# Gravity = warping of spacetime (doesn't make sense)

• Jonnyb42
In summary, the ball on a bedsheet analogy is not a good way to explain gravity because it does not demonstrate why the intrinsic curvature of space-time affects "straight" lines. This analogy is only used because there are no better analogies.
http://img506.imageshack.us/img506/5015/spacetimelu3.jpg

Keep in mind that each of the "gaps" between those lines is the same distance apart, so the region where the lines are curved inwards is "deeper".

Feynman explained it wonderfully (as he always does) by pointing out that if you drew a circle around the curved region, and calculated what the radius would be for that circle, upon actually measuring it you would come up with an excess radius!

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Max™ said:
Feynman explained it wonderfully (as he always does) by pointing out that if you drew a circle around the curved region, and calculated what the radius would be for that circle, upon actually measuring it you would come up with an excess radius!
What you measure really depends on what kind of observer.

In a Schwarzschild solution for a stationary observer at a certain distance from the 'center' the volume of a sphere surrounding it would indeed be larger than as would be expected by measuring the surface of this sphere.

However for a radially free falling (from infinity) observer this would not be the case.

Yeah, but I was trying to avoid causing extra confusion, hence the simplified explanation as provided by Feynman.

:P

Passionflower said:
What you measure really depends on what kind of observer.

In a Schwarzschild solution for a stationary observer at a certain distance from the 'center' the volume of a sphere surrounding it would indeed be larger than as would be expected by measuring the surface of this sphere.

However for a radially free falling (from infinity) observer this would not be the case.

I don't think the red part is true. But I'm not sure how exactly you envision the free faller to perform that measurement, of a sphere around a center that accelerates relative to him.

However, locally the free falling observer can measure tidal forces and thus the curvature of space time.

A.T. said:
I don't think the red part is true. But I'm not sure how exactly you envision the free faller to perform that measurement, of a sphere around a center that accelerates relative to him.

However, locally the free falling observer can measure tidal forces and thus the curvature of space time.
The difference between two Schwarzschild r values is never equal to its proper distance except for a radially free falling observer (who is free falling from infinity, e.g. an observer free falling at escape velocity).

It is true that two free falling points a given distance away will not be able to maintain this distance without proper acceleration but that does not invalidate the above statement.

This is probably the best description I've heard:"I think I'm probably going to inspire more questions than understanding here, but I'll give it a shot anyway.
General relativity is a geometric theory. That means it has to do with geometry more than anything else. John Archibald Wheeler called it "geometrodynamics," as an analogy to "electrodynamics," but the name never really caught on.
So we start by thinking very abstractly about geometry.
Remember your Euclidean geometry, from school? It's a very simple, very idealized geometry, with a basic logic that's easy to wrap your head around. Lines which are parallel at one point are parallel everywhere. The interior angles of a triangle always add up to half a circle. The distance between two points is a function of the relative positions of those two points. And so on.
Well it turns out Euclidean geometry is not the only possible geometry. It's possible to construct, by careful manipulation of the basic geometric postulates, geometries which are entirely consistent, but which are different from Euclidean geometry.
In the 1800s, a variety of mathematical discoveries made it possible to describe arbitrary geometries in a rigorous, consistent way. It's possible for straight lines to curve. It's possible to have a geometry in which lines are parallel at one point, converge at another point, diverge at another point and so on. It's possible to have a geometry in which translating the same triangle from one region to another changes the sum of its internal angles.
This might sound odd, but it really isn't. It's Euclidean geometry — the geometry of the infinite perfect regular plane — that's odd. In the real world, non-Euclidean surfaces are everywhere. The surface of the Earth is non-Euclidean; parallel lines on the Earth's surface inevitably converge and cross. The surface of a bedsheet is even more complexly non-Euclidean, because it has bumps and crinkles.
The point here is that geometry doesn't have to be Euclidean. It can be something else.
This is the insight that Einstein brought to physics. He started with the assumption that the speed of light is the same to all observers, regardless of how they're moving — this was a consequence of Maxwell's theory of light — and began to investigate the way coordinate systems transform between differently moving observers.
What he found was that the way coordinates transform is complex, intricate, counter-intuitive … and entirely consistent and sensible. It's hard to visualize, because we imagine the universe as being Euclidean — straight lines and all that — but it makes sense, and what's more in the decades since it's been directly measured. We now know that the geometry of our universe is not Euclidean.
To get more specific, let's consider the very special case of two observers moving inertially with respect to each other. To move inertially just means to be unaccelerated; an accelerometer carried by an inertial observer will read zero.
If these two observers are moving differently, but they both observe the same ray of light to have the same speed, then their definitions of distance and duration must disagree. This was a very profound insight! Distance and duration are not universal, and depend on how you're moving. This is the source of interesting phenomena like length contraction, time dilation and the relativity of simultaneity.
Einstein then moved on to think about acceleration. He'd cracked the problem of relative inertial motion, but what about accelerated motion? Where he began was with what came to be called the equivalence principle. This principle states that the outcome of a purely local experiment is not dependent on the location of that experiment in spacetime.
In less abstract terms, imagine you're in a small room with no windows. Say you went to sleep the night before and woke up there, with no knowledge of how you arrived. The room is stocked with every piece of scientific equipment you can imagine, from a simple spring scale all the way up to (somehow) a huge particle accelerator.
What experiment can you perform in that room that will tell you whether you're imprisoned somewhere on the surface of the Earth, or out in deep space in a rocketship moving with a constant acceleration of 1 g?
The answer is none. No experiment can tell you which of those is true.
Suddenly, the acceleration pushing your feet to the floor vanishes. You're in free fall. You — and all the expensive equipment in the room — float freely, like an astronaut in orbit.
What experiment can you conduct now that will tell you whether the engines of the spaceship have been turned off, leaving you to coast through deep space far from gravity, or the cables suspending your cell at the top of a tall tower have been cut and you're now plummeting toward the ground?
Again, the answer is none. Do all the experiments you like, and you will never be able to tell whether you're floating or falling.
This is the heart of Einstein's theory of gravitation: Falling is inertial motion. Standing still on the Earth's surface is acceleration.
Gravity, then, is not a force at all. It's a consequence of inertial motion through curved spacetime. The presence of stress-energy — a composite quantity that includes mass, charge, momentum, pressure, sheer stress and so on — changes the fundamental underlying geometry of spacetime. Objects that move through that curved spacetime along entirely mundane, inertial trajectories will be observed, by observers who are at rest relative to the source of gravitation, to accelerate and curve toward the ground, but in fact this is an illusion. The falling object is moving at a constant speed and in a straight line. It's just that in that region of spacetime, where the stress-energy is, straight lines intersect. They intersect at the center of mass of the gravitating body.
As the aforementioned Wheeler so famously put it, matter tells space how to curve, and space tells matter how to move.
That's about as deep into gravitation as I can get without bringing in mathematics. And it's a lot of maths. But that's the essence of it. Everything in the universe that isn't actually accelerating — remembering that acceleration is a purely local phenomenon that can be measured with an accelerometer — moves in a straight line at a constant speed. But depending on where you are, a "straight line" can look like a curved line, and "constant speed" can look like acceleration.
Gravity, in other words, is just an optical illusion."

- RobotRollCall (http://www.reddit.com/user/RobotRollCall)

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