether or not you were moving with respect to a photon, time and space would change for you so that you would always measure the speed of light to be the same, regardless of how fast you moved. Galilean relativity, with one major modification, has been preserved: light is permitted to be in a state of absolute motion, and your measurement of time and space would change so that you could no longer really tell if you were moving with respect to the photon – you would measure the speed of light to be the same in all reference frames, making it impossible to use as a test for absolute motion. It was thus still impossible to determine a state of absolute motion, except unless you excluded light from consideration.

That, then, was the edifice on which special relativity was based on: preserving the idea that absolute motion is a no-no. Yet special relativity is called special for a reason: it only holds if you’re constantly moving at the same velocity. Indeed, all the laws of special relativity held for the special case of when you weren’t accelerating at all. That was Einstein’s problem: how do you preserve the Galilean principle if you’re accelerating?

You see what it means. Acceleration means taking inertia into account: you ‘feel’ a certain force operate on you whenever the car you’re in accelerates or brakes, and you can instantly tell that you’re in motion. True, Newton’s third law states that an equal and opposite force operates on the car; but what if you were accelerating with respect to a house twenty metres away? You couldn’t honestly say that the house felt a similar force: its twenty freaking metres away, for goodness’ sake, you’re nowhere near in contact with it. How do you accommodate the force?

This occupied Einstein’s mind for years. And one day, he got it.

Imagine, for a moment, you’re in an elevator that’s initially moving at a particular speed, say, down. Suddenly, it accelerates: you feel a rushing force as this happens, and must conclude that you are, sighing as you do so, in motion. But wait! Little did you know that, in actual fact, the elevator hasn’t accelerated at all: it’s merely that the mass of the Earth has spontaneously changed (yeah, I know it sounds ridiculous, but bear with me for a moment). Thus, the force of gravity changed – so what you’re actually feeling is simply the force of gravity.

If you think I’m going barmy saying all this, here’s another way to think of it. Would you, as an observer in that elevator, be able to distinguish between the two situations? You could say, on one hand, that you were at rest with respect to the elevator (you’d be moving at the same speed as it is, remember) and that the elevator accelerated. Or you could also say that the elevator was perfectly stationary (at rest with respect to you) and that its (or the Earth’s) mass changed spontaneously, so you felt a force that made you feel as if you were in motion. There is no test to determine which of these situations is correct. 

This was the germ of general relativity: a theory of relativity that could take into account accelerating frames of reference and not just those at a constant velocity. Einstein’s great insight was to realize that a body in acceleration with respect to a stationary observer is virtually indistinguishable (to the observer) from a body that is at rest in a changing gravitational field with an observer that is in motion. Einstein could account for the force now: he linked it to a changing gravitational field. Accelerating and being stationary in a changing gravitational field are indistinguishable. And thus the Galilean principle was saved once more.

Now here’s a thought experiment from special relativity. It’s important to GR, so I’m going to explain it to you.

Imagine you’re in a circular chamber that’s spinning round and round at a constant angular velocity. For some reason you want to measure the value of pi: this is weird, but you’re a mathematician who wants to be a theoretical physicist, so that’s okay. Now what’s pi? The ratio of the circumference to the diameter of this oh-so-wonderful circular chamber you’re in. Ergo, you have to measure both the diameter as well as the circumference to arrive at a value of pi. So you steady up your nerves, ignore your dizziness and set to work.

First, you measure the diameter. So far so good. Because you’re measuring something perpendicular to the direction of the chamber’s motion, Lorentz contraction doesn’t happen: your rod stays exactly the same length, and you manage to arrive at a reading that is exactly what you’d find if the chamber was at rest. Hopes high, you begin to measure the circumference of the chamber. But now you’re in the direction of motion: Lorentz contraction makes your ruler shrink, except you don’t realize this because you’re also moving at the same speed. Naturally, when you finally check your readings, you’re surprised to see that the circumference is actually longer than what you measured it to be at rest. And when you put those two numbers together – a longer circumference divided by the same diameter – you get a value for pi that is no longer 3.14159etc.

The value of pi – your measurement of it - has changed while you were moving at a constant velocity. You can be tempted to ignore it, but this will always be true. What can you conclude from this?

If you are as well-versed in mathematics as I suspect you are, then you probably already know where this is going. Different values of pi are characteristic of regions that are not Euclidean: that are not perfectly flat, so to say, that are curved in one way or another. One example is the curved surface of a (soccer) football, where it is perfectly possible to draw a triangle with three right angles, and other weird things; such spaces are not flat, and are thus not classified as Euclidean. Thus one is forced to conclude that an observer moving at a constant velocity measures events to no longer conform to a Euclidean background: that the events a speeding observer notices is virtually indistinguishable from those that occur on a curved surface. Space and time distort from him in a way that make sense only and only on a curved surface: thus, even in special relativity, one finds evidence that space and time are curved for the observer, and that the observer will accordingly behave as if he’s on a curved surface.

A word of caution here. When I say space and time 'curve' for the observer, I don’t mean they literally curve. Time and space are not, as a friend once told me, ‘fly rods that can be bent over physically’. It is merely that your measurements of time and space are such that they are typical of a curved surface: distances become longer or shorter, the time taken to cover them vary, and so on. Time and space do not ‘curve’: only your measurements of them – distance, length, time – do, so that you could very well conclude you’re moving on a curved surface.

And all this happens when you’re moving at a constant speed: within even the bounds of special relativity. When you’re accelerating – switching from velocity to velocity – your measurements of space and time are going to ‘curve’ more and more: you will measure successively changing values of pi, longer distances, longer times to travel. And since acceleration is indistinguishable from a changing gravitational field, this means that objects that are really in a gravitational field will measure the same things: they will begin to behave exactly as if their measurements of space and time were similarly twisted, so that they too were on curved surfaces that gained more and more curvature as the force of gravity increased. 

Thus, an object in a gravitational field will begin to behave as if it is on a curved surface that is steadily growing curvier (I’m sorry, that word evokes images of bikini babes. Nevertheless, it is all I have). This, I think, is Einstein’s most profoundly insightful idea. Gravity isn’t a force that changes the trajectory of objects around it; what actually happens is that objects within a gravitational field are merely trying to obey Newton’s first law (i.e continue unimpeded with the same velocity in the absence of a force) while on a curved background. Geodesics are straight lines on their surfaces too, remember? There is no ‘force’ involved: merely an object trying to follow Newton’s Euclidean laws in a non-Euclidean world. I always find this magical.

And that’s it, really. 

To summarise: 

Being at rest in a changing gravitational field is indistinguishable from accelerating. When you’re moving at a constant velocity, you are forced to make observations that only make sense if you accept that your space and time are those that are appropriate to a curved surface. Thus, when you’re accelerating, your observations correspond to steadily changing curved surfaces, and you will behave exactly like you were on a curved metric. Since accelerating is indistinguishable from being in a gravitational field, objects that are in a gravitational field behave as if they are on steadily changing curved backgrounds too. They carve out geodesics instead of straight lines because, on a curved surface, geodesics are the only ‘straight’ lines possible. Hence, planets form ellipses around the sun: space and time are so warped for them that their closest conception to a straight line is an ellipse.

That’s all there is to general relativity. It’s probably not the best kind of explanation I could give – I was trying to convey the main ideas the quickest way I could – so do tell me if there’s anything I didn’t clarify enough.

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